Datasets:

RealTimeData

/

arxiv_alltime

like 6

Follow

RealTimeData 9

http://arxiv.org/abs/1701.07589v1

20170126065211

Few-body approach to structure of $\bar{K}$-nuclear quasi-bound states

nucl-th

YITP-17-06 Department of Physics, Hokkaido University, Sapporo 060-0810, Japan Department of Physics, Hokkaido University, Sapporo 060-0810, Japan Department of Physics, Hokkaido University, Sapporo 060-0810, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Structure of light antikaon-nuclear quasi-bound states, which consist of an antikaon (K̅=K^-, K̅^0) and a few nucleons (N=p, n) such as K̅NN, K̅NNN, K̅NNNN and K̅NNNNNN systems, is studied with full three- to seven-body calculations. Employing a realistic K̅N potential based on the chiral SU(3) effective field theory with the SIDDHARTA constraint, we show that the central nucleon densities of these systems increases when the antikaon is injected, by about factor of two at maximum. The K̅NNNN system shows the largest central density, about 0.74 fm^-3 even with the phenomenological K̅N potential, which are not as high as those suggested in previous studies with approximate treatments of the few-body systems. We find the spin of the ground state of the K̅NNNNNN system depends on the strength of the K̅N attraction. Thus, the quantum number of the ground state can be another constraint on the K̅N interaction. Few-body approach to structure of K̅-nuclear quasi-bound states Tetsuo Hyodo December 30, 2023 =============================================================== § INTRODUCTION In recent years, properties of the antikaon(K̅)-nuclear quasi-bound states, so-called kaonic nuclei, have been studied actively. Since the nominal location of the Λ(1405) mass is slightly below the K^-p threshold <cit.>, the Λ(1405) is considered as a K̅N quasi-bound state embedded in the πΣ continuum <cit.>. Motivated by such a picture, phenomenological K̅N interaction models were constructed so that they reproduce the Λ(1405) nominal mass together with two-body scattering data <cit.>. The strong attraction of the phenomenological potential models predicts deeply-bound K̅ states in light nuclei with binding energy larger than 100 MeV, and extremely dense systems about ten times higher than the ordinary nuclear density <cit.>. It should, however, be noted that the few-body problem was not accurately solved to predict such high-density systems, but the optical potential model or the g-matrix approach were adopted. The validity of those approaches should be examined with care, at least in the few-body systems. The K̅N interactions are essential for determining the structure of the kaonic nuclei. The K̅ belongs to a part of the pseudscalar octet of Nambu-Goldstone bosons associated with the spontaneous symmetry breaking of chiral SU(3)_L×SU(3)_R in low-energy QCD. Thus, the chiral SU(3) effective field theory based on the symmetry breaking mechanism is a more systematic framework to obtain the K̅N interaction, and has succeeded in dealing with the K̅N interaction with K̅N-πΣ couplings <cit.>. In fact, including the next-to-leading order (NLO) contributions, the chiral SU(3) approach reproduces all existing experimental data at the level of χ^2/d.o.f ∼ 1 <cit.>. Among others, the precise measurement of the kaonic hydrogen by the SIDDHARTA collaboration <cit.> gives strong constraint at the K̅N threshold, with which the uncertainty in the subthreshold extrapolation of the K̅N amplitude is significantly reduced. The equivalent single-channel K̅N potential to the NLO chiral dynamics including the SIDDHARTA constraint is constructed in Ref. <cit.> based on the framework presented in Ref. <cit.>. Thus, the realistic K̅N potential is now available. The K̅N-πΣ scattering amplitude from the chiral SU(3) dynamics has two poles in the Λ(1405) energy region <cit.>: one is located around 1420 MeV, while the other exhibits a broad resonant structure above the πΣ threshold. The pole located around 1420 MeV corresponds to the K̅N quasi-bound state with the binding energy of 15 MeV, about a half of the binding energy assumed in the phenomenological K̅N interactions. This different pole structure comes from different off-shell properties of the K̅N interactions. The K̅N interaction based on the chiral SU(3) dynamics is energy-dependent, and that in the subthreshold becomes less attractive than the one proposed by the energy-independent phenomenological potential <cit.>. These different off-shell properties also appear in how the Λ(1405) resonance shows up in the differential cross section of the K^-d→πΣ n reaction <cit.>. These differences are further enhanced in the light kaonic nuclei. For the lightest kaonic nuclei so-called strange dibaryons in the K̅NN-π YN (Y=Σ, Λ) coupled system, the energy-dependent potential models <cit.> give resonance energies higher than the energy-independent ones <cit.>. How a possible signature of this strange dibaryon resonance shows up in the resonance production reaction is also of interest as it reflects the two-body dynamics of the K̅N system <cit.>. Given the background described above, we raise three questions to be discussed in this paper; 1) What are the structure of light kaonic nuclei when the reliable NN and K̅N interactions are used? 2) Can the high-density K̅ nuclei be realized within the accurate few-body treatment? 3) How the off-shell dependence of the K̅N interaction affect the few-body systems? To answer these questions, we perform fully microscopic few-body calculations for three- to seven-body systems including an antikaon. Here, the systems with a K̅ and (𝒩-1) nucleons are accurately described by employing the stochastic variational method (SVM) with correlated Gaussian (CG) basis <cit.>. We employ the K̅N interaction based on the chiral SU(3) dynamics with the SIDDHARTA constraint <cit.> as a realistic K̅N force. Combining them with the reliable nuclear forces, we present quantitative predictions of the structure of the light kaonic nuclei. Next, we perform the same few-body calculations with the phenomenological K̅N interaction, so-called Akaishi-Yamazaki (AY) potential <cit.>, in order to examine the validity of the many-body approximations used in the prediction of the high-density states. Furthermore, the comparison of the results with two K̅N potentials serves as a study of the off-shell dependence of the interactions. In this way, we systematically study the structure of kaonic nuclei and discuss how the nuclear structure is changed by K̅. In Sec. <ref>, we briefly review the two-body interactions used in this work. The SVM with the CG for the 𝒩-body systems is explained in Sec. <ref>. We summarize the quantities to analyze the structures of the few-body systems in Sec. <ref>. Numerical results of the properties of the light kaonic nuclei are presented in Sec. <ref>. A summary is given in Sec. <ref>. § TWO-BODY INTERACTIONS §.§ Hamiltonian and expectation values The Hamiltonian for (𝒩-1) nucleons and an antikaon takes the form H =∑_i=1^𝒩T_i-T_cm +∑_i< j^𝒩-1V_ij^(NN)+∑_i=1^𝒩-1V_i𝒩^(K̅N) +∑_i< j^𝒩V^Coul._ij. Here T_i is the kinetic energy of the i-th particle. The particle label, i=𝒩, always indicates an antikaon and the others are for nucleons. T_cm is the energy of the center-of-mass (c.m.) motion T_cm=(∑_i=1^𝒩 p_i )^2/2{(𝒩-1)m_N + m_K̅}, where the isospin-averaged nucleon and antikaon masses, m_N=939 MeV and m_K̅=496 MeV, are used in this paper. V_ij^(NN), V_ij^(K̅N) , and V_ij^ Coul. are the NN, K̅N, and Coulomb interactions between the i- and j-th particles, respectively. The NN and K̅N interactions depend on isospin of two-particles, and they can be written as V_ij = V^I=0_ijP̂^I=0_ij+V^I=1_ijP̂^I=1_ij =1/2(V^I=0_ij+V^I=1_ij)-1/2(V^I=0_ij-V^I=1_ij)P̂_τ^ij, where P̂^I=0_ij=1-τ_i·τ_j/4 and P̂^I=1_ij=3+τ_i·τ_j/4 are isospin-projection operators for I=0 and 1, and P̂_τ^ij=1+τ_i·τ_j/2 is the isospin-exchange operator for the i- and j-th particles. The isospin-exchange operator P̂_τ acts on particle basis as P̂_τ|nn⟩=|nn⟩, P̂_τ|pp⟩=|pp⟩ and P̂_τ|pn⟩=|np⟩ for NN, and P̂_τ|K^-n⟩=|K^-n⟩, P̂_τ|K̅^0p⟩=|K̅^0p⟩, P̂_τ|K^-p⟩=-|K̅^0n⟩ and P̂_τ|K̅^0n⟩=-|K^-p⟩ for K̅N. We have to treat the K^-p-K̅^̅0̅n channel coupling explicitly in the particle basis calculation. The single-channel K̅N potential V^(K̅N) has an imaginary part which represents the decay processes into the lower energy πΣ and πΛ channels. Because of the complex nature of the potential, the Hamiltonian is non-Hermite and can have an eigenstate with a complex eigenvalue, called a quasi-bound state. In order to discuss the structure of the quasi-bound state, we need to evaluate the expectation values of some operators. For a stable bound state, the expectation value of an operator 𝒪̂(x) with the wavefunction Ψ_JMM_T(x) is given by (notation of the wavefunction will be explained in Sec. <ref>) ⟨𝒪̂⟩ ≡∫ d x[Ψ_JMM_T( x)]^*𝒪̂( x)Ψ_JMM_T( x) , with the normalization of the wavefunction 1 =∫ d x|Ψ_JMM_T( x)|^2. However, since the eigenfunctions of a non-Hermite Hamiltonian do not form an orthogonal set <cit.>, we should introduce the Gamow states to treat an unstable state. The expectation value with the Gamow states is ⟨𝒪̂⟩_G ≡∫ d xΨ_JMM_T,G( x) 𝒪̂( x)Ψ_JMM_T,G( x) with the normalization 1 = ∫ d x[Ψ_JMM_T,G( x)]^2. With the normalization of Eq. (<ref>), expectation values are in general obtained as complex numbers, which are not straightforwardly interpreted. In some cases, however, we can extract a real-valued quantity. As explained in Appendix of Ref. <cit.>, for a quasi-bound state whose real part of the eigenenergy is negative, the damping of the wavefunction outside the potential can be extracted from the standard expectation values with the normalization (<ref>). In this paper, we calculate the root-mean-square (rms) distances √(⟨ r^2⟩), density distributions ρ(r), and the probabilities of finding various channels in the wave functions P by using the standard expectation value (<ref>). For the other operators such as Hamiltonian and its decomposition, the expectation values are calculated by using Gamow state normalization (<ref>). §.§ NN interactions As a nucleon-nucleon interaction V_NN we employ the Argonne V4' potential <cit.>. AV4' potential is obtained by simplifying the full AV18 potential by suppressing the small electro-magnetic, the spin-orbit, and the tensor terms and readjusting the central spin- and isospin- dependent interactions. TABLE <ref> lists the binding energies and radii of two- to six-nucleon systems calculated with the AV4' potential. The AV4' potential model reasonably reproduces the properties of light nuclei. §.§ K̅N interactions As a realistic K̅N interaction, V^(K̅N), we employ the SIDDHARTA potential, which is the energy-dependent effective interaction based on the chiral SU(3) dynamics constructed in Ref. <cit.>: V^(K̅N)(r,E) = 1/π^3/2b^3e^-r^2/b^2m_N/2(E+m_N+m_K̅) ×ω_K̅+E_N/ω_K̅E_N[∑_i K_i (E/100 MeV) ^i ], where E, E_N and ω_K̅ are the non-relativistic two-body energy, the energy of the nucleon and the energy of the antikaon: E =√(s)-m_N-m_K̅, E_N =s-m_K̅^2+m_N^2/2√(s), ω_K̅ =s-m_N^2+m_K̅^2/2√(s). The coefficients K_i of the energy dependent strength and the range parameter b are determined so as to reproduce the K̅N amplitude <cit.> calculated based on the NLO chiral SU(3) dynamics (see Ref. <cit.>). The SIDDHARTA potential is the single channel K̅N interaction model where the meson-baryon channel coupling effect with strangeness S=-1 is renormalized, and thus the coefficients K_i are the complex numbers. The origin of the energy dependence is two-fold. The coupled-channel interaction depends on the energy through the time derivatives in the chiral Lagrangians, and the construction of the equivalent single-channel potential introduces additional energy dependence. By solving the Schrödinger equation, the pole positions of the K̅N(I=0) amplitude are found to be 1424-26i and 1381-81i MeV. For the use of the energy-dependent potential, it is necessary to determine the K̅N two-body energies in the 𝒩-body systems. Though the two-body energies in the 𝒩-body systems cannot be determined uniquely, we follow the same way as used in Refs. <cit.> to determine the K̅N two-body energies for practical calculations.[We also examine the prescription of the two-body energy suggested in Ref. <cit.>. The results of the few-body systems turn out to be in between the two choices shown in this paper.] First, we introduce an “antikaon binding energy” B_K̅ as -B_K̅≡⟨ H⟩_G - ⟨ H_N⟩_G , where H_N is the Hamiltonian for (𝒩-1) nucleons defined by H_N=∑_i=1^𝒩-1T_i +∑_i< j^𝒩-1V_ij+ V_Coulomb^NN-T_cm^N with T_cm^N =(∑_i^𝒩-1 p_i)^2/2(𝒩-1)m_N. Note that -B_K̅ is in general complex. We employ the following three types of the K̅N two-body energy as √(s) =m_N+m_K̅+δ√(s), Type I: δ√(s)=-B_K̅, Type II: δ√(s)=-B_K̅/(𝒩-1). Type I corresponds to the picture in which the K̅ field collectively surrounds the (𝒩-1) nucleons, and Type II corresponds to the picture in which the K̅ energy is distributed equally to the (𝒩-1) nucleons <cit.>. The eigenstate is determined in a self-consistent manner; the two-body energy calculated by the expectation values in Eq. (<ref>) should equal to the energy variable in the K̅N interaction in V^(K̅N)(r,E). For comparison, we also examine the Akaishi-Yamazaki (AY) potential. The potential was originally constructed in the coupled-channel K̅N-πΣ-πΛ system by fitting the old data of the scattering lengths in Ref. <cit.> and the nominal pole position of Λ(1405) <cit.>. Here we adopt the single-channel version presented in Ref. <cit.> in which the energy dependence of the potential through the Feshbach projection method is eliminated by hand. We note that the imaginary parts of the SIDDHARTA and AY potentials represent the decay into the πΣ and πΛ channels. In the few-body kaonic nuclei, there are two types of the decay processes, the mesonic decays with a pion emission and the nonmesonic decays with multi-nucleon absorptions. In this work, the imaginary part of the eigenenergy corresponds only to the mesonic decay width, reflecting the imaginary part of the two-body potential. When the nonmesonic decays are taken into account, such effect would increase the decay width of the kaonic nuclei by several tens of MeV <cit.>. § STOCHASTIC VARIATIONAL METHOD WITH CORRELATED GAUSSIAN BASIS We investigate the structure of the kaonic nuclei with a powerful few-body approach, that is, the SVM with the CG <cit.>. The method is flexible to cope with strongly correlated few-particle quantum systems as exemplified in Ref. <cit.>. The wavefunction for the 𝒩-body system is expanded as a combination of the basis functions: Ψ_JMM_T( x)=∑^K_k=1c_kΦ_JMM_T( x,A_k) , where J is the total angular momentum, M (M_T) is the z-component of the total angular momentum (isospin). Since we employ central NN and K̅N interactions, no channel coupling occurs between states with different L. In this paper, we consider total orbital momentum L=0 state by taking the basis functions with total spin J(=S) to have the form Φ_SM_SM_T( x,A) =𝒜{exp (- x A x) χ_SM_Sη_M_T}, where the operator 𝒜 is an antisymmetrizer for the nucleons; M_S(=M) is the z-components of the total spin; x is an (𝒩-1)-dimensional column vector, whose i-th element is a 3-dimensional Jacobi coordinate x_i; the symbol x stands for a transpose of x; A is an (𝒩-1)×(𝒩-1)-positive-definite-symmetric matrix. The Jacobi coordinate, x_i, including the center-of-mass coordinate x_𝒩 are related to the i-th single-particle coordinate r_i by a linear transformation: x_i =∑_j=1^𝒩U_ij r_j with U = [ 1 -1 0 ⋯ 0; 1/2 1/2 -1 ⋯ 0; ⋮ ⋱ ⋱ ⋮; 1/𝒩-1 ⋯ ⋯ 1/𝒩-1 -1; m_N/(𝒩-1)m_N+m_K̅ ⋯ ⋯ m_N/(𝒩-1)m_N+m_K̅ m_K̅/(𝒩-1)m_N+m_K̅ ]  . We can easily apply the CG to the present three- to seven-body model because the CG keeps its functional form under any linear transformation between different coordinate sets for any number of particles. The CG basis in Eq. (<ref>) can only be applicable to states with total orbital angular momentum L=0. It should be noted that the higher partial waves for each coordinate are taken into account through the cross terms, x_i· x_j. The spin wavefunction χ_SM_S is expressed using the basis of successive coupling: χ_SM_S=|[⋯[[1212]_S_1212]_S_123⋯]_SM_S⟩. Here we take all possible intermediate spins (S_12, S_123, …) for a given S. For the isospin wavefunction, η_M_T, we employ the particle basis, which is given as the product of single-particle isospin wavefunctions: η_M_T=η_1/2m_τ_1⋯η_1/2m_τ_𝒩. The sets of the single-particle isospins (m_τ_1, …, m_τ_𝒩) take the values m_τ_k= 1/2 (k=1, …, M_T+𝒩/2-1, 𝒩) -1/2 (otherwise), for the states with K̅^0 and m_τ_k= 1/2 (k=1, …, M_T+𝒩/2) -1/2 (otherwise), for the states with K^-. Each basis function has 𝒩(𝒩-1)/2 nonlinear parameters (A_k)_ij and also spin and isospin quantum numbers. The adequate choice of these parameters is crucial to determine accuracy of the variational calculation. The SVM offers efficient and economical ways to find optimal sets of the variational parameters <cit.>, in which we increase the basis size one-by-one by searching for the best among many random trials for the basis function. For the Hermitian Hamiltonian, the eigenvalues for the trial wavefunctions are larger than or equal to the exact eigenvalue. Since we use the complex K̅N potential in this work, the eigenvalues of the trial wavefunctions give no longer the lower limit. Practically, we apply the SVM for the real part of the Hamiltonian to obtain the energy curve, and then we diagonalize the full Hamiltonian by using the basis optimized for the ground state with the real Hamiltonian. The validity of this method can be confirmed in the two-body sector where the exact value of the pole position can be obtained. Examples of the few-body calculations for K̅NN (K^-pp), K̅NNN (^3_K̅H), K̅NNNN (^4_K̅H) and K̅NNNNNN [^6_K̅He (J^π)] are shown in Fig. <ref> with the AY potential. The eigenvalues with the real part of the Hamiltonian are shown in Fig. <ref> (a). Corresponding complex energy curves of the full Hamiltonian are plotted in Figs. <ref> (b) and (c). We find that if the energy convergence is reached with the real part of the Hamiltonian, the eigenvalues with the full Hamiltonian are also converged. The obtained energies in this method are consistent with other calculations for two- and three-body systems <cit.>. For the K̅NN, K̅NNN, K̅NNNN and K̅NNNNNN systems, the basis sizes are 200, 1000, 4000 and 10000, respectively. The binding energies and widths change less than 0.0001 MeV when the numbers of basis increase by one from these basis numbers. § STRUCTURE OF FEW-BODY SYSTEMS The internal structure of the the K̅ nuclei is reflected in the obtained wavefunction Ψ_JMM_T. Here we define several quantities which are useful to investigate the structure of the few-body systems. We first define the NN root-mean-square (rms) distances √(⟨ r_NN^2 ⟩), K̅N rms distances √(⟨ r_K̅N^2 ⟩), N rms radii √(⟨ r_N^2 ⟩) and K̅ rms radii √(⟨ r_K̅^2 ⟩) by using the following operators: r_NN^2 =∑_i<j^𝒩-12| r_i- r_j|^2/(𝒩-1)(𝒩-2), r_K̅N^2 =∑_i^𝒩-1| r_K̅- r_i|^2/(𝒩-1), r_N^2 =∑_i^𝒩-1| r_i- x_𝒩|^2/(𝒩-1), r_K̅^2 = | r_K̅- x_𝒩|^2, where r_i and r_K̅= r_𝒩 are the single-particle coordinates of the i-th nucleon and the antikaon. The rms distances represent the averaged distance of the two-body subsystems, and the rms radii measure the averaged distance of the particle from the center-of-mass of the total system x_𝒩. As discussed in Sec. <ref>, we calculate the expectation values of these operators √(⟨ r^2⟩) using the standard normalization condition (<ref>). To investigate how the nuclear system shrinks by adding an antikaon, we define the nucleon density distributions ρ_N^Ncm(r) =∑_i=1^𝒩-1∫ d x|Ψ_JMM_T( x)|^2δ( r_i^Ncm- r), r_i^Ncm = r_i-( x_𝒩-1+ r_K̅), where r_i^Ncm denotes the i-th nucleon coordinate measured from the center-of-mass system of nucleons; ρ_N^Ncm(r) is normalized as ∫4π r^2ρ_N^Ncm(r)=𝒩-1. Here, again, we adopt the standard normalization condition (<ref>). The comparison of ρ_N^Ncm(r) with the corresponding quantity of the normal nuclei with 𝒩-1 nucleons shows the effect of the modification of the distribution of the nucleons by the presence of the antikaon. We also calculate the nucleon and antikaon density distribution ρ_N and ρ_K̅ measured from the total center-of-mass system defined as: ρ_N(r) =∑_i=1^𝒩-1∫ d x|Ψ_JMM_T( x)|^2δ( r_i^cm- r), ρ_K̅(r) =∫ d x|Ψ_JMM_T( x)|^2δ( r_K̅^cm- r), where r_i(K̅)^cm= r_i(K̅)- x_𝒩 is i-th nucleon (antikaon) coordinate from the total center-of-mass coordinate x_𝒩. It is also instructive to estimate the fractions of different components in the wavefunctions. We define the projections onto the component with K^- [Eq. (<ref>)] and that with K̅^0 [Eq. (<ref>)] as P̂_K^- =1/2(1-τ_𝒩^(3)), P̂_K̅^0 =1/2(1+τ_𝒩^(3)), with P̂_K^-+P̂_K̅^0=1. By taking the expectation value of Eq. (<ref>), we obtain the probability of finding each component in the wavefunction P_K^- =⟨P̂_K^-⟩ P_K̅^0 =⟨P̂_K̅^0⟩ . The projection operators can also be used to decompose the eigenenergy into different contributions from each term of the Hamiltonian. For this purpose, we use the Gamow state normalization (<ref>). The expectation values of the kinetic energy and potential energy of the diagonal K^- channel ⟨ T⟩^K^-_G and ⟨ V⟩^K^-_G, and of the diagonal K̅^0 channel ⟨ T⟩_G^K̅^0 and ⟨ V⟩_G^K̅^0, and of the off-diagonal K^--K̅^0 channel ⟨ V⟩_G^K^-K̅^0 are given by [ ⟨ T + V⟩^K^-_G ⟨ V⟩_G^K^-K̅^0; ⟨ V⟩_G^K^-K̅^0 ⟨ T + V⟩_G^K̅^0 ] ≡[ ⟨P̂_K^-(T + V)P̂_K^-⟩_G ⟨P̂_K^-VP̂_K̅^0⟩_G; ⟨P̂_K̅^0 VP̂_K^-⟩_G ⟨P̂_K̅^0(T + V)P̂_K̅^0⟩_G ]. With these definitions, the eigenenergy is decomposed as -B-iΓ/2 =⟨ T⟩^K^-_G +⟨ T⟩^K̅^0_G +⟨ V⟩^K^-_G +⟨ V⟩^K̅^0_G +2⟨ V⟩_G^K^-K̅^0 . We also investigate the probability of finding each K̅N isospin component in the wavefunction by using the following expectation values, P_K̅N^I=0 =∑_i^𝒩-1⟨P̂^I=0_i𝒩⟩/𝒩-1, P_K̅N^I=1 =∑_i^𝒩-1⟨P̂^I=1_i𝒩⟩/𝒩-1. § RESULTS AND DISCUSSION §.§ Structure of strange dibaryon resonances K̅NN We proceed now to investigate the structure of the kaonic nuclei. For the three-body systems, we investigate the structure of the I=1/2 quasi-bound states for the strange dibaryon resonances K̅NN, K^-pp-K̅^0pn and K^-pn-K̅^0nn systems, with J^π = 0^- and 1^- in the charge-basis representation. As in the previous studies <cit.>, we find one quasi-bound state below the Λ(1405)+N threshold for J^π=0^-, but we could not find any states below the threshold for J^π=1^-. We summarize the detailed properties of the K^-pp-K̅^0pn and K^-pn-K̅^0nn systems with J^π=0^- in Tables <ref> and <ref>, respectively. We first compare the results with different choices of the two-body energy (Types I and II) discussed in Sec. <ref>. The real part of δ√(s) with Type II is about a half of that of Type I. The binding energy of K^-pp-K̅^0pn system is not sensitive to the choice of the two-body energy and the values are around 27 MeV. Meanwhile, the decay width of Type I (∼ 31 MeV) becomes about a half of that of Type II (∼ 59 MeV). The rms distances √(⟨ r_NN^2 ⟩), √(⟨ r_K̅N^2 ⟩), √(⟨ r_N^2 ⟩), and √(⟨ r_K^2 ⟩) of Type II are slightly smaller than those of Type I. The probabilities of finding several components, P_K^-, P_K̅^0, P_K̅N^I=0 and P_K̅N^I=1 are not sensitive to the choice of the two-body energy. We obtain the binding energies at 20-30 MeV, which are consistent with recent experimental measurement of the ^3 He(K^-,Λ p)n reaction by J-PARC E15 <cit.> and its theoretical analysis <cit.>. The obtained binding energies with the SIDDHARTA potential are 5-10 MeV larger than the values obtained in Refs. <cit.> with the chiral potential constructed in Ref. <cit.>. This difference mainly comes from the different treatment of the imaginary part of the potential as well as the difference of the potential model. In Refs. <cit.>, the Schrödinger equation is solved only with the real part of Hamiltonian and estimate the decay width by taking the expectation value with the imaginary part of the potential, while we solve the Schrödinger equation by direct diagonalization with the full complex Hamiltonian. If we do not take into account the energy dependence of the K̅N potential, the binding energies obtained by direct diagonalization of the complex potential V_K̅N(r,E) become smaller than those obtained by using the real part of the potential Re[V_K̅N(r,E)]. However, the K̅N two-body energy E in the three-body system also becomes a complex value with the complex potential V_K̅N(r,E). Considering the energy dependence of the potential, the K̅N interaction V_K̅N(r,E) becomes more attractive than the K̅N interaction V_K̅N(r,Re[E]) on the real energy axis. This is found in Fig. <ref> which plots the real part of V_K̅N on the complex energy plane. As a result of self-consistent calculation, the binding energies of the complex potential V_K̅N(r,E) become larger than those of only the real part of potential Re[V_K̅N(r,E)]. Therefore, we obtain the binding energies 5-10 MeV larger than those obtained in Refs. <cit.>. We then investigate the origin of the binding using the decomposition in Eq. (<ref>). From Tables <ref> and <ref>, we see that Re ⟨ T⟩^K^-,K̅^0_G almost cancels out Re ⟨ V⟩^K^-,K̅^0_G in both K^- and K̅^0 channels. Therefore the K^--K̅^0 channel coupling is essential for the energy gain. Meanwhile, both of the diagonal and the off-diagonal components contribute to the decay width. It is also instructive to decompose the wavefunction into the isospin components. The dominant isospin component of the K̅NN ground state is considered to have I_NN=1 and total isospin I=1/2. This is because the I_K̅N=0 channel has stronger attraction than that of the I_K̅N=1 channel, and the I_NN=1 channel gives more I_K̅N=0 component than that of the I_NN=0 channel <cit.>. This can easily be explained by recoupling the isospins in the following way: [[η(N)_1/2 η(N)_1/2]_1η(K̅)_1/2]_1/2 =√(3)/2[η(N)_1/2[η(N)_1/2η(K̅)_1/2]_0]_1/2 +1/2[η(N)_1/2[η(N)_1/2η(K̅)_1/2]_1]_1/2, [[η(N)_1/2 η(N)_1/2]_0η(K̅)_1/2]_1/2 =-1/2[η(N)_1/2[η(N)_1/2η(K̅)_1/2]_0]_1/2 +√(3)/2[η(N)_1/2[η(N)_1/2η(K̅)_1/2]_1]_1/2. If the ground state is a pure I=1 NN state, the ratio of the probabilities of finding the K̅N I=0 and I=1 channels is given by P_K̅N^I=0:P_K̅N^I=1= 3:1. We can further decompose this state by the third component of the isospin of the antikaon as [[η(N)_1/2 η(N)_1/2]_1η(K̅)_1/2]_1/2,1/2 =-√(2/3)[η(N)_1/2η(N)_1/2]_1,1η(K̅)_1/2,-1/2 +√(1/3)[η(N)_1/2η(N)_1/2]_1,0η(K̅)_1/2,1/2. where the first (second) term corresponds to K^-pp (K̅^0pn). This leads to the ratio of the probabilities of the K^-pp and K̅^0pn components as P_K^-:P_K̅^0= 2:1. The obtained P_K̅N^Is in Tables <ref> and <ref> well satisfy these relations. The small deviations from the ideal ratios 2:1 and 3:1 come from the contributions of isospin-singlet NN component with odd wave and the Coulomb interaction that induces the isospin mixing. Because the Coulomb interaction is included in our formalism, the energy splitting of the two members of the isospin doublet, K^-pp-K̅^0pn and K^-pn-K̅^0nn, appears. The splitting between these two systems is very small, Δ B=B(K^-pp-K̅^0pn)-B(K^-pn-K̅^0nn) ∼ 0.5 MeV. There are two attractive and one repulsive Coulombic pairs in the K^-pp channel, and one attractive Coulombic pairs in the K^-pn channel. Because P_K^-=0.65 in K^-pp-K̅^0pn is much larger than P_K^-=0.38 in K^-pn-K̅^0nn, the Coulomb interaction affects more attractively the K^-pp-K̅^0pn system than the K^-pn-K̅^0nn system. Next, we show the particle density distributions of the K^-pp-K̅^0pn system. Figure <ref> plots the density distributions of nucleons ρ_N^Ncm(r) defined in Eq. (<ref>). The density distribution of the deuteron is also plotted for comparison. The central nucleon density (r≲ 0.3 fm) is suppressed due to the repulsive core of the nuclear force employed. The nuclear system in the K̅NN system becomes more compact than in the deuteron. The shrinkage effect of nucleons with Type II is slightly stronger than that with Type I. This is because the K̅N interactions with Type II is more attractive than those with Type I due to different δ√(s). In Fig. <ref>, we also show the nucleon and antikaon density distribution ρ_N and ρ_K̅ defined in Eqs. (<ref>) and (<ref>). The nucleon density is suppressed at around the origin, while the antikaon density distribution is not suppressed since there are no repulsive core in the K̅N potential. We also perform the same calculations by employing the phenomenological potential model, so-called Akaishi-Yamazaki (AY) potential <cit.>. The results are shown in Tables <ref> and <ref> and Figs. <ref> and <ref>. The AY potential is more attractive than the SIDDHARTA potential in the subthreshold energy region. The binding energies of the K̅NN systems are approximately twice of those with the SIDDHARTA potential, and the decay widths are Γ∼ 62 MeV. The particle density distributions become more compact, and the rms radii are about 0.85 times smaller than those for the SIDDHARTA potential. Our results for AY potential are comparable with the results in Ref. <cit.>. §.§ Structure of K̅NNN quasi-bound state Next, we investigate the structure of the four-body system, strange tribaryon K̅NNN system with J^π=1/2^-. We find a quasi-bound state in the K^-ppn-K̅^0pnn coupled system (≡ ^3_K̅H). Our results are listed in Table <ref>. We also investigate the K^-ppp-K̅^0ppn coupled system (≡ ^3_K̅He) with J^π=1/2^-, but we do not find any states below the (K̅NN)+N threshold. We see a similar trend, shown in the three-body sector, in the dependence of the choice of the two-body energy (Types I and II). For Type II the real part of δ√(s) is less than a half of those with Type I. Therefore the K̅N attraction with Type II is stronger than that with Type I, and the binding energy with Type II is larger than that with Type I. The decay width with Type II (∼ 69 MeV) becomes around three times as large as that with Type I (∼ 26 MeV). The obtained binding energies by using the SIDDHARTA potential are 15-20 MeV larger than the values obtained in Ref. <cit.>. As discussed in the three-body sector, this difference mainly comes from the different treatment of the imaginary part of the potential as well as the difference of the potential model. Since the number of K̅N pairs is larger, the effects of those differences are stronger in the four-body systems than those in the three-body systems. The rms distances √(⟨ r_NN^2 ⟩), √(⟨ r_K̅N^2 ⟩), √(⟨ r_N^2 ⟩), and √(⟨ r_K^2 ⟩) with Type II are slightly smaller than those with Type I, in accordance with the larger binding. The probabilities of finding the K^-ppn (P_K^-) and K̅^0pnn (P_K̅^0) channels are not sensitive to the choice of the two-body energies. A large contribution to the real part from 2⟨ V⟩_G^K^-K̅^0 indicates that the K^--K̅^0 channel coupling is essential for gaining the binding energy for four-body systems, while the diagonal channels also give contributions to the decay width. The K^-ppn and K̅^0pnn channels are isospin mirror states. Therefore, the probabilities of these two components follow P_K^-:P_K̅^0= 1:1 with the isospin symmetric K̅N and NN interactions. In fact, the numerical results in Table <ref> are consistent with this expectation within a small isospin mixing by the Coulomb interaction. There are two attractive and one repulsive Coulombic pairs in the K^-ppn channel, while there are no Coulomb interacting pair in the K̅^0pnn channel. The Coulomb interaction in total affects attractive in the K^-ppn channel, and the P_K^- becomes slightly larger than the P_K̅^0. Figure <ref> displays ρ_N^Ncm(r) and r^2ρ_N^Ncm(r) of the ^3_K̅H system. In the K̅NNN system, the nuclear system becomes more compact, and the central density becomes about two times larger than that in the ^3He. Since the K̅N interaction with Type II is more attractive than that with Type I due to small magnitude of the real part of δ√(s), the shrinkage effect of nucleons with Type II is slightly stronger than that with Type I. In Fig. <ref>, we show the nucleon and antikaon density distribution, ρ_N and ρ_K̅. Those density distributions are similar to each other, and the antikaon rms radius √(⟨ r_K̅^2⟩) is well comparable with the nucleon rms radius √(⟨ r_N^2⟩). The antikaon moves in the whole region of the nuclear system in order to gain the energy from the strong K̅N interaction. With the AY potential, the binding energy of the K̅NNN system is about 30 MeV larger than the binding energies with the SIDDHARTA potential, and the decay width is Γ∼ 79 MeV. The particle density distributions and the rms radii are similar to those obtained with the SIDDHARTA potential with Type II. It is now clear that the deeply bound (B>100 MeV in Ref. <cit.>) and high density (8.2 times of the normal density shown in Ref. <cit.>) K̅NNN state is not realized in the accurate few-body calculation. Such an extreme result can be regarded as an artifact due to the approximated treatment of the few-body systems. We, however, emphasize that the existence of the bound state is confirmed also with the realistic SIDDHARTA K̅N interaction, and the central density of nucleons can be about two times larger than that in the ^3He. §.§ Structure of K̅NNNN quasi-bound state Next, we investigate the structure of the five-body systems, strange tetrabaryon K̅NNNN systems with J^π=0^-. We find isospin-doublet quasi-bound states in the K^-pppn-K̅^0ppnn (≡ ^4_K̅He) and K^-ppnn-K̅^0pnnn (≡ ^4_K̅H) systems. Our results are listed in Tables <ref> and <ref>. The binding energy with Type II is 5 MeV larger than that with Type I. The decay width with Type II (∼ 75 MeV) is about three times larger than that with Type I (∼ 28 MeV). The rms distances √(⟨ r_NN^2 ⟩), √(⟨ r_K̅N^2 ⟩), √(⟨ r_N^2 ⟩), and √(⟨ r_K^2 ⟩) with Type II are smaller than those with Type I. The probabilities P_K^- and P_K̅^0 are not sensitive to the choice of Types I and II. In contrast to these features, the energy decomposition of the K̅NNNN exhibits different characteristics from the three- and four-body systems. The incomplete cancellation of the kinetic energy and potential energy in the dominant component (⟨ T⟩^K̅^0_G+⟨ K⟩^K̅^0_G in ^4_K̅He and ⟨ T⟩^K^-_G+⟨ K⟩^K^-_G in ^4_K̅H) leaves sizable contributions to the binding energy, which are comparable with the off-diagonal components 2⟨ V⟩_G^K^-K̅^0. For the decay widths, the diagonal channels are also important as in the four-body systems. From the results of P_K^- and P_K̅^0, we see that the dominant component in the ^4_K̅He (^4_K̅H) system is the K̅^0ppnn (K^-ppnn) channel, although the K^-pppn (K̅^0pnnn) channel contains more K̅N I=0 components than the K̅^0ppnn (K^-ppnn) channel. This is because the nucleon contribution of the K̅^0ppnn (K^-ppnn) channel, which can form an α-particle configuration giving the binding energy about 30 MeV, is larger than the nucleon contribution of the K^-pppn (K̅^0pnnn) channel, and thus the K̅^0ppnn (K^-ppnn) channel is favored. It is also for this reason that the K̅^0 (K^-) diagonal component gains the binding energy in the ^4_K̅He (^4_K̅H) system, as we see above. The Coulomb splitting between these two systems is larger than that in the K̅NN systems, Δ B=B(^4_K̅He)-B(^4_K̅H) ∼ 2 MeV. There are one repulsive Coulombic pair in the K̅^0ppnn channel which is the dominant component of the ^4_K̅He system, and two attractive and one repulsive Coulombic pairs in the K^-ppnn channel which is the dominant component of the ^4_K̅H system. Therefore, Coulomb interaction is repulsive in the ^4_K̅He system and attractive in the ^4_K̅H system, and the Coulomb splitting becomes larger than that of the three-body systems. Figures <ref> and <ref> plot the particle density distributions in the ^4_K̅H system. The nucleons in the K̅NNNN system become more compact, and the central density increases to about 1.3-1.5 times higher than that in the ^4He. As in the three- and four-body systems, the shrinkage effect of nucleons with Type II is slightly stronger than that with Type I. The antikaon density distribution is similar to the nucleon density distribution. When we use the AY potential, the binding energies of the K̅NNNN system are about 12-24 MeV larger than those with the SIDDHARTA potential. Because the quasi-bound state appears above the πΣ NNN threshold, it has a sizable decay width of about Γ∼ 87 MeV, in contrast to the narrow state predicted in Ref. <cit.>. The probability P_K^- (P_K̅^0) in the ^4_K̅He (^4_K̅H) system becomes larger than the result with the SIDDHARTA potential because the AY potential model has more attractive K̅N I=0 interaction than that in the SIDDHARTA potential. As a result, P_K̅N^I=0 is enhanced. The rms radius of the antikaon is smaller than the corresponding result of the SIDDHARTA potential with Type I, whereas the nucleon radius is slightly larger. §.§ Nuclear force dependence Here we discuss the NN interaction dependence of our results by comparing the results with the AV4' potential and those with other NN interaction models such as Afnan-Tang S3 (ATS3) <cit.> and Minnesota (MN) potential models <cit.>. These potential models are often used in studying light nuclei, and well reproduce the binding energy of the s-shell nuclei. It is noted that the strengths of the repulsive core are quite different between these three models as displayed in Fig. <ref>. The AV4' potential has the strongest repulsive core, which is comparable to the realistic nuclear forces such as the Argonne V18 potential model <cit.>. The ATS3 potential model has also a strong short-range repulsion at around the origin. The repulsive core of the MN potential is quite soft. Since the K̅N interaction is strongly attractive, and the kaonic nuclei become more compact than ordinary nuclei. Therefore, there is a possibility that these different repulsive cores affect the results of the kaonic nuclei. Figure <ref> displays the binding energies and decay widths of the K^-pp-K̅^0pn (J^π=0^-), _K̅^3H (J^π=1/2^-) and _K̅^4H (J^π=0^-) systems with those three types of nuclear forces. Here, we use the SIDDHARTA potential as the K̅N potential, and the energy dependence of the potential is determined by Type I. The binding energies and decay widths are almost the same in those with three nuclear forces as well as the binding energies of ordinary s-shell nuclei without the antikaon as listed in Table <ref>. The qualitative difference becomes apparent in the density distributions. Figures <ref> and <ref> plot the nucleon and antikaon density distributions of the K^-pp-K̅^0pn (J^π=0^-), ^3_K̅H (J^π=1/2^-) and ^4_K̅H (J^π=0^-) systems. The density distributions with the AV4' potential model are similar to those with the ATS3 model, while the central densities with the MN potential with repulsive core are significantly higher than those for the other potential models. Since the short-range repulsive core of the MN potential is not as strong as those of the other potential models, nucleons can come very close to each other due to the strong K̅N attraction. In the ^3_K̅H system, the central density obtained with the MN potential model becomes ρ_N^Ncm(r=0)∼ 1.2 fm^-3, approximately two times larger than those with the AV4' and ATS3 potential models. The value is close to 1.4 fm^-3 predicted in Ref. <cit.> by using the antisymmetrized-molecular-dynamics calculation with the effective treatment of the K̅N and NN interactions with the g-matrix. Since the K̅N interaction is very strong, the nucleons can be compressed too much with a soft core potential and form such an unrealistically high density state. Use of the realistic nuclear force is necessary in order to avoid such an artificial solution. §.§ K̅NNNNN quasi-bound state For the six-body K̅NNNNN systems, we could not find any states below the strange tetrabaryon and a nucleon (K̅NNNN)+N threshold energy in the L=0 state. Since the ^5He ground state is observed as a resonant state with J^π=3/2^-, the orbital angular momentum of the ground state is expected to be L=1. Investigation with L>0 states is possible by introducing the global vectors that efficiently describe the rotational motion of the system with any L^π <cit.>, but this extension is beyond the scope of this paper. §.§ Structure of K̅NNNNNN quasi-bound state Finally, we investigate the structure of the seven-body systems, strange hexabaryon K̅NNNNNN. The ground state of the six-nucleon systems without an antikaon is ^6Li with J^π=1^+, and the J^π=0^+ isospin-triplet states, ^6He, ^6Li and ^6Be, are the excited states. By adding an antikaon, we can construct two isospin doublets with J^π=1^- and 0^-. We find quasi-bound states in these quantum numbers, while we do not find any states below the (K̅NNNN)+2N threshold for I=3/2 with J^π=0^-, such as K^-ppnnnn-K̅^0pnnnnn (≡ ^6_K̅H) system. Tables <ref>, <ref>, <ref>, and <ref> list our results of K^-ppppnn-K̅^0pppnnn (≡ ^6_K̅Li) system with J^π=0^-, K^-pppnnn-K̅^0ppnnnn (≡ ^6_K̅He) system with J^π=0^-, ^6_K̅Li system with J^π=1^-, and ^6_K̅He system with J^π=1^-, respectively. The binding energies for the J^π=0^- (1^-) states with Type II is 9-10 MeV (7-8 MeV) larger than those with Type I. The decay widths with Type II are three times larger than those with Type I. The rms distances √(⟨ r_NN^2 ⟩), √(⟨ r_K̅N^2 ⟩), √(⟨ r_N^2 ⟩), and √(⟨ r_K^2 ⟩) with Type II are slightly smaller than those with Type I. The probabilities P_K^- and P_K̅^0 are not sensitive to the choice of Types I and II. As in the case of four- and five-body systems, the diagonal channels give important contributions to the decay width. In contrast, both of the diagonal and the off-diagonal components produce about a half of the binding energy. The diagonal components of the channel with ^6Li, that is, the K^- channel in ^6_K̅He and the K̅^0 channel in ^6_K̅Li, are important especially for the J^π=1^- states. The dominant component of the ^6_K̅Li (^6_K̅He) system with J^π=0^- is the K^-ppppnn (K̅^0ppnnnn) channel, while that of the ^6_K̅Li (^6_K̅He) system with J^π=1^- is the K̅^0pppnnn (K^-pppnnn) channel. For the spin-singlet states (J^π=0^-), the core nuclei in ^6_K̅Li and ^6_K̅He are the isospin-triplet states of ^6Be, ^6Li and ^6He, and the channels with larger fraction of the K̅N I=0 components are favored. Meanwhile, for the spin-triplet states (J^π=1^-), the core nucleus with J^π=1^+ (∼ ^6Li) is the isospin-singlet state. The spin-triplet ^6Li is the ground state of the six-nucleon systems, while ^6Be and ^6He with J^π=1^+ are not bound. Therefore, the nucleons in the K̅^0pppnnn (K^-pppnnn) channel feel larger attraction than that in the other channel. This determines the dominant component in the J^π=1^- state. This is also the reason why the K̅^0 (K^-) diagonal component gains the large binding energy in the spin-singlet ^6_K̅Li (^6_K̅He) system as similar to the K̅NNNN system. The Coulomb splitting between ^6_K̅Li and ^6_K̅He in the J^π=0^- channel (0.3-0.8 MeV) is smaller than the splitting in J^π=1^- (2.0-3.2 MeV). In the dominant K^-ppppnn component of the ^6_K̅Li with J^π=0^-, there are four attractive and six repulsive Coulombic pairs, while the dominant K̅^0ppnnnn channel in ^6_K̅He contains one repulsive pair. Therefore, the Coulomb interaction in ^6_K̅Li system is expected to be slightly repulsive than ^6_K̅He system. On the other hand, in the dominant K̅^0pppnnn component of the ^6_K̅Li system with J^π=1^-, there are three repulsive Coulombic pairs, and the dominant K^-pppnnn channel in ^6_K̅He has three attractive and three repulsive pairs. In the ^6_K̅He system, the rms distance √(⟨ r^2_K̅N⟩) is smaller than √(⟨ r^2_NN⟩), and therefore, the K̅N Coulomb attraction is stronger than the NN repulsion, and the Coulomb interaction works in total attractively in the K^-pppnnn channel. Therefore, the Coulomb splitting between ^6_K̅Li and ^6_K̅He in J^π=1^- becomes larger than the splitting in J^π=0^-. Next, we compare the spin-singlet and triplet states. With Type I, the binding energy of ^6_K̅Li with J^π=1^- is 1 MeV larger than the J^π=0^- state, and ^6_K̅He with J^π=1^- is 2.2 MeV larger than in J^π=0^-. With Type II, the binding energy of ^6_K̅Li with J^π=1^- is 2.2 MeV smaller than the J^π=0^- state, and ^6_K̅He with J^π=1^- is 0.8 MeV larger than the J^π=0^- state. Except for ^6_K̅Li with Type II, the binding energies of the spin-triplet states are larger than the spin-singlet states. This is in accordance with the level structure of the six-nucleon systems. With Type II, the magnitude of the real part of δ√(s) used in the two-body K̅N interaction is smaller, and the K̅N interaction becomes more attractive than that with Type I. Because the J^π=0^- state contains larger fraction of the I=0 K̅N components than the J^π=1^- state, as in the case of the strange dibaryon K̅NN systems, the K̅N interaction with Type II is so strong that the binding energy of ^6_K̅Li with J^π=0^- becomes larger than the J^π=1^- state. In other words, by adding an antikaon, the spin of the ground state of the six-nucleon system may change depending on the strength of the K̅N interaction. The inversion of the level structure of the ground state and the first excited state by the antikaon is also seen in the two-nucleon systems (strange dibaryon K̅NN). The ground state of the two-nucleon sector without the antikaon is the spin-triplet deuteron, while the spin-singlet channel is unbound. As discussed in Sec. <ref>, by injecting an antikaon, the ground state is spin singlet which maximizes the fraction of the K̅N(I=0) component. Recalling that the ^6He and ^6Li are well approximated by an α+N+N three-body model (See, for example, Ref. <cit.> and references therein), the difference of J^π=1^- and J^π=0^- states of seven-body systems can be essentially caused by the difference of the K̅NN subsystems. Similar to the three-body systems, the level inversion can take place also in the seven-body systems, which is driven by the balance between the nuclear structure and the attraction in the K̅N system. This property is more pronounced if the strength of the K̅N attraction is further increased. For instance, when we use the AY potential which is more attractive than the SIDDHARTA potential, the binding energies of ^6_K̅Li and ^6_K̅He with J^π=0^- become 10 and 7 MeV larger than those with J^π=1^-. Namely, the stronger K̅N attraction leads to more drastic level inversion in the seven-body systems. In this way, there is a possibility to extract the information on the K̅N interaction not only from the binding energies but also from the ground state quantum number J^π and from the splitting between J^π=0^- and 1^- states. Finally, we show, in Figs. <ref>,  <ref>, <ref>, and <ref>, the particle density distributions of the ^6_K̅He system with J^π=0^- and 1^-. The nucleon density distributions of the ^6He (J^π=0^+) and ^6Li (J^π=1^+) systems are also plotted for comparison. In the K̅NNNNNN system with J^π=0^-, the central nucleon density becomes slightly larger than that with J^π=1^- and about two times larger than that in the ^6Li. Meanwhile, the central antikaon density with J^π=0^- is slightly smaller than that with J^π=1^-. In Fig. <ref>, we summarize the nucleon and antikaon density distributions of various kaonic nuclei from three- to seven-body systems. The central nucleon densities of the seven-body systems become about a half, and antikaon densities become one third of the densities of the four- and five-body systems. The central nucleon densities of the five-body systems are highest in the light kaonic nuclei up to seven-body systems, while the densities are not as high as those suggested by using the effective interaction based on g-matrix approach in Refs. <cit.>. The large nucleon density in the five-body system is mainly caused by the formation of an α-particle configuration, rather than maximizing the K̅N(I=0) pairs. For the antikaon distribution, the central densities become smaller as the number of nucleons increases. Because the antikaon feels attraction from all the nucleons, its spatial extent increases in a large nucleus. § SUMMARY We have studied structure of the light kaonic nuclei, K̅NN, K̅NNN, K̅NNNN, and K̅NNNNNN with a powerful few-body approach, the correlated Gaussian (CG) method. Fully converged three- to seven-body solutions are obtained by the stochastic variational method (SVM). As a realistic K̅N interaction, we employ the SIDDHARTA potential constructed based on the NLO chiral SU(3) dynamics with the SIDDHARTA constraint obtained from Refs. <cit.>. We find one quasi-bound state in the K̅NN, K̅NNN, and K̅NNNN systems, and two quasi-bound states with J^π=0^- and 1^- in the K̅NNNNNN system. All the states are found above the πΣ emission threshold. The central densities of nucleons are enhanced by an injected antikaon, and become about two times larger than those without an antikaon. The central nucleon density reaches its maximum in the K̅NNNN system with J^π=0^-, where the nucleons can form an α-particle configuration. The rms radius of the antikaon increases along with the nucleon rms radius when the mass number is increased. By decomposing the eigenenergy into different contributions, we find that the K^--K̅^0 channel coupling is important for the binding of the light kaonic nuclei. For the K̅NN, K̅NNN and K̅NNNNNN with J^π=0^-, the core nuclei belong to same isospin multiplet, and the mixing between K^- and K̅^0 channels and the energy gains from the off-diagonal components are large. Meanwhile, for the K̅NNNN and K̅NNNNNN with J^π=1^-, the channel with core nucleus ^4He or ^6Li is dominant, and the energy gains from both of the off-diagonal and diagonal components with ^4He or ^6Li become large. To take into account the energy dependence of the SIDDHARTA potential, we examine two methods to determine the K̅N two-body energy in 𝒩-body systems (Types I and II). Quantitatively, the binding energies with Type II become gradually larger than those with Type I as the number of particles increase, and the decay widths with Type II become 2-3 times larger than those with Type I. The qualitative features of the kaonic nuclei are not sensitive to the choice of the method. In order to examine the predictions of deeply-bound and high-density kaonic nuclei <cit.>, we also use the AY potential model. When we employ the AY potential, the binding energies are about 20-30 MeV larger than those with the SIDDHARTA potential for each system, and decay widths become around 60-80 MeV. Even in this case, the obtained binding energies up to seven-body systems except that K̅NNNNNN with J^π=0^- are smaller than 100 MeV predicted by using the optical potential approach <cit.>. The central density of the K̅NNNN system is not as high as those suggested by using effective K̅N and NN interactions based on the g-matrix approach in Ref. <cit.>. The comparison of the two K̅N potential models (SIDDHARTA and AY) leads to interesting results in the seven-body systems. If the K̅N attraction is not so strong, we see the spin-triplet ground (J^π=1^-) and the spin-singlet (J^π=0^-) excited states reflecting the lightest core nucleus, ^6Li with J^π=1^+. If the K̅N interaction is strong enough as the AY potential, the level ordering of the J^π=0^- and J^π=1^- states is inverted. Therefore, it is possible to extract the information on the K̅N interaction from the ground state quantum number J^π as well as the energy splitting between J^π=0^- and 1^- of the seven-body kaonic states. In this work, we employ the single channel K̅N interaction where the πΣ channel coupling effect are renormalized into its imaginary part. This potential model reproduces the two-pole structure of Λ(1405), while we could not find two-pole structure in the kaonic nuclei. One of these poles with the large binding energy and width is originated from the πΣ resonance pole. In order to study the effect of the other pole in the kaonic nuclei, it may be necessary to take into account the channel coupling effect of K̅N-πΣ explicitly. The work in this direction is underway and will be reported elsewhere. The authors thank A. Gal, A. Ohnishi, A. Doté and Y. Ikeda for helpful comments and discussions. The numerical calculation has been performed on supercomputers (NEC SX-ACE) at the Research Center for Nuclear Physics, Osaka University and (CRAY XC40) at the Yukawa Institute for Theoretical Physics, Kyoto University. This work was partly supported by the Grants-in-Aid for Scientific Research on Innovative Areas from MEXT (Grant No. 2404:24105008), by JSPS KAKENHI Grant No. 24740152, and by the Yukawa International Program for Quark-Hadron Sciences (YIPQS). 53 fxundefined [1] ifx#1 fnum [1] #1firstoftwo secondoftwo fx [1] #1firstoftwo secondoftwo noop [0]secondoftwo ref[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0] rl [1]href #1 ifxundefined oi [1]@startlink #1doi:#1@endlink oi [1]@startlink#1@Doi Doi [1]#1@endlink [Patrignani et al.(2016)Patrignani et al.]PDG2016 author author C. Patrignani et al. (collaboration Particle Data Group), 10.1088/1674-1137/40/10/100001 journal journal Chin. Phys., volume C40, pages 100001 (year 2016)NoStop [Dalitz and Tuan(1959)]Dalitz:1959dn author author R. Dalitz and author S. Tuan, 10.1103/PhysRevLett.2.425 journal journal Phys. Rev. Lett., volume 2, pages 425 (year 1959)NoStop [Dalitz and Tuan(1960)]Dalitz:1960du author author R. Dalitz and author S. Tuan, 10.1016/0003-4916(60)90001-4 journal journal Annals Phys., volume 10, pages 307 (year 1960)NoStop [Akaishi and Yamazaki(2002)]Akaishi:2002bg author author Y. Akaishi and author T. Yamazaki, 10.1103/PhysRevC.65.044005 journal journal Phys. Rev., volume C65, pages 044005 (year 2002)NoStop [Shevchenko(2012)]Shevchenko:2011ce author author N. Shevchenko, 10.1103/PhysRevC.85.034001 journal journal Phys. Rev., volume C85, pages 034001 (year 2012), http://arxiv.org/abs/1103.4974 arXiv:1103.4974 [nucl-th] NoStop [Yamazaki and Akaishi(2002)]Yamazaki:2002uh author author T. Yamazaki and author Y. Akaishi, 10.1016/S0370-2693(02)01738-0 journal journal Phys. Lett., volume B535, pages 70 (year 2002)NoStop [Dote et al.(2004)aDote, Akaishi, Horiuchi, and Yamzaki]Dote:2002db author author A. Dote, author Y. Akaishi, author H. Horiuchi, and author T. Yamzaki, 10.1016/j.physletb.2004.03.046 journal journal Phys. Lett., volume B590, pages 51 (year 2004a), http://arxiv.org/abs/nucl-th/0207085 arXiv:nucl-th/0207085 [nucl-th] NoStop [Dote et al.(2004)bDote, Horiuchi, Akaishi, and Yamazaki]Dote:2003ac author author A. Dote, author H. Horiuchi, author Y. Akaishi, and author T. Yamazaki, 10.1103/PhysRevC.70.044313 journal journal Phys.Rev., volume C70, pages 044313 (year 2004b), http://arxiv.org/abs/nucl-th/0309062 arXiv:nucl-th/0309062 [nucl-th] NoStop [Kaiser et al.(1995)Kaiser, Siegel, and Weise]Kaiser:1995eg author author N. Kaiser, author P. Siegel, and author W. Weise, 10.1016/0375-9474(95)00362-5 journal journal Nucl. Phys., volume A594, pages 325 (year 1995), http://arxiv.org/abs/nucl-th/9505043 arXiv:nucl-th/9505043 [nucl-th] NoStop [Oset and Ramos(1998)]Oset:1997it author author E. Oset and author A. Ramos, 10.1016/S0375-9474(98)00170-5 journal journal Nucl. Phys., volume A635, pages 99 (year 1998), http://arxiv.org/abs/nucl-th/9711022 arXiv:nucl-th/9711022 [nucl-th] NoStop [Oller and Meissner(2001)]Oller:2000fj author author J. Oller and author U. G. Meissner, 10.1016/S0370-2693(01)00078-8 journal journal Phys. Lett., volume B500, pages 263 (year 2001), http://arxiv.org/abs/hep-ph/0011146 arXiv:hep-ph/0011146 [hep-ph] NoStop [Hyodo and Jido(2012)]Hyodo:2011ur author author T. Hyodo and author D. Jido, 10.1016/j.ppnp.2011.07.002 journal journal Prog. Part. Nucl. Phys., volume 67, pages 55 (year 2012), http://arxiv.org/abs/1104.4474 arXiv:1104.4474 [nucl-th] NoStop [Ikeda et al.(2011)Ikeda, Hyodo, and Weise]Ikeda:2011pi author author Y. Ikeda, author T. Hyodo, and author W. Weise, 10.1016/j.physletb.2011.10.068 journal journal Phys. Lett., volume B706, pages 63 (year 2011), http://arxiv.org/abs/1109.3005 arXiv:1109.3005 [nucl-th] NoStop [Ikeda et al.(2012)Ikeda, Hyodo, and Weise]Ikeda:2012au author author Y. Ikeda, author T. Hyodo, and author W. Weise, 10.1016/j.nuclphysa.2012.01.029 journal journal Nucl. Phys., volume A881, pages 98 (year 2012), http://arxiv.org/abs/1201.6549 arXiv:1201.6549 [nucl-th] NoStop [Bazzi et al.(2011)Bazzi, Beer, Bombelli, Bragadireanu, Cargnelli et al.]Bazzi:2011zj author author M. Bazzi, author G. Beer, author L. Bombelli, author A. Bragadireanu, author M. Cargnelli, et al., 10.1016/j.physletb.2011.09.011 journal journal Phys. Lett., volume B704, pages 113 (year 2011), http://arxiv.org/abs/1105.3090 arXiv:1105.3090 [nucl-ex] NoStop [Bazzi et al.(2012)Bazzi, Beer, Bombelli, Bragadireanu, Cargnelli et al.]Bazzi:2012eq author author M. Bazzi, author G. Beer, author L. Bombelli, author A. Bragadireanu, author M. Cargnelli, et al., 10.1016/j.nuclphysa.2011.12.008 journal journal Nucl.Phys., volume A881, pages 88 (year 2012), http://arxiv.org/abs/1201.4635 arXiv:1201.4635 [nucl-ex] NoStop [Miyahara and Hyodo(2016)]Miyahara:2015bya author author K. Miyahara and author T. Hyodo, 10.1103/PhysRevC.93.015201 journal journal Phys. Rev., volume C93, pages 015201 (year 2016), http://arxiv.org/abs/1506.05724 arXiv:1506.05724 [nucl-th] NoStop [Hyodo and Weise(2008)]Hyodo:2007jq author author T. Hyodo and author W. Weise, 10.1103/PhysRevC.77.035204 journal journal Phys. Rev., volume C77, pages 035204 (year 2008), http://arxiv.org/abs/0712.1613 arXiv:0712.1613 [nucl-th] NoStop [Jido et al.(2003)Jido, Oller, Oset, Ramos, and Meissner]Jido:2003cb author author D. Jido, author J. Oller, author E. Oset, author A. Ramos, and author U. Meissner, 10.1016/S0375-9474(03)01598-7 journal journal Nucl. Phys., volume A725, pages 181 (year 2003), http://arxiv.org/abs/nucl-th/0303062 arXiv:nucl-th/0303062 [nucl-th] NoStop [Meissner and Hyodo(2016)]Minireview author author U.-G. Meissner and author T. Hyodo, http://pdg.lbl.gov/2015/reviews/rpp2015-rev-lam-1405-pole-struct.pdf journal journal Pole Structure of the Λ(1405) Region in Review of Particle Physics, Chin. Phys., volume C40, pages 100001 (year 2016)NoStop [Ohnishi et al.(2016)Ohnishi, Ikeda, Hyodo, and Weise]Ohnishi:2015iaq author author S. Ohnishi, author Y. Ikeda, author T. Hyodo, and author W. Weise, 10.1103/PhysRevC.93.025207 journal journal Phys. Rev., volume C93, pages 025207 (year 2016), http://arxiv.org/abs/1512.00123 arXiv:1512.00123 [nucl-th] NoStop [Dote et al.(2008)Dote, Hyodo, and Weise]Dote:2008in author author A. Dote, author T. Hyodo, and author W. Weise, 10.1016/j.nuclphysa.2008.02.001 journal journal Nucl. Phys., volume A804, pages 197 (year 2008), http://arxiv.org/abs/0802.0238 arXiv:0802.0238 [nucl-th] NoStop [Dote et al.(2009)Dote, Hyodo, and Weise]Dote:2008hw author author A. Dote, author T. Hyodo, and author W. Weise, 10.1103/PhysRevC.79.014003 journal journal Phys. Rev., volume C79, pages 014003 (year 2009), http://arxiv.org/abs/0806.4917 arXiv:0806.4917 [nucl-th] NoStop [Ikeda et al.(2010)Ikeda, Kamano, and Sato]Ikeda:2010tk author author Y. Ikeda, author H. Kamano, and author T. Sato, 10.1143/PTP.124.533 journal journal Prog. Theor. Phys., volume 124, pages 533 (year 2010), http://arxiv.org/abs/1004.4877 arXiv:1004.4877 [nucl-th] NoStop [Barnea et al.(2012)Barnea, Gal, and Liverts]Barnea:2012qa author author N. Barnea, author A. Gal, and author E. Liverts, 10.1016/j.physletb.2012.04.055 journal journal Phys. Lett., volume B712, pages 132 (year 2012), http://arxiv.org/abs/1203.5234 arXiv:1203.5234 [nucl-th] NoStop [Dote et al.(2015)Dote, Inoue, and Myo]Dote:2014via author author A. Dote, author T. Inoue, and author T. Myo, 10.1093/ptep/ptv039 journal journal PTEP, volume 2015, pages 043D02 (year 2015), http://arxiv.org/abs/1411.0348 arXiv:1411.0348 [nucl-th] NoStop [Yamazaki and Akaishi(2007)]Yamazaki:2007cs author author T. Yamazaki and author Y. Akaishi, 10.1103/PhysRevC.76.045201 journal journal Phys. Rev., volume C76, pages 045201 (year 2007), http://arxiv.org/abs/0709.0630 arXiv:0709.0630 [nucl-th] NoStop [Wycech and Green(2009)]Wycech:2008wf author author S. Wycech and author A. Green, 10.1103/PhysRevC.79.014001 journal journal Phys. Rev., volume C79, pages 014001 (year 2009), http://arxiv.org/abs/0808.3329 arXiv:0808.3329 [nucl-th] NoStop [Shevchenko et al.(2007)aShevchenko, Gal, and Mares]Shevchenko:2006xy author author N. Shevchenko, author A. Gal, and author J. Mares, 10.1103/PhysRevLett.98.082301 journal journal Phys. Rev. Lett., volume 98, pages 082301 (year 2007a), http://arxiv.org/abs/nucl-th/0610022 arXiv:nucl-th/0610022 [nucl-th] NoStop [Shevchenko et al.(2007)bShevchenko, Gal, Mares, and Revai]Shevchenko:2007ke author author N. Shevchenko, author A. Gal, author J. Mares, and author J. Revai, 10.1103/PhysRevC.76.044004 journal journal Phys. Rev., volume C76, pages 044004 (year 2007b), http://arxiv.org/abs/0706.4393 arXiv:0706.4393 [nucl-th] NoStop [Ikeda and Sato(2007)]Ikeda:2007nz author author Y. Ikeda and author T. Sato, 10.1103/PhysRevC.76.035203 journal journal Phys. Rev., volume C76, pages 035203 (year 2007), http://arxiv.org/abs/0704.1978 arXiv:0704.1978 [nucl-th] NoStop [Ikeda and Sato(2009)]Ikeda:2008ub author author Y. Ikeda and author T. Sato, 10.1103/PhysRevC.79.035201 journal journal Phys. Rev., volume C79, pages 035201 (year 2009), http://arxiv.org/abs/0809.1285 arXiv:0809.1285 [nucl-th] NoStop [Ohnishi et al.(2013)Ohnishi, Ikeda, Kamano, and Sato]Ohnishi:2013rix author author S. Ohnishi, author Y. Ikeda, author H. Kamano, and author T. Sato, 10.1103/PhysRevC.88.025204 journal journal Phys. Rev., volume C88, pages 025204 (year 2013), http://arxiv.org/abs/1302.2301 arXiv:1302.2301 [nucl-th] NoStop [Varga and Suzuki(1995)]Varga:1995dm author author K. Varga and author Y. Suzuki, 10.1103/PhysRevC.52.2885 journal journal Phys. Rev., volume C52, pages 2885 (year 1995), http://arxiv.org/abs/nucl-th/9508023 arXiv:nucl-th/9508023 [nucl-th] NoStop [Varga and Suzuki(1997)]Varga:1997xga author author K. Varga and author Y. Suzuki, 10.1016/S0010-4655(97)00059-3 journal journal Comput. Phys. Commun., volume 106, pages 157 (year 1997), http://arxiv.org/abs/nucl-th/9702034 arXiv:nucl-th/9702034 [nucl-th] NoStop [Suzuki and Varga(1998)]suzuki1998stochastic author author Y. Suzuki and author K. Varga, https://books.google.co.jp/books?id=BpyTpm1VLx8C title Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems, series Lecture Notes in Physics Monographs No. number 54 (publisher Springer, year 1998) ISBN isbn 9783540651529NoStop [Hokkyo(1965)]Hokkyo:1965 author author N. Hokkyo, @noop journal journal Prog. Theor. Phys., volume 33, pages 1116 (year 1965)NoStop [Wiringa and Pieper(2002)]Wiringa:2002ja author author R. B. Wiringa and author S. C. Pieper, 10.1103/PhysRevLett.89.182501 journal journal Phys. Rev. Lett., volume 89, pages 182501 (year 2002), http://arxiv.org/abs/nucl-th/0207050 arXiv:nucl-th/0207050 [nucl-th] NoStop [Audi et al.(2002)Audi, Wapstra, and Thibault]Audi:2002rp author author G. Audi, author A. H. Wapstra, and author C. Thibault, 10.1016/j.nuclphysa.2003.11.003 journal journal Nucl. Phys., volume A729, pages 337 (year 2002)NoStop [Angeli and Marinova(2013)]Angeli201369 author author I. Angeli and author K. Marinova, http://dx.doi.org/10.1016/j.adt.2011.12.006 journal journal Atomic Data and Nuclear Data Tables, volume 99, pages 69 (year 2013), ISSN issn 0092-640XNoStop [Friar et al.(1997)Friar, Martorell, and Sprung]PhysRevA.56.4579 author author J. L. Friar, author J. Martorell, and author D. W. L. Sprung, 10.1103/PhysRevA.56.4579 journal journal Phys. Rev. A, volume 56, pages 4579 (year 1997)NoStop [Martin(1981)]Martin:1980qe author author A. D. Martin, 10.1016/0550-3213(81)90247-9 journal journal Nucl. Phys., volume B179, pages 33 (year 1981)NoStop [Sekihara et al.(2009)Sekihara, Jido, and Kanada-En'yo]Sekihara:2009yk author author T. Sekihara, author D. Jido, and author Y. Kanada-En'yo, 10.1103/PhysRevC.79.062201 journal journal Phys. Rev., volume C79, pages 062201 (year 2009), http://arxiv.org/abs/0904.2822 arXiv:0904.2822 [nucl-th] NoStop [Sekihara et al.(2012)Sekihara, Yamagata-Sekihara, Jido, and Kanada-En'yo]Sekihara:2012wj author author T. Sekihara, author J. Yamagata-Sekihara, author D. Jido, and author Y. Kanada-En'yo, 10.1103/PhysRevC.86.065205 journal journal Phys. Rev., volume C86, pages 065205 (year 2012), http://arxiv.org/abs/1204.3978 arXiv:1204.3978 [nucl-th] NoStop [Bayar and Oset(2012)]Bayar:2012rk author author M. Bayar and author E. Oset, 10.1016/j.nuclphysa.2012.04.005 journal journal Nucl. Phys., volume A883, pages 57 (year 2012), http://arxiv.org/abs/1203.5313 arXiv:1203.5313 [nucl-th] NoStop [Mitroy et al.(2013)Mitroy, Bubin, Horiuchi, Suzuki, Adamowicz, Cencek, Szalewicz, Komasa, Blume, and Varga]RevModPhys.85.693 author author J. Mitroy, author S. Bubin, author W. Horiuchi, author Y. Suzuki, author L. Adamowicz, author W. Cencek, author K. Szalewicz, author J. Komasa, author D. Blume, and author K. Varga, 10.1103/RevModPhys.85.693 journal journal Rev. Mod. Phys., volume 85, pages 693 (year 2013)NoStop [Sada et al.(2016)Sada et al.]Sada:2016nkb author author Y. Sada et al. (collaboration J-PARC E15), 10.1093/ptep/ptw040 journal journal PTEP, volume 2016, pages 051D01 (year 2016), http://arxiv.org/abs/1601.06876 arXiv:1601.06876 [nucl-ex] NoStop [Sekihara et al.(2016)Sekihara, Oset, and Ramos]Sekihara:2016vyd author author T. Sekihara, author E. Oset, and author A. Ramos, 10.1093/ptep/ptw166 journal journal PTEP, volume 2016, pages 123D03 (year 2016), http://arxiv.org/abs/1607.02058 arXiv:1607.02058 [hep-ph] NoStop [Afnan and Tang(1968)]Afnan:1968zj author author I. R. Afnan and author Y. C. Tang, 10.1103/PhysRev.175.1337, 10.1103/PhysRevC.1.750.2 journal journal Phys. Rev., volume 175, pages 1337 (year 1968), note [Erratum: Phys. Rev.C1,750(1970)]NoStop [Thompson et al.(1977)Thompson, Lemere, and Tang]Thompson:1977zz author author D. R. Thompson, author M. Lemere, and author Y. C. Tang, 10.1016/0375-9474(77)90007-0 journal journal Nucl. Phys., volume A286, pages 53 (year 1977)NoStop [Wiringa et al.(1995)Wiringa, Stoks, and Schiavilla]Wiringa:1994wb author author R. B. Wiringa, author V. G. J. Stoks, and author R. Schiavilla, 10.1103/PhysRevC.51.38 journal journal Phys. Rev., volume C51, pages 38 (year 1995), http://arxiv.org/abs/nucl-th/9408016 arXiv:nucl-th/9408016 [nucl-th] NoStop [Suzuki et al.(2008)Suzuki, Horiuchi, Orabi, and Arai]Suzuki:2008fg author author Y. Suzuki, author W. Horiuchi, author M. Orabi, and author K. Arai, 10.1007/s00601-008-0200-3 journal journal Few Body Syst., volume 42, pages 33 (year 2008), http://arxiv.org/abs/0804.0067 arXiv:0804.0067 [nucl-th] NoStop [Aoyama et al.(2012)Aoyama, Arai, Suzuki, Descouvemont, and Baye]Aoyama:2011ih author author S. Aoyama, author K. Arai, author Y. Suzuki, author P. Descouvemont, and author D. Baye, 10.1007/s00601-011-0266-1 journal journal Few Body Syst., volume 52, pages 97 (year 2012), http://arxiv.org/abs/1106.3391 arXiv:1106.3391 [nucl-th] NoStop horiuchi07 W. Horiuchi and Y. Suzuki, Phys. Rev. C 76, 024311 (2007).

http://arxiv.org/abs/1701.07847v2

20170126191336

Structural Connectome Validation Using Pairwise Classification

q-bio.NC

Green formulation for studying electromagnetic scattering from graphene–coated wires of arbitrary section Claudio Valencia^1, Máximo A. Riso^2, Mauro Cuevas^3, and Ricardo A. Depine^2,* ^1 Facultad de Ciencias, Universidad Autónoma de Baja California (UABC), Ensenada, BC 22860, México ^2Grupo de Electromagnetismo Aplicado, Departamento de Física, FCEN, Universidad de Buenos Aires and IFIBA, Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Ciudad Universitaria, Pabellón I, C1428EHA, Buenos Aires, Argentina ^3 Facultad de Ingeniería y Tecnología Informática, Universidad de Belgrano, Villanueva 1324, C1426BMJ, Buenos Aires, Argentina and Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) ^*email: rdep@df.uba.ar December 30, 2023 ==================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== In this work, we study the extent to which structural connectomes and topological derivative measures are unique to individual changes within human brains. To do so, we classify structural connectome pairs from two large longitudinal datasets as either belonging to the same individual or not. Our data is comprised of 227 individuals from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) and 226 from the Parkinson's Progression Markers Initiative (PPMI). We achieve 0.99 area under the ROC curve score for features which represent either weights or network structure of the connectomes (node degrees, PageRank and local efficiency). Our approach may be useful for eliminating noisy features as a preprocessing step in brain aging studies and early diagnosis classification problems. machine learning, DWI, structural connectomes § INTRODUCTION Predictive modeling of neurodegenerative diseases using diffusion MR-based structural connectomes has become a popular sub-genre of neuroimaging <cit.>. The great variety of possible pre-processing approaches needed for connectome construction leads to potential challenges in downstream application of the connectomes, for example, in a classification task. Choices of e.g, non-linear registration, parcellation, or tractography, may all have a substantial impact (<cit.>, <cit.>). This state of affairs presents a challenge both in terms of intrinsic connectome reliability, and the degree to which the recovered connectomes are valid, if summary, representations of true brain connectivity (<cit.>, <cit.>, <cit.>, <cit.>). At the same time, the performance of a particular case-control classifier may not suffice as a means of data verification due to small samples and high dimensionality. Alternative, more objective validation may be needed, such as the frequently used Intra-class Correlation Coefficient (ICC) on test-retest data (<cit.>, <cit.>). However, structural connectome ICC is generally low, which complicates method comparison. Also, the parametric constraints of classic ICC <cit.> may not be valid. Non-parametric approaches free of data distribution assumptions may be more suitable. To address this issue, we propose a pairwise classification approach to intrinsically assess connectome utility across time, somewhat in line with a recent method for functional connectomes as well <cit.>. For each set of connectomes C^i_j and features f(C^i_j) in question, we construct all possible pairs (C^i_1_j_1, C^i_2_j_2), where i-indices correspond to images and j-indices correspond to subjects. We then conduct a linear classification on the pairwise differences f(C^i_1_j_1) - f(C^i_2_j_2) of these pairs with respect to the target variable y: y=1 if j_1 = j_2, 0 else. We test this pipeline on structural connectomes derived from two publicly available neuroimaging datasets: ADNI and PPMI. These datasets have scanned subjects multiple times, with at least a one year interval between scans. We achieve 0.99 ROC AUC both for direct connectome measures (bag of edges) and for features representing connectome structure (PageRank), suggesting that the tested data is reliable enough to distinguish subjects by the proposed approach. Similar research was conducted by Yeh et al. <cit.>. Though the authors used a local structural connectome, different features, datasets and connectome construction pipelines, they arrived at similar conclusions. § PAIRWISE CLASSIFICATION We propose the following pipeline for pairwise connectome classification: normalization, building connectome features, building pairwise features based on connectome features. Let's denote a set of connectomes as {C^i_j}, where j is an index of a subject and i is an index of an image. §.§ Normalizations Topological normalization of connectivity matrices may be useful prior to any analysis, because the number of detected streamlines is known to vary from individual to individual and can also be affected by fiber tract length, volume of cortical regions and other factors (<cit.>, <cit.>). There is no consensus on the best normalization approach, so we use the three following topological normalization schemes alongside with pure weights (no normalization at all) – by mean, by maximum and binary normalization with zero threshold: a^b_kl = 1 if a_kl > 0, 0 else where a_kl is a connectome edge. §.§ Network features For each connectome and each normalization we build “bag of edges" vectors from the upper triangle of the symmetric connectivity matrix. In addition, we calculate eight network metrics for each node: weighted degrees, or strength; closeness, betweenness and eigenvector centralities; local efficiency; clustering coefficient; weighted number of triangles around node. We choose these features because they are well-described and reflect different structural properties of connectomes <cit.>. We also calculate PageRank for each node. Introduced in 1998 by Brin and Page <cit.> this metric roughly estimates probability that a person randomly clicking on links in the network will arrive at particular node. §.§ Pairwise features Each normalization and set of features described above defines a mapping from connectome space to feature space C → f(C). Since our goal is to check how well this mapping separates connectomes in it, we propose various pairwise features. For each set of connectome features in question we make all possible pairs of connectome features – (f(C^i_1_j_1), f(C^i_2_j_2)). For each pair, we assign a binary target variable – 1 if connectomes are from the same subject (j_1 = j_2), 0 – if they are from different subjects (j_1 ≠ j_2). Finally, for each pair we build a vector of three features, describing their difference f(C_1) - f(C_2) according to l_1, l_2 and l_∞ norms. §.§ Classifiers and validation We use linear classifiers for pairwise classification: logistic regression (LR), SVM with linear kernel and stochastic gradient descent (SGD) with modified Huber loss. We scale features with standard scaling and apply elastic-net regularization for each of classifiers. Model performance we measure with area under ROC curve (ROC AUC) through a two-step validation procedure. First, for each dataset, we perform hyperparameter grid search based on a 10-fold cross-validation with a fixed random state for reproducibility. For each model we varied overall regularization parameter, l_1-ratio and number of iterations for SGD. Then we evaluate the best parameters on 100 train/test splits with fixed different random states (test size was set to 20% of data). We report the ROC AUC distribution on these 100 test splits for each combination of normalization/base features/diagnostic group in results section. § EXPERIMENTS §.§ Base data We used two datasets for our experiments. Our first dataset, the Alzheimer’s Disease Neuroimaging Initiative (ADNI2), is comprised of 227 individuals (675 scans), mean age at baseline visit 73.1 ± 7.4, 99 females. Each individual had at least 1 brain scan and at most 6 scans. The data include 46 people with AD (111 AD scans), 80 individuals with EMCI (247 MCI scans), 40 people with LMCI (120 LMCI scans) and 61 healthy participants (160 scans). Second, we used imaging data from the Parkinson’s Progression Markers Initiative (PPMI) database. From it we selected subjects with PD (159 subjects) and healthy controls (67 subjects). These included a total of 226 individuals (456 scans), mean age at the baseline 61.0 ± 9.8 years, 79 females. Each individual had at least 1 brain scan and at most 4 scans. §.§ Network construction Inhomogeneity corrected T1-weighted images for ADNI and PPMI data were processed with FreeSufer's <cit.> recon-all pipeline to obtain a triangular mesh of the gray-white matter boundary registered to a shared spherical space, as well as corresponding vertex labels per subject for a cortical parcellation based on the Desikan-Killiany (DK) atlas<cit.>. This atlas includes 68 cortical brain regions; hence, our cortical connectivity matrices were 68×68. In parallel, T1w images were aligned (6-dof) to the 2mm isotropic MNI 152 template. These were used as the template to register the average b_0 of the DWI images, in order to account for EPI related susceptibility artifacts. DWI images were also corrected for eddy current and motion related distortions. Rotation of b-vectors was performed accordingly. Tractography for ADNI data was then conducted using the distortion corrected DWI in 2mm isotropic MNI 152 space. Probabilistic streamline tractography was performed using the Dipy <cit.> LocalTracking module and implementation of constrained spherical deconvolution (CSD) <cit.> with a spherical harmonics order of 6. Streamlines longer than 5mm with both ends intersecting the cortical surface were retained. Edge weights in the original matrices are proportional to the number of streamlines detected by the algorithm. PPMI data were processed in a slightly different fashion to account for variability in the acquisition protocols and to show our method is not dependent on any single processing scheme. Images were initially denoised with an adaptive denoising algorithm <cit.> and two DWI acquisitions from each subject were merged. DWI images were corrected for eddy current and motion related distortions, then non-linearly epi-corrected with ANTs SyN. Rotation of b-vectors was performed accordingly. Tractography for PPMI data was then conducted in 2mm isotropic MNI 152 space, again using the Dipy LocalTracking module. At each voxel, the CSD was fitted recursively <cit.> with a spherical harmonic order of 6. Deterministic streamline tractography was seeded at two random locations in each white matter voxel. Similar to the ADNI data, only streamlines longer than 5mm with both ends intersecting the cortical surface were retained. §.§ Pairwise data For each set of connectomes described above (ADNI, PPMI) we made all possible pairs of connectomes as described in <ref>. Using this technique we obtained 227475 pairs (764 of which were labeled as 1) from ADNI2 data and 152031 pairs from PPMI data (301 of which were labeled as 1). Due to huge imbalance of classes in generated pairs, we used all samples with label 1 and equally sized random subsample of 0. Our result do not depend for different subsamples of 0s, so we report them for a fixed random state. §.§ Pairwise classification performance Figure 1 shows multidimensional scaling (MDS) based on l_2-norm dissimilarity matrix of bag of edges for ADNI subjects (for PPMI data picture is essentially the same, so we omitted it). We see that in most cases images from same subject are near each other in that feature space. We also see that there is no such clear picture with diagnostic groups labels. Figures 2 and 3 quantify this observation for ADNI and PPMI data in terms of ROC AUC distributions depending on normalization and base features. We see that 0.99 ROC AUC can be achieved either for connectome weights themselves, or for features that capture connectome structure. We also see that the choice of normalization greatly affects the accuracy of pairwise classification. Normalizing by the mean is a winner in most cases, with the exception of eigenvector centrality and clustering coefficient features. It is interesting to note that for clustering coefficient and eigenvector centrality, binary normalization performed better than other normalizations even though it preserves somewhat less information. Figures 3-4 show the ROC AUC distributions of pairwise classification depending on the base features and diagnostic group for connectomes normalized by the mean. We see that there is almost no difference in pairwise classification results in different diagnostic groups. We note that interquartile spread is high for diagnostic groups with the smallest number of subjects. § CONCLUSION We have presented a method for structural connectome feature validation through pairwise classification which is free of distribution assumptions. We tested this pipeline on ADNI and PPMI data and obtained high classification performance in terms of ROC AUC suggesting that there are mappings from connectomes to feature spaces that at least differentiate subjects from each other. It is worth noting that pairwise classification is not a feature selection technique for classification tasks. It is possible that a feature distinguish classes in the context of this work, but fails to distinguish subjects, for example in a diagnostic classification task. Our results suggest that pairwise classification may be useful for validating preprocessing pipelines and particular features in terms of how much subject-related signal they preserve. As such, it may be treated as a “first-pass" for connectome features to be used in further studies. There are several limitations. First, we used only one particular pipeline to construct our networks. These results may differ for other tractography algorithms and parcellations. Assessing these effects on pairwise classification is among our future goals. Second, the downsampling of the “different subject pair" class to ensure balanced samples may lead to optimistic accuracy estimates. A detailed look at the multidimensional scaling plot suggests that a number of connectomes from different subjects are “near" each other, though it is unlikely that our downsampling procedure selects them for training or testing. To answer this question, we plan to apply our protocol without downsampling. Finally, due to limitations of the data used here, same-subject pairs were constructed from diffusion images acquired at least one year apart. Such a time scale provides ample opportunity for substantial longitudinal effects, such as those due to neurodegeneration, to affect our features. As we cannot exclude such effects, any conclusions to be drawn about optimal network features and normalizations for further studies must be made with the appropriate reservations. § ACKNOWLEDGMENTS The results of sections 2 and 3 are based on the scientific research conducted at IITP RAS and supported by the Russian Science Foundation (project 14-50-00150). Some of data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database. A complete listing of ADNI investigators as well as data acquisition protocols can be found at <adni.loni.usc.edu>. Additional data were obtained from the Parkinson’s Progression Markers Initiative (PPMI) database. For up-to-date information on the study, visit <www.ppmi-info.org>. IEEEbib

http://arxiv.org/abs/1701.08010v2

20170127105043

Statistical and computational phase transitions in spiked tensor estimation

math.ST

1.1 width=.94 theoremTheorem lemmaLemma propositionProposition corollaryCorollary claimClaim conjectureConjecture ./figures mySymbols mySymbols 00A0  ŁL () [ ]

http://arxiv.org/abs/1701.07527v1

20170126001300

Formation and assembly history of stellar components in galaxies as a function of stellar and halo mass

astro-ph.GA

1Korea Astronomy and Space Science Institute, 776, Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Republic of Korea; syncphy@gmail.com 2Department of Astronomy and Yonsei University Observatory, Yonsei University, Seoul 03722, Republic of Korea Galaxy mass assembly is an end product of structure formation in the ΛCDM cosmology. As an extension of Lee & Yi (2013), we investigate the assembly history of stellar components in galaxies as a function of halo environments and stellar mass using semi-analytic approaches. In our fiducial model, halo mass intrinsically determines the formation and assembly of the stellar mass. Overall, the ex situ fraction slowly increases in central galaxies with increasing halo mass but sharply increases for log M_*/M_⊙≳11. A similar trend is also found in satellite galaxies, which implies that mergers are essential to build stellar masses above log M_*/M_⊙∼11. We also examine the time evolution of the contribution of mass growth channels. Mergers become the primary channel in the mass growth of central galaxies when their host halo mass begins to exceed log M_200/M_⊙∼13. However, satellite galaxies seldom reach the merger-dominant phase despite their reduced star formation activities due to environmental effects. § INTRODUCTION Modern theories for galaxy formation and evolution suggest that galaxies are formed in highly over-dense regions, namely haloes, while cosmological structures are built up via smooth matter accretion or the coalescence of haloes. Thus, some galaxies are eventually involved in hierarchical mergers in the process of structure formation. Galaxy mergers play a pivotal role in the evolution of galaxies by disturbing the kinematics of the stellar and gaseous components, which induces morphology transformation, size growth, star formation, and active galactic nuclei (AGNs) activities <cit.>. Therefore, galaxies, especially massive ones, are the end products of structure formation in the ΛCDM cosmology. Many puzzles in the evolution of galaxies are still waiting to be addressed, and there is no clear consensus even on seemingly simple issues of the assembly history of stellar masses in galaxies in a quantitative aspect. Much effort has been made to systematically investigate these issues in a cosmological context. Semi-analytic models (SAMs) for galaxy formation and evolution have been used to explore the formation and assembly history of stellar components of galaxies in cosmological volumes by taking advantage of their computing efficiency. Empirical downsizing effects <cit.> are reasonably reproduced in SAMs based on the ΛCDM cosmology in which galaxies are born earlier on shorter timescales in denser environments and larger ones are built up through gradual mergers. <cit.>. <cit.> demonstrated that brightest cluster galaxies (BCGs) can have very high fractions of ex situ components (>80%) in terms of total stellar mass. <cit.> examined the assembly history of model galaxies in two clusters, and found that BCGs acquire 35% of their final mass via mergers, and the fractions monotonically decrease with increasing absolute magnitudes of galaxies. <cit.> provide a quantitative prediction of the ex situ fractions of a complete set of galaxies in a cosmological volume as a function of the final stellar mass: 20%, 40%, and 70% for log M_*/M_⊙∼10.75, 11.25, and 11.75 galaxies at z=0, respectively. According to them, in the main progenitors of local massive galaxies (log M_*/M_⊙>11.5 at z=0) galaxy mergers become the leading channel in mass growth at z∼2. For comparison, there is no such transition in smaller galaxies (log M_*/M_⊙<11.0 at z=0). Rapid advances in computing power have enabled us to carry out hydrodynamic simulations with reliable resolution for numerous haloes in cosmological volumes. <cit.> investigated the stellar assembly history of massive galaxies by using zoom-in simulations of 39 haloes. They showed that only ∼20% of the stars in massive galaxies (log M_*/M_⊙≳11.5 at z=0) are formed in situ, and the rest fall into the galaxies via mergers. The galaxies in the zoomed simulations performed by <cit.> have ex situ fractions ∼1/3 of those in <cit.> in similarly massive galaxies. The simulations in the two studies were run without AGN feedback. The effect of AGN feedback on stellar mass growth was investigated using zoomed simulations by <cit.>. AGN feedback effectively suppresses the in situ star formation in massive galaxies, resulting in a 30% higher ex situ fraction than in non-AGN cases. In their zoomed simulations, <cit.> showed that galactic winds play a role similar to the AGN feedback in mass growth. Hydrodynamic simulations for entire cosmological volumes with moderate resolution have become available recently <cit.>. <cit.> examined the contribution of mergers to the galaxy mass assembly history by using a complete set of galaxies identified in a cosmological volume of the Illustris simulation <cit.>. The ex situ fractions of the galaxies in the volume are in good agreement with , i.e., ∼20% for log M_*/M_⊙∼11 and ≳70% for log M_*/M_⊙∼12 at z=0. They also found that the ex situ components are less concentrated than in situ components and half of them flow into galaxies via major mergers in the overall mass range (log M_*/M_⊙>9 at z=0). The aforementioned studies looked into the histories of galaxy assembly primarily as functions of stellar mass. However, the properties of galaxies are governed by halo evolution in the ΛCDM cosmology, as implied by the strong correlation between the stellar and halo masses <cit.>. Furthermore, recent deep imaging observations revealed that a considerable fraction (∼40%) of bright (M_r<20) early-type galaxies have post-merger signatures in both isolated and dense environments <cit.>. Post-merger signatures in satellite galaxies were expected to be rare due to the high peculiar velocities in dense environments. <cit.> argued that the post merger features of non-central galaxies in dense environments may have been pre-processed before becoming satellites. These studies point out that the galaxy assembly history should be traced along with the evolution of environments. The number of neighboring galaxies is widely used to quantify the environments around galaxies in empirical studies, and this is closely connected to the host halo mass <cit.>. Thus, host halo mass can be used as a reasonable proxy of the environments of galaxies. Large cosmological volume simulations are needed to cover a variety of environments. investigated the origin of stellar components only as a function of stellar mass without separating them into centrals and satellites. The size of the cosmological volume used was 99.4Mpc on a side with 512^3 collisionless particles, and less than 30 cluster-scale haloes (log M_200/M_⊙>14) being found in the volume. Up-to-date hydrodynamic cosmological simulations cover a scale of volumes similar to , which are not large enough to harvest many cluster-scale haloes. Therefore, semi-analytic approaches are still effective for investigating galaxy evolution in larger cosmological volumes. As a follow-up of , this study aims to separately scrutinize the mass assembly history of central and satellite galaxies as a function of halo and stellar mass using our own SAM. § MODEL We use in this investigation and provide a summary of it in this section. §.§ Halo catalogue To obtain a sufficient number of haloes in log M_200/M_⊙∼ 10-15, we performed a cosmological volume simulation with 1024^3 collisionless particles in a 284Mpc (200h^-1Mpc) periodic cube using the cosmological simulation code  <cit.>. The initial condition of the simulation was generated using  <cit.>, a parallel version of  <cit.>. We adopted a set of cosmological parameters derived from the seven-year Wilkinson Microwave Anisotropy Probe observations by <cit.>, Ω_ m=0.272, Ω_Λ=0.728, and h=0.704. A total of 125 snapshots were printed out from the volume run, and  <cit.> was used to search for sub-structures in the friends-of-friends groups of the snapshots. The final halo catalogue consists of 118 time steps from z=15.8 to z=0. §.§ Halo merger trees Halo merger trees are the essential backbones of SAMs, and are composed of single or multiple branches. In this study, the branch linking the most massive progenitor among all the progenitors of a halo at each time step is defined as the main branch of a halo merger tree <cit.>. These trees are constructed from the halo catalogue using the tree building code  <cit.>, which traces the most likely descendant or progenitor of a halo by comparing the identifications of particles bound to the haloes in two snapshots. has been updated to provide the number fraction of bound particles exchanged between haloes. We assume that the same fraction of hot gas and diffuse stellar components are transferred along with the collisionless particle exchanges between haloes. §.§ Semi-analytic model allows a halo to form a galaxy in its central region <cit.>. In principle, a halo has one galaxy regardless of whether it is a host or sub. Satellite galaxies are treated as the centrals of subhaloes. performs tree cleaning and repairing processes before planting galaxies onto raw halo merger trees that possibly have problematic branches <cit.>. Halo finding codes sometimes fail to identify haloes embedded in dense environments or close to the mass resolution limit, which eventually results in fragmented trees. removes any host halo branches that disappear without descendants before z=0 and subhalo branches that are identified to be newly formed with no progenitor. analytically calculates the orbits and mass of the subhaloes that merge into host haloes in the raw merger trees before reaching the central regions of their hosts <cit.>. We adopt the prescriptions for gas cooling proposed by <cit.>. The prescription for quiescent star formation was updated from the original prescription in . Star formation in a disk is permitted when the surface density of a cold gas disk is larger than a critical density <cit.>. We use the prescriptions for supernova feedback and merger-induced starbursts proposed by <cit.>. Quasar and radio mode feedback is implemented into as proposed by <cit.> and <cit.>. We also trace the mass loss and chemical enrichment history of individual stellar populations in galaxies. Further details can be found in . §.§ Model calibrations was calibrated for the cosmological volume described in 2.1 by tuning the set of free parameters listed in Table 1. More parameters are used in the prescriptions of . Most of them are, however, fixed and we have mainly adjusted the listed free parameters that regulate feedback and star formation efficiency. Figure <ref> shows the fitting of our fiducial model. The galaxy stellar mass function (GSMF) is adopted as the primary calibration point for . The panel (a) in Figure <ref> displays two empirical GSMFs along with our fiducial model at z=0. The GSMF marked by blue symbols comes from <cit.>, and the red symbols are a composite GSMF of <cit.>, <cit.>, and <cit.>. The two empirical GSMFs are in good agreement overall, even though that of <cit.> has a slightly higher massive end (M_*>10^11M_⊙). Our fiducial model is located in between these two GSMFs. The panel (b) in Figure <ref> shows the evolution of the global star formation density (GSFD) over the cosmic time. The empirical GSFD that was compiled and modified by <cit.> is marked by red crosses. In panel (c), one can see the star formation rate functions (SFRFs) of the empirical data <cit.> and our fiducial model at z∼0.15 for M_*>10^10M_⊙. The GSFD and SFRF are not calibration points in our model, but are used to cross-check whether our model reproduces reliable star formation histories. A tight correlation between the mass of supermassive black holes (SMBHs) and bulge stellar mass has been discovered <cit.>. This relation was used as the secondary calibration point in our model. Three different observations with error bars and P(M_ BH|M_ bulge) of the fiducial model are shown in panel (d) of Figure <ref>. The red, blue, and green crosses indicate the empirical M_ BH-M_ bulge relation derived by <cit.>, <cit.>, and <cit.>, respectively. The P(M_ BH|M_ bulge) distribution of our model is within the observational scatter. A notable feature is that the distribution of the model at log M_ bulge/M_⊙>11 is much narrower than that in the low mass range. This is because the main channel that builds up bulges at the massive end is galaxy mergers, which also feed the SMBHs by injecting gas into very central regions or by inducing SMBH mergers in our model. Dry mergers strengthen the relationship in the later stages of the evolution of massive galaxies. On the other hand, disk instability becomes significant in the growth of (pseudo) bulges for less massive galaxies along with mergers. This results in a less tight M_ BH-M_ bulge relation at log M_ bulge/M_⊙<11, as suggested by the empirical studies <cit.>. This model is used as a fiducial model in this study. § RESULTS All of the galaxies in are born as the central galaxy of each halo. Thus, the final galaxy of a halo merger tree composed of N branches evolves with a maximum of N-1 galaxy mergers. The galaxies that grow along the main branches of the halo merger trees are called the main or direct progenitors of the final galaxies. The stars formed in the main progenitors are classified as in situ components, and those that are formed outside and then come into the main branches are labelled as ex situ components. In this definition, the stellar mass M_* of the galaxy is the sum of the mass of the two components, i.e., M_*=M_ in situ+M_ ex situ, following a conventional nomenclature <cit.>. We simply assume that the formation of in situ components is a self-assembly process. A host halo that habors N subhaloes has one central galaxy and a maximum of N satellite galaxies. §.§ Formation and assembly history of stellar mass We quantify the star formation history of a galaxy by finding the redshifts z_ form by which half of the stellar populations found in the final galaxies have been formed. The assembly time of a galaxy is defined as the epoch z_ assembly at which the stellar mass of the main progenitor reaches half of the final mass. The equivalence z_ form=z_ assembly is made only when galaxies evolve along single-branch merger trees. Figure <ref> shows the distributions of z_ form and z_ form-z_ assembly of the central galaxies in the stellar-to-halo mass plane at z=0. The solid, dashed, and dotted lines mark the median, 16-84th percentile distribution, and 2.5-97.5th percentile distribution of P(M_*|M_ 200). The downsizing trend is clearly shown as a function of the halo mass. This figure implies that the empirical downsizing trend in which more massive galaxies have older stellar ages originates from the dependence of z_ form on the halo mass and stellar-to-halo mass relation. The formation time of very massive galaxies (log M_*/M_⊙>11.5 at z=0) is found at z_ form>2, in agreement with <cit.> who derived the half mass formation time of early type galaxies using the Sloan Digital Sky Survey. found an upsizing trend in the assembly time of stellar mass, which is directly opposite to the downsizing nature of formation time. This result is consistent with recent observations. <cit.> found from the GOODS field that the pair fraction of red massive spheroidals (log M_*/M_⊙ > 11) is higher than that of smaller galaxies at z∼1. They expected that the pairs would eventually merge with each other, and form massive early types. <cit.> showed that the number density of massive galaxies (log M_*/M_⊙∼11.5-12) increases faster than that of less massive galaxies (log M_*/M_⊙∼11-11.5) for 0<z<1. The majority of the massive galaxies appear to be quenched. Consequently, mergers are their most likely mass growth channel during the period of time. In our fiducial model, the centrals of more massive haloes have younger assembly times, which results in the increase of z_ form-z_ assembly with increasing halo mass (the right panel of Figure <ref>). In summary, more massive galaxies are older in terms of formation ages but relatively younger in terms of assembly ages <cit.>, and this trend originates from the halo mass assembly. §.§ Origin of stellar components §.§.§ Ex situ fractions in the stellar-to-halo mass plane We quantified the overall contribution of galaxy mergers to stellar mass growth by measuring the fraction of ex situ components in galaxies, i.e., f_ ex situ=M_ ex situ/M_*. Figure <ref> shows the f_ ex situ distribution of central and satellite galaxies in the stellar-to-host halo mass plane at z=0. The f_ ex situ distribution of the centrals has a strong correlation with M_200, but its dependency on M_* is not clear. This result suggests once again that the increase in f_ ex situ with increasing M_* demonstrated by previous studies <cit.> may be a derivative of the f_ ex situ-M_200 and the stellar to halo mass relations. The stellar-to-halo mass relation of satellites is broken up as haloes are stripped during orbital motion in dense environments <cit.>. Therefore, the f_ ex situ distribution of satellites in the right panel of Figure <ref> is plotted as a function of their host halo mass. As the stellar-to-halo mass relation is shuffled, no correlation was found between the f_ ex situ distribution of satellites and their host halo mass. However, the correlation between f_ ex situ and the mass of satellites appears to be clearer than that of centrals. When they are in the same halo, satellite galaxies are generally less massive than centrals mainly because of low mass growth rates after becoming satellites. Tidal or ram pressure stripping effectively removes gas reservoirs in satellites, eventually quenching star formation activities <cit.>. In addition, there is little opportunity to increase their mass via mergers. We trace the mass growth histories of central galaxies in the stellar-to-halo mass plane. Figure <ref> shows the mean mass evolution tracks of the main progenitors of galaxies binned by final mass. Since we defined M_*=M_ in situ+M_ ex situ, the mass growth rate of a galaxy at an epoch is Ṁ_*=Ṁ_ in situ+Ṁ_ ex situ. The black solid lines represent Ṁ_ in situ≥Ṁ_ ex situ phase and the dotted lines represent the opposite. The color code of the filled circles indicates redshifts. It can be seen that halo mass regulates which channel is dominant in mass growth. The central galaxies of the most massive haloes (log M_200/M_⊙ (z=0)∼14.7) enter the merger-dominant phase at z∼2 while those at log M_200/M_⊙ (z=0)∼13.2 only reach this phase until z=0. This plot shows that galaxies migrate to the merger-dominant phase when their host halo mass begins to exceed ∼10^13M_⊙. As mentioned above, more mergers in the centrals of larger haloes cause earlier quenching and higher ex situ fractions. Galaxies residing in haloes below log M_200/M_⊙∼13 increase their stellar mass in directly proportion to the increase in their halo mass. For example, central galaxies at log M_200/M_⊙ (z=0) ∼12.7 experience 1.4 dex of halo mass growth between z=4-0 and their stellar mass increases to almost the same scale during the same period of time. On the other hand, halo mass growth does not accompany the same rate of stellar mass growth in the most massive group. This is because star formation is suppressed by strong feedback and not all of the stellar mass is concentrated in the central regions. Stars in massive haloes are located in satellite galaxies or intra-cluster <cit.> as well as in the central galaxies of massive haloes. The peak of the stellar-to-halo mass fraction is found at log M_200/M_⊙∼12 as log M_*/M_200∼-1.5 <cit.>. However, the galaxies in the dashed box have notably large fractions. Their number fraction is actually negligible, far outside the 97.5th percentile distribution of P(M_*|M_200). Only 87 galaxies are found in the box out of the 100,000 plus galaxies at log M_200/M_⊙=11.0-11.2 at z=0. We look into how the galaxies gain that particularly large M_*/M_200 ratio. As the averaged evolution track of the galaxies in the box marked by A indicates, they undergo severe halo stripping at z<1. They barely increase their stellar mass during the period of time because of gas loss. Besides, the degree of halo stripping is not harsh enough to significantly strip stellar components in central regions <cit.>. All of the central galaxies are typically located in dense environments where tidal force exerted by neighbouring massive haloes disturbs small systems. §.§.§ Ex situ fractions at a given stellar and halo mass The marginal distribution of f_ ex situ in terms of M_200 or M_* is plotted in Figure <ref>. The grey shading displays the percentile distribution of P(f_ ex situ|M). Panel (a) demonstrates the gradual increase of f_ ex situ of the centrals with increasing M_200. However, in panel (b) the f_ ex situ of the central galaxies stays below 0.1 at log M_*/M_⊙<10.5 and rises sharply in log M_*/M_⊙>11. This is because galaxies with log M_*/M_⊙∼11 at z=0 are hosted by haloes in the wide mass range of log M_200/M_⊙∼12-14.5. The relation between the stellar and halo masses causes the large dispersion of f_ ex situ around log M_*/M_⊙=11. The satellites show a slightly lower f_ ex situ-M_* relation than centrals, but the overall trend is similar, as shown in panel (c). Specifically, the f_ ex situ of the most massive satellites is comparable to that of the centrals. This suggests that mergers are essential to form massive galaxies. Such massive galaxies were centrals until recently, and have only just became satellites. The black solid and dotted lines with colored symbols in the upper panels of Figure <ref> show the averaged evolution tracks of the main progenitors in the f_ ex situ-M planes. The color code and line styles are the same as those in Figure <ref>. The galaxies are binned by 1 dex from log M_200/M_⊙=12 or 0.5 dex from log M_*/M_⊙=10 at z=0. The main progenitors of the galaxies in the different mass bins are marked by different symbols. Panel (a) illustrates that galaxies that become merger-dominant before z=0 are finally hosted by haloes of log M_200/M_⊙>13. In panels (b) and (c), the transition takes place only in the most massive groups of both centrals and satellites. Stochastic effects cause the uneven tracks of the most massive satellites in panel (c). provided a quantitative prediction of the mean f_ ex situ as a function of the final stellar mass without separating galaxies into central and satellites: 20%, 40%, and 70% for the galaxies in log M_*/M_⊙=10.5-11, 11-11.5, and 11.5-12 at z=0, respectively. The red filled circles, stars, and squares, in panels (b) and (c) of Figure <ref> indicate the mean f_ ex situ of the three groups binned by the final stellar mass. Because the majority of the galaxies in the three groups are centrals, the f_ ex situ-M_* relation of the centrals is comparable to that of . The f_ ex situ-M_* relation and the large dispersion at log M_*/M_⊙∼11 are in good agreement with and <cit.>. The bottom panels of Figure <ref> demonstrate the evolution of the f_ ex situ-M relation at z∼0-4. The different redshifts are indicated by the color codes used in panel (a). At a given halo mass, the central galaxies have a higher f_ ex situ at higher redshifts. This is because the haloes that show up at higher redshifts are located in relatively denser environments than those having the same mass at lower redshifts. Therefore, low mass haloes at z∼4 likely have growth histories radically different from local low mass haloes. On the other hand, high redshift galaxies have a low median f_ ex situ in panels (e) and (f). The most massive haloes are always rare and their centrals have stellar masses with larger dispersions at higher redshifts. In other words, the stellar-to-halo mass relation is less tight at higher redshifts. The ex situ fraction begins to rapidly increase at log M_*/M_⊙=11, which implies that galaxy mergers are essential to build up galaxies above log M_*/M_⊙=11. On the contrary, galaxies similar in mass to the Milky Way (log M_*/M_⊙∼10.5) would increase their stellar mass mostly from in situ star formation, regardless of whether they are centrals or satellites. §.§ Evolution history of the two mass growth channels §.§.§ Specific mass growth histories of the main progenitors The previous sections show that the leading mass growth channels change over time, depending on the halo or stellar mass. In order to examine the time evolution of the contribution of mergers and in situ star formation to stellar mass growth, we measure the specific stellar mass accretion rate (SSAR) Ṁ_ ex situ/M_* in the same way that the specific star formation rate (SSFR) is defined. In this study, accretion indicates the infall of stellar components into galaxies only via mergers. Figure <ref> shows the SSFRs and SSARs of the main progenitors of galaxies grouped by final mass and status. In all the panels, the SSFRs are lower in more massive galaxies, while the opposite is true for SSARs, which gently and consistently decrease. However, the SSFRs rapidly decline by z∼4 after which the decay rates decelerate. In all the panels, the two channels decrease similarly in the third and fourth most massive groups at z<3. However, bigger groups experience a sharp decay in the SSFRs. As discussed in 3.2.1, galaxies enter a merger-dominant phase when their host halo mass increases above log M_200/M_⊙∼13 in our model. Central galaxies in larger haloes are effectively quenched by more frequent AGN activities and acquire more stellar mass via more mergers. These two phenomena give rise to a transition of dominant mass growth channels. Mergers become the primary process in mass growth until z=0 for the main progenitors of central galaxies with haloes of log M_200/M_⊙>13. The main progenitors of the most massive centrals in panel (b) enter the merger-dominant phase at z∼1. The SSAR of the second most massive group, however, does not meet its SSFR at all. Galaxies in this group are finally hosted by haloes in a wide mass range of log M_200/M_⊙∼12-14.5. Some of the galaxies residing in haloes above the group scales (log M_200/M_⊙>13) finally settle into the merger-dominant phase, as shown in panel (a). However, the majority of the second massive group are the centrals of the smaller haloes (12<log M_200/M_⊙<13). Thus, the main progenitors of the second most massive galaxies marginally stay in the SSAR≲SSFR phase by z=0. This result is consistent with <cit.>, who found a strong correlation between the quenched fractions of central galaxies and their host halo mass. AGN feedback and decreasing cooling efficiency naturally result in the trend found in . The evolution trend of the specific mass growth rates of satellites in panel (c) appears to be almost the same as that of the centrals in panel (b) in early epochs. This is primarily because they all used to be centrals at high redshifts. As they gradually become satellites with decreasing redshifts, their specific growth rates are suppressed. Because the satellites barely merge with each other, their SSARs are lower than those of the centrals. Environmental effects, such as tidal and ram pressure stripping, suppress star formation activities in satellites by blowing away gas reservoirs. However, AGN feedback is inactive in satellites due to little gas accretion. Therefore, the most and second most massive satellites eventually have slightly lower SSFRs than those of the centrals. The main progenitors of the most massive satellites exhibit behavior similar to those of the most massive centrals. This is mostly because they only recently became satellites. However, the SSARs of the other groups do not even come close to their SSFRs. The transition epochs of the two mass growth channels as a function of the final stellar mass are in good agreement with previous studies that were based on semi-analytic approaches <cit.> and hydrodynamics <cit.>. In Figure <ref>, the redshift transitions between the two channels are plotted in the stellar-to-halo mass plane. The grey shading indicates the mass ranges of the galaxies that stay in the star-formation-dominant phase through z=0 on average in each mass bin. The transitions are found in galaxies that end up in haloes with log M_200/M_⊙≳ 13, as panel (a) of Figure <ref> demonstrates. This suggests that the halo mass evolution plays a key role in the two-phase scenario of massive galaxy formation in which in situ star formation rapidly increases the galaxy stellar mass at early epochs and mergers gradually build it up by z=0 <cit.>. §.§.§ Specific mass growth rates at each epoch The specific mass growth rates of the main progenitors in the previous section indicate how the galaxies chosen at z=0 have evolved over time. Thus, Figure <ref> is plotted based on a theoretical viewpoint. However, observations take a snapshot of the galaxies located at various redshifts . In that sense, Figure <ref> displays the specific mass growth rates of galaxies binned by mass at each epoch. The more massive galaxies or the centrals of more massive haloes have higher SSARs and lower SSFRs. At a fixed stellar mass, the SSFRs decay by two orders of magnitude during cosmic time, as shown in previous studies <cit.>. Cluster-scale haloes (log M_200/M_⊙>14) and very massive galaxies (log M_*/M_⊙>11.5) begin to appear at z≈2 in the co-moving volume of this study. The SSFRs of the most massive groups are comparable to those of less massive groups for a while after emerging in all the panels. The SSFRs are sufficiently high at the moment to increase the stellar mass by a factor of two within a billion years. In our model, galaxies in the most massive ranges at early epochs have been built up by very active star formation, along with galaxy mergers. Rapid halo mass growth leads to violent baryonic accretion into the central regions, eventually inducing very high star formation activities. However, the most massive groups quickly experience a rapid drop in the SSFRs in all the panels. The central galaxies of the haloes in log M_200/M_⊙>13 of panel (a) are in the merger-dominant phase most of the time after they appear in all the panels. The differences between the SSFRs and the SSARs increase in the massive halo groups as cold gas is depleted and cooling is further suppressed by feedback. In the middle panel, the two mass growth channels are almost equally important for the second massive group of centrals (log M_*/M_⊙∼11.0-11.5) all the time. In this panel, 0.5 dex more massive central galaxies have SSARs 0.5 dex higher at z<0.5. Consequently, a ten times larger stellar mass falls into 0.5 dex more massive galaxies via mergers. Star formation contributes an order of magnitude more to mass growth than mergers in log M_*/M_⊙<11. The specific mass growth rates of satellites (panel (c)) are always lower than those of centrals. Environmental effects cause overall low SSFRs. Since satellite galaxies are hardly involved in mergers with other satellites, their SSARs are at least an order of magnitude lower than those of the centrals. Therefore, mergers do not become primary mass growth channels in all groups. In our model, galaxy mergers between satellites mainly occur in sub groups that belong to host haloes that are not yet virialized. Once the sub groups are dynamically dissociated in dense environments, their member galaxies only fall into the central regions of their host haloes in our model. § SUMMARY AND CONCLUSION examined the assembly history of stellar components in galaxies as a function of final stellar mass using semi-analytic approaches. Here we expand in a cosmological context by investigating it in terms of extended parameter spaces using a larger cosmological volume simulation. The size and resolution of the volume were designed to cover a wide range of halo masses log M_200/M_⊙∼10-15. The fiducial model of this study was calibrated to fit a set of empirical data. We labelled the stars formed along the main branches of halo merger trees as in situ components and the rest of the stars falling into galaxies via mergers as ex situ components. The ΛCDM cosmology predicts earlier formation of primordial structures in denser environments and their hierarchical assembly over time. In this framework, the formation and assembly of galaxies are expected to correlate with the evolution of halo environments. In our model, we found that centrals of more massive haloes have older formation times and higher ex situ fractions. The marginal distribution of the ex situ fractions at z=0 gradually increases with increasing halo mass. The ex situ components become the majority in the stellar mass of the central galaxies in log M_200/M_⊙>13. However, the distribution of ex situ fractions sharply rises in terms of stellar mass with large dispersion in log M_*/M_⊙>11. This is because the central galaxies at log M_*/M_⊙∼11 are hosted by a wide range of halo mass log M_200/M_⊙∼12-14.5. As a result, they have similar masses despite their diverse evolution tracks. Satellite galaxies have slightly lower ex situ fractions than centrals but the overall trend is similar. Like massive centrals, massive satellites acquired a considerable fraction of their stellar mass via mergers mainly when they were centrals and have only recently become satellites. The marginal distribution of the ex situ fractions evolves with decreasing redshifts. The centrals of the most massive haloes already reach f_ ex situ∼0.5 at z=4. The ex situ fraction rapidly increases as the galaxy stellar mass begins to exceed log M_*/M_⊙∼11. This can be interpreted to suggest that galaxy mergers are essential for building up galaxies above log M_*/M_⊙∼11, whether they end up becoming centrals or satellites. We examined the time evolution of the specific star formation rates (SSFRs) and specific stellar mass accretion rates (SSARs). First, we traced the specific growth rates of the main progenitors of the galaxies grouped by final halo or stellar mass. The SSFRs of the main progenitors are always lower in more massive groups while the SSARs behave in the opposite manner. In very early epochs, the SSFRs far exceed the SSARs in all galaxies, but this dominance rapidly decreases over time. Furthermore, the SSFRs decay even faster for more massive galaxies, which results in a crossing of the two mass growth channels in some cases. The transition takes place in the main progenitors of the central galaxies that finally reside in the haloes of log M_200/M_⊙>13. In our model, this is the mass range where the two phase scenario for massive early types <cit.> is valid. With the aforementioned results, this suggests that the correlation between the stellar mass and the empirical galaxy formation time <cit.>, and the fraction of ex situ components proposed by theoretical studies <cit.> may be merely the projection of their intrinsic halo mass dependence upon the stellar-to-halo mass relation. When we looked into the SSAR and SSFR evolution of galaxies in an empirical sense, i.e. binning them by given mass ranges at each epoch with no use of progenitor-descendant relations, we found that mergers are a major channel for mass growth at all times in the centrals of log M_*/M_⊙>11 or the centrals of haloes with log M_200/M_⊙>13. However, in satellites, mergers are secondary or even negligible. This study displayed a strong correlation between the formation and assembly of galaxies and halo mass which only represents local environments. In large scales, however, weaker correlations have been found. <cit.> compared the empirical luminosity function of void galaxies with their SAM, concluding that large-scale environments do not significantly affect galaxy properties but halo mass assembly is a decisive factor in galaxy evolution. <cit.> also presented a similar result using that galaxy growth rates are largely insensitive to large-scale environments. So, in sum, the evolution of local halo environments plays a leading role in the formation and assembly of galaxy stellar mass. Central galaxies in dense environments grow with vigorous gas inflow at early epochs and the steady mergers of small structures over time. Some galaxies achieve log M_*/M_⊙∼11 even in low density environments with active in situ star formation, but they come up against a steep barrier to growing further without mergers. Once galaxies are relegated to satellite status, in situ star formation becomes the dominant channel for increasing mass despite the gradual suppression of star formation activities due to environmental effects. Galaxy formation models are, of course, incomplete as yet and are unable to precisely describe the formation of stellar mass <cit.>. Nonetheless, the success of the concordance ΛCDM cosmology gives credibility to the efforts to understand galaxy mass assembly, which is regarded as a consequential process of structure formation in the cosmological framework. This study demonstrates that similarly massive galaxies have a variety of evolution histories, depending on their halo environments. Meanwhile, empirical studies have found a considerable number of luminous galaxies in low density environments as well as dense environments <cit.>. Motivated by these results, we will carry out a comparison study for massive galaxies that end up with similar properties but reside in different local halo environments. § ACKNOWLEDGMENTS We thank Rory Smith for his constructive comments and proofreading. We acknowledge the support from the National Research Foundation of Korea (NRF-2014R1A2A1A01003730). Numerical simulations were performed using the KISTI supercomputer under the programme of KSC-2014-G2-003. S.K.Y. acted as the corresponding author. This work was performed under the collaboration between Yonsei University Observatory and Korea Astronomy and Space Science Institute. 91 natexlab#1#1 [Abadi et al.(1999)Abadi, Moore, & Bower]abadi99 Abadi, M. G., Moore, B., & Bower, R. G. 1999, , 308, 947 [Baldry et al.(2008)Baldry, Glazebrook, & Driver]baldry08 Baldry, I. K., Glazebrook, K., & Driver, S. P. 2008, , 388, 945 [Baldry et al.(2012)Baldry, Driver, Loveday, Taylor, Kelvin, Liske, Norberg, Robotham, Brough, Hopkins, Bamford, Peacock, Bland-Hawthorn, Conselice, Croom, Jones, Parkinson, Popescu, Prescott, Sharp, & Tuffs]baldry12 Baldry, I. K., Driver, S. P., Loveday, J., et al. 2012, , 421, 621 [Behroozi et al.(2010)Behroozi, Conroy, & Wechsler]behroozi10 Behroozi, P. S., Conroy, C., & Wechsler, R. H. 2010, , 717, 379 [Behroozi et al.(2013)Behroozi, Wechsler, & Conroy]behroozi13 Behroozi, P. S., Wechsler, R. H., & Conroy, C. 2013, , 770, 57 [Bertschinger(1995)]bertschinger95 Bertschinger, E. 1995, ArXiv Astrophysics e-prints [Boylan-Kolchin et al.(2008)Boylan-Kolchin, Ma, & Quataert]boylan-kolchin08 Boylan-Kolchin, M., Ma, C.-P., & Quataert, E. 2008, , 383, 93 [Bundy et al.(2009)Bundy, Fukugita, Ellis, Targett, Belli, & Kodama]bundy09 Bundy, K., Fukugita, M., Ellis, R. S., et al. 2009, , 697, 1369 [Chung et al.(2007)Chung, van Gorkom, Kenney, & Vollmer]chung07 Chung, A., van Gorkom, J. H., Kenney, J. D. P., & Vollmer, B. 2007, , 659, L115 [Cimatti et al.(2004)Cimatti, Daddi, Renzini, Cassata, Vanzella, Pozzetti, Cristiani, Fontana, Rodighiero, Mignoli, & Zamorani]cimatti04 Cimatti, A., Daddi, E., Renzini, A., et al. 2004, , 430, 184 [Cowie et al.(1996)Cowie, Songaila, Hu, & Cohen]cowie96 Cowie, L. L., Songaila, A., Hu, E. M., & Cohen, J. G. 1996, , 112, 839 [Cox et al.(2008)Cox, Jonsson, Somerville, Primack, & Dekel]cox08 Cox, T. J., Jonsson, P., Somerville, R. S., Primack, J. R., & Dekel, A. 2008, , 384, 386 [Croton & Farrar(2008)]croton08 Croton, D. J., & Farrar, G. R. 2008, , 386, 2285 [Croton et al.(2005)Croton, Farrar, Norberg, Colless, Peacock, Baldry, Baugh, Bland-Hawthorn, Bridges, Cannon, Cole, Collins, Couch, Dalton, De Propris, Driver, Efstathiou, Ellis, Frenk, Glazebrook, Jackson, Lahav, Lewis, Lumsden, Maddox, Madgwick, Peterson, Sutherland, & Taylor]croton05 Croton, D. J., Farrar, G. R., Norberg, P., et al. 2005, , 356, 1155 [Croton et al.(2006)Croton, Springel, White, De Lucia, Frenk, Gao, Jenkins, Kauffmann, Navarro, & Yoshida]croton06 Croton, D. J., Springel, V., White, S. D. M., et al. 2006, , 365, 11 [Daddi et al.(2005)Daddi, Renzini, Pirzkal, Cimatti, Malhotra, Stiavelli, Xu, Pasquali, Rhoads, Brusa, di Serego Alighieri, Ferguson, Koekemoer, Moustakas, Panagia, & Windhorst]daddi05 Daddi, E., Renzini, A., Pirzkal, N., et al. 2005, , 626, 680 [De Lucia & Blaizot(2007)]delucia07 De Lucia, G., & Blaizot, J. 2007, , 375, 2 [De Lucia et al.(2006)De Lucia, Springel, White, Croton, & Kauffmann]delucia06 De Lucia, G., Springel, V., White, S. D. M., Croton, D., & Kauffmann, G. 2006, , 366, 499 [Dubois et al.(2013)Dubois, Pichon, Devriendt, Silk, Haehnelt, Kimm, & Slyz]dubois13 Dubois, Y., Pichon, C., Devriendt, J., et al. 2013, , 428, 2885 [Elahi et al.(2013)Elahi, Han, Lux, Ascasibar, Behroozi, Knebe, Muldrew, Onions, & Pearce]elahi13 Elahi, P. J., Han, J., Lux, H., et al. 2013, , 433, 1537 [Feldmeier et al.(2002)Feldmeier, Mihos, Morrison, Rodney, & Harding]feldmeier02 Feldmeier, J. J., Mihos, J. C., Morrison, H. L., Rodney, S. A., & Harding, P. 2002, , 575, 779 [Furlong et al.(2015)Furlong, Bower, Theuns, Schaye, Crain, Schaller, Dalla Vecchia, Frenk, McCarthy, Helly, Jenkins, & Rosas-Guevara]furlong15 Furlong, M., Bower, R. G., Theuns, T., et al. 2015, , 450, 4486 [Genel et al.(2014)Genel, Vogelsberger, Springel, Sijacki, Nelson, Snyder, Rodriguez-Gomez, Torrey, & Hernquist]genel14 Genel, S., Vogelsberger, M., Springel, V., et al. 2014, , 445, 175 [Glazebrook et al.(2004)Glazebrook, Abraham, McCarthy, Savaglio, Chen, Crampton, Murowinski, Jørgensen, Roth, Hook, Marzke, & Carlberg]glazebrook04 Glazebrook, K., Abraham, R. G., McCarthy, P. J., et al. 2004, , 430, 181 [Gonzalez et al.(2005)Gonzalez, Zabludoff, & Zaritsky]gonzalez05 Gonzalez, A. H., Zabludoff, A. I., & Zaritsky, D. 2005, , 618, 195 [Gruppioni et al.(2015)Gruppioni, Calura, Pozzi, Delvecchio, Berta, De Lucia, Fontanot, Franceschini, Marchetti, Menci, Monaco, & Vaccari]gruppioni15 Gruppioni, C., Calura, F., Pozzi, F., et al. 2015, , 451, 3419 [Gunn & Gott(1972)]gunn72 Gunn, J. E., & Gott, III, J. R. 1972, , 176, 1 [Guo & White(2008)]guo08 Guo, Q., & White, S. D. M. 2008, , 384, 2 [Häring & Rix(2004)]haring04 Häring, N., & Rix, H.-W. 2004, , 604, L89 [Hirschmann et al.(2015)Hirschmann, Naab, Ostriker, Forbes, Duc, Davé, Oser, & Karabal]hirschmann15 Hirschmann, M., Naab, T., Ostriker, J. P., et al. 2015, , 449, 528 [Hopkins et al.(2010)Hopkins, Bundy, Croton, Hernquist, Keres, Khochfar, Stewart, Wetzel, & Younger]hopkins10a Hopkins, P. F., Bundy, K., Croton, D., et al. 2010, , 715, 202 [Hu(2009)]hu09 Hu, J. 2009, ArXiv e-prints [Ji et al.(2014)Ji, Peirani, & Yi]ji14 Ji, I., Peirani, S., & Yi, S. K. 2014, , 566, A97 [Jiménez et al.(2011)Jiménez, Cora, Bassino, Tecce, & Smith Castelli]jimenez11 Jiménez, N., Cora, S. A., Bassino, L. P., Tecce, T. E., & Smith Castelli, A. V. 2011, , 417, 785 [Jung et al.(2014)Jung, Lee, & Yi]jung14 Jung, I., Lee, J., & Yi, S. K. 2014, , 794, 74 [Karim et al.(2011)Karim, Schinnerer, Martínez-Sansigre, Sargent, van der Wel, Rix, Ilbert, Smolčić, Carilli, Pannella, Koekemoer, Bell, & Salvato]karim11 Karim, A., Schinnerer, E., Martínez-Sansigre, A., et al. 2011, , 730, 61 [Kauffmann & Haehnelt(2000)]kauffmann00 Kauffmann, G., & Haehnelt, M. 2000, , 311, 576 [Kaviraj(2014a)]kaviraj14b Kaviraj, S. 2014a, , 440, 2944 [Kaviraj(2014b)]kaviraj14a —. 2014b, , 437, L41 [Kaviraj et al.(2014)Kaviraj, Huertas-Company, Cohen, Peirani, Windhorst, O'Connell, Silk, Dopita, Hathi, Koekemoer, Mei, Rutkowski, Ryan, & Shankar]kaviraj14c Kaviraj, S., Huertas-Company, M., Cohen, S., et al. 2014, , 443, 1861 [Khim et al.(2015)Khim, Park, Seo, Lee, Smith, & Yi]khim15 Khim, H.-g., Park, J., Seo, S.-W., et al. 2015, , 220, 3 [Kimm et al.(2011)Kimm, Yi, & Khochfar]kimm11 Kimm, T., Yi, S. K., & Khochfar, S. 2011, , 729, 11 [Knebe et al.(2015)Knebe, Pearce, Thomas, Benson, Blaizot, Bower, Carretero, Castander, Cattaneo, Cora, Croton, Cui, Cunnama, De Lucia, Devriendt, Elahi, Font, Fontanot, Garcia-Bellido, Gargiulo, Gonzalez-Perez, Helly, Henriques, Hirschmann, Lee, Mamon, Monaco, Onions, Padilla, Power, Pujol, Skibba, Somerville, Srisawat, Vega-Martínez, & Yi]knebe15 Knebe, A., Pearce, F. R., Thomas, P. A., et al. 2015, , 451, 4029 [Komatsu et al.(2011)Komatsu, Smith, Dunkley, Bennett, Gold, Hinshaw, Jarosik, Larson, Nolta, Page, Spergel, Halpern, Hill, Kogut, Limon, Meyer, Odegard, Tucker, Weiland, Wollack, & Wright]komatsu11 Komatsu, E., Smith, K. M., Dunkley, J., et al. 2011, , 192, 18 [Kormendy & Bender(2013)]kormendy13 Kormendy, J., & Bender, R. 2013, , 769, L5 [Kormendy & Ho(2013)]kormendy13b Kormendy, J., & Ho, L. C. 2013, , 51, 511 [Lackner et al.(2012)Lackner, Cen, Ostriker, & Joung]lackner12 Lackner, C. N., Cen, R., Ostriker, J. P., & Joung, M. R. 2012, , 425, 641 [Lee & Yi(2013)]lee13 Lee, J., & Yi, S. K. 2013, , 766, 38 [Lee et al.(2014)Lee, Yi, Elahi, Thomas, Pearce, Behroozi, Han, Helly, Jung, Knebe, Mao, Onions, Rodriguez-Gomez, Schneider, Srisawat, & Tweed]lee14 Lee, J., Yi, S. K., Elahi, P. J., et al. 2014, , 445, 4197 [Li & White(2009)]li09 Li, C., & White, S. D. M. 2009, , 398, 2177 [Magorrian et al.(1998)Magorrian, Tremaine, Richstone, Bender, Bower, Dressler, Faber, Gebhardt, Green, Grillmair, Kormendy, & Lauer]magorrian98 Magorrian, J., Tremaine, S., Richstone, D., et al. 1998, , 115, 2285 [Marconi & Hunt(2003)]marconi03 Marconi, A., & Hunt, L. K. 2003, , 589, L21 [Matsuoka & Kawara(2010)]matsuoka10 Matsuoka, Y., & Kawara, K. 2010, , 405, 100 [McConnell & Ma(2013)]mcconnell13 McConnell, N. J., & Ma, C.-P. 2013, , 764, 184 [Moster et al.(2010)Moster, Somerville, Maulbetsch, van den Bosch, Macciò, Naab, & Oser]moster10 Moster, B. P., Somerville, R. S., Maulbetsch, C., et al. 2010, , 710, 903 [Muldrew et al.(2011)Muldrew, Pearce, & Power]muldrew11 Muldrew, S. I., Pearce, F. R., & Power, C. 2011, , 410, 2617 [Muldrew et al.(2012)Muldrew, Croton, Skibba, Pearce, Ann, Baldry, Brough, Choi, Conselice, Cowan, Gallazzi, Gray, Grützbauch, Li, Park, Pilipenko, Podgorzec, Robotham, Wilman, Yang, Zhang, & Zibetti]muldrew12 Muldrew, S. I., Croton, D. J., Skibba, R. A., et al. 2012, , 419, 2670 [Murante et al.(2004)Murante, Arnaboldi, Gerhard, Borgani, Cheng, Diaferio, Dolag, Moscardini, Tormen, Tornatore, & Tozzi]murante04 Murante, G., Arnaboldi, M., Gerhard, O., et al. 2004, , 607, L83 [Naab et al.(2009)Naab, Johansson, & Ostriker]naab09 Naab, T., Johansson, P. H., & Ostriker, J. P. 2009, , 699, L178 [Onions et al.(2013)Onions, Ascasibar, Behroozi, Casado, Elahi, Han, Knebe, Lux, Merchán, Muldrew, Neyrinck, Old, Pearce, Potter, Ruiz, Sgró, Tweed, & Yue]onions13 Onions, J., Ascasibar, Y., Behroozi, P., et al. 2013, , 429, 2739 [Oser et al.(2012)Oser, Naab, Ostriker, & Johansson]oser12 Oser, L., Naab, T., Ostriker, J. P., & Johansson, P. H. 2012, , 744, 63 [Oser et al.(2010)Oser, Ostriker, Naab, Johansson, & Burkert]oser10 Oser, L., Ostriker, J. P., Naab, T., Johansson, P. H., & Burkert, A. 2010, , 725, 2312 [Panter et al.(2007)Panter, Jimenez, Heavens, & Charlot]panter07 Panter, B., Jimenez, R., Heavens, A. F., & Charlot, S. 2007, , 378, 1550 [Prunet & Pichon(2013)]prunet13 Prunet, S., & Pichon, C. 2013, MPgrafic: A parallel MPI version of Grafic-1, Astrophysics Source Code Library [Quilis et al.(2000)Quilis, Moore, & Bower]quilis00 Quilis, V., Moore, B., & Bower, R. 2000, Science, 288, 1617 [Rodighiero et al.(2010)Rodighiero, Cimatti, Gruppioni, Popesso, Andreani, Altieri, Aussel, Berta, Bongiovanni, Brisbin, Cava, Cepa, Daddi, Dominguez-Sanchez, Elbaz, Fontana, Förster Schreiber, Franceschini, Genzel, Grazian, Lutz, Magdis, Magliocchetti, Magnelli, Maiolino, Mancini, Nordon, Perez Garcia, Poglitsch, Santini, Sanchez-Portal, Pozzi, Riguccini, Saintonge, Shao, Sturm, Tacconi, Valtchanov, Wetzstein, & Wieprecht]rodighiero10 Rodighiero, G., Cimatti, A., Gruppioni, C., et al. 2010, , 518, L25 [Rodriguez-Gomez et al.(2016)Rodriguez-Gomez, Pillepich, Sales, Genel, Vogelsberger, Zhu, Wellons, Nelson, Torrey, Springel, Ma, & Hernquist]rodriguez-gomez16 Rodriguez-Gomez, V., Pillepich, A., Sales, L. V., et al. 2016, , 458, 2371 [Sani et al.(2011)Sani, Marconi, Hunt, & Risaliti]sani11 Sani, E., Marconi, A., Hunt, L. K., & Risaliti, G. 2011, , 413, 1479 [Schawinski et al.(2006)Schawinski, Khochfar, Kaviraj, Yi, Boselli, Barlow, Conrow, Forster, Friedman, Martin, Morrissey, Neff, Schiminovich, Seibert, Small, Wyder, Bianchi, Donas, Heckman, Lee, Madore, Milliard, Rich, & Szalay]schawinski06 Schawinski, K., Khochfar, S., Kaviraj, S., et al. 2006, , 442, 888 [Schaye et al.(2015)Schaye, Crain, Bower, Furlong, Schaller, Theuns, Dalla Vecchia, Frenk, McCarthy, Helly, Jenkins, Rosas-Guevara, White, Baes, Booth, Camps, Navarro, Qu, Rahmati, Sawala, Thomas, & Trayford]schaye15 Schaye, J., Crain, R. A., Bower, R. G., et al. 2015, , 446, 521 [Sheen et al.(2012)Sheen, Yi, Ree, & Lee]sheen12 Sheen, Y.-K., Yi, S. K., Ree, C. H., & Lee, J. 2012, , 202, 8 [Smith et al.(2016)Smith, Choi, Lee, Rhee, Sanchez-Janssen, & Yi]smith16 Smith, R., Choi, H., Lee, J., et al. 2016, , 833, 109 [Somerville et al.(2008)Somerville, Hopkins, Cox, Robertson, & Hernquist]somerville08 Somerville, R. S., Hopkins, P. F., Cox, T. J., Robertson, B. E., & Hernquist, L. 2008, , 391, 481 [Song et al.(2016)Song, Finkelstein, Ashby, Grazian, Lu, Papovich, Salmon, Somerville, Dickinson, Duncan, Faber, Fazio, Ferguson, Fontana, Guo, Hathi, Lee, Merlin, & Willner]song16 Song, M., Finkelstein, S. L., Ashby, M. L. N., et al. 2016, , 825, 5 [Sparre et al.(2015)Sparre, Hayward, Springel, Vogelsberger, Genel, Torrey, Nelson, Sijacki, & Hernquist]sparre15 Sparre, M., Hayward, C. C., Springel, V., et al. 2015, , 447, 3548 [Springel(2005)]springel05b Springel, V. 2005, , 364, 1105 [Springel et al.(2005)Springel, Di Matteo, & Hernquist]springel05a Springel, V., Di Matteo, T., & Hernquist, L. 2005, , 361, 776 [Springel et al.(2001)Springel, Yoshida, & White]springel01 Springel, V., Yoshida, N., & White, S. D. M. 2001, , 6, 79 [Srisawat et al.(2013)Srisawat, Knebe, Pearce, Schneider, Thomas, Behroozi, Dolag, Elahi, Han, Helly, Jing, Jung, Lee, Mao, Onions, Rodriguez-Gomez, Tweed, & Yi]srisawat13 Srisawat, C., Knebe, A., Pearce, F. R., et al. 2013, , 436, 150 [Steinhauser et al.(2012)Steinhauser, Haider, Kapferer, & Schindler]steinhauser12 Steinhauser, D., Haider, M., Kapferer, W., & Schindler, S. 2012, , 544, A54 [Thomas et al.(2010)Thomas, Maraston, Schawinski, Sarzi, & Silk]thomas10 Thomas, D., Maraston, C., Schawinski, K., Sarzi, M., & Silk, J. 2010, , 404, 1775 [Tonnesen & Bryan(2009)]tonnesen09 Tonnesen, S., & Bryan, G. L. 2009, , 694, 789 [Trujillo et al.(2006)Trujillo, Förster Schreiber, Rudnick, Barden, Franx, Rix, Caldwell, McIntosh, Toft, Häussler, Zirm, van Dokkum, Labbé, Moorwood, Röttgering, van der Wel, van der Werf, & van Starkenburg]trujillo06 Trujillo, I., Förster Schreiber, N. M., Rudnick, G., et al. 2006, , 650, 18 [van Dokkum(2005)]vandokkum05 van Dokkum, P. G. 2005, , 130, 2647 [van Dokkum et al.(2010)van Dokkum, Whitaker, Brammer, Franx, Kriek, Labbé, Marchesini, Quadri, Bezanson, Illingworth, Muzzin, Rudnick, Tal, & Wake]vandokkum10 van Dokkum, P. G., Whitaker, K. E., Brammer, G., et al. 2010, , 709, 1018 [Vogelsberger et al.(2014)Vogelsberger, Genel, Springel, Torrey, Sijacki, Xu, Snyder, Nelson, & Hernquist]vogelsberger14 Vogelsberger, M., Genel, S., Springel, V., et al. 2014, , 444, 1518 [White & Frenk(1991)]white91 White, S. D. M., & Frenk, C. S. 1991, , 379, 52 [White & Rees(1978)]white78 White, S. D. M., & Rees, M. J. 1978, , 183, 341 [Woo et al.(2013)Woo, Dekel, Faber, Noeske, Koo, Gerke, Cooper, Salim, Dutton, Newman, Weiner, Bundy, Willmer, Davis, & Yan]woo13 Woo, J., Dekel, A., Faber, S. M., et al. 2013, , 428, 3306 [Yagi et al.(2010)Yagi, Yoshida, Komiyama, Kashikawa, Furusawa, Okamura, Graham, Miller, Carter, Mobasher, & Jogee]yagi10 Yagi, M., Yoshida, M., Komiyama, Y., et al. 2010, , 140, 1814 [Yi et al.(2013)Yi, Lee, Jung, Ji, & Sheen]yi13 Yi, S. K., Lee, J., Jung, I., Ji, I., & Sheen, Y.-K. 2013, , 554, A122

http://arxiv.org/abs/1701.07936v1

20170127040314

From Abstract Entities in Mathematics to Superposition States in Quantum Mechanics

quant-ph

From Abstract Entities in Mathematics to Superposition States in Quantum Mechanics David Ellerman University of California at Riverside =================================================================================== Given an equivalence relation on a set U, there are two abstract notions of an element of the quotient set U/. The #1 abstract notion is a set S=[ u] of equivalent elements of U (an equivalence class); the #2 notion is an abstract entity u_S that is definite on what is common to the elements of the equivalence class S but is otherwise indefinite on the differences between those elements. For instance, the #1 interpretation of a homotopy type is an equivalence class of homotopic spaces, but the #2 interpretation, e.g., as developed in homotopy type theory, is an abstract space (without points) that has the properties that are in common to the spaces in the equivalence class but is otherwise indefinite. In philosophy, the #2 abstract entities might be called paradigm-universals, e.g., `the white thing' as opposed to the #1 abstract notion of "the set of white things" (out of some given collection U). The paper shows how this #2 notion of a paradigm may be mathematically modeled using incidence matrices in Boolean logic and density matrices in probability theory. Then we cross the bridge to the density matrix treatment of the indefinite superposition states in quantum mechanics (QM). This connection between the #2 abstracts in mathematics and ontic indefinite states in QM elucidates Abner Shimony's literal or objective indefiniteness interpretation of QM. § INTRODUCTION The purpose of this paper is to illuminate the late Abner Shimony's objectively indefinite or `Literal' interpretation of quantum mechanics based on seeing the superposition states as being objectively indefinite. From these two basic ideas alone – indefiniteness and the superposition principle – it should be clear already that quantum mechanics conflicts sharply with common sense. If the quantum state of a system is a complete description of the system, then a quantity that has an indefinite value in that quantum state is objectively indefinite; its value is not merely unknown by the scientist who seeks to describe the system. <cit.> In addition to the Shimony's phrase "objective indefiniteness," other philosophers of physics have used similar phrases for these indefinite states: * Peter Mittelstaedt's "incompletely determined" quantum states with "objective indeterminateness" <cit.>; * Paul Feyerabend's "inherent indefiniteness" <cit.>; * Allen Stairs' "value indefiniteness" and "disjunctive facts" <cit.>; * Steven French and Decio Krause's "ontic vagueness" <cit.>; or * E. J. Lowe's "vague identity" and "indeterminacy" that is "ontic" <cit.>. But how can we understand the notion of an "ontic indefinite state"? § TWO VERSIONS OF ABSTRACTION The claim is that we already have the notion of an indefinite state in the mathematical notion of an entity that abstracts as definite what is common to the distinct elements of a set S and rendering their differences as indefinite. Given an equivalence relation on a set U such as "having the same color" and if u u^' were white, then there are two notions of abstraction: * the #1 version of the abstraction operation takes equivalent entities u u^' to the equivalence class [ u] =[ u^'] of all white entities (in some universe U), and; * the #2 version of the abstraction operation takes all the equivalent entities u u^' to the abstract entity "the white entity" that is definite on what is common in the set of all particular white things but is indefinite on how they differ (e.g., on all the other properties that distinguish them). For instance, there are two notions of an `element' of a quotient set or a quotient group (or any other quotient object in algebra): * a quotient group element as an equivalence class or coset; or * a quotient group element as an abstract entity representing what is common to the equivalence class. Given any property S( u) defined on the elements of U, two abstract objects can be defined: < g r a p h i c s > Figure 1: A property determines two types of abstract objects. Intuitively the #2 abstract object u_S is `the paradigm S-entity' (the blob-sum ⊞ is defined below) which is definite on the S( u) property and indefinite on (i.e., blobs out) the differences between all the u∈ U such that S( u). § AN EXAMPLE STARTING WITH ATTRIBUTES Consider three predicates (binary attributes) P( x), Q( x), and R( x) which could distinguish at most 2^3=8 definite-particular entities: u_1,...,u_8 called eigen-elements and which can be presented in a table like a truth table: c]|c|c|c|c|P( x) Q( x) R( x) u 1 1 1 u_1 1 1 0 u_2 1 0 1 u_3 1 0 0 u_4 0 1 1 u_5 0 1 0 u_6 0 0 1 u_7 0 0 0 u_8 Table 1: Eight entities specified by 3 properties. The general rule is if f,g,h:U→ ℝ are numerical attributes with the number of distinct values as n_f, n_g, and n_h respectively, then those attributes could distinguish or classify n_f× n_g× n_h distinct subsets of U. If the join of the inverse-image partitions is the discrete partition, i.e.,{ f^-1}∨{ g^-1}∨{ h^-1} =1_U <cit.>, then { f,g,h} is a complete set of attributes since they can distinguish or classify the eigen-elements of U. Then we can distinguish the elements of U by their triple of values, i.e., | f( u_j) ,g( u_j) ,h( u_j) ⟩ uniquely determines u_j∈ U. In the example, any subset S⊆ U={ u_1,...,u_8} is characterized by a property S( x), the disjunctive normal form property, common to all and only the elements of S. If S={ u_1,u_4,u_7}, then the DNF property is: S( x) =[ P( x) ∧ Q( x) ∧ R( x) ] ∨[ P( x) ∧¬ Q( x) ∧¬ R( x) ] ∨[ ¬ P( x) ∧¬ Q( x) ∧ R( x) ]. But what are the #1 and #2 abstract entities? 1. The #1 abstract entity is the set S={ u_i∈ U|S( u_i) } ={ u_1 ,u_4,u_7} of all the distinct S( x)-entities; and 2. The #2 abstract entity is the paradigm-universal S( x)-entity symbolized u_S=u_1⊞ u_4⊞ u_7=⊞{ u_i∈ U|S( u_i) } The `superposition' or `blob-sum' of u_1, u_4, and u_7. that is definite on the DNF property S( x) but indefinite on what distinguishes the different S( x)-entities. Thus S( u_S) holds but none of the disjuncts hold since that would make u_S equal to u_1, u_4, or u_7. Hence S( u_S) is a `disjunctive fact' in the sense of Allen Stairs <cit.>. § SOME PHILOSOPHICAL CONCERNS It is best to think of S as the set of definite particular S( x)-entities in some universe U, while u_S is the indefinite paradigm-universal S( x)-entity is the `superposition' u_S=⊞{u_i∈ U|S( u_i) } that is, in general, "one over the many." Only when S={ u_j} is a singleton does the definite description `the S-entity' refer to an element of U, i.e., u_{ u_j}=u_j. Making the "one" u_S=⊞{u_i∈ U|S( u_i) } over the many, i.e., more abstract than the u_i∈ U (for | S| >1) avoids the paradoxes just as the iterative notion of set does in ordinary set theory, i.e., for #1 type of abstractions. Otherwise, if we ignore the given set U, then we can recreate Russell's Paradox for R( u_S) ≡¬ S( u_S) so: u_R=⊞{ u_S|¬ S( u_S) } and thus R( u_R) implies ¬ R( u_R), and ¬ R( u_R) implies R( u_R). But if we define u_R=⊞{ u_S∈ U|¬ S( u_S) }, then assuming u_R∈ U leads to the contradiction so u_R∉ U. The paradigm-universal u_S is not universal `S-ness'. Where S( x) is being white, then u_white= `the white thing`, not `whiteness'. This distinction goes back to Plato: But Plato also used language which suggests not only that the Forms exist separately (χωριστα) from all the particulars, but also that each Form is a peculiarly accurate or good particular of its own kind, i.e., the standard particular of the kind in question or the model (παραδειγμα) to which other particulars approximate. <cit.> Some have considered interpreting the Form as paradeigma as an error. For general characters are not characterized by themselves: humanity is not human. The mistake is encouraged by the fact that in Greek the same phrase may signify both the concrete and the abstract, e.g. λευκ oν (literally "the white") both "the white thing" and "whiteness", so that it is doubtful whether αυτ o τ o λευκ oν (literally "the white itself") means "the superlatively white thing" or "whiteness in abstraction". <cit.> Thus for the abstract property W( u) "whiteness", we have: * the #1 abstraction is the set of white things W={ u∈ U:W( u) }, and; * the #2 abstraction `the white thing' u_W. § RELATIONS BETWEEN #1 AND #2 UNIVERSALS For properties S() defined on U, there is a 1-1 correspondence between the #1 and #2 universals: ∪{{ u} |u∈ U&S( u) } =S ⟷ u_S=⊞{ u_{ u}|u∈ U&S( u) }. In each case, we may extend the definition of the property to the two universals. For T() another property defined on U: S( T) iff ( ∀ u∈ U) ( T( u) ⇒ S( u) ) iff S( u_T). In terms of the #1 universals, S( S) holds by definition and: S( T) iff T⊆ S, and similarly S( u_S) always holds. But what is the #2 universals equivalent of T⊆ S? Intuitively u_S is `the S-thing' that is definite on having the S-property but is otherwise indefinite on the differences between the members of S. If we make more properties definite, then in terms of subsets, that will in general cut down to a subset T⊆ S, so u_T would inherit the paradigmatic property holding on the superset S, i.e., S( u_T). This "process" to changing to a more definite universal u_S⇝ u_T for T⊆ S will be called projection and symbolized: u_T u_S (or u_S u_T) u_T is a "sharpening" or more definite version of u_S. c]|c|c|c|S() defined on U #1 abstraction #2 abstraction Universals for S() S=∪{{ u} |u∈ U&S( u) } u_S=⊞{ u_{ u}|u∈ U&S( u) } T() defined on U S( T) iff T⊆ S iff S( u_T) iff u_T u_S Table 2: Equivalents between #1 and #2 universals In the language of Plato, the projection relation is the relation of "participation" (μεθεξις or methexis). As Plato would say, u_T has the property S() iff it participates in `the S-thing', i.e., S( u_T) iff u_T u_S. Thus there are two theories of abstract objects: * Set theory is the theory of #1 abstract objects, the sets S, where (taking ∈ as the participation relation), sets are never self-participating, i.e., S∉ S; * There is a second theory about the #2 abstract entities, the paradigms u_S, which are always self-participating, i.e., u_S u_S. Like sets S, the #2 abstract entities u_S, the paradigm-universals, are routinely used in mathematics. § EXAMPLES OF ABSTRACT PARADIGMS IN MATHEMATICS There is an equivalence relation A≃ B between topological spaces which is realized by a continuous map f:A→ B such that there is an inverse g:B→ A so the fg:B→ B is homotopic to 1_B (i.e., can be continuously deformed in 1_B) and gf is homotopic to 1_A. Classically "Homotopy types are the equivalence classes of spaces" <cit.> under this equivalence relation. That is the #1 type of abstraction. But the interpretation offered in homotopy type theory is expanding identity to "coincide with the (unchanged) notion of equivalence" <cit.> so it would refer to the #2 homotopy type, i.e., `the homotopy type' that captures the mathematical properties shared by all spaces in an equivalence class of homotopic spaces (wiping out the differences). Note that `the homotopy type' is not one of the classical topological spaces (with points etc.) in the #1 equivalence class of homotopic spaces. While classical homotopy theory is analytic (spaces and paths are made of points), homotopy type theory is synthetic: points, paths, and paths between paths are basic, indivisible, primitive notions. <cit.> Homotopy type theory systematically develops a theory of the #2 type of abstractions that grows out of homotopy theory and type theory in a new foundational theory. From the logical point of view, however, it is a radically new idea: it says that isomorphic things can be identified! Mathematicians are of course used to identifying isomorphic structures in practice, but they generally do so by “abuse of notation”, or some other informal device, knowing that the objects involved are not “really” identical. But in this new foundational scheme, such structures can be formally identified, in the logical sense that every property or construction involving one also applies to the other. <cit.> Our purpose is rather more modest, to model the theory of paradigm-universals u_S and their projections u_T–that is analogous to working with sets and subsets, e.g., in a Boolean algebra of subsets. That is all we will need to show that probability theory can be developed using paradigms u_S instead of subset-events S, and to make the connection to quantum mechanics. Another homotopy example is `the path going once (clockwise) around the hole' in an annulus A (disk with one hole), an abstract entity 1∈π_0( A) ≅ ℤ: < g r a p h i c s > Figure 2: `the path going once (clockwise) around the hole' Note that `the path going once (clockwise) around the hole' has the paradigmatic property of "going once (clockwise) around the hole" but is not one of the particular (coordinatized) paths that constitute the equivalence class of coordinatized once-around paths deformable into one another. In a similar manner, we can view other common #2 abstractions such as: `the cardinal number 5' that captures what is common to the isomorphism class of all five-element sets; `the number 1 mod( n)' that captures what is common within the equivalence class { ...,-2n+1,-n+1,1,n+1,2n+1,...} of integers; `the circle' or `the equilateral triangle'–and so forth. Category theory helped to motivate homotopy type theory for good reason. Category theory has no notion of identity between objects, only isomorphism as `equivalence' between objects. Therefore category theory can be seen as a theory of abstract #2 objects ("up to isomorphism"), e.g., abstract sets, groups, spaces, etc. § THE CONNECTION TO INTERPRETING SYMMETRY OPERATIONS The difference between the #1 abstract set and the #2 abstract entity can also be visually illustrated in a simple example of the symmetry operation (defining an equivalence relation) of reflection on the aA-axis for a fully definite isosceles triangles: < g r a p h i c s > Figure 3: Reflection on vertical axis symmetry operation. Thus the equivalence class of reflective-symmetric figures in the #1 or classical interpretation is the set: < g r a p h i c s > Figure 4: The #1 abstraction of equivalence class. But under the #2 or indefiniteness-abstraction(-quantum) interpretation, the equivalence abstracts to the figure that is definite as to what is the same and indefinite as to what is different between the definite figures in the equivalence class: < g r a p h i c s > Figure 5: The #2 abstraction of indefinite entity. Note that the symmetry operation on the indefinite figure is the identity. As noted in the discussion of homotopy type theory, the movement from the #1 equivalence class S to the #2 abstract-indefinite entity u_S replaces equivalence with identity. That is because the symmetry operation goes from one element in an equivalence class S to another element in S that differs in some definite aspects, but those are precisely the aspects that are removed in the indefinite-abstract u_S–so the symmetry just takes u_S to itself. Since we are later going to relate the #2 entities to the indefinite states of quantum mechanics, the example suggests that while classically a symmetry operation is invariant on an equivalence class S (i.e., takes one definite element in the equivalence class S to another definite element in S), in the #2 quantum case, the symmetry operation on the indefinite entity u_S is the identity. This is illustrated in the transition from the classical Maxwell-Boltzmann statistics to the quantum Bose-Einstein statistics. Suppose we have two particles of the same type which are classically indistinguishable so, following Weyl, we distinguish them as Mike and Ike. If each of the two particles could be in states A, B, or C, then the set of possible states is the set of nine ordered pairs { A,B,C}×{ A,B,C}. Applying the symmetry operation of permuting Mike and Ike, we have six equivalence classes. c]|c|c|Equivalence classes under permutation M-B {( A,B) ,( B,A) } 2/9 {( A,C) ,( C,A) } 2/9 {( B,C) ,( C,B) } 2/9 {( A,A) } 1/9 {( B,B) } 1/9 {( C,C) } 1/9 Table 3: Maxwell-Boltzmann distribution. Since the primitive data are the ordered pairs, we assign the equal probabilities of 1/9 to each pair which results in the Maxwell-Boltzmann distribution for the equivalence classes. But in the quantum case, we don't have an equivalence class S of distinct ordered pairs like {( A,B) ,( B,A) } under the symmetry; we have a single indefinite entity u_{( A,B) ,( B,A) } where the symmetry operation is the identity. Since there are now only six primitive entities, we assign the equal probabilities of 1/6 to each entity and obtain the Bose-Einstein distribution. c]|c|c|Six indefinite states B-E 1|c|u_{( A,B) ,( B,A) } 1/6 1|c|u_{( A,C) ,( C,A) } 1/6 1|c|u_{( B,C) ,( C,B) } 1/6 1|c|u_{( A,A) } 1/6 1|c|u_{( B,B) } 1/6 1|c|u_{( C,C) } 1/6 Table 4: Bose-Einstein distribution. Ruling out repeated states (i.e., the Pauli exclusion principle), there are only three primitive entities and that gives the Fermi-Dirac distribution.[For more of this pedagogical model of QM using sets (where the sets may be given the #2 abstraction u_S interpretation), see <cit.>.] c]|c|c|Three possible indefinite states F-D 1|c|u_{( A,B) ,( B,A) } 1/3 1|c|u_{( A,C) ,( C,A) } 1/3 1|c|u_{( B,C) ,( C,B) } 1/3 Table 5: Fermi-Dirac distribution. § HOW TO MODEL THE #1 AND #2 ABSTRACTS There are simple but different models to distinguish the #1 and #2 interpretations for S⊆ U with a finite U={ u_1 ,...,u_n} such as: < g r a p h i c s > Figure 6: Universe U of figures Ordinarily the set of solid figures S={ u_2,u_3 ,u_4}⊆{ u_1,u_2,u_3,u_4} =U would be represented by a one-dimensional column vector | S⟩ = [ 0; 1; 1; 1 ], but by using a two-dimensional matrix, we can represent the two #1 and #2 versions of S as two types of incidence matrices. * The #1 (classical) representation of S (i.e., set of S-things or set of solid figures) is the diagonal matrix In( Δ S) that lays the column vector | S⟩ along the diagonal: In( Δ S) = [ 0 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1 ] = representation of set S of distinct S-entities. In ( Δ S) is the incidence matrix of the diagonal Δ S⊆ U× U whose entries are the values of the characteristic function χ_Δ S( u_j,u_k) =δ_jkχ_S( u_j). * The #2 (quantum) representation of S (i.e., the S-thing) is the matrix In( S× S) that uses a 1 in the row j, column k cell to mean u_j and u_k are both in S: In( S× S) =| S⟩( | S⟩) ^t= [ 0 0 0 0; 0 1 1 1; 0 1 1 1; 0 1 1 1 ] = representation of one indistinct S-thing, `the solid figure' u_S=u_2⊞ u_3⊞ u_4. In( S× S) is the incidence matrix of the product S× S⊆ U× U (instead of the diagonal Δ S) with the entries χ_S× S( u_j,u_k). Note that for singletons S={ u_j}, In( Δ S) = In( S× S) as expected, and for | S| >1, In( Δ S) ≠ In( S× S). The two representations differ only in the off-diagonal entries. Think of the off-diagonal In( S× S) _j,k=1's as equating, cohering, or `blobbing' together u_j and u_k: In( S× S) = [ 0 0 0 0; 0 1 1 1; 0 1 1 1; 0 1 1 1 ] says [ 0 0 0 0; 0 1 u_2 u_3 u_ 2 u_ 4; 0 u_ 3 u_ 2 1 u_3 u_ 4; 0 u_ 4 u_ 2 u _4 u_3 1 ]. We now can represent the blob-sum #2 operation on entities: u_S =⊞{ u_i∈ U|S( u_i) } as the blob-sum ⊞ of the corresponding incidence matrices: In( S× S) =⊞_u_i∈ S In( { u_i}×{ u_i}) where the blob-sum ⊞ is defined for S_1 ,S_2⊆ U with S=S_1∪ S_2: In( S_1× S_1) ⊞In ( S_2× S_2) := In( S× S) =In( ( S_1∪ S_2) ×( S_1∪ S_2) ) =In( S_1× S_1∪ S_2× S_2∪ S_1× S_2∪ S_2× S_1) =In( S_1× S_1) ∨In ( S_2× S_2) ∨In( S_1× S_2) ∨In( S_2× S_1). Disjunction: In( S_1× S_1) ∨In( S_2× S_2) ∨ blobbing cross-terms.[The disjunction of incidence matrices is the usual entry-wise disjunction: 1∨1=1∨0=0∨1=1 and 0∨0=0, and similarly for conjunction.] For S={ u_2,u_4}, the blob-sum u_S=u_2⊞ u_4 is represented by: In( { u_2}×{ u_2}) ⊞In( { u_4}×{ u_4}) =In ( S× S) where the blob-sum operation ⊞ means `blobbing-out' the distinctions between entities in S (given by the cross-terms in { u_2,u_4}×{ u_2,u_4}): In( S× S) =In( { u_2}×{ u_2}) ⊞In( { u_4}×{ u_4}) = [ 0 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 0 ]⊞ [ 0 0 0 0; 0 0 0 0; 0 0 0 0; 0 0 0 1 ] =In( { u_2,u_4}×{ u_2,u_4}) =In( { u_2}×{ u_2}) ∨In( { u_4}×{ u_4}) ∨In( { u_2}×{ u_4}) ∨In ( { u_4}×{ u_2}) = [ 0 0 0 0; 0 1 0 1; 0 0 0 0; 0 1 0 1 ]. Due to the development of Boolean subset logic and set theory, we are perfectly comfortable with considering the #1 abstractions of sets S of even concrete ur-elements like the set of entities on a table. The representatives In( Δ S) trivially form a BA isomorphic to the BA of subsets ℘( U) . To better understand abstraction in mathematics and indefinite states in QM, we should become as comfortable with paradigms u_S as with sets S. The paradigms u_S for S∈℘( U) form a Boolean algebra isomorphic to ℘( U) under the mapping: for any Boolean operation S#T for S,T∈℘( U), u_S#u_T is the paradigm represented by In( ( S#T) ×( S#T) ). * The union of subsets S∪ T induces the operation on paradigms represented by In( ( S∪ T) ×( S∪ T) ) =In( S× S) ⊞In( T× T), so the union or join of paradigms is the blob-sum u_S∪ T =u_S⊞ u_T (note as expected, for T⊆ S, u_S⊞ u_T=u_S); * The intersection or meet of paradigms u_S∧ u_T=u_S∩ T is represented by In( S∩ T× S∩ T) =In( S× S) ∧In( T× T) (note as expected, for T⊆ S, u_S∧ u_T=u_T); * The negation of a paradigm ¬ u_S=u_S^c is represented by In( S^c× S^c) =⊞{In( { u}×{ u}) |u∉ S} (note as expected, u_S⊞ u_S^c=u_U). § THE PROJECTION OPERATION: MAKING AN INDEFINITE ENTITY MORE DEFINITE Now suppose we classify or partition all the elements of U according to an attribute such as the parity of the number of sides, where a partition is a set of disjoint subsets (blocks) of U whose union is all of U. Let π be the partition of two blocks O={ Odd} ={ u_1,u_3} and E={ Even} ={ u_2,u_4}. The equivalence relation defined by π is *indit( π) =( O× O) ∪( E× E) <cit.> and the disjunction is: In( O× O) ∨In( E× E) =In( *indit( π) ) [ 1 0 1 0; 0 0 0 0; 1 0 1 0; 0 0 0 0 ]∨ [ 0 0 0 0; 0 1 0 1; 0 0 0 0; 0 1 0 1 ] = [ 1 0 1 0; 0 1 0 1; 1 0 1 0; 0 1 0 1 ]. The #1 (classical) operation of intersecting the set of even-sided figures with the set of solid figures to give the set of even-sided solid figures is represented as the conjunction: In( Δ E) ∧In( Δ S) = [ 0 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 1 ]∧ [ 0 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1 ] = [ 0 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 1 ]. The #2 (quantum) operation of `sharpening' or `rendering more definite' `the solid figure' u_S to `the even-sided solid figure' u_{ u_2,u_4}, so u_{ u_2,u_4} u_S (suggested reading: u_{ u_2,u_4} is a projection of u_S) is represented as: In( E× E) ∧In( S× S) = [ 0 0 0 0; 0 1 0 1; 0 0 0 0; 0 1 0 1 ]∧ [ 0 0 0 0; 0 1 1 1; 0 1 1 1; 0 1 1 1 ] = [ 0 0 0 0; 0 1 0 1; 0 0 0 0; 0 1 0 1 ]. But there is a better way to represent `sharpening' using matrix multiplication instead of just the logical operation ∧ on matrices, and it foreshadows the measurement operation in QM. The matrix In ( Δ E) =P_E is a projection matrix, i.e., the diagonal matrix with diagonal entries χ_E( u_i) so P_E| S⟩ =| E∩ S⟩. Then the result of the projection-sharpening can be represented as: | E∩ S⟩( | E∩ S⟩) ^t=P_E| S⟩( P_E| S⟩) ^t=P_E| S⟩( | S⟩) ^tP_E =P_EIn( S× S) P_E=In ( E× E) ∧In( S× S). Under the #2 interpretation, the parity-sharpening, parity-differentiation, or parity-measurement of `the solid figure' by both parities is represented as: In( *indit( π) ) ∧ In( S× S) =P_OIn ( S× S) P_O+P_EIn( S× S) P_E = [ 1 0 1 0; 0 1 0 1; 1 0 1 0; 0 1 0 1 ]∧ [ 0 0 0 0; 0 1 1 1; 0 1 1 1; 0 1 1 1 ] = [ 0 0 0 0; 0 1 0 1; 0 0 1 0; 0 1 0 1 ]. The results are `the even-sided solid figure' u_{ u_2,u_4} and `the odd-sided solid figure' u_{ u_3}=u_3. The important thing to notice is the action on the off-diagonal elements where the action 1⇝0 in the j,k-entry means that u_j and u_k have been deblobbed, decohered, distinguished, or differentiated–in this case by parity: In( S× S) In ( *indit( π) ) ∧ In( S× S) =P_OIn( S× S) P_O+P_EIn ( S× S) P_E [ 0 0 0 0; 0 1 1 deblob0 1; 0 1 deblob0 1 1 deblob 0; 0 1 1 deblob0 1 ]. We could also classify the figures as to having 4 or fewer sides ("few sides") or not ("many sides") so that partition is σ={{ u_1,u_2} ,{ u_3,u_4}} which is represented by: In( *indit( σ) ) = [ 1 1 0 0; 1 1 0 0; 0 0 1 1; 0 0 1 1 ] and In( *indit( σ) ) ∧( In( *indit( π) ) ∧In( S× S) ) = [ 0 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 1 ] = In( Δ S). Thus parity and few-or-many-sides properties suffice to classify the solid figures uniquely and thus to yield the representation In ( Δ S) of the distinct elements of S={ u_2 ,u_3,u_4}. Thus making all the distinctions (i.e., decohering the entities that cohered together in u_S) takes In( S× S) In( Δ S). In QM jargon, the parity and few-or-many-sides attributes constitute a "complete set of commuting operators" (CSCO) so that measurement of `the solid figure' by those observables will take `the solid figure,' to the separate eigen-solid-figures: `the few- and even-sided solid figure' (the square u_2), `the many- and odd-sided solid figure' (the pentagon u_3), and `the many- and even-sided solid figure' (the hexagon u_4). § FROM INCIDENCE TO DENSITY MATRICES The incidence matrices In( Δ S) and In( S× S) can be turned into density matrices by dividing through by their trace: ρ( Δ S) =1/*tr[ In( Δ S) ] In ( Δ S) and ρ( S) =1/*tr[ In( S× S) ] In( S× S). In terms of probabilities, this means treating the outcomes in S as being equiprobable with probability 1/| S|. But now we have the #1 and #2 interpretations of the sample space for finite discrete probability theory. * The #1 (classical) interpretation, represented by ρ( Δ S), is the classical version with S as the sample space of outcomes. For instance, the 6×6 diagonal matrix with diagonal entries 1/6 is "the statistical mixture describing the state of a classical dice [die] before the outcome of the throw" <cit.>; * The #2 (quantum) interpretation replaces the "sample space" with the one indefinite `the sample outcome' u_S represented by ρ( S) (like `the outcome of throwing a die') and, in a trial, the indefinite outcome u_S `sharpens to' or becomes a definite outcome u_i∈ S with probability 1/| S|. Let f:U→ ℝ be a real-valued random variable with distinct values ϕ_i for i=1,...,m and let π={ B_i} _i=1,...,m where B_i=f^-1( ϕ_i), be the partition of U according to the values. The classification of ρ( S) according to the different values is: In( *indit( π) ) ∧ρ( S) which distinguishes the elements of S that have different f-values. If P_B_i is the diagonal (projection) matrix with diagonal elements ( P_B_i) _jj=χ_B_i( u_j), then the probability of a trial returning a u_j with f( u_j) =ϕ_i is: ( ϕ_i|S) =*tr[ P_B_iρ( S) ]. For instance, in the previous example, where f:U→ ℝ gives the parity partition π with the two values ϕ_odd and ϕ_even, then: P_evenρ( S) = [ 0 0 0 0; 0 1 0 0; 0 0 0 0; 0 0 0 1 ] [ 0 0 0 0; 0 1/3 1/3 1/3; 0 1/3 1/3 1/3; 0 1/3 1/3 1/3 ]= [ 0 0 0 0; 0 1/3 1/3 1/3; 0 0 0 0; 0 1/3 1/3 1/3 ] so *tr[ P_evenρ( S) ] =2/3 which is the conditional probability of getting `the even-sided solid figure' starting with `the solid figure' in the #2 (quantum) interpretation. And under the #1 (standard) interpretation, ( ϕ_even|S) =*tr[ P_even ρ( Δ S) ] =2/3 which is the probability of getting an even-sided solid figure starting with the set of solid figures. These two interpretations of finite discrete probability theory extend easily to the case of point probabilities p_j for u_j∈ U, where: * ( ρ( Δ S) ) _jj=χ_S( u_j) p_j/( S), so *tr[ P_evenρ( Δ S) ] = probability of getting an even-sided solid figure starting with the set of solid figures, and * ( ρ( S) ) _j,k=χ_S( u_j) χ_S( u_k) √(p_jp_k)/( S), so *tr[ P_evenρ( S) ] = probability of getting `the even-sided solid figure' starting with `the solid figure.' The whole of finite discrete probability theory can be developed in this manner, mutatis mutandis, for the #2 interpretation paradigms. § DENSITY MATRICES IN QUANTUM MECHANICS The jump to quantum mechanics (QM) is to replace the binary digits like 0,1 in incidence matrices or reals √(p_jp_k) in `classical' density matrices by complex numbers. Instead of the set S represented by a column | S⟩ of 0,1, we have a normalized column |ψ⟩ of complex numbers α_j whose absolute squares are probabilities: |α_j| ^2=p_j, e.g., | S⟩ = [ 0; 1; 1; 1 ]|ψ⟩ = [ α_1; α_2; α_3; α_4 ] where α_1=0 and |α_j| ^2=p_j for j=2,3,4. * The density matrix ρ( Δψ) has the absolute squares |α_j| ^2=p_j laid out along the diagonal. * The density matrix ρ( ψ) has the j,k-entry as the product of α_j and α_k^∗ (complex conjugate of α_k), where p_j=α_j^∗α _j=|α_j| ^2. Thus: ρ( Δψ) = [ 0 0 0 0; 0 p_2 0 0; 0 0 p_3 0; 0 0 0 p_4 ] and ρ( ψ) = [ 0 0 0 0; 0 p_2 α_2α_3^∗ α_2α_4^∗; 0 α_3α_2^∗ p_3 α_3α_4^∗; 0 α_4α_2^∗ α_4α_3^∗ p_4 ]. The] off-diagonal terms of a density matrix...are often called quantum coherences because they are responsible for the interference effects typical of quantum mechanics that are absent in classical dynamics. <cit.> The classifying or measuring operation In( *indit( π) ) ∧ρ( ψ) could still be defined taking the minimum of corresponding entries in absolute value, but in QM it is defined as the Lüders mixture operation <cit.>. If π={ B_1 ,...,B_m} is a partition according to the eigenvalues ϕ _1,...,ϕ_m on U={ u_1,...,u_n} (where U is an orthonormal basis set for the observable being measured), let P_B_i be the diagonal (projection) matrix with diagonal entries ( P_B_i ) _jj=χ_B_i( u_j). Then In ( *indit( π) ) ∧ρ( ψ) is obtained as: ∑_B_i∈πP_B_iρ( ψ) P_B_i The Lüders mixture. The probability of getting the result ϕ_i is: ( ϕ_i|ψ) =*tr[ P_B_i ρ( ψ) ]. § A POP SCIENCE INTERLUDE The popular science version of the simplest case is Schrödinger's cat. < g r a p h i c s > Figure 7: Usual "And" version of Schrödinger's cat. This version of Schrödinger's cat as being "Dead & Alive" is like the usual mis-interpretation of the unobserved particle as going through "Slit 1 & Slit 2" in the double slit experiment. But the cat is not definitely alive and definitely dead at the same time. The quantum version is that the cat is indefinite between those two definite possibilities; it's in cat-limbo. Schrödinger's cat = dead-cat ⊞ live-cat. It would be more accurate to say "Dead or Alive–but neither definitely," a "disjunctive fact" <cit.>. < g r a p h i c s > Figure 8: The disjunctive cat. Technically the state vector is: < g r a p h i c s > Figure 9: Schrödinger's cat state vector. Using density matrices, we would represent Schrödinger's cat as being in the state: ρ( cat) = [ 1/2live 1/2; 1/2 1/2dead ]. § SIMPLEST QUANTUM EXAMPLE Consider a system with two spin-observable σ eigenstates|↑⟩ and |↓⟩ (like electron spin up or down along the z-axis) where the given normalized superposition state is |ψ⟩ =1/√(2) |↑⟩ +1/√(2)|↓⟩ = [ α_↑; α_↓ ] = [ 1/√(2); 1/√(2) ] so the density matrix is ρ( ψ) = [ p_↑ α_↑α_↓^∗; α_↓α_↑^∗ p_↓ ] = [ 1/2 1/2; 1/2 1/2 ] where p_↑=α_↑α_↑^∗ and p_↓=α_↓α_↓^∗. The measurement in that spin-observable σ goes from ρ( ψ) to In( *indit( σ) ) ∧ρ( ψ) = [ 1 0; 0 1 ]∧ [ p_↑ α_↑α_↓^∗; α_↓α_↑^∗ p_↓ ] = [ p_↑ 0; 0 p_↓ ] = [ 1/2 0; 0 1/2 ] =ρ( Δψ). Or using the Lüders mixture operation: P_↑ρ( ψ) P_↑+P_↓ρ( ψ) P_↓ = [ 1 0; 0 0 ] [ p_↑ α_↑α_↓^∗; α_↓α_↑^∗ p_↓ ] [ 1 0; 0 0 ] + [ 0 0; 0 1 ] [ p_↑ α_↑α_↓^∗; α_↓α_↑^∗ p_↓ ] [ 0 0; 0 1 ] = [ p_↑ 0; 0 p_↓ ] = [ 1/2 0; 0 1/2 ] =ρ( Δψ). The two versions of S=U give us two versions of finite discrete probability theory where: #1) U is the sample space or #2) u_U is the sample outcome. * The #1 classical version is the usual version which in this case is like flipping a fair coin and getting head or tails with equal probability. < g r a p h i c s > Figure 10: Outcome set for classical coin-flipping trial. 2. The #2 quantum version starts with the indefinite entity u_U=⊞{ u_i∈ U}, `the (indefinite) outcome', and a trial renders it into one of the definite outcomes u_i with some probability p_i so that u_U could be represented by the density matrix ρ( U) where ( ρ( U) ) _jk=√(p_jp_k). In this case, this is like a coin u_{ H,T} with the difference between heads or tails rendered indefinite or blobbed out, and the trial results in it sharpening to definitely heads or definitely tails with equal probability. < g r a p h i c s > Figure 11: `the outcome state' for quantum coin-flipping trial. Experimentally, it is not possible to distinguish between the #1 and #2 versions by σ-measurements. But in QM the two states ρ( Δψ) and ρ( ψ) can be distinguished by measuring other observables like spin along a different axis <cit.>. Thus we know in QM which version is the superposition (pure) state |ψ⟩ = [ α_↑; α_↓ ]; it is the #2 blob-state ρ( ψ). § CONCLUSIONS Quantum mechanics texts usually mention several interpretations such as the Copenhagen, many-worlds, or hidden-variables interpretations. Now that we have established a bridge from abstraction in mathematics to indefinite states in QM, we may (for fun) cross the bridge in the opposite direction. For instance, in the many-worlds (or many-minds) interpretation, 1∈π_0( A) ≅ ℤ would refer to a different specific coordinatized "once clockwise around the hole" path in each different world (or mind). Shimony, however, suggests the Literal or Objective Indefiniteness Interpretation–which we have seen is suggested by the mathematics itself. But the mathematical formalism ... suggests a philosophical interpretation of quantum mechanics which I shall call "the Literal Interpretation." ...This is the interpretation resulting from taking the formalism of quantum mechanics literally, as giving a representation of physical properties themselves, rather than of human knowledge of them, and by taking this representation to be complete. <cit.> We have approached QM by starting with the logical situation of a universe U of distinct entities. Given a property S( x) on U, we can associate with it: * the #1 abstract object S={ u_i∈ U|S( u_i) }, the set of S( x)-entities, or * the #2 abstract object u_S=⊞{ u_i∈ U|S( u_i) } which is the abstract entity expressing the properties common to the S( x)-entities but "abstracting away from," "rendering indefinite," "cohering together," or "blobbing out" the differences between those entities. We argued that the mathematical formalisms of incidence matrices and then density matrices can be used to formalize the two representations: * #1 representation as In( Δ S) or ρ( Δψ); and * #2 representation as In( S× S) or ρ( ψ). This dove-tailed precisely into usual density-matrix treatment in QM of quantum states |ψ⟩ as ρ( ψ) which, as suggested by Shimony, can be interpreted as objectively indefinite states. Yet since the ancient Greeks, we have the #2 Platonic notion of the abstract paradigm-universal `the S-entity', definite on what is common to the members of a set S and indefinite on where they differ, so the connection that may help to better understand quantum mechanics is: The paradigm u_S, `the S-entity' represented by In( S× S) the superposition state ψ represented by the density matrix ρ( ψ). This recalls Whitehead's quip that Western philosophy is "a series of footnotes to Plato." <cit.> 99 auletta:qmAuletta, Gennaro, Mauro Fortunato, and Giorgio Parisi. 2009. Quantum Mechanics. Cambridge UK: Cambridge University Press. baues:htBaues, Hans-Joachim. 1995. Homotopy Types. In Handbook of Algebraic Topology, edited by I.M. James, 1–72. Amsterdam: Elsevier Science. ell:partitionsEllerman, David 2010. The Logic of Partitions: Introduction to the Dual of the Logic of Subsets. Review of Symbolic Logic. 3 (2 June): 287-350. ell:qm-setsEllerman, David. 2016. Quantum Mechanics over Sets: A Pedagogical Model with Non-Commutative Finite Probability Theory as Its Quantum Probability Calculus. Synthese Online First (August): 34 pages. DOI 10.1007/s11229-016-1175-0. feyerabend:microFeyerabend, Paul 1983 (orig. 1962). Problems of Microphysics. In Frontiers of Science and Philosophy. Robert G. Colodny ed., Lanham MD: University Press of America: 189-283. french:onticvaguenessFrench, Steven and Decio Krause 2003. Quantum Vagueness. Erkenntnis. 59: 97-124. kneales:logicKneale, William, and Martha Kneale. 1962. The Development of Logic. Oxford: Oxford University Press. lowe:vagueidLowe, E. J. 1994. Vague Identity and Quantum Indeterminacy. Analysis. 54 (2): 110-114. mitt:kantMittelstaedt, Peter 1998. The Constitution of Objects in Kant's Philosophy and in Modern Physics. In Interpreting Bodies: Classical and Quantum Objects in Modern Physics. Elena Castellani ed., Princeton NJ: Princeton University Press: 168-180. shim:realityShimony, Abner 1988. The reality of the quantum world. Scientific American. 258 (1): 46-53. shim:viennaShimony, Abner. 1999. Philosophical and Experimental Perspectives on Quantum Physics. In Philosophical and Experimental Perspectives on Quantum Physics: Vienna Circle Institute Yearbook 7. Dordrecht: Springer Science+Business Media: 1-18. stairs:disjfactsStairs, Allen 1983. Quantum Logic, Realism, and Value Definiteness. Philosophy of Science. 50 (4): 578-602. ufp:hottUnivalent Foundations Program. 2013. Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Studies, Princeton. whitehead:pandrWhitehead, Alfred North. 1978. Process and Reality. New York: The Free Press.

http://arxiv.org/abs/1701.07531v4

20170126004315

Design of Improved Quasi-Cyclic Protograph-Based Raptor-Like LDPC Codes for Short Block-Lengths

cs.IT

Design of Improved Quasi-Cyclic Protograph-Based Raptor-Like LDPC Codes for Short Block-LengthsResearch is supported in part by National Science Foundation (NSF) grant CCF-1618272. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF. Research was carried out in part at the Jet Propulsion Laboratory (JPL), California Institute of Technology, under a contract with NASA. This work used computational and storage services associated with the Hoffman2 Shared Cluster provided by UCLA Institute for Digital Research and Education's Research Technology Group. Sudarsan V. S. Ranganathan1, Dariush Divsalar2, and Richard D. Wesel1 1Department of Electrical Engineering, University of California, Los Angeles, Los Angeles, California 90095 2Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109 Email: sudarsanvsr@ucla.edu, Dariush.Divsalar@jpl.nasa.gov, wesel@ucla.edu Received: date / Accepted: date ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ Protograph-based Raptor-like low-density parity-check codes (PBRL codes) are a recently proposed family of easily encodable and decodable rate-compatible LDPC (RC-LDPC) codes. These codes have an excellent iterative decoding threshold and performance across all design rates. PBRL codes designed thus far, for both long and short block-lengths, have been based on optimizing the iterative decoding threshold of the protograph of the RC code family at various design rates. In this work, we propose a design method to obtain better quasi-cyclic (QC) RC-LDPC codes with PBRL structure for short block-lengths (of a few hundred bits). We achieve this by maximizing an upper bound on the minimum distance of any QC-LDPC code that can be obtained from the protograph of a PBRL ensemble. The obtained codes outperform the original PBRL codes at short block-lengths by significantly improving the error floor behavior at all design rates. Furthermore, we identify a reduction in complexity of the design procedure, facilitated by the general structure of a PBRL ensemble. § INTRODUCTION AND BACKGROUND Protograph-based low-density parity-check (LDPC) codes <cit.>, <cit.> are a class of codes amenable to tractable analysis and design procedures. Protograph quasi-cyclic LDPC (QC-LDPC) codes <cit.>, a class of protograph codes, have parity-check matrices composed of circulant permutation matrices (CPMs) and permit very low complexity decoder implementations <cit.>. The presence of CPMs in protograph QC-LDPC codes enables us to understand how the connections in the protograph affect the girth and minimum distance of such codes. Fossorier <cit.>, Karimi and Banihashemi <cit.>, and others analyze the girth of a protograph QC-LDPC code by examining the protograph of the code. More pertinent to this paper are <cit.> and <cit.>. Smarandache and Vontobel <cit.> derive an upper bound on the minimum distance of any QC-LDPC code that can be obtained from a protograph. Butler and Siegel <cit.> extend the results of <cit.> to QC-LDPC codes based on punctured protographs. Protograph-based Raptor-like LDPC codes (PBRL codes) are a class of easily encodable rate-compatible (RC) LDPC code families proposed by Chen, Vakilinia et al. in <cit.>. PBRL code families have an excellent iterative decoding threshold <cit.>, <cit.> and performance across all rates for which they are designed. In <cit.>, the authors design PBRL protographs for long and short block-lengths by optimizing the iterative decoding threshold of the protograph at each rate. They show that PBRL QC-LDPC code families can outperform other RC-LDPC codes in the literature, both at short (≈ 1000 information bits) and long (≈ 16000 information bits) block-lengths. This paper considers the design of RC-LDPC codes for very short block-lengths (≈ 200 information bits). While the iterative decoding threshold is the correct design metric to use for code design at long block-lengths, minimum distance is more important at short block-lengths. One contribution of this paper is a new PBRL design approach. Given a set of design rates, we design protographs for PBRL ensembles by maximizing, at each rate, the upper bounds on the minimum distance that were derived in <cit.> and <cit.>. The resulting PBRL QC-LDPC code families outperform the ones designed by optimizing the iterative decoding threshold at each rate. The complexity of computing the aforementioned upper bounds increases quickly with the size of the protographs. A second contribution of this paper is to leverage the structure of PBRL protographs to identify a significant reduction in the complexity of the design procedure. The paper is organized as follows: Section <ref> provides the design procedure, derives the reduction that is possible in the computational complexity of the design procedure, and discusses design examples. Section <ref> shows simulation results. Section <ref> concludes the paper. § DESIGNING PBRL ENSEMBLES BY MAXIMIZING AN UPPER BOUND ON THE MINIMUM DISTANCE A PBRL ensemble is defined by its protomatrix P that has the following general form: P = [ P_𝖧𝖱𝖢 0; P_𝖨𝖱𝖢 I ]_n_c × n_v Here, 0 and I refer to the all-zeros and identity matrices of appropriate size. The highest-rate code (𝖧𝖱𝖢) of the rate-compatible protomatrix is represented by P_𝖧𝖱𝖢, which is of size n_c_H× n_v_H. The variable nodes of the protomatrix containing the identity matrix in (<ref>) represent the incremental redundancy symbols of P. Some variable nodes of a protomatrix could also be punctured. The design rate is R ≜(n_v-n_c)/n_t for a protomatrix with n_c check nodes, n_v variable nodes, and n_t of the n_v variable nodes that are transmitted. A protomatrix is lifted (see <cit.>, <cit.>) to obtain an LDPC code of block-length that is a multiple of n_t. The use of circulant permutation matrices while lifting yields QC-LDPC codes, which are practical and are the subject of this paper. §.§ Design Method The design of a PBRL protomatrix consists of two steps: First, we choose the 𝖧𝖱𝖢 part, P_𝖧𝖱𝖢, as a protomatrix by itself. Then, we obtain the 𝖨𝖱𝖢 part, P_𝖨𝖱𝖢, one row at a time. In the original work on PBRL codes by Chen, Vakilinia et al. <cit.>, the authors first choose an 𝖧𝖱𝖢 part with a degree distribution and an acceptable iterative decoding threshold. Then they design each row of P_𝖨𝖱𝖢 successively to optimize the iterative decoding threshold of the protomatrix up to that rate while designing the row, keeping all previously obtained rows fixed. The best known families of RC-LDPC codes at both short and long block-lengths are the PBRL codes as designed with the heuristics proposed by Chen, Vakilinia et al. in <cit.>. At short block-lengths, minimum distance can be more important than the iterative decoding threshold as a criterion to use while designing LDPC codes. A key feature of QC-LDPC codes based on protomatrices is that the minimum distance of any such code obtained from a protomatrix is upper bounded by a constant that depends only on the protomatrix. In order to state the upper bounds, which were derived in earlier works, we need the definition of the permanent of a square matrix. Denote the set {1, 2, …, ℓ} by [ℓ]. The permanent of an ℓ×ℓ square matrix A with elements a_i,j, i ∈ [ℓ], j ∈ [ℓ] over some commutative ring is defined as 𝗉𝖾𝗋𝗆 (A) = ∑_σ∏_1 ≤ j ≤ℓ a_j, σ(j) = ∑_σ∏_1 ≤ j ≤ℓ a_σ(j), j, where σ refers to a permutation and the summation is over all permutations of [ℓ]. The permanent, although it looks deceptively similar to the determinant, is harder to compute than the determinant <cit.>. While the arithmetic complexity of computing the determinant is O(ℓ^3), the most efficient algorithm known to compute the permanent of any square matrix, due to Ryser <cit.>, is of complexity Θ(ℓ· 2^ℓ). Let a protomatrix P with a positive design rate and no punctured variable nodes be of size n_c × n_v. If S ⊆[n_v], denote by P_S the sub-matrix of P formed by the columns indexed by elements of S. Then, any QC-LDPC code 𝒞 obtained from the protomatrix P has a minimum distance d_min(𝒞) that is upper bounded as d_min(𝒞) ≤S ⊆ [n_v], |S| = n_c + 1min^∗∑_i ∈ S𝗉𝖾𝗋𝗆(P_S ∖ i), where |·| refers to the cardinality of a set, S ∖ i is shorthand for S ∖{i}, and min^∗ returns the smallest non-zero value in a set of non-negative values with at least one positive value or +∞ if the set is {0}. Note that permanents computed from sub-matrices of a protomatrix are always non-negative. Let a punctured protomatrix P with a positive design rate less than 1 be of size n_c × n_v. Let the set of punctured variable nodes, a subset of [n_v], be denoted 𝒫. Denote any punctured QC-LDPC code that can be obtained from P by 𝒞' and the unpunctured version of the code 𝒞' by 𝒞. Then, provided that 𝒞 and 𝒞' have the same number of codewords in their codebooks (dimensionality), 𝒞' has a minimum distance d_min(𝒞') that is upper bounded as d_min(𝒞') ≤S ⊆ [n_v], |S| = n_c + 1min^∗∑_i ∈ S ∖𝒫𝗉𝖾𝗋𝗆(P_S ∖ i). With Theorems <ref> and <ref> in hand, we propose the following PBRL ensemble search procedure: * Choose an 𝖧𝖱𝖢 matrix of size n_c_H× n_v_H with a desired degree distribution and complexity constraint. A common complexity constraint is to limit the weight of each column in the protomatrix. * 𝖨𝖱𝖢 design: Select the next row of the protomatrix from a set of candidate rows to maximize the upper bound on the minimum distance via Theorem <ref> or <ref> (depending upon whether there are punctured nodes or not). If there are multiple candidates with the best upper bound, then select one at random. * Go to Step 2) if another row of 𝖨𝖱𝖢 is required. Otherwise, terminate the search procedure. It is not known, in general, whether the upper bounds of (<ref>) or (<ref>) are achievable. But our design procedure yields better RC QC-LDPC code families at short block-lengths than the design based on optimizing the iterative decoding thresholds. For a punctured protomatrix, care must be taken to ensure that not too many variable nodes are punctured. Otherwise, the dimensionality requirement in Theorem <ref> may be violated. §.§ Lowering the Complexity of the Design Procedure In this subsection, we leverage the general structure of the protomatrix of a PBRL ensemble in (<ref>) to reduce the complexity of computing the upper bounds in (<ref>) or (<ref>). Assume that we have a PBRL protomatrix of size n_c × n_v with an 𝖧𝖱𝖢 part that is of size n_c_H× n_v_H. Assume also that the protomatrix has no punctured variable nodes[We consider the case when the protomatrix has punctured variable nodes in the Appendix.]. With these assumptions, computation of the upper bound for the protomatrix as given in (<ref>) requires computing n_vn_c + 1·(n_c + 1) permanents, each of size n_c × n_c. The complexity of Ryser's algorithm to compute the required permanents increases quickly while constructing the 𝖨𝖱𝖢 part of a PBRL ensemble. Our following result leads to a significant reduction in both the number of permanents that need to be computed and the size of each permanent to be computed. Let a PBRL protomatrix P of size n_c × n_v with no punctured variable nodes have a positive design rate, i.e. n_v > n_c. Let the 𝖧𝖱𝖢 part be of size n_c_H× n_v_H. Assume that the upper bound in (<ref>) for P is a positive integer. Then, the same upper bound can be obtained with at most n_v_Hn_c_H + 1·(n_c + 1) permanents, each of size at most (n_c_H + 1) ×(n_c_H + 1). Before we provide the proof, we comment on the reduction in complexity of computing (<ref>). The complexity of computing each permanent would now depend only on the number of check nodes in the 𝖧𝖱𝖢 part, n_c_H. Also, the dominating factor in the expression for number of permanents to be computed is the binomial coefficient, which again would now depend only on the size of the 𝖧𝖱𝖢 part, n_c_H× n_v_H, and not on the size of the entire protomatrix. Let us first consider the case when S ⊆[n_v], |S| = n_c + 1 contains the last n_v - n_v_H columns, i.e. the columns that comprise the incremental redundancy variable nodes of the protomatrix and have an identity matrix of size (n_v - n_v_H) ×(n_v - n_v_H). Note that n_v - n_v_H = n_c - n_c_H. The n_c + 1 chosen columns form a sub-matrix with structure that can be written as: P_S = [c_1 c_2 ⋯ c_n_c_H+1 | P_𝖨𝖱], where c_i are the columns chosen from the initial n_v_H columns of P and P_𝖨𝖱 has the following structure: P_𝖨𝖱 = [ 0; I ]_{n_c ×(n_v - n_v_H)} Because each column in P_𝖨𝖱 contains only a single 1, the complexity of computing each of the n_c + 1 required permanents is at most the size of computing the permanent of an (n_c_H+1) ×(n_c_H+1) sub-matrix (when the removed column is from P_𝖨𝖱). When the removed column is not from P_𝖨𝖱, the complexity is the size of computing the permanent of an n_c_H× n_c_H sub-matrix since the product is zero for permutations that select elements not in the 𝖧𝖱𝖢 rows of c_i, 1 ≤ i ≤ n_c_H+1. Furthermore, there are n_v_Hn_c_H + 1 sets S ⊆[n_v] of size n_c + 1 that contain P_𝖨𝖱. Now let us consider the general set S of n_c+1 columns in [n_v]. First, let us assume that ∑_i ∈ S𝗉𝖾𝗋𝗆(P_S ∖ i) > 0, which implies that at least one of the n_c + 1 permanents is positive. Denote by P' one such n_c × n_c sub-matrix of P_S with a positive permanent. There exists a permutation denoted σ^* that has a positive product in (<ref>) when computed for the matrix P'. Assume the following definition of a permanent: 𝗉𝖾𝗋𝗆(P') = ∑_σ∏_1 ≤ j ≤ n_c p'_σ(j), j, where p'_i,j denotes the entries of P'. Consider all columns indexed by j ∈[n_c] such that σ^*(j) > n_c_H. There are n_c - n_c_H = n_v - n_v_H such columns. Replace all these columns by the columns of the sub-matrix P_𝖨𝖱 (whenever possible), in the following manner: Replace column j whose σ^*(j) = j' > n_c_H with the column in P_𝖨𝖱 whose only non-zero element, 1, is present in row j', unless the column from P_𝖨𝖱 is already in the set S of n_c + 1 columns under consideration. Call the newly obtained matrix P'_1. The sub-matrix P'_1 has a permanent that is positive and is at most the value of the permanent of P' due to the following reasons: Permutation σ^* yields a positive product with P'_1 because the replacements (whenever possible) only lead to non-zero entries at locations (σ^*(j),j): σ^*(j) > n_c_H. Furthermore, each permutation σ that yields a positive product ∏_i ∈σ p'_σ(j),j in P' (including σ^*) yields a product with P'_1 that is upper bounded by the product computed with P'. Let us denote the matrix [c_1  c_2  c_3 ⋯ c_n_c_H+1 |  P_𝖨𝖱] by P”, where c_1, c_2, …, c_n_c_H+1 are the columns in P_S chosen from the first n_v_H columns of the protomatrix P and were either in P' and not replaced to obtain P'_1 or was not in P'. Let us denote by S” the columns of P that lead to P”. It is now straight-forward to see from the composition of the matrices P_S and P_S” that ∑_i ∈ S𝗉𝖾𝗋𝗆(P_S ∖ i) ≥∑_i ∈ S”𝗉𝖾𝗋𝗆(P_S”∖ i) > 0. We now consider the final case of the general set of n_c + 1 columns whose n_c + 1 permanents sum to zero. Recall that the statement of the theorem assumes that P has a positive upper bound in (<ref>). Therefore, we may ignore such a case unless those columns contain P_𝖨𝖱, in which case we would compute the sum of n_c + 1 permanents according to the complexity as shown already. This completes the proof as the above shows it suffices to consider subsets S that always contain the columns of P_𝖨𝖱. The requirement that the protomatrix have a finite upper bound on its minimum distance (as assumed in Theorem <ref> and its counterpart in the Appendix for a protomatrix with punctured variable nodes) can be satisfied while designing every row of the 𝖨𝖱𝖢 part via the following observation: Let a PBRL protomatrix P of size n_c × n_v with a positive design rate have an 𝖧𝖱𝖢 part of size n_c_H× n_v_H. Let the 𝖧𝖱𝖢 part, as a protomatrix by itself, have a positive and finite upper bound d_𝖧𝖱𝖢 as computed using (<ref>) or (<ref>). Then the upper bounds for each new row i=n_c_H+1,n_c_H+2,…, n_c added to obtain P, irrespective of the chosen candidates for the rows, are non-decreasing and are lower bounded by d_𝖧𝖱𝖢. Let us consider the design of the first row of the 𝖨𝖱𝖢 part and assume that the protomatrix has no punctured columns. Assume that there is no non-zero integer in the new (n_c_H+1)^th row except the required 1 from the P_𝖨𝖱 part at entry (n_c_H+1, n_v_H +1 ). For any set of n_c_H +2 columns S that does not include the (n_v_H+1)^th column, the sum of the n_c_H +2 permanents is zero. For any other set that includes the (n_v_H+1)^th column, the sum of the permanents is equal to one of the sums of n_c_H+1 permanents computed to find the upper bound for the 𝖧𝖱𝖢 part. Now, if the new row is designed to have non-zero entries in the columns not in P_𝖨𝖱, the upper bound can only increase or remain the same. Similar arguments follow if the protomatrix has punctured columns. This completes the proof as the above arguments can then be successively applied to each new row. §.§ Design Examples In this subsection we design PBRL protomatrices according to the new design method we have proposed. We assume the following 𝖧𝖱𝖢 matrix for all our designs: [ 2 1 2 1 2 1 2 1; 1 2 1 2 1 2 1 2 ] We consider both a punctured and an unpunctured version of the matrix in (<ref>) in our examples. The punctured version has the first variable node punctured. Hence the design rate we start with is either 6/8 (unpunctured) or 6/7 (punctured). Because of Lemma <ref>, choosing an 𝖧𝖱𝖢 with a non-zero and finite upper bound (<ref>) or (<ref>) is sufficient to then use our results on reducing the complexity of search procedure to design the n_c × n_v protomatrix P. The 𝖧𝖱𝖢 matrix in (<ref>) has an upper bound of 12 when none of its variable nodes are punctured and an upper bound of 8 when its first column is punctured. We constrain the last n_c - n_c_H rows of the protomatrix to have a weight of exactly 4 and do not allow any non-zero integer other than 1. These constraints facilitate good performance at short block-lengths because limiting both the density and the number of multiple edges in the protograph helps the resulting LDPC codes have good girth and avoid having too many short cycles. Explicit constraints are necessary because increasing the value of any element at any position of a protomatrix with a finite upper bound of (<ref>) or (<ref>) either results in an increase in the upper bound or the upper bound stays the same. Our design method, which maximizes an upper bound on code minimum distance, does not depend upon the channel over which we deploy the codes. For designing codes for comparison according to the original design method that involves computing iterative decoding thresholds, we assume the binary-input additive white Gaussian noise channel (BI-AWGNC). The threshold values computed in this work are the result of at least 1000 iterations of the reciprocal channel approximation method to computing thresholds (see <cit.>). §.§.§ Unpunctured design We design two unpunctured ensembles using the 𝖧𝖱𝖢 matrix given in (<ref>) with the design constraints in Remark <ref>. The design rates we consider decrease from 6/8 to 6/15. The first ensemble, P_1, is obtained using the new permanent bound design (PBD) method proposed in this paper. For comparison, the second ensemble, P_2, is designed by optimizing the iterative decoding thresholds (referred to as “threshold-based” in results) over BI-AWGNC. The 𝖨𝖱𝖢 parts of P_1 and P_2 that we obtained are: P_1,𝖨𝖱𝖢 = [ 1 1 1 0 0 0 0 0; 0 0 0 1 1 1 0 0; 1 0 0 0 0 0 1 1; 0 1 1 1 0 0 0 0; 1 0 0 0 1 1 0 0; 1 0 1 0 0 0 0 1; 0 1 0 1 0 0 1 0 ], P_2,𝖨𝖱𝖢 = [ 1 1 1 0 0 0 0 0; 0 0 0 1 1 1 0 0; 1 0 0 0 0 0 1 1; 0 0 1 0 1 0 1 0; 1 0 0 1 1 0 0 0; 1 0 1 0 0 1 0 0; 1 0 1 0 0 0 1 0 ] §.§.§ Punctured design Similarly, we design two punctured ensembles using the 𝖧𝖱𝖢 matrix in (<ref>) via the design constraints in Remark <ref>. The first variable node is punctured[A punctured variable node improves the iterative decoding threshold (see <cit.> and <cit.>). Also, following the observations of <cit.> we constrain every row of P_𝖨𝖱𝖢 to connect to the punctured variable node in our punctured designs.], and the design rates decrease from 6/7 to 6/15. The resulting ensembles are called P_3 (PBD) and P_4 (threshold-based). Their 𝖨𝖱𝖢 parts are P_3,𝖨𝖱𝖢= [ 1 1 1 0 0 0 0 0; 1 0 0 1 0 0 1 0; 1 0 0 0 1 0 0 1; 1 0 1 0 0 1 0 0; 1 1 0 1 0 0 0 0; 1 0 0 0 1 1 0 0; 1 0 0 0 1 0 0 1; 1 1 0 0 1 0 0 0 ],P_4,𝖨𝖱𝖢= [ 1 0 1 0 1 0 0 0; 1 0 1 0 0 0 1 0; 1 1 0 0 1 0 0 0; 1 0 0 1 0 0 1 0; 1 0 0 0 1 1 0 0; 1 0 0 0 0 0 1 1; 1 0 0 1 1 0 0 0; 1 1 0 0 0 0 1 0 ] §.§.§ Unconstrained design via original PBRL design method For further comparison, we design an ensemble, called P_5, using the same 𝖧𝖱𝖢 matrix with its first variable node punctured according to the original PBRL design method. The design rates decrease from 6/7 to 6/15. For this ensemble we only have the following restriction in the 𝖨𝖱𝖢 part: We do not permit any integer greater than 1. The design yielded the following 𝖨𝖱𝖢 part: P_5,𝖨𝖱𝖢 = [ 1 0 1 0 1 0 0 0; 1 0 1 0 1 0 1 0; 1 1 1 0 1 0 1 0; 1 1 1 0 1 0 1 0; 1 1 1 0 1 0 1 0; 1 0 1 1 1 0 0 0; 1 1 1 0 1 0 1 0; 1 0 1 1 0 0 0 0 ] We now compare the five ensembles P_i, i ∈ [5] according to two metrics. Fig. <ref> shows the upper bound on the minimum distance obtained at each rate for the five ensembles. P_1, the unpunctured, constrained ensemble obtained via the new PBD method, has the best upper bound at almost every rate. At the other end of the spectrum, P_5, the unconstrained, punctured ensemble designed to optimize the threshold at each rate, has the worst upper bound at every rate. The iterative decoding thresholds at each rate (over BI-AWGNC) for all five ensembles are shown in Fig. <ref>. As expected, ensemble P_5 has the best threshold at each rate. But surprisingly, both the unpunctured and punctured constrained ensembles obtained via the new PBD method, P_1 and P_3, have almost the same threshold at each rate as their counterpart ensembles, P_2 and P_4, which were obtained by optimizing the iterative decoding threshold at each rate. § SIMULATION RESULTS This section presents simulation results of carefully designed RC code families from each of the five ensembles. Codes simulated in this section are all quasi-cyclic. Lifting was performed for the n_c × n_v protomatrix of the lowest rate 6/15 using the circulant-PEG (C-PEG) algorithm combined with the ACE algorithm of Tian et al. <cit.>. The lifting factor used is 33, which resulted in k=198 information bits for all code families. The resulting girth of all code families, at the lowest rate, is 6. Simulation results shown were obtained using a maximum of 100 iterations of full-precision, flooding, LLR-domain belief propagation over BI-AWGNC. At least 100 errors were collected for each frame error rate (FER) point in any simulated E_b/N_0 vs. FER graph. The FER of all five codes are shown in Figs. <ref> and <ref>. The QC-LDPC code family obtained from ensemble P_3 outperforms all the other codes at FERs 10^-4, 10^-5, and 10^-6 and at all rates (even at rates 6/12 and 6/13, which are omitted as the results are similar to the ones shown here). This ensemble has the advantages of a good, if not the best, threshold due to the punctured variable node (Fig. <ref>) and a good upper bound on the minimum distance at all rates (Fig. <ref>). The code family from the ensemble P_1, which has the best minimum distance upper bound at all rates, performs well at lower rates but not at higher rates. The gap to BI-AWGNC capacity at FER of 10^-6 is shown in Fig. <ref>. The code family of ensemble P_3 achieves the best performance at all design rates. The performance of this code is about 1.5 dB away at all rates from the refined normal approximation of <cit.>. § CONCLUSION This paper proposed a new method to design PBRL QC-LDPC codes for short block-lengths. The metric used in the design is an upper bound on the minimum distance of any QC-LDPC code that can be obtained from a protomatrix. By maximizing this upper bound at each design rate of the rate-compatible family of codes, the paper obtained a significant improvement in the error floor region over PBRL codes designed according to the original method of optimizing the iterative decoding threshold. Furthermore, the paper identified a key reduction that is possible in the complexity of the newly proposed design procedure. IEEEtran [Reduction in Complexity of Design Procedure for PBRL Protomatrices With Punctured Nodes] Theorem <ref> provides a considerable reduction in the complexity of the new design procedure to obtain PBRL ensembles with no punctured variable nodes. This appendix explores the case when a protomatrix has punctured variable nodes. Let a PBRL protomatrix P of size n_c × n_v with n_p punctured variable nodes have a positive design rate less than 1, i.e. n_v > n_c and n_t = n_v - n_p > n_v - n_c. Let the 𝖧𝖱𝖢 part be of size n_c_H× n_v_H. Let the set of punctured variable nodes be denoted 𝒫. Assume that the upper bound in (<ref>) for P is a positive integer. Now, if S ⊆[n_v], |S| = n_c + 1, S ∩𝒫 = ϕ, and S does not contain all the columns of P_𝖨𝖱, then the arguments in Theorem <ref> for ignoring such a set S while computing the upper bound for the protomatrix still hold. Similarly, if S includes P_𝖨𝖱 and any subset of columns from the first n_v_H columns of P, the computational complexity arguments for computing the sum of at most n_c + 1 permanents for such a set of columns hold the same way as observed in Theorem <ref>. Now consider the case when S ⊆[n_v], |S| = n_c + 1, S does not contain all the columns of P_𝖨𝖱, and S ∩𝒫ϕ. Let the number of punctured columns in S be n_S_p. The sum ∑_i ∈ S ∖𝒫𝗉𝖾𝗋𝗆(P_S ∖ i) has n_c + 1 - n_S_p permanents that need to be computed, all of size at most n_c × n_c. Every n_c × n_c sub-matrix whose permanent is computed for this sum contains all the n_S_p columns that are punctured. The strategy of replacing n_v - n_v_H columns of a sub-matrix whose permanent is non-zero in (<ref>) by all the columns in P_𝖨𝖱 does not necessarily yield a new summation that is smaller than the summation computed for the columns in S according to (<ref>). We give an example to illustrate this. Let us consider the following PBRL protomatrix whose first column is punctured: [ 1 1 2 1 2 1 0; 0 2 1 2 1 2 0; 1 1 0 0 0 0 1 ] Let S={1,2,3,4}, the first four columns of the protomatrix. The sum in (<ref>) for this set has three terms in it and is equal to 17. The permanent of the 3 × 3 sub-matrix comprised of columns 1, 3, 4 is equal to 5. If we follow the same replacement strategy as in Theorem <ref>, we need to replace column 1 by the only incremental redundancy variable node, i.e. column 7. The new set of columns {2, 3, 4, 7} has no column that is punctured and the summation in (<ref>), which now has four terms, is now equal to 19. Therefore, we observe that when a PBRL protomatrix has n_p punctured variable nodes and a finite upper bound in (<ref>), the complexity of computing the bound can be reduced the following way: * Consider all sets of n_c + 1 columns that contain P_𝖨𝖱. For such a set we need to compute at most n_c + 1 permanents, each of size at most (n_c_H + 1) ×(n_c_H + 1). There are n_v_Hn_c_H + 1 such sets. * Consider all sets of n_c + 1 columns that contain i ≥ 1 punctured variable nodes, and not all columns or no column of P_𝖨𝖱. Compute the summation of the n_c + 1 - i permanents, each of size at most n_c × n_c (if there are columns of P_𝖨𝖱 then the complexity of computing the permanent would decrease). There are ∑_i=1^n_p∑_j=0^n_v - n_v_H - 1n_pin_v_H - n_pn_c + 1 - i - jn_v - n_v_Hj such sets that need to be considered, where the binomial coefficient nk is assumed to be 0 if k ∉{0,1,…,n}.

http://arxiv.org/abs/1701.07654v1

20170126111206

Two Empirical Regimes of the Planetary Mass-Radius Relation

astro-ph.EP

Two Empirical Regimes of the Planetary Mass-Radius Relation Dolev Bashi^1[corresponding author: dolevbashi@gmail.com] , Ravit Helled^1,2, Shay Zucker^1 and Christoph Mordasini^3 ^1School of Geosciences, Tel-Aviv University, Tel-Aviv, Israel ^2 Center for Theoretical Astrophysics & Cosmology, Institute for Computational Science, University of Zurich, Switzerland. ^3Physics Institute, University of Bern, Switzerland. ================================================================================================================================================================================================================================================================================================================================================================================= Today, with the large number of detected exoplanets and improved measurements, we can reach the next step of planetary characterization. Classifying different populations of planets is not only important for our understanding of the demographics of various planetary types in the galaxy, but also for our understanding of planet formation. We explore the nature of two regimes in the planetary mass-radius (M-R) relation. We suggest that the transition between the two regimes of "small" and "large" planets, occurs at a mass of 124 ± 7, M_⊕ and a radius of 12.1 ± 0.5, R_⊕. Furthermore, the M-R relation is R ∝ M^0.55± 0.02 and R ∝ M^0.01±0.02 for small and large planets, respectively. We suggest that the location of the breakpoint is linked to the onset of electron degeneracy in hydrogen, and therefore, to the planetary bulk composition. Specifically, it is the characteristic minimal mass of a planet which consists of mostly hydrogen and helium, and therefore its M-R relation is determined by the equation of state of these materials. We compare the M-R relation from observational data with the one derived by population synthesis calculations and show that there is a good qualitative agreement between the two samples. § INTRODUCTION Exoplanet studies have now reached the level at which planet characterization is possible. There are hundreds of planets with measured masses and radii. Knowledge of these two physical properties provides valuable clues about the planetary composition, through the mass-radius (hereafter M-R) relationship. Traditionally, planets have been divided into two main groups. The first includes the massive, gas-dominated planets, while the second consists of the small, terrestrial planets (e.g., Weidenschilling 1977). In part, this division is inspired by the Solar System, where massive planets are composed of volatile materials (e.g., Jupiter) while the terrestrial planets are small and consist of refractory materials. However, the diversity in masses and radii of exoplanets[See http://exoplanets.org for exoplanet properties.] has taught us that this separation is somewhat arbitrary and may be over-simplistic (see review by Baraffe et al. 2014 and references therein). While the first detected exoplanets had relatively large masses and radii, in the recent few years the number of small exoplanets increased dramatically, due to improvements in technology and detections from space (e.g., CoRoT (Baglin et al. 2006) and Kepler (Borucki et al. 2010)). Of course, since most exoplanets have been detected via radial velocity measurements or transits, there is a difference when defining a "small planet" by mass or by radius. In terms of mass, it is customary to define small planets as planets whose masses are less than ∼ 30 M_⊕ (Mayor et al. 2011, Howard et al. 2010), while in terms of radius, small exoplanets are often those whose radii are smaller than 4 R_⊕ (e.g., Marcy et al. 2014, Weiss & Marcy 2014). These divisions are partially based on the behavior of the planetary mass function of exoplanets. Previous studies that examined the M-R relation have suggested a transition in the M-R relation between 'small' planets (Neptune-like) and 'large' planets (Jovian). Based on visual estimates of the M-R and mass-density relations, Weiss et al. (2013) suggested that the transition point occurs at a mass of ∼ 150 M_⊕. The derived slopes of the M-R relations in the different regimes turned out to be R ∝ M^0.54 for M_p < 150 M_⊕ and R ∝ M^-0.039 for massive planets (M_p > 150 M_⊕). Hatzes & Rauer (2015) have analyzed changes in the slope of the mass-density relation. Using a similar slope criterion, they locate the breakpoint at a mass of ∼ 0.3 M_J≃ 95 M_⊕. In a recent study, Chen & Kipping (2017) presented a detailed forecasting model built upon a probabilistic M-R relation using MCMC. According to their classification the transition between small and large planets occurs at 0.41 ± 0.07 M_J≃ 130 ± 22 M_⊕, corresponding to the transition between Neptunians and Jovians, with slopes of R ∝ M^0.59 and R ∝ M^-0.04 for the low-mass and high-mass planets, respectively. Interestingly, although the studies do not agree exactly on the transition mass between the two regimes, they do agree that it is significantly higher than the traditional cutoff at 20-30 M_⊕. This essentially suggests that the change in the occurrence rate as seen in the mass function of exoplanets (at ∼ 30 M_⊕), i.e., the frequency of planets is not the same as the behavior of the M-R relation, which is linked to the planetary composition. In this paper we present the results of a study we performed in order to empirically characterize the transition point between small and large planets based on their M-R relation. On the one hand our aim was to perform a quantitative straightforward study, that will come up with simple numerical information – the two M-R power law indices, and the transition mass. On the other hand, we opted for a kind of least-square fit, and not an elaborate probabilistic recipe. Our hope was that this would allow a more intuitive yet rigorous characterization of the planetary M-R relation. Finally, we also compare the exoplanet population to formation models and find a qualitative good agreement. § SAMPLE The data we use include only planets with masses and radii that are based directly on the observations, as opposed to being inferred from planetary physics models. Our sample consists of 274 exoplanets queried from http://exoplanets.org on March 2016. The lowest mass planet in our sample is Kepler-138b, with a mass of 0.0667 ± 0.0604 M_⊕, well below Earth mass. The highest planet mass in our sample is that of CoRoT-3b, with a mass of 6945 ± 315 M_⊕(= 21.85 ± 0.99 M_J) – a brown dwarf. For all the planets our sample must include measured masses, radii, and their uncertainties. Thus, we exclude planets with reported masses estimated based on a theoretical M-R relation. All planets in the sample are transiting planets, whose masses have been measured either by RV (238 through RV follow-up, and 9 were first detected by RV), or using TTVs (e.g. Nesvorný & Morbidelli 2008; 27 planets)[It should be noted that almost all the TTV planets are of low mass]. The top panel of Fig. 1 shows the resulting M-R diagram[The list of the planets we use is summarized in Table 2 in the online version.]. § ANALYSIS The model we assume is that of two mass regimes that differ by the M-R power law. In the log-log plane, this translates into a continuous piecewise linear function, with two segments, that we had to fit to the data points. In spite of our ambition to apply the most basic techniques of simple regression to perform this fit, several problems conspire to turn this into a somewhat more complicated problem. First, the two variables – the planetary mass and radius – are both measured with non-negligible errors. If we could assume that only one of them (e.g. the mass) had errors, the problem could have been treated as a standard regression problem. The fact that uncertainties exist for both variables, takes us to the field of 'errors-in-variables' (EIV) problems, which are surprisingly more difficult than standard regression problems (e.g. Durbin 1954), and there is not one agreed approach to analyze them. As difficult as EIV problems are, in our case the complexity is even exacerbated by the fact that we aim to fit not a linear function, but a continuous piecewise-linear function, rendering futile any hope to solve the problem analytically. Even under the assumptions of standard regression, where the so-called explanatory variable has no uncertainty, the problem (dubbed 'segmented regression') is not trivial (e.g. Hinkley 1969). Another difficulty arises because of the nature of our specific sample. A glance at the top panel of Figure 1 reveals that the data points are not scattered evenly across the logarithmic mass range. The points corresponding to the smaller planets seem to be much more sparse than those of the Jovian planets. The same is true for the very large planets, with masses of a few Jupiter's mass. One may say that there seem to be three mass intervals with varying density of sample points. The origin of this differentiation lies beyond the scope of this study, and in any case it may very well be a combination of observational bias and astrophysical processes of formation and evolution. The smaller number of massive planets (above a Jupiter-mass) is a result of the low occurrence rate of such planets (e.g., Cumming et al. 2008), while the clustering of small is probably a result of the massive efforts to detect low-mass planets and their high occurrence rate (e.g., Howard et al. 2012). There are various reasons for this sampling variability, ranging from observational biases to physical effects related to planet formation and evolution. However, the fact remains that for the purposes of regression analysis, the mass affects the sampling. In regression theory this amounts to 'endogenous sampling'. While fitting a simple straight line might be affected only slightly by this imbalanced sampling, it is not guaranteed for a piecewise-linear function. Any fitting procedure should take this imbalance into consideration. To streamline the discussion, let us denote: x = log M_p y = log R_p The choice of logarithm base is irrelevant as long as it consistent throughout the calculation. Eventually the values in linear scale are the important ones, not in logarithm scale. Now our sample, in the log-log plane, consists of a set of ordered pairs (x_i,y_i). Let us further denote by Δ x_i and Δ y_i the corresponding logarithmic uncertainties derived from the uncertainties in the linear scale using the standard transformation. In cases where the transformation led to assymetric uncertainties, we still assigneed symmetric errors, by taking the more conservative (larger error) of the two error estimates. In our quest for the best-fit piecewise-linear function, we chose what is probably the most intuitive approach to EIV: Total Least Squares – TLS (e.g. Markovsky et al. 2010). Similarly to standard regression, in TLS the problem is represented as a minimization problem of a sum of squares. Each data point contributes to the total sum-of-squares its orthogonal distance from the fitted line, measured in units of the two uncertainties. In the simple case where we fit a simple linear function, if we denote the slope and intercept of the line by a and b, the contribution of the point (x_i,y_i) would be: (y_i - a x_i - b)^2/a^2(Δ x_i)^2 + (Δ y_i)^2 , where Δ x_i and Δ y_i are the errors of x_i and y_i. Golub (1973), and Golub & Van Loan (1980) were the first to introduce an algorithm to solve this basic TLS problem, using singular value decomposition. They have also shown that even in this simple linear case a solution is not guaranteed to exist. In our case, where the function we seek consists of two straight lines, we simply calculate, for each point, its weighted orthogonal distances from the two lines, and include the smaller one in the total sum-of-squares: S(a_1,b_1,a_2,b_2) = ∑_i=1^N min{(y_i - a_1 x_i - b_1)^2/a_1^2(Δ x_i)^2 + (Δ y_i)^2 , (y_i - a_2 x_i - b_2)^2/a_2^2(Δ x_i)^2 + (Δ y_i)^2} , where N is the total number of points and a_1,b_1,a_2 and b_2 are the slopes and intercepts of the two straight lines. S is parameterized by four numbers whose meaning is somewhat arbitrary. This is true especially for the two intercepts b_1 and b_2, which are functions of the arbitrary location of the zero point of x. We can instead parameterize S by an alternative, physically more meaningful, quadruple: the two coordinates of the breakpoint (breakpoint mass and corresponding radius), and the two slopes of the separate mass regimes. When we set out to minimize S, it turned out that the solution was numerically unstable. Using diffferent starting points for the optimization algorithm (Nelder-Mead simplex algorithm, see Nelder & Mead 1965) resulted in different solutions. That meant that around the global minimum of S(a_1,b_1,a_2,b_2) there were many local minima. We suspect that this instability resulted from the endogenous sampling problem to which we alluded above (the mass- distribution shown in Fig. 2). In order to rectify this problem we have decided to introduce weights to the definition of S, that will balance the effect each mass range has on the final solution. As is clear from the top panel of Figure 1, there are apparently three intervals: M_p<69 M_⊕, 69 M_⊕≤ M_p < 1660 M_⊕, and 1660 M_⊕≤ M_p[We have decided it is beyond the scope of this study to perform a rigorous clustering analysis. There seems to be a consensus in data-mining literature that at this stage there is not a single clustering algorithm or criterion that is guaranteed to be the best one. Thus, an intuitive division at this stage is completely acceptable (e.g., Estivill-Castro 2002).]. Fig. 2 further demonstrates the differentiation in mass by portraying a histogram of the mass, together with the borders we chose among the three mass ranges. The weighting scheme we applied is known in statistics as Inverse Probability Weighting, which is designed in order to alleviate the implications of endogenous sampling (e.g., Wooldridge 1999). We thus multiplied the contribution of each data point by a weight that was supposed to compensate for the effect of the size of the mass-range set to which the data point belonged to. The weight we assigned was simply proportional to the inverse of the size of the set: N/N_c, where N is the total number of planets and N_c is the size of the set. Table 1 details the three mass-range sets, their sizes and the corresponding weights. The final expression for S is thus: S(a_1,b_1,a_2,b_2) = ∑_i=1^N w_i min{(y_i - a_1 x_i - b_1)^2/a_1^2(Δ x_i)^2 + (Δ y_i)^2 , (y_i - a_2 x_i - b_2)^2/a_2^2(Δ x_i)^2 + (Δ y_i)^2} , where w_i is the weight of each point. After optimizing S, we went on to obtain error estimates for the four variables, using a Monte-Carlo resampling approach. We randomly drew new data points from a Gaussian distribution. The expected values of the Gaussian distribution were the nominal values of x and y, and we used the error bars as the widths (standard deviations) of the Gaussian distribution. We repeated the resampling procedure for 100,000 such random realizations of the data. The resulting random sample yielded the error estimates. § RESULTS Using the approach we outlined in the previous section, we obtained estimates for the two slopes, and the breakpoint. We found the breakpoint at a mass of 124 ± 7 M_⊕, and a radius of 12.1 ± 0.5 R_⊕. The resulting power laws of the two regimes (based on the two slopes in the x-y plane) are: R ∝ M^0.55 ± 0.02 for small planets, and R ∝ M^0.01 ± 0.02 for large planets. The bottom panel of Figure 1 shows the derived relation. It is interesting to note also according to our analysis, Saturn is "a small planet" (e.g., Chen & Kipping 2017; Weiss et al. 2013). Indeed, based on internal structure models, the heavy element fraction is Saturn is estimated to be between ∼ 20% and 40% (e.g., Guillot 2005). Thus, Saturn's mass is not very far from the transition point, and it is important to note that the transition mass at ∼120 M_⊕ must be understood as a statistical quantity. As can be seen in Fig. 1, there is a region near the breakpoint in the fit at 120 M_⊕ that could either be considered as the continuation of the regime where the radius increases with mass to even higher masses, or as an continuation of the high-mass regime (with approximately constant radius) to even lower masses. This transition regime approximately covers a mass range larger than the one derived in the analysis, somewhere between about 80 and 120 M_⊕. Thus, according to the data, the actual transition occurs at the higher end of this mass range. Another point that should be taken into account is that the apparent transition is also affected by stellar irradiation, while Saturn experiences a much lower irradiation than most of the planets that were used in our statistical analysis. Our results are in good agreement with previous studies. The analysis we used to obtain was simple and intuitive and did not rely on subjective estimates. The fact that the transition occurs at a planetary mass larger than that of Saturn's supports the idea that the change in the M-R relation for large planets is due to the dominating composition – in the case of massive planets, a mixture of hydrogen and helium. The data suggest that for planets larger than ∼ 120 M_⊕, the planetary radius is determined by the equation of state of these light elements (e.g., Zapolsky & Salpeter 1969; Fortney et al. 2007). The dominating H-He composition and the compression due to the large mass also naturally explains the weak dependence of the radius on mass for giant planets that consists of mostly hydrogen and helium (e.g., Guillot 2005). Lower-mass planets are less compressed and therefore, have a radius that increases in mass. The relatively large spread of the low-mass planets around the line suggests that in this mass regime, the planets can have various compositions. §.§ Comparison with theoretical calculations In this section we briefly compare the observational data with theoretical results from planet population syntheses based on the core accretion paradigm (Mordasini et al. 2012). These calculations yield the planetary bulk composition (solids and H/He) and the post-formation entropy based on the planets' formation track. Here we use two sets of cores (heavy-element) compositions: silicates and iron or water. These two sets are chosen to assess the impact of various compositions of the solid core on the predicted radii of the synthetic planets. The first core is differentiated, and its composition is assumed to consists in mass of 1/3 iron (inner core) and 2/3 perovskite (outer core), similar to Earth and several low-mass extrasolar planets (e.g., Santos et al. 2015). The second composition corresponds to cores consisting exclusively of water ice. While pure water core are unlikely to exist, these cores represent the limiting case of low-density cores. In all cases, the modified polytropic equation of state is used to derive the core radius, taking into account the pressure exerted by the surrounding envelope (see Mordasini et al. 2012). The star is assumed to be 1 M_⊙. Planets with semi-major axes of 0.01 to 0.5 AU are included in order to have a better comparison with the measurements. The formation model includes the effect of type I and II orbital migration. During the evolutionary phase, no mechanisms that can lead to inflation of the planetary radius (bloating) are included, whereas the effect of atmospheric escape is considered as described in Jin et al. (2014). The planetary opacity used in the formation models is the combination of the ISM opacities (Bell & Lin 1994) reduced by a factor 0.003 plus the grain-free opacities of Freedman et al. (2014). The reduction factor was determined in Mordasini et al. (2014) by comparison with detailed simulations of the grain dynamics by Movshovitz et al. (2010). During the planetary evolution, we assume a grain-free opacity because grains are expected to grow and settle to deeper regions after gas accretion is terminated accretion stops (e.g., Movshovitz & Podolak 2008). The observations and the theoretical data are compared in Figure 3. As can be seen from the figure, the general M-R relation is similar, but there are also important differences. Both data sets show two different regimes. In the low-mass regime, both the observational and synthetic data show a large scatter in the M-R relation, which stems in the synthetic population from different envelope-core mass ratios, which in turn reflect different formation histories. For giant planets, the simulated planets follow a narrow M-R relation, which is clearly a consequence of neglecting bloating, assuming solar opacity, and an internal structure consisting of a pure H/He envelope surrounding a core made of pure heavy elements (i.e., a core+envelope internal structure). This is in contrast to the observations that also contain planets that have significantly larger radii, and probably different compositions and/or internal structures. In addition, the theoretical data correspond to a given age (5 × 10^9 years), while the observed population includes various ages. However, since most of the detected planets are observed around relatively old stars we do not expect a large impact on the goodness of fit to the observed M-R relation. In the giant planet regime, one sees that in the observed exoplanet population, there are both giant planets with significantly larger, but also smaller radii. The large radii can be attributed to bloating, while the smaller planets suggest that there are some planets that contain significantly higher amounts of heavy elements than in the synthetic population. This could be the result of a more efficient accretion of solids during formation, or giant impacts at later times. The effect of bloating on the population of small planets still needs to be studied in detailed although some work on this topic has already been presented (e.g., Lopez et al. 2012; Owen & Wu 2013). At the moment, it is still unclear whether an inflation mechanism is required in order to explain some of the small exoplanets with very low mean densities, since the existence of an (H-He) atmosphere can significantly increase the planetary radius. In addition, unlike massive planets which are expected to be H-He dominated, small planets have a large spread of heavy-elements and various fractions of H-He. This introduces a degeneracy with inflation mechanisms for low-mass planets: an observed M-R relation can probably either be caused by the existence of a more massive H-He envelope without inflation, or alternatively by a physical mechanism that causes the planet to be large, i.e., inflation. A better understanding of inflation and atmospheric loss in small- and intermediate- mass planets is clearly desirable. Clearly, the two data sets should be compared only qualitatively. This is because the observed planets have a variety of ages, atmospheric opacities, and of course, possibly mixed compositions. As a result, the partially strong and tight correlations in the theoretical M-R should not be considered realistic, as they simply represent the composition of pure ice/rock planets (in the case of the bare cores), or the artificially narrow M-R relation of giant planets having all the same atmospheric opacity and lacking bloating mechanism. Nevertheless, there is a rather good agreement in terms of the transition between "small" and "large" planets in the M-R diagram. § DISCUSSION AND CONCLUSIONS Our analysis suggests that the transition between large and small planets occurs at a mass (radius) of 124 ± 7 M_⊕ (12.1 ± 0.5 R_⊕). As expected, we establish two mass-radius relations for exoplanets. For low-mass planets, the radius is increasing with increasing mass, and the M-R relation we derive is: R∝ M^0.55±0.02, whereas for the large planets, the radius is almost independent of the mass, and the M-R relation is R ∝ M^0.01±0.02. Planetary mass and heavy element content almost exclusively determine the radius of low-mass planets < 124 M_⊕. The turnover point at this mass is probably due to the characteristic boundary between planets that are mostly gaseous (H-He dominated) and planets that consist of varying compositions, and therefore, do not have a single M-R relation. When the planet mass reaches > 124 M_⊕ the relation is flattened and is even consistent with a small negative slope, since we are approaching a slope of a compressed hydrogen-helium-dominated planet. This work identifies the transition point between small and large planets based on the M-R relation. This transition point is not the same as the one derived from studies of measured frequency of planets (occurrence rate), although the two might be linked. From standard planet formation models point of view, the transition from a heavy-element-dominated composition to a hydrogen-helium-dominated composition occurs at a mass where the core and envelope mass are similar (crossover mass). Statistical simulations of planet formation have shown (e.g., Mordasini et al. 2015) that this leads to a break in the planetary occurrence rate at about 30 M_⊕, but the actual value can vary significantly depending on the assumed solid-surface density, opacity, accretion rates, etc. Thus, it is interesting to note that there are not many observed planets with masses between 30 and 120 M_⊕ (see Fig. 1; see also Mayor et al. 2011). This may suggest that the two transitions are linked. Finding the link between the two transition points can reveal crucial information on planetary formation and characteristics, and we hope to address this topic in the future. As mentioned earlier, thinking about planetary characterization in terms of M-R relation is useful, but it should be noted that in reality, there is a M-R-flux, or even M-R-flux-time relation for planets. This is because the stellar flux and the time evolution are expected to affect the radius of the planet at a given time. These relations will be better understood in the future when exoplanet detections will include larger radial distances and various ages of stars, as expected from the PLATO mission. §.§ Acknowledgments We thank the anonymous referee for valuable comments and suggestions. R. H. acknowledges support from the Israel Space Agency under grant 3-11485 and from the United States - Israel Binational Science Foundation (BSF) grant 2014112. C. M. acknowledges support from the Swiss National Science Foundation under grant BSSGI0_155816 “PlanetsInTime'. Parts of this work have been carried out within the frame of the National Centre for Competence in Research PlanetS supported by the SNSF. § REFERENCES Baglin, A., Auvergne, M., Boisnard, L., et al., 2006, 36th COSPAR Scientific Assembly, 36, 3749 Baraffe, I., Chabrier, G., Fortney, J. & Sotin, C., 2014, in Protostars and Planets VI, Henrik Beuther, Ralf S. Klessen, Cornelis P. Dullemond, and Thomas Henning (eds.), University of Arizona Press, Tucson, 763 Bell, K. R. & Lin, D. N. C., 1994, ApJ, 427, 987 Borucki, W. J., Koch, D., Basri, G., et al., 2010, Science, 327, 977 Chen, J. & Kipping, D. M., 2017, ApJ, 834, 17 Cumming, A., Butler, R. P., Marcy, G., W. et al., 2008, PASP, 120, 531 Durbin, J., 1954, Rev. Inst. Int. Stat., 22, 23 Estivill-Castro, V., 2002, SIGKDD explor., 4, 65 Fortney, J. J., Marley, M. S. & Barnes, J. W., 2007, ApJ, 659, 1661 Freedman, R. S., Lustig-Yaeger, J., Fortney, J. J., et al., 2014, ApJS, 214, 25 Golub, G., 1973, SIAM rev., 15, 318 Golub, G. & Van Loan, C., 1980, SIAM J. Numer. Anal., 17, 883 Guillot, T., 2005, Annu. Rev. Earth Planet. Sci., 33 Hatzes, A. P. & Rauer, H., 2015, ApJL, 810, L25 Hinkley, D. V., 1969, Biometrika, 56, 495 Howard, A., Marcy, G. W., Johnson, J. A. , et al., 2010, Science, 330, 653 Howard, A. W., Marcy, G. W., Bryson, S. T. et al., 2012, ApJS, 201, 15 Jin, S., Mordasini, C., Parmentier, V., van Boekel, R., Henning, T., Ji, J., 2014, ApJ, 795, 65 Lopez, E. D., Fortney, J. J. & Miller, N., 2012, ApJ, 761, 59 Markovsky, I., Sima, D. M. & Van Huffel, S., 2010, WIREs Comp. Stats. 2, 212 Marcy, G. W., Weiss, L. M., Petigura, E. A., Isaacson, H., Howard, A. W. & Buchhave, L. A., 2014, Proc. Natl. Acad. Sci. USA, 111, 12655 Mayor, M., Marmier, M., Lovis, C. et al., 2011, A&A, submitted. arXiv:1109.2497 Miller, N. & Fortney, J. J., 2011, ApJ, 736, L29 Mordasini, C., Alibert, Y., Georgy, C., et al., 2012, A&A, 547, A112 Mordasini, C., Klahr, H., Alibert, Y., Miller, N. & Henning, T., 2014, A&A, 566, A141 Mordasini, C., Molliére, P., Dittkrist, K. M., Jin, S. & Alibert, Y., 2015, Int. J. Astrobiol., 14, 201 Movshovitz & Podolak, 2008, Icarus, 194, 368 Movshovitz, N., Bodenheimer, P., Podolak, M. & Lissauer, J. J., 2010. Icarus, 209, 616 Nelder, J. A. & Mead, R., 1965, Comput. J., 7, 308 Nesvorný, D. & Morbidelli, A., 2008, ApJ, 688, 636 Owen, J. E. & Wu, Y., 2013, ApJ, 775, 105 Santos, N. C., Adibekyan, V., Mordasini, C. et al., 2015, A&A, 580, L13 Weidenschilling, S. J., 1977, ApSS, 51, 153 Weiss, L. M., Marcy, G. W., Rowe, J. F. et al., 2013, ApJ, 768, 14 Weiss, L. M. & Marcy, G. W., 2014, ApJ, 783, L6 Wooldridge, J. M., 1999, Econometrica, 67, 1385 Zapolsky, H. S. & Salpeter, E. E., 1969, ApJ, 158, 809

http://arxiv.org/abs/1701.07984v1

20170127093721

Weak order in averaging principle for stochastic wave equations with a fast oscillation

math.PR

hbfuhust@gmail.com wanlinju@aliyun.com Research Center of Nonlinear Science, College of Mathematics and Computer Science, Wuhan Textile University, Wuhan, 430073, PR China jcliu@hust.edu.cn xmliu@hust.edu.cn School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, PR China [cor1]Corresponding author at: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China. This article deals with the weak errors for averaging principle for a stochastic wave equation in a bounded interval [0,L], perturbed by a oscillating term arising as the solution of a stochastic reaction-diffusion equation evolving with respect to the fast time. Under suitable conditions, it is proved that the rate of weak convergence to the averaged effective dynamics is of order 1 via an asymptotic expansion approach. Stochastic wave equations, averaging principle, invariant measure weak convergence, asymptotic expansion. MSC: primary 60H15, secondary 70K70 § INTRODUCTION Let D=[0, L]⊂ℝ be a bounded open interval. In the article, for fixed T>0, we consider the following class of stochastic wave equation with fast oscillating perturbation, ∂^2/∂ t^2 U_t^ϵ(ξ)=Δ U_t^ϵ(ξ)+F(U_t^ϵ(ξ), Y_t/ϵ (ξ)) +σ_1Ẇ_̇ṫ^̇1̇(ξ), t∈ [0, T],ξ∈ D, U_t^ϵ(ξ)=0, (ξ, t)∈∂ D× (0, T], U_0^ϵ(ξ)=x_1(ξ), ∂ U_t^ϵ(ξ)/∂ t|_t=0=x_2(ξ), ξ∈ D, where ϵ is positive parameter, Y_t is governed by the stochastic reaction-diffusion equation: ∂/∂ tY_t (ξ)= Δ Y_t (ξ)+ g( Y_t (ξ))+ σ_2Ẇ_̇ṫ^̇2̇(ξ), t∈ [0, T],ξ∈ D, Y_t (ξ)=0, (ξ, t)∈∂ D× (0, T], Y_0 (ξ)=y(ξ), Assumptions on the smoothness of the drift f and g will be given below. The stochastic perturbations are of additive type and W^1_t(ξ) and W^2_t(ξ) are mutually independent L^2(D)-valued Wiener processes on a complete stochastic basis (Ω, ℱ, ℱ_t, ℙ), which will be specified later. The noise strength coefficients σ_1 and σ_1 are positive constants and the parameter ϵ is small, which describes the ratio of time scale between the process X^ϵ_t(ξ) and Y_t/ϵ(ξ). With this time scale the variable X_t^ϵ(ξ) is referred as slow component and Y_t/ϵ (ξ) as the fast component. The equation (<ref>) is an abstract model for a random vibration of a elastic string with a fast oscillating perturbation. More generally, the nonlinear coupled wave-heat equations with fast and slow time scales may describe a thermoelastic wave propagation in a random medium <cit.>, the interactions of fluid motion with other forms of waves <cit.>, wave phenomena which are heat generating or temperature related <cit.>, magneto-elasticity <cit.> and biological problems <cit.>. Averaging principle plays an important role in the study of asymptotic behavior for slow-fast dynamical systems. It was first studied by Bogoliubov<cit.> for deterministic differential equations. The theory of averaging for stochastic ordinary equations may be found in <cit.>, the works of Freidlin and Wentzell <cit.>, Veretennikov <cit.>, and Kifer <cit.>. Further progress on averaging for stochastic dynamical systems with non-Gaussian noise in finite dimensional space was studied in <cit.>. Concerning the infinite dimensional case, it is worth quoting the paper by Cerrai <cit.>, Bréhier <cit.>, Wang <cit.>, Fu <cit.> and Bao <cit.>. In our previous article <cit.>, the asymptotic limit dynamics (as ϵ tends to 0) of system (<ref>) was explored within averaging framework. Under suitable conditions, it can be shown that a reduced stochastic wave equation, without the fast component, can be constructed to characterize the essential dynamics of (<ref>) in a pathwise sense, as it is done in <cit.> for stochastic partial equations of parabolic type and for stochastic ordinary differential equations <cit.>. In the present paper, we are interested in the rate of weak convergence of the averaging dynamics to the true solution of slow motion U^ϵ_t(ξ). Namely, we will determine the order, with respect to timescale parameter ϵ, of weak deviations between original solution of slow equation and the solution of the corresponding averaged equation. To our knowledge, up to now this problem has been treated only in the case of deterministic reaction diffusion equations in dimension d=1 subjected with a random perturbation evolving with respect to the fast time t/ϵ (to this purpose we refer to the paper by Bréhier <cit.>). Once the noise is included in slow variable, the method in <cit.> used to obtain the weak order 1-ε for arbitrarily small ε>0 will be more complicated due to the lack of time regularity for slow solution. In the situation we are considering, an additive time-space white noise is included in the slow motion and the main results show that order 1 for weak convergence can be derived, which can be compared with the order 1-ε in <cit.>. Under dissipative assumption on Eq. (<ref>), the perturbation process Y_t admits a unique invariant measure μ with mixing property. Then, by averaging the drift coefficient of the slow motion Eq. (<ref>) with respect to the invariant measure μ, the effective equation with following form can be established: ∂^2 /∂ t^2U̅_t(ξ)=ΔU̅_t(ξ)+F̅(U̅_t(ξ))+σ_1Ẇ_̇ṫ^̇1̇(ξ), U̅_t(ξ)=0, (ξ, t)∈∂ D× (0, T], U̅_0(ξ)=x_1(ξ), ∂U̅_t(ξ)/∂ t|_t=0= x_2(ξ), ξ∈ D, where for any u,y∈ H:=L^2(D), F̅(u):=∫_HF(u,y)μ(dy), u∈ H. We prove that, under a smoothness assumption on drift coefficient in the slow motion equation, an error estimate of the following form |𝔼ϕ( U^ϵ_t)-𝔼ϕ(U̅_t)|≤ Cϵ for any function ϕ with derivatives bounded up to order 3. In order to prove the validity of above bound, we adopt asymptotic expansion schemes in <cit.> to decompose 𝔼ϕ(U^ϵ_t) with respect to the scale parameter ϵ in form of 𝔼ϕ(U^ϵ_t)=u_0+ϵ u_1+r^ϵ, where the functions u_0 has to coincide with 𝔼ϕ(U̅_t) by uniqueness discuss, as it can be shown that they are governed by the same Kolmogorov equation via identification the powers of ϵ. Due to solvability of the Poisson equation associated with generator of perturbation process Y_t, an explicit expression of u_1 can be constructed such that its boundedness is based on a priori estimates for the Y_t and smooth dependence on initial data for averaging equation. The next step consist in identifying r^ϵ as the solution of a evolutionary equation and showing that |r^ϵ|≤ Cϵ. The proof of bound for r^ϵ is based on estimates on du_1/dt and ℒ_2u_1, where ℒ_2 is the Kolmogorov operator associated with the slow motion equation. We would like to stress that this procedure is quite involved, as it concerns a system with noise in infinite dimensional space, and the diffusion term leading to quantitative analysis on higher order differentiability of 𝔼ϕ(U̅_t) with respect to the initial datum. Let us also remark that asymptotic expansion of the solutions of Kolmogorov equations was studied in <cit.> and <cit.>. The rest of the paper is arranged as follows. Section 2 is devoted to the general notation and framework. The ergodicity of fast process and the averaging dynamics of system (<ref>) is introduced in Section 3. Then the main results of this article, which is derived via the asymptotic expansions and uniform error estimates, is presented in Section 4. In the final section, we state and prove technical lemmas applied in the preceeding section. Throughout the paper, the letter C below with or without subscripts will denote generic positive constants independent of ϵ, whose value may change from one line to another. § PRELIMINARY To rewrite the systems (<ref>) and (<ref>) as the abstract evolution equations, we present some notations and some well-known facts for later use. For a fixed domain D=[0, L], we use the abbreviation H:=L^2(D) for the space of square integrable real-valued functions on D. The scalar product and norm on H are denoted by (·, ·)_H and ·, respectively. We recall the definition of the Wiener process in infinite space. For more details, see <cit.>. Let {q_i,k(ξ)}_k∈ℕ be H-valued eigenvectors of a nonnegative, symmetric operator Q_i with corresponding eigenvalues {λ_i, k}_k∈ℕ, for i=1, 2, such that Q_iq_i,k(ξ)=λ_i, k q_i, k(ξ), λ_i, k>0, k∈ℕ. For i=1, 2, let W_t^i(ξ) be an H-valued Q_i-Wiener process with operator Q_i satisfying TrQ_i=∑_k=1^+∞λ_i, k< +∞. Then W^i_t(ξ)=∑_k=1^+∞λ^1/2_i, kβ_i, k(t)q_i,k(ξ), t≥ 0, where {β_i, k(t)}^i=1, 2_k∈ℕ are mutually independent real-valued Brownian motions on a probability base (Ω, ℱ, ℱ_t, ℙ). For the abbreviation, we will sometimes omit the spatial variable ξ in the sequel. Let { e_k(ξ)}_k∈ℕ denote the complete orthornormal system of eigenfunctions in H such that, for k = 1,2,…, -Δ e_k=α_ke_k, e_k(0)=e_k(L)=0 , with 0<α_1≤α_2≤⋯α_k≤⋯. Here we would like to recall the fact that e_k(ξ)=sinkπξ/L and α_k=-k^2π^2/L^2 for k = 1,2,⋯. Let A be the realization in H of the Laplace operator Δ with zero Dirichlet boundary condition, which generates a strong continuous semigroups {E_t}_t≥ 0, defined by, for any h∈ H, E_th=∑_k=1^+∞ e^-α_kte_k(e_k, h)_H. It is straightforward to check that {E_t}_t≥0 are contractive semigroups on H. For s∈ℝ, we introduce the space H^s:=D((-A)^s/2), which equipped with inner product ⟨ g,h⟩_s:=((-A)^s/2g,(-A)^s/2h)_H=∑_k=1^+∞α^s_i(g,e_k)(h,e_k)_H, g,h∈ H^s and the norm φ_s={∑_k=1^+∞α_k^s(φ, e_k)_H^2}^1/2 for φ∈ H^s. It is obvious that H^0=H and H^α⊂ H^β for β≤α. We note that in the case of s>0, H^-s can be identified with the dual space (H^s)^*, i.e. the space of the linear functional on H^s which are continuous with respect to the topology induced by the norm ·_s. We shall denote by ℋ^α the product space H^α× H^α-1, α∈ℝ, endowed with the scalar product (x, y)_ℋ^α=⟨ x_1,y_1⟩_α+⟨ x_2,y_2⟩_α-1, x=(x_1, x_2)^T, y=(y_1, y_2)^T, and the corresponding norm |x|_α={x_1^2_α+x_2^2_α-1}^1/2, x=(x_1,x_2)^T. If α=0 we abbreviate H^0× H^-1=ℋ and |·|=|·|_0. To consider (<ref>) as an abstract evolution equation, we set V^ϵ_t=d/dtU^ϵ_t and let X^ϵ_t=[ [ U^ϵ_t; V^ϵ_t; ]] with X_0^ϵ :=x=[ [ x_1; x_2; ]]. The systems (<ref>) and (<ref>) can be rewritten as an abstract form dX^ϵ_t=𝒜 X^ϵ_tdt+F(X^ϵ_t,Y^ϵ_t)dt+BdW^1_t, dY^ϵ_t=1/ϵA Y^ϵ_tdt+1/ϵg(Y^ϵ_t)dt+σ/√(ϵ)dW^2_t, X^ϵ_0=x, Y^ϵ_0=y, where 𝒜:=[ [ 0 I; A 0 ]], F(x, y):=[ [ 0; F(Π_1∘ x,y) ]], B:=[ [ 0; I ]], with 𝒟(𝒜)={X=(x_1,x_2)^T∈ℋ:𝒜X=[ [ x_2; Ax_1 ]]∈ℋ}=ℋ^1, here A is regarded as an operator from H^1 to H^-1, and Π_1 denotes the canonical projection ℋ→ H. It is well known that the operator 𝒜 is the generator of a strongly continuous semigroup {𝒮_t}_t≥0 on ℋ with the explicit form 𝒮_t=e^𝒜t=[ [ C(t) (-A)^-1/2S(t); - (-A)^1/2S(t) C(t) ]], t≥ 0, where C(t)=cos((-A)^1/2 t) and S(t)=sin((-A)^1/2 t) are so-called cosine and sine operators with the expression in term of the orthonormal eigenpairs {α_1, e_i}_i∈ℕ of A: C(t)h=cos((-A)^1/2 t)h=∑_k=1^+∞cos{√(α_k)t}(e_k, h)_H· e_k, S(t)h=sin((-A)^1/2 t)h=∑_k=1^+∞sin{√(α_k)t}(e_k, h)_H· e_k. Moreover, it is easy to check that |𝒮_t x|≤|x| for t≥ 0, x∈ℋ. In order to ensure existence and uniqueness of the perturbation process Y_t we shall assume throughout this paper that: (Hypothesis 1) For the mapping g:H → H, we require that there exists a constant L_g>0 such that g(u_1)-g(u_2)≤ L_g(u_1-u_2), u, v∈ H. moreover, we assume that L_g< α_1. Concerning the coefficient F we impose the following conditions: (Hypothesis 2) For the mapping F:H× H → H, we assume that there exists a constant L_F>0 such that F(u_1,v_1)-F(u_2,v_2)≤ L_F(u_1-u_2+v_1-v_2), u_1,u_2, v_1,v_2∈ H. Also suppose that for any u∈ H, the mapping F(u,·): H→ H is of class C^2, with bounded derivatives. Moreover, we require that there exists a constant L such that for any u,v,w,y,y'∈ H its directional derivatives are well-defined and satisfy D_uF(u,y)· w≤ L w, D^2_uuF(u,y)· (v,w)≤ L v·w. [D_uF(u,y)-D_uF(u,y')]· w≤ L y-y'·w D^2_uu[F(u,y)-F(u,y')]· (v,w)≤ L y-y'·v·w. A simple example of the dirft coefficient F is given by F(u,y)=F_1(u)+F_2(y), here F_1, F_2: H→ H are of class C^2 with uniformly bounded derivatives up to order 2. According to conditions (<ref>) and (<ref>), system (<ref>) admits a unique mild solution. Namely, as discussed in <cit.>, for any y∈ H there exists a unique adapted process Y (y)∈ L^2(Ω,C([0, T];H) such that Y _t(y)=E_t y + ∫_0^tE_t-s g( Y_s (y))ds+ σ_2∫_0^tE_t-s dW^2_s, By arguing as in the proof of <cit.>, Theorem 7.2, it is possible to show that there exists a constant C>0 such that 𝔼Y_t(y)^2≤ C(1+y^2), t>0, and in correspondence of such Y_t(y), for any ϵ>0 and x=(x_1, x_2)^T∈ℋ there exists a unique adapted process X^ϵ(x,y)∈ L^2(Ω,C([0,T ];ℋ)) such that X^ϵ_t(x,y)=𝒮_tx+∫_0^t𝒮_t-sF(X_s^ϵ(x,y), Y _s/ϵ(y))ds+σ_1∫_0^t𝒮_t-sBdW^1_s. We point out that if x=(x_1,x_2)^T is taken in D(𝒜)=ℋ^1, then X^ϵ_t values in ℋ^1 for t>0 (see <cit.>) and satisfies 𝔼|X^ϵ_t(x,y)|^2_1≤ C(1+y^2+x^2_1) for some constant C>0. Moreover, we present an estimate for the ℋ-norm of 𝒜X_t^ϵ, which is uniform with respect to ϵ>0. Let X^ϵ_t(x, y)=(U_t^ϵ(x, y), V_t^ϵ(x, y))^T be the solution to the problem (<ref>), where the initial value satisfies X^ϵ_0=x=(x_1, x_2)^T∈ℋ^1, and the function F satisfies (<ref>). Then it holds that 𝔼|𝒜X^ϵ_t(x, y)|^2≤ C (1+y^2+|x|_1^2). We have 𝒜X^ϵ_t(x,y)=( [ 0 I; A 0 ]) ( [ U_t^ϵ(x,y); V_t^ϵ(x,y) ])= ( [ V_t^ϵ(x,y); A(U_t^ϵ(x,y)) ]), so that |𝒜X^ϵ_t(x, y)|^2 = V_t^ϵ(x,y)^2+A(U_t^ϵ(x,y))^2_-1 = V_t^ϵ(x,y)^2+A^1/2(U_t^ϵ(x,y))^2 . Let us start to estimate the norm of A^1/2(U_t^ϵ(x,y)) and consider the expression A^1/2U_t^ϵ(x,y) = A^1/2C(t)x_1-S(t)x_2-∫_0^tS(t-s)F(U_s^ϵ(x,y),Y _s/ϵ(y))ds + σ_1∫_0^tS(t-s)dW^1_s. Directly, we have A^1/2C(t)x_1^2+S(t)x_2^2 ≤ C(x_1^2_1+x_2^2). In view of the assumptions on F given in (<ref>), we obtain 𝔼∫_0^tS(t-s)F(U_s^ϵ(x,y),Y_s/ϵ(y))ds^2 ≤ C_1+C_2∫_0^t𝔼 [U_s^ϵ(x,y)^2+Y_s/ϵ(y)^2]ds ≤ C_1+C_2∫_0^t𝔼 [A^1/2U_s^ϵ(x,y)^2+Y_s/ϵ(y))^2]ds, and then, thanks to (<ref>), we have 𝔼∫_0^tS(t-s)F(U_s^ϵ(x,y),Y_s/ϵ(y))ds ≤ C_1(1+y^2)+C_2∫_0^t 𝔼A^1/2U_s^ϵ(x,y)^2ds. Notice that in view of Ito's isometry, we have 𝔼∫_0^tS(t-s)dW^1_s^2≤ C_3 and then, combining this estimate with (<ref>) and (<ref>), we have 𝔼A^1/2U_t^ϵ(x,y)^2 ≤ C_1(1+y^2+x_1_1^2+x_2^2) + C_2∫_0^t𝔼A^1/2U_s^ϵ(x,y)^2ds. From the Gronwall's lemma, this gives 𝔼A^1/2U_t^ϵ(x,y)^2 ≤ C_1(1+y^2+x_1_1^2+x_2^2). In an analogous way, we can prove that 𝔼V_t^ϵ(x,y)^2≤ C_1(1+y^2+x_1_1^2+x_2^2). Thanks to (<ref>), the two inequalities above yield (<ref>). If 𝒳 is a Hilbert space equipped with inner product (·,·)_𝒳, we denote by C^1(𝒳,ℝ) the space of all real function ϕ:𝒳→ℝ with continuous Fréchet derivative and use the notation Dϕ(x) for the differential of a C^1 function on 𝒳 at the point x. Thanks to Riesz representation theorem, we may get the identity for x,h∈𝒳: Dϕ(x)· h=(Dϕ(x), h)_𝒳. We define C_b^2(𝒳, ℝ) to be the space of all real-valued, twice Fréchet differential function on 𝒳, whose first and second derivatives are continuous and bounded. For ϕ∈ C_b^2(𝒳, ℝ), we will identify D^2ϕ(x) with a bilinear operator from 𝒳×𝒳 to ℝ such that D^2ϕ(x)· (h,k)=(D^2ϕ(x)h,k)_𝒳, x,h,k∈𝒳. On some occasions, we also use the notation ϕ',ϕ” instead of Dϕ or D^2ϕ. § ERGODICITY OF Y_T AND AVERAGING DYNAMICS Now, we consider the transition semigroup P_t associated with perturbation process Y_t(y) defined by equation (<ref>), by setting for any ψ∈ℬ_b(H) the space of bounded functions on H, P_tψ(y)=𝔼ψ(Y_t(y)). By arguing as <cit.>, we can show that 𝔼Y_t(y)^2≤ C(e^-(α_1-L_g)ty^2+1), t>0 for some constant C>0. This implies that there exists an invariant measure μ for the Markov semigroup P _t associated with system (<ref>) in H such that ∫_HP_t ψ dμ =∫_Hψ dμ , t≥ 0 for any ψ∈ℬ_b(H) (for a proof, see, e.g., <cit.>, Section 2.1). Then by repeating the standard argument as in the proof of Proposition 4.2 in <cit.>, the invariant measure has finite 2-moments: ∫_Hy^2μ(dy)≤ C. Let Y_t(y') be the solution of (<ref>) with initial value Y_0=y', it can be check that for any t≥0, 𝔼Y_t(y)-Y_t( y')^2≤y-y'^2e^-η t with η=(α_1-L_g)>0, which implies that μ is the unique invariant measure for P _t. Then, by averaging the coefficient F with respect to the invariant measure μ, we can define a H-valued mapping F̅(u):=∫_HF(u,y)μ(dy), u∈ H, and then, due to condition (<ref>), it is easily to check that F̅(u_1)-F̅(u_2)≤ Lu_1-u_2, u_1, u_2∈ H. Now we will consider the effective dynamics system ∂^2/∂ t^2U̅_t(ξ)=ΔU̅_t(ξ)+ F̅(U̅_t(ξ))+σ_1Ẇ_t^1, (ξ,t)∈ D×[0, T], U̅_t(ξ)=0, (ξ, t)∈∂ D× [0, +∞), U̅_0(ξ)=x_1(ξ), ∂/∂ tU̅_t(ξ)|_t=0=x_2(ξ), ξ∈ D. Following the same notation as in Section 2, the problem (<ref>) can be transferred to a stochastic evolution equation: dX̅_t=𝒜X̅_tdt+ F̅(X̅_t)dt+BdW^1_t, X_0=x, where X̅_t=[ [ U̅_t; V̅_t; ]] with V̅_t=d/dtU̅_t and F̅(x):=[ [ 0; F̅(Π_1∘ x) ]]=[ [ 0; F̅(u) ]]. The mild form for system (<ref>) is given by X̅_t(x)=𝒮_tx+∫_0^t𝒮_t-sF̅(X̅_s(x))ds+σ_1∫_0^t𝒮_t-sBdW^1_s. By arguing as before, for any x=(x_1, x_2)^T∈ℋ the above integral equation admits a unique mild solution in L^2(Ω,C([0,T ];ℋ)) such that 𝔼|X̅_t(x)|≤ C(1+|x|), t∈ [0, T]. § ASYMPTOTIC EXPANSIONS Let ϕ∈ C_b^2( H , ℝ) and define a function u^ϵ: [0, T]×ℋ× H→ℝ by u^ϵ(t, x,y)=𝔼ϕ(U_t^ϵ(x,y)). Let Π_1 be the canonical projection ℋ→ H. Define the function Φ: ℋ→ℝ by Φ(x):=ϕ(Π_1 x)=ϕ(x_1) for x=(x_1, x_2)^T∈ℋ. Clearly, we have u^ϵ(t, x,y)=𝔼Φ(X_t^ϵ(x,y)). We now introduce two differential operators associated with the systems (<ref>) and (<ref>), respectively: ℒ_1φ(y) = (Ay+g(y), D_yφ(y))_H +1/2σ_2^2Tr(D^2_yyφ(y)Q_2(Q_2)^*), φ(y)∈ C_b^2(H,ℝ), ℒ_2Ψ(x) = (𝒜x+F(x,y), D_xΨ(x))_ℋ +1/2σ_1^2Tr(D^2_xxΨ(x)BQ_1(BQ_1)^*), Ψ(x)∈ C_b^2(ℋ,ℝ ). It is known that u^ϵ is a solution to the forward Kolmogorov equation: d/dtu^ϵ(t, x, y)=ℒ^ϵ u^ϵ(t, x, y), u^ϵ(0, x,y)=Φ(x), where ℒ^ϵ=1/ϵℒ_1+ℒ_2. Also recall the Kolmogorov operator for the averaging system is defined as ℒ̅Ψ(x) = (𝒜x+F̅(x), D_xΨ(x))_ℋ +1/2σ_1^2Tr(D^2_xxΨ(x)BQ_1(BQ_1)^*), Ψ(x)∈ C_b^2(ℋ, ℝ ). If we set u̅(t, x)=𝔼ϕ(U̅_t(x))=𝔼Φ(X̅_t(x)), we have d/dtu̅(t, x)=ℒ̅u̅(t, x), u̅(0, x)=Φ(x). Then the weak difference at time T is equal to 𝔼ϕ(U̅_T)-𝔼ϕ(U^ϵ_T)=u^ϵ(T, x,y)-u̅(T,x). Henceforth, when there is no confusion, we often omit the temporal variable t and spatial variables x and y. For example, for u^ϵ(t, x, y), we often write it as u^ϵ. Our aim is to seek matched asymptotic expansions for the u^ϵ(T, x,y) of the form u^ϵ=u_0+ϵ u_1+r^ϵ, where u_0 and u_1 are smooth functions which will be constructed below , and r^ϵ is the remainder term. With the above assumptions and notation we have the following result, which is a direct consequence of Lemma <ref>, Lemma <ref> and Lemma <ref>. Assume that x∈ℋ^1, y∈ D(A). Then, under Hypotheses 1 and 2, for any any T>0 and ϕ∈ C_b^3(H), there exist a constant C_T,ϕ,x,y such that |𝔼ϕ(U^ϵ_T(x,y))-𝔼ϕ(U̅_T(x))|≤ C_T,ϕ,x,yϵ. §.§ The leading term Let us first determine the leading terms. Now, substituting (<ref>) into (<ref>) yields du_0/dt+ϵdu_1/dt+dr^ϵ/dt = 1/ϵℒ_1u_0+ℒ_1u_1+1/ϵℒ_1r^ϵ +ℒ_2u_0+ϵℒ_2u_1+ℒ_2r^ϵ. By comparing coefficients of powers of ϵ, we obtain ℒ_1u_0=0, du_0/dt=ℒ_1u_1+ℒ_2u_0. It follows from (<ref>) that u_0 is independent of y, which means u_0(t,x, y)=u_0(t,x). We also impose the initial condition u_0(0,x)=Φ(x). Since μ is the invariant measure of a Markov process with generator ℒ_1, we have ∫_Hℒ_1u_1(t,x,y)μ(dy)=0, which, by invoking (<ref>), implies du_0/dt(t,x) = ∫_Hdu_0/dt(t,x)μ(dy) = ∫_Hℒ_2u_0(t,x)μ(dy) = (𝒜u_0(t,x)+∫_HF(x,y)μ(dy), D_xu_0(t,x))_ℋ +1/2σ_1^2Tr(D^2_xxu_0(t,x)BQ_1(BQ_1)^*) = ℒ̅u_0(t,x), so that u_0 and u̅ satisfies the same evolution equation. By using a uniqueness argument, such u_0 has to coincide with the solution u̅ and we have the following lemma: Assume Hypotheses 1 and 2. Then for any x∈ D(𝒜), y∈ D(A) and T>0, we have u_0(T,x,y)=u̅(T,x). §.§ Construction of u_1 Let us proceed to carry out the construction of u_1. Thanks to Lemma <ref> and (<ref>), the equation (<ref>) can be rewritten ℒ̅u̅=ℒ_1u_1+ℒ_2u̅, and hence we get an elliptic equation for u_1 with form ℒ_1u_1(t,x,y)=(F̅(x)- F(x,y), D_xu̅(t,x))_ℋ:=-ρ(t,x,y), where ρ is of class C^2 with respect to y, with uniformly bounded derivative. Moreover, it satisfies for any t≥ 0 and x∈ℋ^1, ∫_Hρ(t,x,y)μ(dy)=0. For any y∈ D(A) and s>0 we have d/dsP_sρ(t,x,y) = (Ay+g(x,y),D_y(P_sρ(t,x,y)))_H + 1/2σ_2^2Tr[D^2_yy(P_sρ(t,x,y))Q_2Q_2^*], here P_sρ(t,x,y)=𝔼ρ(t, x,Y_s(y)) satisfying lim_s→+∞𝔼ρ(t, x,Y_s(y))=∫_Hρ(t,x,z)μ(dz)=0. Indeed, by the invariant property of μ and Lemma <ref>, |𝔼ρ(t, x,Y_s(y))-∫_Hρ(t,x,z)μ(dz)| =|∫_H𝔼[ρ(t, x,Y_s(y))- ρ(t, x,Y_s(z))μ(dz)]| ≤∫_H|𝔼(F(x,Y_s(z)- F(x,Y_s(y), D_xu̅(t,x))_ℋ|μ(dz) ≤ C∫_H 𝔼Y_s(z)- Y_s(y)μ(dz). This, in view of (<ref>) and (<ref>), yields |𝔼ρ(t, x,Y_s(y))-∫_Hρ(t,x,z)μ(dz)| ≤ Ce^-η/2s, which implies the equality (<ref>). Therefore, we get (Ay+g(x,y),D_y∫_0^+∞P_sρ(t,x,y) ds)_H +1/2σ_2^2Tr[D^2_yy∫_0^+∞(P_sρ(t,x,y))Q_2Q_2^*]ds =∫_0^+∞d/dsP_sρ(t,x,y)ds =lim_s→+∞𝔼ρ(t, x,Y_s(y))-ρ(t,x,y) =∫_Hρ(t,x,y)μ(dy)-ρ(t,x,y) =-ρ(t,x,y), which means ℒ_1(∫_0^+∞P_sρ(t,x,y) ds)=-ρ(t,x,y). Therefore, we can set u_1(t,x,y)=∫_0^+∞𝔼ρ(t,x,Y_s(y))ds. Assume Hypotheses 1 and 2. Then for any x∈ D(𝒜), y∈ D(A) and T>0, we have |u_1(t,x,y)|≤ C_T(1 +y), t∈[0, T]. As known from (<ref>), we have u_1(t,x,y)=∫_0^+∞𝔼(F̅(x)- F(x,Y_s(y)), D_xu̅(t,x))_ℋds. This implies that |u_1(t,x,y)| ≤ ∫_0^+∞|F̅(x)- 𝔼[F(x,Y_s(y))]| ·|D_xu̅(t,x)| ds. Then, in view of Lemma <ref> and (<ref>), this implies : |u_1(t,x,y)| ≤ C_T(1 +y)∫_0^+∞e^-η/2 sds ≤ C_T(1 +y). §.§ Determination of remainder r^ϵ Once u_0 and u_1 have been determined, we can carry out the construction of the remainder r^ϵ. It is known that (∂_t-ℒ^ϵ)u^ϵ=0, which, together with (<ref>) and (<ref>), implies (∂_t-ℒ^ϵ)r^ϵ = -(∂_t-ℒ^ϵ) u_0-ϵ(∂_t-ℒ^ϵ)u_1 = -(∂_t-1/ϵℒ_1-ℒ_2)u_0-ϵ(∂_t-1/ϵℒ_1-ℒ_2)u_1 = ϵ(ℒ_2u_1-∂_tu_1). In order to estimate the remainder term r^ϵ we need the following crucial lemmas. Assume Hypotheses 1 and 2. Then for any x∈ D(𝒜), y∈ D(A) and T>0, we have |du_1/dt(t,x,y)|≤ C (1+|x|_1 )y, t∈ [0, T]. According to (<ref>), we have du_1/dt(t,x,y)=∫_0^+∞𝔼(F̅(x)- F(x,Y_s(y)), d/dtD_xu̅(t,x))_ℋds. For any h=(h_1,h_2)^T∈ℋ^1, D_xu̅(t,x)· h = D_x[𝔼ϕ(Π_1∘X̅(t,x))] = 𝔼[ϕ'(Π_1∘X̅(t,x))· (Π_1∘η_t^h,x)], and hence d/dt(D_xu̅(t,x)· h) = 𝔼[ϕ”(Π_1∘X̅(t,x))·(Π_1∘η_t^h,x, d/dt(Π_1∘X̅(t,x)) )] + 𝔼[ϕ'(Π_1∘X̅(t,x))·d/dt(Π_1∘η_t^h,x)], so that, due to the fact of ϕ∈ C_b^2(H, ℝ), we obtain |d/dt(D_xu̅(t,x)· h)| ≤ C[𝔼Π∘η_t^h,x^2]^1/2·[𝔼d/dt(Π_1∘X̅(t,x)^2]^1/2 + 𝔼d/dt(Π_1∘η_t^h,x ≤ C |h|_1·[𝔼d/dt(Π_1∘X̅(t,x)^2]^1/2 + 𝔼d/dt(Π_1∘η_t^h,x, where we used the estimate (<ref>) such that Π_1∘η_t^h,x≤ C|h| ≤ C|h|_1. Now, as X̅_t(x) is the mild solution of averaging equation with initial data x=(x_1,x_2)^T∈ℋ^1, we have Π_1∘X̅_t(x) = U̅_t(x) = C(t) x_1+(-A)^-1/2S(t) x_2+∫_0^t (-A)^-1/2S(t-s)F̅(U̅_s(x))ds +σ_1∫_0^t (-A)^-1/2S(t-s)dW^1_s with d/dt[Π_1∘X̅_t(x)] = -(-A)^1/2S(t) x_1+C(t) x_2+∫_0^t C(t-s)F̅(U̅_s(x))ds +σ_1∫_0^t C(t-s)dW^1_s, By straightforward computation, we have -(-A)^1/2S(t) x_1^2≤x_1_1^2 and C(t) x_2^2≤x_2^2. According to the Lipschitz continuity of F̅ and (<ref>), we have 𝔼∫_0^t C(t-s)F̅(U̅_s(x))ds^2 ≤ C_T𝔼∫_0^t(1+U̅_s(x)^2)ds ≤ C_T (1+|x|_1). Now, from (<ref>)-(<ref>) it follows 𝔼d/dt[Π_1∘X̅_t(x)]^2≤ C(1+|x|^2_1) Now, we prove uniform bounds for time derivative of Π_1∘η_t^h,x with respect to x. Clearly, we have d/dt(Π_1∘η_t^h,x) = -(-A)^1/2S(t)h_1+C(t) h_2 + ∫_0^t C(t-s)[Π_1∘(F̅'(X̅_s(x))·η_t^h,x)]ds. Note that for any h=(h_1, h_2)^T∈ℋ^1, (-A)^1/2S(t)h_1^2+C(t) h_2^2≤|h|_1^2. In view of (<ref>) and (<ref>), we obtain ∫_0^t C(t-s)[Π_1∘(F̅'(X̅_s(x))·η_t^h,x)]ds ≤ C∫_0^t|η_s^h,x|ds ≤ C_T |h|_1. Then thanks to (<ref>) and (<ref>), we obtain d/dt(Π_1∘η_t^h,x)^2≤ C |h|_1^2. So, if we plug the above estimate and estimate (<ref>) into (<ref>), we get |d/dtD_xu̅(t,x)· h|≤ C|h|_1(1+|x|_1 ), which, together with (<ref>), implies |du_1/dt(t,x,y)| ≤ C(1+|x|_1 )∫_0^+∞|F̅(x)- 𝔼F(x,Y_s(y))|_1ds ≤ C (1+|x|_1)y∫_0^+∞e^-η/2 sds ≤ C (1+|x|_1 )y. Hence the assertions is completely proved. Assume that all conditions in Lemma <ref> are fulfilled. Then we have |ℒ_2u_1(t,x,y)|≤(1+|𝒜x|+|x|_1+y)(1+y), t∈ [0, T]. As known, for any x∈ D(𝒜), ℒ_2u_1(t,x,y) = (𝒜x+ F(x, y), D_xu_1(t,x,y) )_ℋ + 1/2σ_2^2Tr(D^2_xxu_1(t,x,y)(BQ_1)(BQ_1)^*). We will carry out the estimate of |ℒ_2u_1(t,x,y)| in two steps. (Step 1) Estimate of (𝒜x+ F(x, y), D_xu_1(t,x,y) )_ℋ. For any k∈ℋ, we have D_xu_1(t,x,y)· k = ∫_0^+∞𝔼(D_x(F̅(x)- F(x,Y_s))· k,D_xu̅(t,x))_ℋds +∫_0^+∞𝔼(F̅(x)- F(x,Y_s),D^2_xxu̅(t,x)· k)_ℋds := I_1(t,x,y,k)+I_2(t,x,y,k). According to the invariant property of measure μ, (<ref>) and (<ref>) we have |I_1(t,x,y,k)| ≤ ∫_0^+∞|𝔼(D_x(F̅(x)- F(x,Y_s(y)))· k,D_xu̅(t,x))_ℋ| ds = ∫_0^+∞| 𝔼∫_H (D_x[F(x,z)- F(x,Y_s(y))]· k,D_xu̅(t,x))_ℋμ(dz)|ds = ∫_0^+∞| 𝔼∫_H (D_x[F(x,Y_s(z))- F(x,Y_s(y))]· k,D_xu̅(t,x))_ℋμ(dz)|ds ≤ C|k| ·|D_xu̅(t,x)| ·∫_0^+∞[∫_H𝔼Y_s(z)-Y_s(y)μ(dz)]ds By making use of (<ref>) and (<ref>), the above yields |I_1(t,x,y,k)| ≤ C|k| ·|D_xu̅(t,x)| ·∫_0^+∞ e^-η/2 s (1+y)ds ≤ C|k| ·|D_xu̅(t,x)| ≤ C|k|, where we used Lemma <ref> in the last step. By Lemma <ref> and (<ref>), we have |I_2(t,x,y,k)| ≤ ∫_0^+∞|(F̅(x)- 𝔼F(x,Y_s(y)),D^2_xxu̅(t,x)· k)_ℋ|ds ≤ C|k| ∫_0^+∞|F̅(x)- 𝔼F(x,Y_s(y))| ds ≤ C|k| (1 +y)∫_0^+∞ e^-η/2 s ds ≤ C|k| (1 +y). Together with (<ref>) , this yields |D_xu_1(t,x,y)· k| ≤ C|k| (1+ y) which means |( 𝒜x+ F(x, y), D_xu_1(t,x,y) )_ℋ| ≤ C(1+|𝒜x|+|x|_1+y)(1+y). (Step 2) Estimate of Tr(D^2_xxu_1(t,x,y)(BQ_1)(BQ_1)^*). Note that we have D_xxu_1(t,x,y)· (h, k) =∫_0^+∞𝔼(D^2_xx(F̅(x)- F(x,Y_s(y)))· (h,k),D_xu̅(t,x))_ℋds +∫_0^+∞𝔼(D_x(F̅(x)- F(x,Y_s(y)))· h,D^2_xxu̅(t,x)· k)_ℋds +∫_0^+∞𝔼(D_x(F̅(x)- F(x,Y_s(y)))· k,D^2_xxu̅(t,x)· h)_ℋds +∫_0^+∞𝔼(F̅(x)- F(x,Y_s(y)), D^3_xxxu̅(t,x)· (h,k))_ℋds :=∑_i=1^4J_i(t,x,y,h,k). In view of (<ref>) and invariant property of measure μ we have |J_1(t,x,y,h,k)| ≤ ∫_0^+∞|𝔼(D^2_xx(F̅(x)- F(x,Y_s(y)))· (h,k),D^2_xu̅(t,x))_ℋ| ds = ∫_0^+∞| 𝔼∫_H (D^2_xx[F(x,z)- F(x,Y_s(y))]· (h,k),D_xu̅(t,x))_ℋμ(dz)|ds = ∫_0^+∞| 𝔼∫_H (D^2_xx[F(x,Y_s(z))- F(x,Y_s(y))]· (h,k),D_xu̅(t,x))_ℋμ(dz)|ds By taking (<ref>) and Lemma <ref> into account, we can deduce |J_1(t,x,y,h,k)| ≤ C|h|·|k| ·∫_0^+∞[∫_H𝔼Y_s(z)-Y_s(y)μ(dz)]ds ≤ C|h|·|k| ·∫_0^+∞ e^-η/2 s (1+y)ds ≤ C|h|·|k| ≤ C|h|·|k| (1+y), Again, by (<ref>) and invariant property of measure μ, we have |J_2(t,x,y,h,k)| ≤ ∫_0^+∞| 𝔼∫_H (D_x[F(x,Y_s(z))- F(x,Y_s(y))]· h,D^2_xxu̅(t,x)· k)_ℋμ(dz)|ds, which, by Lemma <ref> and condition (<ref>), implies |J_2(t,x,y,h,k)| ≤ C|h|·|k| ·∫_0^+∞[∫_H𝔼Y_s(z)-Y_s(y)μ(dz)]ds ≤ C|h|·|k| ·∫_0^+∞ e^-η/2 s (1+y)ds ≤ C|h|·|k|(1+y). Parallel to (<ref>), we can obtain the same estimate for J_3(t,x,y, h,k), that is, |J_3(t,x,y,h,k)| ≤ C|h|·|k|(1+y). By proceeding again as in the estimate for J_1(t,x,y,h,k) we have |J_4(t,x,y,h,k)| ≤ ∫_0^+∞| 𝔼∫_H (F(x,Y_s(z))- F(x,Y_s(y)) ,D^3_xxxu̅(t,x)·(h,k))_ℋμ(dz)|ds and then thanks to Lemma <ref> and (<ref>), we get |J_4(t,x,y,h,k)| ≤ C|h|·|k| ·∫_0^+∞[∫_H𝔼Y_s(z)-Y_s(y)μ(dz)]ds ≤ C|h|·|k|(1+y). Collecting together (<ref>), (<ref>), (<ref>) and (<ref>), we obtain |D^2_xxu_1(t,x,y)· (h, k)|≤ C |h|·|k|(1+y), which means that for fixed y∈ H and t∈ [0, T], D^2_··u_1(t,·,y) _L(ℋ×ℋ,ℝ)≤ C(1+y), so that, as the operator Q_1 has finite trace, we get Tr(D^2_xxu_1(t,x,y)(BQ_1)(BQ_1)^*) ≤D^2_xxu_1(t,x,y)Tr( (BQ^1)(BQ^1)^*) ≤ C(1+y). Finally, by taking inequalities (<ref>) and (<ref>) into account, we can conclude the proof of the lemma. As a consequence of Lemma <ref> and <ref>, we have the following fact for the remainder term r^ϵ. Under the conditions of Lemma <ref>, for any T>0, x∈ D(𝒜), y ∈ H, we have |r^ϵ(T,x,y)|≤ Cϵ(1+|x|+y)(1+|𝒜x|+|x|_1). By a variation of constant formula, we have r^ϵ(T,x,y) = 𝔼[r^ϵ(0,X^ϵ_T(x,y),Y_T/ϵ( y)] + ϵ𝔼[∫_0^T(ℒ_2u_1-∂ u_1/∂ s)( X^ϵ_T-s(x,y),Y_T-s/ϵ( y)) ds]. Since u^ϵ and u_0=u̅ has the same initial condition Φ(x), we have |r^ϵ(0, x,y)| = |u^ϵ(0,x,y)-u̅(0,x)-ϵ u_1(0,x,y)| = ϵ |u_1(0,x,y)|, so that, from (<ref>) and (<ref>) we have 𝔼[r^ϵ(0,X^ϵ_T(x,y),Y_T/ϵ(y)]≤ Cϵ(1+ y). Thanks to Lemma <ref> and Lemma <ref>, we have 𝔼[(ℒ_2u_1-∂ u_1/∂ s)( X^ϵ_T-s(x,y),Y_T-s/ϵ(y))] ≤ C 𝔼[(1+|𝒜X^ϵ_T-s(x,y)| +|X^ϵ_T-s(x,y)|_1+Y_T-s/ϵ(y)) ·(1+Y_T-s/ϵ(y))], and, according to (<ref>), (<ref>), (<ref>) and the Hölder inequality, this implies that 𝔼[∫_0^T(ℒ_2u_1-∂ u_1/∂ s)( X^ϵ_T-s(x,y),Y_T-s/ϵ) ds] ≤ C (1+ y)(1+|𝒜x|+|x|_1+y). This, together with (<ref>), implies |r^ϵ(T,x,y)|≤ Cϵ(1+ y) (1+|𝒜x|+|x|_1+y), which completes the proof. § APPENDIX In this section, we state and prove some technical lemmas needed in the former sections. For any x∈ℋ and y∈ H, there exists a constant C>0 such that |F̅(x)-𝔼[F(x,Y_t(y))]|^2_1 ≤ Ce^-η t(1 +y^2), where η= α_1-L_g >0. According to the invariant property of μ, (<ref>) and hypothesis (<ref>), we have |F̅(x)-𝔼[F(x,Y_t(y))]|^2_1 = F̅(Π_1x)- 𝔼[F(Π_1x,Y_t(y))]^2 = ∫_HF(Π_1 x,z)μ(dz)- 𝔼[F(Π_1x,Y_t(y))]^2 = ∫_H 𝔼[F(Π_1x, Y_t(z))-F(Π_1x,Y_t(y))]μ(dz)^2, so thanks to (<ref>) and (<ref>), we have |F̅(x)-𝔼[F(x,Y_t(y))]|_1^2 ≤ C∫_H𝔼Y_t(y)-Y_t(z)^2μ (dz) ≤ Ce^-η t∫_Hy-z^2μ (dz) ≤ Ce^-η t(1 +y^2). Next, we introduce the following regularity results of averaging function F̅. For any w∈ H, the function (F̅(·), w)_H: H→ℝ is Gâteaux differential and for any v∈ H, it hold that (DF̅(u)· v, w)_H=∫_H(D_u F(u,y)· v, w)_Hμ(dy), w∈ H. For any λ≠0 we have (∫_H1/λ[F(u+λ v,y)-F(u,y)]μ(dy),w)_H-∫_H(D_uF(u,y)· v,w)_Hμ(dy) =∫_H (1/λ[F(u+λ v,y)-F(u,y)]-D_uF(u,y)· v,w)_Hμ(dy). and then |(∫_H1/λ[F(u+λ v,y)-F(u,y)]μ(dy),w)_H-∫_H(D_uF(u,y)· v,w)_Hμ(dy)| ≤∫_H |(1/λ[F(u+λ v,y)-F(u,y)]-D_uF(u,y)· v,w)_H|μ(dy) ≤w∫_H 1/λ[F(u+λ v,y)-F(u,y)]-D_uF(u,y)· vμ(dy). Now, since F(·, y): H→ H is Gâteaux differentiable in H, for any h∈ H, we obtain lim_λ→ 01/λ[F(u+λ v,y)-F(u,y)]-D_uF(u,y)· v=0. Moreover, by mean value theorem, 1/λ[F(u+λ v,y)-F(u,y)]-D_uF(u,y)· v =∫_0^1[D_u F(u+λθ v)-D_uF(u,y)]· v dθ so that, due to the boundedness of D_uF(u,y), we get 1/λ[F(u+λ v,y)-F(u,y)]-D_uF(u,y)· v≤ C v. and then, by using the dominated convergence theorem, taking (<ref>) into account, we can conclude lim_λ→ 0(∫_H1/λ[F(u+λ v,y)-F(u,y)]μ(dy),w)_H =∫_H(D_uF(u,y)· v,w)_Hμ(dy), which implies that (DF̅(u)· v, w)_H=∫_H(D_u F(u,y)· v,w)_Hμ(dy). As a as a consequence of Lemma <ref>, it is easily to check that (D_xF̅(x)· h, k)_ℋ=∫_H(D_x F(x,y)· h,k)_ℋμ(dy), h,k∈ℋ, and this yields |D_xF̅(x)· h| ≤ ∫_H|D_x F(x,y)· h|μ(dy) = ∫_HA^-1/2[(D_uF)(Π_1∘ x,y)·(Π_1∘ h)]μ(dy) ≤ C∫_H (D_uF)(Π_1∘ x,y)·(Π_1∘ h) μ(dy). Moreover, by invoking conditions (<ref>), we get |D_xF̅(x)· h| ≤ C |h|, h∈ℋ. As far as the higher order derivative are concerned, by proceeding as in the proof of above lemma, we can show that (D_uu^2F̅(u)·(v,w), ν)_H=∫_H(D_uu F(u,y)·(v,w),ν)_Hμ(dy), v,w,ν∈ H. As a consequence, we obtain (D_xx^2F̅(x)·(h,k), l)_ℋ=∫_H(D_xx^2 F(x,y)·(h,k),l)_ℋμ(dy), h,k,l∈ℋ and |D_xx^2F̅(x)·(h,k)≤ C|h|·|k|, h,k∈ℋ. For any T>0, there exists C_T>0 such that for any x∈ℋ and t∈ [0, T], we have D_xu̅(t,x)≤ C_T. Note that for any h∈ℋ, D_xu̅(t,x)· h = 𝔼[DΦ(X̅_t(x))·η ^h,x_t] = 𝔼(Φ'(X̅_t(x)),η^h,x_t)_ℋ, where η^h,x_t is the mild solution of dη ^h,x_t=(𝒜η ^h,x_t+DF̅(X̅_t(x))·η ^h,x_t)dt η ^h,x(0)=h. This means that η ^h,x_t is the solution of the integral equation η^h,x_t=𝒮_th+∫_0^t𝒮_t-s[DF̅(X̅_s(x))·η ^h,x_t]ds, and then thanks to (<ref>), we get |η ^h,x_t| ≤|h| +C∫_0^t|η ^h,x_s| ds. Then by Gronwall lemma it follows that |η ^h,x_t| ≤ C_T|h|, t∈ [0, T], which means |D_xu̅(t,x)· h|≤ C_Tsup_z∈ℋ|Φ(z)|·|h|, so that |D_xu̅(t,x)| ≤ C_T. For any T>0, there exists C_T>0 such that for any x,h,k∈ℋ and t∈ [0, T], we have |D^2_xxu̅(t,x)·(h,k)|≤ C_T,ϕ|h|·|k|. For any h, k ∈ℋ, we have D^2_xxu̅(t,x)·(h,k) = 𝔼[Φ”(X̅_t(x))·(η^h,x_t,η^k,x_t) + Φ'(X̅_t(x))·ζ^h,k,x_t], where ζ^h,k,x is the mild solution of equation dζ^h,k,x_t=[𝒜ζ^h,k,x_t+ F̅”(X̅_t(x))·(η^h,x_t,η^k,x_t)+F̅'(X̅_t(x))·ζ^h,k,x_t]dt ζ^h,k,x_0=0. This means that ζ^h,k,x_t is the solution of the integral equation ζ^h,k,x_t= ∫_0^t𝒮_t[F̅”(X̅_s(x))·(η^h,x_s,η^k,x_s)+F̅'(X̅_s(x))·ζ^h,k,x_s]ds. Thus, by (<ref>) and (<ref>) we have |ζ^h,k,x_t| ≤ C∫_0^t(|η^h,x_s|·|η^k,x_s|+|ζ^h,k,x_s|)ds ≤ C|h|·|k| +C∫_0^t|ζ^h,k,x_s|ds. By applying the Gronwall lemma we have |ζ^h,k,x_t|≤ C_T|h|·|k|, t>0. Returning to (<ref>), we can get |D_xxu̅(t,x)·(h,k)|≤ C |h|·|k|. By proceeding again as in the proof of above lemma, we have the following result. For any T>0, there exists C_T>0 such that for any x,h,k,l∈ℋ and t∈ [0, T], we have D^3_xxxu̅(t,x)·(h,k,l)≤ C_T,ϕ|h|·|k|·|l|. § ACKNOWLEDGMENTS We would like to thank Professor Jinqiao Duan for helpful discussions and comments. Hongbo Fu is supported by CSC scholarship (No. [2015]5104), NSF of China (Nos. 11301403, 11405118, 11271295) and Foundation of Wuhan Textile University 2013. Li Wan is supported by NSF of China (No. 11271295) and Science and Technology Research Projects of Hubei Provincial Department of Education (No. D20131602). Jicheng Liu is supported by NSF of China (Nos. 11271013, 10901065). Xianming Liu is supported by NSF of China (No. 11301197) 99 Bao J. H. Bao, G. Yin, C. G. Yuan, Two-time-scale stochastic partial differential equations driven by α-stable noises: Averaging principles, Bernoulli 23 (1) (2017) 645-669. Bogoliubov N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear Oscillations, Gordon & Breach Science Publishers, New York, 1961. Brehier C. E. Bréhier, Strong and weak orders in averging for SPDEs, Stochastic Process. Appl. 122 (2012) 2553-2593. Cardetti F. Cardetti, Y. S. Choi, A parabolic-hyperbolic system modelling a moving cell, Electron. J. Differential Equations 95 (2009) 1-11. Cerrai1 S. Cerrai, M. I. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Proba. Theory Related Fields. 144 (2009) 137-177. Cerrai2 S. Cerrai, A Khasminkii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab. 19 (3) (2009) 899-948. Cerrai-Siam S. Cerrai, Averaging principle for systems of reaction-diffusion equations with polynomial nonlinearities perturbed by multiplicative type noise, SIAM J. Math. Anal. 43(6) (2011) 2482-2518. Choi Y. S. Choi, Craig Miller, Global existence of solutions to a coupled parabolic-hyperbolic system with moving boundary, Proc. Amer. Math. Soc. 139 (2011) 3257-3270 Chow-1 P. L. Chow, Thermoelastic wave propagation in a random medium and some related problems, Int. J. Eng. Sci. 11 (1973) 253-971. Chow P. L. Chow, Stochastic Partial Differential Equations, 2nd ed., Chapman & Hall/CRC, New York, 2015. Freidlin-Wentzell1 M. I. Freidlin, A. D. Wentzell, Random Perturbation of Dynamical Systems, 2nd ed., Springer-Verlag, New York, 1998. Freidlin-Wentzell2 M. I. Freidlin, A. D. Wentzell, Long-time behavior of weakly coupled oscillators, J. Stat. Phys 123 (2006) 1311-1337. Fu-Liu H. Fu, L. Wan, Y. Wang, J. Liu, Strong convergence rate in averaging principle for stochastic FitzHug-Nagumo system with two time-scales, J. Math. Anal. Appl. 416 (2014) 609-628. Fu-Liu-2 H. Fu, L. Wan, J. Liu, Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales. Stochastic Process. Appl. 125 (2015) 3255-3279. Givon1 D. Givon, I. G. Kevrekidis, R. Kupferman, Strong convergence of projective integration schemes for singular perturbed stochastic differential systems, Comm. Math. Sci. 4 (2006) 707-729. Khas R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations (Russian), kibernetika 4 (1968) 260-279. Khas2 R. Z. Khasminskii, G. Yin, Limit behavior of two-time-scale diffusions revisited, J. Differential Equations 212 (1) (2005) 85-113. Khas3 R.Z. Khasminskii, G. Yin, On Averaging Principles: An Asymptotic Expansion Approach, SIAM J. Math. Anal 35(6) 1534-1560. Kifer1 Y. Kifer, Some recent advance in averaging, Modern dynamical systems and applications, Cambridge University Press, Cambridge, UK, 2004, pp. 385-403. Kifer2 Y. Kifer, Diffusion approximation for slow motion in fully coupled averaging, Proba. Theor. Relat. Fields, 129 (2004) 157-181. Kifer3 Y. Kifer, Another proof of the averaging principle for fully coupled dynamical systems with hyperbolic fast motions, Discrete Contin. Dyn. Syst. 13 (2005) 1187-1201. Leung-0 A. W. Leung, Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations, J. Differential Equations 187 (2003) 184-200. Leung A. W. Leung, Stable invariant manifolds for coupled Navier-Stokes and second-order wave systems, Asymptotic Analysis 43 (2005) 339-357. LiuDi D. Liu, Strong convergence of principle of averaging for multiscale stochastic dynamical systems, Commun. Math. Sci. 8 (4) (2010) 999-1020. Daprato G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, 2nd ed., Cambridge University Press, Cambridge, 2014. Rivera J. E. M. Rivera, R. Racke, Smoothing properties, decay and global existence of solutions to nonlinear coupled systems of thermoelasticity type, SIAM J. Math. Anal. 26 (1995) 1547-1563. Vere1 A. Y. Veretennikov, On the averaging principle for systems of stochastic differential equations, Mathematics of the USSR-Sbornik 69 (1991) 271-284. Vere2 A. Y. Veretennikov, On large deviations in the averaging principle for SDEs with full dependence, Ann. Probab. 27 (1999) 284-296. Wainrib G. Wainrib, Double averaging principle for periodically forced slow-fast stochastic systems, Electron. Commun. Probab. 18(51) (2013) 1-12. wangwei W. Wang, A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations. J. Differential Equations 253 (2012) 1265-1286 Weinan E. Weinan, D. Liu, E. Vanden-Eijnden, Analysis of multiscale methods for stochastic differential equations, Comm. Pure Appl. Math. 58 (11) (2005) 1544-1585. S.H. Wu S. H. Wu, H. Chen, W. X. Li, The local and global existence of the solutions of hyperbolic-parabolic system modeling biological phenomena, Acta Mathematica Scientia 28B(1) (2008) 101-116. Xu Y. Xu, J. Duan, W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Physica D 240(17) (2011) 1395-1401. Xu1 Y. Xu, B. Pei, Y. Li, An averaging principle for stochastic differential delay equations with fractional Brownian motion, Abstr. Appl. Anal. (2014) ID 479195. Xu2 Y. Xu, R. Guo, D. Liu, H. Zhang, J. Duan, Stochastic averaging principle for dynamical systems with fractional Brownian motion, Discrete Contin. Dynam. Systems (Series B) 19(4) (2014) 1197-1212. Xu3 Y. Xu, B. Pei, Y. Li, Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Math. Methods Appl. Sci. 4 (1) (2014) DOI: 10.1002/mma.3208. Xu4 Y. Xu, R. Guo, W. Xu, A limit theorem for the solutions of slow-fast systems with fractional Brownian motion, Theor. Appl. Mech. Lett. 4 (2014) ID 3-013003. X Zhang X. Zhang, E. Zuazua, Long-time behavior of a coupled heat-wave system arising in fluid-structure interaction, Arch. Rational Mech. Anal. 184(1) (2007) 49-120.

http://arxiv.org/abs/1701.08167v2

20170127190005

Conformal partition functions of critical percolation from $D_3$ Thermodynamic Bethe Ansatz equations

cond-mat.stat-mech

=10000 equationsection figuresection #1 #1#1 #1#1 plain LemmeLemma[section] TheoremeTheorem PropositionProposition ConjectureConjecture Lemma[Lemme]Lemma Corollaire[Lemme]Corollary DefinitionDefinition[section] Proof. 0.5em0.5em unit=0.4cmunit=0.3cm0.25exχ#1#2[fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt]#1#2 [gridlabels=0pt,subgriddiv=1]#1#2#1#2http://arxiv.org/abs/#1 i𝖾 dω mod trtrABXY 𝖳𝖫ℰ𝖯𝖳𝖫𝖵𝖨𝖶Z_d^(N)(q)Z_d^(N)(q,q̅)^(N)Z_cyl^(M,N)Z_cylZ^(N odd)_cylZ^(N even)_cylZ_torus^(M,N)Z_torusZ^(N odd)_torusZ^(N even)_torusZ^(M even,N even)_torusZ^(M even,N odd)_torusZ^(M odd,N even)_torusZ^(M odd,N odd)_torusZ^(M odd, N even)_torus, CDPZ^(M even, N odd)_torus, CDPZ^(M odd, N even)_torus, CDPZ^(M odd, N odd)_torus, CDPZ^(N even)_torus, CDPZ^(N odd)_torus, CDP #1#2[fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt]#1#2 [gridlabels=0pt,subgriddiv=1]#1#2#1#2[fillstyle=solid,fillcolor=lightblue,linewidth=0pt]#1#2 [gridlabels=0pt,subgriddiv=1]#1#2[linewidth=.25pt](0,0)(1,1) [linewidth=1.5pt,linecolor=blue](1,0).590180[linewidth=1.5pt,linecolor=blue](0,1).5-900[linewidth=.25pt](0,0)(1,1) [linewidth=1.5pt,linecolor=blue](0,0).5090[linewidth=1.5pt,linecolor=blue](1,1).5180270[shift=-0.2](-0.07,0)(0.07,0.6) [linestyle=dashed,dash=1pt 1pt]-(0,0)(0,0.6) [shift=-0.1](-0.07,0)(0.07,0.4) [linestyle=dashed,dash=1pt 1pt]-(0,0)(0,0.4) [linewidth=0.5pt,linecolor=black,fillstyle=solid,fillcolor=white](0,0).15[linewidth=0.5pt,linecolor=black,fillstyle=solid,fillcolor=black](0,0).15[linewidth=0.5pt,linecolor=black,fillstyle=solid,fillcolor=gray](0,0).15#1#2𝒜^#1_#2#1#2#3𝒟^#1_#2,#3#1#2𝒜̅^#1_#2Å#1#2#3#4#5𝒟^#1,#2_#3,#4,#5#1#2#3#4#5#6𝒟^#1,#2 (#6)_#3,#4,#5#1#2#3#4#5𝒜^#1,#2_#3,#4,#5#1#2#3#4#5#6𝒜^#1,#2 (#6)_#3,#4,#5#1#2⟨ #1 #2 ⟩#1#2⟨[ #1; #2 ]⟩#1#2[ #1 #2 ]_q#1#2[ [ #1; #2 ]]_q#1#2[ #1 #2 ]_q#1#2[ #1 #2 ]_q̅#1#2[ [ #1; #2 ]]_q̅#1#2[ #1 #2 ]_q̅#1#2#3{ #1 #2, #3 }_q#1#2#3{[ #1; #2, #3 ]}_q -15mm 05mm Conformal partition functions of critical percolation from D_3 Thermodynamic Bethe Ansatz equations Markus J. Aschwanden^1 ========================================================================================================= Alexi Morin-Duchesne^∗, Andreas Klümper^†, Paul A. Pearce^ ^∗Institut de Recherche en Mathématique et Physique Université catholique de Louvain, Louvain-la-Neuve, B-1348, Belgium ^†Fachbereich C Physik, Bergische Universität Wuppertal, 42097 Wuppertal, Germany ^†School of Mathematics and Statistics, University of Melbourne Parkville, Victoria 3010, Australia alexi.morin-duchesne @ uclouvain.be kluemper @ uni-wuppertal.de papearce @unimelb.edu.au Using the planar Temperley-Lieb algebra, critical bond percolation on the square lattice can be reformulated as a loop model. In this form, it is incorporated as LM(2,3) in the Yang-Baxter integrable family of logarithmic minimal models LM(p,p'). We consider this model of percolation in the presence of boundaries and with periodic boundary conditions. Inspired by Kuniba, Sakai and Suzuki, we rewrite the recently obtained infinite Y-system of functional equations. In this way, we obtain nonlinear integral equations in the form of a closed finite set of TBA equations described by a D_3 Dynkin diagram. Following the methods of Klümper and Pearce, we solve the TBA equations for the conformal finite-size corrections. For the ground states of the standard modules on the strip, these agree with the known central charge c=0 and conformal weights Δ_1,s for s∈ℤ_≥ 1 with Δ_r,s=((3r-2s)^2-1)/24. For the periodic case, the finite-size corrections agree with the conformal weights Δ_0,s, Δ_1,s with s∈1/2ℤ_≥ 0. These are obtained analytically using Rogers dilogarithm identities. We incorporate all finite excitations by formulating empirical selection rules for the patterns of zeros of all the eigenvalues of the standard modules. We thus obtain the conformal partition functions on the cylinder and the modular invariant partition function (MIPF) on the torus. By applying q-binomial and q-Narayana identities, it is shown that our refined finitized characters on the strip agree with those of Pearce, Rasmussen and Zuber. For percolation on the torus, the MIPF is a non-diagonal sesquilinear form in affine u(1) characters given by the u(1) partition function Z_2,3(q)=Z_2,3^Circ(q). The u(1) operator content is N_Δ,Δ̅=1 for Δ=Δ̅=-1/24, 35/24 and N_Δ,Δ̅=2 for Δ=Δ̅=1/8, 1/3, 5/8 and (Δ,Δ̅)=(0,1), (1,0). This result is compatible with the general conjecture of Pearce and Rasmussen, namely Z_p,p'(q)=Z^Proj_p,p'(q)+n_p,p' Z^Min_p,p'(q) with n_p,p'∈ℤ, where the minimal partition function is Z^Min_2,3(q)=1 and the lattice derivation fixes n_2,3=-1. Keywords: percolation, solvable lattice models, conformal field theory. § INTRODUCTION In 1957, Broadbent and Hammersley <cit.> introduced a lattice percolation model as a mathematical model of the physical process of a fluid flowing through a random medium. Most importantly, they showed that their model exhibits a phase transition characterized by a critical probability threshold p=p_c. Comprehensive reviews of percolation theory can be found in <cit.>. One of the challenging goals in percolation is to understand the precise thermodynamic behavior of the model in the vicinity of the critical point. This behavior is believed to be conformally invariant and universal. Invariance under conformal maps implies invariance under translation, rotation and local scaling transformations. Universality implies that the critical behavior, characterized by critical exponents, depends on the lattice dimensionality but is otherwise insensitive to the details of the lattice model (for example the lattice structure or the choice of site versus bond percolation). In this paper, we view critical bond percolation on the two-dimensional square lattice as a Yang-Baxter integrable <cit.> loop model and solve exactly the associated D_3Y-system for the conformal spectra to establish how it fits into the framework of logarithmic conformal field theory <cit.>. In bond percolation on the square lattice, the bonds j of the lattice are open or occupied (σ_j=1) with a probability p and closed or empty (σ_j=0) with a probability 1-p. A typical bond configuration is shown in <ref>. In this description, the configuration space representing the local degrees of freedom is Ω={0,1}^ℤ^2. The “spins" σ_j are independent identically distributed random variables. As a consequence, the usual observables given by the correlations of these spins factorize and are trivial. Accordingly, the statistical weight W(σ) of a configuration σ is W(σ)=p^# bonds(1-p)^# empty bonds, Z=∑_σ W(σ)=1 and the partition function Z is trivial. In fact, the interesting physical behavior resides in the properties of connected clusters. The probability P(p) that the origin is part of an infinite connected cluster is called the percolation probability. For p<p_c, the open bonds are sparsely distributed at random throughout the lattice with no large clusters and P(p)=0. For p>p_c, the percolation probability is strictly positive: P(p)>0. The percolation probability is thus the order parameter for an order-disorder phase transition. For bond percolation on the square lattice, it has been proved that P(p_c)=0 <cit.> and that the critical threshold is precisely p_c=1/2 <cit.>. More generally, the interesting physical observables <cit.> include the probabilities P(j_1,j_2,…,j_n) that the bonds j_1,j_2,…,j_n all lie in the same connected cluster. The behavior of connected clusters is captured by introducing degrees of freedom in the form of planar non-crossing loop segments representing non-local connectivities. Mathematically, the local properties of loop segments are encoded in the planar Temperley-Lieb algebra <cit.>. In critical percolation, the loop segments can close to form loops with an assigned statistical weight or fugacity β=1. This description of percolation is sometimes referred to as hull percolation <cit.>. On the square lattice, there is a one-to-one mapping between bond configurations and loop configurations. This is illustrated by an example in <ref>. On rectangular lattices, each connected cluster is surrounded by loop segments, and crucially for crossing probabilities <cit.>, a connected cluster spans the lattice if and only if the surrounding loop segments also span the lattice. The critical point p=p_c of percolation marks a second order phase transition. From the viewpoint of statistical mechanics, the universality class of such a phase transition is characterized by critical exponents. The first few critical exponents considered for percolation are related to the number of clusters per site, the percolation probability, the truncated mean cluster size, the cluster volume and the correlation length respectively: α =2Δ_t-1/Δ_t-1=-2/3, β=Δ_h/1-Δ_t=5/36, γ=2Δ_h-1/Δ_t-1=43/18, δ =1-Δ_h/Δ_h=91/5, ν=1/2(1-Δ_t)=4/3. For example, as p→ p_c^+, the percolation probability behaves as P(p)∼ (p-p_c)^β. Only two of these exponents are independent. The others are related by scaling relations <cit.> to the thermal and magnetic conformal weights Δ_t=Δ_2,1=5/8, Δ_h=Δ_1/2,0=5/96. The values of these critical exponents were originally conjectured by den Nijs <cit.> and Nienhuis, Riedel and Schick <cit.> based on Coulomb gas arguments <cit.> by viewing percolation as the Q→ 1 limit of the critical Q-state Potts model. The Q-state Potts model with Q=1 is indeed trivial. It has a unique frozen state and the partition function is trivially Z=1, so a Q→ 1 limit is needed <cit.> to recover the critical exponents. In general, the critical line of the Coulomb gas maps onto the critical line of the six-vertex model and is parameterised by the crossing parameter λ∈ (0,π). It is related to the loop fugacity by β=√(Q)=2cosλ, with percolation corresponding to λ=π/3. The statistical behavior of percolation shares many commonalities with the model of critical dense polymers, which has a loop fugacity β=0. In polymers, the non-local degrees of freedom are extended segments of polymer chains which are not allowed to form closed loops. The study of polymers and percolation as Conformal Field Theories (CFTs) began with Saleur and Duplantier <cit.> in the mid-eighties. A CFT is a continuum theory that describes directly the universal properties of a critical statistical system (characterized by a linear system size N, a lattice spacing a and a continuum coordinate R) in the continuum scaling limit (N→∞, a→ 0, Na→ R). The conformal symmetry of percolation is described by a Virasoro algebra with central charge c=0. As a CFT, the Coulomb critical line is an su(2) affine Wess-Zumino-Witten CFT with effective central charge c_eff=1. The nature of a critical point on this line is very different if λ/π is rational compared to generic points where λ/π is irrational. If λ/π=p'-p/p' is rational, the theory admits a higher symmetry algebra described by the sl_2 loop algebra <cit.>. These points are characterized by two integers p,p' satisfying 1≤ p<p', (p,p')=1 and are dense along the critical line, with each point representing a different CFT. At each of these points, there are additional eigenvalue degeneracies and the theory is logarithmic. Percolation corresponds to the point with (p,p')=(2,3). To set it in context, the loop model of critical bond percolation is the (p,p')=(2,3) member of the family of logarithmic minimal models LM(p,p') <cit.> with conformal data consisting of the central charge c and Virasoro Kac conformal weights Δ_r,s c= 1 - 6(p'-p)^2/pp', Δ_r,s=(p' r-p s)^2-(p'-p)^2/4pp'. Since these loop models are defined in terms of the diagrammatic action of local operators on a vector space of link states, the logarithmic minimal models are intrinsically quantum in nature. The choice of the vector space of link states is an integral part of the definition of the model. But this space of states is not a Hilbert space since the inner product is not positive-definite. The infinitely extended Virasoro Kac table of conformal weights for percolation is shown in <ref> for r, s ∈ℤ_> 0. In this paper, we will encounter conformal weights Δ_r,s with r=0,1 and s ∈1/2ℤ_>0. Additional physical conformal weights are given by allowing r or s or both in this Kac formula to be half-integers <cit.> or even possibly to take values in ℚ<cit.>. The central charge and conformal weights, given by the Kac formula, vary continuously with the parameter λ∈ (0,π). In analogy to the rational minimal models M(m,m'), the logarithmic minimal models LM(p,p') are coset CFTs <cit.>. This analogy is the origin of the name but, in contradistinction to the unitary minimal models M(m'-1,m'), the logarithmic minimal models are all nonunitary. In particular, LM(2,3) is a nonunitary coset CFT with c=0, Δ_min=-1/24, c_eff=c-24Δ_min=1, Δ_r,s=(3r-2s)^2-1/24. Tellingly, since λ/π=13, critical percolation is a logarithmic CFT <cit.> and not a rational CFT <cit.>. The infinitely extended Virasoro Kac table of percolation in <ref> displays the conformal weights of an infinite number of Virasoro scaling operators. If a theory is rational, there can only be a finite number of conformal weights associated with a finite number of scaling operators and the associated Virasoro (or extended) representations must be irreducible and close among themselves under fusion. In contrast, logarithmic CFTs are characterized <cit.> by the existence of reducible yet indecomposable representations of the Virasoro algebra. On the strip, there is a single copy of the Virasoro algebra but, on a torus, there are two chiral copies of the Virasoro algebra and conformal invariance extends <cit.> to include invariance under the modular group. For simple rational CFTs, such as the c<1A-D-E models <cit.>, conformal and modular invariance together suffice to uniquely determine the conformal torus partition function. This is not the case for general logarithmic minimal models. Strikingly, conformality was only rigorously established in 2001 by Smirnov <cit.> for critical site percolation on the triangular lattice. This mathematical approach, which is built on random conformally invariant fractal curves, entails the identification of the models (<ref>) with _κ (Schramm-Loewner Evolution) with κ = 4p'/p. For percolation, with (p,p')=(2,3) and κ=6, the fractal dimensions d=2(1-Δ) of various fractal geometric curves are known <cit.> including those of chordal SLE paths, hulls (H), cluster mass (C), external perimeter (EP) and red bonds (RB): (Δ_path^SLE,Δ_H,Δ_C,Δ_EP,Δ_RB) =(Δ_p,p'±1,Δ_p,p'± 1,Δ_1/2(p±1),1/2p',Δ_p±1,p',Δ_p,p'±2) =(18,18,596,13,58), (d_path^SLE,d_H,d_C,d_EP,d_RB) =(74,74,9148,43,34). The value d_EP=4/3 was conjectured by Mandelbrot <cit.> and much later proved by Lawler, Schramm and Werner <cit.>. The value d_path^SLE=7/4 was proved by Beffara <cit.>. The incorporation of critical dense polymers LM(1,2) <cit.> and critical percolation LM(2,3) into the framework of the family of logarithmic minimal models LM(p,p') <cit.> establishes that these models are Yang-Baxter integrable. The transfer matrices of the logarithmic minimal models are built from so called transfer tangles of the planar Temperley-Lieb algebra <cit.>, which we respectively denote by (u) and (u) for the boundary and the periodic cases. The finite-size corrections to the eigenvalues D(u) and T(u) of the transfer matrices provide a direct way to access the central charge and conformal weights analytically. Indeed, for large horizontal system size N, the leading eigenvalues of the transfer matrices behave as -ln D(u) = 2N f_b(u) + f_s(u) + 2 πNsin(π uλ) (- c 24 + Δ + k ) + o(1N), -ln T(u) = N f_b(u) + 2 πN(sin(π uλ)(Δ + Δ̅+ k + k̅- c 12) + cos(π uλ) (Δ - Δ̅+ k - k̅) ) + o(1N), where f_b(u) and f_s(u) are the non-universal bulk and surface free energies, and k,k̅ are integers. Yang-Baxter integrability on the lattice means that f_b(u) and f_s(u) can be calculated exactly. In addition, by solving T- and Y-systems <cit.> satisfied by the commuting transfer matrices, one can calculate analytically the 1/N term to obtain universal quantities such as the central charge, conformal weights and conformal partition functions. The T-system takes <cit.> the form of a bilinear Hirota equation and is the master equation of integrability. Two key steps <cit.> in the process of solving the system are, first, to derive the Y-system from the T-system and, second, to use analyticity properties of the eigenvalues to convert the Y-system of functional equations into non-linear integral equations in the form of Thermodynamic Bethe Ansatz (TBA) equations <cit.>. These latter works on TBA focused on the ground state. The approach of Klümper and Pearce <cit.>, which we follow closely here, applies to all finite excitations and enables the analytic calculation of conformal partition functions. The T-system is non-universal but the Y-system, which relates to the conformal spectra, is universal <cit.> in the sense that it holds for all boundary conditions and topologies. So these calculations can be carried out with periodic boundary conditions or in the presence of boundaries on the strip <cit.>. This program has been carried to completion <cit.> for prototypical c<1A-type rational minimal models. Within the lattice approach, our longer term goal is to extend these calculations, based on functional equations and TBA, to the general logarithmic minimal models LM(p,p'). The T- and Y-systems for the general LM(p,p') models were obtained recently in <cit.>. These hierarchies of functional equations are infinite but the Y-system can be truncated to a finite D-type Y-system following the methods of <cit.>. In this paper, we start with critical percolation LM(2,3) as a prototypical example with p>1. This model admits a set of TBA equations encoded by a D_3≃ A_3 Dynkin diagram. Our specific goals are to calculate analytically, for critical percolation, the following quantities: (i) the central charge and conformal weights using dilogarithm identities; (ii) the finitized characters on the strip for half-arc boundary conditions and an arbitrary number of defects; (iii) the cylinder conformal partition functions with half-arc boundary conditions; (iv) the modular invariant partition function (MIPF) on the torus. This program has been completed <cit.> for critical dense polymers LM(1,2). In this case, the task was simplified because the transfer matrices satisfy a trivial Y-system in the form of an inversion identity similar to that of the (free-fermionic) Ising model <cit.>. The analysis of critical dense polymers introduced combinatorial constructs to enumerate patterns of zeros, namely single- and double-column diagrams and q-Narayana polynomials. Remarkably, these reappear in generalizing the calculations to critical percolation. Similarly, because of the occurrence of non-contractible loops and winding on the cylinder, a modified trace <cit.> (analogous to the Markov trace <cit.> on the strip) is needed to obtain the MIPF of critical percolation as was the case for critical dense polymers. We also stress that, as for critical dense polymers, the MIPF that we find for ℒℳ(2,3) is obtained from the scaling limit of the loop model on a torus of size M× N, with M and N even, where each non-contractible loop is weighted by a fugacity α = 2. [linewidth=.25pt](0,0)(1,1) [linewidth=1.5pt,linecolor=blue](1,0).590180[linewidth=1.5pt,linecolor=blue](0,1).5-900[linewidth=.25pt](0,0)(1,1) [linewidth=1.5pt,linecolor=blue](0,0).5090[linewidth=1.5pt,linecolor=blue](1,1).5180270[linewidth=.25pt](0,0)(1,1) [linewidth=1.5pt,linecolor=blue](1,0).590180[linewidth=1.5pt,linecolor=blue](0,1).5-900[linewidth=3pt,linecolor=blue](0,0)(1,1) [linewidth=.25pt](0,0)(1,1) [linewidth=1.5pt,linecolor=blue](1,0).590180[linewidth=1.5pt,linecolor=blue](0,1).5-900[linewidth=3pt,linecolor=red](0,0)(1,1) [linewidth=.25pt](0,0)(1,1) [linewidth=1.5pt,linecolor=blue](0,0).5090[linewidth=1.5pt,linecolor=blue](1,1).5180270[linewidth=3pt,linecolor=blue](1,0)(0,1) [linewidth=.25pt](0,0)(1,1) [linewidth=1.5pt,linecolor=blue](0,0).5090[linewidth=1.5pt,linecolor=blue](1,1).5180270[linewidth=3pt,linecolor=red](1,0)(0,1) #1-1.5[shift=0](0,0)(4,4) [linewidth=.25pt,fillstyle=solid,fillcolor=lightlightblue](0,0)(4,0)(4,4)(0,4) [linewidth=2pt,linecolor=blue](4,0)290180[linewidth=2pt,linecolor=blue](0,4)2-900(2,2)#1#1-1.5[shift=0](0,0)(4,4) [linewidth=.25pt,fillstyle=solid,fillcolor=lightlightblue](0,0)(4,0)(4,4)(0,4) [linewidth=2pt,linecolor=blue](0,0)2090[linewidth=2pt,linecolor=blue](4,4)2180270(2,2)#11mu1pt7pt.2mu 4pt.2mu7pt.1mu For logarithmic minimal models LM(p,p') with p>1, the MIPF is not uniquely determined by conformal and modular invariance. The conjectured form <cit.> for these MIPFs is Z_p,p'(q) =Z^Proj_p,p'(q)+n_p,p' Z^Min_p,p'(q), =12(1+n_p,p')Z_1,pp'^Circ(q)+12(1-n_p,p')Z_p,p'^Circ(q), n_p,p'∈ℤ, where the integer n_p,p' is undetermined and the projective partition function Z^Proj_p,p'(q) is defined in <cit.>. The u(1) modular invariant partition functions, corresponding to a compactified boson on S^1 with radius R=√(%s/%s)2p'p, are Z^Circ_p,p'(q)=∑_j=0^2n-1ϰ^n_j(q)ϰ^n_ω_0 j(q̅) where n=pp' and the u(1) characters ϰ_j^n(q) are given by (<ref>). The Bezout number ω_0 is defined by ω_0=r_0p'+s_0p () in terms of the Bezout pair (r_0,s_0) which is uniquely determined by the conditions r_0p'-s_0p=1, 1≤ r_0≤ p-1, 1≤ s_0≤ p'-1, p s_0<p'r_0. For p=1, Z_1,p'(q)=Z^Circ_1,p'(q) is the diagonal u(1) partition function Z^Circ_1,p'(q) = ∑_j=0^p' d^p'_j |ϰ_j^p'(q)|^2, d_j^n= {[ 1, j=0, n,; 2, ]. and Z^Min_p,p'(q)=12(Z^Circ_1,pp'(q)-Z^Circ_p,p'(q)) implies Z^Min_1,p'(q)=0. For p>1, Z^Circ_p,p'(q) is a non-diagonal u(1) partition function. For critical percolation, Z^Min_2,3(q)=1 and our analytic derivation of the MIPF from the lattice model shows that n_2,3 = -1. The MIPF of critical percolation is therefore given by the non-diagonal u(1) partition function Z_2,3(q)=Z^Circ_2,3(q) where the Bezout number giving the Bezout conjugation is ω_0=5. The layout of the paper is as follows. <ref> recalls the conformal data for bond percolation which is referred to in the rest of the paper. <ref> contains our computations and results for bond percolation on the strip with vacuum boundary conditions. We recall the definition of the Temperley-Lieb algebra _N and the transfer tangle (u) in <ref> and review the standard modules over this algebra in <ref>. In <ref>, we give the definition of the fused transfer matrices and present the fusion hierarchy relations. We write down the corresponding T- and Y-systems in <ref>. In <ref>, we analyse the analyticity properties of the eigenvalues of the transfer matrices in terms of their patterns of zeros. In <ref>, we transform the T- and Y-systems into TBA equations and solve for the finite-size corrections of the finite excitations characterized by their patterns of zeros. The results are expressed in terms of sums of Rogers dilogarithms which are evaluated in <ref>. In <ref>, we specialise the result to the ground states of the standard modules and reproduce the conformal weights in the (1,s) column of the Kac table. In <ref>, we review the construction of single- and double-column diagrams which were previously introduced in the analysis of critical dense polymers. In <ref>, we formulate a set of empirical selection rules which describe, in terms of column diagrams, the patterns of zeros for the full set of eigenvalues of the standard modules. We use these to write down explicit expressions for the finite-size characters. These are simplified to the known finitized Kac characters in <ref> using identities derived in <ref>. In <ref>, we combine the partition functions of the standard modules using the Markov trace to obtain the conformal cylinder partition function. In <ref>, we present our results for periodic boundary conditions. <ref> follow the same presentation as <ref>, presenting the corresponding results for the periodic case. In <ref>, we write down empirical selection rules that describe the full set of finite excitations in the standard modules over the periodic Temperley-Lieb algebra _N(α, β). These allow us to write down explicit expressions for the spectrum generating functions, which are collected in <ref>. Also in <ref>, the behavior of these generating functions in the scaling limit is extracted using the identities derived in <ref>. In <ref>, we combine the previous results using the equivalent of the Markov trace for the torus, compute the modular invariant/covariant partition functions and write the result in terms of u(1) characters. <ref> presents a discussion of our results and an overview of future avenues to be explored. The torus partition functions for critical dense polymers and sample patterns of zeros for critical percolation are collected in Appendices C and D respectively. § CONFORMAL DATA OF CRITICAL PERCOLATION For critical percolation, the central charge c and the Virasoro Kac conformal weights Δ_r,s are given by c = 0, Δ_r,s = (3r-2s)^2 -1/24, with r,s ∈ℤ_>0. These are organised in the infinitely extended Kac table in the left panel of <ref>. The conformal weights with r=0,1 and s taking half-integer are given in the right panel. In terms of the lattice data, the modular nome is given by q = exp(- 2πδsin(3u)) for the boundary case and by q = exp(- 2πδ ^-3 u) for the periodic case, where the aspect ratio is δ = lim_M,N→∞M/N. The finitized Kac characters are given by _r,s(q) = q^Δ_r,s-c/24(NN-s+r/2-q^rsNN-s-r/2) and yield the conformal Kac characters in the scaling limit: _r,s(q) = lim_N→∞_r,s(q) = q^Δ_r,s-c/24 (1-q^rs)/(q)_∞ where (q)_∞ = ∏_i=1^∞(1-q^i). The u(1) characters are given by ϰ_j^n(q) = ϰ_j^n(q,1), ϰ_j^n(q,z)=Θ_j,n(q,z)/q^1/24(q)_∞=q^-1/24/(q)_∞∑_k∈ℤ z^kq^(j+2kn)^2/4n, with n = pp' = 6 for percolation. § CRITICAL PERCOLATION WITH STRIP BOUNDARY CONDITIONS §.§ The transfer tangle and the Temperley-Lieb algebra The dense loop model of critical percolation is a Temperley-Lieb model described in terms of the elementary face operator unit=.9cm[shift=-.42](1,1) (0,0)(1,1)[linewidth=0.025]-(0,0)0.16090(.5,.5)u = s_1(-u) [shift=-.45](1,1) (0,0)(1,1)[bl](0,0) + s_0(u)[shift=-.42](1,1) (0,0)(1,1)[bl](0,0) where s_k(u)=sin (u+kλ)/sinλ. Here λ is the crossing parameter and is related to the loop fugacity β by β = 2 cosλ. For critical percolation, λ = π/3 and therefore β =1. The double-row transfer tangle is defined as (u)= (-1)^N unit=0.9[shift=-1.7](-0.5,-0.8)(5.5,2.0) (0,0)(5,2)[linewidth=0.025]-(0,0)0.16090[linewidth=0.025]-(0,1)0.16090[linewidth=0.025]-(1,0)0.16090[linewidth=0.025]-(1,1)0.16090[linewidth=0.025]-(4,0)0.16090[linewidth=0.025]-(4,1)0.16090(2.5,0.5)…(2.5,1.5)…(3.5,0.5)…(3.5,1.5)…[linewidth=1.5pt,linecolor=blue]-(0,1)0.590-90[linewidth=1.5pt,linecolor=blue]-(5,1)0.5-9090(0.5,.5)u(0.5,1.5)λ-u(1.5,.5)u(1.5,1.5)λ-u(4.5,.5)u(4.5,1.5)λ-u(2.5,-0.5)_N where u is the spectral parameter. The tangle (u) is a linear combination of connectivity diagrams and therefore an element of the Temperley-Lieb algebra <cit.>_N(β) at β =1: _N(β)=⟨ I, e_j ; j=1,…,N-1⟩, I= [shift=-0.55](0.0,-0.65)(2.0,0.45) [fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.35)(2.0,-0.35)(2.0,0.35)(0,0.35) (1.4,0.0)...[linecolor=blue,linewidth=1.5pt]-(0.2,0.35)(0.2,-0.35)(0.2,-0.55)_1[linecolor=blue,linewidth=1.5pt]-(0.6,0.35)(0.6,-0.35)(0.6,-0.55)_2[linecolor=blue,linewidth=1.5pt]-(1.0,0.35)(1.0,-0.35)(1.0,-0.55)_3[linecolor=blue,linewidth=1.5pt]-(1.8,0.35)(1.8,-0.35)(1.8,-0.55)_N , e_j= [shift=-0.55](0.0,-0.65)(3.2,0.45) [fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.35)(3.2,-0.35)(3.2,0.35)(0,0.35) (0.6,0.0)...(2.6,0.0)...[linecolor=blue,linewidth=1.5pt]-(0.2,0.35)(0.2,-0.35)(0.2,-0.55)_1[linecolor=blue,linewidth=1.5pt]-(1.0,0.35)(1.0,-0.35) [linecolor=blue,linewidth=1.5pt]-(2.2,0.35)(2.2,-0.35) [linecolor=blue,linewidth=1.5pt]-(3.0,0.35)(3.0,-0.35)(3.0,-0.55)_N[linecolor=blue,linewidth=1.5pt]-(1.6,0.35)0.21800(1.35,-0.55)_j[linecolor=blue,linewidth=1.5pt]-(1.6,-0.35)0.20180(1.85,-0.55)_j+1 . The algebra _N(β) is a unital associative algebra whose defining relations are e_j^2=β e_j, e_j e_j±1 e_j = e_j, e_i e_j = e_j e_i (|i-j|>1). The transfer tangle satisfies a number of relations, in particular the crossing symmetry (λ - u) = (u), the periodicity symmetry (u+π) = (u), the commutativity property [(u),(v)] = 0 and the initial condition (u=0) = (-1)^N. We sometimes denote the identity connectivity using the bold letter . The braid transfer matrix is also an element of _N(β). It is defined by _∞= unit=0.9[shift=-1.7](-0.5,-0.8)(5.5,2.0) (0,0)(5,2)(2.5,0.5)…(2.5,1.5)…(3.5,0.5)…(3.5,1.5)…[linewidth=1.5pt,linecolor=blue]-(0,1)0.590-90[linewidth=1.5pt,linecolor=blue]-(5,1)0.5-9090(0,0)[linewidth=1.5pt,linecolor=blue]-(0.0,0.5)(1,0.5) [linewidth=1.5pt,linecolor=blue]-(0.5,0)(0.5,0.35) [linewidth=1.5pt,linecolor=blue]-(0.5,0.65)(0.5,1)(1,0)[linewidth=1.5pt,linecolor=blue]-(0.0,0.5)(1,0.5) [linewidth=1.5pt,linecolor=blue]-(0.5,0)(0.5,0.35) [linewidth=1.5pt,linecolor=blue]-(0.5,0.65)(0.5,1)(4,0)[linewidth=1.5pt,linecolor=blue]-(0.0,0.5)(1,0.5) [linewidth=1.5pt,linecolor=blue]-(0.5,0)(0.5,0.35) [linewidth=1.5pt,linecolor=blue]-(0.5,0.65)(0.5,1)(0,1)[linewidth=1.5pt,linecolor=blue]-(0.5,0.0)(0.5,1) [linewidth=1.5pt,linecolor=blue]-(0,0.5)(0.35,0.5) [linewidth=1.5pt,linecolor=blue]-(0.65,0.5)(1,0.5)(1,1)[linewidth=1.5pt,linecolor=blue]-(0.5,0.0)(0.5,1) [linewidth=1.5pt,linecolor=blue]-(0,0.5)(0.35,0.5) [linewidth=1.5pt,linecolor=blue]-(0.65,0.5)(1,0.5)(4,1)[linewidth=1.5pt,linecolor=blue]-(0.5,0.0)(0.5,1) [linewidth=1.5pt,linecolor=blue]-(0,0.5)(0.35,0.5) [linewidth=1.5pt,linecolor=blue]-(0.65,0.5)(1,0.5)(2.5,-0.5)_N where the elementary braid operators are given by unit=.9cm[shift=-.4](1,1) (0,0)(1,1)[linewidth=1.5pt,linecolor=blue]-(0,0.5)(1,0.5) [linewidth=1.5pt,linecolor=blue]-(0.5,0)(0.5,0.35) [linewidth=1.5pt,linecolor=blue]-(0.5,0.65)(0.5,1) = e^-π-λ2 [shift=-.4](1,1) (0,0)(1,1)[bl](0,0) + e^π-λ2 [shift=-.4](1,1) (0,0)(1,1)[bl](0,0) , unit=.9cm[shift=-.4](1,1) (0,0)(1,1)[linewidth=1.5pt,linecolor=blue]-(0,0.5)(0.35,0.5) [linewidth=1.5pt,linecolor=blue]-(0.65,0.5)(1,0.5) [linewidth=1.5pt,linecolor=blue]-(0.5,0)(0.5,1) =e^π-λ2 [shift=-.4](1,1) (0,0)(1,1)[bl](0,0) + e^-π-λ2 [shift=-.4](1,1) (0,0)(1,1)[bl](0,0) . The braid transfer matrix is obtained as the u →∞ limit of (u): _∞ = lim_u →∞(^(π-λ)/s_0(u)^2)^N (-1)^N(u). We note that _∞ is also obtained by taking the limit u→ -∞ of (u). §.§ Standard modules The representation theory of the Temperley-Lieb algebra was investigated by Jones <cit.>, Martin <cit.>, Goodman and Wenzl <cit.> and Westbury <cit.> and was recently reviewed by Ridout and Saint-Aubin <cit.>. In the following, we study the action of (u) on a family of finite-dimensional modules over _N(β): the standard modules _N^d. These modules are constructed on the vector spaces generated from link states with d defects, with 0 ≤ d ≤ N and d ≡ N mod 2, and have dimension _N^d = NN-d/2-NN-d-2/2. For example, for N = 6 and d=2, there are nine link states: [ unit=0.8cm[shift=-0.0](-0.0,0)(2.4,0.5) -(0,0)(2.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.4,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.2,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.8,0)(1.8,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(2.2,0)(2.2,0.5) , [shift=-0.0](-0.0,0)(2.4,0.5) -(0,0)(2.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.4,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.6,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.0,0)(1.0,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(2.2,0)(2.2,0.5) , [shift=-0.0](-0.0,0)(2.4,0.5) -(0,0)(2.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.4,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(2.0,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.0,0)(1.0,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(1.4,0)(1.4,0.5) , [shift=-0.0](-0.0,0)(2.4,0.5) -(0,0)(2.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.8,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.6,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.2,0)(0.2,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(2.2,0)(2.2,0.5) , [shift=-0.0](-0.0,0)(2.4,0.5) -(0,0)(2.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.8,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(2.0,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.2,0)(0.2,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(1.4,0)(1.4,0.5) ,; unit=0.8cm[shift=-0.0](-0.0,0)(2.4,0.5) -(0,0)(2.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.2,0)(0.2,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(0.6,0)(0.6,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(1.2,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(2.0,0)0.20180 , [shift=-0.0](-0.0,0)(2.4,0.5) -(0,0)(2.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.8,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.2,0)(0.2,0.7)(1.4,0.7)(1.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(1.8,0)(1.8,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(2.2,0)(2.2,0.5) , [shift=-0.0](-0.0,0)(2.4,0.5) -(0,0)(2.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(1.2,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.6,0)(0.6,0.7)(1.8,0.7)(1.8,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.2,0)(0.2,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(2.2,0)(2.2,0.5) , [shift=-0.0](-0.0,0)(2.4,0.5) -(0,0)(2.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(1.6,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.0,0)(1.0,0.7)(2.2,0.7)(2.2,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.2,0)(0.2,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(0.6,0)(0.6,0.5) . ] The standard modules are defined by the defect-preserving action of the Temperley-Lieb connectivity diagrams on the link patterns. To compute this action, one draws the link state above the connectivity diagram and reads the new link state from the bottom nodes. A multiplicative factor of β is then added for each closed loop. The result is set to zero if the number of defects of the new link state is smaller than that of the original link state. Here are examples to illustrate: [shift=-0.55](0,-0.65)(1.6,0.95) [fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.35)(1.6,-0.35)(1.6,0.35)(0,0.35) [linecolor=blue,linewidth=1.5pt]-(0.2,0.35)(0.2,-0.35) [linecolor=blue,linewidth=1.5pt]-(0.6,0.35)(0.6,-0.35) [linecolor=blue,linewidth=1.5pt]-(1.2,0.35)0.21800[linecolor=blue,linewidth=1.5pt]-(1.2,-0.35)0.20180-(0,0.35)(1.6,0.35) [linecolor=darkgreen,linewidth=1.5pt]-(0.8,0.35)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.2,0.35)(0.2,1.05)(1.4,1.05)(1.4,0.35) = [shift=0.0](0,0.35)(1.6,0.95) -(0,0.35)(1.6,0.35) [linecolor=darkgreen,linewidth=1.5pt]-(0.4,0.35)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.2,0.35)0.20180 , [shift=-0.55](0,-0.65)(1.6,0.95) [fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.35)(1.6,-0.35)(1.6,0.35)(0,0.35) [linecolor=blue,linewidth=1.5pt]-(0.2,0.35)(0.2,-0.35) [linecolor=blue,linewidth=1.5pt]-(0.6,0.35)(0.6,-0.35) [linecolor=blue,linewidth=1.5pt]-(1.2,0.35)0.21800[linecolor=blue,linewidth=1.5pt]-(1.2,-0.35)0.20180-(0,0.35)(1.6,0.35) [linecolor=darkgreen,linewidth=1.5pt]-(1.2,0.35)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.2,0.35)(0.2,0.85) [linecolor=darkgreen,linewidth=1.5pt]-(0.6,0.35)(0.6,0.85) = β [shift=0.0](0,0.35)(1.6,0.95) -(0,0.35)(1.6,0.35) [linecolor=darkgreen,linewidth=1.5pt]-(1.2,0.35)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.2,0.35)(0.2,0.85) [linecolor=darkgreen,linewidth=1.5pt]-(0.6,0.35)(0.6,0.85) , [shift=-0.55](0,-0.65)(1.6,0.95) [fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.35)(1.6,-0.35)(1.6,0.35)(0,0.35) [linecolor=blue,linewidth=1.5pt]-(0.2,0.35)(0.2,-0.35) [linecolor=blue,linewidth=1.5pt]-(0.6,0.35)(0.6,-0.35) [linecolor=blue,linewidth=1.5pt]-(1.2,0.35)0.21800[linecolor=blue,linewidth=1.5pt]-(1.2,-0.35)0.20180-(0,0.35)(1.6,0.35) [linecolor=darkgreen,linewidth=1.5pt]-(0.4,0.35)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.0,0.35)(1.0,0.85) [linecolor=darkgreen,linewidth=1.5pt]-(1.4,0.35)(1.4,0.85) = 0. The standard modules play a key role in the representation theory of _N(β). In particular, they generate a complete set of irreducible modules for generic values of β. The case β = 1 is however not generic: The standard modules remain indecomposable, but depending on d some of them are reducible. Let us refer to the integers d ≡ 2 mod 3 in the set {0, …, N} as critical integers and to the maximal such integer as d̂. For d critical, the standard module _N^d is irreducible: _N^d ≃_N^d. For d ≡ 0,1 mod 3, _N^d typically has two composition factors: an irreducible submodule R_N^d and an irreducible quotient _N^d. The submodule R_N^d is isomorphic to _N^d' where d' = d+4 for d ≡ 0 mod 3 and d' = d+2 for d ≡ 1 mod 3. In other words, d' is the integer obtained by reflecting d with respect to the next critical integer. The structure of _N^d in this case is easily understood from its Loewy diagram, which we write as _N^d ≃_N^d →_N^d'. The arrow indicates that the states in _N^d' can be obtained from those in _N^d by the action of _N(β = 1), but not the other way around. If d'>d̂, then R_N^d is trivial and the corresponding standard module is irreducible: _N^d≃_N^d. The modules _N^d, with 0 ≤ d ≤ N and d ≡ N mod 2, form a complete list of non-isomorphic irreducible modules of _N(β =1). Their dimensions are given by _N^d = {[ _N^d d ≡ 2 mod 3,; ∑_k≥ 0_N^d+6k - ∑_k≥ 0_N^d'+6k d ≡ 0,1 mod 3, ]. where it is understood that _N^d = 0 for d>N. These dimensions are displayed in <ref> for 1≤ N ≤ 10. We note that _N^0 and _N^1 are always one-dimensional. §.§ Fused transfer matrices and the fusion hierarchy Starting from the transfer tangle ^1(u) = (u), one can construct a family of fused transfer tangles ^n(u) satisfying the fusion hierarchy relations ^n_0^1_n=s_n-3(2u)s_2n(2u)/s_n-2(2u)s_2n-1(2u) f_n ^n-1_0 + s_n-1(2u)s_2n-2(2u)/s_n-2(2u)s_2n-1(2u)f_n-1 ^n+1_0, n ≥ 0, where _k^n = ^n(u + k λ), _0^0 = f_-1, ^-1_k = 0, f_k = (-1)^Ns_k(u)^2N. These transfer tangles were constructed in terms of fused face operators in <cit.>. They commute as elements of _N(β): [^m(u),^n(v)] = 0. The fusion hierarchy relations were proven using the diagrammatic calculus of the Temperley-Lieb algebra. Of particular relevance for our investigation in later sections is the fused transfer tangle with fusion label n=2. For β 0, it is constructed from the Wenzl-Jones projector on two sites, [shift=-0.05](0,-0.15)(1.0,0.15) [fillstyle=solid,fillcolor=pink](0,-0.15)(1.0,-0.15)(1.0,0.15)(0,0.15)(0,0.15) (0.5,0)_2 = [shift=-0.25](-0.0,-0.35)(0.8,0.35) [linecolor=blue,linewidth=1.5pt]-(0.2,0.35)(0.2,-0.35) [linecolor=blue,linewidth=1.5pt]-(0.6,0.35)(0.6,-0.35) - 1/β[shift=-0.25](-0.,-0.35)(0.8,0.35) [linecolor=blue,linewidth=1.5pt]-(0.4,0.35)0.21800[linecolor=blue,linewidth=1.5pt]-(0.4,-0.35)0.20180 , and the 1 × 2 fused face operator unit=.9cm[shift=-0.4](0,0)(1,1) [fillstyle=solid,fillcolor=lightlightblue](0,0)(1,0)(1,1)(0,1)(0,0) [linewidth=0.025]-(0,0)0.16090(0.5,0.75)_1× 2(0.5,0.5)u = 1/s_0(u) [shift=-0.9](-0.3,0)(1.3,2) (0,0)(1,2)(0.5,0.55)u(0.5,1.55)u+λ[fillstyle=solid,fillcolor=pink](0,0.1)(0,1.9)(-0.3,1.9)(-0.3,0.1)(0,0.1)(-0.15,1)_2[fillstyle=solid,fillcolor=pink](1,0.1)(1,1.9)(1.3,1.9)(1.3,0.1)(1,0.1)(1.15,1)_2[linewidth=0.025]-(0,0)0.16090[linewidth=0.025]-(0,1)0.16090 = s_1(-u) [shift=-0.9](-0.3,0)(1.3,2) (0,0)(1,2)(0,0)(0,1)[fillstyle=solid,fillcolor=pink](0,0.1)(0,1.9)(-0.3,1.9)(-0.3,0.1)(0,0.1)(-0.15,1)_2[fillstyle=solid,fillcolor=pink](1,0.1)(1,1.9)(1.3,1.9)(1.3,0.1)(1,0.1)(1.15,1)_2 + s_1(u) [shift=-0.9](-0.3,0)(1.3,2) (0,0)(1,2)(0,0)(0,1)[fillstyle=solid,fillcolor=pink](0,0.1)(0,1.9)(-0.3,1.9)(-0.3,0.1)(0,0.1)(-0.15,1)_2[fillstyle=solid,fillcolor=pink](1,0.1)(1,1.9)(1.3,1.9)(1.3,0.1)(1,0.1)(1.15,1)_2 . The normalisation is different from that appearing in <cit.> and is instead chosen such that the fusion hierarchy relations (<ref>) are identical to those of the rational models, see for instance <cit.>. The 1× 2 fused transfer tangle is defined as unit=.9cm^2(u)= [shift=-1.6](-0.7,-0.7)(5.7,2) [linewidth=6pt,linecolor=blue]-(0,1)0.590-90[linewidth=4pt,linecolor=white]-(0,1)0.590-90[linewidth=6pt,linecolor=blue]-(5,1)0.5-9090[linewidth=4pt,linecolor=white]-(5,1)0.5-9090(0,0)(5,2)[linewidth=0.025]-(0,0)0.16090[linewidth=0.025]-(0,1)0.16090[linewidth=0.025]-(1,0)0.16090[linewidth=0.025]-(1,1)0.16090[linewidth=0.025]-(4,0)0.16090[linewidth=0.025]-(4,1)0.16090(0.5,0.75)_1× 2(0.5,1.75)_1× 2(1.5,0.75)_1× 2(1.5,1.75)_1× 2(4.5,0.75)_1× 2(4.5,1.75)_1× 2(2.5,0.5)…(2.5,1.5)…(3.5,0.5)…(3.5,1.5)…(0.5,.5)u(0.52,1.45)-u(1.5,.5)u(1.52,1.45)-u(4.5,.5)u(4.52,1.45)-u(2.5,-0.5)_N . It satisfies the relation ^2(-u)=^2(u), the periodicity property (u+π)=(u), the commutativity property [^2(u),^n(v)] = 0, n=1,2, and most importantly the fusion hierarchy relation (<ref>) with n=1. The construction of ^n(u) for n>2 in terms of diagrams uses similar ideas. For rational values of the crossing parameter, that is for λ =(p'-p)π/p' with (p,p') a pair of integers satisfying (<ref>), it was shown in <cit.> that the fused transfer tangles satisfy the closure relation ^p'_0 = ^p'-2_1 + 2(-1)^p'-p f_-1. Critical percolation corresponds to (p,p') = (2,3), in which case this is just ^3_0 = ^1_1 - 2 f_-1. The proof given in <cit.> relies on diagrammatic manipulations performed on the tangles ^n(u) as elements of _N(β). If follows that (<ref>) holds for all modules, and in particular for _N^d. §.§ The T-system and the Y-system Using the fusion hierarchy relations and a recursive argument <cit.>, one can show that the transfer tangles satisfy a set of functional relations known as the T-system: ^n_0^n_1 = s_-2(2u)s_2n(2u)/s_n-2(2u)s_n(2u) f_-1 f_n + s_n-1(2u)^2/s_n-2(2u)s_n(2u)^n+1_0^n-1_1, n ≥ 0. The T-system holds for generic values of β. By defining ^n(u) = s_n-1(2u)^2/s_-2(2u)s_2n(2u)_1^n-1_0^n+1/f_-1f_n, n ≥ 0, and ^n_k = ^n(u+k λ), one finds that these tangles satisfy a set of non-linear equations known as the (universal) Y-system: ^n_0 ^n_1=(+ ^n-1_1)(+ ^n+1_0), n ≥ 1, where ^0_k = 0. For rational values of λ, this set of non-linear relations closes finitely. Indeed, it was found in <cit.> that the tangles ^n_k satisfy a linear, four-term closure relation. After a careful analysis, we find that this closure relation is not convenient for extracting eigenvalue solutions for (u). Inspired by ideas applied to vertex models <cit.>, one instead defines the tangle _0 = (-1)^p'-p/f_-1_1^p'-2 and, using (<ref>) and (<ref>), finds the following alternative closure relations + ^p'-1_0 = (+_0)^2, _0_1 = +_1^p'-2. The closed Y-system thus consists of the relations (<ref>) for n = 1, …, p'-2 along with the relations (<ref>). This truncates the initial infinite Y-system, which corresponds to a one-sided A_∞ Dynkin diagram, to a finite Y-system described by a Dynkin diagram of type D_p', see <ref>. For (p,p')=(2,3), the Dynkin diagram is D_3 ≃ A_3 and the finite Y-system only involves three tangles: the tangles and ^1, which are scalar multiples of ^1(u) and ^2(u) respectively, and the identity . It takes the form _0^1_1^1 = (+_0)^2, _0_1 = + _1^1. By defining ^1(x) = _0^1( x 3), ^2(x) = _0( x 3- π 6), ^1(x) = +^1(x), ^2(x) = (+^2(x))^2, the Y-system is written in a symmetric form: ^1(x-π2)^1(x+π2) = ^2(x), ^2(x-π2)^2(x+π2) = ^1(x). §.§ Properties of the eigenvalues The functional relations given in the previous section were derived quite generally in <cit.> using the diagrammatic calculus of the Temperley-Lieb algebra. The eigenvalues of the corresponding tangles are solutions to these relations in any given representation. In <ref>, we use these relations to extract the finite-size corrections for the eigenvalues of (u) in the modules _N^d. The analysis is based on some properties of the eigenvalues of ^n(u), ^1(u) and (u) in _N^d, which we respectively denote by D^n(u), d^1(u) and K(u). This section details these properties. Razumov-Stroganov eigenvalues. A simple solution to (<ref>) is K_0 = -1 and d_0^1 = 0, which corresponds to D^1(u) = f_-2 = (-1)^N (sin (u+π/3)/sinπ/3)^2N, D^2(u) = 0. This solution appears once, as the ground state, in the spectrum of the standard module _N^d for d=0 if N is even, and for d=1 if N is odd. This is the celebrated Razumov-Stroganov eigenvalue for the loop model with strip boundary conditions <cit.>. It corresponds to the unique eigenvalue of the trivial rational model of percolation, or alternatively to the one-dimensional irreducible representations of _N(β = 1), _N^0 or _N^1, see <ref>. In this case, the expansion (<ref>) can be computed exactly and the 1/N finite-size corrections are exactly zero, consistent with c= Δ = 0. Patterns of zeros and analyticity strips. For the other eigenvalues, the exact solutions to the Y-system are unknown, but as shown in <ref>, it is possible to compute the finite-size corrections. This requires knowledge about the analytic behavior of the eigenvalues, which we formulate empirically based on exact computations for small system sizes. Our computer implementation produces the zeros and analyticity data of the eigenvalues in the standard modules up to N=12. We find that the leading eigenvalues have the following analyticity strips: D^1(u): -π/6< Re(u) < π/2, D^2(u): -π/3< Re(u) < π/3. Let us be more precise as to what this means. From the definitions (<ref>) and (<ref>), D^1(u) and D^2(u) are Laurent polynomials in the variable z = ^ u, with minimal and maximal power -2N and 2N. Their eigenvalues share this property and thus as functions of z have at most 4N zeros, and no poles except at z = 0. Due to the property ^n(u+π)=^n(u), in the complex u plane, there are at most 2N zeros in any vertical strip of width π and the pattern is repeated periodically. For ^1(u), the leading eigenvalues in each _N^d have a finite number of zeros inside the analyticity strip. The other zeros are located either on the edges of this strip, that is for Re(u) = -π/6 and π/2, or outside the analyticity strip at Re(u) =-π/3,2π/3. As N grows, the number of zeros of these leading eigenvalues inside the analyticity strip of ^1(u) remains unchanged, whereas the number of zeros on the edges increases. For ^2(u), the analyticity strip also contains finitely many zeros, and as N increases, the extra zeros accumulate outside the analyticity strip on the lines Re(u) = ±π/2. Two examples of patterns of zeros are given in <ref>. We observe that all the eigenvalues of D^2(u) share single real zeros at u = ±π/6,±π/3. These zeros can be understood from the fusion hierarchy equation (<ref>) for n=1. We also see, for instance in <ref>, that D^1(u) has a zero near u = π/2. Its location is however not exactly at u = π/2. Indeed, by specializing (<ref>) to n=1 and u = π/2, we find ^1(π2)^2 = (13)^2N. Using a diagrammatic argument, one can show that ^1(π 2) =(-13)^N. Thus, D^1(u) has a zero near but not directly at u = π/2. Its location in fact varies slightly for each eigenvalue. The other zeros of D^1(u) and D^2(u) are not common to all the eigenvalues and come in complex conjugate pairs. From this observation, we infer that these eigenvalues are real for Im(u) = 0. Empirically, we also find that pairs of complex zeros inside the analyticity strips all lie on the central vertical line, that is respectively at Re(u) = π/6 and Re(u) = 0 for D^1(u) and D^2(u). The degeneracy of these zeros is always one for D^1(u), but can be one or two for D^2(u). For instance, the eigenvalue whose zeros are shown in the right panel of <ref> has one pair of double zeros with Re(u) = 0. In all cases, the patterns of zeros are symmetric with respect to a reflection about the central vertical line of the analyticity strip. It follows that the eigenvalues are real on the central line of the analyticity strips. Because the zeros are symmetric under a reflection about the real axis, we depict an eigenvalue with a pattern diagram that includes only the zeros of the lower-half plane. In these diagrams, we omit the zeros on the real axis as they are common to all eigenvalues. For instance, the eigenvalues in <ref> are represented by the patterns [ [shift=-1.2](-0.4,-2.4)(1.4,0.5) -(0,-1.9)(0,0)(1,0)(1,-1.9) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-0.9) [dotsize=0.09cm](0.5,-1.2) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-0.9) [shift=-1.2](-0.4,-2.4)(1.4,0.5) -(0,-1.9)(0,0)(1,0)(1,-1.9) ] and [ [shift=-1.2](-0.4,-2.4)(1.4,0.5) -(0,-1.9)(0,0)(1,0)(1,-1.9) [dotsize=0.09cm](0,-0.3)(0,-0.9) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2)(0.5,-1.5) [dotsize=0.09cm](1,-0.3)(1,-0.9) [shift=-1.2](-0.4,-2.4)(1.4,0.5) -(0,-1.9)(0,0)(1,0)(1,-1.9) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.95) ] . Black and gray circles respectively denote single and double zeros, and the analyticity strip is delimited by the vertical segments. More examples are given in <ref>. The analyticity properties of K(u) and d^1(u) follow readily from those of D^1(u) and D^2(u). From the definition (<ref>), we see that K(u) has a pole of order 2N at u = π/3, whereas d^1(u) has poles of order 2N at u =-π/3, π/3, a zero of order 2N+2 at u=0 and neither poles nor zeros at u = ±π/6. The resulting analyticity strips K(u): -π/2< Re(u) < π/6, d^1(u): -π/3< Re(u) < π/3 have width 2 λ. A similar analysis on small system sizes reveals that the functions 1+K(u) and 1+d^1(u) are analytic and non-zero in the following strips of width λ: 1+K(u): -π/3< Re(u) < 0, 1+d^1(u): -π/6< Re(u) < π/6. In these cases, the analyticity strips are entirely free of zeros and poles. Finally, in terms of the variables defined in (<ref>), the analyticity strips take the following elegant forms: 𝔞^1(x): - π < Im(x) < π , 𝔞^2(x): -π < Im(x) < π, 𝔄^1(x): -π/2< Im(x) < π/2, 𝔄^2(x): -π/2< Im(x) < π/2. In terms of x, the patterns of zeros are rotated by 90 degrees, the central line of the analyticity strip coincides with the real axis and the pairs of complex zeros inside the analyticity strips lie on this axis. Braid limit. The braid limit ^n_∞ is obtained by multiplying ^n(u) by a suitable trigonometric function and taking the limit u→∞. For n=1, ^1_∞≡_∞ is defined in (<ref>). For n=2, ^2_∞ = lim_u →∞(-^2(π-λ)/s_1(u)^2)^N ^2(u) = (_∞^1)^2 - . The last equality is obtained by applying the braid limit to (<ref>) for n=1. Likewise, the braid tangles ^1_∞, _∞, ^1_∞ and ^2_∞ are defined as ^1_∞ = lim_u →±∞^1(u) = _∞^2, _∞ = lim_u →±∞(u) = - _∞, ^n_∞ = lim_x →±∞^n(x). For generic β, the matrix representatives of the braid transfer tangles on _N^d are scalar multiples of the identity matrix <cit.>, and the corresponding scalars depend only on d. On a given standard module _N^d, each eigenvalue of ^1(u) has the same braid limit, and likewise for (u). For β = 1, the braid behavior is d ≡ 0,1 mod 3: D^1_∞ = 1, D^2_∞ = 0, d ≡ 2 mod 3: D^1_∞ = -2, D^2_∞ = 3, which implies that d ≡ 0,1 mod 3: d^1_∞ = 𝔞^1_∞= 0, K_∞ = 𝔞^2_∞= -1, d ≡ 2 mod 3: d^1_∞ =𝔞^1_∞= 3, K_∞ = 𝔞^2_∞= 2. One can readily check that these satisfy (<ref>). The analysis of the finite-size corrections in <ref> in the case d ≡ 0,1 3 requires a refinement of (<ref>) to two subcases, characterised by the rate of convergence of 1+𝔞^2(x) to zero as x→±∞. From our numerical investigation, we find the following subcases:[We note that the Razumov-Stroganov eigenvalue does not fit in any of the two subcases. The analysis presented here holds for all the other eigenvalues.]Subcase A: lim_x →±∞^2|x|/3(1+ 𝔞^2(x)) = κ, Subcase B: lim_x →±∞^4|x|/3(1+ 𝔞^2(x)) = κ', where κ,κ' are non-zero real constants. We argue that κ,κ'>0. Indeed, as noted earlier, on the real x axis, the function 1+ 𝔞^2(x) is real but never zero. Our numerics reveal that there are always one or more values x=x^j where 𝔞^2(x) vanishes, implying that 1+𝔞^2(x^j) = 1. From the previous observations, 1+𝔞^2(x) is positive everywhere on the real x axis, so κ,κ' >0. Together, (<ref>) and (<ref>) determine the rate of convergence of 𝔞^1(x) to zero as x→±∞: Subcase A: lim_x →±∞^2|x|/3𝔞^1(x) = -κ, Subcase B: lim_x →±∞^4|x|/3𝔞^1(x) = κ'. For d≡ 0,1 3, we include the letter A or B at the bottom of the patterns of zeros to indicate which subcase the corresponding eigenvalue belongs to, see for example <ref>. Bulk behavior. The bulk limit of the eigenvalues is obtained by increasing the system size N while keeping the spectral parameter u finite and within the analyticity strip. Importantly, it comes into play in <ref> as the asymptotic behavior at x = -∞ of the scaling functions, defined in (<ref>). In this limit, d^1(u) and K(u) converge to constants: d^1_bulk(u) = 0, K_bulk(u) = -1. Indeed, for finite values of u near the origin, the behavior of d^1(u) is governed by the zero of order 2N+2 at u = 0, so the function is approximately zero for large N. In the bulk limit, (<ref>) becomes K_bulk(u)^2 = 1, so K_bulk(u) ∈{+1, -1}. For a given pattern of zeros, one can deduce the values of K_bulk(u) from the braid value K_∞ and the number of zeros in the central line of the analyticity strip. For instance, let us consider the eigenvalue of <ref>. The braid limit is K_∞ = 2. On the central line of the analyticity strip (namely Re(u) = π/6 for D^1(u), corresponding to Re(u) =-π/6 for K_0(u)), there are three zeros with Im(u)>0. We conclude that K(-π/6) is negative and that K_bulk(u) = -1. Applying the same logic to the pattern of zeros of <ref> would also yield K_bulk(u) = -1. The same reasoning can be repeated for each eigenvalue. Empirically, we find that every pattern of zeros has the same asymptotic behavior: K_bulk(u) = -1. §.§ Finite-size corrections The eigenvalues D(u) can be factored into a bulk, a surface and a finite-size correction as D(u) = D_b(u) D_s(u) D_f(u). From (<ref>), the bulk and surface contributions satisfy the inversion relations D_b(u)D_b(u+λ) = f_-1f_1, D_s(u)D_s(u+λ) = s_-2(2u)s_2(2u)/s_-1(2u)s_1(2u), whereas the finite-size correction satisfies D_f(u)D_f(u+λ) = 1+d^1(u). The solutions to (<ref>) give the bulk and surface free energies and were obtained in <cit.> for generic β. For β =1, the right-hand side of the inversion identity for D_s(u) equals 1 and the solution is D_s(u)=1. In this section, we derive the finite-size corrections of D_f(u) in the standard modules _N^d. We do so by using the methods developed in <cit.>. The solution holds for any finite excitation and works for all three cases, namely d ≡ 0,1 mod 3 (A), d ≡ 0,1 mod 3 (B) and d ≡ 2 mod 3. For convenience, we work with the functions 𝔞^1(x), 𝔞^2(x), 𝔄^1(x) and 𝔄^2(x) in terms of which the Y-system is symmetric. We also define 𝔟(x) = D_f( x 3+ π 6), so that (<ref>) becomes 𝔟(x - π2) 𝔟(x + π2) = 𝔄^1(x). We note that (<ref>) defines 𝔟(x) up to a sign. We choose this sign to be such that 𝔟_∞ = D^1_∞. TBA equations. Let us denote by w^1, w^2, …, w^t^1 the values of x where 𝔞^1(x)=0 is zero on the half-line Im(x)=0, Re(x)>0. The same pattern is repeated symmetrically on the negative part of the real axis: 𝔞^1(± w^i) = 0, i = 1, …, t^1. Likewise for 𝔞^2(x), we denote the positions of its zeros on the positive real x axis by x^1, x^2, …, x^t^2. The pattern is repeated symmetrically on the negative part of the real axis: 𝔞^2(± x^j) = 0, j = 1, …, t^2. For convenience, we label the zeros such that w^1≥ w^2≥…≥ w^t^1 and x^1> x^2 > …> x^t^2, recalling that the zeros of 𝔞^2(x) are all distinct whereas those of 𝔞^1(x) can be twofold degenerate. We introduce the finite-size correction functions ℓ^1(x) and ℓ^2(x) by writing 𝔞^1(x) = tanh^2N+2( x 2)·(∏_i = 1^t^1tanh(x-w^i2)tanh(x+w^i2) )·ℓ^1(x), 𝔞^2(x) = (∏_j = 1^t^2tanh(x-x^j2)tanh(x+x^j2) )·ℓ^2(x). The functions ℓ^1(x), ℓ^2(x), 𝔄^1(x) and 𝔄^2(x) are analytic and non-zero in their corresponding analyticity strips, and satisfy ℓ^1(x-π2)ℓ^1(x+π2) = 𝔄^2(x), ℓ^2(x-π2)ℓ^2(x+π2) = 𝔄^1(x). From (<ref>), ℓ^1(x), ℓ^2(x), 𝔄^1(x) and 𝔄^2(x) have constant asymptotics for d ≡ 2 mod 3. One can therefore define the Fourier transforms of their logarithmic derivatives. From (<ref>), for d ≡ 0,1 mod 3, ℓ^2(x), 𝔄^1(x) and 𝔄^2(x) also have non-zero asymptotics, but ℓ^1(x) does not. According to (<ref>), ℓ^1(x) behaves respectively as ^-2|x|/3 and ^-4|x|/3 as x→±∞ for the subcases A and B, so we instead define the Fourier transform of the second logarithmic derivative. To treat d ≡ 0,1,2 mod 3 simultaneously, we consider the second logarithmic derivative of ℓ^1(x) in all cases: L^1(k) = 1/2 π∫_-∞^∞ x (lnℓ^1(x))”^- k x, (lnℓ^1(x))” = ∫_-∞^∞ k L^1(k) ^ k x, L^2(k) = 1/2 π∫_-∞^∞ x (lnℓ^2(x))'^- k x, (lnℓ^2(x))' = ∫_-∞^∞ k L^2(k) ^ k x, A^n(k) = 1/2 π∫_-∞^∞ x (ln𝔄^n(x))'^- k x, (ln𝔄^n(x))' = ∫_-∞^∞ k A^n(k) ^ k x, n=1,2. Applying the Fourier transform to (<ref>), the reverse transform and then integrating with respect to x, we obtain the TBA equations: ln𝔞^1(x) = lntanh^2N+2( x 2) + ∑_i = 1^t^1ln(-tanh(x-w^i2)tanh(x+w^i2) ) + K ∗ln𝔄^2 + ϕ x + ϕ^1, ln𝔞^2(x) = ∑_j = 1^t^2ln(-tanh(x-x^j2)tanh(x+x^j2) ) + K ∗ln𝔄^1 + ϕ^2, where ϕ, ϕ^1 and ϕ^2 are the integration constants. The kernel K(x) is given by K(x) = 1/2πcosh x and the convolution of two functions is defined as (f ∗ g) (x) = ∫_-∞^∞ y f(x-y) g(y) = ∫_-∞^∞ y f(y) g(x-y). The constants ϕ, ϕ^1 and ϕ^2 are evaluated below from the braid limits. Scaling TBA equations. In the non-linear integral equations, the dependence on N appears only algebraically, in the function lntanh^2N+2(x/2). If x is of order ln N and N is large, this function has the following behavior: lim_N →∞tanh^2N+2(± 12(x + ln N)) = exp(-4 ^-x). To compute the finite-size corrections, we assume that the following scaling limits also exist: 𝖺^n(x) = lim_N →∞𝔞^n(± (x + ln N)), 𝖠^n(x) = lim_N →∞𝔄^n(± (x + ln N)), n = 1,2. Because the patterns of zeros are symmetric in the upper and lower parts of the complex u plane, the functions 𝔞^n(x) and 𝔄^n(x) are even in x. In (<ref>), the limits are therefore independent of the choice of the signs ±. Let us denote the zeros of 𝖺^1(x) and 𝖺^2(x) by z^i and y^j, namely z^i = w^i - ln N, y^j = x^j - ln N. In the scaling limit, the non-linear equations become ln𝖺^1(x) = -4 ^-x + ∑_i=1^t^1ln(-tanh (x-z^i2) ) + K ∗ln𝖠^2 + ϕ x + ϕ^1, ln𝖺^2(x) = ∑_j=1^t^2ln(-tanh (x-y^j2) )+ K ∗ln𝖠^1 + ϕ^2. Evaluation of the constants. For d ≡ 2 mod 3, we fix the branch cuts of the logarithms by fixing ln𝖺^1(x)ln 3, ln𝖺^2(x)ln 2, ln(-tanh(x-y2))π. With this choice, the constants ϕ^1 and ϕ^2 are evaluated using the braid limit. Indeed, using 1/2π∫_-∞^∞ x/cosh x = 1/2, we find K ∗ln𝖠^1 ln 2, K ∗ln𝖠^2 ln 3 and d ≡ 2 mod 3: ϕ = 0, ϕ^1 = -π t^1, ϕ^2 = -π t^2. For d ≡ 0,1 mod 3 (A), according to (<ref>), the large x behavior of 𝔞^1(x) and 𝔞^2(x) is given by 𝔞^1(x) -κ ^-2x/3, 1+𝔞^2(x) κ ^-2x/3, with κ>0. The branches of the logarithms are fixed using the convention ln𝖺^1(x) -2x/3 + lnκ + π, ln𝖺^2(x) π. The braid behavior of K ∗ln𝖠^2 can be evaluated explicitly: K ∗ln𝖠^2 1/2π∫_-∞^∞ y ln(κ^2 ^-4(x-y)/3)/cosh y = lnκ - 2x/3. We find that the constants are d ≡ 0,1 mod 3 (A): ϕ = 0, ϕ^1 = -π (t^1-1), ϕ^2 = -π(t^2-1). Finally, for d ≡ 0,1 mod 3 subcase B, according to (<ref>), the large x behavior of the eigenvalues is 𝔞^1(x) κ' ^-4x/3, 1+𝔞^2(x) κ' ^-4x/3, with κ'>0. The branches of the logarithms are fixed using the conventions ln𝔞^1(x) -4x/3 + lnκ', ln𝔞^2(x) π, the braid behavior of K ∗ln𝖠^2 is K ∗ln𝖠^2 1/2π∫_-∞^∞ y ln((κ')^2 ^-8(x-y)/3)/cosh y = lnκ' - 4x/3 and the integration constants are evaluated to d ≡ 0,1 mod 3 (B): ϕ = 0, ϕ^1 = -π t^1, ϕ^2 = -π(t^2-1). Finite-size corrections. To apply the Fourier transform and its inverse to (<ref>), one needs to remove the zeros of 𝔟(x) on the real x axis by dividing by ∏_j = 1^t^2tanh(x-x^j2)tanh(x+x^j2). Because 𝔟(x) has constant asymptotics, a single derivative of its logarithm is required. The result is ln𝔟(x) = ∑_j = 1^t^2ln(-tanh(x-x^j2)tanh(x+x^j2) ) + K ∗ln𝔄^1 + ψ where ψ is the integration constant. With the branch choices ln𝔟(x) {[ 0 d ≡ 0,1 mod 3,; ln 2 + π d ≡ 2 mod 3, ]. the constant ψ is evaluated using the braid limit and found to be -π t^2 and -π (t^2-1) for d ≡ 0,1 mod 3 and d ≡ 2 mod 3 respectively. Because ψ is independent of N, it does not contribute to the finite-size corrections. These are written in terms of the scaling functions as follows: ln𝔟(x) - ψ = ∑_j = 1^t^2ln(-tanh(x-x^j2)tanh(x+x^j2) ) + 1/2π∫_-ln N^∞ y ( ln𝔄^1(y+ln N)/cosh(x-y-ln N)+ln𝔄^1(-y-ln N)/cosh(x+y+ln N)) ≃ -2 cosh x/N(2 ∑_j=1^t^2^-y^j- 1/π∫_-∞^∞ y ^-yln𝖠^1(y)). Zeros of 𝔞^1(x) and 𝔞^2(x). The next step is to rewrite ^-y^j in terms of integrals involving the scaling functions. From (<ref>), we find that 𝔞^2(x^j) = 0 ⇒ 𝔞^1(x^j-π2) = -1 ⇒ 𝖺^1(y^j-π2) = -1. Taking the logarithm and using (<ref>), we find (2 k^j-1)π = ln𝖺^1 (y^j - π2) = 4 ^-y^j + ∑_i=1^t^1ln(-tanh12(y^j-z^i- π2 )) + 1/2π∫_-∞^∞ y ln𝖠^2(y)/sinh(y-y^j) + ϕ^1 where the k^j are integers. Isolating ^-y^j from this equation and replacing it in (<ref>) produces an expression for the finite-size corrections wherein the zeros of 𝖺^1(x) and 𝖺^2(x) appear in terms of the expression ln(-tanh12(y^j-z^i- π2 )). We can also write these in integral form: 𝔞^1(w^i) = 0 ⇒ 𝔞^2(w^i-π2) = -1 ⇒ 𝖺^2( z^i-π2) = -1 and (2 ℓ^i-1)π = ln𝖺^2 (z^i - π2) = ∑_j=1^t^2ln(-tanh12(z^i-y^j -π2)) + 1/2 π∫_-∞^∞ y ln𝖠^1(y)/sinh(y-z^i)+ ϕ^2 where the ℓ^i are integers. From our choice of branches for the logarithms, we have ln(-tanh12(z^i-y^j - π2) ) = - π - ln(-tanh12(y^j-z^i - π2) ) which we use to isolate the expression ln(-tanh12(y^j-z^i- π2 )) in (<ref>). We obtain ln𝔟(x) - ψ≃ -2 πcosh x/N( ∑_j=1^t^2(k^j-12- ϕ^12 π) + ∑_i=1^t^1(ℓ^i-12 + t^22- ϕ^22 π) - 1/π^2∫_-∞^∞ y ^-yln𝖠^1(y) - 1/4π^2∑_i=1^t^1∫_-∞^∞ y ln𝖠^1(y)/sinh(y-z^i) - 1/4π^2∑_j=1^t^2∫_-∞^∞ y ln𝖠^2(y)/sinh(y-y^j)). Dilogarithm technique. To evaluate (<ref>), we consider the integral 𝒥 = ∫_-∞^∞ y ((ln𝖺^1)' ln𝖠^1 -ln |𝖺^1| (ln𝖠^1)' ) + ∫_-∞^∞ y ((ln𝖺^2)' ln𝖠^2 -ln |𝖺^2| (ln𝖠^2)' ) where ln |𝖺^1| and ln |𝖺^2| are real for all x and are thus given by ln |𝖺^n|(x) = ln𝖺^n(x) + θ^n(x), n = 1,2. Here, the θ^n(x) are step functions defined for x ∈ℝ. Starting from x=+∞ and moving to the left on the x axis, the θ^n(x) decrease by π each time a zero of the corresponding type is crossed (z^i for θ^1(x) and y^j for θ^2(x)). The values at the right endpoints are θ^1(x) = 0 for x > z_1 and θ^2(x) = 0 for x > y_1, consistent with our choice of branches for the logarithms. The integral 𝒥 can be evaluated in two ways. For the first, one uses the non-linear integral equations (<ref>) and the symmetries of K(x) to obtain 𝒥 = 4∫_-∞^∞ y ^-y(ln𝖠^1 + (ln𝖠^1)') + ∫_-∞^∞ y ∑_i=1^t^1[ln(-tanh (y-z^i2)) ]' ln𝖠^1 - ∫_-∞^∞ y (∑_i=1^t^1ln(-tanh (y-z^i2)) + ϕ^1 - θ^1(y) ) (ln𝖠^1)' + ∫_-∞^∞ y ∑_j=1^t^2[ln(-tanh (y-y^j2)) ]' ln𝖠^2 - ∫_-∞^∞ y (∑_j=1^t^2ln(-tanh (y-y^j2)) + ϕ^2- θ^2(y) ) (ln𝖠^2)'. The integrals involving derivatives of ln𝖠^1 and ln𝖠^2 are transformed using integration by parts. For each one, it can be argued using the non-linear integral equations that the surface terms are zero. This yields 𝒥 = 8 ∫_-∞^∞ y ^-yln𝖠^1(y) + 2 ∑_i=1^t^1∫_-∞^∞ y ln𝖠^1(y)/sinh(y-z^i) + 2∑_j=1^t^2∫_-∞^∞ y ln𝖠^2(y)/sinh(y-y^j), which is precisely the combination of integrals needed to compute the finite-size corrections: ln𝔟(x) - ψ≃ -2 πcosh x/N( ∑_j=1^t^2(k^j-12- ϕ^12 π) + ∑_i=1^t^1(ℓ^i-12 + t^22- ϕ^22 π) - 𝒥/8 π^2). The second way of performing the integrals consists in changing the integration variable from y to 𝖺. For the integral involving 𝖺^1(x), we obtain ∫_-∞^∞ y (( ln𝖺^1)' ln𝖠^1 -ln |𝖺^1| (ln𝖠^1)' ) = ∫_-∞^∞ y 𝖺^1/ y ( ln(1+ 𝖺^1)/𝖺^1 - ln |𝖺^1|/1+ 𝖺^1) = [∫_-∞^z_t^1+∫_z_t^1^z_t^1-1+ … + ∫_z_2^z_1+ ∫_z_1^∞] y 𝖺^1/ y ( ln(1+ 𝖺^1)/𝖺^1 - ln |𝖺^1|/1+ 𝖺^1) = [∫_0^0+∫_0^0+ … + ∫_0^0+ ∫_0^𝖺^1(∞)] 𝖺 ( ln(1+ 𝖺)/𝖺 - ln |𝖺|/1+ 𝖺) = 2L_+(𝖺^1(∞)) = 2 L(𝖺^1(∞)1+𝖺^1(∞)) where the Rogers dilogarithm functions are given by L(x) =-1/2∫_0^x y (ln (1-y)/y + ln y/1-y), L_+(x) = 1/2∫_0^x y (ln (1+y)/y - ln y/1+y)=L(x/1+x). Recalling that 𝖠^2(x) = (1 + 𝖺^2(x))^2, the integral involving 𝖺^2(x) is computed with the same arguments: ∫_-∞^∞ y (( ln𝖺^2)' ln𝖠^2 -ln |𝖺^2| (ln𝖠^2)' ) = 2∫_-∞^∞ y 𝖺^2/ y ( ln(1+ 𝖺^2)/𝖺^2 - ln |𝖺^2|/1+ 𝖺^2) = 2[∫_-∞^y_t^2+∫_y_t^2^y_t^2-1+ … + ∫_y_2^y_1+ ∫_y_1^∞] y 𝖺^2/ y ( ln(1+ 𝖺^2)/𝖺^2 - ln |𝖺^2|/1+ 𝖺^2) = 2[∫_-1^0+∫_0^0+ … + ∫_0^0+ ∫_0^𝖺^2(∞)] 𝖺 ( ln(1+ 𝖺)/𝖺 - ln |𝖺|/1+ 𝖺) = 4L_+(𝖺^2(∞)) +4L(1) = 4 L(𝖺^2(∞)1+𝖺^2(∞)) + 4 L(1). Putting these results together, we find 𝒥 = 4 L(1) + 2 L(𝖺^1(∞)1+𝖺^1(∞)) + 4 L(𝖺^2(∞)1+𝖺^2(∞)) = {[ 𝒦_-1(2π3) = 0 d ≡ 0,1 mod 3,; 𝒦_-1(0) =4 π^2/3 d ≡ 2 mod 3, ]. where the integral 𝒦_-1(γ) is computed in <ref>. From (<ref>), the final result for the finite-size corrections is ln𝔟(x) - ψ≃ -2 πcosh x/N( ∑_j=1^t^2 k^j+∑_i=1^t^1ℓ^i + τ), τ = {[ - 12t^1t^2 d ≡ 0,1 mod 3 (A),; - 12(t^2+t^1t^2) d ≡ 0,1 mod 3 (B),; - 12(t^1+t^2+t^1t^2)-1/6 d ≡ 2 mod 3, ]. where we used the values of ϕ^1 and ϕ^2 given in (<ref>), (<ref>) and (<ref>). Comparing with (<ref>) specialised to c=0, we find that the conformal dimension of the corresponding conformal state is the content of the parenthesis, namely: Δ = ∑_j=1^t^2k^j+∑_i=1^t^1ℓ^i + τ. §.§ Solution for the ground states For the ground state eigenvalue in _N^d, our experimentations on small system sizes reveal that the corresponding pattern of zeros is characterised by t^1 = {[ d-3/3 d ≡ 0 mod 3,; d-4/3 d ≡ 1 mod 3,; d-2/3 d ≡ 2 mod 3, ]. t^2 = {[ 2d/3 d ≡ 0 mod 3,; 2d-2/3 d ≡ 1 mod 3,; 2d-1/3 d ≡ 2 mod 3, ]. and by N-d/2 (pairs of) zeros lying on its boundary lines for D^1(u), see <ref>. Each single zero of D^2(u) is joined by a pair of single zeros, sitting at the same height on the edges of the analyticity strip. For d ≡ 0 mod 3 and d ≡ 1 mod 3, we find empirically that the ground state eigenvalue respectively belongs to the subcase A and B. The integers k^j and ℓ^i are not fixed by the technique used in <ref>. They can instead be estimated from (<ref>) and (<ref>) using exact diagonalisation on small system sizes. For the ground state of _N^d, we find, again empirically, that the k^j and ℓ^i are given by k^j = {[ j-1 d ≡ 0 mod 3,; j d ≡ 1 mod 3,; j d ≡ 2 mod 3, ]. ℓ^i = i. <ref> features the patterns of zeros for the ground states for N=14,15.[Note that for N=14,15, our program can only produce the pattern diagrams for d ≥ 8. For d<8, the corresponding pattern diagrams given in <ref> were inferred from our understanding of the patterns for smaller system sizes.] For d=0 and d=1, the second analyticity strip is colored in gray, indicating that D^2(u)=0 for all u for the Razumov-Stroganov eigenvalue. The integers k^j and ℓ^i given there are those obtained using the method described above. To correctly compute the conformal weight from (<ref>), we recall that t^1 counts the zeros in the second analyticity strip, and t^2 counts the zeros in the first. This is because, up to prefactors and shifts in the arguments, 𝔞^1(x) corresponds to D^2(u) whereas 𝔞^2(x) corresponds to D^1(u). Carrying out the sum in (<ref>) with the data given above, we find in all three cases (d ≡ 0,1,2 3) that the conformal weight of the ground state of _N^d is Δ = d(d-1)/6 = Δ_1,d+1. §.§ Single- and double-column diagrams Our classification of the patterns of zeros for the excited states in <ref> uses the column diagrams introduced in <cit.>. Here we recall the definitions and some basic results in a self-consistent manner. Single-column diagrams. A single-column diagram in the set Mm is a vertical array of M sites of which m are occupied and the other M-m are unoccupied. We draw occupied and unoccupied sites in black and white respectively, as in the example of <ref>. The sites are assigned the height labels 1, …, M starting from the bottom. The signatureS = {S_1, …, S_m} of a single-column diagram in Mm is the list of the heights of its occupied sites in decreasing order. The energy of a single-column diagram is E = ∑_j=1^m S_j and its weight is q^E. An example is given in the left panel of <ref>. The generating function Mm is defined as the sum of the weights over the single-column diagrams in Mm. It is a polynomial in q and can be computed using the following recursive argument. By removing the lowest site of a configuration in Mm, we obtain a new configuration containing M-1 sites, with either m or m-1 occupied sites depending on whether the lowest site of the original configuration was occupied. Moreover, the labels of the new configuration range from 2 to M. The generating function then satisfies the recursion relation Mm = q^m (M-1m+M-1m-1). Here, the factor q^m corrects for the energy difference arising due to the relabelling of the sites. For the second term, it also includes a factor of q^1 for the energy contribution of the lowest site of the original single-column diagram. Along with the boundary conditions M0=1 and MM=q^1/2M(M+1), the recursion relation fixes Mm completely to Mm = q^1/2m(m+1)Mm where the Gaussian polynomial (or q-binomial) is defined as Mm = [M][M-1] ⋯ [M-m+1]/[m][m-1]⋯[1], [m] = 1-q^m/1-q. The factor q^1/2m(m+1) in (<ref>) is identified as the energy of the single-column diagram in Mm with all occupied sites at the bottom. We note that if the height labels of the single-column diagram are a, …, M+a-1 instead of 1, …, M, then the energies of all the eigenvalues are shifted by m(a-1) and the generating function is q^m(a-1)Mm = q^1/2m(m+2a-1)Mm. A single-column diagram is mapped to a pattern of zeros of D^1(u) using the following rule: (i) an occupied site at height j produces a zero of order one in the center of the analyticity strip; (ii) an unoccupied site produces two zeros, each of order one, lying on the edges of the analyticity strip. This map is illustrated in the left panel of <ref>. Double-column diagrams. We define Mmn to be the set of double-column diagrams that satisfy the condition of dominance. Such diagrams are made of two single-column configurations of M sites drawn side by side, with respectively m and n occupied sites in the left and right columns. An example is given in the right panel of <ref>. We respectively denote by L and R the signatures of the left and right column. The energy of a double-column diagram is E=∑_i=1^m L_i + ∑_j=1^n R_j and its weight is q^E. A double-column configuration in Mmn satisfies the condition of dominance if L_i ≤ R_i, j = 1, …, m. This of course presupposes that 0≤ m≤ n≤ M. The criterion (<ref>) can be translated in terms of a diagrammatic rule for the double-column diagram. One draws m non intersecting lines pairing the top m occupied sites of each column starting from the top. The double-column diagram satisfies dominance if the slope of each line is non-negative. The generating function for Mmn is denoted Mm,n. It is the sum of the weights of the diagrams in Mmn. By removing the lowest row of a given configuration in Mmn, we obtain a new configuration of height M-1 in which the occupation numbers are (m,n), (m-1,n), (m,n-1) or (m-1,n-1). Crucially, the resulting column configuration satisfies dominance in all cases. This is easy to see either from the definition (<ref>) or from the diagrammatic rule. As a result, the generating function satisfies the recursion relation Mm,n = q^m+n(M-1m,n+M-1m-1,n+M-1m,n-1+M-1m-1,n-1), where the prefactor q^m+n compensates for the relabelling of the height labels and the energy contribution of the lowest sites. With the conditions M0,0= 1, M0,M= q^1/2M(M+1), MM,0= 0, MM,M= q^1/2M(M+1)+1/2N(N+1), the recursion relation fixes the generating functions entirely. The result is Mm,n = q^1/2m(m+1)+1/2n(n+1)Mmn where Mmn are the generalised q-Narayana numbers Mmn =q^-M+n(MmM+1n+1-M+1mMn+1). The factor q^1/2m(m+1)+1/2n(n+1) in (<ref>) is the weight of the double-column diagram in Mmn with minimal energy. A double-column diagram is mapped to a pattern of zeros of D^2(u) using the following rule: (i) if both sites at height j are occupied, a zero of order two sits in the center of the analyticity strip; (ii) if both sites at height j are unoccupied, a pair of double zeros is inserted on the edges of the analyticity strip; (iii) if one site is occupied and the other is not, this yields three zeros of order one, one of which is inserted in the center of the analyticity strip whereas the two others are inserted on the edges. An example is given in the right panel of <ref>. §.§ Solution for all the eigenvalues In this section, we find closed expressions for the finitized spectrum generating functions , defined by = ∑_eigenstates of D^1(u) in _N^d q^Δ, where q is the modular nome. We note that the sum is over the finite set of eigenvalues in _N^d, characterized by their patterns of zeros, whereas Δ is the conformal weight of the corresponding pattern of zeros in the scaling limit. Selection rules for the patterns of zeros. Our computation of is based on a conjecture for the selection rules for the eigenvalues in _N^d which we now formulate. These empirical rules give the patterns of zeros and the values taken by the integers k^j and ℓ^i for each eigenvalue. Similar selection rules for the model of critical dense polymers on the strip were conjectured in <cit.> and later proven in <cit.>. The conjectured selection rules given below are supported by data produced with our computer implementation of the transfer matrices for N ≤ 12. For a given eigenvalue, our program outputs the corresponding patterns of zeros of D^1(u) and D^2(u). To illustrate, the data corresponding to all the eigenstates in _8^2 and _8^4 is given in <ref>. The selection rules for the patterns of zeros are described in terms of the single- and double-column diagrams discussed in <ref>. Let (σ,σ') be a pair of column diagrams[We use the letter M for both the number of sites of a column diagram and the vertical width of the lattice. It should be clear from the context which one is referred to, and likewise in <ref>.] with σ∈Mm and σ' ∈Lnℓ. We denote the set of such pairs by ÅMLmnℓ. An example is given in <ref>. For d ≡ 0,1 3, we indicate whether the patterns of zeros of the corresponding set belong to subcases A or B by writing MLmnℓ or MLmnℓ. For _8^2 and _8^4, <ref> reveal that the patterns of zeros of D^1(u) and D^2(u) are encoded by the following sets: _8^2 : Å40100 ∪ Å41300 ∪ Å51311 ∪ Å52511 ∪ Å62522 ∪ Å73733, _8^4 : 40200 ∪ 41400 ∪ 51411 ∪ 62622 ∪ 51401 ∪ 62612. In general, for d>1, we conjecture that the full set of patterns of zeros in _N^d is given by the following sets: d=3t : ⋃_i=0^N-d/2⋃_j=0^⌊1/2(N-d/2-i)⌋N+t/2 + ii+j+t-12(i+j+t)ii+t-1∪⋃_i=0^N-d-4/2⋃_j=0^⌊1/2(N-d-4/2-i)⌋N+t/2 + ii+j+t2(i+j+t)+2ii+t , d=3t+1 : ⋃_i=0^N-d/2⋃_j=0^⌊1/2(N-d/2-i)⌋N+t-1/2 + ii+j+t-12(i+j+t)ii+t-1∪⋃_i=0^N-d-2/2⋃_j=0^⌊1/2(N-d-2/2-i)⌋N+t+1/2 + ii+j+t2(i+j+t)+2ii+t , d=3t+2 : ⋃_i=0^N-d/2⋃_j=0^⌊1/2(N-d/2-i)⌋ÅN+t/2 + ii+j+t2(i+j+t)+1ii+t . For d = 0 and d=1, (<ref>) and (<ref>) are ill-defined because some indices are negative. In these cases, we instead have the following selection rules: d=0 : Å00000∪⋃_i=0^N-4/2⋃_j=0^⌊1/2(N-4/2-i)⌋N/2 + ii+j2(i+j)+2ii , d=1 : Å00000∪⋃_i=0^N-3/2⋃_j=0^⌊1/2(N-3/2-i)⌋N+1/2 + ii+j2(i+j)+2ii . where Å00000 is the set that contains a unique element: the pattern of zeros corresponding to the Razumov-Stroganov eigenvalue. Selection rules for the integers. As part of the conjectured selection rules, we also provide the prescription for the values taken by the numbers k^j and ℓ^i for each eigenvalue. For N≤12, we have obtained these integers for each pattern of zeros by evaluating (<ref>) and (<ref>) with the finite-size spectra produced by our computer implementation. The precision of the approximate values of the k^j and ℓ^i obtained in this way is remarkably good even for small system sizes: The error is less than 0.1 in almost all the cases. In <ref>, the values of these integers are given alongside the corresponding patterns of zeros. The prescription is as follows: for an element in ÅMLmnℓ, (i) the heights of the single-column diagram (corresponding to the k^j) are labelled from 0 to M-1 for d≡ 0,1 3 (A), and from 1 to M for d≡ 0,1 3 (B) and d≡ 2 3; (ii) the heights of the double-column diagram (corresponding to the ℓ^i) are labelled from 1 to L in all cases. Finitized spectrum generating functions. The conformal weight Δ corresponding to a given pattern of zeros is given in (<ref>). The selection rules and the prescription for the integers allow us to write explicit expressions for the finitized spectrum generating functions. These are obtained as sums of the generating functions of the sets of column diagrams given in (<ref>). To do so, we compute the minimal conformal weights Δ_min for ÅMLmnℓ, MLmnℓ and MLmnℓ in terms of the energies E, E^() and E^() of the corresponding minimal configurations. Using (<ref>) and the above prescription (i) and (ii), we find: d ≡ 0,1 3 (): Δ_min = E^()= 12(m^2+n^2+ℓ^2 - m + n+ℓ - mn-mℓ), d ≡ 0,1 3 (): Δ_min = E^()= 12(m^2+n^2+ℓ^2 + n+ℓ - mn-mℓ), d ≡ 2 3: Δ_min = E-16 = 12(m^2+n^2+ℓ^2- mn-mℓ)-16. The spectra generating functions are then obtained by summing q^Δ_minMmLnℓ over the sets ÅMLmnℓ given by (<ref>). In doing so, we note that the indices i and j run over all possible values for which the sets are well-defined. Equivalently, they run over all values such that the q-binomials in the generating functions have positive arguments, with the top one larger or equal to the bottom one. We therefore omit the indices of the sums over i and j, understanding these sums as running over ℤ, with only finitely many contributions. We obtain: d = 3t: = q^d(d-1)/6∑_i,j q^i^2+2j(i+j)+t(2i+3j)N+t/2+i2(i+j+t)i+j+t-1ii+t-1 + q^(d+3)(d+4)/6∑_i,j q^i(i+3)+2j(i+j+2)+t(2i+3j)N+t/2+i2(i+j+t)+2i+j+tii+t, d = 3t+1: = q^d(d-1)/6∑_i,j q^i(i+1)+2j(i+j+1/2)+t(2i+3j)N+t-1/2+i2(i+j+t)i+j+t-1ii+t-1 + q^(d+1)(d+2)/6∑_i,j q^i(i+2)+2j(i+j+3/2)+t(2i+3j)N+t+1/2+i2(i+j+t)+2i+j+tii+t, d = 3t+2 : = q^d(d-1)/6∑_i,j q^i(i+1)+2j(i+j+1)+t(2i+3j)N+t/2+i2(i+j+t)+1i+j+tii+t. For d=0 and d=1, (<ref>) and (<ref>) must be modified so that the first line is replaced by 1. In <ref>, we show using q-binomial identities that these complicated expressions for can be simplified to yield the finitized Kac characters, namely: =q^d(d-1)/6(NN-d/2-q^d+1NN-d-2/2) = _1,d+1(q). This holds for d≡ 0,1,2 mod 3. Scaling limit. The behavior of the finitized character in the scaling limit is easily extracted using NN-d/21/(q)_∞ which holds for fixed d. We recall that (q)_∞ is defined in (<ref>). This yields _1,d+1(q). As a final remark, we note that (<ref>) provides expressions for finitized characters of the irreducible Virasoro representations 𝖨_1,s. This is trivially true for d ≡ 2 mod 3 because _1,d+1≃𝖨_1,d+1 is already irreducible. For d ≡ 0 mod 3, the first and second lines of (<ref>) are respectively finitized characters for 𝖨_1,d+1 and 𝖨_1,d+5. For d ≡ 1 mod 3, the first and second lines of (<ref>) are respectively finitized characters for 𝖨_1,d+1 and 𝖨_1,d+3. §.§ Cylinder partition functions The partition function of the loop model on the cylinder of size 2M× N is obtained from the traces of (u)^M over _N^d: = ∑_0 ≤ d ≤ N d ≡ N mod 2 U_d(α2) (u)^M. This is the so-called Markov trace <cit.>, used to embed the Temperley-Lieb algebra on a cylinder. This was also used by Jacobsen and Richard <cit.> in the context of the Q-state Potts model. Here, α is the weight of the non-contractible loops and U_k(x) is the k-th Chebyshev polynomial of the second type. The conformal cylinder partition function is obtained by removing the non-universal energy contributions and by taking the scaling limit: ^2MN f_b(u)+M f_s(u)(q) where the ratio M/N is taken to converge to a real number δ, and the modular nome q is given in terms of the lattice data in (<ref>). The scaling limit is taken separately for the two parities of N. We denote by (q) and (q) the corresponding partition functions. We then have (q) = lim_N→∞∑_0 ≤ d ≤ N d ≡ N mod 2U_d(α2) = ∑_d ≥ 0 d ≡ N mod 2U_d(α2)_1,d+1(q). For α = 1, (<ref>) simplifies to (q)|_α = 1= (q)|_α = 1=1 which is the trivial cylinder partition function of the rational model of percolation. For α = 2, U_d(1) = d+1 and the partition functions can be written in terms of the u(1) characters and their derivatives. Writing ϰ_j(q,z) = ϰ^6_j(q,z) and ϰ_j(q) = ϰ^6_j(q), we find (q) + (q)|_α = 2 = 1/(q)_∞∑_d≥ 0(d+1) (q^Δ_1,d+1-q^Δ_1,d+4) = 3/(q)_∞∑_d ≥ 1q^Δ_1,d+1 = 3/2(q)_∞∑_d ∈ℤq^Δ_1,d+1 = 3/2(q)_∞∑_r=0^5 ∑_k∈ℤ q^Δ_1,6k+r+1 = 3(ϰ_1(q)+ ϰ_3(q)+ ϰ_5(q)) and (q) - (q)|_α = 2 = 1/(q)_∞∑_d≥ 0(-1)^d(d+1) (q^Δ_1,d+1-q^Δ_1,d+4) = 1/(q)_∞∑_d ≥ 1(-1)^d(2d-1)q^Δ_1,d+1 = 1/(q)_∞∑_d ∈ℤ(-1)^d d q^Δ_1,d+1 = 1/(q)_∞∑_r=0^5 (-1)^r∑_k∈ℤ(6k+r) q^Δ_1,6k+r+1 = -ϰ_1(q)+ 3ϰ_3(q)- 5ϰ_5(q) -12/ z(ϰ_1(q,z) - 3ϰ_3(q,z) + ϰ_5(q,z) )|_z=1 . The final result is (q)|_α = 2 = 2ϰ_1(q) + 4 ϰ_5(q) +6/ z(ϰ_1(q,z) - 3ϰ_3(q,z) + ϰ_5(q,z) )|_z=1 , (q)|_α = 2 = ϰ_1(q) + 3ϰ_3(q) - ϰ_5(q) -6/ z(ϰ_1(q,z) - 3ϰ_3(q,z) + ϰ_5(q,z) )|_z=1 . § CRITICAL PERCOLATION WITH PERIODIC BOUNDARY CONDITIONS §.§ The transfer tangle and the enlarged periodic Temperley-Lieb algebra The transfer tangle with periodic boundary conditions is defined as (u)= ^N unit=0.9[shift=-1.1](-0.2,-0.7)(5.2,1.0) (0,0)(5,1)[linewidth=0.025]-(0,0)0.16090[linewidth=0.025]-(1,0)0.16090[linewidth=0.025]-(4,0)0.16090[linewidth=1.5pt,linecolor=blue]-(0,0.5)(-0.2,0.5) [linewidth=1.5pt,linecolor=blue]-(5,0.5)(5.2,0.5) (2.5,0.5)…(3.5,0.5)…(0.5,.5)u(1.5,.5)u(4.5,.5)u(2.5,-0.5)_N , with the elementary face operator given in (<ref>). Loop models on periodic geometries are usually described by the so-called periodic Temperley-Lieb algebra. This algebra was studied by Levy <cit.> and Martin and Saleur <cit.> in the context of the Potts model. Its representation theory was also studied by Graham and Lehrer <cit.>, Green <cit.> and Erdmann and Green <cit.>. The terminology and precise definitions of this algebra tend to vary from author to author, and here we will work with the enlarged periodic Temperley-Lieb algebra _N(α, β), defined <cit.> as _N(α, β) = ⟨Ω, Ω^-1, e_j; j=1,…,N⟩. It is a unital algebra, with the identity element I obtained from the product of the generators Ω and Ω^-1. The connectivities I and e_j for 1≤ j≤ N-1 are given by the diagrams in (<ref>), but drawn on a rectangle with periodic boundary conditions in the horizontal direction. The other generators are depicted as e_N= [shift=-0.45](0,-0.55)(2.4,0.35) [fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.35)(2.4,-0.35)(2.4,0.35)(0,0.35) (0.2,-0.55)_1(0.6,-0.55)_2(1.0,-0.55)_3(2.2,-0.55)_N(1.4,0.0)...[linecolor=blue,linewidth=1.5pt]-(0.0,0.35)0.2-900[linecolor=blue,linewidth=1.5pt]-(0.0,-0.35)0.2090[linecolor=blue,linewidth=1.5pt]-(0.6,0.35)(0.6,-0.35) [linecolor=blue,linewidth=1.5pt]-(1.0,0.35)(1.0,-0.35) [linecolor=blue,linewidth=1.5pt]-(1.8,0.35)(1.8,-0.35) [linecolor=blue,linewidth=1.5pt]-(2.4,-0.35)0.290180[linecolor=blue,linewidth=1.5pt]-(2.4,0.35)0.2180-90[fillstyle=solid,linecolor=white,linewidth=0pt](-0.1,-0.4)(0,0.4) [fillstyle=solid,linecolor=white,linewidth=0pt](2.4,-0.4)(2.5,0.4) , [shift=-0.45](-0.7,-0.55)(2.0,0.35) (0.2,-0.55)_1(0.6,-0.55)_2(1.0,-0.55)_3(1.4,-0.55)...(1.8,-0.55)_N[fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.35)(2.0,-0.35)(2.0,0.35)(0,0.35) (0,0)(0.4,0)6[linecolor=blue,linewidth=1.5pt]-(-0.2,-0.35)(-0.2,-0.0)(0.2,0.0)(0.2,0.35)[fillstyle=solid,linecolor=white,linewidth=0pt](-0.3,-0.4)(0,0.4) [fillstyle=solid,linecolor=white,linewidth=0pt](2.0,-0.4)(2.4,0.4) (-0.55,0.042)Ω= , [shift=-0.45](-1.1,-0.55)(2.0,0.35) (0.2,-0.55)_1(0.6,-0.55)_2(1.0,-0.55)_3(1.4,-0.55)...(1.8,-0.55)_N[fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.35)(2.0,-0.35)(2.0,0.35)(0,0.35) (0,0)(0.4,0)6[linecolor=blue,linewidth=1.5pt]-(-0.2,0.35)(-0.2,-0.0)(0.2,0.0)(0.2,-0.35)[fillstyle=solid,linecolor=white,linewidth=0pt](-0.3,-0.4)(0,0.4) [fillstyle=solid,linecolor=white,linewidth=0pt](2.0,-0.4)(2.4,0.4) (-0.75,0.07)Ω^-1= . The defining relations of _N(α, β) include (<ref>), wherein the indices are in the set {1, …, N} and are taken modulo N, as well as ΩΩ^-1= Ω^-1Ω = I, Ω e_i Ω^-1 = e_i-1, Ω^N e_N = e_N Ω^N, (Ω^± 1 e_N)^N-1 = Ω^± N (Ω ^± 1 e_N). For N even, there are extra relations which remove the non-contractible loops in favor of weights α: E Ω^± 1 E = α E where E= e_2e_4… e_N-2e_N. We refer to this algebra as _N(α, β) for both parities of N, even if the parameter α does not come into play for N odd. The transfer tangle (u) is an element of _N(α, β). We are interested in the case λ=π/3 and therefore β = 1 corresponding to the model of critical percolation. The parameter α remains free until <ref>, whereafter the case α=2 is our main focus. As an element of _N(α,β), (u) satisfies a number of properties, namely: (i) the periodicity property (u+π) = (-1)^N(u), (ii) the commutativity property [(u),(v)] = 0, and (iii) the specialisations (u=0) = ^NΩ and (u=λ) = ^NΩ^-1. The braid transfer matrices are also elements of _N(α,β). They are defined as = unit=.9cm[shift=-1.1](-0.2,-0.7)(5.2,1.0) (-0,0)(5,1)[linewidth=1.5pt,linecolor=blue]-(-0.2,0.5)(2,0.5) [linewidth=1.5pt,linecolor=blue]-(0.5,0)(0.5,0.35) [linewidth=1.5pt,linecolor=blue]-(0.5,0.65)(0.5,1) [linewidth=1.5pt,linecolor=blue]-(1.5,0)(1.5,0.35) [linewidth=1.5pt,linecolor=blue]-(1.5,0.65)(1.5,1) (2.53,.51)…(3.53,.51)…[linewidth=1.5pt,linecolor=blue]-(4,0.5)(5.2,0.5) [linewidth=1.5pt,linecolor=blue]-(4.5,0)(4.5,0.35) [linewidth=1.5pt,linecolor=blue]-(4.5,0.65)(4.5,1) (2.5,-0.5)_N , = [shift=-1.1](-0.2,-0.7)(5.2,1.0) (-0,0)(5,1)(2.53,.51)…(3.53,.51)…[linewidth=1.5pt,linecolor=blue]-(0,0.5)(-0.2,0.5) [linewidth=1.5pt,linecolor=blue]-(5,0.5)(5.2,0.5) [linewidth=1.5pt,linecolor=blue]-(0.5,0)(0.5,1) [linewidth=1.5pt,linecolor=blue]-(1.5,0)(1.5,1) [linewidth=1.5pt,linecolor=blue]-(4.5,0)(4.5,1) [linewidth=1.5pt,linecolor=blue]-(0,0.5)(0.35,0.5) [linewidth=1.5pt,linecolor=blue]-(0.65,0.5)(1.35,0.5) [linewidth=1.5pt,linecolor=blue]-(1.65,0.5)(2,0.5) [linewidth=1.5pt,linecolor=blue]-(4,0.5)(4.35,0.5) [linewidth=1.5pt,linecolor=blue]-(4.65,0.5)(5,0.5) (2.5,-0.5)_N , with the braid face operators defined in (<ref>). The two braid transfer matrices are not equal in general. They are obtained as the u →±∞ limits of (u) as follows: = lim_u →∞(^(π-λ)/2/s_0(u))^N (-)^N(u), = lim_u → -∞(^(π-λ)/2/s_1(-u))^N (-)^N(u). §.§ Standard modules The standard modules _N^d over _N(α, β) are defined on the vector space generated by the span of periodic link states on N nodes with d defects, where N ≡ d mod 2. The construction is similar to the one for _N(β) in <ref>. One difference is that the link states are drawn on a line segment with the left and right ends identified, and the loop segments can connect via the back (virtual cut) of the cylinder. The dimensions of these vector spaces are _N^d = NN-d/2. For example, the link states that generate _5^1 are [ unit=0.8cm[shift=-0.0](-0.0,0)(2.0,0.5) -(0,0)(2.0,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.4,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.2,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.8,0)(1.8,0.5) , [shift=-0.0](-0.0,0)(2.0,0.5) -(0,0)(2.0,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.4,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.6,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.0,0)(1.0,0.5) , [shift=-0.0](-0.0,0)(2.0,0.5) -(0,0)(2.0,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.8,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.6,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.2,0)(0.2,0.5) , [shift=-0.0](-0.0,0)(2.0,0.5) -(0,0)(2.0,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.0,0)0.2090[linecolor=darkgreen,linewidth=1.5pt]-(2.0,0)0.290180[linecolor=darkgreen,linewidth=1.5pt]-(1.2,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.6,0)(0.6,0.5) , [shift=-0.0](-0.0,0)(2.0,0.5) -(0,0)(2.0,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.0,0)0.2090[linecolor=darkgreen,linewidth=1.5pt]-(2.0,0)0.290180[linecolor=darkgreen,linewidth=1.5pt]-(0.8,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.4,0)(1.4,0.5) ,; unit=0.8cm[shift=-0.0](-0.0,0)(2.0,0.5) -(0,0)(2.0,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.2,0)(0.2,0.7)(1.4,0.7)(1.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.8,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.8,0)(1.8,0.5) , [shift=-0.0](-0.0,0)(2.0,0.5) -(0,0)(2.0,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.6,0)(0.6,0.7)(1.8,0.7)(1.8,0) [linecolor=darkgreen,linewidth=1.5pt]-(1.2,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.2,0)(0.2,0.5) , [shift=-0.0](-0.0,0)(2.0,0.5) -(0,0)(2.0,0) [linecolor=darkgreen,linewidth=1.5pt]-(1.0,0)(1.0,0.7)(2.2,0.7)(2.2,0) [linecolor=darkgreen,linewidth=1.5pt]-(-0.2,0.54)(-0.12,0.43)(0.2,0.4)(0.2,0) [linecolor=darkgreen,linewidth=1.5pt]-(1.6,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.6,0)(0.6,0.5) [fillstyle=solid,linecolor=white,linewidth=0pt](2.0,-0.1)(2.4,0.9) [fillstyle=solid,linecolor=white,linewidth=0pt](0.0,-0.1)(-0.4,0.9) , [shift=-0.0](-0.0,0)(2.0,0.5) -(0,0)(2.0,0) [linecolor=darkgreen,linewidth=1.5pt]-(-0.12,0.53)(-0.07,0.545)(0.6,0.545)(0.6,0) [linecolor=darkgreen,linewidth=1.5pt]-(2.12,0.53)(2.07,0.545)(1.4,0.545)(1.4,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.0,0)0.2090[linecolor=darkgreen,linewidth=1.5pt]-(2.0,0)0.290180[linecolor=darkgreen,linewidth=1.5pt]-(1.0,0)(1.0,0.5) [fillstyle=solid,linecolor=white,linewidth=0pt](2.0,-0.1)(2.4,0.9) [fillstyle=solid,linecolor=white,linewidth=0pt](0.0,-0.1)(-0.4,0.9) , [shift=-0.0](-0.0,0)(2.0,0.5) -(0,0)(2.0,0) [linecolor=darkgreen,linewidth=1.5pt]-(2.2,0.54)(2.12,0.43)(1.8,0.4)(1.8,0) [linecolor=darkgreen,linewidth=1.5pt]-(-0.2,0)(-0.2,0.7)(1.0,0.7)(1.0,0) [linecolor=darkgreen,linewidth=1.5pt]-(0.4,0)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.4,0)(1.4,0.5) [fillstyle=solid,linecolor=white,linewidth=0pt](2.0,-0.1)(2.4,0.9) [fillstyle=solid,linecolor=white,linewidth=0pt](0.0,-0.1)(-0.4,0.9) . ] The standard action of _N(α, β) on _N^d is defect-preserving. For d=0, the resulting representations depend on β and α. For d>0, they depend on β and a twist parameterω. To compute the action of a connectivity on a link state, one draws the link state above the connectivity and reads the new link state from the bottom nodes. If some defects have annihilated pairwise, the result is set to zero. Otherwise, a multiplicative factor of β is inserted for each contractible loop. For d=0, a multiplicative factor of α is also inserted for each non-contractible loop. For d>0, a multiplicative factor of ω or ω^-1 is inserted each time a defect crosses the virtual cut of the cylinder: ω if the defect travels to the left and ω^-1 if it travels to the right. Here are examples of the standard action of _N(α, β): [shift=-0.55](0,-0.65)(1.6,1.05) [fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.5)(1.6,-0.5)(1.6,0.5)(0,0.5) [linecolor=blue,linewidth=1.5pt]-(1.2,-0.5)0.20180[linecolor=blue,linewidth=1.5pt]-(0.4,0.5)0.21800[linecolor=blue,linewidth=1.5pt]-(0.2,-0.5)(0.2,0)(1.0,0)(1.0,0.5) [linecolor=blue,linewidth=1.5pt]-(0.6,-0.5)(0.6,0)(1.4,0)(1.4,0.5) -(0,0.5)(1.6,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(0,0.5)0.2090[linecolor=darkgreen,linewidth=1.5pt]-(0.6,0.5)(0.6,1.2)(1.8,1.2)(1.8,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(1.2,0.5)0.20180[fillstyle=solid,linecolor=white,linewidth=0pt](-0.4,-0.5)(0,1.2) [fillstyle=solid,linecolor=white,linewidth=0pt](1.6,-0.5)(2.0,1.2) = α [shift=0.0](0,0.35)(1.6,0.95) -(0,0.35)(1.6,0.35) [linecolor=darkgreen,linewidth=1.5pt]-(0.4,0.35)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(1.2,0.35)0.20180 , [shift=-0.55](0,-0.65)(1.6,1.05) [fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.5)(1.6,-0.5)(1.6,0.5)(0,0.5) [linecolor=blue,linewidth=1.5pt]-(0.8,0.5)0.21800[linecolor=blue,linewidth=1.5pt]-(1.6,0.5)0.2180270[linecolor=blue,linewidth=1.5pt]-(0,0.5)0.22700[linecolor=blue,linewidth=1.5pt]-(0.8,-0.5)0.20180[linecolor=blue,linewidth=1.5pt]-(0.2,-0.5)(0.2,0.2)(1.4,0.2)(1.4,-0.5) [fillstyle=solid,linecolor=white,linewidth=0pt](-0.4,-0.5)(0,0.5) [fillstyle=solid,linecolor=white,linewidth=0pt](1.6,-0.5)(2.0,0.5) -(0,0.5)(1.6,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(0.2,0.5)(0.2,1.0) [linecolor=darkgreen,linewidth=1.5pt]-(0.6,0.5)(0.6,1.0) [linecolor=darkgreen,linewidth=1.5pt]-(1.2,0.5)0.20180 = 0 , [shift=-0.55](0,-0.65)(1.6,1.05) [fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.5)(1.6,-0.5)(1.6,0.5)(0,0.5) [linecolor=blue,linewidth=1.5pt]-(0.8,-0.5)0.20180[linecolor=blue,linewidth=1.5pt]-(0,0.5)0.22700[linecolor=blue,linewidth=1.5pt]-(1.6,0.5)0.2180270[linecolor=blue,linewidth=1.5pt]-(0.2,-0.5)(0.2,0)(1.0,0)(1.0,0.5) [linecolor=blue,linewidth=1.5pt]-(-0.2,-0.5)(-0.2,0)(0.6,0)(0.6,0.5) [linecolor=blue,linewidth=1.5pt]-(1.4,-0.5)(1.4,0)(2.2,0)(2.2,0.5) -(0,0.5)(1.6,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(0.6,0.5)(0.6,1) [linecolor=darkgreen,linewidth=1.5pt]-(1.0,0.5)(1.0,1) [linecolor=darkgreen,linewidth=1.5pt]-(0,0.5)0.2090[linecolor=darkgreen,linewidth=1.5pt]-(1.6,0.5)0.290180[fillstyle=solid,linecolor=white,linewidth=0pt](-0.4,-0.5)(0,1.2) [fillstyle=solid,linecolor=white,linewidth=0pt](1.6,-0.5)(2.25,1.2) = ωβ [shift=0.0](0,0.35)(1.6,0.95) -(0,0.35)(1.6,0.35) [linecolor=darkgreen,linewidth=1.5pt]-(0.8,0.35)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.2,0.35)(0.2,0.85) [linecolor=darkgreen,linewidth=1.5pt]-(1.4,0.35)(1.4,0.85) , [shift=-0.55](0,-0.65)(1.6,1.05) [fillstyle=solid,fillcolor=lightlightblue,linewidth=0pt](0,-0.5)(1.6,-0.5)(1.6,0.5)(0,0.5) [linecolor=blue,linewidth=1.5pt]-(0.6,-0.5)(0.6,0)(-0.2,0)(-0.2,0.5) [linecolor=blue,linewidth=1.5pt]-(1.0,-0.5)(1.0,0)(0.2,0)(0.2,0.5) [linecolor=blue,linewidth=1.5pt]-(1.4,-0.5)(1.4,0)(0.6,0)(0.6,0.5) [linecolor=blue,linewidth=1.5pt]-(1.8,-0.5)(1.8,0)(1,0)(1,0.5) [linecolor=blue,linewidth=1.5pt]-(2.2,-0.5)(2.2,0)(1.4,0)(1.4,0.5) [linecolor=blue,linewidth=1.5pt]-(0.2,-0.5)(0.2,0)(-0.6,0)(-0.6,0.5) -(0,0.5)(1.6,0.5) [linecolor=darkgreen,linewidth=1.5pt]-(1.4,0.5)(1.4,1) [linecolor=darkgreen,linewidth=1.5pt]-(1.0,0.5)(1.0,1) [linecolor=darkgreen,linewidth=1.5pt]-(0.4,0.5)0.20180[fillstyle=solid,linecolor=white,linewidth=0pt](-0.65,-0.5)(0,1.2) [fillstyle=solid,linecolor=white,linewidth=0pt](1.6,-0.5)(2.25,1.2) = ω^-2 [shift=0.0](0,0.35)(1.6,0.95) -(0,0.35)(1.6,0.35) [linecolor=darkgreen,linewidth=1.5pt]-(1.2,0.35)0.20180[linecolor=darkgreen,linewidth=1.5pt]-(0.6,0.35)(0.6,0.85) [linecolor=darkgreen,linewidth=1.5pt]-(0.2,0.35)(0.2,0.85) . For generic values of β, α and ω, the modules _N^d are irreducible modules over _N(α,β). For percolation, the indecomposable structures of _N^0 with α=1 and _N^d>0 with ω^N=1 are discussed in <cit.>. §.§ Fused transfer matrices and the fusion hierarchy Using the Wenzl-Jones projectors for generic values of β, one can define a sequence of commuting fused transfer tangles ^n(u), starting with ^1(u) = (u), which satisfy the fusion hierarchy relations ^n_0^1_n = h_n ^n-1_0 + h_n-1^n+1_0, n ≥ 0, where _k^n = ^n(u + k λ), _0^0 = h_-1, ^-1_k = 0, h_k = s_k(u)^N. These tangles indeed commute as elements of _N(α,β): [^m(u),^n(v)] = 0. In particular, for β 0, the fused transfer tangle ^2(u) is constructed from the 1×2 fused face operator, see (<ref>), as follows: ^2 (u)= (-1)^N unit=0.9[shift=-1.1](-0.2,-0.7)(5.2,1.0) (0,0)(5,1)[linewidth=0.025]-(0,0)0.16090[linewidth=0.025]-(1,0)0.16090[linewidth=0.025]-(4,0)0.16090[linewidth=1.5pt,linecolor=blue]-(0,0.5)(-0.2,0.5) [linewidth=1.5pt,linecolor=blue]-(5,0.5)(5.2,0.5) (2.5,0.5)…(3.5,0.5)…(0.5,.5)u(0.5,0.75)_1× 2(1.5,.5)u(1.5,0.75)_1× 2(4.5,.5)u(4.5,0.75)_1× 2(2.5,-0.5)_N . For the rational values λ = π(p'-p)/p', see (<ref>), the hierarchy of fused transfer matrices closes <cit.> at n=p': ^p'_0 = ^p'-2_1 + 2(-)^N(p'-p) h_-1, = T_p'(12 ), where T_k(x) in the definition of is the k-th Chebyshev polynomial of the first kind. For percolation, this becomes ^3_0 = ^1_1 + 2(-)^N h_-1, = T_3(12 ) = 12 ()^3-32. We note that the closure relation could equivalently be written in terms of the other braid transfer matrix _-∞. §.§ The T-system and the Y-system The T-system for the periodic transfer matrices is found using the fusion hierarchy relations (<ref>) and a recursive argument. It takes the form ^n_0 ^n_1 = h_-1 h_n +^n+1_0 ^n-1_1, n≥ 0. By defining ^n(u) =_1^n-1_0^n+1/h_-1h_n, n ≥ 0, and ^n_k = ^n(u+k λ), we find that the tangles ^n(u) satisfy precisely the same Y-system as on the strip, namely: ^n_0 ^n_1=(+ ^n-1_1)(+ ^n+1_0), n ≥ 1. For rational values of λ, this Y-system can be written in terms of finitely many objects. Indeed, by defining (u) and as follows: (u) = ^N(p'-p)/h_-1_1^p'-2, = cos, we find the closure relations: + ^p'-1_0 = (+ ^Λ_0)(+ ^-Λ_0), _0_1 = 1+_1^p'-2. The finite Y-system consists of the relations (<ref>) for n = 1, …, p'-2 and (<ref>). It is described by the Dynkin diagram D_p'-2 in <ref>, with the new feature that the endpoint nodes to the right are distinguished by the factors ^±Λ. For percolation, the Y-system involves only the tangles ^1(u), (u), and and is written as _0^1_1^1 = (+ ^Λ_0)(+ ^-Λ_0), _0_1 = + _1^1. By defining ^1(x) = _0^1( x 3), ^2(x) = (-1)^N_0( x 3- π 6), ^1(x) = +^1(x), ^2(x) = (+(-1)^N^Λ^2(x))(+(-1)^N^-Λ^2(x)), the Y-system takes a symmetric form: ^1(x-π2)^1(x+π2) = ^2(x), ^2(x-π2)^2(x+π2) = ^1(x). §.§ Properties of the eigenvalues Y-system for the eigenvalues. The tangles _±∞ are not proportional to the identity tangle, but nevertheless act as multiples of the identity on the standard modules. For α=2 and ω =1, the (unique) eigenvalues of _+∞ and _-∞ on _N^d are identical and were computed in <cit.> for generic values of λ. The generalisation to arbitrary α and ω is straightforward. For λ=π/3, we obtain T_±∞ = 2 cos(±γ - π d3), J = (-1)^d cos (3 γ), ^Λ = (-1)^d ^3 γ, where we parameterise α and ω in terms of a single parameter γ as follows: α = 2 cosγ, ω = ^γ. The Y-system satisfied by the eigenvalues in _N^d is then given by (<ref>) with 𝔄^2(x) = (1+^3 γ𝔞^2(x))(1+^-3 γ𝔞^2(x)). The cancellation of the factors of (-1)^d and (-1)^N here explains the choice made in (<ref>) to include an extra (-1)^N in the definition of ^2(x). Razumov-Stroganov eigenvalues. For periodic boundary conditions, the Razumov-Stroganov eigenstate appears in two situations: (i) for d=0 with γ = ±π/3,±2π/3, and (ii) for d=1 with γ = 0, π. This is consistent with the spin-chain results of <cit.>. Indeed, a simple solution to the Y-system (<ref>) in these cases is K_0 = -1 and t_0^1 = 0, which corresponds to T^1(u) = -h_-2/^N = -^N(sin (u+π/3)/sinπ/3)^N, T^2(u) = 0. In both cases (i) and (ii), the Razumov-Stroganov eigenvalue is non-degenerate and acts as the ground state of the corresponding standard module. Patterns of zeros and analyticity strips. The eigenvalues T^1(u) and T^2(u) are centered Laurent polynomials in the variable z=^ u with minimal and maximal powers -N and N. Due to the periodicity properties of ^n(u), the patterns of zeros are periodic with period π in the complex u plane and each vertical strip of width π has at most N zeros. Examples of patterns of zeros are given in <ref>. Empirically, we find that for the leading eigenvalues, only a finite number of these zeros are in the following analyticity strips: T^1(u): -π/6< Re(u) < π/2, T^2(u): -π/3< Re(u) < π/3. As opposed to the strip boundary conditions, none of the zeros are common to all the eigenvalues. This is due to the absence of surface terms in the fusion hierarchy relations (<ref>). In general, the zeros have three possible locations: (i) in the center of the analyticity strip, (ii) on one of the two edges of the analyticity strip, or (iii) outside the analyticity strip. We will be interested in zeros of type (i) and (ii). The zeros that lie in the center of the analyticity strip of T^1(u) are always of order one. For γ = 0, the zeros in the center of the analyticity strip of T^2(u) can be of order one or two. This degeneracy is lifted away from γ = 0, but may occur again for special values of γ. For generic values of γ, all zeros of T^2(u) are of order one. We observe that all the patterns of zeros are symmetric under a reflection with respect to the central line of the analyticity strips. This implies that the corresponding eigenvalues are real on this central line. In contrast, some patterns of zeros, for example the one in <ref>, are not symmetric with respect to a reflection about the real u axis. In general, these eigenvalues are therefore complex for Im(u)=0. We shall see in <ref> that, for the finite-size corrections, the zeros in the upper and lower parts of the u plane are tied to contributions to Δ and Δ̅ respectively. For small values of N, we find that some zeros are very close to the real axis, and even coincide with this axis in certain instances. For the periodic boundary conditions, the line dividing the plane into two chiral halves need not necessarily coincide with the real axis. In fact, it can be chosen arbitrarily, as long as the convention does not impact the finite-size corrections for the leading eigenvalues. This is discussed further in <ref>. We encode the content of zeros in both the upper- and lower-half planes together in a pattern diagram. For the eigenvalues corresponding to <ref>, the pattern diagrams are [shift=-1.2](-0.4,-1.4)(1.4,1.4) -(0,-1.4)(0,1.4)-(1,-1.4)(1,1.4)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,0.3)(0,0.6) [dotsize=0.09cm](0.5,-0.9)(0.5,0.9) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,0.3)(1,0.6) [shift=-1.2](-0.4,-1.4)(1.4,1.4) -(0,-1.4)(0,1.4)-(1,-1.4)(1,1.4)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) and [shift=-1.2](-0.4,-1.4)(1.4,1.4) -(0,-1.4)(0,1.4)-(1,-1.4)(1,1.4)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(0,-0.3)(0,0.6) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9)(0.5,0.3)(0.5,0.9) [dotsize=0.09cm](1,0)(1,-0.3)(1,0.6) [shift=-1.2](-0.4,-1.4)(1.4,1.4) -(0,-1.4)(0,1.4)-(1,-1.4)(1,1.4)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0) . We work with the following convention. For a given eigenvalue, if the first analyticity strip has a site, occupied or unoccupied, lying on the dashed separation line, it is considered to belong to the upper half-plane. The same convention was used in <cit.> for critical dense polymers. The pattern of zeros in the second strip is described below as a pair of single-column configurations. If the second analyticity strip has a site lying on the dashed separation line, then the corresponding site in the first column is considered to be in the upper-half of the plane and the site in the second column is considered to be in the lower-half. We will come back to this in <ref> and <ref>. The analyticity properties of K(u) and t^1(u) are deduced from those of T^1(u) and T^2(u). From (<ref>), we see that K(u) has a pole of order N at u = π/3. Likewise, t^1(u) has poles of order N at u = -π/3, π/3 and a zero of order N at u=0. The analyticity strips have width 2λ: K(u): -π/2< Re(u) < π/6, t^1(u): -π/3< Re(u) < π/3. A similar analysis on small system sizes reveals that the functions 1+^±ΛK(u) and 1+t^1(u) are analytic and non-zero in the following strips: 1+^±ΛK(u): -π/3< Re(u) < 0, 1+t^1(u): -π/6< Re(u) < π/6. We observe that these analyticity strips have width λ and are entirely free of zeros and poles. Finally, in terms of the variables defined in (<ref>), the analyticity strips take the following elegant forms: 𝔞^1(x): - π < Im(x) < π , 𝔞^2(x): -π < Im(x) < π, 𝔄^1(x): -π/2< Im(x) < π/2, 𝔄^2(x): -π/2< Im(x) < π/2. Braid limit. The braid transfer tangles ^n_±∞ are obtained as the u→±∞ limit of the transfer tangle ^n(u). For n=1, ^1_±∞≡_±∞ are defined in (<ref>). For n=2, we have ^2_+∞ = lim_u →∞(-^(π-λ)/s_1(u))^N ^2(u) = ()^2 - , ^2_-∞ = lim_u → - ∞(^(π-λ)/s_1(-u))^N ^2(u) = ()^2 - . The last equalities are obtained by taking the proper u →±∞ limits of (<ref>) for n=1. The braid limits of _±∞ and ^1_±∞ are likewise given by _±∞ = (-1)^N^2_±∞= lim_u →±∞(u) = ^1_±∞, ^1_±∞ = ^1_±∞= lim_u →± ∞^1(u) = ^2_±∞= (^1_±∞)^2-. The matrix realisations on _N^d of all these braid tangles are scalar multiples of the identity matrix. For β = 1, the result for T^1_±∞ is given in (<ref>) and yields 𝔞^1_±∞ = 4 cos^2(±γ+2π d3)-1, 𝔞^2_±∞ = 2 cos(±γ+2π d3). The eigenvalues thus have different braid behaviors in the lower- and upper-half planes for generic γ. For γ = 0 and d ≡ 0,1 3, we have 𝔞^1_±∞ = 0 and 𝔞^2_±∞ = -1. As for the strip case, the convergence of 1+𝔞^2(x) to zero can follow different subcases. In the periodic case however, the convergence can be different for x →∞ and x → -∞. From our numerical data, we find the following four subcases: Subcase : lim_x → -∞^-2x/3(1+ 𝔞^2(x)) = κ_-, Subcase : lim_x → -∞^-4x/3(1+ 𝔞^2(x)) = κ'_-, Subcase A: lim_x → +∞^2x/3(1+ 𝔞^2(x)) = κ_+, Subcase B: lim_x → +∞^4x/3(1+ 𝔞^2(x)) = κ'_+. The constants κ_± and κ'_± are real and strictly positive, for all eigenvalues except the Razumov-Stroganov eigenvalues. The argument leading to this conclusion is identical to the one given below (<ref>). From (<ref>) and (<ref>), the rate of convergence of 𝔞^1(x) to zero as x →±∞ is Subcase : lim_x → -∞^-2x/3𝔞^1(x) = -κ_-, Subcase : lim_x → -∞^-4x/3𝔞^1(x) = κ'_-, Subcase : lim_x → +∞^2x/3𝔞^1(x) = -κ_+, Subcase : lim_x → +∞^4x/3𝔞^1(x) = κ'_+. An eigenstate is thus described by one of these four pairs: (,), (,), (,) or (,). In the pattern diagrams, this is indicated by small letters at the bottom and top of the analyticity strips, as in <ref>. Bulk behavior. The eigenvalues of ^1(u) and (u) have the following bulk behaviors: t^1_bulk(u) = 0, K_bulk(u) = σ, σ∈{-1,1}. Indeed, the behavior of t^1(u) near the origin is governed by the zero of order N at u = 0, so the function is approximately zero in this neighborhood. From this remark, it follows that (<ref>) is simply K_bulk(u)^2 = 1 in this neighborhood. For a given pattern of zeros, one can deduce the values of σ from the braid value K_+∞, and the number of zeros on the central line of the analyticity strip. For instance, let us consider the ground state of _10^2 for ω =1, whose pattern is displayed in <ref>. The braid behavior is K_+∞ = -1. In the upper half-plane, there is a single zero on the central line of the analyticity strip, so K(-π/6) is positive. We deduce that σ=1. In contrast, applying the same reasoning to the eigenvalue corresponding to <ref>, we find σ=-1. For periodic boundary conditions, both values of σ are possible. §.§ Finite-size corrections The eigenvalues T(u) can be factored as T(u) = T_b(u) T_f(u). Compared to (<ref>), the surface term is absent because of the periodic boundary conditions. The bulk and finite contributions satisfy T_b(u)T_b(u+λ) = h_-1h_1, T_f(u)T_f(u+λ) = 1+t^1(u). The solution to the first relation in (<ref>) gives a bulk free energy identical to the one for the strip. Below, we compute the finite-size correction to T_f(u) for any eigenvalue in _N^d. We use the functions 𝔞^1(x), 𝔞^2(x), 𝔄^1(x) and 𝔄^2(x) for the computation, as well as 𝔟(x) = T_f( x 3+ π 6). The relation for T_f(u) in (<ref>) then becomes 𝔟(x - π2) 𝔟(x + π2) = 𝔄^1(x). Since (<ref>) defines 𝔟(x), only up to a sign, we choose the convention that 𝔟_±∞ = T^1_±∞. The rest of this section computes the finite-size corrections to 𝔟(x) for any eigenvalue. The results could be obtained for the full interval γ∈ [-π,π], but for simplicity we consider γ in a neighborhood of γ = 0, namely γ∈ (-π/3,π/3) for d ≡ 0 3 and γ∈ (-π/6,π/6) for d ≡ 1,2 3. TBA equations. For a given eigenstate of 𝔞^1(x) and 𝔞^2(x), we respectively denote by t^1_+ and t^1_- the numbers of zeros w_±^i of 𝔞^1(x) in the upper and lower part of the plane. These zeros are ordered such that w_±^i≥ w_±^i+1. Likewise t^2_+ and t^2_- count the zeros x_±^j of 𝔞^2(x) in the upper and lower half-planes, which are ordered as x_±^j≥ x_±^j+1. Depending on γ, some of the zeros may be degenerate. The finite-size correction functions ℓ^1(x) and ℓ^2(x) are defined as 𝔞^1(x) = tanh^N( x 2)·(∏_i = 1^t_+^1tanh(x-w^i_+2) )(∏_i = 1^t_-^1tanh(x-w^i_-2))·ℓ^1(x), 𝔞^2(x) = (∏_j = 1^t^2_+tanh(x-x^j_+2) )(∏_j = 1^t^2_-tanh(x-x^j_-2))·ℓ^2(x). The functions ℓ^1(x), ℓ^2(x), 𝔄^1(x) and 𝔄^2(x) are analytic and free of zeros in their corresponding analyticity strips, and satisfy ℓ^1(x-π2)ℓ^1(x+π2) = 𝔄^2(x), ℓ^2(x-π2)ℓ^2(x+π2) = 𝔄^1(x). For γ generic, the four functions have constant asymptotics, so we can define the Fourier transforms of their logarithmic derivatives: L^n(k) = 1/2 π∫_-∞^∞ x (lnℓ^n(x))'^- k x, A^n(k) = 1/2 π∫_-∞^∞ x (ln𝔄^n(x))'^- k x, (lnℓ^n(x))' = ∫_-∞^∞ k L^n(k) ^ k x, (ln𝔄^n(x))' = ∫_-∞^∞ k A^n(k) ^ k x. Applying the Fourier transform and its inverse to (<ref>), and subsequently integrating with respect to x, we find ln𝔞^1(x) = lntanh^N( x 2) + ∑_i = 1^t^1_+ln(tanh(x-w^i_+2) ) + ∑_i = 1^t^1_-ln(tanh(x-w^i_-2)) + K ∗ln𝔄^2 + ϕ^1, ln𝔞^2(x) = ∑_j = 1^t^2_+ln(tanh(x-x^j_+2) ) +∑_j = 1^t^2_-ln(tanh(x-x^j_-2) ) + K ∗ln𝔄^1 + ϕ^2, where ϕ^1 and ϕ^2 are the integration constants. The kernel K(x) and the convolution are defined as for the strip, see (<ref>) and (<ref>). We note that for special values of γ, the asymptotics of the functions ℓ^1(x), ℓ^2(x), 𝔄^1(x) and 𝔄^2(x) can be zero. In this case, one should instead define the Fourier transform of the second logarithmic derivative, which results in extra linear terms of the form ϕ x in (<ref>). However, this care turns out to be superfluous because the constants ϕ are evaluated to zero using the braid limit. Scaling TBA equations. In the scaling regime, the function tanh^N(x/2) has the following behavior: lim_N →∞tanh^N(± 12(x + ln N)) = (-1)^Nexp(-2 ^-x). We assume that the following scaling limits also exist: 𝖺^n_±(x) = lim_N →∞𝔞^n(± (x + ln N)), 𝖠^n_±(x) = lim_N →∞𝔄^n(± (x + ln N)), n = 1,2. Because the functions 𝔞^n(x) are not symmetric under the transformation x → -x, the functions 𝖺^n_+(x) and 𝖺^n_-(x) are not equal. We denote the zeros of 𝖺^1_±(x) and 𝖺^2_±(x) by z_±^i and y_±^j, namely z_±^i = w_±^i - ln N, y_±^j = x_±^j - ln N. In the scaling regime, the non linear integral equations become ln𝖺_±^1(x) = -2 ^-x + ∑_i=1^t^1_±ln(-tanh (x-z_±^i2) ) + K ∗ln𝖠_±^2 +ϕ^1_±, ln𝖺_±^2(x) = ∑_j=1^t^2_±ln(-tanh (x-y^j_±2) )+ K ∗ln𝖠_±^1 + ϕ^2_±. Here, we consider the non linear integral equations to be independent in the upper and lower part of the planes, thus allowing two different sets of constants, (ϕ^1_-,ϕ^2_-) and (ϕ^1_+,ϕ^2_+). Evaluation of the constants. For d ≡ 0 3, we consider γ∈ (-π/3,π/3). On this interval, the braid limits (<ref>) are both positive. We fix the branches of the logarithms by imposing ln(-tanh(x-y2))π and ln𝖺_±^1(x)ln (4 cos^2 γ-1), ln𝖺_±^2(x)ln (2 cosγ). Applying the braid limit to (<ref>), we find that d ≡ 0 3: ϕ_±^1 = -π t_±^1, ϕ_±^2 = -π t_±^2. For d ≡ 1,2 3, the interval γ∈ (-π/6, π/6) splits into three cases: γ∈ (-π/6,0), γ∈ (0,π/6) and γ = 0. Here, 𝖺_±^2(∞) is negative in all three cases. Fixing the branches of the logarithms fixes the constants ϕ^2_±: ln𝖺_±^2(x)ln(-2 cos (±γ+2π d3)) + π, ϕ^2_± = -π(t^1_±-1). In stark contrast, 𝖺_±^1(∞) takes opposite signs in the first and second interval and is zero for γ = 0. For d ≡ 1 3, γ∈ (-π6,0) as well as for d ≡ 2 3, γ∈ (0,π6), we have [ ln𝖺^1_-(x) ln (4 cos^2 (-γ+2π d3)-1), ϕ^1_- = -π t^1_-,; ln𝖺^1_+(x) ln(1-4 cos^2 (γ+2π d3)) + π, ϕ^1_+ = -π (t^1_+-1). ] Likewise, for d ≡ 1 3, γ∈ (0,π6) and d ≡ 2 3, γ∈ (-π6,0), we have [ ln𝖺^1_-(x) ln(1-4 cos^2 (-γ+2π d3)) + π, ϕ^1_- = -π(t^1_–1),; ln𝖺^1_+(x) ln(4 cos^2 (γ+2π d3)-1), ϕ^1_+ = -π t^1_+. ] For γ = 0, the result depends on the subcases or and or : Subcase : ln𝖺^1_-(x) -2x3 + lnκ_- + π ϕ^1_- = -π (t^1_–1), Subcase : ln𝖺^1_-(x) -4x3 + lnκ'_- ϕ^1_- = -π t^1_-, Subcase : ln𝖺^1_+(x) -2x3 + lnκ_+ + π ϕ^1_+ = -π (t^1_+-1), Subcase : ln𝖺^1_-(x) -4x3 + lnκ'_+ ϕ^1_+ = -π t^1_+. Finite-size corrections. We apply the Fourier transform to (<ref>) by first removing the zeros of 𝔟(x) on the real axis. This yields ln𝔟(x) = ∑_j = 1^t_+^2ln(-tanh(x-x^j_+2) ) + ∑_j = 1^t_-^2ln(tanh(x-x_-^j2)) + K ∗ln𝔄^1 + ψ. The constant ψ can be evaluated from the braid limits to be a multiple of π. It does not contribute to the finite-size corrections. These are written in terms of the scaling functions and their zeros as follows: ln𝔟(x) - ψ = ∑_j = 1^t^2_-ln(tanh(x-x_-^j2) )+∑_j = 1^t^2_+ln(-tanh(x-x^j_+2) ) + 1/2π∫_-ln N^∞ y ( ln𝔄^1(y+ln N)/cosh(x-y-ln N)+ln𝔄^1(-y-ln N)/cosh(x+y+ln N)) ≃ - 1/N(2 ∑_j=1^t_-^2^-x-y_-^j + 2 ∑_j=1^t_+^2^x-y_+^j-^-x/π∫_-∞^∞ y ^-yln𝖠_-^1(y)- ^x/π∫_-∞^∞ y ^-yln𝖠_+^1(y)). Zeros of 𝔞^1(x) and 𝔞^2(x). To rewrite ^-y_±^j in terms of integrals involving the scaling functions, we use (<ref>) and find 𝔞^2(x^j_±) = 0 ⇒ 𝔞^1(x^j_±-π2) = -1 ⇒ 𝖺^1_±(y_±^j-π2) = -1. Using (<ref>), we find (2 k_±^j-1)π = ln𝖺^1_± (y^j_± - π2) = 2 ^-y_±^j + ∑_i=1^t_±^1ln(-tanh12(y_±^j-z_±^i- π2 )) + 1/2π∫_-∞^∞ y ln𝖠_±^2(y)/sinh(y-y_±^j) + ϕ^1_± where the k_±^j are integers. The terms ln(-tanh12(y_±^j-z_±^i- π2 )) in this last expression are rewritten using (<ref>): 𝔞^1(w^i_±) = 0 ⇒ 𝔞^2(w^i_±-π2) = -1 ⇒ 𝖺^2_±( z^j_±-π2) = -1. Similarly, (2 ℓ_±^i-1)π = ln𝖺^2_± (z_±^i - π2) = ∑_j=1^t_±^2ln(-tanh12(z_±^i-y_±^j -π2)) + 1/2 π∫_-∞^∞ y ln𝖠^1_±(y)/sinh(y-z_±^i)+ ϕ_±^2 where the ℓ_±^i are integers. We can then apply (<ref>) to each of the logarithms in (<ref>). By combining (<ref>), (<ref>) and (<ref>), we obtain ln𝔟(x) - ψ ≃ -2 π/N(^x [ ∑_j=1^t_+^2( k_+^j -12 - ϕ^1_+2π) + ∑_i=1^t^1_+(ℓ_+^i-12+t^2_+2 - ϕ^2_+2π) - 1/2π^2∫_-∞^∞ y ^-yln𝖠_+^1(y) - 1/4π^2∑_i=1^t_+^1∫_-∞^∞ y ln𝖠_+^1(y)/sinh(y-z_+^i)- 1/4π^2∑_j=1^t_+^2∫_-∞^∞ y ln𝖠_+^2(y)/sinh(y-y_+^j)] +^-x[ ∑_j=1^t_-^2( k_-^j -12 - ϕ^1_-2π) + ∑_i=1^t^1_-(ℓ_-^i-12+t^2_-2 - ϕ^2_-2π)- 1/2π^2∫_-∞^∞ y ^-yln𝖠_-^1(y) - 1/4π^2∑_i=1^t_-^1∫_-∞^∞ y ln𝖠_-^1(y)/sinh(y-z_-^i) - 1/4π^2∑_j=1^t_-^2∫_-∞^∞ y ln𝖠_-^2(y)/sinh(y-y_-^j)] ). Dilogarithm technique. To evaluate (<ref>), we consider the integrals 𝒥_± = ∫_-∞^∞ y ((ln𝖺_±^1)' ln𝖠_±^1 -ln |𝖺_±^1| (ln𝖠_±^1)' ) + ∫_-∞^∞ y ((ln𝖺_±^2)' ln𝖠_±^2 -ln |𝖺_±^2| (ln𝖠_±^2)' ) where ln |𝖺_±^1| and ln |𝖺_±^2| are real for all x and given by ln |𝖺_±^n|(x) = ln𝖺^n_±(x) + θ_±^n(x), n = 1,2. The functions θ_+^n(x) and θ_-^n(x) are step functions defined for x ∈ℝ with the following behavior: starting at x = ∞ and progressing to the left on the real x axis, they decrease by π each time a zero of the corresponding type is crossed (z_±^i for θ^1_±(x) and y_±^j for θ^2_±(x)). The values at the endpoints are θ_±^1(x) = 0 for x > z_±^1, θ_±^2(x) = 0 for x > y_±^1. The integrals 𝒥_± can be evaluated in two ways. For the first, one uses the non-linear integral equations (<ref>) and the symmetries of K(x) and obtains 𝒥_± = 2∫_-∞^∞ y ^-y(ln𝖠_±^1 + (ln𝖠_±^1)') + ∫_-∞^∞ y ∑_i=1^t_±^1[ln(-tanh (y-z_±^i2)) ]' ln𝖠_±^1 - ∫_-∞^∞ y (∑_i=1^t_±^1ln(-tanh (y-z_±^i2)) - θ_±^1(y) - π t_±^1) (ln𝖠_±^1)' + ∫_-∞^∞ y ∑_j=1^t_±^2[ln(-tanh (y-y_±^j2)) ]' ln𝖠_±^2 - ∫_-∞^∞ y (∑_j=1^t_±^2ln(-tanh (y-y_±^j2)) - θ_±^2(y) - π t_±^2) (ln𝖠_±^2)'. The integrals involving derivatives of ln𝖠_±^1 and ln𝖠_±^2 are transformed using integration by parts. For each one, it can be argued using the non-linear integral equations that the surface terms are zero. This yields 𝒥_± = 4 ∫_-∞^∞ y ^-yln𝖠_±^1(y) + 2 ∑_i=1^t_±^1∫_-∞^∞ y ln𝖠_±^1(y)/sinh(y-z_±^i) + 2∑_j=1^t_±^2∫_-∞^∞ y ln𝖠_±^2(y)/sinh(y-y_±^j), which is precisely the combination of integrals needed to compute the finite-size corrections: ln𝔟(x) -ψ≃ -2π/N( ^x [∑_j=1^t_+^2( k_+^j -12 - ϕ^1_+2π) + ∑_i=1^t^1_+(ℓ_+^i-12+t^2_+2 - ϕ^2_+2π) - 𝒥_+/8 π^2] + ^-x[ ∑_j=1^t_-^2( k_-^j -12 - ϕ^1_-2π) + ∑_i=1^t^1_-(ℓ_-^i-12+t^2_-2 - ϕ^2_-2π) - 𝒥_-/8 π^2]). The second way of performing the integrals consists in changing the integration variable from y to 𝖺_±. The asymptotic behavior of the scaling functions is 𝖺^1_±(-∞) = 0, 𝖺^2_±(-∞) = σ, 𝖺^1_±(∞) = 4 cos^2(±γ+2π d3)-1, 𝖺^2_±(∞) = 2 cos (±γ+2π d3) with σ∈{-1,+1}. For the integrals involving 𝖺^1_±, we find ∫_-∞^∞ y (( ln𝖺_±^1)' ln𝖠_±^1 -ln |𝖺_±^1| (ln𝖠_±^1)' ) = ∫_-∞^∞ y 𝖺_±^1/ y ( ln(1+ 𝖺_±^1)/𝖺_±^1 - ln |𝖺_±^1|/1+ 𝖺_±^1) = [∫_-∞^z^±_t_±^1+∫_z^±_t_±^1^z^±_t_±^1-1+ … + ∫_z^±_2^z^±_1+ ∫_z^±_1^∞] y 𝖺_±^1/ y ( ln(1+ 𝖺_±^1)/𝖺_±^1 - ln |𝖺_±^1|/1+ 𝖺_±^1) = [∫_0^0+∫_0^0+ … + ∫_0^0+ ∫_0^𝖺^1_±(∞)] 𝖺 ( ln(1+ 𝖺)/𝖺 - ln |𝖺|/1+ 𝖺) =∫_0^𝖺^1_±(∞)𝖺 ( ln(1+ 𝖺)/𝖺 - ln |𝖺|/1+ 𝖺). Recalling that 𝖠^2_± = (1 + ^3 γ𝖺^2_±)(1 + ^-3 γ𝖺^2_±), the integrals involving 𝖺^2_± are obtained with the same arguments. We find ∫_-∞^∞ y ((ln𝖺_±^2)' ln𝖠_±^2 -ln |𝖺_±^2| (ln𝖠_±^2)' ) = ∫_σ^𝖺^2_±(∞)𝖺 ( ln(1+ ^3 γ𝖺)/𝖺 - ^3 γln |𝖺|/1+ ^3 γ𝖺) + ∫_σ^𝖺^2_±(∞)𝖺 ( ln(1+ ^-3 γ𝖺)/𝖺 - ^-3 γln |𝖺|/1+ ^-3 γ𝖺). The sum of the integrals (<ref>) and (<ref>) is expressed as 𝒥_± = {[ 𝒦_σ (γ) d ≡ 0 3,; 𝒦_σ (2π/3±γ) d ≡ 1 3,; 𝒦_σ (2π/3∓γ) d ≡ 2 3,; ]. where 𝒦_σ (γ) is defined and evaluated in <ref>. Comparing with (<ref>) specialised to c=0, we find that the first and second bracket in (<ref>) are identified with Δ and Δ̅: Δ = ∑_j=1^t_+^2( k_+^j -12 - ϕ^1_+2π) + ∑_i=1^t^1_+(ℓ_+^i-12+t^2_+2 - ϕ^2_+2π) - 𝒥_+/8 π^2, Δ̅ = ∑_j=1^t_-^2( k_-^j -12 - ϕ^1_-2π) + ∑_i=1^t^1_-(ℓ_-^i-12+t^2_-2 - ϕ^2_-2π) - 𝒥_-/8 π^2, where the integers k_±^j and ℓ_±^i are given in terms of the eigenvalue and its zeros by (<ref>) and (<ref>). Specialisation to γ = 0. For γ=0, we have 𝒥_+ = 𝒥_- and Δ = ∑_j=1^t^2_+k_+^j+∑_i=1^t^1_+ℓ_+^i + τ_+ + ι(σ), Δ̅= ∑_j=1^t^2_-k_-^j+∑_i=1^t^1_-ℓ_-^i + τ_- + ι(σ) with τ_± = {[ -1/2(t^1_± + t^2_± + t^1_± t^2_±) d≡ 0 3,; -1/2t^1_± t^2_± d ≡ 1,2 3 ( or ),; -1/2(t^2_± + t^1_± t^2_±) d ≡ 1,2 3 ( or ) ]. ι(σ) = -𝒥_±/8π^2 = {[ -1/24 d≡ 0 3, σ = 1,; -1/6 d≡ 0 3, σ = -1,; 1/8 d≡ 1,2 3, σ = 1,; 0 d≡ 1,2 3, σ = -1.; ]. §.§ Solution for the ground states In this section, we study the conformal weights of the ground states of each _N^d for γ = 0. For the ground state of _N^d, we observe that the pattern of zeros is symmetric with respect to the horizontal axis, implying that Δ = Δ̅. The analysis of the bulk behavior of these eigenvalues (using the method discussed below (<ref>)) reveals that σ = 1 for N even and σ = -1 for N odd. Moreover, for d ≡ 1,2 mod 3, we observe that the ground state eigenvalue respectively belongs to the subcases (,) and (,). For N even, the zero patterns are characterised by t^1_± = {[ d/6 d ≡ 0 mod 6,; d-2/6 d ≡ 2 mod 6,; d-4/6 d ≡ 4 mod 6, ]. t^2_± = {[ d/3 d ≡ 0 mod 6,; d+1/3 d ≡ 2 mod 6,; d-1/3 d ≡ 4 mod 6. ]. For N odd, we work with the convention described in <ref> that the single zeros on the axis are in the upper-half plane. We stress however that the opposite convention produces the same conformal weights. With this convention, we have t^1_+ = {[ d-1/6 d ≡ 1 mod 6,; d+3/6 d ≡ 3 mod 6,; d+1/6 d ≡ 5 mod 6, ]. t^1_- = {[ d-7/6 d ≡ 1 mod 6,; d-3/6 d ≡ 3 mod 6,; d-5/6 d ≡ 5 mod 6, ]. t^2_± = {[ d-1/3 d ≡ 1 mod 6,; d/3 d ≡ 3 mod 6,; d+1/3 d ≡ 5 mod 6. ]. In both cases, there are also N-d/2 pairs of zeros on the boundary edges of the analyticity strip of T^1(u), split between the upper and lower halves, see the examples in <ref>. Each zero in the center of the analyticity strip of T^2(u) is of order one and is joined, at the same height, by a pair of single zeros sitting on the edges of the analyticity strip. The integers k^j_± and ℓ^i_± are not fixed by the technique of <ref>. We estimate them using our computer program by evaluating (<ref>) and (<ref>) on small system sizes. For the ground state of _N^d, we find: k^j_± = {[ j d ≡ 0 mod 3,; j d ≡ 1 mod 3,; j-1 d ≡ 2 mod 3, ]. ℓ^i_± = i. The patterns of zeros for the ground states for N=12,13 are given in <ref>. In this figure, for d=1, the analyticity strip of T^2(u) for the Razumov-Stroganov eigenstate is colored in gray, indicating that T^2(u)=0. The zeros on the separation lines are divided between the upper and lower half-planes using the prescription discussed in <ref>. Inserting the data (<ref>) and (<ref>) into (<ref>), we find that in all cases, the ground state conformal weights are given by Δ = Δ̅= d^2-1/24 = Δ_0,d/2. §.§ Solution for all the eigenvalues In this section, we describe the full spectrum in the standard modules _N^d for γ = 0. We compute the finitized spectrum generating functions = ∑_eigenstates of T^1(u) in _N^dε q^Δq̅^Δ̅, where Δ and Δ̅ are the conformal weights of the limiting conformal states, ε is the overall sign of the eigenvalue and the sum is over eigenstates characterised by the patterns of zeros. Indeed, for periodic boundary conditions, the eigenvalues can have either a positive or a negative overall sign. This sign is crucial in <ref> for the partition function on the M× N torus with M odd. Concretely, ε is obtained from the bulk limit K_bulk = σ and the vertical lattice width M as ε = σ^M. As for the strip, our derivation is built on a set of conjectured selection rules for the patterns of zeros in _N^d. This conjecture is supported by data produced with our computer implementation of the transfer matrices and the computation of the eigenvalues and patterns of zeros, for 1 ≤ N ≤ 12. We present data for _6^0 and _6^2 in <ref>. Selection rules for the patterns of zeros. The selection rules are expressed in terms of triples (σ, σ',σ”) of single-column diagrams, with σ∈Mm, σ' ∈Ln and σ”∈Lℓ. We denote by MLmnℓ the set of such triples. In the selection rule, the configuration σ describes the content in zeros of the first strip, whereas the pair (σ', σ”) describes the content of the second strip. Crucially, in contrast with the boundary case, the constraint of dominance is not imposed on the pair (σ', σ”). The cardinality of MLmnℓ is therefore simply given by the product of the cardinalities of Mm, Ln and Lℓ: |MLmnℓ| = MmLnLℓ. For instance, 42311 contains 16 configurations. These are precisely the configurations appearing in Å42311 given in <ref>, except that those on the second row are twice degenerate. For d ≡ 1,2 3, we include (, ), (, ), (, ) or (, ) in the upper labels of MLmnℓ to specify which subcase the corresponding eigenvalues belong to. We give a conjecture for the full content of zeros in the analyticity strips, with the separation between the upper and lower planes discussed below. For example, the patterns for _6^0 and _6^2 given in <ref> are encoded by the following sets of column configurations: _6^0 : 30000 ∪ 31200 ∪ 41211 ∪ 42411 ∪ 52422 ∪ 63633, _6^2 : 40200, ∪ 41400, ∪ 51411, ∪ 62622, ∪ 41401, ∪ 41401,. More generally, for d≠ 1, we conjecture that the full set of patterns of zeros in _N^d is described by the following sets: d=3t : ⋃_i=0^N-d/2 ⋃_j=0^⌊1/2(N-d/2-i)⌋N+t/2 + ii+j+t2(i+j+t)ii+t , d=3t+1 : ⋃_i=0^N-d/2 ⋃_j=0^⌊1/2(N-d/2-i)⌋N+t-1/2 + ii+j+t-12(i+j+t)ii+t-1,∪⋃_i=0^N-d-2/2 ⋃_j=0^⌊1/2(N-d-2/2-i)⌋N+t+1/2 + ii+j+t2(i+j+t+1)ii+t, ∪⋃_i=0^N-d-2/2 ⋃_j=0^⌊1/2(N-d-2/2-i)⌋N+t+1/2 + ii+j+t2(i+j+t+1)ii+t,∪⋃_i=0^N-d-4/2 ⋃_j=0^⌊1/2(N-d-4/2-i)⌋N+t+3/2 + ii+j+t+12(i+j+t+2)ii+t+1, , d=3t+2 : ⋃_i=0^N-d/2 ⋃_j=0^⌊1/2(N-d/2-i)⌋N+t+2/2 + ii+j+t2(i+j+t+1)ii+t,∪⋃_i=0^N-d-4/2 ⋃_j=0^⌊1/2(N-d-4/2-i)⌋N+t+2/2 + ii+j+t+12(i+j+t+2)ii+t+1, ∪⋃_i=0^N-d-4/2 ⋃_j=0^⌊1/2(N-d-4/2-i)⌋N+t+2/2 + ii+j+t+12(i+j+t+2)ii+t+1,∪⋃_i=0^N-d-8/2 ⋃_j=0^⌊1/2(N-d-8/2-i)⌋N+t+2/2 + ii+j+t+22(i+j+t+3)ii+t+2, . For d=1, we instead have d=1: 00000∪⋃_i=1^N-1/2 ⋃_j=1^⌊1/2(N-1/2-i)⌋N-1/2 + ii+j-12(i+j)ii-1,∪⋃_i=0^N-3/2 ⋃_j=0^⌊1/2(N-3/2-i)⌋N+1/2 + ii+j2(i+j+1)ii, ∪⋃_i=0^N-3/2 ⋃_j=0^⌊1/2(N-3/2-i)⌋N+1/2 + ii+j2(i+j+1)ii,∪⋃_i=0^N-5/2 ⋃_j=0^⌊1/2(N-5/2-i)⌋N+3/2 + ii+j+12(i+j+2)ii+1, . Using (<ref>), one can use binomial identities to show that the sum of the cardinalities of the above sets equals _N^d in all cases. Separation between upper and lower half-planes. Writing down explicit expressions for the spectrum generating functions requires understanding how the zeros are split between the upper and lower halves of the plane. Let us denote by (M_1m_1M_2m_2) the set of single-column diagrams in M_1+M_2m_1+m_2 for which the subconfigurations of the lower and upper half planes respectively belong to M_1m_1 and M_2m_2. The bar in M_1m_1 is a reminder that this factor contributes powers of q̅ in . We first describe the splitting of the heights in the first strip. A configuration of zeros of the first strip is described by a single-column diagram of Mm. From the selection rules, we see that for the leading eigenvalues, M is a number that grows linearly with the system size N, whereas m remains small.[Here M is the number of sites of the column diagram in Mm, not the vertical width of the M× N lattice.] The zeros of these leading eigenvalues lie at a distance ∼ln N from the real axis and the separating line. There is thus some arbitrariness in the choice of the position of the separation. One can choose Mm → ⋃_k_1 = 0^m(⌊M/2⌋ - ϵk_1⌊M+1/2⌋ + ϵm-k_1) where ϵ is an arbitrary fixed number that is much smaller than M. The union over k_1 allows for the zeros to be split between the upper and lower half-planes in any possible way. Depending on ϵ, some eigenvalues are treated differently in terms of the contribution of their zeros to Δ and Δ̅. Indeed for finite N, the numbers t^2_- and t^2_+ of zeros in each half-plane depends on ϵ, and likewise for the resulting conformal weights (<ref>). However, this does not occur for the leading eigenstates, namely those that correspond to the conformal states in the scaling limit. The finitized partition functions given below depend on ϵ, but by varying ϵ (while keeping ϵ≪ M), the resulting expressions only change by powers of q and/or q̅ that are linear in N. For ϵ≪ M, the scaling limits of these partition functions are therefore independent of ϵ. We note that the data presented in <ref> corresponds to ϵ = 0. Crucially, the bulk behavior K_bulk=σ is a function of the variable k_1, which controls the separation between upper and lower halves in (<ref>): σ = {[ (-1)^k_1 d ≡ 0 3,; (-1)^k_1+1 d ≡ 1,2 3.; ]. The value of k_1 therefore dictates both the selection of ι(σ) in (<ref>) and the sign ε in (<ref>). The second strip does not allow as much arbitrariness in the choice of the separation line. This is because for the leading eigenvalues, the number of sites in the second strip does not scale with the system size N. Let us denote by [(M_1m_1M_2m_2)(L_1n_1L_2m_2)(P_1ℓ_1P_2ℓ_2)], with L_1+L_2 = P_1 + P_2, the set of configurations (σ, σ', σ”) in M_1+M_2L_1+L_2m_1+m_2n_1+n_2ℓ_1+ℓ_2 for which σ∈ (L_1n_1L_2m_2), σ' ∈ (L_1n_1L_2m_2) and σ”∈ (P_1ℓ_1P_2ℓ_2). We discuss the cases d ≡ 0 3 and d ≡ 1,2 3 separately. For d ≡ 0 3, we note that all sets in the selection rule (<ref>) are of the form ML2Lnℓ. The number of occupied sites in the first strip is twice the number of sites in the second strip. From our numerical data, we find that the splitting of the zeros in the second strip follows the same rule: If σ has k_1 zeros in the lower half of the first strip, then the total number of sites in the lower half-plane of the second strip (namely those of σ' plus those of σ”) is also equal to k_1. The splitting is therefore as follows: ML2Lnℓ → ⋃_k_1 = 0^2L⋃_k_2 = 0^n⋃_k_3 = 0^ℓ[(⌊M/2⌋ - ϵk_1⌊M+1/2⌋ + ϵ2L-k_1) (⌊k_1/2⌋k_2L-⌊k_1/2⌋n-k_2) (⌊k_1+1/2⌋k_3L-⌊k_1+1/2⌋ℓ-k_3)]. Thus, if k_1 is even, the sites are split evenly between σ' and σ”. If k_1 is odd, we choose the lower half-plane of σ” to have an extra site compared to the lower half-plane of σ'. Had we chosen the opposite convention, the finitized characters given below would be slightly different, but their scaling limits would be identical. For d ≡ 1,2 3, the sets in the selection rules (<ref>) and (<ref>) are of the form ML2(L+1)nℓ, with , ∈{, }. In this case, the splitting is such that the total number of sites in the second strip equals the number of occupied sites in the lower half of the first strip, minus one: ML2(L+1)nℓ, → ⋃_k_1 = 0^2(L+1)⋃_k_2 = 0^n⋃_k_3 = 0^ℓ[(⌊M/2⌋ - ϵk_1⌊M+1/2⌋ + ϵ2(L+1)-k_1) (⌊k_1-1/2⌋k_2L-⌊k_1-1/2⌋n-k_2) (⌊k_1/2⌋k_3L-⌊k_1/2⌋ℓ-k_3)]. In computing , each [(M_1m_1M_2m_2)(L_1n_1L_2m_2)(P_1ℓ_1P_2ℓ_2)] will contribute q̅^Δ̅_minq^Δ_minM_1m_1L_1n_1P_1ℓ_1M_2m_2L_2n_2P_2ℓ_2 where Δ_min and Δ̅_min are the conformal weights of the minimal configurations. Selection rules for the integers and spectrum generating functions. Remarkably, the prescription for the integers k^j_± and ℓ^i_± is very similar to the one found for the strip. We formulate it based on data collected from our computer implementation, namely approximations to k^j_± and ℓ^i_± obtained using (<ref>) and (<ref>), for all the eigenvalues in _N^d for N≤12. In <ref>, the values of these integers are displayed with each pattern of zeros for _6^0 and _6^2. The prescription is as follows. For any configuration in MLmnℓ, (i) the heights of the single-column diagram (corresponding to the k^j_±) are labelled from 0 to M-1 for d≡ 1,2 3 () or (). They are labeled from 1 to M for d≡ 0 3, and likewise for d≡ 1,2 3 () or (); (ii) the heights of the double-column diagram (corresponding to the ℓ^i_±) are labelled from 1 to L in all cases. Using the selection rules and the formula (<ref>) for the conformal weights, we can formulate expressions for by writing down the generating function for each set in (<ref>) and taking their sum over i and j. Because of the similarities between the current prescription and the one in <ref> for the strip, the energies of the minimal configuration of each set in (<ref>) is obtained from (<ref>) under the suitable specifications of m, n and ℓ. The resulting expressions are complicated power-law series in q and q̅, involving products of three q-binomials and three q̅-binomials, with sums over five integers: i, j, k_1, k_2 and k_3. Moreover for d ≡ 1,2 3, there are four contributions to the generating functions depending on the subcases or and or . The resulting expressions are collected in <ref>. Scaling behavior. Unlike in the boundary case, for the periodic case we are unable to directly simplify the expressions for the finitized conformal partition functions. We are nevertheless able to perform such simplifications in the scaling limit: Z_d(q,q̅). Indeed, in <ref>, we use the q-binomial identities of <ref> to extract the following expressions for Z_d(q,q̅): d ≡ 0 3 : Z_d(q,q̅) = 1/(q)_∞(q̅)_∞∑_ℓ∈ℤ(q^Δ_0,3ℓ -d q̅^Δ_0,3 ℓ + (-1)^M q^Δ_1,3ℓ - dq̅^Δ_1,3 ℓ), d ≡ 1 3 : Z_d(q,q̅) = 1/(q)_∞(q̅)_∞∑_ℓ∈ℤ( q^Δ_0,3ℓ -d +2q̅^Δ_0,3 ℓ+2 + (-1)^Mq^Δ_1,3ℓ - d+2q̅^Δ_1,3 ℓ+2), d ≡ 2 3 : Z_d(q,q̅) = 1/(q)_∞(q̅)_∞∑_ℓ∈ℤ(q^Δ_0,3ℓ -d +1q̅^Δ_0,3 ℓ+1 + (-1)^Mq^Δ_1,3ℓ - d+1q̅^Δ_1,3 ℓ+1). §.§ Torus partition functions Computing the partition function of a model described by the periodic Temperley-Lieb algebra on the M× N torus requires a proper understanding of the representation theory of this algebra. This was achieved for the Q-Potts model by Jacobsen and Richard <cit.> and by Aufgebauer and Klümper for quantum spin chains <cit.>. For the loop model, we set the weight of a non-contractible loop to α, independent of its winding numbers around the torus. For d >0, let us denote by _d,j the coefficients of the trace of (u)^M on _N^d in an expansion in ω: (u)^M|__N^d = ∑_j ∈ℤ_d,jω^j. For finite M, the trace of (u)^M is a Laurent polynomial in ω, so there are finitely many contributions in the sum on the right-hand side of (<ref>). The partition function of the loop model for arbitrary α and β then reads <cit.> = δ_N≡ 0 mod 2 (u)^M|__N^0 +2 ∑_1 ≤ d ≤ N d ≡ N 2∑_j∈ℤ T_j ∧ d(α 2) _d,j , where T_k(x) is the k-th Chebyshev polynomial of the first kind and j ∧ d is the greatest common divisor of j and d. Even if our current derivation does not reveal this, we know that |_α=1=1 for all parities of M and N. For α = 2, a remarkable simplification occurs: T_j∧ d(1) = 1 and |_α = 2 = ∑_0 ≤ d ≤ N d ≡ N mod 2(2-δ_d,0) (u)^M|__N^d, ω = 1 . The torus conformal partition function is then obtained from the scaling limit of the finite-size partition function: ^MN f_b(u)|_α = 2(q,q̅) = ∑_d≥ 0 d ≡ N mod 2(2-δ_d,0) with q given in terms of the lattice data in (<ref>). From here, the computation splits between the odd and even parity of N. From (<ref>), we extend the definition of Z_d(q,q̅) to d<0. Using the relation Z_d(q,q̅) = Z_-d(q,q̅), we find = ∑_d ∈ 2 ℤ Z_d(q,q̅), = ∑_d ∈ 2 ℤ + 1 Z_d(q,q̅). These can in turn be written in terms of the u(1) characters ϰ_j(q) = ϰ^6_j(q) given in (<ref>). Defining y = 1/(q)_∞(q̅)_∞, we find the following identities for N odd: y∑_d ∈ 6 ℤ+3∑_ℓ∈ℤ q^Δ_0,3ℓ -d q̅^Δ_0,3 ℓ= ϰ_0(q)ϰ_6(q̅) + ϰ_6(q)ϰ_0(q̅), y∑_d ∈ 6 ℤ+3∑_ℓ∈ℤ q^Δ_1,3ℓ -d q̅^Δ_1,3 ℓ= 2 |ϰ_3(q)|^2, y(∑_d ∈ 6 ℤ+1∑_ℓ∈ℤ q^Δ_0,3ℓ -d+2q̅^Δ_0,3 ℓ+2 + ∑_d ∈ 6 ℤ+5∑_ℓ∈ℤ q^Δ_0,3ℓ -d+1 q̅^Δ_0,3 ℓ+1)= 2ϰ_2(q)ϰ_4(q̅) + 2ϰ_4(q)ϰ_2(q̅), y(∑_d ∈ 6 ℤ+1∑_ℓ∈ℤ q^Δ_1,3ℓ -d+2q̅^Δ_1,3 ℓ+2 + ∑_d ∈ 6 ℤ+5∑_ℓ∈ℤ q^Δ_1,3ℓ -d+1 q̅^Δ_1,3 ℓ+1)= 2|ϰ_1(q)|^2 + 2|ϰ_5(q)|^2. For N even, we find y∑_d ∈ 6 ℤ∑_ℓ∈ℤ q^Δ_0,3ℓ -d q̅^Δ_0,3 ℓ= |ϰ_0(q)|^2 + |ϰ_6(q)|^2, y∑_d ∈ 6 ℤ∑_ℓ∈ℤ q^Δ_1,3ℓ -d q̅^Δ_1,3 ℓ= 2 |ϰ_3(q)|^2, y∑_d ∈ 6 ℤ+2∑_ℓ∈ℤ q^Δ_0,3ℓ -d+1 q̅^Δ_0,3 ℓ+1 + y ∑_d ∈ 6 ℤ+4∑_ℓ∈ℤ q^Δ_0,3ℓ -d+2q̅^Δ_0,3 ℓ+2 = 2|ϰ_2(q)|^2 + 2|ϰ_4(q)|^2, y∑_d ∈ 6 ℤ+2∑_ℓ∈ℤ q^Δ_1,3ℓ -d+1 q̅^Δ_1,3 ℓ+1 + y∑_d ∈ 6 ℤ+4∑_ℓ∈ℤ q^Δ_1,3ℓ -d+2q̅^Δ_1,3 ℓ+2 = 2ϰ_1(q)ϰ_5(q̅) + 2ϰ_5(q)ϰ_1(q̅). These relations are obtained using simple algebraic manipulations. To illustrate, (<ref>) is proved as follows: y∑_d ∈ 6 ℤ∑_ℓ∈ℤ q^Δ_0,3ℓ -d q̅^Δ_0,3 ℓ = y ∑_k,ℓ∈ℤ q^Δ_0,3ℓ -6kq̅^Δ_0,3 ℓ = y ∑_r=0,1∑_j,k ∈ℤ q^Δ_0,6(j-k)+3rq̅^Δ_0,6 j + 3r = y ∑_r=0,1∑_i,j∈ℤ q^Δ_0,6i+3rq̅^Δ_0,6 j + 3r = y | ∑_i∈ℤ q^Δ_0,6i|^2 + y | ∑_i∈ℤ q^Δ_0,6i+3|^2 = |ϰ_0(q)|^2 + |ϰ_6(q)|^2. For N even, (<ref>) is the only seemingly anti-diagonal contribution. It can be changed to a diagonal contribution using the relation ϰ_1(q)ϰ_5(q̅) + ϰ_5(q)ϰ_1(q̅) = |ϰ_1(q)|^2 + |ϰ_5(q)|^2-1, which itself follows from ϰ_1(q)-ϰ_5(q) = 1. Using (<ref>), one can also write (<ref>) as an anti-diagonal contribution. Our final expressions for the conformal torus partition functions are =ϰ_0(q)ϰ_6(q̅) + 2(-1)^M ϰ_1(q)ϰ_5(q̅) + 2 ϰ_2(q)ϰ_4(q̅) + 2(-1)^M |ϰ_3(q)|^2 + 2 ϰ_4(q)ϰ_2(q̅) + 2(-1)^M ϰ_5(q)ϰ_1(q̅) + ϰ_6(q)ϰ_0(q̅) + 2(-1)^M, = |ϰ_0(q)|^2+ 2(-1)^M |ϰ_1(q)|^2 + 2 |ϰ_2(q)|^2+2(-1)^M |ϰ_3(q)|^2 +2 |ϰ_4(q)|^2+ 2(-1)^M |ϰ_5(q)|^2 + |ϰ_6(q)|^2 -2(-1)^M. These can be written in compact form: = ∑_j=0^6 ((-1)^M j d_j^6 ϰ_j(q)ϰ_6-j(q̅)) + 2(-1)^M, = ∑_j=0^6 ((-1)^M j d_j^6 |ϰ_j(q)|^2) - 2(-1)^M, with d_j^n defined in (<ref>). In terms of the u(1) characters, the odd and even parities of N thus correspond to different mixtures of diagonal and anti-diagonal sectors. For M and N even, the result agrees with the conjecture (<ref>) of Pearce and Rasmussen<cit.> with n_2,3 = -1 so that Z_p,p'(q)=Z^Circ_p,p'(q). It should also be noted that the partition functions (<ref>) for M odd are not genuine conformal partition functions, since some coefficients in the q,q̅ expansions are negative. Modular invariance and covariance. The u(1) characters behave as follows under the T and S transformations of the modular group: T: ϰ^n_j(^2 π(τ+1)) = exp(2 π (j^24n-124)) ϰ^n_j(^2 πτ), S: ϰ^n_j(^-2 π/τ) = ∑_k=0^2n-1 S_jk ϰ^n_k(^2 πτ) = 1/√(2n)∑_k=0^2n-1^-π j k/nϰ^n_k(^2 πτ). It follows that diagonal and anti-diagonal terms in (<ref>) transform trivially under T: |ϰ^n_j(^2 π(τ+1))|^2 = |ϰ^n_j(^2 π(τ))|^2, ϰ^n_j(^2 π(τ+1))ϰ^n_n-j(^-2 π(τ+1)) = (-1)^n ϰ^n_j(^2 πτ)ϰ^n_n-j(^-2 πτ), where we recall that n = p p' = 6 for percolation. The partition functions and are thus invariant under the action of T for all parities of M and N. The S matrix describing the transformations of the u(1) characters is obtained from (<ref>) and ϰ^n_j(q) =ϰ^n_2n-j(q). For n= pp' = 6, in the basis {ϰ_j(q), j = 0, …, 6}, we have S = 1/2 √(3)( [ 1 2 2 2 2 2 1; 1 √(3) 1 0 -1 -√(3) -1; 1 1 -1 -2 -1 1 1; 1 0 -2 0 2 0 -1; 1 -1 -1 2 -1 -1 1; 1 -√(3) 1 0 -1 √(3) -1; 1 -2 2 -2 2 -2 1 ]). Applying the S transform to (<ref>), we find that the torus partition functions transform as follows: (^-2 π/τ) = (^2 πτ), (^-2 π/τ) = (^2 πτ), (^-2 π/τ) = (^2 πτ). Therefore the partition functions for M,N both odd or both even are modular invariant, whereas the mixed cases are modular covariant, as they map to one another under the S transformation. § CONCLUSION In this paper, we derive and analyse truncated T- and Y-systems of functional relations satisfied by the transfer matrix eigenvalues of critical bond percolation considered as the loop model LM(2,3). Using analyticity properties of the eigenvalues and empirically based selections rules, we solve non-linear integral equations in the form of D_3 TBA equations to obtain exact expressions for the finite-size corrections and conformal data. These calculations are carried out for both the geometry of the strip with the boundaries consisting of simple half-arcs, and for periodic boundary conditions. Formulating selection rules encoding the patterns of zeros for all finite excitations, we give explicit expressions for finite-size spectrum generating functions in the various sectors. On the strip, our refined finitized characters reproduce the finitized Kac characters of <cit.>. Additionally, fixing the weight of the non-contractible loops to α=2, we obtain in the continuum scaling limit the conformal cylinder and torus partition functions. Our expressions for the conformal cylinder partition functions are to be compared to those found using Coulomb gas arguments in <cit.>. In particular, the partition function given in <cit.> with g = 2/3 almost coincides with our result (<ref>), with the difference that (<ref>) considers separately the partition functions for the N odd and N even cases, whereas the expression given in <cit.> is the sum of the two. The Coulomb gas argument is easily fixed by noting that for the dense loop model, the number of contractible loops has the same parity as N, and this produces the correct partition functions. Our expressions for the conformal torus partition functions depend on the parities of the lattice dimensions M and N. In each of the four cases, (q,q̅) is a simple sesquilinear form (<ref>) in the u(1) characters ϰ_j^6(q). Somewhat surprisingly, for M,N both even, the MIPF Z_2,3(q)=Z^Circ_2,3(q)= |ϰ_0(q)|^2+ 2 |ϰ_2(q)|^2+2 |ϰ_3(q)|^2 +2 |ϰ_4(q)|^2+ 2[ϰ_1(q)ϰ_5(q̅) + ϰ_5(q)ϰ_1(q̅)] + |ϰ_6(q)|^2 is non-diagonal and not of diagonal A-type. Consequently, it differs from the diagonal A-type MIPF of the triplet model <cit.>. Nevertheless, it is of the form conjectured by Pearce and Rasmussen <cit.> as in (<ref>): Z_p,p'(q)=Z^Proj_p,p'(q)+n_p,p' Z^Min_p,p'(q), n_p,p'∈ℤ since our lattice derivation shows that n_2,3 = -1. Based on the assumption of a diagonal A-type MIPF, Pearce and Rasmussen instead conjectured n_p,p'=2 for all p,p'. Knowing what we now know about the action of the modified trace, the result n_2,3 = -1 can be understood by observing that, if Z_2,3(q) has the form in (<ref>), its q,q̅ expansion starts out as Z_2,3(q)=(qq̅)^-1/24 + (1+n_2,3) + 2 (qq̅)^1/8 + 2 (qq̅)^1/3 + ⋯ The coefficient n_2,3 only appears in the constant term, with (1+n_2,3) counting the number of states with conformal weights Δ = Δ̅= 0, namely the Razumov-Stroganov eigenstates. The MIPF found from the loop model with α = 2 and M, N both even is in fact equal to the partition function of the six-vertex model on a torus with no twist. Indeed, the spin-chain representation of _n(α,β) with magnetisation m has the same spectra <cit.>, including degeneracies, as the standard module _n^d with d=m/2. Because the representations with magnetisation ± m are isomorphic, taking the normal trace of (u) in the full spin-chain representation precisely produces (<ref>): The sector with m=0 is singly degenerate, whereas all others are doubly degenerate. Since there are no Razumov-Stroganov eigenstates for N even in the untwisted case <cit.>, the expansion (<ref>) implies that n_2,3 = -1. Our result (<ref>) for Z_d(q,q̅) should then be compared with the XXZ partition function <cit.>. An explicit comparison with <cit.> specialised to h=3/8, Q = d/2, l = 0 and z=q reveals that the partition functions coincide for M even. For M odd, however, (<ref>) incorporates factors of (-1)^M to give the correct conformal partition functions for M odd. Ultimately, one of our goals is to derive the conformal torus partition functions for all the logarithmic minimal models ℒℳ(p,p'), for all parities of M and N. It is then natural to compare the results found here for percolation with those for critical dense polymers, corresponding to ℒℳ(1,2). For M even, the conformal torus partition functions of this model were found in <cit.> and expressed in terms of 𝒲-irreducible characters. In <ref>, we present the conformal partition functions for all parities of M and N in terms of u(1) characters. For M and N both even, we note that the torus partition function is just Z^Circ_1,2(q), hinting at the possibility that for ℒℳ(p,p'), the torus partition function for M and N even may equal Z^Circ_p,p'(q). For the other parities, the results for the two models are quite different and it is not possible to identify a general pattern. Clearly, more data is required before the full picture can emerge describing the conformal partition function in this case, and the structure of the indecomposable representations in general. From the lattice perspective, the study of the latter was indeed initiated in <cit.> and <cit.>. This has become the subject of an extensive investigation in the following years, and yet much is still to be understood about the indecomposable structures <cit.> of bulk logarithmic conformal field theories including percolation. This work leaves open a plethora of avenues for future research on an analytic approach to the logarithmic minimal models LM(p,p'). In particular, in this paper, we only consider boundary conditions on the strip corresponding to conformal Kac modules with highest weight Δ_1,s. However, the methods of this paper should extend to the more general Kac modules with highest weight Δ_r,s, which are realised on the lattice by including a seam on the boundary <cit.>. These methods should also extend to boundary conditions described by the one- and two-boundary Temperley-Lieb algebra <cit.>. In all these cases, it is expected that the transfer matrix eigenvalues will satisfy the same universal Y-system, encoded by the Dynkin diagram D_p'. However, the analyticity properties will differ in the various cases, leading to the different Kac conformal weights describing the finite-size corrections. Ultimately, it would also be of interest to obtain the modular invariants of the dilute loop models. Needless to say, the analytic and combinatorial classification problems in all these challenges promise to be formidable. §.§ Acknowledgments AMD was supported by the Belgian Interuniversity Attraction Poles Program P7/18 through the network DYGEST (Dynamics, Geometry and Statistical Physics) and by the FNRS fellowship CR28075116. AMD and AK acknowledge the support of the ERC grant Loop models, integrability and combinatorics and are grateful for the kind hospitality of Paul Zinn-Justin and the LPTHE where early stages of this work were done. AMD and PAP acknowledge the hospitality of the University of Wuppertal and are grateful for the kind hospitality of Holger Frahm at the ITP in Hannover where later stages of this work were done. All authors were supported by DFG through the program FOG 2316. PAP thanks the APCTP, Pohang for hospitality during the writing of this paper. The authors thank Paul Zinn-Justin, Jø rgen Rasmussen and Yacine Ikhlef for useful discussions. § INTEGRALS INVOLVING ROGERS DILOGARITHMS In this section, we study the following integrals: ℐ_1 = ∫_0^𝖺^1(∞)𝖺(ln(1+ 𝖺)/𝖺 - ln |𝖺|/1+𝖺), ℐ_2 = ∫_𝖺^2(-∞)^𝖺^2(∞)𝖺(ln(1+ ^3γ𝖺)/𝖺 - ^3γln |𝖺|/1+^3γ𝖺), ℐ_3 = ∫_𝖺^2(-∞)^𝖺^2(∞)𝖺(ln(1+ ^-3γ𝖺)/𝖺 - ^-3γln |𝖺|/1+^-3γ𝖺), with 𝖺^1(∞) = 4 cos^2γ-1, 𝖺^2(∞) = 2 cosγ, and consider two cases, namely 𝖺^2(-∞) = σ=± 1. All the integrals coming from the dilogarithm technique in <ref> can be written in terms of 𝒦_σ(γ): 𝒦_σ(γ)=ℐ_1+ℐ_2+ℐ_3. Because 𝒦_σ(γ) = 𝒦_σ(-γ) = 𝒦_σ(γ+ 2π), we restrict γ to the interval [0,π]. We perform the calculations by taking derivatives of 𝒦_σ=𝒦_σ(γ) with respect to γ. This removes all integrals and allows for calculations with explicit rational functions and logarithms. In some instances, we need to distinguish between the two cases σ=± 1. First we perform integration by parts on the second integrand in each integral and obtain ℐ_1 = 2∫_0^𝖺^1(∞)𝖺ln(1+ 𝖺)/𝖺 -ln(1+𝖺)ln |𝖺||_0^𝖺^1(∞) , ℐ_2 = 2∫_𝖺^2(-∞)^𝖺^2(∞)𝖺ln(1+ ^3γ𝖺)/𝖺 - ln(1+^3γ𝖺)ln |𝖺||_𝖺^2(-∞)^𝖺^2(∞) , ℐ_3 = 2∫_𝖺^2(-∞)^𝖺^2(∞)𝖺ln(1+ ^-3γ𝖺)/𝖺 - ln(1+^-3γ𝖺)ln |𝖺||_𝖺^2(-∞)^𝖺^2(∞) . Next we use a substitution of the variable of integration 𝖺↦^± 3γ𝖺 which moves the γ dependence from the integrand to the terminals, and use explicit expressions for the terminals in terms of =^γ: ℐ_1 = 2∫_0^^2+1+^-2𝖺ln(1+ 𝖺)/𝖺 -ln(1+𝖺)ln |𝖺||_0^^2+1+^-2 , ℐ_2 = 2∫_σ^3^^2+^4𝖺ln(1+𝖺)/𝖺 - ln(1+^3𝖺)ln |𝖺||_σ^+^-1 , ℐ_3 = 2∫_σ^-3^^-2+^-4𝖺ln(1+𝖺)/𝖺 - ln(1+^-3𝖺)ln |𝖺||_σ^+^-1 . Note that the explicit log-log terms vanish at the lower terminals. The derivatives with respect to read /𝒦_σ= +22 ^2-2 ^-2/^2+1+^-2ln(^2+2+^-2) +22^2+4^4/^2+^4ln(1+^2+^4) -6ln(1+σ^3) +2-2^-2-4^-4/^-2+^-4ln(1+^-2+^-4) +6ln(1+σ^-3) -2 ^2-2 ^-2/^2+2+^-2ln|^2+1+^-2| -2 ^2-2 ^-2/^2+1+^-2ln(^2+2+^-2) -2 ^2+4 ^4/1+^2+^4ln|+^-1| -- ^-1/+^-1ln(1+^2+^4) --2 ^-2-4 ^-4/1+^-2+^-4ln|+^-1| --^-1/+^-1ln(1+^-2+^-4). Grouping terms and using ln(^2+2+^-2)=ln(+^-1)^2=2ln|+^-1|, we find /𝒦_σ= 7+5^-1/+^-1ln(1+^2+^4) -5+7^-1/+^-1ln(1+^-2+^-4) -2-^-1/+^-1ln|^2+1+^-2| +6ln(1+σ^-3)-6ln(1+σ^3). Next we perform simplifications where care is taken when using the functional relations for the logarithm: ln(1+^2+^4) =ln[(^2+1+^-2)^2] =ln|^2+1+^-2| + 2ln - n π, ln(1+^-2+^-4) =ln[(^2+1+^-2)^-2] =ln|^2+1+^-2| - 2ln + n π, ln(1+σ^-3)-ln(1+σ^3) =-3ln+m π, where the integers m and n are given by n = {[ 0 γ∈ (0,π/3) ,; 1 γ∈ (π/3,2π/3) ,; 2 γ∈ (2π/3,π) , ]. m = {[ 0 γ∈ (0,π/3), σ = +1,; 2 γ∈ (π/3,π), σ = +1,; 1 γ∈ (0,2π/3), σ = -1,; 3 γ∈ (2π/3,π), σ = -1. ]. These integers are determined in the following way. The logarithms on the left-sides of (<ref>)-(<ref>) are the same as in (<ref>) and have imaginary parts in (-π,π] because of our choice of the branches. On the right-sides of (<ref>)-(<ref>), a real logarithm of a positive real number and ln=γ appear. The argument of the logarithms on the left-side of (<ref>) and (<ref>) is zero for γ = π/3, 2π/3. The range (0, π) therefore splits into the three subintervals on which n takes constant integer values: (0,π/3), (π/3,2π/3) and (2π/3, π). It is easy to check that the integer n has to be chosen in the above specified manner in order to have the imaginary part of the right-sides of (<ref>) and (<ref>) in (-π,π]. Exponentiating both sides of (<ref>), we find that the integer m is even for σ = 1 and odd for σ = -1. The arguments of the logarithms on the left-side of (<ref>) assumes the value 0 for γ=π/3 (2π/3) for σ=+1 (-1). We treat σ=+1 first. The integer valued m is constant in [0,π/3) and in (π/3,π). Inserting γ=0 and γ=2π/3 into (<ref>) we find m=0 and 2 respectively in these intervals. Next we treat σ=-1. The integer valued m is constant in (0,2π/3) and in (2π/3,π]. Inserting γ=π/3 and γ=π into (<ref>) we find m=1 and 3 in these intervals. Simplifying (<ref>) by use of (<ref>) yields /𝒦_σ=6ln+(6m-12 n)π. For the two cases, we find explicitly /𝒦_+ = {[ 6ln; -12π+6ln ]. [ 0 ≤γ≤2 π/3,; 2 π/3≤γ≤π, ] /𝒦_- = {[ 6π+ 6ln; -6π+ 6ln ]. [ 0 ≤γ≤π/3,; π/3≤γ≤π. ] Note that ln=γ and /γ=/. We next integrate with respect to γ: 𝒦_+ = {[ C_+-3γ^2; C_+-8π^2+12πγ-3γ^2 ]. [ 0 ≤γ≤2 π/3,; 2 π/3≤γ≤π, ] 𝒦_- = {[ C_–6πγ-3γ^2; C_–4π^2+6πγ-3γ^2 ]. [ 0 ≤γ≤π/3,; π/3≤γ≤π, ] where C_± are integration constants and the required continuity at γ=π/3 and γ=2π/3 has been imposed. The constants are determined quite easily by noting that 𝒦_1(π/3) = 𝒦_-1(2π/3) = 0, as for these values of γ the upper and lower terminals of all integrals coincide. This yields C_+=π^2/3 and C_-=4π^2/3. We finally find 1/8π^2𝒦_+(γ) = {[ 1/24 - 3/8(γ/π)^2; -23/24 + 3/2(γ/π)- 3/8(γ/π)^2 ]. [ 0 ≤γ≤2 π/3,; 2 π/3≤γ≤π, ] 1/8π^2𝒦_-(γ) = {[ 1/6 - 3/4(γ/π) - 3/8(γ/π)^2; -1/3 + 3/4(γ/π)- 3/8(γ/π)^2 ]. [ 0 ≤γ≤π/3,; π/3≤γ≤π. ] § SPECTRUM GENERATING FUNCTIONS AND CHARACTERS §.§ Identities for q-binomials The goal of this section is to derive the relation (<ref>) for the product of two Gaussian polynomials, which is useful in <ref> and <ref>. The q=1 specialisations of the identities derived below are proven in <cit.>, and here we follow the same ideas. From the definition (<ref>) of the q-binomial, nm is non zero for integers m,n with 0 ≤ m ≤ n. The definition of the q-binomial is extended to n<0 by using the definition (<ref>). This yields -ab= (-1)^b q^-1/2b(b-1) -aba+b-1b, a∈ℤ_> 0. It follows that nm is non zero for n<0, m∈ℤ_≥ 0. The q-binomials satisfy the relations ab = aa-b abbc = aca-cb-c ab = a-1b+q^a-ba-1b-1=q^ba-1b+a-1b-1 as well as the q-Vandermonde identity: a+bc = ∑_j ac-jbj q^j(a-c+j). Here and below, sums without bounds indicate that the variable is summed from -∞ to ∞, with the summand being non-zero on a finite range only. We first derive sum formulas for single q-binomials. A first relation is obtained as follows: n-pm(<ref>)=∑_k nm-k-pkq^k(n-m+k)(<ref>)=∑_k(-1)^k nm-kp+k-1kq^k(n-m+k)-1/2k(k-1)-pk. A second relation is given by n-pm (<ref>)= (-1)^m q^-1/2m(m-1)-m(p-n)p-n+m-1m (<ref>)= (-1)^m q^-1/2m(m-1)-m(p-n)∑_k pm-k-n+m-1kq^k(p-m+k) (<ref>)=q^-1/2m(m-1)-m(p-n)∑_k (-1)^m+kpm-kn-m+kkq^k(p-m+k)-1/2k(k-1)-k(n-m+1) = ∑_j(-1)^j n-jm-jpj q^1/2j(j+1)+j(n-m-p), where we substituted k = m-j at the last step. By replacing n-p with n, this identity can be rewritten as nm =∑_k(-1)^k n+p-km-kpkq^1/2k(k+1)+k(n-m) = ∑_k (-1)^k n+p-kn-m-kpkq^1/2k(k+1)+km (<ref>)=∑_k (-1)^k n+p-km+ppkq^1/2k(k+1)+km where we used nm = nn-m at the second equality to replace m by n-m. Using these relations, we now derive an identity for the product of two Gaussian polynomials: mpnr = m-n+r+n-rpnr(<ref>)=∑_k m-n+rp-kn-rknr q^k(m-n+r-p+k) (<ref>)=∑_k m-n+rp-kr+krnr+k q^k(m-n+r-p+k) (<ref>)=∑_k m-n+rp-kr+krq^k(m-n+r-p+k)∑_j(-1)^jn+p-k-jp+rp-kjq^1/2j(j+1)+j(r+k) = ∑_k m-n+rp-kr+krq^k(m-n+r-p+k)∑_i(-1)^p-k-in+ip+rp-kiq^1/2(p-k-i)(p+k-i+1+2r) (<ref>)=∑_i n+ip+r∑_k(-1)^p-k-ir+kkm-n+rp-kp-kiq^k(m-n+r-p+k)+1/2(p-k-i)(p+k-i+1+2r) (<ref>)=∑_i n+ip+rm-n+ri∑_k(-1)^p-k-ir+kkm-n+r-ip-i-kq^k(m-n+r-p+k)+1/2(p-k-i)(p+k-i+1+2r) (<ref>)=∑_i n+ip+rm-n+ri(-1)^p-im-n-i-1p-iq^1/2i(i-1)+1/2p(p+1)+pr-pi-ri (<ref>)=∑_i n+ip+rm-n+rin-m+pp-iq^1/2i(i-1)+1/2p(p+1)+pr-pi-ri-1/2(p-i)(p-i-1)+(p-i)(m-n-i-1) (<ref>)=∑_i n+ip+rm-n+rin-m+pn-m+i q^i^2+i(n-m-p-r)+mp-np+pr. The last line is an identity that is used multiple times in <ref>. §.§ Character identities for the boundary case In this section, we show that the finitized partition functions given in (<ref>) are equal to the finitized Kac characters _1,d+1^(N)(q). Let t ∈ℤ with t ≡ N 2. We define X^1_t = ∑_k≥ 0∑_i=0^k q^i^2 + 2k (k-i+1/2)+t(-t+i-k-1)N+t/2+i2k+1kik+1i+t+1, X^2_t = ∑_k≥ 1∑_i=0^k-1 q^i(i+1) + 2k (k-i-1/2)+t(-t+i-k+1)N+t/2+i2kk-1ik+1i+t+1, X^3_t = ∑_k≥ 0∑_i=0^k q^-1+i(i+1) + 2k (k-i-1/2)+t(-t+i-k-2)N+t/2+i2kkiki+t+1. These quantities are useful because of the following identities: d=3t: = q^d(d-1)/6(X^2_t - q^d+1 X^3_t + δ_d,0), d=3t+1: = q^d(d-1)/6(X^3_t-1 - q^d+1 X^2_-t-1), d = 3t+2: = q^d(d-1)/6(X^1_t - q^d+1 X^1_-t-2). In proving (<ref>), we start with the case d ≡ 2 3 which is easiest. By using (<ref>) and substituting j by k=i+j+t in (<ref>), we find = q^d(d-1)/6 ∑_i,k q^i^2 + 2k (k-i+1/2)+t(-t+i-k-1)N+t/2+i2k+1(kik+1i+t+1 - k+1iki+t+1). The positive part is readily identified as X^1_t. After substituting i for j=i+t+1 and setting t =-s-2 in the negative part, it is found to equal -q^-3(1+s)X^1_s = -q^d+1X^1_-t-2, ending the proof of (<ref>). For d = 3t and t>0, applying (<ref>) to (<ref>) yields four double-sums: = q^d(d-1)/6( ∑_i,k q^i(i+1)+2k(k-i-1/2)+t(-t+i-k+1)N+t/22k(k-1iki+t-kik-1i+t) + ∑_i,ℓ q^1+i(i+2)+2ℓ(ℓ-i-1/2)+t(-t+i-ℓ+2)N+t/22ℓ(ℓ-1iℓi+t+1-ℓiℓ-1i+t+1)), where the sum on the second line was obtained by substituting ℓ = k+1. Using the identity (<ref>), the first terms and second terms of each line of (<ref>) respectively combine, resulting in =q^d(d-1)/6∑_i,k q^i(i+1)+2k(k-i-1/2)+t(-t+i-k+1)N+t/22k(k-1ik+1i+t+1-kiki+t+1) = q^d(d-1)/6(X^2_t-q^d+1 X^3_t), ending the proof of (<ref>). The case d=0 is computed separately and the proof of (<ref>) in this case is a simple exercise in q-binomials, with the Razumov-Stroganov eigenvalue responsible for the extra factor δ_d,0. The proof of (<ref>) uses the ideas of the previous two cases and is straightforward. The identity (<ref>) derived in <ref> allows us to simplify X^1_t, X^2_t and X^3_t. The parameters m, n, p and q are specialised as follows: for X^1_t: m=N-t-22, n=N+t2, p=k-t, r=k+t+1, for X^2_t: m=N-t-22, n=N+t2, p=k-t, r=k+t, for X^3_t: m=N-t-22, n=N+t2, p=k-t-1, r=k+t+1. Starting from (<ref>), for X^1_t we find X^1_t (<ref>)=∑_k≥ 0 q^k(k+1)-t(t+1)N+t/2k+t+1N-t-2/2k-t = ∑_j q^j(2t+1+j)N+t/2j+2t+1N-t-2/2j (<ref>)=∑_j q^j(2t+1+j)N+t/2N-3t-2/2-jN-t-2/2j(<ref>)=N-1N-3t-2/2. Likewise for X^2_t: X^2_t (<ref>)=∑_k≥ 1 q^k^2-t^2N+t/2k+tN-t-2/2k-t = (∑_k≥ 0 q^k^2-t^2N+t/2k+tN-t-2/2k-t) - δ_t,0 (<ref>)=(∑_j q^j(2t+j)N+t/2j+2tN-t-2/2j) - δ_t,0 (<ref>)=( ∑_j q^j(2t+1+j)N+t/2N-3t-2/2-jN-t-2/2j) - δ_t,0(<ref>)=N-1N-3t-2/2- δ_t,0 where we used q^-t^2N+t/2tN+t/2-t= δ_t,0 at the second equality. The steps are identical for X^3_t: X^3_t (<ref>)=∑_k q^k^2-(t+1)^2N+t/2k+t+1N-t-2/2k-t-1 = ∑_j q^j(j+2t+2)N+t/2j+2t+2N-t-2/2j(<ref>)=N-1N-3t-4/2. Combining (<ref>), (<ref>) and (<ref>) with (<ref>), we find = q^d(d-1)/6(N-1N-d/2-q^d+1N-1N-d-4/2)= q^d(d-1)/6(NN-d/2-q^d+1NN-d-2/2) = _1,d+1(q), which holds for d ≡ 0,1,2 3. The second equality is obtained (from right to left) using (<ref>). §.§ Partition functions for the periodic case Explicit expressions. The finitized partition functions defined in (<ref>) are obtained by writing down the generating functions for each set in (<ref>) and summing over the corresponding values of i and j. We note that the indices i and j in the selection rules always run over all possible values for which the corresponding sets MLmnℓ are well defined. This is also true for k_1, k_2 and k_3 in (<ref>). For ease of notation, we omit to write the bounds of the sums over i,j,k_2 and k_3 and interpret the corresponding indices as running over ℤ, understanding that only finitely many terms are non-zero. The same applies to k_1, except that the sum is split between the odd and even values, as they correspond to different values of σ, see (<ref>). For d ≡ 0 3, we use the prescription (<ref>) for the separation between upper and lower halves and write down the generating function (<ref>) for each set in (<ref>). Summing over these sets, we find Z_d=3t^(N)(q,q̅) = ∑_i,j,k_2,k_3(∑_k_1 even (q q̅)^-1/24+ ∑_k_1 odd(-1)^M (q q̅)^-1/6) (qq̅)^E q^E_0⌊1/2(N+t/2+i)⌋ - ϵk_1⌊k_1/2⌋k_2⌊k_1+1/2⌋k_3 ×⌊1/2(N+t/2+i+1)⌋ + ϵ2(i+j+t)-k_1i+j+t-⌊k_1/2⌋i-k_2i+j+t-⌊k_1+1/2⌋i+t-k_3 where E and E_0 are obtained from the proper specialisations of (<ref>): E = 12(k_1^2+k_2^2+k_3^2-k_1k_2 - k_1 k_3), E_0 = i^2 - i k_1+ 2j(i+j-k_1+12 k_2+12k_3) + t(2i+3j - 32 k_1 + k_2 + 32 t). For d ≡ 1,2 3, from (<ref>), (<ref>) and (<ref>), is split between contributions coming from the different subcases: Z_d=3t+1^(N)(q,q̅) = Z_1^, + Z_1^, + Z_1^, + Z_1^, + (-1)^M δ_d,1, Z_d=3t+2^(N)(q,q̅) = Z_2^, + Z_2^, + Z_2^, + Z_2^, . Each Z_k^,, with ∈{, }, ∈{, } and k = 1,2, is obtained by using the prescription (<ref>) for the separation between upper and lower halves, writing down the corresponding generating function (<ref>), and summing over the corresponding sets given by the selection rules: Z_k^, = ∑_i,j,k_2,k_3(∑_k_1 even (-1)^M+ ∑_k_1 odd (q q̅)^1/8) (qq̅)^E() q^E_k(,) F_k(, ) with F_1(, ) = ⌊1/2(N+t-1/2+i)⌋ - ϵk_1⌊k_1-1/2⌋k_2⌊k_1/2⌋k_3 ×⌊1/2(N+t-1/2+i+1)⌋ + ϵ2(i+j+t)-k_1i+j+t-1-⌊k_1-1/2⌋i-k_2i+j+t-1-⌊k_1/2⌋i+t-1-k_3, F_1(, ) = F_2(, ) = ⌊1/2(N+t+1/2+i)⌋ - ϵk_1⌊k_1-1/2⌋k_2⌊k_1/2⌋k_3 ×⌊1/2(N+t+1/2+i+1)⌋ + ϵ2(i+j+t+1)-k_1i+j+t-⌊k_1-1/2⌋i-k_2i+j+t-⌊k_1/2⌋i+t-k_3, F_1(, ) = ⌊1/2(N+t+3/2+i)⌋ - ϵk_1⌊k_1-1/2⌋k_2⌊k_1/2⌋k_3 ×⌊1/2(N+t+3/2+i+1)⌋ + ϵ2(i+j+t+2)-k_1i+j+t+1-⌊k_1-1/2⌋i-k_2i+j+t+1-⌊k_1/2⌋i+t+1-k_3, and F_2(, ) = ⌊1/2(N+t+2/2+i)⌋ - ϵk_1⌊k_1-1/2⌋k_2⌊k_1/2⌋k_3 ×⌊1/2(N+t+2/2+i+1)⌋ + ϵ2(i+j+t+1)-k_1i+j+t-⌊k_1-1/2⌋i-k_2i+j+t-⌊k_1/2⌋i+t-k_3, F_2(, ) = F_2(, ) = ⌊1/2(N+t+2/2+i)⌋ - ϵk_1⌊k_1-1/2⌋k_2⌊k_1/2⌋k_3 ×⌊1/2(N+t+2/2+i+1)⌋ + ϵ2(i+j+t+2)-k_1i+j+t+1-⌊k_1-1/2⌋i-k_2i+j+t+1-⌊k_1/2⌋i+t+1-k_3, F_2(, ) = ⌊1/2(N+t+2/2+i)⌋ - ϵk_1⌊k_1-1/2⌋k_2⌊k_1/2⌋k_3 ×⌊1/2(N+t+2/2+i+1)⌋ + ϵ2(i+j+t+3)-k_1i+j+t+2-⌊k_1-1/2⌋i-k_2i+j+t+2-⌊k_1/2⌋i+t+2-k_3. The energies for the and subcases are obtained from (<ref>) under the proper specialisations of m, n and ℓ. The minimal energies for the and subcases are likewise obtained from (<ref>). We find E() = 12 (k_1^2+k_2^2+k_3^2-k_1+k_2+k_3-k_1 k_2-k_1 k_3), E() = 12 (k_1^2+k_2^2+k_3^2+k_2+k_3-k_1 k_2-k_1 k_3), and q^E_1(, ) =q^1 + i(i+1) - i k_1-1/2k_1 - k_2 + 2j(i+j-k_1+1/2 k_2+1/2k_3+1/2) + t(2i+3j - 3/2 k_1 + k_2 + 3/2 t+1/2), q^E_1(, ) =q^2 + i(i+3) - i k_1- 3/2k_1 + 2j(i+j-k_1+1/2 k_2+1/2k_3+2) + t(2i+3j - 3/2 k_1 + k_2 + 3/2 t+7/2), q^E_1(, ) =q^1 + i(i+2) - i k_1- 3/2k_1 + 2j(i+j-k_1+1/2 k_2+1/2k_3+3/2) + t(2i+3j - 3/2 k_1 + k_2 + 3/2 t+5/2), q^E_1(, ) =q^5 + i(i+4) - i k_1-5/2k_1+k_2 + 2j(i+j-k_1+1/2 k_2+1/2k_3+3) + t(2i+3j - 3/2 k_1 + k_2 + 3/2 t+11/2), q^E_2(, ) =q^1 + i(i+2) - i k_1-k_1 + 2j(i+j-k_1+1/2 k_2+1/2k_3+3/2) + t(2i+3j - 3/2 k_1 + k_2 + 3/2 t+5/2), q^E_2(, ) =q^7 + i(i+5) - i k_1- 3k_1 +k_2+ 2j(i+j-k_1+1/2 k_2+1/2k_3+7/2) + t(2i+3j - 3/2 k_1 + k_2 + 3/2 t+13/2), q^E_2(, ) =q^5 + i(i+4) - i k_1-3k_1 +k_2+ 2j(i+j-k_1+1/2 k_2+1/2k_3+3) + t(2i+3j - 3/2 k_1 + k_2 + 3/2 t+11/2), q^E_2(, ) =q^15 + i(i+7) - i k_1-5k_1+2k_2 + 2j(i+j-k_1+1/2 k_2+1/2k_3+5) + t(2i+3j - 3/2 k_1 + k_2 + 3/2 t+19/2). Scaling behavior for d ≡ 03. We explicitly derive the formula (<ref>) for the scaling limit of Z_d=3t^(N)(q,q̅). We perform the calculation separately for the odd and even k_1 contributions in (<ref>), namely we write Z_d=3t^(N)(q,q̅) = Z_0,even + (-1)^M Z_0,odd. We start with the even case. As discussed in <ref>, for ϵ≪ N, the scaling behavior of (<ref>) is independent of ϵ. In fact, ϵ can be chosen to depend on i, whose values are indeed much smaller than N for the leading eigenvalues. For the same reason, one could also choose ϵ to depend on j, k_1, k_2 and k_3. Here we make a special choice of ϵ that allows the computation to go forward, namely we choose ϵ such that the first q̅-binomial in (<ref>) does not depend on i. This allows us to perform the sums over i and j first, noting that their summands involve only powers of q and not of q̅. Defining N_t = N+t, we get Z_0,even=(q q̅)^-1/24∑_k_2,k_3∑_k_1 even (q q̅)^E⌊N_t/4⌋k_1k_1/2k_2k_1/2k_3 S_0,even where S_0,even = ∑_i,j q^E_0⌊N_t/4+1/2⌋+i2(i+j+t)-k_1i+j+t-k_1/2i-k_2i+j+t-k_1/2i+t-k_3 = ∑_k,ℓ q^E_0⌊N_t/4+1/2⌋+k_2+k2ℓ-k_1ℓ-k_1/2kℓ-k_1/2k+t+k_2-k_3. At the last equality, we substituted first ℓ = i+j+t and then k = i-k_2. In the last expression, the sums over k and ℓ still run over ℤ. Although the notation does not make it explicit, the expression (<ref>) for E_0 is understood as changing with every change of summation variables. It remains quadratic in the various parameters. The next step consists in using (<ref>) with m = ⌊N_t4 + 12⌋ + k_3 - t, n = ⌊N_t4 + 12⌋ + k_2, p = ℓ - k_1/2 - k_2 + k_3 - t, r = ℓ - k_1/2 + k_2 - k_3 + t. This yields S_0,even = ∑_ℓ q^E_0'⌊N_t4 + 12⌋ + k_3 - tℓ - k_1/2 - k_2 + k_3 - t⌊N_t4 + 12⌋ + k_2ℓ - k_1/2 + k_2 - k_3 + t where E_0' = -14 k_1^2 + 12 k_1 k_2 + 12 k_1 k_3 - k_2 k_3 + ℓ^2 - ℓ k_1 + 12 t^2 + t k_2 - t k_3. This remaining sum can be evaluated using (<ref>): S_0,even = ∑_j q^E_0'⌊N_t4 + 12⌋ + k_3 - tj + 2k_3 - 2k_2 - 2t⌊N_t4 + 12⌋ + k_2j = q^E_0”2⌊N_t4 + 12⌋ + k_2+k_3 - t⌊N_t4 + 12⌋ + 2k_2 - k_3 + t. Remarkably, E_0” satisfies E_0” + E = 32(k_3 -k_2-t)^2 = Δ_0,3(k_3-k_2-t)+ 124 with E given in (<ref>). The right-side depends only on the difference between k_3 and k_2. Changing the summation variables to k_3 = ℓ + k_2 and k_1 = 2i, we obtain Z_0,even=q̅^ -1/24∑_ℓ q^Δ_0,3(ℓ-t)∑_i,k_2q̅^E⌊N_t/4⌋2iik_2ik_2 + ℓ2⌊N_t4 + 12⌋ + 2k_2+ℓ - t⌊N_t4 + 12⌋ + k_2 - ℓ + t. The expression is thus reduced to sums of products of four binomials, with only one depending on q. While one may wish to reduce this expression further to a single sum with one binomial of each kind, this appears not to be feasible because the arguments of the remaining q-binomial involve both ℓ and k_2. However, both entries of this q-binomial scale linearly with N. We consider standard modules where the number d of defects remains small as N →∞, namely values of d such that t ≪ N. Recalling that k_2, ℓ≪ N_t for large N, in the scaling limit we have 2⌊N_t/4 + 1/2⌋ + 2k_2+ℓ - t⌊N_t4 + 12⌋ + k_2 - ℓ + t1/(q)_∞ and therefore Z_0,even≃q̅^ -1/24/(q)_∞∑_ℓ q^Δ_0,3(ℓ-t)∑_i,k_2q̅^E⌊N_t/4⌋2iik_2ik_2 + ℓ. Here, X ≃ Y means that X and Y are equal up to terms which go to zero in the scaling limit. The sums over i and k_2 involve only powers of q̅. The expression is not suitable for us to use (<ref>) to remove the sum over i, nor (<ref>) for the sum over k_2 because E is not of the correct form. We however note that ⌊N_t/4⌋2i≃⌊N_t/4⌋+k_22i and ∑_i,k_2q̅^E⌊N_t/4⌋+k_22iik_2ik_2 + ℓ = ∑_i q̅^E_0”'⌊N_t/4⌋-ℓi-ℓ⌊N_t/4⌋i+ℓ = q̅^3/2ℓ^22⌊N_t/4⌋ - ℓ⌊N_t/4⌋+ℓ≃q̅^3/2ℓ^2/(q̅)_∞ where we use (<ref>) and (<ref>) at the first and second equalities. Using Δ_0,3 ℓ = 3/2ℓ^2 - 1/24, we find Z_0,even≃1/(q)_∞ (q̅)_∞∑_ℓ q^Δ_0,3 (ℓ-t)q̅^Δ_0,3 ℓ. This is the first term in (<ref>). The derivation of the scaling behavior of Z_0,odd follows the same steps with only few modifications: Z_0,odd=(q q̅)^-1/6∑_k_2,k_3∑_k_1 odd (q q̅)^E⌊N_t/4⌋k_1k_1/2k_2k_1/2k_3 S_0,odd with S_0,odd = ∑_i,j q^E_0⌊N_t/4+1/2⌋+i2(i+j+t)-k_1i+j+t-k_1-1/2i-k_2i+j+t-k_1+1/2i+t-k_3 = ∑_k,ℓ q^E_0⌊N_t/4+1/2⌋+k_2+k2ℓ-k_1ℓ-k_1-1/2kℓ-k_1+1/2k+t+k_2-k_3 = ∑_ℓ q^E_0'⌊N_t4 + 12⌋ + k_3 - tℓ - k_1+1/2 - k_2 + k_3 - t⌊N_t4 + 12⌋ + k_2ℓ - k_1-1/2 + k_2 - k_3 + t = q^E_0”2⌊N_t4 + 12⌋ + k_2+k_3 - t⌊N_t4 + 12⌋ + 2k_2 - k_3 + t + 1≃q^E_0”/(q)_∞ with E_0' and E_0” adapted accordingly. In particular, E_0” satisfies E_0” + E = Δ_1,3(k_3-k_2-t)+ 16. Changing the summation variables to k = 2i+1 and k_3 = k_2 +ℓ, we find Z_0,odd ≃q̅^ -1/6/(q)_∞∑_ℓ q^Δ_1,3(ℓ-t)∑_i,k_2q̅^E⌊N_t/4⌋2i+1ik_2i+1k_2 + ℓ ≃q̅^ -1/6/(q)_∞∑_ℓ q^Δ_1,3(ℓ-t)∑_i,k_2q̅^E⌊N_t/4⌋+k2i+1ik_2i+1k_2 + ℓ = q̅^ -1/6/(q)_∞∑_ℓ q^Δ_1,3(ℓ-t)∑_iq̅^E_0”'⌊N_t/4⌋-ℓi+1-ℓ⌊N_t/4⌋i+ℓ = 1/(q)_∞∑_ℓ q^Δ_1,3(ℓ-t)q̅^Δ_1,3ℓ2⌊N_t/4⌋ - ℓ⌊N_t/4⌋+ℓ-1≃1/(q)_∞ (q̅)_∞∑_ℓ q^Δ_1,3 (ℓ-t)q̅^Δ_1,3 ℓ. This ends the proof of (<ref>). Scaling behavior for d ≡ 1,23. These cases are more complicated because of the various contributions to in (<ref>). For d ≡ 1 3, we do the calculations only for d>1 for simplicity. The case d=1 uses the same arguments, with extra care given to the bounds of the sums and the contribution from the Razumov-Stroganov eigenvalue. Writing d = 3t+1 with t>0, each contribution to (<ref>) splits into an even and an odd part as Z_1^, = (-1)^M Z_1,even^, + Z_1,odd^, , ∈{, }, ∈{, }. Starting with the even contributions, as before, we choose ϵ such that the q̅-binomials do not depend on i. Defining N_t = N+t+1, we obtain Z^,_1,even= ∑_k_2,k_3∑_k_1 even (q q̅)^E()⌊N_t/4⌋k_1k_1/2-1k_2k_1/2k_3 S^,_1,even with S^, _1,even = ∑_i,j q^E_1(, )⌊N_t/4+1/2⌋+i-12(i+j+t)-k_1i+j+t-k_1/2i-k_2i+j+t-k_1/2-1i+t-k_3-1 = ∑_k,ℓ q^E_1(, )⌊N_t/4+1/2⌋+k_2+k-12ℓ-k_1ℓ-k_1/2kℓ-k_1/2-1k+t+k_2-k_3-1, S^, _1,even = ∑_i,j q^E_1(, )⌊N_t/4+1/2⌋+i2(i+j+t+1)-k_1i+j+t+1-k_1/2i-k_2i+j+t-k_1/2i+t-k_3 = ∑_k,ℓ q^E_1(, )⌊N_t/4+1/2⌋+k_2+k-12ℓ-k_1ℓ-k_1/2k-1ℓ-k_1/2-1k+t+k_2-k_3-1, S^, _1,even = ∑_i,j q^E_1(, )⌊N_t/4+1/2⌋+i2(i+j+t+1)-k_1i+j+t+1-k_1/2i-k_2i+j+t-k_1/2i+t-k_3 = ∑_k,ℓ q^E_1(, )⌊N_t/4+1/2⌋+k_2+k2ℓ-k_1ℓ-k_1/2kℓ-k_1/2-1k+t+k_2-k_3, S^, _1,even = ∑_i,j q^E_1(, )⌊N_t/4+1/2⌋+i+12(i+j+t+2)-k_1i+j+t+2-k_1/2i-k_2i+j+t-k_1/2+1i+t-k_3+1 = ∑_k,ℓ q^E_1(, )⌊N_t/4+1/2⌋+k_2+k2ℓ-k_1ℓ-k_1/2k-1ℓ-k_1/2-1k+t+k_2-k_3. It is not possible to simplify each S^, _1,even individually using (<ref>). One instead combines S^, and S^, using E_1(, ) - E_1(, ) = j and (<ref>): S^, _1,even + S^, _1,even = ∑_k,ℓ q^E_1(, )⌊N_t/4+1/2⌋+k_2+k-12ℓ-k_1ℓ-k_1/2+1kℓ-k_1/2-1k+t+k_2-k_3-1, = ∑_ℓ q^E_1'(, )⌊N_t4 + 12⌋ + k_3 - tℓ - k_1/2 - k_2 + k_3 - t⌊N_t4 + 12⌋ + k_2-1ℓ - k_1/2 + k_2 - k_3 + t = q^E_1”(, )2⌊N_t4 + 12⌋ + k_2+k_3 - t-1⌊N_t4 + 12⌋ + 2k_2 - k_3 + t≃q^E_1”(, )/(q)_∞ with E_1”(, ) satisfying E_1”(, )+E() = Δ_1,3(k_3-k_2-t)+1. The relations (<ref>) and (<ref>) were used for the second and third equality. Changing the summation variables in (<ref>) to k_1 = 2i and k_3 = k_2 + ℓ, we obtain Z_1, even^, + Z_1, even^,≃1/(q)_∞∑_ℓ q^Δ_1,3(ℓ-t)+1∑_i,k_2q̅^E()⌊N_t/4⌋2ii-1k_2ik_2+ℓ. Likewise for S^, and S^,, we use E_1(, ) - E_1(, ) =j and find after simplification: S^, _1,even + S^, _1,even = q^E_1”(, )2⌊N_t4 + 12⌋ + k_2+k_3 - t⌊N_t4 + 12⌋ + 2k_2 - k_3 + t + 2≃q^E_1”(, )/(q)_∞ with E_1”(, )+E() = Δ_1,3(k_3-k_2-t-1)+1. With k = 2i, k_2 = k_2'-1 and k_3 = k_2' + ℓ, we get Z_1, even^, + Z_1, even^,≃1/(q)_∞∑_ℓ q^Δ_1,3(ℓ-t)+1∑_i,k'_2q̅^E()⌊N_t/4⌋2ii-1k'_2-1ik'_2+ℓ. Putting these results together, we find Z_1, even^, + Z_1, even^, + Z_1, even^, + Z_1, even^,≃1/(q)_∞∑_ℓ q^Δ_1,3(ℓ-t)+1∑_i,k_2q̅^E_1()⌊N_t/4⌋2iik_2ik_2 + ℓ ≃1/(q)_∞∑_ℓ q^Δ_1,3(ℓ-t)+1∑_i,k_2q̅^E_1()⌊N_t/4⌋+k_22iik_2ik_2 + ℓ = … = 1/(q)_∞∑_ℓ q^Δ_1,3(ℓ-t)+1q̅^Δ_1,3ℓ+22⌊N_t/4⌋ - ℓ⌊N_t/4⌋+ℓ≃1/(q)_∞ (q̅)_∞∑_ℓ q^Δ_1,3 (ℓ-t)+1q̅^Δ_1,3 ℓ+2 where the last steps follow those in (<ref>). The result produces the second term in (<ref>). The odd contributions are treated similarly. We thus only write down the intermediate results. Each contribution is written as Z^,_1,odd= (q q̅)^1/8∑_k_2,k_3∑_k_1 odd (q q̅)^E()⌊N_t/4⌋k_1k_1-1/2k_2k_1-1/2k_3 S^,_1,odd with S^, _1,odd = ∑_k,ℓ q^E_1(, )⌊N_t/4+1/2⌋+k_2+k-12ℓ-k_1ℓ-k_1+1/2kℓ-k_1+1/2k+t+k_2-k_3-1, S^, _1,odd = ∑_k,ℓ q^E_1(, )⌊N_t/4+1/2⌋+k_2+k-12ℓ-k_1ℓ-k_1+1/2k-1ℓ-k_1+1/2k+t+k_2-k_3-1, S^, _1,odd = ∑_k,ℓ q^E_1(, )⌊N_t/4+1/2⌋+k_2+k2ℓ-k_1ℓ-k_1+1/2kℓ-k_1+1/2k+t+k_2-k_3, S^, _1,odd = ∑_k,ℓ q^E_1(, )⌊N_t/4+1/2⌋+k_2+k2ℓ-k_1ℓ-k_1+1/2k-1ℓ-k_1+1/2k+t+k_2-k_3. These combine pairwise: S^, _1,odd + S^, _1,odd≃q^Δ_0,3(k_3-k_2-t)+1-1/8-E()/(q)_∞, S^, _1,odd + S^, _1,odd≃q^Δ_0,3(k_3-k_2-t)-2-1/8-E()/(q)_∞. The partial partition functions then satisfy Z_1, odd^, + Z_1, odd^, ≃q̅^1/8/(q)_∞∑_ℓ q^Δ_0,3(ℓ-t)+1∑_i,k_2q̅^E()⌊N_t/4⌋2i+1ik_2ik_2+ℓ, Z_1, odd^, + Z_1, odd^, ≃q̅^1/8/(q)_∞∑_ℓ q^Δ_0,3(ℓ-t)+1∑_i,k_2q̅^E()⌊N_t/4⌋2i+1ik_2-1ik_2+ℓ, and the final result is Z_1, odd^, + Z_1, odd^, + Z_1, odd^, + Z_1, odd^,≃1/(q)_∞ (q̅)_∞∑_ℓ q^Δ_0,3 (ℓ-t)+1q̅^Δ_0,3 ℓ+2, as announced in (<ref>). For d ≡ 2 3, the derivation is done using the same ideas. One considers separately the odd and even k_1 contributions. For the even case (and likewise for the odd case), one chooses ϵ so only the q-dependent part depends on the sum label i. One writes down the S^, _2,even corresponding to each Z^,_2,even, and then combines S^, _2,even with S^, _2,even and S^, _2,even with S^, _2,even. Their scaling limits are evaluated using (<ref>) and (<ref>) as the ratio of a power of q with (q)_∞. Combining the four contributions and using (<ref>) and (<ref>) one last time, the final result is Z_2, even^, + Z_2, even^, + Z_2, even^, + Z_2, even^, ≃1/(q)_∞ (q̅)_∞∑_ℓ q^Δ_1,3 (ℓ-t)-1q̅^Δ_1,3 ℓ+1, Z_2, odd^, + Z_2, odd^, + Z_2, odd^, + Z_2, odd^, ≃1/(q)_∞ (q̅)_∞∑_ℓ q^Δ_0,3 (ℓ-t)-1q̅^Δ_0,3 ℓ+1, consistent with (<ref>). § TORUS PARTITION FUNCTIONS OF CRITICAL DENSE POLYMERS For critical dense polymers, namely ℒℳ(1,2), the torus conformal partition functions for even M are written in terms of 𝒲-irreducible characters in <cit.>. Here we express these results in terms of the u(1) characters and present the results for M odd. In this case, one must keep track of the overall sign ε of each eigenvalue. In the scaling limit, the partition function in each standard module _n^d then reads Z_d(q,q̅) = (q q̅)^1/12/(q)_∞(q̅)_∞∑_ℓ∈ℤ (-1)^M ℓ q^Δ_2ℓ +d/2q̅^Δ_2ℓ -d/2. This holds for the odd and even parities of N and for all values of d. The torus partition of the loop model with α = 2 is computed from the modified trace (<ref>). After simplification, we find that it can be written in terms of the u(1) characters ϰ^8_j(q): = ∑_j=1,3,5,7 d_j^8 ( |ϰ^8_j(q)|^2 + (-1)^M ϰ^8_j(q)ϰ^8_8-j(q̅)), = ∑_j=0,2,4,6,8 d_j^8 ( |ϰ^8_j(q)|^2 + (-1)^M ϰ^8_j(q)ϰ^8_8-j(q̅)). From the transformation laws (<ref>), the cases with M and N of the same parity are found to be fully modular invariant, whereas the cases with opposite parities are invariant under the action of T and covariant under the action of S, as is the case for percolation in (<ref>). Using ϰ_2j^4n(q) ±ϰ_4n-2j^4n(q) = ϰ_j^n(q,± 1), the torus partition function for critical dense polymers can also be written in terms of the u(1)-characters ϰ^2_j(q,±1), with j integer and half-integer for N even and odd respectively: = 2 ϰ^2_1/2(q,1)ϰ^2_1/2(q̅,(-1)^M) + 2 ϰ^2_3/2(q,1)ϰ^2_3/2(q̅,(-1)^M), = ϰ^2_0(q,1)ϰ^2_0(q̅,(-1)^M) +2 ϰ^2_1(q,1)ϰ^2_1(q̅,(-1)^M) + ϰ^2_2(q,1)ϰ^2_2(q̅,(-1)^M). For M and N both even, the resulting conformal torus partition function is Z^Circ_1,2(q). § EXAMPLES OF PATTERNS OF ZEROS §.§ Strip boundary conditions [ [ d=0 d=2 d=4 d=6 d=8 d=10 d=12 d=14; [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [shift=-1.0](-0.4,0)(1.4,-3.7) [fillstyle=solid,fillcolor=lightgray,linecolor=lightgray](0,-3)(0,0)(1,0)(1,-3) -(0,-3)(0,0)(1,0)(1,-3) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-0.9)(0,-1.2)(0,-1.5)(0,-1.8) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-0.9)(1,-1.2)(1,-1.5)(1,-1.8) [dotsize=0.09cm](0.5,-2.1) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-0.9)(0,-1.2)(0,-1.5) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-0.9)(1,-1.2)(1,-1.5) [dotsize=0.09cm](0.5,-1.8)(0.5,-2.1) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) (-0.4,-3.3)B [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-0.9)(0,-1.2) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-0.9)(1,-1.2) [dotsize=0.09cm](0.5,-1.5)(0.5,-1.8)(0.5,-2.1)(0.5,-2.4) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.6)(0.5,-0.6)(1,-0.6) (-0.4,-3.3)A [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-0.9) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-0.9) [dotsize=0.09cm](0.5,-1.2)(0.5,-1.5)(0.5,-1.8)(0.5,-2.1)(0.5,-2.4) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.6)(0,-1.2) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) [dotsize=0.09cm](1,-0.6)(1,-1.2) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3)(0,-0.6) [dotsize=0.09cm](1,-0.3)(1,-0.6) [dotsize=0.09cm](0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8)(0.5,-2.1)(0.5,-2.4) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.6)(0,-1.2) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) [dotsize=0.09cm](1,-0.6)(1,-1.2) (-0.4,-3.3)B [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](1,-0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8)(0.5,-2.1)(0.5,-2.4)(0.5,-2.7) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.6)(0,-1.2)(0,-1.8) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2)(0.5,-1.8) [dotsize=0.09cm](1,-0.6)(1,-1.2)(1,-1.8) (-0.4,-3.3)A [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8)(0.5,-2.1)(0.5,-2.4)(0.5,-2.7) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.6)(0,-1.2)(0,-1.8)(0,-2.4) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2)(0.5,-1.8)(0.5,-2.4) [dotsize=0.09cm](1,-0.6)(1,-1.2)(1,-1.8)(1,-2.4); k^j: * 1 1,2 0,1,2,3 1, …, 5 1, …, 6 0, …, 7 1, …, 9; ℓ^i: * * * 1 1,2 1,2 1,2,3 1,2,3,4; q^Δ: 1 q^1/3 q^2 q^5 q^28/3 q^15 q^22 q^91/3 ]; ; [ d=1 d=3 d=5 d=7 d=9 d=11 d=13 d=15; [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [shift=-1.0](-0.4,0)(1.4,-3.7) [fillstyle=solid,fillcolor=lightgray,linecolor=lightgray](0,-3)(0,0)(1,0)(1,-3) -(0,-3)(0,0)(1,0)(1,-3) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-0.9)(0,-1.2)(0,-1.5)(0,-1.8) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-0.9)(1,-1.2)(1,-1.5)(1,-1.8) [dotsize=0.09cm](0.5,-2.1)(0.5,-2.4) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) (-0.4,-3.3)A [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-0.9)(0,-1.2)(0,-1.5) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-0.9)(1,-1.2)(1,-1.5) [dotsize=0.09cm](0.5,-1.8)(0.5,-2.1)(0.5,-2.4) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](1,-0.6) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-0.9)(0,-1.2) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-0.9)(1,-1.2) [dotsize=0.09cm](0.5,-1.5)(0.5,-1.8)(0.5,-2.1)(0.5,-2.4) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](1,-0.6) (-0.4,-3.3)B [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-0.9) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-0.9) [dotsize=0.09cm](0.5,-1.2)(0.5,-1.5)(0.5,-1.8)(0.5,-2.1)(0.5,-2.4)(0.5,-2.7) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.6)(0,-1.2) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) [dotsize=0.09cm](1,-0.6)(1,-1.2) (-0.4,-3.3)A [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3)(0,-0.6) [dotsize=0.09cm](1,-0.3)(1,-0.6) [dotsize=0.09cm](0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8)(0.5,-2.1)(0.5,-2.4)(0.5,-2.7) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.6)(0,-1.2)(0,-1.8) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2)(0.5,-1.8) [dotsize=0.09cm](1,-0.6)(1,-1.2)(1,-1.8) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](1,-0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8)(0.5,-2.1)(0.5,-2.4)(0.5,-2.7) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.6)(0,-1.2)(0,-1.8) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2)(0.5,-1.8) [dotsize=0.09cm](1,-0.6)(1,-1.2)(1,-1.8) (-0.4,-3.3)B [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8)(0.5,-2.1)(0.5,-2.4)(0.5,-2.7)(0.5,-3.0) [shift=-1.0](-0.4,0)(1.4,-3.7) -(0,-3)(0,0)(1,0)(1,-3) [dotsize=0.09cm](0,-0.6)(0,-1.2)(0,-1.8)(0,-2.4) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2)(0.5,-1.8)(0.5,-2.4) [dotsize=0.09cm](1,-0.6)(1,-1.2)(1,-1.8)(1,-2.4) (-0.4,-3.3)A; k^j: * 0,1 1,2,3 1,2,3,4 0,…,5 1, …, 7 1, …, 8 0, …, 9; ℓ^i: * * 1 1 1,2 1,2,3 1,2,3 1,2,3,4; q^Δ: 1 q q^10/3 q^7 q^12 q^55/3 q^26 q^35; ] ] figureThe patterns of zeros for the ground states in _14^d and _15^d. [ [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-0.9) [dotsize=0.09cm](0.5,-1.2) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-0.9) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-1.2) [dotsize=0.09cm](0.5,-0.9) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3)(0,-0.6) [dotsize=0.09cm](0.5,-0.9)(0.5,-1.2)(0.5,-1.5) [dotsize=0.09cm](1,-0.3)(1,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3)(0,-0.9)(0,-1.2) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](1,-0.3)(1,-0.9)(1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3)(0,-0.9) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2)(0.5,-1.5) [dotsize=0.09cm](1,-0.3)(1,-0.9) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6)(0,-0.9)(0,-1.2) [dotsize=0.09cm](0.5,-0.3) [dotsize=0.09cm](1,-0.6)(1,-0.9)(1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3)(0,-1.2) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9)(0.5,-1.5) [dotsize=0.09cm](1,-0.3)(1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](1,-0.3) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,-0.6) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](1,-0.6); k^j: 1 2 1,2,3 3 1, 2,4 4 1,3,4 1,2,3; ℓ^i: * * 1,1 * 1,1 * 1,1 *; q^Δ: q^1/3 q^4/3 q^7/3 q^7/3 q^10/3 q^10/3 q^13/3 q^13/3 ] [ [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6)(0,-0.9) [dotsize=0.09cm](0.5,-0.3)(0.5,-1.2)(0.5,-1.5) [dotsize=0.09cm](1,-0.6)(1,-0.9) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3)(0,-1.5) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](1,-0.3)(1,-1.5) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6)(0,-1.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.9)(0.5,-1.5) [dotsize=0.09cm](1,-0.6)(1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](1,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,-0.6) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](1,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8) [dotsize=0.09cm](1,-0.3) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6)(0,-1.5) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](1,-0.6)(1,-1.5) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.9)(0,-1.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-1.5) [dotsize=0.09cm](1,-0.9)(1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.9) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-1.2) [dotsize=0.09cm](1,-0.9) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,-0.6) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](1,-0.6); k^j: 1,2,5 2,3,4 1,3,5 1,2,4 1,2,3,4,5 2,3,5 1,4,5 1,3,4; ℓ^i: 1,1 1,1 1,1 * 1,1,2,2 1,1 1,1 *; q^Δ: q^13/3 q^16/3 q^16/3 q^16/3 q^19/3 q^19/3 q^19/3 q^19/3 ] [ [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8) [dotsize=0.09cm](1,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.9)(0,-1.5) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-1.2) [dotsize=0.09cm](1,-0.9)(1,-1.5) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-1.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9) [dotsize=0.09cm](1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,-0.6) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](1,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.9) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8) [dotsize=0.09cm](1,-0.9) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-1.2)(0,-1.5) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9) [dotsize=0.09cm](1,-1.2)(1,-1.5) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,-0.6) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-1.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](1,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-1.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.5)(0.5,-1.8) [dotsize=0.09cm](1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,-1.2); k^j: 1,2,3,4,6 2,4,5 2,3,4 1,2,3,5,6 3,4,5 1,2,3,4,5 1,2,4,5,6; ℓ^i: 1,1,2,2 1,1 * 1,1,2,2 1,1 1,1 1,1,2,2; q^Δ: q^22/3 q^22/3 q^22/3 q^25/3 q^25/3 q^25/3 q^28/3 ] [ [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6)(0,-1.2) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) [dotsize=0.09cm](1,-0.6)(1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-1.5) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.8) [dotsize=0.09cm](1,-1.5) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,-1.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-1.8) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5) [dotsize=0.09cm](1,-1.8) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8)(0.5,-2.1) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,-1.2)(0.5,-1.8); k^j: 1,2,3,4,5 1,3,4,5,6 1,2,3,4,5 2,3,4,5,6 1,2,3,4,5,6,7; ℓ^i: 1,2 1,1,2,2 2,2 1,1,2,2 1,1,2,2,3,3; q^Δ: q^28/3 q^31/3 q^31/3 q^34/3 q^37/3; ]figureThe 28 patterns of zeros for _8^2. [ [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3)(0,-0.6) [dotsize=0.09cm](0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](1,-0.3)(1,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3)(0,-0.9) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) [dotsize=0.09cm](1,-0.3)(1,-0.9) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3)(0,-1.2) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9) [dotsize=0.09cm](1,-0.3)(1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](1,-0.6)(1,-0.9) [dotsize=0.09cm](0.5,-0.3)(0.5,-1.2) [dotsize=0.09cm](0,-0.6)(0,-0.9) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](1,-0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5) [dotsize=0.09cm](0,-0.3) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](1,-0.6) (-0.4,-2.5)A [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6)(0,-1.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.9) [dotsize=0.09cm](1,-0.6)(1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5) [dotsize=0.09cm](1,-0.3) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](1,-0.6) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5) [dotsize=0.09cm](0,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](1,-0.6) (-0.4,-2.5)A; k^j: 1,2 1,3 2,3 1,4 0,1,2,3 2,4 1,2,3,4 0,1,2,4; ℓ^i: * * * * 1 * 1,1 1; q^Δ: q^2 q^3 q^4 q^4 q^5 q^5 q^6 q^6; ] [ [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.9)(0,-1.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6) [dotsize=0.09cm](1,-0.9)(1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5) [dotsize=0.09cm](1,-0.6) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.9) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-1.2)(0.5,-1.5) [dotsize=0.09cm](1,-0.9) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](1,-0.6) (-0.4,-2.5)A [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.9) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-1.2)(0.5,-1.5) [dotsize=0.09cm](1,-0.9) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-1.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.5) [dotsize=0.09cm](1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](1,-0.6) (-0.4,-2.5)A [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,-0.6) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](1,-0.6) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-1.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.5) [dotsize=0.09cm](1,-1.2) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-1.5) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](1,-1.5) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](1,-0.6) (-0.4,-2.5)A; k^j: 3,4 1,2,3,5 0,1,3,4 1,2,4,5 0,2,3,4 1,2,3,4 1,3,4,5 1,2,3,4; ℓ^i: * 1,1 1 1,1 1 * 1,1 1; q^Δ: q^6 q^7 q^7 q^8 q^8 q^8 q^9 q^9; ] [ [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-1.5) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](1,-1.5) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) (-0.4,-2.5)B [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](1,-0.6) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-1.2) (-0.4,-2.5)A [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0,-1.2) [dotsize=0.09cm](0.5,-1.2) [dotsize=0.09cm](1,-1.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) (-0.4,-2.5)A [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2)(0.5,-1.5)(0.5,-1.8) [shift=-1.5](-0.4,-3.2)(1.4,0.5) -(0,-2.2)(0,0)(1,0)(1,-2.2) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) (-0.4,-2.5)B; k^j: 2,3,4,5 0,1,2,3,4,5 0,1,2,3,4,5 1,2,3,4,5,6; ℓ^i: 1,1 1,1,2 1,2,2 1,1,2,2; q^Δ: q^10 q^10 q^11 q^12; ]figureThe 20 patterns of zeros for _8^4 §.§ Periodic boundary conditions [ [ d=0 d=2 d=4 d=6 d=8 d=10 d=12; [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.3)(0,0.6)(0,0.9) [dotsize=0.09cm](0,-0.3)(0,-0.6)(0,-0.9) [dotsize=0.09cm](1,0.3)(1,0.6)(1,0.9) [dotsize=0.09cm](1,-0.3)(1,-0.6)(1,-0.9) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(1,0) [dotsize=0.09cm](0,0.3)(0,0.6) [dotsize=0.09cm](0,-0.3)(0,-0.6) [dotsize=0.09cm](1,0.3)(1,0.6) [dotsize=0.09cm](1,-0.3)(1,-0.6) [dotsize=0.09cm](0.5,0.9)(0.5,-0.9) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) (-0.4,-2.0)(-0.4,2.0)A [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.3)(0,0.6) [dotsize=0.09cm](0,-0.3)(0,-0.6) [dotsize=0.09cm](1,0.3)(1,0.6) [dotsize=0.09cm](1,-0.3)(1,-0.6) [dotsize=0.09cm](0.5,0.9)(0.5,-0.9) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) (-0.4,-2.0)(-0.4,2.0)B [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(1,0) [dotsize=0.09cm](0,0.3) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9) [dotsize=0.09cm](0.5,0.6)(0.5,0.9) [dotsize=0.09cm](1,0.3) [dotsize=0.09cm](1,-0.3) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0,0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](0.5,0.6) [dotsize=0.09cm](1,-0.6) [dotsize=0.09cm](1,0.6) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.3) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](0.5,0.6)(0.5,0.9)(0.5,1.2) [dotsize=0.09cm](1,0.3) [dotsize=0.09cm](1,-0.3) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0,0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](0.5,0.6) [dotsize=0.09cm](1,-0.6) [dotsize=0.09cm](1,0.6) (-0.4,-2.0)(-0.4,2.0)A [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(1,0) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9) [dotsize=0.09cm](0.5,0.3)(0.5,0.6)(0.5,0.9) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0,0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](0.5,0.6) [dotsize=0.09cm](1,-0.6) [dotsize=0.09cm](1,0.6) (-0.4,-2.0)(-0.4,2.0)B [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](0.5,0.3)(0.5,0.6)(0.5,0.9)(0.5,1.2) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6)(0,-1.2) [dotsize=0.09cm](0,0.6)(0,1.2) [dotsize=0.09cm](0.5,-0.6)(0.5,-1.2) [dotsize=0.09cm](0.5,0.6)(0.5,1.2) [dotsize=0.09cm](1,-0.6)(1,-1.2) [dotsize=0.09cm](1,0.6)(1,1.2); σ: 1 1 1 1 1 1 1; k_-^j k_+^j: ** 00 11 1,21,2 0,1,20,1,2 1,2,31,2,3 1,2,3,41,2,3,4; ℓ_-^jℓ_+^j: ** ** ** 11 11 11 1,21,2; q^Δq̅^Δ̅: (qq̅)^-1/24 (qq̅)^1/8 (qq̅)^5/8 (qq̅)^35/24 (qq̅)^21/8 (qq̅)^33/8 (qq̅)^143/24 ]; ; [ d=1 d=3 d=5 d=7 d=9 d=11 d=13; [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [shift=-1.2](-0.4,-2.5)(1.4,2.5) [fillstyle=solid,fillcolor=lightgray,linecolor=lightgray](0,-1.7)(0,1.7)(1,1.7)(1,-1.7) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(1,0) [dotsize=0.09cm](0,0.3)(0,0.6) [dotsize=0.09cm](0,-0.3)(0,-0.6) [dotsize=0.09cm](1,0.3)(1,0.6) [dotsize=0.09cm](1,-0.3)(1,-0.6) [dotsize=0.09cm](0.5,0.9)(0.5,-0.9) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(0.5,0)(1,0) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.3)(0,0.6) [dotsize=0.09cm](0,-0.3)(0,-0.6) [dotsize=0.09cm](1,0.3)(1,0.6) [dotsize=0.09cm](1,-0.3)(1,-0.6) [dotsize=0.09cm](0.5,0.9)(0.5,1.2)(0.5,-0.9)(0.5,-1.2) [shift=-1.2](-0.4,-2.5)(1.4,2.5) [dotsize=0.09cm](0,0)(0.5,0)(1,0) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) (-0.4,-2.0)(-0.4,2.0)A [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(1,0) [dotsize=0.09cm](0,0.3) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9) [dotsize=0.09cm](0.5,0.6)(0.5,0.9) [dotsize=0.09cm](1,0.3) [dotsize=0.09cm](1,-0.3) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(0.5,0)(1,0) (-0.4,-2.0)(-0.4,2.0)B [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.3) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](0.5,0.6)(0.5,0.9)(0.5,1.2) [dotsize=0.09cm](1,0.3) [dotsize=0.09cm](1,-0.3) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(0.5,0)(1,0) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0,0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](0.5,0.6) [dotsize=0.09cm](1,-0.6) [dotsize=0.09cm](1,0.6) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(1,0) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](0.5,0.3)(0.5,0.6)(0.5,0.9)(0.5,1.2) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(0.5,0)(1,0) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0,0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](0.5,0.6) [dotsize=0.09cm](1,-0.6) [dotsize=0.09cm](1,0.6) (-0.4,-2.0)(-0.4,2.0)A [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6)(0.5,-0.9)(0.5,-1.2) [dotsize=0.09cm](0.5,0.3)(0.5,0.6)(0.5,0.9)(0.5,1.2) [shift=-1.2](-0.4,-2.5)(1.4,2.5) -(0,-1.7)(0,1.7)-(1,-1.7)(1,1.7)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(0.5,0)(1,0) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0,0.6) [dotsize=0.09cm](0.5,-0.6) [dotsize=0.09cm](0.5,0.6) [dotsize=0.09cm](1,-0.6) [dotsize=0.09cm](1,0.6) (-0.4,-2.0)(-0.4,2.0)B; σ: -1 -1 -1 -1 -1 -1 -1; k_-^j k_+^j: ** 11 0,10,1 1,21,2 1,2,31,2,3 0,1,2,30,1,2,3 1,2,3,4 1,2,3,4; ℓ_-^jℓ_+^j: ** *1 *1 *1 11,2 11,2 11,2; q^Δq̅^Δ̅: 1 (qq̅)^1/3 (qq̅)^1 (qq̅)^2 (qq̅)^10/3 (qq̅)^5 (qq̅)^7 ] ] figureThe patterns of zeros for the ground states in _12^d and _13^d. [ [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.3)(0,0)(0,0.3) [dotsize=0.09cm](1,-0.3)(1,0)(1,0.3) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.3)(0,0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,0.6) [dotsize=0.09cm](1,-0.3)(1,0.3) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0) [dotsize=0.09cm](0.5,-0.3)(0.5,0.3) [dotsize=0.09cm](1,0) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,0)(1,0) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.3)(0,-0.6) [dotsize=0.09cm](0.5,0.3)(0.5,0.6) [dotsize=0.09cm](1,-0.3)(1,-0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.3)(0,0.6) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3) [dotsize=0.09cm](1,0.3)(1,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0.3)(0.5,0.6) [dotsize=0.09cm](1,0) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.3)(0,0.6) [dotsize=0.09cm](0.5,-0.6)(0.5,0.3) [dotsize=0.09cm](1,-0.3)(1,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6)(0,0.3) [dotsize=0.09cm](0.5,-0.3)(0.5,0.6) [dotsize=0.09cm](1,-0.6)(1,0.3) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0); k^j_- k^j_+: * * 1 1 1 1 * 1,2 1,2 * 1,2 1,2 1 2 2 1; ℓ^i_-ℓ^i_+: * * 1 1 * * * 1,1 1,1 * 1,1 1,1 1 1 1 1; q^Δq̅^Δ̅: q^-1/24q̅^-1/24 q^1/3q̅^1/3 q^1/3q̅^1/3 q^23/24q̅^-1/24 q^-1/24q̅^23/24 q^23/24q̅^23/24 q^4/3q̅^1/3 q^1/3q̅^4/3; σ: 1 -1 -1 1 1 1 -1 -1; ] [ [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6)(0,0.6) [dotsize=0.09cm](0.5,-0.3)(0.5,0.3) [dotsize=0.09cm](1,-0.6)(1,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.3) [dotsize=0.09cm](0.5,-0.3)(0.5,0) [dotsize=0.09cm](1,0.3) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,0)(1,0) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](0.5,0.3)(0.5,-0) [dotsize=0.09cm](1,-0.3) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,0.6)(1,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0.3)(0.5,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6)(0.5,-0.6)(1,-0.6)(0,0.6)(0.5,0.6)(1,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0.3)(0.5,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6)(0.5,-0.6)(1,-0.6)(0,0.6)(0.5,0.6)(1,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,-0)(0.5,0.6) [dotsize=0.09cm](1,0.3) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,0.0)(0.5,0.3)(0.5,0.6) [dotsize=0.09cm](1,-0.3) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0.6)(0.5,0) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0.3)(0.5,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,-0.6)(0.5,0.6)(1,-0.6); k^j_- k^j_+: 2 2 1 2 * 1,2 1,2 1,2 1,2 1,2 1,2 1,3 1 1,2,3 1,2 1,2; ℓ^i_-ℓ^i_+: 1 1 * * * * 1 1 1 1 1,1 1,1 1 1,1,2 * 1,1; q^Δq̅^Δ̅: q^4/3q̅^4/3 q^4/3q̅^1/3 q^47/24q̅^-1/24 q^35/24q̅^35/24 q^35/24q̅^35/24 q^47/24q̅^23/24 q^7/3q̅^1/3 q^23/24q̅^47/24; σ: -1 -1 1 1 1 1 -1 1; ] [ [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0.3)(0.5,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,0.6)(0.5,-0.6)(1,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.6) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0)(0.5,0.3) [dotsize=0.09cm](1,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.3)(0.5,0.0)(0.5,0.3)(0.5,0.6) [dotsize=0.09cm](1,-0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0)(0.5,0.6) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0.5,-0.9)(0.5,-0.6)(0.5,-0.3)(0.5,0.3)(0.5,0.6)(0.5,0.9) [shift=-0.4](-0.4,-1.0)(1.4,1.0) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,0)(0.5,0.6); k^j_- k^j_+: 1,2 1,2 1,2 2,3 2 1,2,3 1,2,3 1,2,3; ℓ^i_-ℓ^i_+: 1,1 * 1,1 1,1 1 1,1,2 1,1,2 1,1,2; q^Δq̅^Δ̅: q^47/24q̅^23/24 q^71/24q̅^23/24 q^7/3q̅^4/3 q^7/3q̅^7/3; σ: 1 1 -1 -1; ]figureThe 20 patterns of zeros for _6^0 for α = 2 and the corresponding data for ϵ = 0 [ [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.3)(0,0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,0.6) [dotsize=0.09cm](1,-0.3)(1,0.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.3)(0,-0.6) [dotsize=0.09cm](0.5,0.3)(0.5,0.6) [dotsize=0.09cm](1,-0.3)(1,-0.6) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.3)(0,0.6) [dotsize=0.09cm](0.5,-0.3)(0.5,-0.6) [dotsize=0.09cm](1,0.3)(1,0.6) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6)(0,0.3) [dotsize=0.09cm](0.5,-0.3)(0.5,0.6) [dotsize=0.09cm](1,-0.6)(1,0.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.6)(0,-0.3) [dotsize=0.09cm](0.5,0.3)(0.5,-0.6) [dotsize=0.09cm](1,0.6)(1,-0.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0.3)(0.5,0.6) [dotsize=0.09cm](1,0) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0.3)(0.5,0.6) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0,0)(1,0) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6)(0,0.6) [dotsize=0.09cm](0.5,-0.3)(0.5,0.3) [dotsize=0.09cm](1,-0.6)(1,0.6) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) (-0.4,1.3)A(-0.4,-1.3); σ: 1 -1 -1 1 1 -1 -1 1; k^j_- k^j_+: 0 0 * 0,1 0,1 * 1 0 0 1 0,1 0,1 0,1 0,1 1 1; ℓ^i_-ℓ^i_+: * * * * * * * * * * 1 1 * * * *; q^Δq̅^Δ̅: q^1/8q̅^1/8 q q̅ q^1/8q̅^9/8 q^9/8q̅^1/8 qq̅ qq̅ q^9/8q̅^9/8 ] [ [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.3) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0)(0.5,0.6) [dotsize=0.09cm](1,0.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.3) [dotsize=0.09cm](0.5,0.6)(0.5,0.3)(0.5,0)(0.5,-0.6) [dotsize=0.09cm](1,-0.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0.6) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0.3)(0.5,0.6) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(0.5,0)(1,0) (-0.4,1.3)B(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0.3)(0.5,0.6) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0)(0.5,0)(1,0) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,0.6) [dotsize=0.09cm](0.5,-0.6)(0.5,-0.3)(0.5,0)(0.5,0.3) [dotsize=0.09cm](1,0.6) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0,-0.6) [dotsize=0.09cm](0.5,-0.3)(0.5,0)(0.5,0.3)(0.5,0.6) [dotsize=0.09cm](1,-0.6) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,0.6) (-0.4,1.3)A(-0.4,-1.3) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [dotsize=0.09cm](0.5,-0.9)(0.5,-0.6)(0.5,-0.3)(0.5,0.3)(0.5,0.6)(0.5,0.9) [shift=-1.2](-0.4,-1.6)(1.4,1.6) -(0,-1)(0,1)-(1,-1)(1,1)[linestyle=dashed,dash=1pt 1pt]-(0,0)(1,0) [linecolor=black,fillcolor=lightgray,dotstyle=o,dotsize=0.09cm](0.5,-0.6)(0.5,0.6) (-0.4,1.3)A(-0.4,-1.3); σ: -1 1 -1 -1 -1 1 1; k^j_- k^j_+: 0,1 0,2 0 0,1,2 0,1 1,2 1,2 0,1 0,1 1,2 1 0,1,2 0,1,2 0,1,2; ℓ^i_-ℓ^i_+: 1 1 * 1,1 * 1 * 1 1 1 * 1,1 1,1 1,1; q^Δq̅^Δ̅: q^2q̅^1 q^17/8q̅^1/8 q^2 q̅ q q̅^2 q^3q̅ q^17/8q̅^9/8 q^17/8q̅^17/8 ]figureThe 15 patterns of zeros for _6^2 for ω = 1 and the corresponding data for ϵ = 0 99 BroadHamm57 S. Broadbent, J. Hammersley, Percolation processes I. Crystals and mazes, Proc. Camb. Phil. Soc. 53 (1957) 629. KestonPerc82 H. Kesten, Percolation theory for mathematicians, Springer Science (1982). Stauffer92 D. Stauffer, A. Aharony, Introduction to Percolation Theory, Taylor and Francis (1992). Grimmet97 G. Grimmett, Percolation and disordered systems, Lectures on Probability Theory and Statistics, Springer (1997). Saberi16 A.A. Saberi, Recent advances in percolation theory and its applications, Phys. Rep. 578 (2015) 1–32, 1504.02898[cond-mat.stat-mech]. BaxBook R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press (1982). FMS P. Di Francesco, P. Mathieu, D. Sénéchal, Conformal Field Theory, Springer (1997). Gurarie V. Gurarie, Logarithmic operators in conformal field theory, Nucl. Phys. B410 (1993) 535–549, hep-th/9303160. SpecialIssue A. Gainutdinov, D. Ridout, I. Runkel (Guest Editors), Special issue on logarithmic conformal field theory, J. Phys. A: Math. Theor. 46(49) (2013). Harris T.E. Harris, A lower bound for the critical probability in a certain percolation process, Proc. Cambridge Philos. Soc. 56 (1960) 13–20. Kesten80 H. Kesten, The critical probability of bond percolation on the square lattice equals 1/2, Commun. Math. Phys. 74 (1980) 41–59. VJS12 R. Vasseur, J.L. Jacobsen, H. Saleur, Logarithmic observables in critical percolation, J. Stat. Mech. (2012) L07001, 1206.2312[cond-mat.stat-mech]. DPSV13 G. Delfino, M. Picco, R. Santachiara, J. Viti, Spin clusters and conformal field theory, J. Stat. Mech. (2013) P11011, 1307.6123[cond-mat.stat-mech]. TL71 H. Temperley, E. Lieb, Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: Some exact results for the “percolation” problem, Proc. Roy. Soc. London Ser.  A322 (1971) 251–280. Jones V.F.R. Jones, Planar algebras I, math/9909027[math.QA]. HullPerc88 S. Roux, E. Guyon, D. Sornette, Hull percolation, J. Phys. A 21 (1988) L475–L482. DuplantierHull B. Duplantier, Hull percolation and standard percolation, J. Phys. A 21 (1988) 3969–3973. Cardy92 J. Cardy, Critical percolation in finite geometries, J. Phys. A 25 (1992) L201–L206, hep-th/9111026; Conformal invariance and percolation, math-ph/0103018. LPSA94 R. Langlands, P. Pouliot, Y. Saint-Aubin, Conformal invariance in two-dimensional percolation, Bull. Am. Math. Soc. 30 (1994) 1–61, math/9401222[math-ph]. CardyRGScaling J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics, Cambridge University Press (1996). denNijs79 M.P.M. den Nijs, A relation between the temperature exponents of the eight-vertex model and q-state Potts model, J. Phys. A 12 (1979) 1857–68. NienhuisRS80 B. Nienhuis, E.K. Riedel, M. Schick, Magnetic exponents of the two-dimensional q-state Potts model, J. Phys. A 13 (1980) L189–92. CoulombGas1 B. Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Phys. Rev. Lett. 49 (1982) 1062–1065. CoulombGas2 B. Nienhuis, Critical behaviour of two-dimensional spin models and charge asymmetry in the Coulomb gas, J. Stat. Phys. 34 (1984) 731–761. FSZ87 P. di Francesco, H. Saleur, J.-B. Zuber, Relations between the Coulomb gas picture and conformal invariance of two-dimensional critical models, J. Stat. Phys. 49 (1987) 57–79; Generalized Coulomb-gas formalism for two-dimensional critical models based on SU(2) coset construction, Nucl. Phys. B300 (1988) 393–432. Delfino1 G. Delfino, J. Cardy, Universal amplitude ratios in the two-dimensional q-state Potts model and percolation from quantum field theory, Nucl. Phys. B519 (1998) 551–578, hep-th/9712111. Delfino2 G. Delfino, J. Viti, J. Cardy, Universal amplitude ratios of two-dimensional percolation from field theory, J. Phys. A 43 (2010) 152001, 1001.5424[hep-th]. Saleur86 H. Saleur, New exact exponents for the two-dimensional self-avoiding walks, J. Phys. A: Math. Gen. 19 (1986) L807–L810. Saleur87a H. Saleur, Conformal invariance for polymers and percolation, J. Phys. A: Math. Gen. 20 (1987) 455–470. Saleur87b H. Saleur, Magnetic properties of the two-dimensional n = 0 vector model, Phys. Rev. B35 (1987) 3657–3660. Duplantier86 B. Duplantier, Exact critical exponents for two-dimensional dense polymers, J. Phys. A: Math. Gen. 19 (1986) L1009–L1014. DupSaleur86 B. Duplantier, H. Saleur, Exact surface and wedge exponents for polymers in two dimensions, Phys. Rev. Lett. 57 (1986) 3179–3182. SaleurDup87 H. Saleur, B. Duplantier, Exact determination of the percolation hull exponent in two dimensions, Phys. Rev. Lett. 58 (1987) 2325–2328. Saleur92 H. Saleur, Polymers and percolation in two dimensions and twisted N=2 supersymmetry, Nucl. Phys. B382 (1992) 486–531, hep-th/9111007. DeguchiFabMcCoy T. Deguchi, K. Fabricius, B.M. McCoy, The sl_2 loop algebra symmetry of the six-vertex model at roots of unity, J. Stat. Phys. 102 (2001) 701–736, cond-mat/9912141[cond-mat.stat-mech]. Deguchi T. Deguchi, The sl_2 loop algebra symmetry of the twisted transfer matrix of the six-vertex model at roots of unity, J. Phys. A 37 (2004) 347–358, cond-mat/0306498[cond-mat.stat-mech]; Regular XXZ Bethe states as highest weight vectors of the sl_2 loop algebra at roots of unity, J. Phys. A 40 (2007) 7473–7508, cond-mat/0212217[cond-mat.stat-mech]. PRZ06 P.A. Pearce, J. Rasmussen, J.-B. Zuber, Logarithmic minimal models, J. Stat. Mech. (2006) P11017, hep-th/0607232. PRT14 P.A. Pearce, J. Rasmussen, I.Y. Tipunin, Critical dense polymers with Robin boundary conditions, half-integer Kac labels and ℤ_4 fermions, Nucl. Phys. B889 (2014) 580–636, 1405.0550[hep-th]. BPT2016 J.-E. Bourgine, P.A. Pearce, E. Tartaglia, Logarithmic minimal models with Robin boundary conditions, J. Stat. Mech. (2016) 063104, 1601.04760[hep-th]. CJV17 R. Couvreur, J.L. Jacobsen, R. Vasseur, Non-scalar operators for the Potts model in arbitrary dimension, 1704.02186[cond-mat.stat-mech]. PRcosetBranch13 P.A. Pearce, J. Rasmussen, Coset construction of logarithmic minimal models: branching rules and branching functions, J. Phys. A 46 (2013) 355402, 1305.7304[hep-th]. MathieuRidout P. Mathieu, D. Ridout, From percolation to logarithmic conformal field theory, Phys. Lett. B29 (2007) 120–129, 0708.0802[hep-th]. MooreSeiberg G. Moore, N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177–254. MDSA2011 A. Morin-Duchesne, Y. Saint-Aubin, The Jordan structure of two-dimensional loop models, J. Stat. Mech. (2011) P04007, 1101.2885[math-ph]. MDSA13no2 A. Morin-Duchesne, Y. Saint-Aubin, Jordan cells of periodic loop models, J. Phys. A: Math. Theor. 46 (2013) 494013, 1302.5483[math-ph]. CardyModInv J.L. Cardy, Operator content of two-dimensional conformally invariant theories, Nucl. Phys. B270 (1986) 186–204. CIZ A. Cappelli, C. Itzykson, J.-B. Zuber, Modular Invariant Partition Functions in Two Dimensions, Nucl. Phys. B280 (1987) 445–465. Smirnov01 S. Smirnov, Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits, Comptes Rendus de l'Académie des Sciences, Series I - Mathematics, 333 (2001) 239–244. AAMRH J. Asikainen, A. Aharony, B.B. Mandelbrot, E.M. Rauch, J.-P. Hovi, Fractal geometry of critical Potts clusters, Euro. Phys. J. B34 (2003) 479–487, cond-mat/0212216[cond-mat.dis-nn]. JS2005 W. Janke, A.M.J. Schakel, Fractal structure of spin clusters and domain walls in the two-dimensional Ising model, Phys. Rev. E71 (2005) 036703, cond-mat/0410364[cond-mat.stat-mech]. JS2006 W. Janke, A.M.J. Schakel, Two-dimensional critical Potts and its tricritical shadow, Braz. J. Phys. 36 (2006) 708–716, cond-mat/0612650[cond-mat.stat-mech]. SAPR2009 Y. Saint-Aubin, P.A. Pearce, J. Rasmussen, Geometric exponents, SLE and logarithmic minimal models, J. Stat. Mech. (2009) P02028, 0809.4806[cond-mat.stat-mech]. Mandelbrot B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Co. (1982). LSW01 G.F. Lawler, O. Schramm, W. Werner, The dimension of the Brownian frontier is 4/3, Math. Rev. Lett. 8 (2001) 13–24, math/0010165[math.PR]. Beffara V. Beffara, Hausdorff dimensions for _6, Ann. Prob. 32 (2004) 2606–2629, math/0204208[math.PR]. PR07 P.A. Pearce, J. Rasmussen, Solvable critical dense polymers, J. Stat. Mech. (2007) P02015, hep-th/0610273. PRVcyl2010 P.A. Pearce, J. Rasmussen, S.P. Villani, Solvable dense polymers on the cylinder, J. Stat. Mech. (2010) P02010, 0910.4444[hep-th]. MD11 A. Morin-Duchesne, A proof of selection rules for critical dense polymers, J. Phys. A: Math. Theor. 44 (2011) 495003, 1109.6397[math-ph]. PRV1210 P.A. Pearce, J. Rasmussen, S.P. Villani, Infinitely extended Kac table of solvable critical dense polymers, J. Phys. A: Math. Theor. 46 (2013) 175202, 1210.8301[math-ph]. MDPR13 A. Morin-Duchesne, P.A. Pearce, J. Rasmussen, Modular invariant partition function of critical dense polymers, Nucl. Phys. B874 (2013) 312–357, 1303.4895[hep-th]. PRcosetGraphs11 P.A. Pearce, J. Rasmussen, Coset graphs in bulk and boundary logarithmic minimal models, Nucl. Phys. B846 (2011) 616–649, 1010.5328[hep-th]. MDPR14 A. Morin-Duchesne, P.A. Pearce, J. Rasmussen, Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models, J. Stat. Mech. (2014) P05012, 1401.7750[math-ph]. PTC2015 P.A. Pearce, E. Tartaglia, R. Couvreur, Kac boundary conditions of the logarithmic minimal models, J. Stat. Mech. (2015) P01018, 1410.0103[hep-th]. MDRD2015 A. Morin-Duchesne, J. Rasmussen, D. Ridout, Boundary algebras and Kac modules for logarithmic minimal models, Nucl. Phys. B899 (2015) 677–769, 1503.07584[hep-th]. BR1989 V.V. Bazhanov, N. Reshetikhin, Critical RSOS models and conformal field theory, Int. J. Mod. Phys. A4 (1989) 115–142. Zam1991a A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B253 (1991) 391–394. Zam1991b A.B. Zamolodchikov, Thermodynamic Bethe ansatz for RSOS scattering theories, Nucl. Phys. B358 (1991) 497–523. PK91 P.A. Pearce, A. Klümper, Finite-size corrections and scaling dimensions of solvable lattice models: an analytic method, Phys. Rev. Lett. 66 (1991) 974–977. KP92 A. Klümper, P.A. Pearce, Conformal weights of RSOS lattice models and their fusion hierarchies, Physica A183 (1992) 304–350. KunibaNS9309 A. Kuniba, T. Nakanishi, J. Suzuki, Functional relations in solvable lattice models I: functional relations and representation theory, Int. J. Mod. Phys. A9 (1994) 5215–5266, hep-th/9309137. KunibaNS9310 A. Kuniba, T. Nakanishi, J. Suzuki, Functional relations in solvable lattice models II, Int. J. Mod. Phys. A9 (1994) 5267–5312, hep-th/9310060. KunibaNS1010 A. Kuniba, T. Nakanishi, J. Suzuki, T-systems and Y-systems in integrable systems, J. Phys. A: Math. Theor. 44 (2011) 103001, 1010.1344[hep-th]. KLWZ1997 I. Krichever, O. Lipan, P. Wiegmann, A. Zabrodin, Quantum integrable models and discrete classical Hirota equations, Commun. Math. Phys. 188 (1997) 267–304, hep-th/9604080. YangYang C.N. Yang, C.P. Yang, Thermodynamics of a one-dimensional system of bosons with repulsive delta-function interaction, J. Math. Phys. 10 (1969) 1115. T71 M. Takahashi, One-dimensional Heisenberg model at finite temperature, Prog. Theor. Phys. 46 (1971) 401. G71 M. Gaudin, Thermodynamics of the Heisenberg-Ising ring for Δ≥ 1, Phys. Rev. Lett. 26 (1971) 1301. Zam90 A.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models: Scaling 3-state Potts and Lee-Yang models, Nucl. Phys. B342 (1990) 695–720. Zam91 A.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for ADE reflectionless scattering theories, Phys. Lett. B253 391–394; Thermodynamic Bethe ansatz for RSOS scattering theories, Nucl. Phys. B358 (1991) 497–523; From tricritical Ising to critical Ising by thermodynamic Bethe ansatz, Nucl. Phys. B358 (1991) 524–546; TBA equations for integrable perturbed SU(2)_k× SU(2)_1/SU(2)_k+1 coset models, Nucl. Phys. B366 (1991) 122–132. CMP02 C. Chui, C. Mercat, P.A. Pearce, Integrable boundaries and universal TBA functional equations, Prog. Math. Phys. 23 (2002) 391–413, hep-th/0108037. Sklyanin E.K. Sklyanin, Boundary conditions for integrable quantum systems, J. Phys. A21 (1988) 2375–2389. BPO96 R.E. Behrend, P.A. Pearce, D.L. O'Brien, Interaction-round-a-face models with fixed boundary conditions: The ABF fusion hierarchy, J. Stat. Phys. 84 (1996) 1–48, hep-th/9507118. KP91 A. Klümper, P.A. Pearce, Analytic calculation of scaling dimensions: Tricritical hard squares and critical hard hexagons, J. Stat. Phys. 64 (1991) 13–76. OPW97 D.L. O'Brien, P.A. Pearce, S.O. Warnaar, Analytic calculation of conformal partition functions: Tricritical hard squares with fixed boundaries, Nucl. Phys. B501 (1997) 773–799. BDP15 Z. Bajnok, O. el Deeb, P.A. Pearce, Finite-volume spectra of the Lee-Yang model, JHEP 04 (2015) 73. KSS98 A. Kuniba, K. Sakai, J. Suzuki, Continued fraction TBA and functional relations in XXZ model at root of unity, Nucl. Phys. B525 (1998) 597–626, math/9803056[math.QA]. OPW1996 D.L. O'Brien, P.A. Pearce, S.O. Warnaar, Finitized conformal spectrum of the Ising model on the cylinder and torus, Physica A 228 (1996) 63–77. BGN01 M.T. Batchelor, J. de Gier, B. Nienhuis, The quantum symmetric XXZ chain at Δ=-1/2, alternating sign matrices and plane partitions, J. Phys. A: Math. Gen. 34 (2001) L265, cond-mat/0101385[cond-mat.stat-mech]. MBNM04 S. Mitra, B. Nienhuis, J. de Gier, M.T. Batchelor, Exact expressions for correlations in the ground state of the dense O(1) loop model, J. Stat. Mech. (2004) P09010, cond-mat/0401245[cond-mat.stat-mech]. Jones83 V. Jones, Index for subfactors, Invent. Math. 72 (1983) 1–25. M91 P. Martin, Potts models and related problems in statistical mechanics, World Scientific, Singapore (1991). GW93 F. Goodman, H. Wenzl, The Temperley-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993) 307–334. W95 B. Westbury, The representation theory of the Temperley-Lieb algebras, Math. Zeit. 219 (1995) 539–565. RSA14 D. Ridout, Y. Saint-Aubin, Standard modules, induction and the structure of the Temperley-Lieb algebra, Adv. Theo. Math. Phys. 18 (2014) 957–1041, 1204.4505[math-ph]. RJ06 J.F. Richard, J.L. Jacobsen, Character decomposition of Potts model partition functions. I. Cyclic geometry, Nucl. Phys. B750 (2006) 250–264, math-ph/0605016[math-ph]. L91 D. Levy, Algebraic structure of translation-invariant spin-1/2 XXZ and q-Potts quantum chains, Phys. Rev. Lett. 67 (1991) 1971–1974. MS93 P. Martin, H. Saleur, On an algebraic approach to higher dimensional statistical mechanics, Commun. Math. Phys. 158 (1993) 155–190, hep-th/9208061. GL98 J.J. Graham, G.I. Lehrer, The representation theory of affine Temperley-Lieb algebras, Enseign. Math. 44 (1998) 173–218. G98 R.M. Green, On representations of affine Temperley-Lieb algebras, CMS Conf. Proc. 24 (1998) 245–261. EG99 K. Erdmann, R.M. Green, On representations of affine Temperley-Lieb algebras, II, Pac. J. Math. 191 (1999) 243–274, math/9811017[math.RT]. RS01 A.V. Razumov, Y.G. Stroganov, Spin chains and combinatorics, J. Phys. A: Math. Theor. 34 (2001) 3185–3190, cond-mat/0012141[cond-mat.stat-mech]. RS01no2 A.V. Razumov, Y.G. Stroganov, Spin chains and combinatorics: twisted boundary conditions, J. Phys. A: Math. Theor. 34 (2001) 5335–5340, cond-mat/0102247[cond-mat.stat-mech]. S01 Y.G. Stroganov, The importance of being odd, J. Phys. A 34 (2001) L179–L185, cond-mat/0012035[cond-mat.stat-mech]. RJ07 J.F. Richard, J.L. Jacobsen, Eigenvalue amplitudes of the Potts model on a torus, Nucl. Phys.  B769 (2007) 256–274, math-ph/0608055[math-ph]. AK10 B. Aufgebauer, A. Klümper, Quantum spin chains of Temperley-Lieb type: periodic boundary conditions, spectral multiplicities and finite temperature, J. Stat. Mech. (2010) P05018, 1003.1932[cond-mat.stat-mech]. SB89 H. Saleur, M. Bauer, On some relations between local height probabilities and conformal invariance, Nucl. Phys. B320 (1989) 591–624. C06 J. Cardy, The O(n) model on the annulus, J. Stat. Phys. 125 (2006) math-ph/0604043. GRW10 M.R. Gaberdiel. I. Runkel, S. Wood, A modular invariant bulk theory for the c=0 triplet model, J. Phys. A 44 (2010) 1008.0082[hep-th]. MDSA13no1 A. Morin-Duchesne, Y. Saint-Aubin, A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra, J. Phys. A: Math. Theor. 46 (2013) 285207, 1203.4996[math-ph]. ABGR88 F.C. Alcaraz, M. Baake, U. Grimm, V. Rittenberg, Operator content of the XXZ chain, J. Phys. A: Math. Gen. 21 (1988) L117–120. AGR89 F.C. Alcaraz, U. Grimm, V. Rittenberg, The XXZ Heisenberg chain, conformal invariance and the operator content of c<1 systems, Nucl. Phys. B316 (1989) 735–768. RS07 N. Read, H. Saleur, Associative-algebraic approach to logarithmic conformal field theories, Nucl. Phys. B777 (2007) 316–351, hep-th/0701117. GR08 M.R. Gaberdiel. I. Runkel, From boundary to bulk in logarithmic CFT, J. Phys. A: Math. Theor. 41 (2008) 075402, 0707.0388[hep-th]. RGW14 I. Runkel, M.R. Gaberdiel, S. Wood, Logarithmic bulk and boundary conformal field theory and the full centre construction, in Bai et al. (eds) Conformal Field Theories and Tensor Categories, Math. Lect. from Peking Univ., Springer (2014), 1201.6273[hep-th]. GRSV15 A.M. Gainutdinov, N. Read, H. Saleur, R. Vasseur, The periodic sℓ(2|1) alternating spin chain and its continuum limit as a bulk logarithmic conformal field theory at c=0, J. High Energ. Phys. 114 (2015), 1409.0167[hep-th]. JS08 J.L. Jacobsen, H. Saleur, Conformal boundary loop models, Nucl. Phys. B788 (2008) 137–166, math-ph/0611078. DJS09 J. Dubail, J.L. Jacobsen, H. Saleur, Conformal two-boundary loop model on the annulus, Nucl. Phys. B813 (2009) 430–459, 0812.2746. R79 J. Riordan, Combinatorial identities, Robert E. Krieger Pub. Co (1979).

http://arxiv.org/abs/1701.08198v1

20170127212857

Adversarial Evaluation of Dialogue Models

cs.CL

[NO \title GIVEN] [NO \author GIVEN] December 30, 2023 ====================== The recent application of RNN encoder-decoder models has resulted in substantial progress in fully data-driven dialogue systems, but evaluation remains a challenge. An adversarial loss could be a way to directly evaluate the extent to which generated dialogue responses sound like they came from a human. This could reduce the need for human evaluation, while more directly evaluating on a generative task. In this work, we investigate this idea by training an RNN to discriminate a dialogue model's samples from human-generated samples. Although we find some evidence this setup could be viable, we also note that many issues remain in its practical application. We discuss both aspects and conclude that future work is warranted. § INTRODUCTION Building machines capable of conversing naturally with humans is an open problem in language understanding. Recurrent neural networks (RNNs) have drawn particular interest for this problem, typically in the form of an encoder-decoder architecture. One network ingests an incoming message (a Tweet, a chat message, etc.), and a second network generates an outgoing response, conditional on the first network's final hidden state. This sort of approach has been shown to improve significantly over both a statistical machine translation baseline <cit.> and traditional rule-based chatbots <cit.>. However, evaluating dialogue models remains a significant challenge. While perplexity is a good measure of how well a model fits some data, it does not measure performance at a particular task. N-gram-based measures such as BLEU, while useful in translation, are a poor fit for dialogue models because two replies may have no n-gram overlap but equally good responses to a given message. Human evaluation may be ideal, but does not scale well, and can also be problematic in applications like Smart Reply <cit.>, where data cannot be viewed by humans. This work investigates the use of an adversarial evaluation method for dialogue models. Inspired by the success of generative adversarial networks (GANs) for image generation (<cit.>, and others), we propose that one measure of a model's quality is how easily its output is distinguished from a human's output. As an initial exploration, we take a fully trained production-scale conversation model deployed as part of the Smart Reply system (the "generator"), and, keeping it fixed, we train a second RNN (the "discriminator") on the following task: given an incoming message and a response, it must predict whether the response was sampled from the generator or a human. Our goal here is to understand whether an adversarial setup is viable either for evaluation. We find that a discriminator can in fact distinguish the model output from human output over 60% of the time. Furthermore, it seems to uncover the major weaknesses that humans have observed in the system: an incorrect length distribution and a reliance on familiar, simplistic replies such as "Thank you". Still, significant problems with the practical application of this method remain. We lack evidence that a model with lower discriminator accuracy (i.e., that fools it) necessarily would be better in human evaluation as well. We present here the details of our analysis, as well as further discussion of both merits and drawbacks of an adversarial setup. We conclude that additional investigation is warranted, and lay out several suggestions for that work. §.§ Related work Much recent work has employed RNN encoder-decoder models to translate from utterance to response (<cit.>, <cit.>, <cit.>). Work in <cit.> has used policy gradient but the rewards are manually defined as useful conversational properties such as non-redundancy. Evaluation remains a significant challenge <cit.>. The adversarial setup we describe is inspired by work on GANs for image generation <cit.>; however, we apply the concept to dialogue modeling, which raises the challenges of sequential inputs/outputs and conditional generation. To support our aim of understanding the discriminator, we also do not train the generator and discriminator jointly. An adversarial loss for language understanding is also used in <cit.> as a means of evaluation; however, the metric is not applied to any real world task, nor are the properties of the discriminator itself explored and evaluated, as we will do in this work. § MODEL Like a GAN, our architecture consists of a generator and a discriminator; however, these are two separate models which are not trained to a single objective. The generator is a sequence-to-sequence model, consisting of an RNN encoder and an RNN decoder. Given a corpus of message pairs (𝐨, 𝐫) where 𝐨, the original message, consists of tokens {o_1, ..., o_n} and 𝐫, the response message, consists of tokens {r_1, ..., r_m}, this model is trained to maximize the total log probability of observed response messages, given their respective original messages: ∑_(𝐨, 𝐫)log P(r_1, ..., r_m | o_1, ..., o_n) The discriminator is also an RNN, but has only an encoder followed by a binary classifier. Given a corpus of message pairs and scores (𝐨, 𝐫, y) , where y = 1 if 𝐫 was sampled from the training data and 0 otherwise, this model is trained to maximize: ∑_(𝐨, 𝐫, y)log P(y| o_1, ..., o_n, r_1, ..., r_m ) § EXPERIMENTS §.§ Data and training We investigate the proposed adversarial loss using a corpus of email reply pairs (𝐨, 𝐫). The generator is trained on the same data and in the same manner as the production-scale model that is deployed as part of the Smart Reply feature in Inbox by Gmail <cit.>. [In particular, from <cit.>: "All email data (raw data, preprocessed data and training data) was encrypted. Engineers could only inspect aggregated statistics on anonymized sentences that occurred across many users and did not identify any user."] The discriminator is then trained on a held out set of the email corpus. For half the pairs (𝐨, 𝐫) in the held out set, we leave the example unchanged and assign a 1. For the other half we replace 𝐫 with a message 𝐫' that has been sampled from the generator, and assign the pair (𝐨, 𝐫') a score of 0. Then the discriminator is trained as described in the previous section. §.§ Discriminator performance We observe that the discriminator can distinguish between generator samples and human samples, conditional on an original message, 62.5% of the time. This in itself may be somewhat unexepcted: one may expect that since the discriminator is only as powerful as the generator, it would not be able to distinguish its distribution from the training distribution. A full precision-recall curve is shown in Figure  <ref>. §.§ Comparison with perplexity A qualitative analysis shows that the discriminator objective favors different features than the generator objective. To demonstrate this, we sample 100 responses from the generator for each of 100 donated email messages. These are then ranked according to both the discriminator score and the generator's assigned log likelihood, and the two rankings are compared. First, we see that the discriminator's preferences are strongly correlated with length (Figure  <ref>). This is relevant because it has been previously documented that sequence-to-sequence models have a length bias <cit.>. The discriminator relies too heavily on this signal, favoring longer responses even when they are not internally coherent. Still, it is noteworthy that it identifies something humans have documented as a key weakness of the model <cit.>. The discriminator does not assign equal probability to all responses of the same length. When comparing responses of the same length, we find that it has a significantly different ranking than the likelihood assigned by the generator, with an average Spearman's correlation of -0.02. Broadly speaking we find that the discriminator has less preference for the most common responses produced by the generator, things like "Thank you!" and "Yes!" (Table  <ref>). The lack of diverse generated language has been documented as a weakness of these dialogue models in <cit.> and <cit.>, both of which incorporate significant post-processing and re-ranking to overcome this noted weakness. As with length, the discriminator's preference for rarer language does not necessarily mean it is favoring better responses; it is noteworthy only in that it shows signs of detecting the known weakness of the generator. Future work might incorporate minibatch discrimination <cit.> to more explicitly address the diversity weakness. § DISCUSSION In this research note we investigated whether the discriminator in GANs can be employed for automatic evaluation of dialogue systems. We see a natural progression towards using discriminators: * Ask humans to evaluate each single system published in a consistent manner. Though ideal, it would also be time consuming and prohibitively expensive. * Annotate a large dataset of dialogues, learn a “critic” (e.g., a neural network), and use it to score any new system (so that extra human labour would not be required). However, this critic would likely not perform well when evaluated off-policy, and overfitting could occur, as researchers may naturally find its weaknesses. * Use the discriminator of a GAN as a proxy for giving feedback akin to a human. Since training such a critic would be simple for any dialogue system, each research group could provide theirs and any new system could be evaluated with a variety of discriminators. The last item is the simplest, and it is what we have explored in this work. Our preliminary work suggests that the critic we trained on a production-quality dialogue system is able to automatically find some of the previously identified weaknesses when employing probabilistic models – sequence length and diversity. It also succeeds in identifying real vs generated responses of a highly tuned system. However, as with GANs, more needs to be understood and using discriminators alone won’t solve the evaluation challenges of dialogue systems. Despite the fact that GANs do not use any extra information than what’s already present in the training dataset, some have argued that it is a better loss than likelihood <cit.>. Still, there remains a tension between what we train the discriminator on (samples) and what we typically use in practice (the maximally likely response, or some approximation of it). Discriminators have a harder time when sampling versus using beam search, but this conflicts with human observations that some amount of search typically is useful to get the highest quality responses. Further work is required to understand if and how discriminators can be applied in this domain. abbrv

http://arxiv.org/abs/1701.07982v2

20170127093343

Star Cluster Formation in a Turbulent Molecular Cloud Self-Regulated by Photo-Ionisation Feedback

astro-ph.GA

firstpage–lastpage * Luke Hutton and Tristan Henderson December 30, 2023 ===================================== Most stars in the Galaxy are believed to be formed within star clusters from collapsing molecular clouds. However, the complete process of star formation, from the parent cloud to a gas-free star cluster, is still poorly understood. We perform radiation-hydrodynamical simulations of the collapse of a turbulent molecular cloud using the RAMSES-RT code. Stars are modelled using sink particles, from which we self-consistently follow the propagation of the ionising radiation. We study how different feedback models affect the gas expulsion from the cloud and how they shape the final properties of the emerging star cluster. We find that the star formation efficiency is lower for stronger feedback models. Feedback also changes the high mass end of the stellar mass function. Stronger feedback also allows the establishment of a lower density star cluster, which can maintain a virial or sub-virial state. In the absence of feedback, the star formation efficiency is very high, as well as the final stellar density. As a result, high energy close encounters make the cluster evaporate quickly. Other indicators, such as mass segregation, statistics of multiple systems and escaping stars confirm this picture. Observations of young star clusters are in best agreement with our strong feedback simulation. galaxies: star clusters: general - galaxies: star clusters: individual: (NGC 3603 YC, Arches) - stars: formation - stars: kinematics and dynamics - H ii regions - ultraviolet: stars § INTRODUCTION Establishing a full and consistent theory of star cluster formation remains an open task for the scientific community. The most widely adopted view is that star clusters form from the collapse of giant molecular clouds. On a timescale of a few millions years, a cloud undergoes gravitational collapse and converts part of its gas into many dense molecular cores, each core leading to the formation of one or a few proto-stellar objects <cit.>. These protostars can continue accreting material from their surroundings, and eventually become proper stellar, main sequence objects, whose stellar luminosity is high enough to inject considerable amounts of energy into their parent cloud. This stellar feedback modifies the properties of the cloud and the star formation process itself and as a result regulates the properties of the emerging star cluster, such as its dynamical state, the mass distribution and the fate of its stellar population. Understanding the impact of stellar feedback on the star cluster properties, and the transition from the initial turbulent GMC to the final gas-free association of stars (such as observed open clusters, embedded clusters or even globular clusters) is at the moment one of the most intriguing fields of research in astrophysics, mainly because of the numerous and complex physical processes at play during the entire history of the star cluster formation. A classic reference is the work of <cit.>, which states that 90% of stars are likely to form in star clusters . In <cit.>, star clusters are defined as groups of at least 35 stars and with a stellar mass density of at least 1 pc^-3. These numbers can be derived by requiring that the evaporation timescale of the star cluster is longer than 100 Myr. A more recent study by <cit.> revealed how the fraction of stars in the solar neighbourhood forming in clusters is strongly dependent on the adopted definition for star clusters , with values ranging between 45 and 90%. They concluded that stars form within a broad and smooth distribution of surface densities, which is consistent with star formation proceeding hierarchically, within the turbulent, hierarchical structure of the parent molecular cloud, where denser regions are systematically embedded in less dense regions <cit.>. Defining what is a truly bound cluster or an unbound stellar association is indeed not straightforward, especially when the system is young. It is only after these stellar structures have dynamically evolved, that they are easier to distinguish from their environment. The identification of the fraction of stars residing within these older stellar systems is more reliable, and is observed to be around 10-30% <cit.>. <cit.> shows that the cluster-formation efficiency varies from 1-70% depending on the galactic gas surface densities at which the cluster forms. It is also very important to establish what is the fraction of stars which formed in star clusters but do not reside there anymore today. This is usually referred as star clusters infant mortality, outlining the fact that, when we compare the fraction of stars in young, embedded star clusters with the fraction of stars in older, open clusters, most of the clusters seems to have been disrupted during this transition from embedded to exposed <cit.>. Note that this interpretation assumes that the fraction of stars in star clusters is the rather old one presented in <cit.>. The commonly adopted picture for the cause of this infant mortality is the fast expulsion of the initial gas, leading to the rapid expansion and disruption of the star cluster. Only clusters with a star formation efficiency (SFE, i.e. the fraction of gas converted into stars) higher than 30% are believed to survive the gas removal and stay bound <cit.>. Yet, the star formation efficiency is not the only parameter that can decide whether a star cluster will survive gas expulsion. Two other important factors are: 1-the timescale of gas removal and 2-the actual dynamical state of the star cluster right before expulsion. Regarding the first point, it has been shown for example that systems with star formation efficiency as low as 10% can remain bound, as long as the gas is removed slowly and adiabatically <cit.>. The second factor has been pointed out by <cit.>, showing a strong dependence of the star cluster mass loss (hence survival) on the virial ratio of the emerging star cluster. Indeed, if the system is sub-virial before gas is expelled, it can survive even with SFE lower than 30%. Conversely, an initially super-virial system, even with a SFE as high as 50%, will be at edge of survivability <cit.>. <cit.> questions the importance of gas expulsion in determining the fate of the star cluster and justifies the observed poor number of bound clusters as direct result of the star formation process. According to the author, most of the natal cloud is characterised by low SFE and will therefore form dispersed structure, while only the few sites of high SFE will give birth to bound star clusters. The SFE within star forming molecular clouds is poorly understood from theoretical grounds. Simple models based only on self-gravitating turbulence predict a very high SFE, higher than 90%, meaning that star formation occurs during one free-fall time of the parent cloud, in contradiction with observational constraints <cit.>. Stellar feedback has been invoked to reduce the SFE by terminating star formation in giant molecular clouds <cit.>. Stellar feedback is a broad term that refers to the injection of mass, momentum and energy by stars and protostars into the star forming gas itself. The different mechanisms of stellar feedback are photoionisation from massive main sequence stars, infrared and optical radiation from accreting protostars, radiation pressure associated to these various types of radiation, proto-stellar jets, stellar winds from main sequence or post-main sequence stars, supernovae explosions. Although all these ingredients are likely to play an important role in regulating the star formation efficiency and in setting the properties of the emerging star clusters, they act on different spatial and temporal scales, and are associated with stars of different masses. During the first Myrs of a star cluster life, before the first OB stars form, feedback modes from pre-main sequence stars play a significant role. These include jets, deuterium-burning and accretion feedback. Pre-main sequence feedback is generally not effective on large-scale and does not drive the process of gas clearing, however it has been shown to be able to sustain turbulence and reduce the conversion rate of gas into stars <cit.>. Moreover radiation focusing in the direction of outflow cavities prevents the formation of radiation pressure-supported gas bubbles, diminishing the radiative heating and outward radiation force exerted on the infalling cloud gas <cit.>, resulting in higher mass accretion onto the protostar. Disk fragmentation is also suppressed as a result of thermal feedback from protostars <cit.>, affecting the multiplicity of stellar systems. On the observational side, several surveys can be used to cast light on the star cluster formation process. The MYSTiX survey <cit.>, for example, is targeting massive star forming regions and has revealed that star clusters are frequently divided into sub-clusters <cit.>. We now have evidence that these sub-clusters are expanding or merging, with clear signs of ongoing dynamical relaxation. For example, we observe mass segregation (see Section <ref> for a definition) down to 1.5 <cit.>. Similarly, <cit.> have studied the morphology and the dynamical state of the Orion Nebula Cluster. They concluded that the core appears rounder and smoother than the outskirts, which is consistent with ongoing dynamical processing. The Gaia-ESO Survey <cit.> has recently discovered several kinematically distinct populations in the young star cluster Gamma Velorum, surrounding the γ^2 Velorum binary in the Vela OB2 association. According to <cit.>, the first component of Gamma Velorum is a bound remnant of an initially larger cluster, formed in a dense region of the Vela OB2 association, that has been partially disrupted by gas expulsion. The second component consists of a scattered population of unbound stars born later (as indicated by lithium depletion) in less dense regions. The gas surrounding this second population was probably evaporated by the radiation coming from the first one, quenching the star formation episode quite abruptly. In general, very young star clusters, sometimes still embedded in their parent gas cloud, are ideal laboratories to study the effect and phenomenology of stellar feedback and gas expulsion. In the Milky way, the so-called “starburst star clusters” (e.g. NGC 3603 YC, Quintuplet, Arches, Westerlund 1 and 2) represent the youngest (< 5 Myr) and more actively star forming clusters <cit.>. NGC 3603 YC, for example, is only ∼ 1 Myr old, and is surrounded by glowing interstellar gas and obscuring dust <cit.>. The Arches, the second youngest with an age of ∼ 2.5 Myr, is already free of any gas in its centre <cit.> with a clear X-ray signature of hot outflowing gas <cit.>. These newborn star clusters are characterised by the presence of strongly UV-radiation from O and B stars that ionises the nebula and disperses the gas <cit.>. On the theoretical side, the challenge of modelling star clusters is due to the lack of a complete theory of star formation. This is an inherently multi-scale, multi-physics problem, with a central role played by feedback mechanisms. We point to the reviews by <cit.> and <cit.> for a detailed presentation of the problem. Here we present only a few selected earlier studies, relevant for our work which focuses specifically on the star cluster formation process. <cit.> and <cit.> modelled fractal clouds by means of 3D smoothed particle hydrodynamics simulations and explored the effect of a ionising O-star at the centre of a 10^4 cloud. They found that some global properties, such as the total outflow rate, the distribution of gas into high- and low-density and the injected kinetic energy are all independent of the fractal dimension, while the statistical properties of the triggered star formation events and the shell morphology both appear to correlate with the cloud fractal dimension. <cit.>, <cit.> and <cit.> used direct N-body simulations, starting from initial conditions drawn from the results of previous smoothed particle hydrodynamics (SPH) simulations of turbulent molecular clouds. Because the adopted SPH resolution was relatively low (∼ 0.1 pc), the authors could not resolve the formation of individual stars, but could still capture the clumpy structure of the gas. After one free-fall time of the initial gas cloud, they stopped the hydro simulation and replaced dense enough gas particles with stellar particles, assuming a star formation efficiency (or gas to star conversion factor) depending on the local gas density. The remaining gas particles were removed instantaneously and the stellar particles dynamics was integrated further in time using a direct N-body code. They derived that the initial properties of the parent cloud (mass, density) determine the characteristics of the emerging cluster, whether it will become an association, an open cluster or a dense massive one. Moreover, to form massive clusters, they claimed that a local star formation efficiency >50% is needed. Using a more elaborate methodology, <cit.>, <cit.>, <cit.>, <cit.>, <cit.> and <cit.> studied in a series of papers the effects of photo-ionisation feedback on embedded clusters and its disruptive impact on clouds of different masses (from 10^4 to 10^6 M_⊙) and sizes (from 2 to 220 pc), either initially bound or unbound. In <cit.>, the authors added stellar winds to photoionisation feedback and studied how the overall star formation efficiency, the average star formation rate (SFR) and the fraction of unbound gas varied with the initial cloud properties. Their methodology was based on SPH simulations of turbulent molecular clouds, with an initial shallow Gaussian density profile. The velocity field was initialised as a turbulent, divergence-free Gaussian random field, with a power spectrum to P(k) ∝ k^-4 consistent with isothermal supersonic turbulence. The cloud was evolved using self-gravity and cooling, and star formation was modelled using sink particles. The mass and spatial resolution was also relatively low, with 10^6 particles per cloud, but using 100 neighbours in the smoothing kernel, so only 10^4 independent resolution elements <cit.>. Radiative transfer of the photo-ionising photons was performed using a Strömgren sphere filling technique <cit.>. Using the same set of simulations, <cit.> focused on the properties of the stellar populations of the star clusters formed. They found that the star formation efficiency is lowered by the presence of feedback, however they stressed how the disruptive effect of feedback depends on the cloud properties, especially the escape velocity. Natal gas from massive clouds with elevated escape velocities is expelled only in minimal part. Winds are found to have little impact on the dynamics of gas compared to ionising feedback. Moreover, in these simulations the number of stars unbound by feedback is very modest and is not related to the fraction of gas expelled. Along the same lines as in <cit.>, the longer term evolution of these star clusters was finally investigated in another series of paper by <cit.>. They concluded that clusters formed in simulations with feedback tend to remain sub-structured longer than in the non-feedback cases. Moreover, at the end of the pure N-body evolution, the authors found that simulations with feedback contain fewer bound stars than in the control run. In terms of mass segregation, they do not provide a unique conclusion, because different analysis return contrasting results. More recently, several papers have addressed the problem of star cluster formation from a realistic, gaseous, turbulent environment using grid-based simulation techniques. Using the RAMSES code, <cit.> studied the conditions required in the parent cloud to obtain a bound star cluster. The authors aimed to examine the properties of the gaseous proto-cluster born from the collapse of a 10^4 molecular cloud. To achieve this they performed magnetohydrodynamics simulations, without stellar feedback and varying the initial level of turbulent support. Prestellar cores were followed using the same sink particles algorithm adopted in our work. The typical mass of a sink was 10 . The proto-cluster turned out to be in virial equilibrium, with turbulence and rotation supporting the collapse. The virial status and size of the proto-cluster were considered to be directly imprinted by the parent cloud, therefore they concluded that the study of the gaseous proto-cluster phase is a fundamental step in the context of stellar cluster formation. Using the FLASH code, coupled to a ray tracing code, <cit.> studied the effect of various cloud initial conditions, then subjected to the ionising radiation of massive stars, on the final properties of the star cluster system. This study focused on giant 10^6 M_⊙ molecular clouds, with different initial virial parameters (α), ranging from bound (α=0.5) to unbound (α=5). The main goal was to study how feedback and the virial status affect the formation of star clusters and subsequent evolution of the cloud. In this case sink particles represented single star clusters and star formation within each cluster is implemented with a subgrid model, by randomly sampling the IMF. Their conclusion was that the initial virial parameter strongly influences the SFE, with more bound clouds having higher efficiency, while radiative feedback did not play a major role, lowering the previous values only by few percent. They also found that the number of star clusters formed depends on the boundedness of the cloud: the more bound the cloud, the fewer the star clusters. Moreover, the clusters from unbound clouds were gas poorer and star richer than the ones formed from bound clouds. In this work, we model the collapse of a ∼ 2.5 × 10^4 M_⊙ turbulent cloud with photo-ionisation feedback from massive stars at extremely high resolution (smallest cell size ∼ 500 AU), and study how the star cluster forms and emerges from its parent cloud. Our radiative transfer technique is based on the moment method with the M1 closure <cit.> and allows to model an arbitrary number of photon sources, much faster than traditional ray tracing schemes. We consider two different feedback scenarios (strong and weak) and a reference simulation without any feedback. We subsequently analyse how the different feedback scenarios affect the properties of our new born star clusters, using various observables related to the stellar mass function, its spatial distribution, the mass segregation, the distribution of escaping stars and the stellar multiplicity function. The paper is organised as follows: in Section <ref>, we describe the numerical methods we have used for our simulations. In Section <ref>, we analyse the properties of the star clusters we have obtained, and finally, in Section <ref>, we discuss our findings in light of previous studies, both theoretical and observational. § NUMERICAL METHODS We now describe in details the numerical techniques we use to model the collapse of a turbulent molecular cloud and the formation of massive stars, following the effects of ionising radiation on the cloud itself. §.§ Initial Conditions We first perform a decaying turbulence simulation in a periodic box sampled with 1024^3 cells. This simulation is initialised with a uniform gas density ρ_0=1 (in arbitrary units) and a Gaussian random velocity field with a power spectrum P(k) ∝ k^-4, where k is the wavenumber. P(k) is normalised so that the 3D velocity dispersion in the full box was set to σ_ 3D = M c_s, where the sound speed is c_s=1 in arbitrary units and the initial Mach number is set to M=20. After one turbulence crossing time, t_ turb = L / σ_ 3D (where the box size was also set to 1 in arbitrary units), the kinetic energy has decayed by a factor of two, and the actual Mach number by a factor of √(2). At that time, the turbulence is fully developed, with density fluctuations following a clear log-normal distribution function and the variance in logρ reaching its peak value. <cit.> found that whether turbulence is initially fully developed or not has significant impact on the results. We then use this final snapshot as a template for the initial turbulent cloud. We first set up the physical scales of our problem. The cloud is considered to be fully composed of molecular gas Hydrogen with temperature T_0=10 K and isothermal sound speed c_s = 0.2 km/s. The mean density in the box is set to n_H=10^3 H/cc and the periodic box length to 20 pc. We carve out of the periodic box a sphere of radius 5 pc, centred on a large filament resulting from a large compressive mode. As a result, the mean density in the spherical cloud is larger than the mean density in the original box, and the Mach number in the cloud is smaller than in the original box (by another factor of √(2)) with M≃10. The final cloud properties are the following: radius R=5 pc, mass M ≃ 2.5×10^4 M_⊙ and velocity dispersion σ_ 3D≃ 2km/s. Note that, because we have adopted a velocity dispersion at the low end of values found in observations of clouds of a similar size, our cloud virial parameter α_ vir = 5 σ^2_ 3D R/3 G M≃ 0.3, is small enough to ensure a fast collapse, i.e. the free-fall time is ∼ 1 Myr. The simulations are then run to t=2Myr. Such a choice for the virial parameter was meant to explore the stabilising effect of feedback. We chose, in fact, an extreme situation to investigate the action range of photoionisation even in very bound and dense environments, characterised by a high degree of dynamical interactions and escaping stars. Moreover, cloud disruption driven by large scale turbulence (see works by Dale et al.) is not effective for our cloud. We intend to relax such an extreme condition in a follow-up paper. §.§ Refinement strategy Our initial coarse grid corresponds to a minimum refinement level ℓ_ min=10 with cell size Δ x_ max≃0.02 pc, which allows us to resolve our sonic scale l_s ≃ 0.08 pc, i.e. the scale at which our scale-dependent 3D velocity dispersion is equal to the sound speed. During the course of the simulation, we refine this initial grid level using a quasi-Lagrangian refinement criterion. Our maximum resolution is fixed to our maximum refinement level ℓ_ max=13, which corresponds to a minimum cell size of Δ x_ min≃ 500 AU. Assuming for the isothermal sound speed c_s = 0.2 km/s, and requiring for the Jeans length λ_ J=c_s √(π/G ρ) > 4 Δ x_ min, this gives us the constraint that ρ < ρ_ J≃ 2 × 10^-17 g/cc. This maximum density corresponds also to a Jeans mass m_ J = 4 π/3ρ_ J( λ_ J/2)^3 ≃ 0.14 M_⊙. We require to resolve this Jeans mass with at least 64 resolution elements, which gives us a mass resolution of m_ res≃ 2 × 10^-3 M_⊙. Our refinement strategy is thus the following: if a cell has accumulated a gas mass larger than m_ res, then it is refined individually into 8 new children cells, up to the maximum refinement level. Note that with our adopted initial coarse level and our quasi-Lagrangian strategy, we also automatically satisfy the additional criterion that the Jeans length is always refined by at least 4 cells for any gas density smaller than ρ_ J. §.§ Sink Particles When the gas density exceeds ρ_ J, we violate our requirement to always resolve the Jeans length with 4 cells and the Jeans mass with 64 resolution elements. Therefore we adopt this criterion to form sink particles, using the technique developed in <cit.>. We first detect density peaks in our 3D density field using the PHEW clump finder <cit.>. The density threshold is set to ρ_ threshold=2× 10^-18 g/cc, or 10% of the Jeans density. After we have identified a discrete set of peak patches delimited by either the isosurface at the density threshold or the saddle surface with a neighbouring peak patch, we draw a sphere, 4 cell size in radius, around the density maximum. If the density at the maximum exceeds the Jeans density, if the sphere is contracting and if its virial parameter is less than 1, we form a sink with a seed mass equal to m_ J≃ 0.14 M_⊙ <cit.>. In our simulations one sink corresponds to a single star. The sink particle is then treated like a point mass. We follow the sink particles dynamics by a leap-frog, direct N-body integrator, using a softened 1/r^2 acceleration (with softening length 0.5 Δ x_ min) between sinks, and also between the sinks and the gas. Only the self-gravity of the gas is based on the grid-based Poisson solver in RAMSES. Gas accretion onto the sink particles is modelled through what is described as “flux accretion" in <cit.>. §.§ Radiative Processes In this paper, we model the emission and the propagation of ionising, ultra-violet (UV) radiation, together with associated heating and cooling processes. We used the RAMSES-RT radiative transfer module developed by <cit.>, using one photon group, with energies between 13.6 eV and 24.6 eV. We do not account for photon energies below 13.6 eV, namely optical and infrared radiation, as the scope of the paper is to study the effect of photo-ionisation heating on the molecular cloud. We will study these other sources of radiation in a follow-up paper. Details in the adopted photo-absorption cross section, chemistry and cooling processes are available in <cit.>. Metal cooling prescriptions are based on <cit.> for temperatures above 10^4 K and on <cit.> for metal fine-structure cooling below 10^4 K. We extended the cooling function by <cit.> down to 10 K, to account for CO and fine structure cooling due to CII, OI, according to prescriptions of <cit.>. Following <cit.>, the photon group energy and cross-section are derived sampling the blackbody spectral energy distribution of a 20 star. The frequency-dependent ionisation cross sections are taken from <cit.> and <cit.>. A reduced speed of light of 10^-4 c is used. This is done to improve the efficiency of our simulations, since the speed of light affects the tilmestep calculation, through the Courant factor. The UV radiation emitted by the sink particles is modelled using the following simple strategy. We implemented two feedback regimes, namely strong and weak. For the strong feedback case, we basically consider all the energy emitted from the sink/star (even optical and infrared) as ionising radiation. To derive the energy associated with every sink we assume a power-law luminosity-mass relation, L = L_⊙ (M/M_⊙)^3.5, where L_⊙ and M_⊙ are the solar luminosity and solar mass, respectively . The number of photons emitted per second, Q_HI, was then obtained by dividing this luminosity by a mean value of photon energy in the ionisation range (13.6eV-24.6 eV). For the weak feedback case, we computed an analytical fit of photon emission rates presented in <cit.>, obtained through radiation-driven wind atmosphere models of OB stars. We derived the following analytic expression of the number of emitted ionising photons per second as a function of the stellar mass: log [Q_HI (M)] = 48.65+log(M/M_⊙) - 2.4/log(M/M_⊙-8)^1.9 . This formula was applied to calculate emission rates for all sinks with M > 10 M_⊙. For stars with lower mass we assume there are no ionising photons. Figure <ref> compares the resulting Q_HI from the two feedback models considered. § ANALYSIS In this section we focus on the analysis of the simulations. In particular, we study the structural characteristics of the star cluster (such as mass function, virial status, mass segregation, escapers, binaries) in the three different runs, to understand the role of feedback (FB) in shaping the star cluster itself. Figure <ref> shows ratios of kinetic to potential energies of sinks (upper panel), cluster sizes and the SFE (lower panel) as a function of time. Focusing first on the SFE, the ionising radiation clearly has a major effect in suppressing star formation. In Figures <ref> and <ref> we demonstrate the effects of the radiation qualitatively, plotting time-sequences of gas density and temperature maps, to compare the strong, weak and no feedback cases. The initial phase of the cloud collapse proceeds identically in the three cases. The cloud gravitationally contracts and starts forming filaments, where local overdensities allow the creation of stars, here represented by sinks (in yellow or turquoise, depending on the map). In the no-feedback case this contraction proceeds without resistance until, eventually all the gas is converted into stellar objects; from Figure <ref> we can see how even in the latest snapshot the amount of dense gas is still high and by the end of the simulation time (2 Myr) the fraction of total mass still available in gas is ∼10%. In general, we can notice how the final shape of the star cluster becomes more and more spherical with the simulation progressing. The gas temperature in the no-feedback case does not show huge changes throughout the collapse. In the weak-feedback case, stars emit ionising radiation and we now follow the photo-chemistry of Hydrogen. Differences with the no-feedback case start being visible around already 0.4 Myr in the temperature map, when the most massive stars in the lower part of the filament start emitting UV photons and cause the gas temperature to increase locally. This bubble of hot gas becomes more and more extended since more stars are formed, accreting more gas. The neutral HI gets dissipated, due to the quick expansion of the HII region. At the end of the simulation, the star cluster is completely free of dense and neutral gas. The strong-feedback case is analogous to the weak-feedback case but the process of photoionisation and gas expulsion is much more rapid and violent, so as a result the star cluster is devoid of gas already at 1.2 Myr. §.§ Virial properties In the top panel of Figure <ref> we show the evolution over time of the virial ratio of the star cluster, E_k/E_p (where E_k and E_p are respectively the total kinetic and potential energy of the sinks) for the three simulations. We do not consider snapshots before 0.5 Myr because before this time there is still a large amount of gas mass which will become sinks, and therefore the stellar cluster cannot yet be treated as isolated system. As seen in the figure, the case without feedback is clearly super-virial, hence expanding. The two cases with feedback, instead, result in virial or even sub-virial state. This can be explained as a result of feedback, which halted the collapse of the cloud, ionising and dispersing the neutral gas. This determined the formation of a much less dense aggregation of stars than in the control simulation. In the run without feedback, the collapse proceeds unhindered and the new-born stars are immersed in a dense, highly-collisional environment, experiencing very strong close interactions. This inevitably leads to the ejections of many sinks and expansion of the cluster. The middle panel of Fig. <ref> clarifies the evolution of the size of the star cluster, considering both the half-mass radius (dashed line) and the global size obtained as √(I/M), where I is the moment of inertia of the cluster and M is the total mass. From the plot it can be seen that the reference run is the more extended one, but at the same time half of its mass is very concentrated at the center. The expansion is then due to the escaping (massive) stars, not to a generally unbound cluster (a similar case was presented in ). However, when feedback is included, its effect is to oppose this runaway collapse and allow the onset of a lower density regime, where the stellar distribution finds a stable configuration. It is interesting to notice how this result goes against traditional predictions (see the Introduction), which argue for a complete disintegration of the star cluster after a violent expulsion of gas. However, these often assume a fully formed star cluster still embedded in gas, which at some point gets ejected. In our case, stars are created while the gas is expelled in a self-regulating fashion. Therefore the virial status of the emerging star cluster changes along with the collapse. The outcome of our simulations results from the interplay between the highly subvirial initial virial ratio and the strength of the feedback adopted: a very subvirial cloud produces a cluster too dense to survive, unless feedback slows down the collapse. We also conclude that the star formation efficiency alone is not a good indicator of the survivability, as it is usually believed. In the lower part of Figure <ref> we show the fraction of gas transformed into stars. Stellar feedback is very efficient in stopping the collapse and lowering the SFE. In fact in the case with the strongest feedback the SFE halts at ∼20% (while virtually unity for the control simulation). For a weaker feedback, we get a higher fraction. Despite the fact that in the simulation without feedback all gas is eventually transformed into stars, we stress that the outcome of the simulation is the dispersal of the emerging star cluster, while for the strong feedback case, which results in a very low star formation efficiency, the outcome is a stable (or even subvirial) star cluster. §.§ Mass function In Figure <ref>, we plot the stellar mass function for all the feedback cases we have considered and at different times. In the run without feedback, our mass function peaks at a relatively large mass of ∼10 and shows a strong accumulation of very massive stars at the high mass end, with the mean sink mass being around 15 and the most massive sink reaching 460 . This is due partly to our limited resolution (see later) and to the lack of feedback to limit the maximum stellar mass. In the weak feedback scenario, the maximum mass is lower, around 250 and the mass function flattens, with a slight increase of very low mass stars (close to our resolution limit of 0.1 ). The trend gets even clearer if we look at the case with strong feedback, where there is a significant peak of stars with mass around 0.1 (corresponding to the sink seed mass) and the most massive star is now around 120 . We observe in the simulation that this excess of low mass stars close to the resolution limit is caused by the fragmentation of the outer dense shells of HII regions. Looking at the mass function at earlier times (specifically, t=0.25 Myr and t=0.5 Myr, paler lines in figure 3), it is clear that the onset of the sink mass function proceeds similarly in the three cases. It is mainly the final mass distribution that shows visible differences between the feedback and no-feedback cases. To summarise, these are 1) the high-mass cut-off due to feedback effects that stop accretion onto massive sinks, 2) a peak at the low-mass end, due to fragmentation of dense gas around HII regions. §.§.§ Comparison to observations It is very instructive to compare the results of our simulations to available observations. We choose to consider NGC 3603 YC and the Arches cluster, since they are among the youngest (< 2 Myr) well-studied star clusters, part of large, still gas-rich, regions. NGC 3603 YC (also known as HD 97950) is a very compact and massive young star cluster at the centre of the vast homonym HII region. It is composed of three Wolf-Rayet stars and around 40 O-type stars, a dozen of which resides in the very central part of the core, within less than 1 ly from the centre <cit.>. <cit.> estimated the total mass to be between 1 and 2 × 10^4 . The H-R diagram in <cit.> reveals the presence of at least 15 stars with masses greater than 60 . The most massive stars in the cluster seem to be coeval with ages between 1 and 2 Myr <cit.>. However, the age spread between the pre-main-sequence stars <cit.> and the slightly older stars in the cluster outskirts <cit.> suggests a possible extended star formation scenario. The Arches cluster is considered to be the densest cluster in our Galaxy. It also falls in the category of so-called starburst star clusters. It is located near the Galactic centre and its age is estimated to be around 2 Myr. Its total mass is estimated to be around 2 × 10^4 <cit.> and it contains 160 O-stars and 13 Wolf-Rayet[This is about 5% of all known Wolf-Rayet stars in the Milky Way <cit.>.] <cit.>. For NGC 3603 YC, we considered the mass function results published by <cit.> and for the Arches the one published by <cit.>. To derive the mass function of NGC 3603 YC, the authors considered stars in absolute V-magnitude bins and then derived the correspondent masses using the isochrone models from <cit.> for high mass stars and <cit.> for low mass stars. Their mass bins have a logarithmic size of 0.2. The data were corrected both for incompleteness and foreground stars contamination and include all stars within 60” (∼ 2 pc). <cit.> derived the present day mass function of the Arches cluster by converting the K-band magnitudes from the corrected color-magnitude diagram into masses using a 2 Myr Geneva main-sequence solar metallicity isochrone from <cit.>. They also binned their data using logarithmic intervals of size 0.2 and they computed the mass function 10 times, each time shifting the bins by 0.02. The final present-day mass function was created by averaging all the points from these 10 mass functions and takes into account all stars within 0.4 pc. Comparing these observational data to our simulations is not trivial, since we do not know the SFE of the parent clouds of both NGC 3603 YC and the Arches. The targeted clusters have about twice the mass of our simulated ones from the feedback runs, but roughly equal to the one in our no-feedback simulation. If the true SFEs of the observed star clusters were very low, say 10%, this would imply that the original clouds would be as massive as 10^5 M_⊙, which is computationally too expensive to simulate at the current resolution and with our radiation solver. Therefore, we decided to re-normalise the observations. The normalisation factors are computed requiring that the mass bin at 15 M_⊙ in the two observational datasets have the same value, equal to that of our simulated data set. The normalisation coefficients for the Arches dataset are 0.4, 0.5 and 1.1 with respect to the strong, weak and no feedback cases, respectively, while the normalisation coefficients for the NGC 3603 dataset are 1.2, 1.5 and 3.3 with respect to the strong, weak and no feedback cases, respectively In Figure <ref> we compare these renormalised observed mass functions to our simulated ones. Renormalised observational data are showed with red triangles (NGC 3603) and green circles (Arches). The best agreement, especially at the high-mass end, is obtained with the strong feedback (after renormalisation). The weak and no-feedback runs clearly produce too many very high-mass stars. The agreement is worse at lower masses, especially below 10 M_⊙. As we explain below, we believe this is due to our limited resolution. §.§.§ Slope of the mass function The previous analysis was carried out considering all the sinks in the simulation box. We now study the mass function dependency with radius. In Fig. <ref>, we show the mass function taking into account only sinks within specific radii[Unless otherwise stated, the radius is always considered respect to the centre of density of the system defined as in <cit.>.] , namely 1, 3 and 5 pc, and for all three feedback regimes. The last radial bin contains 92%, 74% and 88% of the simulated sinks respectively for strong, weak, no feedback. The solid curve corresponds to the whole box, or a radius of 10 pc. Although the mass function appears to be independent of radius for the no feedback case, it looks clearly flatter in the inner parts and steeper in the outer parts for the two feedback cases. <cit.> showed that a similar effect is present in NGC 3603: the slope of the mass function steepens with radius, indicating that the most massive stars are mostly concentrated in the centre. This feature is generally explained by mass segregation. We will develop this topic in the next section. If we now quantify the slope of the mass function, we found that all our simulations show a slope (Γ) much flatter than that of the Salpeter IMF (i.e. Γ=-1.35), depending sensitively on the range of masses used to compute it (see Fig.  <ref>), which means the mass function is probably not a power-law all in all. A shallower slope than the Salpeter is also the case for observed young and embedded star clusters. NGC 3603, for example, has Γ = -0.88 ± 0.15, considering only log(M/) > 0.6 for completeness reason. For the Arches, <cit.> detected a change in the slope of the mass function at about 6 M_⊙, hence they fitted the mass function in the range log(M/) > 0.8. The resulting value was measured to be Γ=-0.86 ± 0.15. Both these clusters have slopes flatter than the Salpeter slope, which seems to be in general a distinguishing feature of young starburst clusters. The origin of this discrepancy from the Salpeter slope is probably due to many reasons. On the simulation side, <cit.> showed that the simulated IMF can be affected by resolution, with the peak or turn-over mass depending directly on it. The higher the resolution, the lower the turn-over mass, which implies a progressive steepening of the mass function with increasing resolution. These authors estimated that the peak mass is roughly ∼ 30 times the minimum Jeans mass, which is our case corresponds to about 4.5 M_⊙, and agrees quite well with our no-feedback case. Studying the formation of low-mass protostars in radiative feedback simulations, <cit.> obtained IMF profiles with slopes compatible with the Salpeter prescription. So, resolution effects are likely a cause of the low value for Γ in our simulations in the intermediate mass range log(M/) < 1. Moreover, feedback inevitably plays a role in all this, lowering the number of stars in the intermediate-high mass range, therefore contributing to an even shallower slope. At larger masses, on the other hand, recent theories of turbulent cloud collapse argue for an asymptotic Salpeter slope <cit.>. This could be consistent with our simulated star clusters, but also with the observed ones, without being very conclusive, reminding us that the story is probably not so simple. §.§ Mass segregation We have already introduced mass segregation in the previous section to explain a steepening of the slope of the mass function as a function of radius. We now analyse our simulations with more traditional tools to quantify mass segregation in star clusters. A star cluster is considered to be mass segregated when the massive stars are more centrally concentrated than the lower mass stars. The main question related with mass segregation is whether it has a primordial or a dynamical origin. Mass segregation can indeed be the result of two or three body interactions between stars (dynamical) or the direct outcome of the star formation process within the gas cloud itself (primordial). Our simulations are ideal experiments to try and answer this question. The problem of comparing the mass function for different radii to characterise mass segregation is that we need to define unambiguously the centre of the star cluster, which is a difficult task. <cit.> introduced the Minimum Spanning Tree (MST) to quantify the degree of mass segregation in a star cluster. The MST is defined as the shortest path connecting all points, which does not contain any closed loop. We used the routine included in the csgraph module of scipy, which implements the MST according to Kruskal's algorithm <cit.>. We followed <cit.> prescription to quantify mass segregation using the MST. We computed the length, L_ massive, of the MST of the N most massive stars and compared this to the average length of the MST of N random stars in the cluster, or L_ random. L_ random was calculated by picking 1000 random sets of N stars, in order to have a small error on the dispersion σ. Mass segregation is quantified using the Minimum Spanning Tree Ratio Λ_MSTR defined by <cit.> as Λ_MSTR = L_random/L_massive±σ/L_massive. For Λ_MSTR∼ 1, the distribution of massive stars is comparable to that of all stars. For Λ_MSTR > 1, massive stars are more concentrated, a clear sign of mass segregation. The larger Λ_MSTR, the more pronounced is the mass segregation. This method was already adopted by <cit.> to analyse the dynamical evolution of star forming regions, starting from the final states of the SPH simulations by <cit.>. Using Λ_MSTR for their N=10 most massive stars, they found in their no-feedback simulation a strong primordial mass segregation with Λ_MSTR≃ 5, which disappears after 3 Myr due to stellar evolution and reappears at the same level after 8 Myr due to dynamical interactions between the cluster members. However, in their feedback simulations that include winds and photoionisation, they did not detect any mass segregation, with Λ_MSTR≃ 1 at all times. In Figure <ref>, we plot Λ_MSTR as a function of N_MST, the number of stars we use for the spanning tree, at t = 2Myr. We include in our analysis all stars up to an outer radius of 7.7 pc, 9.3 pc and 9.8 pc, corresponding to the distance from the centre of the cluster of the most external bound star, in the strong, weak, and no feedback cases respectively. This is done to prevent extreme outlier stars to dominate the calculation of the random spanning tree. Our data point with N=10 corresponds to the estimator used in <cit.>. All three cases show some degree of mass segregation. Our no-feedback case is strongly mass segregated for N=10 with Λ_MSTR≃ 10, and is still significantly segregated for N=20 with Λ_MSTR≃ 5. The signal however disappears for N≥30. The strong feedback case shows the weakest mass segregation for N=10 with Λ_MSTR≃ 2, but the segregation signal is still detectable up to N=60. The weak feedback case lies in between the two other cases. The two crucial pieces of information Figure <ref> provides are 1) the degree of mass segregation of the cluster, namely the value of Λ_MSTR and 2) the extent of mass segregation, namely the maximum number of stars that are mass segregated. From our results, two different situations emerge. In the no feedback case (and to some extent in the weak feedback case), only a handful of super-massive stars are tightly concentrated at the centre. Only those most massive stars are mass segregated. The high stellar density is supported by the high measured values of Λ_MSTR. This population of massive stars forms effectively a sub-cluster at the centre of the main cluster, that keeps contracting and decouples dynamically from the rest, transferring its kinetic energy to less massive stars that are ejected (see next Section). On the contrary, in the strong feedback case, photo-ionisation feedback is efficient enough to halt the collapse of the gas, limiting the number density of massive stars. This prevents the formation of an independent self-gravitating system within the cluster itself. This translates into a lower degree of mass segregation and at the same time a higher number of stars being mass segregated. In order to compare with observations, we plot Λ_MSTR as a function of the stellar mass (Fig. <ref>). Following <cit.>, we sort the stars by their mass and then consider blocks of 20 stars moving in steps of 10 stars, such that the data partially overlap. For example, the first 20 stars in the weak feedback case (magenta line in Figure <ref>) cover the range 200 to 80 in mass, the second mass group goes from 130 to 60, etc. The mass interval considered is indicated by horizontal bars in the plot. For every bar a marker denotes the mean mass of the interval. The three profiles of Λ_MSTR versus mass in Figure <ref> look qualitatively similar, but they are shifted to higher and higher masses with increasing feedback strength. The no feedback case shows mass segregation only in the first bin (M>200) with an amplitude much larger than unity. For the weak feedback case, only stars down to a mass of 60  are weakly segregated, with an amplitude of 2, and for the strong feedback case, the transition goes down to 30 . In Figure <ref>, we compare our simulations to the data of <cit.> on NGC 3603 (yellow points). A very good agreement is obtained with the strong feedback case. In Figure <ref>, we plot only the strong feedback case and the observations using a linear scale in mass to allow a better comparison and to outline the very good quantitative match between our model and the observed segregation, both in terms of amplitude and of transition mass. Despite being young, NGC 3603 shows already a clear signal of mass segregation. This is not an isolated case. There is also strong evidence of mass segregation in the Orion Nebula clusters, but also in the Arches, NGC 6611, NGC 2244 and NGC 6530, to name a few <cit.>. The origin of the mass segregation in these clusters is still an open question (primordial or dynamical). <cit.> proposes for NGC 3603 a dynamical origin. Using analytical arguments, they show that the cluster dense core could dynamically segregate in one crossing time down to a mass of 30 . To test this hypothesis, we have performed our clustering analysis at earlier times and find no indication of mass segregation for massive stars. We have estimated the local two-body relaxation timescale of the densest part of the cluster (r < 2 pc) and find it to be less than 0.5 Myr for all 3 cases, supporting our claim that dynamical friction can cause mass segregation after 1 Myr. To quantify further the structure and morphology of our star clusters , we have used another statistical indicator called the Q parameter <cit.>. Q is defined as the ratio between the normalised mean edge length m of the MST of all stars in the cluster and the normalised correlation length s of the same stars[The correlation length is defined as the mean separation between stars in the cluster.]. These parameters taken separately cannot distinguish between a smooth, radially concentrated distribution and an extended, fractal distribution, but their ratio can <cit.>. A cluster with Q > 0.8 is smooth and centrally concentrated, while if Q < 0.8, it is extended with a fractal distribution. In Figure <ref>, we show the evolution of the Q parameter with time. In all our simulations, the star cluster is initially fragmented and extended. The no feedback case rapidly evolves towards a more spherical and centrally concentrated distribution with Q ≃ 1.5, while in the two other cases, the transition is slightly slower, supporting a longer survival of substructures, and reaches a smaller maximum value with Q ≃ 1.1 and 1.2. This supports a scenario in which gravitational collapse together with stellar dynamical interactions progressively erase the initial conditions in the gas cloud and build up a dense and spherical star cluster. In this context, feedback acts as a delay mechanism, favouring lower stellar densities with a longer relaxation timescale, allowing the longer survival of the initial substructure and a more extended final distribution. §.§ Stellar dynamics In this section, we focus on the dynamics of individual stars and study the influence of the star cluster formation scenario. Our interest is on escaping stars, due to various dynamical interactions in the densest regions of the star cluster. We then study binary stars, as they are the most likely source of escaping stars during the early phase of the life of the star cluster. §.§.§ Escaping stars Escaping stars are particularly interesting when they are massive: they can travel long distances in the galaxy and eventually explode as supernova (SN) in a location far from their original birthplace, typically in the diffuse ISM. In the kiloparsec scale simulations of <cit.> and <cit.>, the global star formation rate in the Galaxy was reproduced if supernovae were allowed to explode up to 20 pc from their natal cloud, while “homebred” supernova explosions were much weaker in suppressing star formation . Similarly, <cit.> showed that allowing SN to explode at random positions, rather than at density peaks significantly changes the properties of the ISM, resulting in a hot gas filled volume ISM in the first case and a filamentary, hot gas deprived ISM in the second one. Thus, being able to predict the correct number of escaping massive stars to be used as input in galactic scale simulations is of vital importance. Escaping stars (or for short “escapers”) are usually categorised into “walkaway stars” and “runaway stars”[Hypervelocity stars are here considered an extra category, which is not treated in this work. These stars are thought to have a Galactic centre origin <cit.>, probably resulting from close encounters between binary systems and the central supermassive black hole. They reach velocities of ∼1000 km/s, and hence they are actually unbound from the Galaxy. The current fraction of known hypervelocity stars is ∼ 10^-8% of all stars in our Galaxy <cit.>. ]. Runaway stars (RS) are defined as stars with velocities larger than 30 km/s <cit.>, produced either by supernova explosion in a tight binary system, during which the companion star of the supernova gets expelled <cit.> or through dynamical ejection due to very close, three body encounters with massive stars <cit.>. In this section, we focus only on the latter mechanism, while the former can be thought of as a direct consequence of the multiplicity function which we will discuss in the next section. Walkaway stars (WS , velocities lower than 30 km/s) are normally defined as “slow escapers", since these are slowly moving stars ejected though normal relaxation processes, such as evaporating stars though distant two-body encounters with other single stars or soft binaries <cit.>. In Figure <ref>, we plot the modulus of velocity versus position of all stars in the cluster. The size of the symbols is proportional to the mass of the star. Filled symbols indicate single stars, while open circles denote stars which are part of a multiple system (binary, triple or more). The escape velocity is computed as a function of radius (green solid line in Fig. <ref>), assuming spherical symmetry, which is a good approximation at t=2Myr (see Fig. <ref>), by averaging over the individual escape velocities at different positions within the same spherical shell. The no feedback case exhibits the highest number of RS candidates[It is important to clarify that in Figure <ref> for binaries, triple systems and more, we plot the true velocity, not the velocity of the centre of mass of the multiple system. Thus, some very high velocity binary members are actually still bound. In the computation of the number of RS we did not correct for this, hence we prefer to talk about RS “candidates”, meaning that some are probably not unbound yet, but very likely to be, due to frequent interactions with other particles.], namely 31, or about 2 % of the total number of stars in the cluster. In the cases with feedback, the number of RS is lower, only 1 and 3 in the strong and weak feedback respectively, accounting only for 0.1% and 0.4% of the total number of stars. The RSs in our simulations are generally only massive stars (38, 229, 132, 2) in the feedback cases, while in the no feedback they cover the whole mass spectrum, going from 0.15 to 417 . The fact that RS are close in mass to the most massive stars in the cluster is easily explained considering the mechanism through which these fast stars formed. Indeed, RS are originated as escaping members of perturbed binary systems, which in our case are mostly composed by massive stars. Due to three body interactions, the lighter member of the binary can escape. RS will therefore have very high masses, close in mass but still lighter than the original massive companion. Regarding WS, the fraction changes slightly depending on the exact definition used. A first possibility is to take all stars with velocity higher than the escape velocity at a given radius and lower than 30 km/s. This gives us a percentage of WS similar in all simulations, around 30%. If we remove stars in multiple systems that are still bound (see Footnote 2), then the fraction is reduced to 20%. The final option is to consider WS only in the outskirts of the star cluster, in order to avoid counting stars that are only momentarily unbound. If we call R_ esc the radius at which the escape velocity becomes comparable to the average stellar velocity at that radius, we can impose the extra-requirement to be at a distance greater than R_ esc≃ 5 pc from the centre of the star cluster. In this case, we get a very conservative estimate of the fraction around 15% of the total number of members of the cluster. Table <ref> gives an overview of the statistics for escaping stars and multiple systems. We also report the fractions of bound and unbound stars, derived by calculating the kinetic and potential energy for every star, and then verifying whether the sum of the two energies is negative and positive, respectively. In all simulations the fraction of bound stars is about the same, around 60%. Comparing the populations of RS and WS in the three simulations, we find that the run without feedback produces much more fast escaping stars than the two feedback cases. This is consistent with our conclusions in the previous sections, of a very dense star cluster hosting a central clump of tight multiple systems of fast massive stars. Three-body interactions can cause the violent ejection of a member of a binary, of the perturber or of the entire binary system (see Fig. <ref>). In the feedback cases the central densities are lower, and therefore RS stars are rare events. The number of WS follow the same trend, with the strong feedback case having slightly less WS stars than the weak and no feedback cases. Strong feedback leads to the less frequent interactions, owing to the lower stellar density, which slows down the evaporation of the stars. We also notice that the different conditions in the three runs have an effect on the typical velocity and mass of WS. In the strong feedback case, they don't reach velocities higher than 10 km/s and are mostly low mass stars, probably escaping due to several repeated low energy kicks, typical of evaporation, while in the no feedback case both low- and high-mass stars can reach velocities close to the RS limit of 30 km/s, as a result of direct ejection. §.§.§ Multiple systems We focus now on the analysis of multiple stellar systems. We identify candidate multiple systems by analysing all possible pairs of stars from the cluster. For each pair we calculate the internal energy, as the total energy of the system in the frame of their centre of mass <cit.>, E=1/2μv^2_12 - G m_1 m_2/r_12, where m_1 and m_2 are the masses of the two stars, μ=m_1 m_2 / (m_1+m_2) is the reduced mass, v_12 is the relative velocity, r_12 the relative distance between the two stars, and G is the gravitational constant. We define the two stars as a binary when E < 0. We consider all the binary connections as edges in a graph, whose nodes are all the stars involved in multiple systems. We use graph reduction algorithms to extract which edges share the same nodes, and we group the nodes together, defining triple, quadruple or quintuple systems in this way. For example, two binary systems, (i, j) and (j, k), which share one node, are considered a triple system. A slightly different technique was used by <cit.> to identify multiple systems. They replaced the binary systems by a virtual star sitting at the centre of mass and with mass equal to the sum of the two masses. They then searched for isolated stars with a negative binding energy with these virtual stars. The same procedure was iterated only up to quadruple systems. An advantage of our graph-based method is that we can easily identify systems with multiplicity larger than 4. However, in most cases the two algorithms will produce the same catalogue of multiple systems, since, in our case, most multiple systems include a massive star, which dominates the gravitational potential of the system (see Fig. <ref>). In Table <ref>, we report on the statistics of binary, triple and more than 3-body systems for all three simulations. We note that the fractions of stars in multiple systems, also known as the multiplicity fraction, correlates with the strength of feedback, with overall percentages spanning from 11% (no feedback) to 19% (strong feedback). If we exclude stars with mass lower than 1 , the multiplicity fraction differentiates even more between the three feedback regimes and rises to 12%, 20% and 31% for no, weak and strong feedback respectively. For stars, with mass greater than 10 , the fraction goes up to 24%, 39% and 55%. Due to the adopted sink density threshold, fragmentation is not fully resolved for low-mass stars, which might contribute to lower the multiplicity fraction of low-mass stars. A more detailed study focused on the multiplicity of low-mass pre-stellar cores was performed by <cit.>. The observed multiplicity fraction is around 20%, when one considers field stars and low mass stars, but reaches 60% for OB and massive stars <cit.>. These values are well reproduced by our strong feedback case, while our no feedback run underestimates the number of stars in multiple system, when compared to observations, especially for massive stars. Observations also reveal that the binary fraction is higher in lower density star forming regions, like in our strong feedback case, while denser clusters exhibit multiplicity fractions comparable to the field or low mass stars, like in our no feedback case <cit.>. In Figure <ref>, we plot the distribution of multiple systems in terms of position versus velocity. Here, we consider the positions and the velocities of the centres of masses, explaining why velocities are lower than in Figure <ref>. In general, we observe that in the feedback simulations binary, triple and more than 3-body systems are uniformly distributed throughout the cluster, while the no feedback case shows many systems with very high multiplicity in the very inner part of the cluster, while binaries and triple systems occupy the outskirts. In all cases, we see many ejections of binary systems. In the same plot, we also indicate the exact count of multiple systems, in particular for groups with more than 3 bodies. We notice that the maximum multiplicity reaches a much higher value in absence of feedback, due to the very high stellar density. With feedback, the most crowded multiple systems have 5 or 6 members, while in the no feedback run we have systems with as many as 9, 12 and 21 members. All these high multiplicity systems are highly unstable and they will be destroyed during the subsequent dynamical evolution of the cluster. As a matter of fact, we do not observe such systems in real star clusters . In the strong feedback case, the lower stellar density will also guarantee the survival of the binary systems, which otherwise, like in the no feedback case, aggregate in bigger associations or are destroyed in three-body interactions. In that context, it is useful to divide binaries into two categories, soft binaries and hard binaries. Soft binaries are systems for which |E|< K, while hard binaries have |E|>K, where K is the typical kinetic energy of the stars in the cluster <cit.>. We use here the median kinetic energy. According to this definition, for the two feedback cases, we have 50% hard and 50% soft binaries, while the no feedback case shows only 30% hard and 70% soft binaries, which support even more our conclusion that binaries will survive longer in the strong feedback case. In Figure <ref>, we plot the time evolution of the number of binary, triple and more than 3-body systems. In all three models, the number of triple (or more) systems is almost constant. This is not the case for the number of binaries. In the strong feedback case, it increases sharply during cloud collapse and after the gas has been dispersed around 1 Myr, it slowly decreases. No additional stars are created and the soft binaries get destroyed through ejection or evaporation. In the no feedback case, the number of binaries keeps increasing since star formation continues until the end of the simulation. The weak feedback case shows an intermediate behaviour, with a mild initial increase, followed by a almost constant evolution. § SUMMARY AND DISCUSSION In this work, we have performed radiation hydrodynamics simulations of a collapsing turbulent molecular cloud with the adaptive mesh refinement code RAMSES. We have studied in detail the emergence of the star cluster from the parent gas cloud with and without the influence of photoionisation feedback. Stars are modelled using a sink particle algorithm. Photo-ionising radiation is included with two different regimes: weak and strong feedback. We also perform a reference simulation without any feedback. Our main focus is the emerging properties of the star cluster, both from a structural and a dynamical point of view. The main effect of photo-ionisation feedback is to reduce considerably the stellar density of the star cluster and to limit the accretion on very massive stars. This has a large impact on the dynamical properties of the final star cluster. As a result of the reduced stellar density, the star cluster can settle in virial (or even sub-virial) state, while in the absence of feedback, strong and frequent close interactions in a highly collisional environment lead to the disruption of the cluster. This is in contrast with the traditional view that strong feedback is responsible for the star cluster early mortality, by rapidly removing gas from the emerging cluster <cit.>. The star formation efficiency can be reduced down to 20%, without affecting the virial equilibrium of the star cluster. The stellar mass function is also affected at the high mass end, with a clear self-regulating role played by feedback, limiting the mass of the most massive stars by a factor of 4 compared to the no feedback case. As a result, our mass function with strong feedback compares favourably with observations of two starburst clusters (NGC 3603 and Arches) but only after re-normalising the data and for masses larger than 10 M_⊙. We also use mass segregation statistics to test our model. In absence of feedback, the higher stellar density causes an unrealistically too high degree of mass segregation for a few very massive stars. When including strong feedback, we obtain a more extended star cluster with a degree of mass segregation consistent with the one observed in NGC 3603. We have also computed the number of ejected stars, which anti-correlates with the feedback strength: for weaker feedback, we get a higher stellar density and more stars are escaping, both as runaway and walkaway stars. This result has profound implications for galactic evolution, when supernovae will start exploding at later time in a large variety of galactic environments. Our statistics of multiple systems of stars supports the same conclusion: in a denser environment, the fraction of stable binary systems is lower, and most stars tend to either cluster into unstable many-body systems, or are ejected. On the other hand, in the strong feedback case, the lower stellar density guarantees the survival of a higher fraction of binaries, in better agreement with observations. Our results are in line with the findings of <cit.>, which showed that photo-ionisation feedback effectively lowers the star formation efficiency, and, for low-mass clouds like ours, can expel most of the gas within 3 Myr, before the first supernova can explode. <cit.> also observed that photo-ionisation feedback reduces the stellar density in the emerging cluster, which allows substructures to survive longer than in a scenario without feedback. However, in contrast with <cit.> who did not find any mass segregation in the feedback case, we do see a weak mass segregation signature, which is well in agreement with observations. Interestingly, although <cit.> found that a local star formation efficiency of at least 50% is necessary for the formation of young massive clusters, we could reach a value as low as 20% and still form a bound star cluster. Our goal in this work is to better understand the transition from a gas cloud to a stellar cluster, or in other words, from gas dynamics to stellar dynamics. In that context, our direct N-body integrator, a second order leap frog scheme is probably accurate enough for our relatively short time integration, but its accuracy is far below the required standards in stellar dynamics for longer time scales. This sets the limit on the runtime of our simulations to a couple of Myr. This explains why, in comparison to <cit.>, who were able to investigate the long term evolution of the star cluster, we are forced to limit our study to the first 2 Myr. We have also decided in this work to focus exclusively on photo-ionisation radiation. We have therefore neglected magnetic fields and other radiation processes, but also other important physical processes that could be relevant. Supernovae explosions, for example, are ignored, but, given the cloud mass we have adopted, all the gas is removed from the star cluster after only 2 Myr and they are therefore irrelevant. For larger cloud masses, however, this would not be the case. We have also ignored the possible role of stellar winds, but these have been shown to be negligible compared to photo-ionisation feedback <cit.>. We have also ignored the effect of the UV radiation force (or UV radiation pressure) on the gas dynamics. It has been shown that momentum transfer from photo-absorptions is only relevant for ultra compact HII regions, with densities larger than 10^-15 g/cc and sizes smaller than 10^-3 pc, completely unresolved in our simulations <cit.>. More relevant would be the inclusion of lower energy photons, in the optical and infrared range. These propagate from accreting stars through dust grains, and are scattered into new infrared photons. Inside the HII regions we can probably ignore these effects as dust is quickly sublimated at 10^4 K, however, infrared and optical radiation can play a role before massive stars form. <cit.> have shown that infrared radiation has very little impact on the gas removal and on the cloud destruction for realistic values of the dust opacity. Infrared radiation is likely to play a more important role on the fragmentation of molecular cores, but at scales we also do not resolve in our simulations. In summary, we are able to simulate the collapse of a molecular cloud and the emergence of a star cluster, whose properties are tightly connected to the gas dispersal process. Comparing our results to two observed, very young and still active star cluster, NGC 3603 and Arches, we conclude that an initially sub-virial molecular cloud with a star formation efficiency lower than 30% can reproduce observations fairly well. Our analysis provides useful insights also for simulations on galactic scales. Star clusters are indeed the building blocks of galaxy formation and evolution. Understanding in details their properties, such as mass segregation, mass and multiplicity functions and escaping stars statistics, just after they emerged from their parent cloud, is of primary importance for their longer term dynamical evolution, but also for the evolution of their host galaxies. § ACKNOWLEDGEMENTS We thank the anonymous referee for their thoughtful comments and helpful suggestions. This work is supported by the STARFORM Sinergia Project funded by the Swiss National Science Foundation. We would like to thank Sam Geen, Patrick Hennebelle and Michela Mapelli for useful discussions. mnras

http://arxiv.org/abs/1701.07998v1

20170127101810

Similarity and diversity of black holes - view from the Very High Energies

astro-ph.HE

Quasi-homography Warps in Image Stitching Nan Li, N. Li is with the Center for Applied Mathematics, Tianjin University, Tianjin 300072, China. E-mail: nan@tju.edu.cn.Yifang Xu^*, and Chao Wang Y. Xu is with the Center for Combinatorics, Nankai University, Tianjin 300071, China. Email: xyf@mail.nankai.edu.cn. C. Wang is with the Department of Software, Nankai University, Tianjin 300071, China. Email: wangchao@nankai.edu.cn. January 27, 2017 ==================================================================================================================================================================================================================================================================================================================================================================================================== Active galactic nuclei, hosting supermassive black holes and launching relativistic jets, are the most numerous objects on the gamma-ray sky. At the other end of the mass scale, phenomena related to stellar mass black holes, in particular gamma-ray bursts and microquasars, are also seen on the gamma-ray sky. While all of them are thought to launch relativistic jets, the diversity even within each of these classes is enormous. In this review, I will discuss recent very high energy gamma-ray results that underline both the similarity of the black hole systems, as well as their diversity. § INTRODUCTION The known black hole systems cover both stellar mass and supermassive black holes. In Active Galactic Nuclei, the activity is driven by the accretion of matter to supermassive black hole in the centre of the galaxy. In inactive galaxies the black hole may occassionally shine up due to tidal disruption events, which occur when a star gets too close to the supermassive black hole and is bulled apart by the black hole's tidal forces. In extreme and rare cases (only two have been observed <cit.>) this launches a collimated jet of particles. On stellar black hole mass scales, the microquasar phenomenon consists of a black hole feeding from companion star and launching a jet. Finally, we assume that long gamma-ray bursts are death crowls of massive stars, creating a black hole and collimated jets of particles. The common astrophysical ingredients of these systems, the spinning black hole, the accretion disk, and the collimated jets of particles have led several scientists to look for similarities among the different systems. What properties simply scale with the black hole mass and could this be signature of something fundamental? For example, in the early work by <cit.> a fundamental plane of black hole activity was established, when the authors found that active galaxies and galactic black holes lie on a plane in three dimensional space (radio luminosity, X-ray luminosity and mass of the black hole). More recently, <cit.> found that jets created by black holes maintain the same coupling between the total power carried by the jet and power radiated away. In our work, we have investigated if the jets of blazars and microquasars are similar in terms of outbursting mechanism and jet parameters that can be derived from the decomposition of the radio to optical light curves (<cit.>). We found that indeed the outburst of the microquasars and quasars are well described by shock-in-jet model, and that the jet parameters derived were rather similar for both types of systems (<cit.>). In recent years, astroparticle physics has opened new observational window to black hole systems and during our symposium we heard several interesting presentations coming from this community. Very High Energy (VHE, E>100 GeV) γ-ray experiments have revealed >170 astrophysical sources, and black hole systems are well represented among these sources. IceCube has started the era of neutrino astronomy with the discovery of neutrinos of astrophysical origin (<cit.>) and while their origin is still unknown, black hole systems are certainly among the candidates. Finally, the LIGO experiment measured gravitational waves from merging black holes (<cit.>). In this paper, I concentrate on the new observations of black hole system in the very high energies, covering the observations of active galactic nuclei, tidal disruption events, microquasars and gamma-ray bursts. § SUPERMASSIVE BLACK HOLES AT VERY HIGH ENERGIES §.§ Active Galactic Nuclei Active galactic nuclei (AGN) are the most numerous sources on the extragalactic VHE γ-ray sky. However, there is quite some diversity among this population. The most numerous ones are blazars, a type of active galactic nuclei, where the relativistic jet points very close to our line of sight. The blazars divide into several subcategories, all of which are seen at the VHE γ-ray sky. Furthermore, also several radio galaxies are detected. Blazar spectral energy distribution (SED) shows two bumps, the low energy bump extends from radio to ultraviolet–X-rays while the high energy bump extends from X-rays to VHE γ-rays. The low energy emission is synchrotron emission by the electrons spiralling in the magnetic field of the jet, while the high energy emission is in most cases inverse Compton emission. The location of the synchrotron peak is used to divide the blazars in sub-categories. The sources having the peak at UV-X-ray energies are the most numerous sources in the extragalactic VHE γ-ray sky, but nowdays also low synchrotron peaking objects have been detected. Blazars are variable in all wavelengths from radio to VHE γ-rays in timescales ranging from minutes to years. While short timescale variability is expected, as the jet is pointing so close to our line of sight, it becomes challenging when the timescales are as short as minutes. However, current instruments have observed minute-scale variability in VHE γ-rays from several sources, from all blazar classes and also from a radio galaxy (<cit.>). The minute-scale variability is challenging for models, as a huge energy has to be radiated within an extremely compact region. The time scale is actually shorter than the scale expected from the central black hole's horizon (which for 10^9M_sun black hole is an order of one hour). Furthermore, it is a challenge for particle acceleration and emission models. Many solutions to this dilemma has been suggested, such as strong recollimation of the jet or very compact region embedded within large scale jets (spine-sheath, jets-in-jet, ring-of-fire) (e.g. <cit.>). In many cases the fast VHE γ-ray variability is detected during periods when the source has been showing enhanced flux levels in all wavebands already prior to the detection of the fast variability. It is therefore evident that these events somehow connect to the overall activity within the relativistic jet. In the particular case of flat spectrum radio quasars, it is also evident that the VHE γ-ray emission cannot originate very close to the central black hole, as it is surrounded by broad line emission clouds (at the distance of ∼1 parsec) that are very efficient in absorbing VHE γ-rays via pair production (e.g. <cit.> and references therein). One particularly interesting case of fast variability is radio galaxy IC 310 in Perseus cluster. MAGIC Telescopes detected a huge flare in the VHE γ-ray band in November 2013. The flux reached several Crab Units, and the light curve revealed variability with doubling timescales faster than 4.8 minutes (<cit.>). As the viewing angle of the jet is rather well constrained to 10-20 degrees, the general solution of introducing Γ>50 to explain the fast variability does not work. In general it was concluded that models, where such a bright flare with such fast variability would be produced within the jet, were not viable. Therefore it was suggested that the flare actually originated in the magnetosphere of the black hole and it was dubbed a black hole lightning. The magnetospeheric model for IC 310 was discussed in detail in <cit.>, who concluded that it is feasible to produce the observed flare if the black hole at the time of the flare was accreting at very low rate. The feasibility of this model was discussed also during the symposium (see Barkov, this volume). Beyond the fast variability of VHE γ-rays, also the slower timescale variability of the VHE emission, the shape of the VHE γ-ray spectrum and in more general terms the population of active galaxies that we see at the highest energies tell us important stories about supermassive black holes. Looking at the variability of the light curves in different bands as well as the snap-shot spectral energy distributions have shown that the emission takes place in multiple emission regions within the relativistic jets (e.g. <cit.>). It has also been shown that, at least occasionally, the main energy dissipation region must move further out in the jet, at least to distances beyond the broad line region clouds (<cit.>). §.§ Tidal Disruption Events In addition to AGN, the VHE γ-ray telescopes have been pointed to a normally inactive supermassive black hole, to the famous tidal disruption event Sw 1644+57. <cit.> and <cit.> argued that this event arose from the activation of a beamed jet that was hypothesized to be the result of a tidal disruption of a star by a ∼10^6-10^7M_⊙ black hole. The VERITAS Telescopes started observing Sw J1644+57 approximately 22.5 hours after the first BAT trigger and followed it for 18 days (<cit.>). The MAGIC telescopes observed the Sw J1644+57 during the flaring phase, starting observations nearly 2.5 days after the trigger time (<cit.>). Neither of the telescopes found evidence for emission above the energy threshold of 100 GeV and the upper limits were in agreement both with the synchrotron and inverse Compton scattering scenarios for the X-ray emission (<cit.>). However, in principle the VHE γ-ray observations can be very constraining for the conditions of the jet in such tidal disruption events, in particular for the synchrotron origin of the X-ray emission and for the lorentz factor of the newly formed jet. In addition to Sw 1644+57, only one other possible case of jetted tidal disruption event is known (<cit.>), but to my knowledge it was not followed by the VHE γ-ray telescopes. §.§ Supermassive black hole in the centre of our Galaxy Galactic center is a strong and well established source of VHE γ-rays (<cit.>). It was already early suggested that the point source in the galactic centre would correspond to a central black hole (<cit.>), but as there are also other candidates, such as supernova remnant Sgr A East, the question of direct emission of the black hole in the center of our galaxy is not yet resolved. Very recently H.E.S.S. Collaboration showed, using 10 years of VHE γ-ray observations of the galactic centre, that the black hole in the centre of our galaxy is the first established PeVatron, i.e particle accelerator that can accelerate particles up to PeV energies (<cit.>). § STELLAR MASS BLACK HOLES AT VERY HIGH ENERGIES §.§ Microquasars X-ray binaries are binary systems with a compact object (a black hole or a neutron star) feeding from a companion. In microquasars this feeding launches a relativistic jet and as an analogy to more massive quasars, these jets could accelarate particles to energies high enough to emit VHE γ-rays. There were two detections of VHE γ-ray emission from binary systems that were first considered as possible microquasars, LS5039 and LSI +61 303 (<cit.>), but later observations have supported binary pulsar model for these sources (e.g. <cit.>). Up to date, the hint of VHE γ-ray emission from a microquasar Cyg X-1 (<cit.>) is the only indication of VHE emission, while other observations have resulted only in upper limits (<cit.>). The reason can be that the conditions are not favorable to emission of VHE γ-rays, given that it always requires acceleration of particles to extreme energies and typically also presence of photons to be up-scattered. Other possible explanation is an observational bias, as it is evident that the high energy emission from the microquasar jets is a transient phenomena. Cyg X-1 and Cyg X-3 are both detected in the lower gamma-ray energies by the Fermi satellite and the origin of this emission seems to be the jet (<cit.>). Therefore, it is to be expected that during flares the emission would extend also to energies >100 GeV, unless the gamma-rays are always emitted very close to jet base, where strong photon field absorbing VHE γ-rays is present. If the microqusars were to be analogies to quasars, one would expect that occassionally the energy dissipation region would move further out and VHE γ-rays could escape. To catch such a epoch, which is assumably an order of an hour in duration or shorter, with ground-based telescopes (limited duty cycle and limited field of view) is of course challenging. §.§ Gamma-ray bursts Gamma-ray bursts (GRBs) launch short lived, but extremely fast and luminous jets. As the duration of the prompt emission from GRBs is typically an order of a minute or less, fast pointing to the location of the GRBs is in a key role. Up to now there is no detection of VHE γ-ray emission from GRBs by the Imaging Air Cherenkov Telescopes (<cit.>), but the detection of high energy photons by the Fermi-LAT from some GRBs (e.g. <cit.>) certainly gives hope also to ground-based experiments. VHE γ-ray observations are a powerful tool for emission processes and physical conditions in GRBs and could also give important insight on the similarity of the emission processes in GRB and blazar jets. § SUMMARY Phenomena related to black holes at VHE γ-rays are mostly transient or extremely variable and therefore a huge observational challenge. As this window to black hole systems is still rather new, there are many open questions and also significant discovery potential in this energy range. The most prominent black holes at VHE γ-rays are the active galactic nuclei, where we have already started to see diversity as the number of known sources is steadily increasing (currently >70). For the rest; tidal disruption events, microquasars and gamma-ray bursts, we will have to wait for further observations at VHE γ-rays, with the current generation of telescopes and with the future Cherenkov Telescope Array, to be able to say something about the similarity of the all black hole systems at very high energies. [Aartsen 2013]IceCube M. G. Aartsen, R. Abbasi, Y. Abdou, (IceCube Coll.) 2013, Science, 342, 6161 [Abbott 2016]ligo B. P. Abbott, R. Abbott, T.D. Abbott (LIGO Coll.) 2016, PhysRevLett, 116, id.061102 [Abdo 2009]fermi_cygx3 A. A. Abdo, M. Ackermann, M. Ajello (Fermi-LAT Coll.) 2009, Science, 326, 1512 [Abdalla 2016]hess_ul H. Abdalla, A. Abramowski, F. Aharonian (H.E.S.S. Coll.) 2016, A&A, accepted, arxiv:1607.04613 [Abramowski 2016]hess_pev A. Abramowski, F. Aharonian, F. A. Benkhali (H.E.S.S. Coll.) 2016, Nature, 531, 476 [Acciari 2011]acciari11 V. A. Acciari, E. Aliu, T. Arlen (VERITAS Coll.) 2011, ApJ, 743, 62 [Ackermann 2014]lat_grb M. Ackermann, M. Ajello, K. Asano (Fermi-LAT Coll.) 2014, Science, 343, 42 [Aharonian 2004]hess_gc F. Aharonian, A. G. Akhperjanian, K.-M. Aye (H.E.S.S. Coll.) 2004, A&A, 425, 13 [Aharonian 2005]ls5039 F. Aharonian, A. G. Akhperjanian, K.-M. Aye (H.E.S.S. Coll.) 2005, Science, 309, 746 [Aharonian 2007]2155 F. Aharonian, A. G. Akhperjanian, A. R. Bazer-Bachi (H.E.S.S. Coll.) 2007, ApJ, 664, 71 [Aharonian 2009]aharonian09 F. Aharonian, A. G. Akhperjanian, U. Barres (H.E.S.S. Coll.) 2009, ApJ, 690, 1068 [Ahnen 2015]pks1441 M. Ahnen, S. Ansoldi, L. A. Antonelli (MAGIC Coll.) 2015, ApJ, 815, 23 [Albert 2006a]magic_gc J. Albert, E. Aliu, H. Anderhub (MAGIC Coll.) 2006a, ApJ, 638, 101 [Albert 2006b]lsi J. Albert, E. Aliu, H. Anderhub (MAGIC Coll.) 2006b, Science, 312, 1771 [Albert 2006c]albert06 J. Albert, E. Aliu, H. Anderhub (MAGIC Coll.) 2006c, ApJ, 641, 9 [Albert 2007a]mrk501 J. Albert, E. Aliu, H. Anderhub (MAGIC Coll.) 2007a, ApJ, 669, 862 [Albert 2007b]cygx1 J. Albert, E. Aliu, H. Anderhub (MAGIC Coll.) 2007b, ApJ, 665, 51 [Albert 2007c]albert07 J. Albert, E. Aliu, H. Anderhub (MAGIC Coll.) 2007c, ApJ, 667, 358 [Albert 2008]magic_lsi08 J. Albert, E. Aliu, H. Anderhub (MAGIC Coll.) 2008, ApJ, 684, 1351 [Aleksic 2010a]magic_cygx3 J. Aleksic, L. A. Antonelli, P. Antoranz (MAGIC Coll.) 2010a, ApJ, 721, 843 [Aleksic 2010b]aleksic10 J. Aleksic, L. A. Antonelli, P. Antoranz (MAGIC Coll.) 2010b, A&A, 517, 5 [Aleksic 2011]pks1222 J. Aleksic, L. A. Antonelli, P. Antoranz (MAGIC Coll.) 2011, ApJ, 730, 8 [Aleksic 2013]tde J. Aleksic, L. A. Antonelli, P. Antoranz (MAGIC Coll.) 2013, A&A, 552, 112 [Aleksic 2014a]ic310 J. Aleksic, S. Ansoldi, L. A. Antonelli (MAGIC Coll.) 2014, Science, 346, 1080 [Aleksic 2014b]1424 J. Aleksic, S. Ansoldi, L. A. Antonelli (MAGIC Coll.) 2014, A&A, 567, 135 [Aleksic 2014c]pks1510 J. Aleksic, S. Ansoldi, L. A. Antonelli (MAGIC Coll.) 2014, A&A, 569, 46 [Aleksic 2014d]aleksic14 J. Aleksic, S. Ansoldi, L. A. Antonelli (MAGIC Coll.) 2014d, MNRAS, 437, 3103 [Aleksic 2015a]mrk421 J. Aleksic, S. Ansoldi, L. A. Antonelli (MAGIC Coll.) 2015, A&A, 578, 22 [Aliu 2011]veritas_tde E. Aliu, T. Arlen, T. Aune (VERITAS Coll.) 2011, ApJ, 738, 30 [Arlen 2013]bllac T. Arlen, T. Aune, M. Beilicke (VERITAS Coll.) 2013, ApJ, 762, 92 [Archer 2014]veritas_gc A. Archer, A. Barnacka, M. Beilicke (VERITAS Coll.) 2014, ApJ, 790, 149 [Bloom (2011)]bloom J. Bloom, D. Giannios, B. Metzger 2011, Science, 333, 203 [Burrows (2011)]burrows D. N. Burrows, J. A. Kennea, G. Ghisellini 2011, [Böttcher & Els 2016]bottcher16 M. Böttcher & P. Els 2016, ApJ, 821, 102 [Cenko 2012]cenko S. B. Cenko, H. A. Krimm, A. Horesh 2012, ApJ, 753, 77 [Ghisellini 2005]ghisellini05 G. Ghisellini, F. Tavecchio, M. Chiaberge 2005, A&A, 432, 401 [Ghisellini 2013]ghisellini13 G. Ghisellini, F. Tavecchio, L. Foschini 2013, MNRAS, 432, 66 [Giannios 2009]giannios D. Giannios, D. Uzdensky, M. Begelman 2009, MNRAS, 395, 29 [Hirotani & Pu (2016)]hirotani K. Hirotani & H. Pu 2016, ApJ, 818, 50 [Komossa 2015]komossa S. Komossa 2015, Journal of High Energy Astrophysics, 7, 148 [Lindfors 2016]Lindfors E. Lindfors, T. Hovatta, K. Nilsson , 2016 A&A, 593, 98 [MacDonald 2015]macdonald N. MacDonald, A. Marscher, S. Jorstad , 2015 ApJ, 804, 111 [Merloni (2003)]merloni A. Merloni, S. Heinz, T. di Matteo, 2003 MNRAS, 345, 1057 [Nemmen (2012)]nemmen R. S., Nemmen, M. Georganopoulos, S. Guiriec 2012, Science, 338, 1445 [Tam 2006]tam06 P. H. Tamm, S. J. Wagner, G. Pühlhofer (on behalf of H.E.S.S. Coll.) 2006, Il Nuovo Cimento B, 121, 1595 [Türler 1999]turler99 M. Türler, T. J.-L. Courvoisier & S. Paltani 1999, A&A, 349, 45 [Türler & Lindfors 2007]turler07 M. Türler & E. Lindfors 2007, Proceedings of IAU Symposium #238, Cambridge, UK: Cambridge University Press, 238, 305 [Zanin 2016]zanin R. Zanin, A. Fernandez-Barral, E. de Ona-Wilhelmi 2016, A&A, accepted, arxiv:1605.05914 [Zdziarski 2016]zdziarski A. A. Zdziarski, D. Malyshev, M. Chernyakova 2016, MNRAS, submitted, arxiv:1607.05059

http://arxiv.org/abs/1701.07793v1

20170126175427

Wurtzite spin lasers

cond-mat.mes-hall

São Carlos Institute of Physics, University of São Paulo, 13566-590 São Carlos, São Paulo, Brazil Department of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA Department of Physics, National Taiwan University, Taipei 106, Taiwan São Carlos Institute of Physics, University of São Paulo, 13566-590 São Carlos, São Paulo, Brazil Department of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA zigor@buffalo.edu Department of Physics, University at Buffalo, State University of New York, Buffalo, New York 14260, USA Semiconductor lasers are strongly altered by adding spin-polarized carriers. Such spin lasers could overcome many limitations of their conventional (spin-unpolarized) counterparts. While the vast majority of experiments in spin lasers employed zinc-blende semiconductors, the room temperature electrical manipulation was first emonstrated in wurtzite GaN-based lasers. However, the underlying theoretical description of wurtzite spin lasers is still missing. To address this situation, focusing on (In,Ga)N-based wurtzite quantum wells, we develop a theoretical framework in which the calculated microscopic spin-dependent gain is combined with a simple rate equation model. A small spin-orbit coupling in these wurtzites supports simultaneous spin polarizations of electrons and holes, providing unexplored opportunities to control spin lasers. For example, the gain asymmetry, as one of the key figures of merit related to spin amplification, can change the sign by simply increasing the carrier density. The lasing threshold reduction has a nonmonotonic depenedence on electron spin polarization, even for a nonvanishing hole spin polarization. Wurtzite spin lasers Igor Žutić ==================== § I. INTRODUCTION Introducing spin-polarized carriers in semiconductor lasers offer an alternative path to realize spintronic applications, beyond the usually employed magnetoresistive effects <cit.>. Through carrier recombination, the angular momentum of the spin-polarized carriers is transferred to photons, thus leading to the circularly polarized emitted light <cit.>. Such spin lasers provide opportunity to extend the functionality of spintronic devices, as well to exceedthe performance of conventional (spin-unpolarized) lasers, from reducing the lasing threshold to improving their dynamical performance and digital operation <cit.>. Almost all spin lasers have been based on zinc-blende (ZB) semiconductors, such as GaAs or InAs, in which spin-dependent optical transitions were extensively studied for over 45 years <cit.>. However, a lone exception of a spin laser with an gain (active) region made of a wurtzite (WZ) semiconductor has so far also been the only case of an electrically manipulated spin laser at room temperature <cit.>. Unlike many theoretical studies of ZB spin lasers <cit.>, a theoretical description for WZ spin lasers is still missing. Focusing on WZ GaN-based quantum wells (QWs) as the gain region, we develop the first microscopic description of WZ spin lasers. The significance of WZ materials for optical devices has been recognized by the 2014 Nobel prize in physics for an efficient blue light emitting diodes (LEDs). WZ-based optical devices using a direct band gap GaN and its In and Al alloys are ubiquitous in our daily lives, from efficient lightning to blue-ray disc readers. Due to their high electron saturation velocities and high breakdown voltages, GaN-based semiconductors are also promising for high-speed/high-power electronic devices <cit.>. However, for spin-dependent optical properties, WZ GaN does not appear encouraging, leading to only a negligibly small degree of a circular polarization of an emitted light which can be attributed to a rather weak spin-orbit coupling (SOC) <cit.>. Therefore, the realization of the first electrically manipulated spin laser at room temperature using GaN-based gain region came as a surprise. To better understand the differences between employing ZB and WZ semiconductors in optical devices, in Figs. <ref>(a) and (b) we show their bulk band structure and possible band edge optical transitions within the conventional 8×8 k·p Hamiltonians, using the typical notation: conduction band (CB), heavy holes (HH), light holes (LH) and spin-orbit split-off holes (SO) for ZB <cit.> and CB, HH, LH and crystal-field split-off hole (CH) for WZ <cit.>. Each of the marked dipole transitions has a different amplitude for specific spins that apply both to radiative recombinations and excitations. If we denote the photon density of positive (negative) helicity by S^+ (S^-), we can describe the relevant helicity in each of the transitions. For example, in the CB-HH transition spin up (down) leads to S^- (S^+), in CB-LH spin up (down) leads to aS^+ (aS^-), while in CB-SO for ZB, or CB-CH for WZ, spin up (down) leads to bS^+ (bS^-). For ZB the amplitude of helicity contributions are fixed: a=1/3, b=2/3. The electron spin polarization in terms of spin up (down) electron density n_+ (n_-), P_n=(n_+-n_-)/(n_+ + n_-), arising from optical spin injection (HH/LH-CB) yields P_n= (1-1/3)(1+1/3) = 50%, a well-known result at the band gap, neglecting electron spin relaxation <cit.>. In contrast, for WZ the corresponding amplitudes depend on the materials parameters related to the SOC <cit.>, a = E^2_+/(E^2_+ + 2Δ^2_3), b=2Δ^2_3 / (E^2_+ + 2Δ^2_3), where the energy E_+ is expressed as, E_+ = (Δ_1 - Δ_2)/2 + √((Δ_1 - Δ_2)^2/4 + 2Δ^2_3) , in terms of the crystal field splitting energy Δ_1 and SOC splitting energies Δ_2,3. With removed HH and LH degeneracy at the wavevector k=0 (Γ-point) in WZ semiconductors [see Fig. <ref>(b)], one would expect P_n → 100% optical spin injection at the band gap, overcoming the 50% limitation of ZB materials <cit.>. However, due to the relatively weak SOC in nitride-based materials <cit.>, the energy separation for the topmost valence bands is typically ∼10-20 meV, comparable to the energy of the broadening effects by impurities and room temperature, ultimately leading to inefficient optical spin injection <cit.>. In GaN-based spin LEDs only a small circular polarization of electroluminescence was detected at 200 K <cit.> as well as at 300 K in the applied magnetic field <cit.>. These limitations could be overcome in electrical spin injection or extraction, as shown (In,Ga)N/GaN-based nanodiscs and nanorods covered by Fe_3O_4 nanoparticles <cit.>. In this study, we investigate WZ spin lasers with In_0.1Ga_0.9N/GaN QWs as their gain region using microscopic k·p band structure calculations. While a weak SOC retains desirable spin-dependent properties of optical gain, it also necessitates simultaneous consideration of electron and hole spin polarizations, largely overlooked in the previous studies. By combining macroscopic rate equations with microscopic gain calculations based on a k·p method, we establish a versatile method to describe spin lasers which extends the strengths of the two complementary approaches. In Sec. II we describe the k·p method to evaluate the electronic structure of (In,Ga)N QW which is used in Sec. III to calculate microscopic spin-dependent optical gain. In Sec. IV we combine these microscopic gain calculations with simple rate equations, suitable to describe various dynamical phenomena in spin lasers. In Sec. V we discuss future opportunities to apply our theoretical framework to other systems. § II. QUANTUM WELL ELECTRONIC STRUCTURE An important consequence of the atomic arrangement of WZ materials along the [0001] direction is the presence of the polarization fields. A relative displacement between cations and anions in the unit cell leads to the spontaneous polarization along the growth direction in QWs. Under external applied strain this cation-anion displacement is modified and also yields piezoelectric polarization <cit.>. Such polarization fields are schematically shown in Fig. <ref>(c) for a WZ QW. The response of the quantum confined states to the static electric field is known as the quantum confined Stark effect and recognized as a very efficient mechanism to tune the optical transitions in semiconductor nanostructures <cit.>. These polarization fields modify both electronic levels as well as change the spatial electron-hole separation and thus the overlap integral between CB and VB wave functions. Within the k·p method combined with the envelope function approximation, and including the polarization effects, the total Hamiltonian of the QW system is, H_QW(z) = H_kp(z) + H_st(z) + H_O(z) + H_pol(z) , with the growth axis along the z direction (the c axis, or [0001] direction, of the WZ structure). The Hamiltonian H_kp(z) denotes the k·p term, H_st(z) describes the strain, H_O(z) includes the band-offset at the interface that generates the QW energy profile, and H_pol(z) includes the potential profile due to spontaneous and piezoelectric polarizations. In this study, we considered the 8×8 k·p Hamiltonian for WZ materials with explicit interaction between CB and VB which gives rise to the dipole coupling for optical transitions. The specific definitions of these Hamiltonians are described in Refs. <cit.>. In order to numerically solve the resulting system of coupled differential equations from Eqs. (<ref>), we apply the plane wave expansion discussed in Refs. <cit.>. For the gain region of the laser we consider a 3 nm thick single strained In_0.1Ga_0.9N QW, surrounded by 6 nm GaN barriers, the typical lengths and composition of (In,Ga)N-based vertical cavity surface emitting lasers (VCSELs) <cit.>. The bulk InN and GaN parameters are obtained from Ref. <cit.>, we use their linear interpolation for the alloy In_0.1Ga_0.9N and the bowing parameter for the band gap, E_g <cit.>. The interface band offsets are Δ E_C = 0.7Δ E_g and Δ E_V = 0.3Δ E_g <cit.>. We choose E_g at T = 300 K with Varshni parameters and the refractive indexes from Refs. <cit.>. To develop some intuition about the relevant SOC parameters in (In,Ga)N QWs, we recall that in GaAs, as the representative ZB semiconductor, at the Γ-point HH and LH are degenerate and separated by Δ_SO = 0.341 eV <cit.> from the SO band. It is helpful to think of ZB GaAs as a WZ structure without crystal-field splitting energy (Δ_1 = 0) and a much larger SOC that yields Δ_2 = Δ_3 = Δ_SO/3 ≈ 114 meV <cit.>. For the GaN barrier, Δ_1 = 10 meV and Δ_2 = Δ_3 = 5.7 meV, and for the QW material In_0.1Ga_0.9N, Δ_1 = 13 meV and Δ_2 = Δ_3 = 5.3 meV. In the bulk case, such values of Δ_1,2,3 provide an energy difference at the Γ-point in GaN of ∼5.1 meV for HH-LH and 21.9 meV for HH-CH. For In_0.1Ga_0.9N the energy differences are ∼6 meV and ∼22.9 meV for HH-LH and HH-CH, respectively. QW confinement and polarization fields can provide larger energy separations for the different wave functions (no nodes, one node, etc). However, the typical HH-LH QW states separation with same number of nodes remains similar to the bulk energy values. The resulting band structure of In_0.1Ga_0.9N/GaN QW, is presented in Fig. <ref>(a), showing the two confined conduction subbands, CB1 and CB2, and the top four confined valence subbands, HH1, LH1, HH2 and LH2, labeled according to the dominant component of the total envelope function <cit.>. Each subband is twofold degenerate in k=0 and for nonzero k values the effect of the asymmetric polarization field creates small spin splittings in the valence subbands <cit.>, lifting Kramers degeneracy <cit.>. Considering optical transitions at room temperature (k_BT∼25 meV), the spin splittings are negligible as if the bands were twofold degenerate. Furthermore, because of the energy separation of ∼150 (80) meV from CB1 (LH1) to CB2 (HH2) subbands, we can expect the emission range of the gain spectra to be ruled by CB1-HH1 (2.963 eV) and CB1-LH1 (2.973 eV) transitions. The corresponding density of states (DOS) shown in Fig. <ref>(b), confirms that spin-resolved DOS has equal contributions for spin up and spin down. § III. MICROSCOPIC SPIN-DEPENDENT GAIN Obtained electronic structure with the corresponding carrier populations provides the starting point to microscopically calculate the optical gain depicted in Fig. <ref>, the hallmark of lasers. The resulting gain coefficient (or gain spectrum) is the negative value of the absorption coefficient and is calculated as <cit.>, g^a_i(ω)=C_0c,v, k∑| p^a_cv k|^2 (f_c k-f_v k) δ[ħω_cv k-ħω], where the summation indices c and v label the conduction and valence subbands, respectively p^a_cv is the interband dipole transition amplitude for the polarization of light α, f_c(v) k is the Fermi-Dirac distribution for the electron occupancy in the conduction (valence) subbands, ħ is the Planck's constant, ω_cvk is the interband transition frequency, and δ is the Dirac delta-function, which is often replaced to include broadening effects for finite lifetimes <cit.>. In the constant C_0 = 4π^2 e^2/(ε_0 c_l n_r m_0^2ωΩ), ε_0 is the vacuum permittivity, c_l is the speed of light (to distinguish it from the CB index), n_r is the dominant real part of the refractive index of the material, e is the electron charge, m_0 is the free electron mass, and Ω is the QW volume. Similar to ZB GaAs-based spin lasers <cit.>, the dipole selection rules for the interband optical transitions are spin-conserving, i. e., the dipole matrix element does not change spin. Therefore, the gain coefficient for the light polarization α includes independent contributions of spin-up and spin-down carriers, g^a(ω) = g^a_+(ω) + g^a_-(ω), denoted by the subscripts + and -, respectively. To develop intuition and understand the role of SOC in the optical transitions, we first illustrate the gain spectra on the example of conventional lasers. This implies injecting vanishing electron and hole spin polarization, P_n=P_p=0, where the expression for P_p is analogous to Eq. (<ref>). In Fig. <ref>(a) we show such a gain spectra as function of photon energy for various carrier densities. For calculated gain spectra in (In,Ga)N QWs it is customary to include various broadening effects. In addition to the homogeneous broadening, frequently used in ZB QWs <cit.>, parametrized here by sech with 10 meV full width at half-maximum (FWHM), we also consider an inhomogeneous Gaussian broadening, attributed to compositional and potential fluctuations. Our choice of Gaussian broadening with 20 meV FWHM is consistent with a decreased broadening for smaller emission wavelengths in (In,Ga)N QW lasers and reported values relevant for wavelengths of ∼415 nm <cit.> which corresponds to the typical energy of the gain peak in our calculations. Because of the broadening effects, the individual CB1-HH1 and CB1-LH1 transitions that dominate the gain spectra cannot be distinguished [HH1 and LH1 are 10 meV apart, see Fig. <ref>(a)]. On the other hand, by analyzing the spin-resolved gain we can identify different contributions of CB1-HH1 and CB1-LH1 transitions. In Figs. <ref>(b) and <ref>(c) we show the gain spectra decomposed in spin up and spin down transitions at n = 6 × 10^12 cm^-2 for S^+ and S^- light polarization, respectively. For the total gain we have g^+=g^- which requires g^+_-=g^-_+ and g^+_+=g^-_- <cit.>, as could be seen in Figs. <ref>(b) and <ref>(c). Due to the small SOC energy in nitrides, the S^- (S^+) gain peak of spin up (down) CB1-HH1 transition is twice as large as the spin down (up) CB1-LH1 transition. For a larger SOC energy, this ratio would increase. For example, in ZB GaAs spin laser <cit.>, this ratio is ∼6 (for a SOC energy of Δ_2 = Δ_3 ≈ 114 meV, compared to Δ_2 = Δ_3 ∼ 10-20 meV in nitrides). We next turn to the gain properties in spin-lasers where injected carriers are spin-polarized. Guided by the typical spin dynamics for ZB semiconductors in which hole spin relaxes nearly instantaneously, previous studies have largely focussed on spin lasers with nonzero P_n, but vanishing P_p. However, since the degeneracy of HH and LH in bulk WZ semiconductors is lifted by the crystal field potential, the spin relaxation times of holes in GaN could be comparable to those of electrons <cit.>. This is in stark contrast to bulk GaAs where at 300 K the hole spin relaxation time is three to four orders of magnitude shorter than for electrons <cit.>. We therefore also consider the effect of nonzero P_p, excluded in the two prior microscopic studies of gain spectra in spin lasers <cit.>. The gain for WZ spin lasers is shown in Fig. <ref> as a function of photon energy and carrier density. These results confirm that the gain becomes helicity-dependent, g^+≠ g^-, as known from ZB spin lasers. However, the role of simultaneous presence of nonvanishing P_n and P_p requires further attention. With fixed P_n=0.2 we see that a change from P_p=0 to P_p=0.2 [panels (a) and (b)] enhances the difference between the gain contribution for S^- and S^+ , while a change from P_p=0 to P_p=-0.2 [panels (a) and (c)] reduces such a difference. Since equal but opposite electron spin polarizations [P_n=-P_p, Fig. <ref>(c)] describe the vanishing total spin in the gain region, it is helpful to note another realization of a vanishing total spin in Fig. <ref>(a). Nevertheless, the gain spectra in these two cases are slightly different which can be attributed to the different features of CB and VB including their curvature, number of confined bands, and DOS. Thus, the difference between the gain contribution for S^- and S^+ cannot be eliminated for P_n=-P_p. A complementary information about the calculated gain is given with its density dependence in Figs. <ref>(d)-(f). The results are shown for photon energies, corresponding to the CB1-HH1 and CB1-LH1 transitions [recall Fig. <ref>] which can be individually favored by the cavity design in a single-mode VCSEL <cit.>. Several trends can be inferred. For example, a nonlinear gain-dependence on density is different for the two photon energies. With an increased carrier density, CB1-LH1 transition provides larger gain values than as compared to CB1-HH1. While a reference curve (long dashed) for the gain of a conventional laser is lower than g^- for P_p=0 and P_p=0.2 [panels (d) and (e)], the situation is reversed above the gain threshold (green horizontal line) for P_p=-0.2 [panel (f)] where at larger density g^+ > g^- is possible. To better understand the helicity-dependent gain, it is useful to calculate the corresponding gain asymmetry, g_asy(ω) = g^-(ω)- g^+(ω). an important figure of merit in spin lasers. Considering that lasers have nonlinear light-injection characteristics, such gain asymmetry could enable robust spin filtering or spin amplification <cit.>. Close to the lasing threshold even a small carrier spin polarization in the gain region can lead to completely circularly polarized emitted light <cit.>. The results for the gain asymmetry, extracted from Fig. <ref>, are shown in Fig. <ref> as a function of the photon energy and the carrier density. While a large |g_asy| is desirable, it is crucial that it corresponds to the g>0 regime. For example, the largest |g_asy| in Figs. <ref>(a) and (c) is found for photon energies of 125 - 140 meV above the band gap. However, as seen in Figs. <ref>(a) and (c), this range corresponds to the absorption regime (g<0) and such g_asy does not influence the emitted light. As P_p and n vary, the largest useful |g_asy| is found slightly above the gap. As shown in Figs. <ref>(d)-(f), to enhance |g_asy| a lower density and CB1-LH1 are slightly better. An interesting deviation from these trends is seen in Fig. <ref> for P_n = -P_p = 0.2. Near the band gap, an increase in n leads to the sign change of g_asy and its maximum magnitude in the g<0 regime for the larges shown carrier density. This behavior points to yet unexplored opportunities to optimize the operation of spin lasers with a simultaneous spin polarization of electrons and holes. Our results show that despite the small SOC energy of WZ nitrides, considered detrimental for optical spin injection, the gain asymmetry remains robust. Another important figure of merit of spin lasers is their threshold reduction, the lasing operation could be attained at lower injected carrier density than in conventional lasers. We will analyze this behavior in the next section. § IV. RATE EQUATIONS Here we briefly review a complementary approach based on rate equations (REs) and discuss how its understanding can be enhanced from our microscopic gain calculations. REs have been successfully used to describe both conventional and spin lasers <cit.>. An advantage of this approach is its simplicity. REs can provide a direct relation between material characteristics and device parameters, as well as often allowing analytical solutions and an effective method to elucidate many trends in the operation of lasers <cit.>. With notation widely used for conventional lasers <cit.>, generalized to include spin- and helicity-resolved quantities, we can write REs as <cit.>, dn_±/dt = J^n_±-g_±(n_±,p_±,S)S^∓-(n_±-n_∓)/τ_sn-R_sp^± dp_±/dt = J^p_±-g_±(n_±,p_±,S)S^∓-(p_±-p_∓)/τ_sp-R_sp^± dS^±/dt = Γ g_∓(n_∓,p_∓,S)S^±-S^±/τ_ph+βΓ R_sp^∓. In the gain term, g_±(n_±,p_±,S)= g_0(n_±+p_±-n_tran)/(1+ϵ S), n_tran is the transparency density, and ϵ is the gain compression factor <cit.>, ensuring that the output light S does not increase indefinitely with injection J, g_0 is the gain parameter, and Γ is the optical confinement factor. The electron spin relaxation is given by (n_±-n_∓) /τ_sn, where τ_sn is the electron spin relaxation time (τ_sp for holes) <cit.>. The carrier recombination R_sp^± can have various dependences on carrier density <cit.> and be characterized by a carrier recombination time τ_r. β is the fraction of the spontaneous recombination producing light coupled to the resonant cavity, and τ_ph is the photon lifetime, to model optical losses <cit.>. While the k·p method does not include spin relaxation (τ_sn/τ_r, τ_sp/τ_r →∞), similar dynamical effects are easily included in REs. However, REs rely on various input parameters that can be obtained from experiments or microscopic calculations. A more complete description of spin lasers can be therefore developed by combining the k·p method and the macroscopic RE model. We illustrate this approach by focusing on the optical gain in WZ spin lasers. Specifically, the gain parameter and the transparency density in the gain term in REs, can be obtained by fitting, for each P_n and P_p, the carrier density dependence of the calculated microscopic gain presented in Figs. <ref>(d)-(f). Following the REs for spin lasers <cit.> we use a simple linear dependence of gain on the carrier density to provide a better comparison with the published work. This is illustrated in Fig. <ref> for calculated gain of a conventional WZ laser. In the linear fit, the slope of the gain at n_tran (where g=0) in REs is matched with the slope of the calculated gain. However, we note that the logarithmic gain model, often used in conventional QW lasers <cit.>, would be a better fit. Another difference between REs and the calculated gain is the helicity-dependent gain coefficient (recall Figs. <ref> and  <ref>) and we include that behavior by fitting the RE gain for each helicity separately. To follow the k·p method we choose τ_sn/τ_r, τ_sp/τ_r ≫ 1, rather than seeking the best possible fit between the two methods. Likewise, we choose ϵ=0 even though the gain compression could give a better agreement at larger n. The remaining RE parameters are assigned from the previous work <cit.>. Unlike a single lasing threshold, J_T, in conventional lasers, with spin-dependent gain, there are two lasing thresholds in spin lasers, J_T1≤ J_T2 which delimits three operating regimes <cit.> : (i) For J ≤ J_T1 a spin LED regime, (ii) For J_T1≤ J ≤ J_T2 there is a spin-filtering regime and the lasing with only one helicity. (iii) For J≥ J_T2, there is a lasing with both helicities. It is then convenient to define the threshold reduction, r=1-J_T_1/J_T, as an important figure of merit that influences both the steady-state and dynamical operation of spin lasers <cit.>. In Fig. <ref> we compare the threshold reduction as a function of electron spin polarization calculated using the microscopic method and REs, for several values of hole spin polarization. Despite noticeable differences between the two methods, they both show an overall nonmonotonic dependence of r on P_n, preserved for each P_p. It is instructive to note that previously studied REs with P_p=0 and fixed g_0 yield a monotonic increase in r with P_n, from r=0 at P_n=0 to r=1/2 at P_n=1 <cit.>. However, using REs with a linear fit of the gain for P_p=0 at each P_n shows in Fig. <ref> a much closer agreement to the microscopic gain results and, by constructions, the two methods coincide at P_n=P_p=0. Including the hole spin polarization, the disagreement between the two methods is more pronounced for P_p=-0.2, than for P_p=0.2. The corresponding RE results largely fail to capture the calculated threshold increase (r<0, reported also in ZB lasers <cit.>) and are not properly defined for P_n<0.2. To explore why the RE results for P_p=-0.2 are worse, it is useful to recall the dipole optical selection rules for transitions sketched in Fig. <ref>. In our notation that means that both spontaneous and stimulated recombination (optical gain) involve only electrons and holes of the same spin. For example, spontaneous radiative recombination has terms n_+p_+ or n_-p_- <cit.>. However, in Eqs. (<ref>) or (<ref>) the gain term does not accurately respect these selection rules. For a sufficiently large carrier density the lasing would occur, even if the carrier spins are not compatible with the selection rules. When P_n and P_p have the opposite sign there are more carriers having a “wrong spin” to satisfy the selection rules leading to a worse agreement with the microscopic results. Such a disagreement would be less pronounced for shorter spin relaxation times, allowing “wrong spin carriers” to recombine while respecting the selection rules. It is also possible to address the missing RE data for P_n<0.2. In the steady-state, Eq. (<ref>) implies, n_∓+p_∓= n_tran +1/ (Γ g_0 τ_ph) - β R^∓_sp/(g_0 S^±). In the operating regime (iii): J>J_T2, both helicities lase and S^± are large, which yields, n_∓+p_∓≈ n_tran + 1/(Γ g_0 τ_ph), Therefore, n_++p_+≈ n_-+p_-. Together with the charge neutrality, we have p_+=n_- and p_-=n_+, which means that P_n=-P_p is guaranteed in the regime (iii). This is relevant for the case P_p=-0.2 and P_n<0.2, because emitted S^- is associated with minority instead of majority spin, such that the lasing of S^- is in the regime (iii). The required P_n=-P_p in the third regime thus reduces the freedom of a realizable spin polarization in REs. § V. CONCLUSIONS Our framework of combining microscopic gain calculations with simple rate equations provides predictive and computationally inexpensive materials-specific approach to explore spin lasers. The choice of wurtzite lasers was guided by the first realization of an electrically manipulated spin laser at room temperature <cit.> and the absence of any prior theoretical work. In contrast to zinc-blende GaAs, in wurtzite GaN there is a much smaller spin-orbit coupling, usually considered as a detrimental situation for optical spin injection. We have shown that even such a small spin-orbit coupling in wurtzites yields robust signatures of a spin-dependent gain, including the gain asymmetry, desirable for spin lasers. With the presence of nonvanishing electron and hole spin polarization, largely overlooked in the previous studies, the gain asymmetry can even change its sign by simply increasing the carrier density. The lasing threshold reduction has a nonmonotonic depenedence on electron spin polarization, even for a nonvanishing hole spin polarization. While a weak spin-orbit coupling is expected to lead to an enhanced spin relaxation times, this is not the case for GaN which has a defect dominated spin-relaxation and electron spin relaxation times about an order of magnitude shorter than in GaAs <cit.>. Although materials advances could enable longer spin relaxation times in GaN, the current values are already suitable for digital and high-frequency operation of spin lasers <cit.>. The present framework can be adapted for other materials and laser geometries. With an increasing interest in non-nitride III-V wurtzite materials with large spin-orbit coupling <cit.>, we expect they could facilitate optically-injected spin lasers at room temperature. While we have focused on spin VCSELs, our approach would also be useful for vertical external cavity surface emitting lasers (VECSELs)<cit.>. They enable depositing a thin-film ferromagnet just 100-200 nm away from the gain region for spin injection at the room temperature. Various spin and phonon lasers can also be implemented using intraband transitions within the conduction band <cit.> or in metallic systems <cit.>. It would be interesting to develop a suitable description for them by combining microscopic gain calculations and simple rate equations. An important materials challenge for the advances in wurtzite spin lasers would be to establish magnetic regions and their detailed theoretical description for robust electrical spin injection into the gain region. In zinc-blende semiconductors a number of such materials are already available <cit.>. In addition to demonstrating that Fe_3O_4 nanomagnets are suitable for wurtzite spin lasers <cit.>, many other opportunities could be explored. For example, ferromagnetic semiconductors provide electrically- and optically-controlled magnetic properties <cit.>, while supporting ultrafast optical processes <cit.>. With a thin barrier region, even simple ferromagnets may enable tunable carrier spin polarization relying on gate-controlled magnetic proximity effects <cit.>. We thank N. C. Gerhardt for valuable discussions of the optical gain. This work was supported by NSF ECCS-1508873, U.S. ONR N000141310754, FAPESP Grants No. 2012/05618-0, 2011/19333-4, CNPq Grants No. 304289/2015-9, 246549/2012-2, CAPES-CsF 88887.125287/2015-00, and MOST-105-2811-M-147, Taiwan, Republic of China. 99 Hallstein1997:PRB S. Hallstein, J. D. Berger, M. Hilpert, H. C. Schneider, W. W. Rühle, F. Jahnke, S. W. Koch, H. M. Gibbs, G. Khitrova, and M. Oestreich, Phys. Rev. B 56, R7076 (1997). Ando1998:APL H. Ando, T. Sogawa, and H. Gotoh, Appl. Phys. Lett. 73, 566 (1998). Rudolph2003:APL J. Rudolph, D. Hägele, H. M. Gibbs, G. Khitrova, and M. Oestreich, Appl. Phys. Lett. 82, 4516 (2003); J. Rudolph, S. Döhrmann, D. Hägele, M. Oestreich, and W. Stolz, Appl. Phys. Lett. 87, 241117 (2005). Holub2007:PRL M. Holub, J. Shin, D. Saha, and P. Bhattacharya, Phys. Rev. Lett. 98, 146603 (2007). Hovel2008:APL S. Hövel, A. Bischoff, and N. C. Gerhardt, M. R. Hofmann, and T. Ackemann, A. Kroner, and R. Michalzik, Appl. Phys. Lett. 92, 041118 (2008). Basu2008:APL D. Basu, D. Saha, C. C.Wu, M. Holub, Z. Mi, and P. Bhattacharya, Appl. Phys. Lett. 92, 091119 (2008). Basu2009:PRL D. Basu, D. Saha, and P. Bhattacharya, Phys. Rev. Lett. 102, 093904 (2009). Saha2010:PRB D. Saha, D. Basu, and P. Bhattacharya, Phys. Rev. B 82, 205309 (2010). Fujino2009:APL H. Fujino, S. Koh, S. Iba, T. Fujimoto, and H. Kawaguchi, Appl. Phys. Lett. 94, 131108 (2009). Jahme2010:APL M. Li, H. Jähme, H. Soldat, N. C. Gerhardt, M. R. Hofmann, and T. Ackemann, A. Kroner, and R. Michalzik Appl. Phys. Lett. 97, 191114 (2010). Gerhardt2011:APL N. C. Gerhardt, M. Y. Li, H. Jähme, H. Höpfner, T. Ackemann, and M. R. Hofmann, Appl. Phys. Lett. 99, 151107 (2011). Iba2011:APL S. Iba, S. Koh, K. Ikeda, and H. Kawaguchi, Appl. Phys. Lett. 98, 081113 (2011). Frougier2013:APL J. Frougier, G. Baili, M. Alouini, I. Sagnes, H. Jaffrès, A. Garnache, C. Deranlot, D. Dolfi, and J.-M. George, Appl. Phys. Lett. 103, 252402 (2013). Frougier2015:OE J. Frougier, G. Baili, I. Sagnes, D. Dolfi, J.-M. George, and M. Alouini, Opt. Expr. 23, 9573 (2015). Alharthi2015:APL S. S. Alharthi, A. Hurtado, V.-M. Korpijarvi, M. Guina, I. D. Henning, and M. J. Adams, Appl. Phys. Lett. 106, 021117 (2015). Alharthi2015:APLb S. S. Alharthi, J. Orchard, E. Clarke, I. D. Henning, and M. J. Adams, Appl. Phys. Lett. 107, 151109 (2015). Hsu2015:PRB F.-k. Hsu, W. Xie, Y.-S. Lee, S.-D. Lin, and C.-W. Lai, Phys. Rev. B 91, 195312 (2015). Sinova2012:NM J. Sinova and I. Žutić, Nature Mater. 11, 368 (2012). Lee2010:APL J. Lee, W. Falls, R. Oszwałdowski, and I. Žutić, Appl. Phys. Lett. 97, 041116 (2010). Boeris2012:APL G. Boéris, J. Lee, K. Výborný, and I. Žutić, Appl. Phys. Lett. 100, 121111 (2012). Lee2012:PRB J. Lee, R. Oszwałdowski, C. Gøthgen, and I. Žutić, Phys. Rev. B 85, 045314 (2012). Lee2014:APL J. Lee, S. Bearden, E. Wasner, and I. Žutić, Appl. Phys. Lett. 105, 042411 (2014). Wasner2015:APL E. Wasner, S. Bearden, J. Lee, and I. Žutić, Appl. Phys. Lett. 107, 082406 (2015). FariaJunior2015:PRB P. E. Faria Junior, G. Xu, J. Lee, N. C. Gerhardt, G. M. Sipahi, and I. Žutić, Phys. Rev. B 92, 075311 (2015). Hopfner2014:APL H. Höpfner, M. Lindemann, N. C. Gerhardt, and M. R. Hofmann, Appl. Phys. Lett. 104, 022409 (2014). Lindemann2016:APL M. Lindemann, T. Pusch, R. Michalzik, N. C Gerhardt, and M. R. Hofmann, Appl. Phys. Lett. 108, 042404 (2016). Pusch2015:EL T. Pusch, M. Lindemann, N. C. Gerhardt, M. R. Hofmann, and R. Michalzik, Electron. Lett. 51, 1600 (2015). Meier:1984 Optical Orientation, edited by F. Meier and B. P. Zakharchenya (North-Holland, New York, 1984). Chen2014:NN J.-Y. Chen, T.-M. Wong, C.-W. Chang, C.-Y. Dong, and Y.-F Chen, Nat. Nanotech. 9, 845 (2014). Zutic2014:NN I. Žutić and P. E. Faria Junior, Nat. Nanotech. 9, 750 (2014). Gothgen2008:APL C. Gøthgen, R. Oszwałdowski, A. Petrou, and I. Žutić, Appl. Phys. Lett. 93, 042513 (2008). Holub2011:PRB M. Holub and B. T. Jonker, Phys. Rev. B 83, 125309 (2011). Akasaki2015:RMP I. Akasaki, Rev. Mod. Phys. 87, 1119 (2015). Amano2015:RMP H. Amano, Rev. Mod. Phys. 87, 1133 (2015). Nakamura2015:RMP S. Nakamura, Rev. Mod. Phys. 87, 1139 (2015). Chen2005:APL W. M. Chen, I. A. Buyanova, K. Nishibayashi, Kayanuma, K. Seo, A. Murayama, Y. Oka, G. Thaler, R. Frazier, C. R. Abernathy, F. Ren, S. J. Pearton, C.-C. Pan, G.-T. Chen and J.-I Chyi, Appl. Phys. Lett. 87, 192107 (2005). Winkler:2003 R. Winkler, Spin-orbit Coupling Effects in Two-Dimensional Electron and Hole Systems, (Springer, New York, 2003). Lee2014:PRB J. Lee, K. Výborný, J. E. Han, and I. Žutić, Phys. Rev. B 89, 045315 (2014). Chuang1996:PRB S. L. Chuang and C. S. Chang, Phys. Rev. B 54, 2491 (1996). Pierce1975:PLA D. Pierce, F. Meier and P. Zürcher, Phys. Lett. A 51, 465 (1975). Zutic2004:RMP I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). note:GaN In stark contrast to very dilute ZB nitrides with strong SOC, such as Ga_0.67In_0.33N_0.016As_0.984 QW-based spin lasers, K. Schires, R. A. Seyab, A. Hurtado, V.-M. Korpijärvi, M. Guina, I. D. Henning, and M. J. Adams, Opt. Expr. 20, 3550 (2012). Banerjee2013:APL D. Banerjee, R. Adari, S. Sankaranarayan, A. Kumar, S. Ganguly, R. W. Aldhaheri, M. A. Hussain, A. S. Balamesh, and D. Saha, Appl. Phys. Lett. 103, 242408 (2013). Chen2010:APL L.-C. Chen, C.-H. Tien, and C.-S. Mu, Opt. Expr. 18, 2302 (2010). Chen2014:NL J. Y. Chen, C. Y. Ho, M. L. Lu, L. J. Chu, K. C. Chen, S. W. Chu, W. Chen, C. Y. Mou, and Y.-F. Chen, Nano Lett. 14, 3130 (2014). note:pols See, for example, Chapter 3 of Nitride Semiconductor Devices: Principles and Simulation, edited by J. Piprek (Wiley, Weinheim, 2007). Chuang:2009 S. L. Chuang, Physics of Optoelectronic Devices, 2^nd Edition (Wiley, New York, 2009). FariaJunior2012:JAP P. E. Faria Junior and G. M. Sipahi, J. Appl. Phys. 112, 103716 (2012). Miao2012:PRL M. S. Miao, Q. Yan, C. G. Van de Walle, W. K. Lou, L. L. Li, and K. Chang, Phys. Rev. Lett. 109, 186803 (2012). FariaJunior2014:JAP P. E. Faria Junior, T. Campos, and G. M. Sipahi, J. Appl. Phys. 116, 193501 (2014). Sipahi1996:PRB G. M. Sipahi, R. Enderlein, L. M. R. Scolfaro and J. R. Leite, Phys. Rev. B 53, 9930 (1996). Lu2008:APL T.-C. Lu, C.-C. Kao, H.-C. Kuo, G.-S. Huang and S.-C. Wang, Appl. Phys. Lett. 92, 141102 (2008). Lu2010:APL T.-C. Lu, S.-W. Chen, T.-T. Wu, P.-M. Tu, C.-K. Chen, C.-H. Chen, Z.-Y. Li, H.-C. Kuo and S.-C. Wang, Appl. Phys. Lett. 97, 071114 (2010). Kasahara2011:APE D. Kasahara, D. Morita, T. Kosugi, K. Nakagawa, J. Kawamata, Y. Higuchi, H. Matsumura and T. Mukai, Appl. Phys. Expr. 4, 072103 (2011). Lin2014:LPL B. C. Lin, Y. A. Chang, K. J. Chen, C. H. Chiu, Z. Y. Li, Y. P. Lan, C. C. Lin, P. T. Lee, Y. K. Kuo, M. H. Shih, H. C. Kuo, T. C. Lu and S. C. Wang, Laser Phys. Lett. 11, 085002 (2014). Vurgaftman2001:JAP I. Vurgaftman, J. R. Meyer and L. R. Ram-Mohan, J. Appl. Phys. 89, 5815 (2001). Dong2013:JAP L. Dong, J. V. Mantese, V. Avrutin, Ü Özgür, H. Morkoç and S. P. Alpay, J. Appl. Phys. 114, 043715 (2013). note:web http://www.ioffe.ru/SVA/NSM/Semicond/ Chuang1996:APL S. L. Chuang and C. S. Chang, Appl. Phys. Lett. 68, 1657 (1996). note:spinsplittings The VB spin splitting appears because of the off-diagonal SOC energy Δ_3 in the Hamiltonian that mediates the coupling between the asymmetric potential and k-dependent terms. Within this k·p model the CB has no SOC interaction with other bands, therefore the spin splitting is not seen in the CB structure. Park2000:JAP S. H. Park and S. L. Chuang, J. Appl. Phys. 87, 353 (2000). Haug:2004 H. Haug and S. W. Koch, Quantum Theory of Optical and Electronic Properties of Semiconductors, 4^th Edition (World Scientific Publishing, Singapore, 2004). Chow:1999 W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials, (Springer, New York, 1999). Funato2013:APE M. Funato, Y. S. Kim, Y. Ochi, A. Kaneta, Y. Kawakami, T. Miyoshi and S. Nagahama, Appl. Phys. Express 6, 122704 (2013). Brimont2009:JAP C. Brimont, M. Gallart, A. Gadalla, Olivier Crégut, Bernd Hönerlage, and P. Gilliot, J. Appl. Phys. 105, 023502 (2009). Michalzik:2013 VCSELs Fundamentals, Technology and Applications of Vertical-Cavity Surface-Emitting Lasers, edited by R. Michalzik (Springer, Berlin, 2013). Coldren:2012 L. A. Coldren, S. W. Corzine, and M. L. Mašović, Diode Lasers and Photonic Integrated Circuits, 2^nd Edition (Wiley, Hoboken, 2012). Zutic:2011 I. Žutić, R. Oszwałdowski, C. Gøthgen, and J. Lee, Semiconductor Spin-Lasers, in Handbook of Spin Transport and Magnetism, edited by E. Y. Tsymbal and I. Žutić (CRC Press/Taylor&Francis, New York, 2011), pp. 731-746. Zutic2003:APL I. Žutić, J. Fabian, and S. Das Sarma, Appl. Phys. Lett. 82, 221 (2003). Arif2008:IEEEJQE R. A. Arif, H. Zhao, Y.-K. Ee, and N. Tansu, IEEE J. Quant. Electron. 44, 573 (2008). note:recombine Within the rate-equation description, including a widely-used spin-flip model [M. San Miguel, Q. Feng, and J.V. Moloney, Phys. Rev. A 52, 1728 (1995)], the n_+p_+ recombination only gives S^- helicity of the emitted light. This means that there is only one type of holes within a four-band model (CB and VB with a twofold spin degeneracy). Bhattacharya2016:APL A. Bhattacharya, M. Z. Baten, and P. Bhattacharya, Appl. Phys. Lett. 108, 042406 (2016). De2010:PRB A. De and C. E. Pryor, Phys. Rev. B 81, 155210 (2010). Cheiwchanchamnangij2011:PRB T. Cheiwchanchamnangij and W. R. Lambrecht, Phys. Rev. B 84, 035203 (2011). FariaJunior2016:PRB P. E. Faria Junior, T. Campos, C. M. O. Bastos, M. Gmitra, J. Fabian and G. M. Sipahi, Phys. Rev. B 93, 235204 (2016). Gmitra2016:PRB M. Gmitra and J. Fabian, Phys. Rev. B 94, 165202 (2016). Khaetskii2013:PRL A. Khaetskii, V. N. Golovach, X. Hu, and I. Žutić, Phys. Rev. Lett. 111, 186601 (2013). Korenivski2013:EPL V. Korenivski, A. Iovan, A. Kadigrobov, and R. I. Shekhter, Europhys. Lett. 104, 27011 (2013). Naidyuk2012:NJP Y. G. Naidyuk, O. P. Balkashin, V. V. Fisun, I. K. Yanson, A. Kadigrobov, R. I. Shekhter, M. Jonson, V. Neu, M. Seifert, S. Andersson, and V. Korenivski, New J. Phys. 14, 093021(2012). Fabian2007:APS J. Fabian, A. Mathos-Abiague, C. Ertler, P. Stano, and I. Žutić, Acta Phys. Slov. 57, 565 (2007). Hanbicki2002:APL A. T. Hanbicki, B. T. Jonker, G. Itskos, G. Kioseoglou, and A. Petrou, Appl. Phys. Lett. 80, 1240 (2002). Zega2006:PRL T. J. Zega, A. T. Hanbicki, S. C. Erwin, I. Žutić, G. Kioseoglou, C. H. Li, B. T. Jonker, and R. M. Stroud, Phys. Rev. Lett. 96, 196101 (2006). Salis2005:APL G. Salis, R. Wang, X. Jiang, R. M. Shelby, S. S. P. Parkin, S. R. Bank, and J. S. Harris, Appl. Phys. Lett. 87, 262503 (2005). Koshihara1997:PRL S. Koshihara, A. Oiwa, M. Hirasawa, S. Katsumoto, Y. Iye, C. Urano, H. Takagi, and H. Munekata, Phys. Rev. Lett. 78, 4617 (1997). Ohno2000:N H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl , Y. Ohno and K. Ohtani, Nature 408, 944 (2000). Dietl2014:RMP T. Dietl and H. Ohno, Rev. Mod. Phys. 86, 187 (2014). DiasCabral2011:PRB E. Dias Cabral, M. A. Boselli, R. Oszwaldowski, I. Žutić, and I. C. da Cunha Lima, Phys. Rev. B 84, 085315 (2011); L. D. Anh, P. N. Hai, Y. Kasahara, Y. Iwasa, and M. Tanaka, ibid. bf 92, 161201(R) (2015). Wang2006:JPCM J. Wang, C. Sun, Y. Hashimoto, J. Kono, G. A. Khodaparast, L. Cywiński, L. J. Sham, G. D. Sanders, C. J. Stanton, and H. Munekata, J. Phys.: Condens. Matter 18, R501 (2006). Lazic2016:PRB P. Lazić, K. D. Belashchenko, and I. Žutić, Phys. Rev. B 93, 241401(R) (2016).

http://arxiv.org/abs/1701.08162v3

20170127190000

Low-mass White Dwarfs with Hydrogen Envelopes as a Missing Link in the Tidal Disruption Menu

astro-ph.HE

Jamie Law-Smith lawsmith@ucsc.edu Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA NASA Einstein Fellow School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA Harvard-Smithsonian Center for Astrophysics, The Institute for Theory and Computation, 60 Garden Street, Cambridge, MA 02138, USA Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA We construct a menu of objects that can give rise to bright flares when disrupted by massive black holes (BHs), ranging from planets to evolved stars. Through their tidal disruption, main sequence and evolved stars can effectively probe the existence of otherwise quiescent supermassive BHs and white dwarfs can probe intermediate mass BHs. Many low-mass white dwarfs possess extended hydrogen envelopes, which allow for the production of prompt flares in disruptive encounters with moderately massive BHs of 10^5–10^7 M_—masses that may constitute the majority of massive BHs by number. These objects are a missing link in two ways: (1) for probing moderately massive BHs and (2) for understanding the hydrodynamics of the disruption of objects with tenuous envelopes. A flare arising from the tidal disruption of a 0.17 M_ white dwarf by a 10^5 M_ BH reaches a maximum between 0.6 and 11 days, with a peak fallback rate that is usually super-Eddington and results in a flare that is likely brighter than a typical tidal disruption event. Encounters stripping only the envelope can provide hydrogen-only fallback, while encounters disrupting the core evolve from H- to He-rich fallback. While most tidal disruption candidates observed thus far are consistent with the disruptions of main sequence stars, the rapid timescales of nuclear transients such as Dougie and PTF10iya are naturally explained by the disruption of low-mass white dwarfs. As the number of observed flares continues to increase, the menu presented here will be essential for characterizing nuclear BHs and their environments through tidal disruptions. § INTRODUCTION When a star wanders too close to a massive black hole (MBH), it can be ripped apart by the hole's tidal field <cit.>. In a typical disruption, half of the material will be ejected on hyperbolic trajectories and half of the material will remain bound to the MBH; the accretion of this material gives rise to a transient usually referred to as a tidal disruption event (TDE). With accurate theoretical modeling, TDEs allow us to uncover the mass of the black hole, the characteristics of the surrounding stellar population, the dynamics of the galactic nucleus, and the physics of black hole accretion under well-defined conditions <cit.>. TDEs can also provide a direct and unambiguous probe of the MBH occupation fraction of low-mass galaxies, which is crucial for constraining MBH seed formation efficiency at high redshifts—a dominant mechanism of initial galaxy formation <cit.>. The opportunity to study BHs in the local universe through TDEs is important, because for every actively accreting BH, there are ∼ 170 quiescent BHs <cit.>. TDEs are observationally identified by a combination of a dramatic increase in brightness, proximity to a non-active host galaxy's center, and weak or no color evolution at optical/UV wavelengths, with a decay in luminosity that is theoretically predicted to follow a t^-5/3 law <cit.>. The most compelling events are those in which the rise, peak, and decay of the optical/UV transient are observed with frequent cadence, as each of these phases of a TDE contain vital information about the disruption, and can be used to constrain the properties of the host black hole and the object that was disrupted <cit.>. Taken in a statistical sense, the observed rates of tidal disruption and, in particular, the relative rates of disruptions of different stellar objects, will hold tremendous distinguishing power in terms of both the dynamical mechanisms operating in galactic centers and the properties of the populations of stars themselves <cit.>. A central objective of this work is to understand the menu of all possible TDEs about massive BHs—i.e., which objects produce tidal disruption flares for which BH masses, and how they dictate the properties of the fallback accretion rate onto the BH. An object of mass M and radius R can be torn apart if it crosses the tidal radius, r_t = (M_bh/M)^1/3 R, of a BH with mass M_bh. Therefore, the characteristics of a particular stellar object hold information about the nature of its disruption—whether it occurs near the BH's innermost bound circular orbit, and, if so, how relativistic the encounter is. BHs with masses ≳ 10^7 M_ are well probed by MS stars, evolved stars, and planets, but the debris could be ineffective at circularizing for BHs with masses ≲ 10^6 M_, as shown by semi-analytic results in <cit.>, as well as the Newtonian and relativistic hydrodynamic simulations of <cit.> and <cit.>. BHs with masses ≲ 10^5 M_ could be probed by typical white dwarfs <cit.>. Thus far, most observed TDE candidates come from host galaxies with inferred BH masses of ≳ 10^6 M_. Even though survey selection effects make seeing TDEs from lower-mass BHs less likely <cit.>, we should expect to observe them with future surveys if the BH mass function is not truncated below 10^6 M_. Tidal disruption flares are potentially a powerful probe of the galaxy occupation fraction of these BHs, and could help discriminate between BH mass functions that are flat, rising (as extrapolated from the M–σ relation), and/or truncated at low masses. Our ability to use TDEs as direct probes of black hole demographics necessitates a detailed understanding of how the observability of TDEs depends on the properties of the disrupted star. Constructing a complete menu of stellar tidal disruption simulations—as we do in this work—is an important step in addressing these questions. Theoretical studies of stellar structure and fallback rate began with Lagrangian <cit.> and Eulerian <cit.> calculations, and have evolved to include detailed studies of MS stars <cit.>, giant planets <cit.>, white dwarfs <cit.>, and giant stars <cit.>. A finding common to all calculations is that a more centrally concentrated object has a quicker-peaking fallback rate and requires a deeper encounter for full disruption than a less centrally concentrated object. Here, “deeper" is in relation to the tidal radius definition, which relates to the average density. The presence of a core is also important in determining the fallback rate; in giant stars, the massive core plays a key role yet typically remains intact, while in giant planets, the lighter core is much more vulnerable. These considerations are crucial, as we expect the stellar structure to be imprinted on the luminosity evolution of the flare. In many of the observed events, the luminosity evolution closely follows the predicted mass fallback onto the BH <cit.>. This preservation of the fallback rate implies that circularization of the debris is prompt in these cases; the mass feeding rate is primarily determined by fallback and is not significantly delayed by viscous effects. Flares can be delayed if the amount of energy dissipated per orbit—or “viscosity”—is small. When the stream's self-intersection point is relatively close to the BH, energy dissipation is large, allowing the debris to circularize quickly <cit.>. Once the disk is formed, the viscous transport timescale (i.e., the time it takes material to accrete) at the circularization radius is much shorter than the peak fallback timescale. When the stream's self-intersection point is much farther from the BH than the periapse distance, however, circularization is not effective, and a highly elliptical disk is formed <cit.>. In this case, the viscous timescale can be significantly longer than the peak fallback timescale <cit.>. Stellar structure in tidal disruption calculations has thus far been implemented using polytropic profiles, with the simplest examples being the single-polytrope models of MS stars and WDs. Evolved stars and planets with cores are not well described by a single polytrope; these objects have been studied using a nested polytrope in which the envelope is a significant fraction of the total mass <cit.>. In this work we perform the first tidal disruption calculations for objects where the atmosphere has a small mass relative to the core, with our primary motivating physical example being a low-mass He WD with a hydrogen envelope—though we note that this structure could potentially also be used to model hot Jupiters or very evolved stars. Any WD below ≈ 0.46 M_ has a helium core, and possesses a hydrogen envelope that, despite its comparatively low mass, can extend to several times the core's radius <cit.>. In this work, we calculate the disruption of these objects and predict their observational properties. We argue that these objects are a missing link in two ways: (1) for probing moderately massive BHs, and (2) for understanding the hydrodynamics of the disruption of objects with tenuous envelopes, as such structures have not yet been studied. We find that these low-mass WDs with hydrogen envelopes offer prompt flares at higher-mass BHs than their more typical WD counterparts, and occupy a unique parameter space in time and luminosity at peak. In Section <ref>, we develop the tidal disruption menu, which is our motivation for the hydrodynamical simulations of this paper. In Section <ref>, we discuss the particulars of He WDs. In Section <ref>, we outline our hydrodynamical setup for disrupting these objects, and in Section <ref> we present numerical results from these simulations. In Section <ref>, we present an overview of tidal disruption flare demographics in terms of peak timescales and fallback rates. In Section <ref>, we summarize our findings and show that fast-rising events such as Dougie and PTF10iya are naturally explained by the disruption of an He WD. § TIDAL DISRUPTION MENU To determine whether an object is disrupted or swallowed by a black hole, we need to compare the tidal radius, r_t, to the innermost bound circular orbit of the black hole, r_ibco = 2GM_bh/c^2(1 - a_∗/2 + √(1-a_∗)), where a_∗ = a/M, a = J_∗/M_∗ c, M = GM_∗ / c^2, and M_∗ and J_∗ are the mass and angular momentum of the BH, respectively <cit.>. For a non-spinning BH, r_ibco=4GM_bh/c^2, and for a maximally spinning BH, r_ibco=GM/c^2. If r_t > r_ibco, disruption is possible. Otherwise, the object is swallowed whole <cit.>. For simplicity we assume here that disruption is only possible when the impact parameter β = r_t/r_peri≥ 1; more accurately, disruption is a smooth function of β. For a non-spinning BH, we therefore require M_bh≤ M_bh, lim = R_⋆^3/2/M_⋆^1/2(c^2/4G)^3/2∝ρ_⋆^-1/2 for disruption. The mass-radius relationship, then, determines whether an object will be disrupted at a given BH mass. Denser objects such as WDs can only be disrupted by lower-mass BHs while more tenuous objects such as MS or evolved stars can be disrupted by higher-mass BHs. We can calculate the upper limit for the disruption of a class of objects by using the above relation. We show this menu of BH-object combinations for a non-spinning BH, along with a prompt circularization condition explained below, in Figure <ref>. We use mass-radius relations for WDs, MS stars, evolved stars, and sub-stellar objects. We find that He WDs with hydrogen envelopes play a special role in this menu, as, similar to evolved stars, they can have a wide range of radii at a given mass, depending on their age. Compared to the relatively tight mass-radius relation for typical white dwarfs, these objects allow access to a higher range of BH masses. More details on He WDs and our stellar evolution calculations of their structure are given in Section <ref>. Our choice of representative masses and ages is justified there. Many of the tidal disruption candidates observed thus far show a luminosity time evolution that closely follows the mass fallback rate from the star to the BH <cit.>. This suggests that current observations may select for events in which debris circularization is prompt. Recent work suggests that prompt circularization occurs predominantly for encounters where general relativistic effects are important <cit.>. We take a “circularization condition” of r_t < 10 GM_bh/c^2 in order to select encounters in this regime. Following <cit.> and <cit.>, this corresponds to a de Sitter apsidal precession of Ω≳ 54^∘ for non-spinning BHs. Note that more weakly plunging encounters will still circularize some fraction of the time, and that they may also be observable as events where the luminosity evolution is viscously delayed; our condition is meant as a guideline for where we can expect to see predominantly prompt circularization events for a given disruptee. Note also that most events in the X-rays appear to be viscously delayed <cit.>. For a non-spinning BH, our condition for prompt flares is then 4GM_bh/c^2 < r_t < 10 GM_bh/c^2. WDs can only be disrupted by BHs with masses ≲ 10^5 M_, while MS and evolved stars only obey our prompt flare condition for BHs with masses ≳ 10^6 M_. Because of their extended radius, low-mass WDs with hydrogen envelopes can serve as a missing link between these two regimes. Their envelope can be disrupted and stripped by higher BH masses than allowed for by typical WDs. These BH masses offer a relatively smaller fraction of prompt flares from MS stars due to their inefficient circularization here. The constraints derived for He WDs, here assumed to be at least 1 Gyr after formation, could be extended to higher-mass BHs for younger He WDs, which have significantly more extended envelopes. For example, a 100 Myr old 0.17 M_ He WD can have a radius of 0.5 R_, allowing it to be disrupted by a 10^8 M_ BH. Low-mass WDs can thus extend the range of BH masses available to the higher-mass, single-star evolution WDs through tidal disruption.[There is some evidence now mounting for observational candidates of WD disruptions by intermediate mass BHs. In particular, an emerging class of ultra-long gamma-ray burst (ULGRB) sources share similar timescales and luminosities to WD disruptions <cit.>.] While these objects make up a small fraction of the stellar population, they deserve to be examined in more detail because of their unique location in our prompt circularization menu, which, as we argue, makes their emerging flares more favorable to detection. § HELIUM-CORE HYDROGEN-ENVELOPE WDS §.§ Properties Since WDs have an inverse mass-radius relationship, the lowest mass WDs will be able to probe the highest mass BHs. Let us estimate the lowest mass WD available through single-star evolution. Setting the main sequence lifetime equal to the age of the universe <cit.> using an analytic formula for the MS lifetime from <cit.> gives M_i≈ 0.9 M_. Using this mass in an empirical initial–final mass relation for WDs from <cit.> for M_i < 2.7 M_, M_f = (0.096 ± 0.005) M_i + (0.429 ± 0.015), we find that the minimum WD mass possible through single-star evolution is M_WD≈ 0.5 M_. WDs less massive than roughly half a solar mass will have formed through binary interactions, barring cases of extreme metallicity <cit.>. Low-mass WDs can be formed either through stable Roche-lobe overflow mass transfer or common-envelope evolution <cit.>. A helium-core WD forms if one component of the binary loses its hydrogen envelope before helium burning. This object has a degenerate helium core and is formed with an extended hydrogen envelope supported by a thin hydrogen burning layer. The maximum mass of an He WD is approximately 0.46 M_, and only He WDs are formed below this mass <cit.>. The final mass of the He WD depends on the mass of the progenitor and the binary orbital properties <cit.>. The progenitor star needs a zero-age main sequence mass below 2.3 M_, as more massive stars do not form helium cores. The strict minimum timescale for formation of an He WD is therefore the MS lifetime of a 2.3 M_ star, t_MS≈ 1.16 Gyr <cit.>. <cit.> performed calculations of He WD formation via stable mass transfer; we quote some results below. After detachment from Roche-lobe overflow, the progenitor star enters a “bloated” proto-WD phase where much of the hydrogen in the envelope is burned in stable hydrogen shell burning. The mass of hydrogen left after Roche-lobe detachment is on the order of 10^-2 M_, yet this can fuel a proto-WD phase lasting up to 2.5 Gyr for the lowest mass (M ≲ 0.20 M_) WDs. <cit.> derived a timescale for hydrogen burning, Δ t_proto≃ 400 Myr(0.20 M_/M_WD)^7, which describes the star's contraction from Roche-lobe detachment to its maximum effective temperature on the cooling track. <cit.> also defined a cooling timescale, t_cool, L_-2, which is the time from detachment to reaching log (L/L_)=-2 on the cooling track. This timescale is set primarily by the mass of the hydrogen envelope left at the end of the proto-WD phase. Generally, a shorter orbital period at the onset of mass transfer leads to a lower proto-WD mass and a higher final envelope mass. There is a growing body of observations of these low-mass objects: the targeted survey for extremely low-mass (ELM; M < 0.3 M_) WDs has found 76 binaries to date, with a median primary mass of ≈ 0.18 M_ <cit.>. Many of these WDs appear to be bloated, and this bloated state can persist for a long time: <cit.> find that roughly half of these systems will still be burning hydrogen when they merge. One object in this sample is the binary system NLTT 11748 <cit.>, which contains a helium-core hydrogen-envelope WD of mass 0.17 M_ and radius 0.043 R_, whereas a standard WD mass-radius relation for this mass would give a radius of 0.02 R_. This object's bloated size allows it to be disrupted by a BH of up to 3.8 × 10^6 M_. This WD has a cooling age of 1.6–1.7 Gyr; younger He WDs can have much more extended envelopes, allowing them to be disrupted by even 10^7 M_ or 10^8 M_ BHs. As an example of this more extreme bloating, observations and astroseismological studies of the eclipsing binary J0247–25 find a He WD with mass 0.186 ± 0.002 M_ and radius 0.368 ± 0.005 R_ <cit.>. Note that this He WD has a larger radius than a MS star of its mass. In a study of the Galactic WD binary population, <cit.> found that roughly half of WDs in binaries are He WDs, and that the probability density distribution for He WDs is relatively flat below 0.4 M_. It is difficult to estimate the typical age of a He WD upon disruption by a MBH, as these objects are formed from a range of progenitor masses and undergo a binary interaction of uncertain timescale. We do know that nuclear star clusters exhibit a wide range of stellar ages. For example, observations of the nearby S0 galaxy NGC 404 show that half of the mass of the nuclear star cluster is from stars with ages of ≈1 Gyr, while the bulge is dominated by much older stars <cit.>. In our own Galactic center, roughly 80% of the stars formed over 5 Gyr ago and the remaining 20% formed in the last 0.1 Gyr <cit.>. In addition, TDEs have so far been found preferentially in post-starburst galaxies, with significant 1 Gyr old or younger stellar populations <cit.>. Another consideration is that in a study of a population of He WDs in the globular cluster NGC 6397, <cit.> found that the progenitor binaries of the He WDs very likely underwent an exchange interaction within the last Gyr. Finally, we note that the two-body relaxation time is ≈ 0.1 Gyr for a 10^5 M_ BH and ≈ 1.8 Gyr for a 10^6 M_ BH. Motivated by the above considerations, in our disruption simulations we take the radius of the He WD at 1 Gyr after formation (i.e., since Roche-lobe detachment). For the tidal disruption calculations in this work, we construct a 0.17 M_ He WD consisting of a 0.16 M_ degenerate helium core and a 0.01 M_ hydrogen envelope using the MESA stellar evolution code <cit.>. This envelope mass is consistent with theoretical predictions of hydrogen retention <cit.>. The left panel in Figure <ref> shows the relative abundance of helium and hydrogen as a function of radius for this object. The hydrogen envelope extends to roughly 10 times the radius of the core, and is supported by a thin hydrogen burning shell. This snapshot is at 1 Gyr after formation. We also calculate the radius as a function of time since formation (through a binary interaction) for several He WDs in MESA. This is shown for our 0.17 M_ object in the right panel of Figure <ref>. We show the radius of a core-only WD of the same mass for comparison in dashed blue (here we show a fixed radius that does not evolve with time). In a similar calculation for a 0.15 M_ WD, we find that a very extended envelope persists for >10 Gyr. §.§ Disruption and Flaring Rates The particular tidal disruption rates of different types of objects depend on the detailed dynamics and evolution of the dense stellar system surrounding the central BH. Given these uncertainties, here we make a simple estimate of the relative rate of He WD disruptions. We find that several factors could increase the rate from that suggested by these objects' low population fraction. We can decompose the observed rate into (1) the fractional disruption rate and (2) the rate of luminous flares. §.§.§ Disruption First, the fractional disruption rate. This can be written as f_disrupted = f_pop× f_rel, where f_pop is the fraction of the stellar population that are He WDs, and f_rel is the specific likelihood of an He WD being disrupted. First we estimate f_pop. Modeling the Galactic population of double WDs, <cit.> found a Galactic birth rate of close double white dwarfs of 0.05 yr^-1 and a formation rate of planetary nebulae of 1 yr^-1. They found that 63% of the stars in these pairs are He WDs. This implies that the production rate of He WDs is approximately 0.05 × 0.63 ≈ 0.03 times that of single stellar evolution WDs. Choosing an age of 10 Gyr for the Galactic disk gives a turnoff mass of approximately 1 M_ for the stars in our Galaxy. We estimate the WD fraction by dividing the number of stars with masses of 1–8 M_ (those that evolve to leave WD remnants) by the number with masses of 0.1–8 M_ using a <cit.> IMF; this gives a WD fraction of approximately 0.16. The population fraction of He WDs is then f_pop≈ 0.16 × 0.03 ≈ 0.005. There is a concern that mass segregation might limit f_ pop in central cluster regions. In clusters, low-mass stars are evaporated from the central regions as above-average mass objects settle deeper in a trend toward energy equipartition on the cluster relaxation time <cit.>. However, binaries containing an He WD, even though the He WD mass is low, will not be evaporated from the central regions as their total mass is on average higher than the average mass of a typical stellar population. Indeed, in a study of the central regions of globular cluster NGC 6397, <cit.> found a sample of He WDs with masses of 0.2–0.3 M_. These objects show strong Hα absorption lines (indicating that they still retain their hydrogen envelopes), and are significantly more concentrated in the cluster center than either the CO WDs or the turnoff stars. We therefore expect that mass segregation either enhances f_ pop or, at least, does not reduce it in nuclear star clusters. This population fraction could also be larger due to the fact that in dense stellar systems, the rate of dynamically assembled compact binaries is observed to be enhanced by a factor of 10–100 when compared to the field <cit.>. We might expect similar enhancements in the dense and dynamical nuclear region surrounding an MBH. Note that the separation of He WDs from their companions is observed to be 10^10 cm < a < 3 × 10^11 cm in the ELM survey, making these binaries stable against ionization for typical nuclear cluster conditions. For f_rel, we follow <cit.> and scale the specific likelihood of disruption as f_rel∝ r_t^1/4. Relative to an MS star, this is f_rel = (R_He/R_MS)^1/4(M_MS/M_He)^1/12, which is of order unity for our 0.17 M_ He WD and a ∼ 0.5 M_ MS star. This gives us a conservative total fractional disruption rate of f_disrupted = f_pop× f_rel≈ 0.005. For a hydrogen-depleted He WD, f_rel is closer to 1/2. As a potential comparison, simulations of star clusters by <cit.> found that the relative fraction of WD disruptions is ≈0.15. Multiplying this by the relative production rate of He WDs <cit.> suggests a fractional disruption rate of f_disrupted≈ 0.005, consistent with our above estimate. However, as mentioned, mass segregation and dynamical assembly effects can enhance our above estimate. The estimate using star cluster simulations may also be low, as these simulations include very low-mass BHs and a population of single stars. These calculations therefore model the disruption of only single WDs, which also follow the substantially more compact typical WD mass radius relation. We lack a proper N-body simulation of the relative disruption rates for binary systems such as those that produce He WDs. §.§.§ Flaring Second, we consider the relative rate of luminous flares arising from the disruption of He WDs. One consideration is that He WD disruptions will produce a higher peak luminosity relative to MS stars, simply because they are more compact. For 0.5 M_ WDs, <cit.> showed that their disruption rate Ṅ is lower than that of MS stars, but that, when weighted by their luminosities, the total number of observed transients is higher for these WDs than MS stars for M_bh≲ 10^5 M_, as the observing volume grows with luminosity. For He WDs, one can similarly expect their luminosity-weighted rates to be higher relative to MS stars than their pure fractional rates estimated above. The fraction of prompt versus delayed flares is also important here. As suggested earlier, prompt flares occur when general relativistic effects are important. <cit.> showed that MS stars are ineffectively circularized for lower BH masses, leading to viscously delayed luminosity evolution. For 10^5 < M_bh/M_ < 10^6, the fraction of prompt events from MS stars is ≈ 13% (if we include events that are viscously slowed only as they rise to peak, this fraction is ≈ 17%). Because He WDs are disrupted in the strongly relativistic regime, these objects should be rapidly circularized for these BH masses, as shown in Figure <ref>. As a result, He WD disruptions should make up a higher fraction of prompt flares than their population fraction suggests. This effect becomes especially important at lower BH masses, for which the occupation fraction remains unconstrained <cit.>. As we will see, even partial disruptions of He WDs with hydrogen envelopes can provide super-Eddington fallback onto the BH. These partial disruptions are also favorably prompt compared to MS disruptions, and could further enhance the relative rate of flares from He WD disruptions. § NUMERICAL SETUP §.§ MESA Calculations Using the MESA stellar evolution code, we construct a 0.17 M_ white dwarf with a 0.16 M_ degenerate helium core and a 0.01 M_ hydrogen envelope. As noted in the previous section, there is a growing population of observed objects in this mass range. In these low-mass objects, the extended envelope lasts for a long time, as Δ t_proto∝ M_WD^-7 (Equation <ref>). We might therefore be more likely to see flares from the stripped envelopes of objects close to this mass, as they exist in a bloated state for longer than their higher-mass cousins. We approximate the core and envelope as nested polytropes <cit.>, using polytropic indices n_core=1.5 and n_env=3.8. Figure <ref> shows the density versus radius profile of this object from MESA as well as from the nested polytrope that we matched. We use this nested polytrope as an input to our hydrodynamical simulations as it provides a reasonable description of the object's structure, and makes possible comparisons with non-hydrogen-envelope WD disruption calculations using polytropic equations of state <cit.>. A single polytrope is unstable to small variations in pressure p_0 and volume V_0 if (∂ p/∂ V)_0 is positive, and this occurs for polytropic indices of n>3. However, it is difficult to derive simple stability criteria for our nested polytrope structure, as it is not differentiable across the core-envelope discontinuity. We instead ensure that two heuristic tests of stability are satisfied: (1) the entropy increases with radius, or ∂ S/∂ r>0, and (2) the star does not contract or relax significantly when placed on our hydrodynamical grid structure for 20 dynamical timescales of the full star. The dynamical timescale for the full star is t_dyn^full≃√(R^3/GM) = 535 s. In this work we will often refer to the dynamical timescale of the He core of this WD for comparison; this is t_dyn^core=22.5 s. §.§ Hydrodynamical Setup Our simulations of tidal disruption are performed with the basic framework and code described in detail in <cit.>, <cit.>, <cit.>, and <cit.>. We use FLASH <cit.>, a 3D adaptive mesh grid-based hydrodynamics code including self-gravity. Hydrodynamics equations are solved using the using the piecewise parabolic method <cit.>. We refine the grid mesh on the value of the density, and derefine by one level every decade in density below ρ=10^-4 g cm^-3. All of the simulations presented here are resolved by at least R_⋆/Δ r_min>130, where Δ r_min is the size of the smallest cells. We note that adaptive mesh refinement is well suited for disruption calculations of an object with this core and envelope structure, as the envelope occupies a large volume yet has a very low mass fraction. We perform our calculations in the rest-frame of the star to avoid introducing artificial diffusivity by moving the star rapidly across the grid structure. We solve the self-gravity of the star using a multipole expansion about the center of mass of the star with l_max=10. We then evolve the orbit based on the center of mass of the star and the position of a point-mass black hole <cit.>. We use Newtonian gravity for the black hole, which is a reasonable approximation as our star's closest approach in any of our simulations is >10 r_g, in the weak field regime. <cit.> showed that general relativistic effects in tidal disruption simulations should be small is this regime. Note that, because we use Newtonian gravity, by construction, the encounters we simulate are outside of our rapid circulation condition defined in Section <ref>. The effect of relativistic encounters is discussed in Section <ref>. We run our simulations using the 0.17 M_ He WD described above and a 10^5 M_ BH. We input the MESA profile, matched as a nested polytrope (Figure <ref>), into FLASH. We use two different fluids in the simulation: one for the helium core and one for the hydrogen envelope. Both have the same equation of state, with a γ_fluid=5/3. This setup has an envelope composition of 100% hydrogen. More accurately, the envelope has a residual helium abundance that will migrate toward the core over time depending on the relative strength of mixing and gravitational settling. We relax the object onto the grid for 5 t_dyn before sending the BH toward it. We use an eccentricity e≈1, as most disrupted stars originate from orbits scattered from the sphere of influence <cit.>. As discussed in <cit.>, for a given stellar structure, we can understand the vast majority of disruptions by surveying in impact parameter β=r_t/r_p as all other parameters obey simple scaling relations when relativistic effects are unimportant. Similar to the dynamical timescale, we can define β with respect to the tidal radius of the full star or the degenerate core. We survey in β_full from 1 to 10 in 12 runs. This corresponds to β_core of ≈ 0.1 to 1.2. We run our simulations for 21 t_dyn^full=500 t_dyn^core, well into the self-similar decay portion of the mass fallback rate. § NUMERICAL RESULTS §.§ Phenomenology: Core versus envelope Figure <ref> shows the time evolution of the star for a β_core=0.7 encounter in 2D slices in density through the 3D simulation box, zoomed in on the star. Time is labeled in terms of the dynamical time of the core. In this moderately plunging encounter, the star is distorted through pericenter, evolving into a surviving remnant and two tidal tails—one bound and one unbound from the BH. As we increase the impact parameter, the star is perturbed closer to its center. For mildly plunging encounters, only the hydrogen envelope is stripped, while the core survives intact. For more deeply plunging encounters, both the core and envelope are disrupted and fed to the BH. We can see this qualitatively in Figure <ref>, where we show slices through the simulation box zoomed in on the star for β_core=0.5, 0.7, and 0.9 encounters. We plot the ratio of the core material to envelope material density. The different spatial distributions of core and envelope material will result in different fallback times to the BH, which will result in observed light curves dominated by material of different compositions at different times. Because of their different structures—the envelope has a steeper density gradient than the core—these two fluids react to losing mass in characteristically different ways, as we will see below. §.§ Mass lost Figure <ref> shows the mass lost from the star as a function of impact parameter, calculated at the last timestep of our simulations. We run our simulations long enough so that the mass lost calculated from this final timestep is asymptotically close to the final mass lost. Note that half of the lost mass will return to the black hole and half is ejected as an unbound debris stream. The object is smoothly disrupted with the impact parameter, albeit with two components from the envelope and the core. This is different from giant star disruptions <cit.>, where the core is never disturbed, and likely arises because the density contrast between the core and envelope is in general larger for giants than it is for He WDs. A fitting formula from <cit.> for a Γ=5/3 polytrope fits the mass lost from the core well. This is expected, as once the core has been penetrated, the envelope has negligible dynamical effect, and the disruption will proceed as if for a typical WD. Full disruption occurs at β_core≈ 0.9. This n=1.5 polytrope has a lower critical β (for full disruption) compared to higher index polytropes, as the mass is distributed more evenly. In addition to this, a n=1.5 polytrope has an inverse mass-radius relation, and so expands when mass is removed, making the object more vulnerable to disruption. We model the envelope, on the other hand, as an n=3.8 polytrope, which reacts to mass removal by contracting—“protecting” itself. Because the envelope has a steeper density gradient, its critical β is higher than for a Γ=5/3 polytrope. We can see this in the shallower slope of Δ M/M versus β for envelope material relative to core material. §.§ Spread in binding energy and mass fallback rate We calculate the spread in binding energy of the star's material to the BH, dM/dE versus E, over time. We compute the specific binding energy of the material in each cell of the simulation, which depends on its distance and velocity relative to the center of mass of the star and to the black hole. Details of the calculation are presented in <cit.>. Only material that is bound to the BH and not bound to the star will contribute to the mass fallback onto the BH. We compute the specific binding energy of the material in each cell of the simulation, which depends on its distance and velocity relative to the center of mass of the star and to the black hole. Figure <ref> shows the spread in dM/dE over time for β_core=0.5, 0.7, and 0.9 encounters, with the contribution from material unbound to the star in solid black and contributions from the core and envelope of the remnant in red and blue. We see that impact parameter drastically changes the spread in binding energy through and following disruption, both for the bound and unbound material. Grazing encounters leave the core relatively unperturbed and are able to retain more envelope material, while deeper encounters leave a compact remnant that has been all but stripped of its envelope. Given dM/dE and a pericenter distance, we can calculate the mass fallback rate onto the BH by Kepler's third law, dM/dt = dM/dEdE/dt = (dM/dE) 1/3(2π GM_bh)^2/3 t^-5/3. The left panel of Figure <ref> shows the spread in specific binding energy dM/dE versus E at the last timesteps of our simulations for all impact parameters. We verify that the binding energy has effectively “frozen in,” or converged to its final distribution, by this timestep. The right panel shows dM/dE mapped onto dM/dt across time for the same impact parameters, with the Eddington limit for this BH shown in dashed black. We take Ṁ_Edd=0.02 (η/0.1) (M_bh/10^6 M_) M_/yr with η=0.1. Feeding rates peak at t_peak∼ 5 × 10^4 to 10^6 s≈ 0.6 to 11 d depending on β. Weakly plunging encounters peak later, while deeply plunging encounter peak earlier. Note that t_peak evolves strongly with β (it spans more than an order of magnitude), in contrast to single polytrope solutions where the evolution in t_peak is much more gradual <cit.>. This means that the He WD disruptions—and disruptions of other objects with this core and extended envelope structure—probe a much wider range of potential transient characteristics for a given BH mass. We see that even for very weakly plunging encounters, for which only a fraction of the envelope is stripped (see mass lost in Figure <ref>), the mass fallback rate is super-Eddington. Encounters only stripping the envelope appear to have a shallower slope in early-time mass fallback and smoother evolution near peak than encounters penetrating the core. This is due to their different polytropic structures. <cit.> presented a fitting formula for the peak fallback rate of material onto the BH, where Ṁ_peak = f(M_bh, β, γ). Figure <ref> shows Ṁ_peak values from our simulations of the disruption of an 0.17 M_ He WD compared with those from this fitting formula for a Γ=5/3 non-hydrogen-envelope WD with a mass of 0.155 M_, the mass of the core of the He WD. We expect this functional form to match for disruptions that penetrate the core. In low β encounters, the hydrogen envelope provides mass return rates that are unavailable to WDs without envelopes. §.§ Composition of debris We track the core and envelope material separately in our simulations, which allows us to track the composition of the debris falling onto the BH. In Figure <ref> we show Ṁ as a function of time for β_core=0.5, 0.6, and 0.8, with absolute and fractional contributions from the helium core in red and the hydrogen envelope in blue. The mass fallback rate from weakly plunging encounters can be super-Eddington and hydrogen-dominated. In more deeply plunging encounters, the early rise of the mass fallback rate is fed almost entirely by the hydrogen envelope, while the peak and late time evolution are fed by the helium core; the nature of this transition depends on β. Note that the disruption turns the star inside out: the material that is removed first accretes first, and is then buried underneath the material that is removed last and accretes last. The diffuse envelope material feeds a qualitatively slower rise in the mass fallback curve compared to the core material. The first evidence that a range of stellar or spectral properties might be represented in TDEs was the discovery of a helium-rich TDE, PS1-10jh <cit.>. Gezari et al. explained its hydrogen-free spectrum as the result of the tidal disruption of the helium-rich core of a star, similar in structure to an He WD progenitor. <cit.> noted that TDEs observed thus far show a continuum of helium-rich to hydrogen-rich spectral features; there is an ongoing debate over the origin of the strong helium emission. <cit.> found that stellar evolution can play a role in producing this spectral diversity. <cit.> modeled the emission from TDEs through an extended, optically thick envelope formed from stellar debris. They find that due to optical depth effects, hydrogen Balmer line emission is often strongly suppressed relative to helium line emission. For MS stars, for example, it is possible for the hydrogen emission lines to be absent. Having said this, the specific composition of the material is expected to have consequences on the detailed line ratios. Line diagnostics from disruptions of He WDs could transition from hydrogen to helium smoothly with β. An encounter stripping only the envelope could provide a rare, (nearly) pure hydrogen-powered mass fallback. If an optically thick reprocessing envelope exists, however, observational evidence of this type of encounter could be variable. § TDE DEMOGRAPHICS Here we explore the tidal disruption menu of BHs and disrupted objects in terms of the peak fallback rate and its associated peak timescale, and place our He WDs in context. Through Kepler's third law, we can write scalings of the peak mass fallback rate and its associated time of peak, Ṁ_peak ∝ M_bh^-1/2 M_⋆^2 R_⋆^-3/2 t_peak ∝ M_bh^1/2 M_⋆^-1 R_⋆^3/2, where the Ṁ_peak∝ M_⋆^2 scaling results when we assume that a constant fraction of the star's mass is lost in the disruption. <cit.> found fitting parameters for these scaling relations that depend on the polytropic Γ and impact parameter β. We use these below. In Figure <ref>, we show Ṁ_peak versus t_peak values for the He WD disruptions presented in this work, as well as for several representative disruptions of other objects: a 0.6 M_ non-He WD, a 0.6 M_ MS star, a 50 M_Jup brown dwarf (BD), a 1 M_Jup planet, and a 1.4 M_, 10 R_ red giant (RG). We use fitting parameters from Equations A1 and A2 of <cit.> to calculate Ṁ_peak and t_peak for the other objects, and to scale with BH mass. We use a polytropic Γ of 5/3 for the WD, MS star, BD, and planet (the values are similar if we use 4/3 for the MS star), and 4/3 for the RG. We show impact parameters that remove from Δ M/M_⋆=0.01 to 1 from each object. As in the tidal disruption menu shown in Figure <ref>, we only show encounters with BHs obeying our prompt circularization condition, 4GM_bh/c^2 < r_t < 10 GM_bh/c^2. Here, flares resulting from the fallback of material onto the BH are both visible (disruption occurs outside the innermost bound circular orbit) and predominantly prompt (circularization of the debris is efficient). We color the encounters by BH mass. There is a huge variety in the timescales and fallback rates with which stars feed MBHs following TDEs. Prompt flares separate into different timescale classes based on the stellar type and BH mass combination. Prompt flares also show relatively unique timescale/BH mass combinations—i.e., a timescale and a prompt flare can imply not only a stellar type but also a BH mass. The clean separations blur slightly if we allow for (1) the full distribution of masses and radii available for different classes of objects, which is especially important for He WDs, and (2) the effects of viscous delay, which smear the effective timescales and mass fallback rates. In Figure <ref> we show a fallback rate curve for each of the objects in Figure <ref>, scaled to disruptions with a 10^6 M_ BH for comparison. Note that the 0.6 M_ WD disruption would occur inside the event horizon for this BH mass. We show fallback from a 1.4 M_ RG at two different points along the giant branch: ascending the RG branch (RG1; R ≈ 10 R_) and the tip of RG branch (RG2; R ≈ 100 R_), from <cit.>. We show a β=0.9 encounter (full disruption) for the non-He WD, the MS star, and the planet, and a β=1.5 encounter for the giant stars. We show two Ṁ curves for the He WD: one for a full disruption (β_core=0.9) and one for an envelope-stripping encounter (β_core=0.5). For a given BH mass, these objects offer distinct fallback rates and characteristic timescales. Converting these fallback rates into luminosities is not straightforward. In this paper we have focused on rapidly circularized TDEs, where the accretion rate (and so the luminosity) is expected to closely follow the fallback rate. This is predicted to be true for emission both from the disk <cit.> and from stream collisions <cit.>, and is observed to be the case in the best-sampled, non-beamed UV/optical events <cit.>. However, it is not evident that the luminosity will always follow the fallback rate, in particular when circularization is inefficient or for BHs accreting at highly super-Eddington rates <cit.>. For example, the event Sw J1644+57 <cit.> did not appear to follow a t^-5/3 luminosity evolution during its prompt decline phase. In addition, jetted emission may not be Eddington limited; its strength depends on the radiative efficiency of the (relativistic) flow. Even in the absence of a jet, <cit.> show that the radiative efficiency of super-Eddington accretion flows can be high under certain circumstances. While most full disruptions are expected to provide super-Eddington accretion rates (Figure <ref>), the observed peak luminosities of UV/optical TDEs appear to be Eddington limited, or sub-Eddington <cit.>. Two possible solutions to this are (1) that the most commonly observed events are partial disruptions, where the fallback rate can be significantly lower <cit.> or (2) that the radiative efficiency is low <cit.>. Constructing this menu—which spans many orders of magnitude in BH mass, fallback timescale, and fallback rate—is nonetheless a key step toward making meaningful comparisons with observations. We have only shown a few representative objects; the full phase space of luminosities and timescales, the effects of viscous delay, and a comparison to observations will be explored in future work. § DISCUSSION §.§ Possible Candidates for He WD Disruption Here we compare t_peak values from simulations to those of two particularly rapidly rising TDE candidates, Dougie <cit.> and PTF10iya <cit.>, accounting for luminosity and BH mass constraints. <cit.> estimate Dougie's peak bolometric luminosity as L_peak≈ 5(±1) × 10^44 erg s^-1 and its rise time as t_rise∼10 d. They estimate a central BH mass of a few 10^6 to 10^7 M_ for Dougie's host galaxy. <cit.> estimated 10iya's peak bolometric luminosity as L_peak≈ (1-5) × 10^44 erg s^-1 and place a limit on its rise time of t_rise<5 d. They constrain the central BH mass via the observed bulge luminosity versus BH mass relation as log M_BH/M_≲ 7.5. In order to constrain the kinds of disruptions that can produce such rapid flares, we construct a histogram of t_peak for the 0.17 M_ He WD disruptions presented in this work as well as for regular WDs, MS stars, BDs, and planets. We model regular WDs, MS stars, BDs, and planets with M < 0.3 M_ as Γ=5/3 polytropes. We model MS stars with M > 0.3 M_ as Γ=4/3 polytropes. The mass at which we transition from 5/3 to 4/3 does not affect our conclusions significantly, as their t_peak values overlap. Giant star disruptions have longer timescales than we are interested in here. We draw from flat distributions in M_obj, with white dwarf masses of 0.2 M_ < M_WD < 1 M_, MS star masses of 0.085 M_ < M_MS < 3 M_, BD masses of 13 M_Jup < M_BD < 0.085 M_, and planet masses of 1 M_Jup < M_pl < 13 M_Jup. We use only the one 0.17 M_ He WD mass. We draw from a flat distribution in BH mass with 10^6 < M_bh/M_ < 10^7, roughly the BH mass constraints for Dougie and 10iya. We draw from a flat distribution in β, discarding encounters where r_p<r_ibco. We estimate the peak luminosity from each encounter as L_peak = min(0.1 Ṁ_peak c^2, L_Edd) for the given BH, as these events were observed in the optical/UV and we expect accretion luminosity to be Eddington limited. We discard encounters with L_peak < 3 × 10^44 erg s^-1, which is comfortably below the errors in Dougie's peak luminosity. In Figure <ref>, we show the outcome of the above exercise. We find that the only objects that satisfy the luminosity requirement are MS stars, BDs, and our prototypical He WD. MS stars and BDs, however, cannot reproduce the rapid timescales of Dougie and 10iya from Ṁ alone. Thermal TDEs such as PS1-10jh show a good correspondence between the observed luminosity and the fallback rate <cit.>. This simplicity makes the disruption of He WDs an appealing explanation for rapidly rising nuclear transients. In order to explain Dougie as a MS star disruption, models require a strong wind component with a functional form that may not directly reflect Ṁ <cit.>. A wind that carries a significant amount of kinetic and thermal energy may be produced if the accretion rate onto the BH exceeds its Eddington limit <cit.>. While this scenario could explain Dougie and other rapidly rising TDEs such as PTF10iya, their timescales can be naturally explained by the Ṁ from He WD disruptions. We note that <cit.> found that Dougie appears offset ≈3.9 kpc from the photometric center of its host galaxy. This initially seems to disfavor a TDE interpretation. However, the photometric center of a galaxy is not necessarily its dynamical center. Vinkó et al. also noted that lower-mass off-center BHs are rare yet not unprecedented <cit.>, making the TDE hypothesis tenable. §.§ Caveats Our study focuses on a single example of the disruption of a prototypical 0.17 M_ He WD. However, as we saw in Section <ref>, these objects can have a wide range of masses and radii, and the radius evolution even for a single mass is appreciable (see Figure <ref>). The inclusion of hydrogen-bearing He WDs with a larger range of core masses and envelope masses could potentially explain events with shorter or longer timescales than the prototypical encounters presented here. In this work, we model the interaction between only a single He WD and a BH. We expect, however, that many He WDs will be in binary systems as they approach the BH, composed of either two He WDs or one He WD and one CO/ONe WD. This suggests that some disruptions of He WDs involve two stars instead of one <cit.>. The interaction of the binary with the BH can shift the distribution in binding energy of the debris, and cause the time of peak accretion to occur either earlier or later depending on the sign of the energy shift <cit.>. In extreme cases, this interaction can bind all of the material to the BH (as opposed to just half), allowing the BH to accrete the whole star; alternatively, all of the material can become unbound, preventing any accretion onto the BH. If the binary separation is of order the tidal radius, double tidal disruptions are possible <cit.>. However, our single-star calculations are still applicable for double disruptions, as the hydrodynamics of the disruption are independent for each of the components of the binary. In cases where the outgoing debris streams from the two disrupted stars do not interact with one another, the fallback resulting from a binary disruption can be mimicked by applying simple shifts to the binding energy distribution of the debris of the single-star case. We do not consider general relativistic effects in our disruption calculations—the gravitational potential of our point mass is purely Newtonian. <cit.> investigated relativistic effects on the fallback rate of debris. For highly relativistic encounters, they found a more gradual rise and delayed peak of the fallback compared to the Newtonian result. For a 1 M_, 1 R_ MS star encounter with a 10^7 M_ BH, where r_p/r_g≈ 10, they found a difference in Ṁ_peak of ≈ 18% and a difference in t_peak of ≈ 10% between Newtonian and relativistic simulations. For a 0.6 M_ WD encounter with a 10^5 M_ BH, where r_p/r_g≈ 4.6, the difference in Ṁ_peak is ≈ 69% and the difference in t_peak is ≈ 49%. For the He WD encounters presented in this work, the critical β of full disruption has r_p/r_g≈ 12, and the transition between an envelope-stripping encounter and one penetrating the core occurs at r_p/r_g≈ 19. Thus, relativistic corrections to our results should be small. In scaling to higher BH masses, however, our errors will increase. However, this will not weaken (and will in fact strengthen) our conclusions regarding the ability of He WDs to achieve the peak timescales of rapidly rising TDE candidates such as Dougie and PTF10iya though Ṁ alone, as the relativistic effect is to lengthen the peak fallback timescale. We use a nested polytrope matched to a MESA profile of the He WD as the initial condition in our disruption calculations. We also track only two fluids—one for the core and one for the envelope—in the simulation, and make the simple choice to model the core as fully helium and the envelope as fully hydrogen. A more realistic treatment might use the MESA profile directly in the disruption calculations, and track the composition of the object more fully. For the particular object used in the simulations in this work, however, the gains in accuracy (aside from composition information) in using the MESA profile directly may be minimal, as the nested polytrope profile is very close to the true profile. §.§ Conclusions We have modeled the tidal disruption of a new class of object: the low-mass He WD with an extended hydrogen envelope. These objects are a missing link both hydrodynamically and in terms of BH masses probed through prompt tidal disruption flares. In summary, we find that: * Because of their lower density cores and extended envelopes, these objects extend the potential BH masses probed by single-star evolution WDs. In general, their peak fallback timescales will be longer that those of typical WDs and shorter than those of MS stars. * Grazing encounters that strip only the envelope will be hydrogen dominated, and—for a very small amount of mass removed—can provide high and often super-Eddington fallback. * Encounters penetrating the core generally have a fallback rate that is hydrogen-dominated in its rise and helium-dominated in its peak and decline, with relative composition versus time a function of impact parameter. * The typical peak accretion rate of He WD disruptions is a few times larger than that of a typical MS disruption. This likely makes these disruptions observable to larger distances, which would make them a larger fraction of the observed total than suggested by their relative population. These objects are perhaps the last missing piece of a theoretical tidal disruption menu that includes WDs, MS stars, planets, and evolved stars. Constructing this menu is key to better understanding tidal disruptions. The reader is referred to Figures <ref>, <ref>, and <ref> for a summary of the phase space of the menu. This work may have particular bearing on two puzzling observational aspects of TDEs that have emerged in the past few years. The first is their rates. There is a great deal of uncertainty in the properties of the nuclear star clusters from which stars are fed into disruptive orbits. Most calculations make standard assumptions of a spherical single-mass nuclear star cluster that feeds stars to the BH by a two-body relaxation-driven random walk in angular momentum space. These calculations predict disruption rates of ≳ 10^-4 yr^-1 per galaxy <cit.>, and are in general in tension with the lower observationally derived rates of roughly 10^-5 yr^-1 <cit.>. However, there can be several complicating effects—such as secular relaxation, or the presence of a triaxial potential, rings or disks of stars, and/or a second massive body—and there is a lack of understanding of their relative importance in local galaxies. In addition, we need to better understand the mass spectrum of disrupted stars, in particular given mass segregation <cit.>. The second puzzling observation is that a significant fraction of tidal disruptions may arise from unique stellar populations. We are learning that tidal disruption flares may occur preferentially in post-starburst galaxies <cit.>, and that these types of galaxies are overrepresented as TDE hosts. This remains a mystery. Post-starburst galaxies are elliptical-type galaxies that have experienced a star formation burst that has stopped within the past ∼ 1 Gyr, leaving these galaxies with both old and very young stars. If only certain types of stars (which are a small fraction of the population) produce prompt flares for BH masses of ∼ 10^6 M_ due to circularization effects, this could alleviate some of the tension in the observed flaring versus disruption rate. As we have argued in Section <ref>, the rate of luminous flare production can be distinct from the disruption rate itself. We have shown in Section <ref> and Section <ref> that different stellar types probe distinct islands of BH mass when we consider prompt flares. This is strong evidence for a connection between stellar population details and the disruption flare rates. The post-starburst galaxy preference may be due to the production of particular stellar species in the nuclei of these galaxies, rather than in the dynamics of their nuclei. We caution, however, that the stellar population of a galaxy as a whole does not necessarily reflect its nuclear population. In this work, we have argued that to effectively use TDEs to constrain the mass function of BHs, we need to acknowledge that not all disruptions produce luminous flares. Moving forward likely involves understanding the intersection of nuclear region stellar dynamics, stellar populations, and stellar evolution, along with the hydrodynamics of the disruptions themselves. Targeting the observational characteristics of certain TDEs might offer a way to identify BHs at the low end of the supermassive BH mass range. We thank the members of the 2015 Jerusalem Workshop on TDEs for useful comments and discussions. We thank the anonymous referee for constructive comments. The calculations for this research were carried out in part on the UCSC supercomputer Hyades, which is supported by National Science Foundation (award number AST-1229745) and UCSC. M.M. is grateful for support from NASA through Einstein Postdoctoral Fellowship grant number PF6-170155 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. J.G. is grateful for support from Einstein grant number PF3-140108. P.M. is grateful for support from the NSF Graduate Research Fellowship and the Eugene Cota-Robles Graduate Fellowship. E.R.R. is grateful for support from the Packard Foundation and NASA ATP grant NNX14AH37G. This work is also supported by NSF grant AST-1615881. aasjournal

http://arxiv.org/abs/1701.07789v2

20170126174520

Calabi-Yau Structures, Spherical Functors, and Shifted Symplectic Structures

math.AG

Wurtzite spin lasers Igor Žutić ==================== A categorical formalism is introduced for studying various features of the symplectic geometry of Lefschetz fibrations and the algebraic geometry of Tyurin degenerations. This approach is informed by homological mirror symmetry, derived noncommutative geometry, and the theory of Fukaya categories with coefficients in a perverse Schober. The main technical results include (i) a comparison between the notion of relative Calabi-Yau structures and a certain refinement of the notion of a spherical functor, (ii) a local-to-global gluing principle for constructing Calabi-Yau structures, and (iii) the construction of shifted symplectic structures and Lagrangian structures on certain derived moduli spaces of branes. Potential applications to a theory of derived hyperkähler geometry are sketched. § INTRODUCTION This paper develops a program aimed at studying various features of Picard-Lefschetz theory using the language and methods of higher category theory and derived geometry, with a view toward applications in algebraic geometry, symplectic geometry, and homological mirror symmetry. Some of the potential applications to fundamental questions in classical geometry, and hyperkähler geometry in particular, are sketched in section <ref>, where we outline the contours of a twistorial approach to a theory of derived hyperkähler geometry. The main results of this paper are summarized in subsection <ref> below. Before giving a precise statement of the results, we begin with a leisurely informal discussion of the motivation and background for these results. §.§ Categorical Picard-Lefschetz theory: spherical functors and monodromy. The categorical approach to Picard-Lefschetz theory derives from viewing Picard-Lefschetz theory through the lens of symplectic geometry. Arnol'd <cit.> observed that the monodromy transformations of Picard-Lefschetz theory are in fact symplectomorphisms, leading to a shift in perspective, from a topological view of Lefschetz fibrations, to the richer and more refined symplectic viewpoint. The theory of symplectic Lefschetz fibrations was introduced by Donaldson <cit.>, and many of the categorical ideas are implicit in his work. Following seminal ideas of Donaldson, Fukaya and Kontsevich, Seidel introduced and developed a beautiful and far-reaching theory of Fukaya-type categories associated to Lefschetz fibrations <cit.>, generalizing many features of classical Picard-Lefschetz theory to the realm of symplectic topology. The point of view that we will take in this paper, following Seidel, is that the symplectic topology of a Lefschetz fibration is captured by the interaction between the Fukaya-Seidel category of the fibration and the Fukaya category of its generic fiber. As we will show below, this interaction is encoded in the data of certain spherical functors. Let w: X →ℂ be a Lefschetz fibration. For simplicity and concreteness, let us assume that X is a quasi-projective variety over ℂ, and that w is a proper holomorphic map. The function w has a finite set {p_1,p_2,...,p_n} of critical values. On ℂ-{p_1,...,p_n}, w defines a locally trivial fiber bundle. To a point t ∈ℂ-{p_1,...,p_n}, we can associate the Fukaya category (X_t) <cit.> of the fiber X_t := w^-1(t). This is an ∞-category that is linear over the Novikov field ℂ((t^ℝ)). The objects of this category are, roughly speaking, Lagrangian submanifolds of X_t equipped with unitary local systems. The space of morphisms between two objects is given by Lagrangian Floer cohomology. As t varies, we obtain a local system of ∞-categories over the complement of the set of critical values of w: indeed, if γ: [0,1] →ℂ -{p_1,...,p_n} is a path, then symplectic parallel transport gives rise to a symplectomorphism X_γ(0)→ X_γ(1), which in turn induces an equivalence of categories (X_γ(0)) →(X_γ(1)). On the other hand, we have the Fukaya-Seidel category (X,w) of the Lefschetz fibration. Objects of this category are, roughly speaking, Lagrangian submanifolds of X decorated with unitary local systems, with the property that the intersection of w(L) with the complement of some compact set is contained in the positive real axis. If we fix a smooth fiber Y:= X_t of w, then there is a natural functor ∩: (X,w) →(Y) given by sending a Lagrangian L in X to L ∩ X_t. This leads naturally to the following: What extra structure does the functor ∩: (X,w) →(Y) carry? A partial answer to this question is given by the notion of spherical functor introduced by Anno and Logvinenko <cit.>. A functor F: → between stable ∞-categories is spherical if - F admits a left adjoint F^* and a right adjoint F^!. - The homotopy cofiber T_ of the counit F ∘ F^! →𝕀_ is an autoequivalence of - The homotopy fiber T_ of the unit 𝕀_→ F^! ∘ F is an autoequivalence of . As originally observed by Kapranov and Schechtman <cit.>, this structure should be viewed as a natural categorification of the structure of a perverse sheaf on the unit disc D in ℂ, with Whitney stratification consisting of the strata {0} and D - {0}. Indeed, to a perverse sheaf on D one can associate a pair of vector spaces: the vanishing cycles Φ() of , and the nearby cycles Ψ(). These vector spaces come with a pair of maps Φ() @<.5ex>[r]^u Ψ() @<.5ex>[l]^v with the property that T_Ψ := 𝕀_Ψ - uv and T_Φ := 𝕀_Φ - vu are invertible morphisms of vector spaces. Furthermore, the classification of perverse sheaves on D <cit.> says that the datum of a perverse sheaf on D is equivalent to the data of pair of vector spaces and maps satisfying the condition above. Theorems <ref> and <ref> together imply that the functor ∩: (X,w) →(Y) is spherical, under hypotheses that are expected to hold in great generality. We refer the reader to Section <ref> for the precise statement. This fact has been anticipated by the experts, and a different proof was outlined by Abouzaid in <cit.>. The emphasis of our investigation is on the interaction of the spherical functor with Calabi-Yau structures, to which we turn now. Let be an ∞-category linear over some field k. Recall that a Serre functor <cit.> for is an autoequivalence S_: → with the property that for any two objects x, y in we have (x,y) ≃(x, S_y)^∨ where (x,y) is the k-module spectrum of maps from x to y. For example, if = (X) is the category of perfect complexes on a smooth projective variety X of dimension d, then S_ = (-) ⊗ω_X[d] where ω_X is the canonical line bundle. A Serre functor is unique up to isomorphism if it exists, and a Serre functor exists for any smooth and proper category. A weak Calabi-Yau structure of dimension d on is an equivalence of functors between the Serre functor S_ of and 𝕀_[d]. If we are given a Lefschetz fibration w: X →ℂ as above, with w proper, then the generic smooth fiber Y is a compact oriented manifold. Poincare duality on Y is reflected in the fact that the category := (Y) carries a Calabi-Yau structure <cit.>, and in particular S_≃𝕀_[d], where 2d is the real dimension of Y. On the other hand, the Serre functor S_ of := (X,w) is non-trivial. This leads naturally to the following question: How is the Serre functor of (X,w) related to the data of the spherical adjunction (X,w) ⇆(Y) and the monodromy of the Lefschetz fibration w: X →ℂ? The monodromy around infinity of the fibration w gives rise to an autoequivalence σ_* of (X,w). In <cit.>, Seidel observed, using symplectic geometry, that there is a natural map σ_* →𝕀. In the language of spherical functors that we have introduced above, σ_* is the functor T_ := (𝕀_→∩^! ∘∩). Seidel attributes to Kontsevich the idea that σ_* ≃ S_[-(d+1)]. Motivated by these considerations, we introduce the notion of what we will call a compatible spherical functor, building on the notion of spherical functor introduced by Anno and Logvinenko <cit.> and a subsequent addition by Katzarkov, Kontsevich and Pantev <cit.> (see Definition <ref>). Roughly speaking, given an arbitrary linear ∞-category , a d-Calabi-Yau category , and a spherical functor F: →, we say that F is compatible with the Calabi-Yau structure, or simply that F is a compatible spherical functor, if we are given an equivalence T_≃ S_[-(d+1)] satisfying a compatibility condition. This is a relative version of the notion of a weak Calabi-Yau structure in the sense that it reduces to the latter when = 0. Theorems <ref> and <ref> imply that the functor ∩: (X,w) →(Y) is in fact a spherical functor that is compatible with the natural Calabi-Yau structure on (Y), under certain technical hypotheses. §.§ Gluing spherical functors: perverse Schobers and categorical surgery. The monodromy around infinity of the Lefschetz fibration can be computed from the monodromy around the individual singular fibers. From the discussion of the last paragraph of <ref> above, we should expect that this manifests itself categorically as a “decomposition” of the Serre functor of (X,w). Let us try to formulate this more precisely. Let the notation be as in <ref>. Choose a disc U_i centered at each of the critical values p_i ∈ℂ of w: X →ℂ, such that the discs are pairwise disjoint. Let X_i := w^-1(U_i) and let w_i: X_i → U_i be the restriction of w to X_i. Then we can define a Fukaya-Seidel category _i = (X_i,w_i) associated to (X_i, w_i) exactly as we did for (X,w). More precisely, if we choose a point q_i^∞ on ∂ U_i, then the objects of (X_i, w_i) are decorated Lagrangians that, outside some compact set containing p_i, project to the ray joining p_i to q_i^∞. The morphisms are defined exactly as for (X,w) (see <cit.>). Let us call (X_i, w_i) the local Fukaya-Seidel category at p_i. This is a categorification of the space of vanishing cycles at p_i. Can the Fukaya-Seidel category (X,w) be reconstructed from the data of the local categories (X_i, w_i) and the collection of spherical functors ∩_i: (X_i, w_i) →(Y), where Y is a generic smooth fiber of the Lefschetz fibration? Kontsevich has proposed a powerful sheaf-theoretic framework for Fukaya-type categories <cit.>, which provides a natural setting within within which to pose and study such local-to-global questions. Given a topological fibration π: X → B, we can compute the cohomology of X as the derived global sections of a sheaf on B whose stalks are the cohomology of the fibers: H^*(X, ℤ) = RΓ(B, Rπ_*ℤ). In the same vein, if π: X → B is a symplectic fibration of symplectic manifolds, then Kontsevich proposes that one should recover Fukaya-type categories associated to X as a certain global object associated to a sheaf of categories over B. More precisely, Kontsevich proposes that one should introduce a notion of a perverse sheaf of linear ∞-categories. A symplectic fibration π: X → B should give rise to such a “perverse sheaf” 𝔛, whose generic stalk is the Fukaya category of the generic fiber of π. By passing to a suitable additive invariant, such as periodic cyclic homology, one should obtain a perverse sheaf on B, in the usual sense of the term. Given a singular Lagrangian ℒ→ B, one should have an induced constructible sheaf of ∞-categories 𝔛_ℒ on ℒ, and we define (ℒ, 𝔛) to be the global sections of 𝔛_ℒ (there is a dual version involving cosheaves; for simplicity we stick to the case of sheaves here). The semi-classical Fukaya category of B with coefficients in 𝔛 can then be defined to be ^sc(B, 𝔛) = _ℒ→ B(ℒ, 𝔛) The Fukaya category (B, 𝔛) of B with coefficients in 𝔛 should then be a certain deformation of ^sc(B, 𝔛) by pseudo-holomorphic discs in B. Conjecturally, when 𝔛 comes from a symplectic fibration π: X → B, (B, 𝔛) should recover an appropriate version of the partially wrapped Fukaya category of X. In the special case that this paper is concerned with, the base has real dimension 2. In this situation, the notion of a perverse sheaf of categories has been given a concrete mathematical incarnation under the moniker “perverse Schobers” by Kapranov and Schechtman in <cit.>. Let us restrict, as we did above, to the case where the base is topologically ℂ. Then a perverse Schober 𝔛 on ℂ with singularities at {p_1,...,p_n} is essentially given by the data of categories , _1,..., _n, and spherical functors F_i: _i → (see 2 of <cit.>). The particular presentation of the Schober depends on a “system of cuts” in the base; we refer to loc. cit for details about how the presentation depends on this choice. The category is the stalk of the local system of categories 𝔛_|ℂ- { p_1,...,p_n } at some fixed point t ∈ℂ-{p_1,...,p_n}. When the Schober “arises from” a Lefschetz fibration, then we have _i = (X_i, w_i), = (Y) = (X_t) and F_i = ∩_i. Note that in this situation, a singular Lagrangian is just a ribbon graph Γ in ℂ. The sheaf 𝔛_Γ associates to - a generic smooth point of Γ the category , - an n-valent vertex located at point in ℂ- { p_1,...,p_n } the category ⊗(A_n-1) - a 1-valent vertex that coincides with the critical point p_i of w the category _i Conjecturally, we should have (ℂ, 𝔛) ≃(X,w) With this background in mind we can formulate * Monodromy around infinity induces an autoequivalence of the generic fiber of the perverse Schober 𝔛. How can describe this autoequivalence in terms of the presentation of the Schober by spherical functors F_i: _i →? In other words, can we effectively compute the Serre functor of the global sections of the Schober in terms of the Serre functors S__i and the spherical functors F_i? * Can (compatible) spherical functors be glued together to produce new (compatible) spherical functors? * Let 𝔛 be a perverse Schober, such that the generic fiber is Calabi-Yau, and all of the spherical functors in some presentation are compatible with the Calabi-Yau structure (see Definition <ref>). Is there a natural compatible spherical structure on the map from the global sections of this Schober to the generic fiber? The phenomena that this question seeks to address have been studied using a different language in <cit.> and <cit.>. In <cit.>, Seidel introduces the notion of a noncommutative divisor to capture the structures associated with the symplectic geometry of a Lefschetz fibration in the language of A_∞-algebras. Essentially the same algebraic structure was introduced earlier in a different context by Tradler and Zeinalian in <cit.>, under the moniker V_∞-algebra. Independently, Kontsevich and Vlassopoulos have introduced a generalization of this structure in <cit.>. Our approach differs in that, whereas the aforementioned papers rely on explicit formulas and resolutions within the A_∞-formalism, here we seek to formulate everything intrinsically, and in a manifestly model-independent manner. It would be an interesting question to compare the structures and constructions we introduce here with those in loc. cit. As in those papers, our constructions are most naturally viewed from the perspective of derived noncommutative geometry, to which we turn now. §.§ Noncommutative oriented cobordisms and Tyurin degenerations. Derived noncommutative geometry, pioneered by Kontsevich and developed in <cit.>, is a powerful paradigm in which to formulate and study various dualities predicted by string theory, such as mirror symmetry. The fundamental underlying principle is that it is useful to identify a “space” with the physical theory it defines. In the context of topological string theory, the relevant physical theory is an extended 2d-topological field theory (TFT). According to the cobordism hypothesis <cit.>, such a TFT is determined by the k-linear ∞-category that it assigns to a framed point. With this as motivation, one defines a derived noncommutative space, or -space for short, to be a k-linear stable ∞-category. This point of view leads naturally to the homological/categorical interpretation of physical mirror symmetry introduced in <cit.>. The central problem of -geometry is to extract the partition function of the TFT on various manifolds from the data of the category, and to describe various geometric operations on the target spaces for the A-model TFT and B-model TFT in purely categorical terms. A smooth and proper category determines an extended 2d-TFT defined on framed manifolds. In order to obtain a theory that is defined on oriented manifolds, the category must be equipped with an additional structure: namely, a Calabi-Yau structure. The notion of a weak Calabi-Yau structure introduced in <ref> (an isomorphism between the Serre functor and a shift of the identity functor), turns out to be insufficient for this purpose. To rectify this, Kontsevich and Soibelman introduced <cit.> the notion of a Calabi-Yau structure of dimension d on , which is a class of degree -d _*() → k[-d] in the dual of the cyclic homology _∗() ≃_∗()_hS^1 of which induces an isomorphism of the Serre functor with 𝕀_[d] under the identification _*()^∨≃(𝕀_, S_) upon forgetting the S^1 equivariance data. Kontsevich-Soibelman <cit.> and Costello <cit.> showed how to construct an oriented topological field theory from the data of a category equipped with a Calabi-Yau structure. Lurie's classification of topological field theories <cit.> places these results within a very general context. In loc. cit., he introduces the notion of a Calabi-Yau object in an arbitrary symmetric monoidal (∞,2)-category, and shows that Calabi-Yau objects in a certain (∞,2)-category of k-linear categories can be identified with Calabi-Yau categories in the sense of <cit.>. There is another sense in which the term orientation is associated with Calabi-Yau structures. If X is a compact oriented C^∞-manifold of real dimension d, then capping with the fundamental class defined by the orientation defines a map of degree -d ∩ [X]: C^*(X,k) → k[-d] from the space of k valued cochains on X to the base field k. This map implements Poincaré duality in the sense that the composite map C^*(X,k) ⊗ C^*(X,k) [r]^(0.6)∪ C^*(X,k) [r]^∩ [X] k[-d] defines a perfect pairing on C^*(X,k). One can view the a manifold X, or more generally any topological space X, as a constant derived stack. From this point of view, C^*(X,k) can be viewed as the space of functions on X, i.e., the (derived) global section Γ(X, 𝒪_X). Now, if X is an arbitrary derived stack satisfying certain finiteness conditions, then one defines <cit.> an 𝒪-orientation of dimension d on X to be a map or: Γ(X, 𝒪_X) → k[-d] satisfying properties analogous to those satisfied by ∩ [X] above. If X is a projective variety of dimension d, then it is straightforward to see that an 𝒪-orientation of dimension d on X is a Calabi-Yau structure of dimension d on X. The formal similarity between the definition of a Calabi-Yau structure on a category and an 𝒪-orientation on a derived stack is manifest. Furthermore, it is true that a projective variety X admits an 𝒪-orientation of dimension d if and only if the linear ∞-category (X) admits a Calabi-Yau structure of dimension d. In manifold theory, it is essential to understand how manifolds can be glued out of more elementary manifolds with boundary. Tyurin <cit.> has emphasized that the following two situations should be seen as analogous: - a smooth oriented C^∞-manifold X with boundary ∂ X ⊂ X, as encoded by the pullback map C^*(X,k) → C^*(∂ X, k). Note that the boundary inherits a natural orientation. - an anticanonical divisor D:= ∂ X in a Fano variety X, and the associated pullback map on derived global sections Γ(X, 𝒪_X) →Γ(∂ X, 𝒪_∂ X). Note that since ∂ X is anticanonical, it acquires a natural Calabi-Yau structure. Broadening our world of spaces to include all derived stacks, these two examples can be placed on a common footing. Motivated by the theory of Lagrangian structures in shifted symplectic geometry, Calaque has introduced the notion of a nondegenerate boundary structure <cit.>, which is essentially an 𝒪-orientation on a “derived stack with boundary”. Roughly speaking, it consists of an orientation Γ(∂ X, 𝒪_∂ X) → k[-d] together with a homotopy commutative diagram Γ(X,𝒪_X) [r] [d] Γ(∂ X, 𝒪_∂ X) [d] 0 [r] k[-d] witnessing a nullhomotopy of the pullback of the orientation, that is required to satisfy a certain nondegeneracy condition. Tyurin's fundamental insight was that, just as one can understand an oriented manifold by viewing it as being obtained by gluing two oriented manifolds with boundary (i.e., by splitting along a codimension one submanifold), one can study a Calabi-Yau variety X by degenerating it to the union of two Fano varieties X_1, X_2 glued along a common anticanonical divisor Z: X ⇝ X_1 ∪_Z X_2. From the perspective of the B-model, this corresponds to a deformation of the category (X_1) ×_(Z)(X_2) to the category (X). Since the latter category is Calabi-Yau, it is natural to expect that the former category carries a natural Calabi-Yau structure. To better understand this situation, we introduce the notion of a relative Calabi-Yau structure on a functor F: → (Definition <ref>); this is a noncommutative analogue of the nondegenerate boundary structures of <cit.>. Our definition builds on a suggestion made by Toën in <cit.>. Expanding on Toën's suggestion and ideas that emerged initially in discussions between Dyckerhoff and the second named author, Brav and Dyckerhoff have also introduced and studied notions of relative Calabi-Yau structures in <cit.>. Roughly speaking, a relative Calabi-Yau structure on F: → consists of a Calabi-Yau structure ϕ: _*() → k[-d], together with “isotropy data” given by a functor Δ^1 ×Δ^1 →(k) _*() [r] [d] _*() [d]^ϕ 0 [r] k[-d] satisfying a certain nondegeneracy condition. In <cit.>, it is shown that, in the situation above, the restriction functor i^*: (X_i) →(Z) carries a natural relative Calabi-Yau structure. Now, the mirror to this functor is the cap functor ∩: (X^∨_i,w_i) →(Z^∨) for a Landau-Ginzburg model (X^∨_i,w_i) mirror to X_i. Thus we are led to the following question, to which an affirmative answer is provided by Theorem <ref> (see <ref>): How can we construct a natural relative Calabi-Yau structure on the cap functor ∩: (X,w) →(Y) of a Landau-Ginzburg model without appealing to mirror symmetry (i.e., using symplectic geometry)? As mentioned earlier, a key feature of ∩: (X,w) →(Y) is that it is a spherical functor. Given a spherical functor F: → with Calabi-Yau target, there is natural compatibility structure that one can ask for, relating the Calabi-Yau structure to F (Definition <ref>). What is the relationship between the notion of a compatible spherical functor and the notion of a relative Calabi-Yau structure? Theorem <ref> if a functor F admits both left and right adjoints then a weak Calabi-Yau structure on F are the same thing as the structure of a compatible spherical functor. It is implied by a conjecture in <cit.>(Conjecture 10.2.8) that given a weak Calabi-Yau structure, there exists a strong Calabi-Yau structure for which the underlying equivalence S_≃𝕀_[d] is the given weak CY-structure. The relative analogue of this conjecture would be the statement that every compatible spherical functor can be lifted to a relative Calabi-Yau structure. In light of this, it seems interesting to investigate the relationship between compatible spherical functors and strong relative Calabi-Yau structures. This investigation is particularly germane to the problem of constructing Calabi-Yau structures on perverse Schobers <cit.>, and showing that such a structure gives rise to a Calabi-Yau structure on the Fukaya category of the base manifold with coefficients in the Schober, as is expected <cit.>. We return to the problem of constructing a Calabi-Yau structure on the fiber product category (X_1) ×_(Z)(X_2). Let be a category equipped with a d-Calabi-Yau structure ϕ_ and be equipped with a d-Calabi-Yau structure ϕ_. By an oriented noncommutative cobordism from (, ϕ_) to (, ϕ_), we mean a functor F: →×, together with a relative Calabi-Yau structure on F for which the Calabi-Yau structure on × is given by (- ϕ_, ϕ_). The discussion in the paragraphs above motivates the following: Can noncommutative oriented cobordisms be glued together? More precisely: * If _i → carry relative Calabi-Yau structures, then is _1 ×__2 equipped with a natural Calabi-Yau structure? * If each of the functors defining a perverse Schober is equipped with a relative Calabi-Yau structure, does the global sections of the Schober over a Lagrangian skeleton inherit a (relative) Calabi-Yau structure? One of our main results, Theorem <ref>, is an affirmative answer to the first part of this question. The remaining parts will be the subject of a future investigation. The notion of a Pre-CY structure introduced by Kontsevich and Vlassopoulos <cit.> long predates the concept of relative Calabi-Yau structures, and appears to be very closely related to the latter. They construct in loc. cit a TQFT starting from the data of a Pre-CY structure. It seems very plausible that the gluing constructions involved in the construction of their TQFT are related to the gluing construction of Theorem <ref>. It would be very interesting to understand the precise relationship between the two approaches. One may also consider the inverse problem: given two Fano varieties X_1, X_2 glued along a common anticanonical divisor Z, when may we consider X_1∪_Z X_2 a degeneration of a smooth Calabi-Yau X? For sufficiently nice X_i and Z, Kawamata and Namikawa <cit.> show this is equivalent to the criterion that the normal bundles N_Z/X_1 and N_Z/X_2 are inverse. This criterion is then the same as certain spherical twists being inverses. Following Doran, Harder, and Thompson <cit.> we note that this situation is entirely analogous to the case where we have two Landau-Ginzburg models w_i:X_i→ with the same generic fibre, which we can then glue to form a single model w:X→. If the monodromies on the fibre of each X_i are inverses, then the monodromy at infinity of w is trivial, so we can extend this to some w̃:X̃→ℙ^1. We may consider (X̃) to be a deformation of (X) by instanton corrections; this latter category is obtained by gluing the Fukaya-Seidel categories of each X_i. In <cit.> the authors conjecture that the two cases are equivalent under homological mirror symmetry, and show several cases where this is true. We may consider all these cases within the realm of noncommutative geometry, which lets us treat them on an equal footing. Is there a noncommutative version of the theory of Tyurin degenerations and Friedman-Kawamata-Namikawa smoothings in the language of noncommutative cobordisms? §.§ From noncommutative orientations to shifted symplectic structures. Shifted symplectic structures are analogues of symplectic structures in the world of derived geometry. The main novel features in the derived context are: - For a differential form (such as a pre-symplectic form) to be closed is a property/condition in ordinary geometry, while it is an extra structure in derived geometry. - A n-shifted symplectic structure induces an equivalence 𝕋_X→𝕃_X[n] between the tangent complex and a shift of the cotangent complex. Shifted symplectic structures were first introduced in the context of supermanifolds in <cit.> in order to construct certain topological field theories. This theory has been vastly generalized in <cit.> to the world of derived ∞-stacks. As in that paper, we will work in an algebro-geometric context; thus, the shifted symplectic structures in this paper are analogues of holomorphic symplectic structures, rather than C^∞ ones. Throughout this paper, we will use the powerful language developed in <cit.>, and we refer the reader to that paper for a more detailed discussion of shifted symplectic geometry. There are two main ways one construct new shifted symplectic stacks from old ones: - Forming the derived mappping stack from an oriented stack to a shifted symplectic stack - Forming the derived intersection of two Lagrangians in a shifted symplectic stack One of the motivating goals of this paper is to formulate and exploit noncommutative analogues of these constructions. After recalling these constructions, we will explain how the results proven here contribute toward this goal. Let X and Y be derived Artin stacks, with Y having an n-shifted symplectic form. A d-orientation [X] on X is a “fundamental class” [X]:Γ(X,_X)→ k[-d] satisfying certain nondegeneracy properties. Given such an orientation, we can construct a symplectic form on (X,Y): Let Y be a derived Artin stack, and let X be an -compact derived stack with a d-orientation [X]. Assume the derived mapping stack (X,Y) is a derived Artin stack locally of finite presentation over k. Then we have a transgression map: ∫_[X]^*(-):(Y,n)→((X,Y),n-d). Let Y be a derived Artin stack equipped with an n-shifted symplectic structure. Let X be an -compact derived stack with a d-orientation [X], and let i: X → X' be a morphism equipped with a nondegenerate boundary structure. Assume that the mapping stacks (X,Y) and (X',Y) are derived Artin stacks locally of finite presentation over k. Then the pullback map: i^*: (X,Y') →(X,Y) is equipped with a natural Lagrangian structure. Several examples of orientations are given in <cit.>, following Theorem 2.5. Here is one: Let X be a smooth and compact Calabi-Yau variety. If X has complex dimension d and we have an isomorphism ω_X≃_X, then projection of Γ(X,_X) onto the degree d cohomology ^d(X,_X)[-d], followed by the isomorphism ^d(X,_X)≃^d(X,ω_X)≃ k provides a map [X]:Γ(X,_X)→ k[-d]. The fact that this map defines a d-orientation is essentially the content of Serre duality on X. By Theorem <ref>, (X) = (X, ) is equipped with a natural (2-d)-shifted symplectic structure, since carries a canonical 2-shifted symplectic structure. What can be said in Example <ref> if we drop the hypothesis that X is compact? Theorem <ref> is the answer that we offer in this paper. It says that the moduli stack of perfect complexes with compact support carries a shifted symplectic structure, provided that the variety can be compactified in a manner compatible with the trivialization of the canonical bundle. What are the noncommutative analogues of Theorem <ref> and Theorem <ref>? The expected answer to this question is the following: if a functor F: → is equipped with a (strong) relative Calabi-Yau structure, and the moduli spaces of objects in these categories are locally geometric stacks, then the induced map on moduli spaces of objects carries a natural Lagrangian structure. We do not prove this statement here. However, we give evidence for this statement, by adding to the following to the list of relative Calabi-Yau functors for which the statement is true: - the functor _c(U) → 0 for U a “non-compact Calabi-Yau” (Theorem <ref>). - the functor i_*: (D) →(X) for D a smooth divisor in a smooth and compact Calabi-Yau variety X (Theorem <ref>). Strictly speaking, we only prove that i_*: (D) →(X) is a compatible spherical functor, which in turn implies that it carries a weak relative Calabi-Yau structure. We expect that this can be promoted to a strong structure. One can also interpret Theorem <ref> as providing evidence for this claim. The next theorem implies, in particular, that the derived critical locus Crit(f) of any function f carries a shifted symplectic structure, since Crit(f) is the intersection of the zero section with the graph of df in the cotangent bundle. Let (X,ω) be a derived stack with n-shifted symplectic structure, and let L → X and L' → X be morphisms of derived stacks equipped with Lagrangian structures. Then L ×_X L' is equipped with a natural (n-1)-shifted symplectic structure. It is natural to ask whether this theorem has a noncommutative analogue. In fact, we have already encountered this question before (Question <ref>), in a slightly different context. Theorem <ref>, which says that the fiber product of relative Calabi-Yau functors carries a Calabi-Yau structure, is a noncommutative analogue of Theorem <ref>. §.§ Main results The main results of this paper are partial answers to the questions raised in sections <ref> through <ref> above. Our first result, which partially answers Question <ref>, can be interpreted as expressing a close relationship between the notion of relative orientation in the sense of nc-geometry on the one hand, and the monodromy of Lefschetz fibrations as captured by the notion of a compatible spherical functor (Definition <ref>). Let be a Calabi-Yau category, and let F: → be a functor that admits left and right adjoints. Then F has a weak relative right Calabi-Yau structure if and only if it has the structure of a compatible spherical functor. The remaining sections are devoted to providing evidence for the existence of relative Calabi-Yau structures on compatible spherical functors arising from symplectic geometry and algebraic geometry. In Section <ref>, we prove the following theorem, giving one possible answer to Question <ref>, which was raised at the beginning of this introduction: Let w: X →ℂ be a admissible Landau-Ginzburg model (Definition <ref>) with generic fiber Y. Then the cap functor ∩: (X,w) →(Y) carries a natural relative Calabi-Yau structure (see Theorem <ref> for a precise statement). It follows from Theorem <ref> and the existence of an adjoint to ∩ given by the Orlov functor ∪, that the functor ∩: (X,w) →(Y) carries a natural compatible spherical structure with respect to the natural Calabi-Yau structure on (Y). Zachary Sylvan has independently arrived at a similar result in the context of partially wrapped Fukaya categories <cit.>. The next theorem is central to this paper. It is an answer to Question <ref>, a noncommutative analogue of Theorem <ref>, and a major step toward the construction of derived noncommutative version of Weinstein's category of Lagrangian spans. This theorem plays an important role in our discussion of Calabi-Yau structures on perverse Schobers in <ref> (see e.g. Remark <ref>). Suppose ,, and 𝒵 are categories with right d-Calabi-Yau structures, and suppose 𝒰→× and 𝒱→×𝒵 have relative Calabi-Yau structures. Let 𝒲=𝒰×_𝒱 be the pullback. Then 𝒲→×𝒵 has a natural relative Calabi-Yau structure. The derived moduli stacks of objects in d-Calabi-Yau categories carry (2-d)-shifted symplectic structures. Furthermore, relative Calabi-Yau structures on categories induce Lagrangian structures on the moduli space of objects. These statements provide the link between the results about relative Calabi-Yau structures proven in this paper, and the main results of <cit.>. The proofs of these statements are similar to the proof of the statement that the moduli of perfect complexes on a Calabi-Yau variety carries a shifted symplectic structures <cit.>, and will appear elsewhere. In this paper, we will content ourselves with proving the following two theorems about the existence of shifted symplectic structures and Lagrangian structures, which provide evidence for the general statement above, as explained in <ref>. Let X be a smooth d-dimensional variety, D⊂ X a divisor, and U=X\ D. Let α be a meromorphic section of _X(K_X) which is holomorphic nonvanishing on U. Then α induces a (2-d)-shifted symplectic structure on _c(U). Since _c(U) is a d-Calabi-Yau category, this is a special case of the existence of shifted symplectic structures on moduli spaces of objects in categories. Let D be a smooth divisor in a smooth and proper Calabi-Yau manifold X, and let i: D → X denote the inclusion. The map on moduli spaces of perfect complexes induced by the pushforward functor i_*: (D) →(X) carries a Lagrangian structure. §.§ Organization of the paper This document is organized in a modular fashion, and for the most part, the individual sections can be read independently of each other. Section <ref> is devoted to giving the definitions of compatible spherical functors (Definition <ref>) and weak relative Calabi-Yau structures (Definition <ref>), and proving Theorem <ref>, which says that these two notions are in fact essentially equivalent. Readers who have familiarized themselves with these two definitions and the statement of Theorem <ref>, can skip to other sections, which are all logically independent of each other. Section <ref> explains how compatible spherical functors can be glued together to produce Calabi-Yau structures on global sections of perverse Schobers. The key result of this section is Theorem <ref>, to which the entire subsection <ref> is devoted. <ref> explains how this theorem can be used to construct CY-structures on global sections of perverse Schobers (Remark <ref>). The rest of <ref> is devoted to illustrating how “categorical surgery” (the modification of a perverse Schober by changing the gluing data) can be carried out, and the effect this has on monodromy, through explicit computations. In particular, we show in Example <ref> how the Kronecker quiver with n arrows can be obtained via this procedure, starting from the mirror of ℙ^2. The sole purpose of Section <ref> is outline a proof of Theorem <ref>, which asserts the existence of a relative Calabi-Yau structure on the natural functor ∩: (X,w) →(Y), where Y is a generic smooth fiber of a good Picard-Lefschetz fibration w: X →ℂ. Section <ref> is devoted to shifted symplectic structures on certain derived stacks, and more specifically, to the proofs of Theorem <ref> and Theorem <ref>. The brief review of shifted symplectic structures at the beginning of this section is intended mainly to fix notation, and the reader is referred to <cit.> for the necessary background on the subject. The final section, <ref>, is devoted to directions for further research inspired by the current paper. It outlines an approach to a theory of derived hyperkähler geometry and categorical hyperkähler geometry using the twistor approach to hyperkähler manifolds. §.§ Notation and conventions Throughout the paper, we will use the following notation: - k is a fixed ground field. - , , ... denote small k-linear stable ∞-categories. - X, Y, ... usually denote derived stacks over k or symplectic manifolds. - _*() and _*() denote, respectively, the chain complexes computing the Hochschild and cyclic homologies of . - (X,Y) is the mapping space in an ∞-category. - (X,Y) is the internal hom in a cartesian closed ∞-category. - (x,y) is the (k-module) spectrum valued internal hom in a (k-linear) stable ∞-category. In particular, Ω^∞((x,y)) ≃(x,y). - (X) denotes the ∞-category of perfect complexes on a derived stack X, while (X) := (X, ) is the moduli stack of objects in (X). - Unless explicitly stated otherwise, category means k-linear stable ∞-category, and all constructions, such as limits and colimits, should be understood in the ∞-categorical sense. §.§ Acknowledgments We would like to thank Mohammed Abouzaid, Denis Auroux, Damien Calaque, Sheel Ganatra, Andrew Harder, Maxim Kontsevich, Tony Pantev, Nick Rozenblyum, Carlos Simpson, Zachary Sylvan, Hiro Lee Tanaka and Yiannis Vlassopoulos for useful conversations related to the subject matter of this paper. We are especially grateful to Denis Auroux, Mohammed Abouzaid and Sheel Ganatra for patiently answering our many questions about the A-model, and to Tony Pantev and Carlos Simpson for a careful reading of our outline of derived hyperkähler geometry, and for numerous invaluable comments and suggestions about the same. The authors were supported by Simons research grant, NSF DMS 150908, ERC Gemis, DMS1265230, DMS1201475 OISE1242272 PASI, Simons collaborative Grant - HMS and HSE Megagrant. The first named author is supported by a Simons Investigators Award, and is partially supported by the Laboratory of Mirror Symmetry NRU HSE, RF Government grant, ag. No.14.641.31.0001. § COMPATIBLE SPHERICAL FUNCTORS AND RELATIVE CALABI-YAU STRUCTURES The purpose of this section is twofold. First, we give the definitions of spherical functors compatible with a given Calabi-Yau structure (Definition <ref>) and weak relative right Calabi-Yau structures (Definition <ref>). Second, we prove that these two notions are equivalent; this is the content of Proposition <ref> and Theorem <ref>. We begin by recalling some relevant definitions. Our notation and the discussion that follows closely mirrors that in <cit.>, to which we refer the reader for further details. Let k be a field, and let be a k-linear ∞-category. * The k-linear ∞ category (, _k) = _(, _k) of functors from to the ∞-category of k-module spectra is called the category of left -modules, and denoted _. * The category of right modules is obtained by replacing with ^op in the definition above. * We will denote by ^e the ∞-category ⊗^op. The category of left ^e modules is also called the category of --bimodules. * We will write _(M,N) for _(, _k)(M,N) = __(M,N) when the meaning is clear from context. Let be a k-linear ∞-category. * The diagonal bimodule or identity bimodule _Δ is the functor _Δ: ^op⊗→_k defined by _Δ(x,y) := _(x,y) where the right hand side is the k-module spectrum of maps from x to y in the category . * The right dual of the identity bimodule is the functor ^∨: ⊗^op→_k defined by ^∨(x,y) := _(x,y)^∨ = _k(_(x,y),k) for all objects x and y in . A k-linear ∞-category is locally proper if for all objects x, y in , the mapping spectrum (x,y) is a compact object of _k, i.e., if (x,y) ∈(k). If is locally proper, then a functor representing the bimodule ^∨ of Definition <ref> is the same thing as a Serre functor for . Indeed, suppose that the bimodule ^∨ is representable by a functor S_: →, i.e., we have an equivalence of functors ^∨(-,-) ≃(-, S_(-)) Then by definition of ^∨, we have an equivalence (x,y)^∨≃(y, S_(x)) for all x, y in . Using the fact that (x,y) is a perfect module, we have (x,y) ≃(x,y)^∨∨≃(y, S_(x))^∨ showing that S_ is a Serre functor. Recall that the Hochschild homology complex _*() of a small k-linear ∞-category can be computed by the formula _*() ≃_Δ⊗_⊗^op_Δ Using this formula for _*(), and the standard tensor-hom adjunction, we have the following string of equivalences of chain complexes: _*()^∨≃_k( _Δ⊗_⊗^op_Δ, k) ≃_⊗^op(_Δ, _k(_Δ,k)) ≃_⊗^op(_Δ, ^∨) The bimodule _Δ is representable by the identity functor on . If ^∨ is representable, then it is representable by the Serre functor S_, so we have _⊗^op(_Δ, ^∨) ≃_(, )(𝕀_, S_). Thus we see that the data of map 𝕀_[d] → S_ is equivalent to the data of a map _*() → k[-d]. The Hochschild chain complex _*() of a category carries a natural S^1-action that is manifest in the definition of Hochschild homology via the cyclic bar complex, or via topological field theories <cit.>. An S^1-action on a chain complex over k is the same thing as the structure of a module over C^*(S^1,k) = k[B]/B^2; in other words a differential of homological degree 1. The circle action on _*() is given by Connes B-operator. In order to have a definition of Calabi-Yau structures that is adequate for producing oriented TFTs, and for producing shifted symplectic structures on the moduli of objects, it is necessary to incorporate the circle action into the definition of a Calabi-Yau structure. Recall the definition of a Calabi-Yau structure introduced in <cit.>, as described in <cit.>: Let be a locally proper category. Let Ξ: _*()^∨→_⊗^op(_Δ, ^∨) be the natural morphism described in Remark <ref>. * A weak d-dimensional right Calabi-Yau structure is map ϕ:_*()→ k[-d] such that Ξ(ϕ):_Δ[d]→^∨ is a weak equivalence. * A d-dimensional right Calabi-Yau structure on is a morphism ϕ̃:_*()_S^1→ k[-d] such that the composite map _*() →_*()_S^1→ k[-d] is a weak Calabi-Yau structure. Note that _k(_*()_S^1,k)≃_S^1(_*(),k), where on the right hand side k has the trivial S^1 structure. Thus we may consider ϕ̃ as an S^1-equivariant map _*()→ k[-d]. If is smooth and proper, we can identify ^∨ with the Serre functor S_, and a (weak) right Calabi-Yau ϕ structure is equivalent to the data of the equivalence of functors 𝕀_[d] ≃ S_ corresponding to Ξ(ϕ) under the identification _⊗^op(_Δ, ^∨) ≃_(, )(𝕀_, S_). Note that the natural map from _*() to the homotopy orbits HC_*() := _*()_S^1 has a highly non-trivial cofiber in general. In particular, the choosing a lift of a weak Calabi-Yau structure to a Calabi-Yau structure involves giving an “infinite tower of higher coherence data”. We can now formulate the definition of a relative Calabi-Yau structure, which is a noncommutative analogue of the notion of boundary structure introduced by Calaque <cit.>. Our definition follows a suggestion of Toën <cit.>. Let and be locally proper k-linear ∞-categories and let F → be an ∞-functor. * Suppose that is equipped with a weak right Calabi-Yau structure ϕ: _*() → k[-d] (Definition <ref>, (1)). A weak isotropy structure for F with respect to ϕ is a functor Δ^1 ×Δ^1 →(k) as follows: *5pc5pc_*() [d] [r]^_*(F) _*() [d]^ϕ 0 [r] k[-d] witnessing a nullhomotopy of ϕ∘_*(F). * Suppose that is equipped with a right Calabi-Yau structure ϕ̃: _*():= _*()_hS^1→ k[-d] (Definition <ref>, (2)). A (strong) isotropy structure for F with respect to ϕ is a 2-cell witnessing the commutativity of the following diagram: **5pc5pc_*() [d] [r]^_*(F) _*() [d]^ϕ̃ 0 [r] k[-d] Every strong isotropy structure has an underlying weak isotropy structure obtained by forgetting the S^1-equivariance data in the nullhomotopy. Or, to say the same thing differently, one obtains the underlying weak isotropy structure by concatenating the diagram (<ref>) with the natural diagram ***5pc5pc_*() [r]^_*(F)[d] _*() [d] _*() [r]__*(F) _*() to obtain a functor Δ^2 ×Δ^1 →(k) whose outer square is the required diagram Δ^1 ×Δ^1 →(k) of the form (<ref>). The vertical maps in (<ref>) are the canonical quotient maps _*(-) →_*(-)_hS^1. Let and be locally proper k-linear ∞-categories and let F → be an ∞-functor. For arbitrary objects x,y in , consider the diagram (<ref>). †5pc5pc_(x,y) ⊗_(y,x) [d]_𝕀[r]^F ⊗𝕀 _(Fx, Fy) ⊗_(y,x) [d]^𝕀⊗ F _(x,y) ⊗_(y,x) [d]_m_[r]^F ⊗ F _(Fx, Fy) ⊗_(Fy, Fx) [d]^m_ _(x,x) [r]^F[d]__x _(Fx,Fx) [d]^_Fx _*() [r]^_*(F) _*() Here the vertical maps in the middle square are given by multiplication in the category, and the vertical maps in the bottom square are the universal trace maps (boundary-bulk maps). The upper square can clearly be promoted to a commutative square, i.e., a homotopy (𝕀⊗ F) ∘ (F ⊗𝕀) ≃ (F ⊗ F) ∘ (𝕀⊗𝕀). Since F is an ∞-functor, for any choice of multiplication maps in and , the middle square is promoted to a commutative square in a canonical way. Similarly, functoriality of and the boundary-bulk map give us a 2-cell witnessing the commutativity of the bottom square. In fact, the diagram above can be promoted to a functor Δ^3 ×Δ^1 →(k). Now suppose that the functor F: → is equipped with a weak isotropy structure. Gluing the commutative square (<ref>) to the outer commutative square of the diagram (<ref>), one obtains a commutative square ††5pc_(x,y) ⊗_(y,x) [r]^F ⊗𝕀[d] _(Fx, Fy) ⊗_(y,x) [d] 0 [r] k[-d] By the adjunction between ⊗ and in _k, this gives rise to a commutative square †††5pc_(x,y) [r]^F[d] _(Fx, Fy) [d] 0 [r] _(y,x)^∨[-d] The right hand vertical map in (<ref>) sits in a diagram 5pc_(Fx.Fy) [r]^ϕ^♯_[dr] _(Fy, Fx)^∨[-d] [d]^F^∨ _(y.x)^∨[-d] realizing it as a composition of the k-linear dual of F and the Serre duality equivalence on . Let and be locally proper k-linear stable ∞-categories, and let F: → be an ∞-functor. * A weak relative Calabi-Yau structure of dimension d on F is a weak isotropy structure (<ref>) (Definition <ref>) for which the induced diagram (<ref>) given by Construction <ref> is a pushout-pullback square (in which case we will sat that the isotropy data satisfies the non-degeneracy condition). * A strong relative Calabi-Yau structure of dimension d on F is a strong isotropy structure (<ref>) (Definition <ref>) for which the underlying weak isotropy structure of Remark <ref> defines a weak relative Calabi-Yau structure. We now recall the definition of a spherical functor as given in <cit.>, and describe an additional structure that may exist on spherical functors. Let and be categories and F:→ a functor, with left and right adjoints F^* and F^!. Let ρ:F^!F→ T be the cofiber map of the unit η_F^!F: 𝕀_→ F^! ∘ F of the adjunction F ⊣ F^!, and let η_FF^* be the unit of the adjunction F^* ⊣ F. Then F is spherical if: * The cofiber of the unit T:=(𝕀_→ F^!∘ F) is an autoequivalence of . * The composition τ=(ρ F^*)∘ F^!η_FF^* 5pc F^![r]^F^!η_FF^*[dr]_τ F^!∘ F∘ F^* [d]^ρ F^* T∘ F^* is an isomorphism of functors. Furthermore, suppose has a Calabi-Yau structure ϕ of dimension d, and has a Serre functor S_. By general results on adjoint functors, we have F^!≃ S_F^*S_^-1. Let κ denote the isomorphism F^!≃ S_F^*S_^-1≃ S_[-d]F^*. Then a ϕ-compatible structure on F is an isomorphism α:T≃ S_[-d] such that the map F^! TF^* S_[-d]F^* is homotopic to κ. F^![r]@=[d] F^!FF^*[r] TF^*[d]^α F^* F^![r]^(0.40)∼ S_F^*S_^-1[r]^∼ S_[-d]F^* A spherical functor with ϕ-compatible structure will be called an ϕ-compatible spherical functor. When ϕ is understood we will say compatible spherical functor. The definition of spherical functors above coincides with that in <cit.>. The first compatibility condition appears in <cit.>, as part of their definition of spherical functors. If F is a compatible spherical functor, the composition F^!F→ T≃ S_[-d] is uniquely determined. By the proof of Theorem 2.13 in <cit.>, the morphism F^!F→ S_[-d] is the composition F^!F S_F^*F S_. Thus α is determined by a homotopy between the composition 𝕀_→ F^!F→ S_ and 0. Let be a Calabi-Yau structure of dimension d. For a general spherical functor F:→ we have T∘ F^*≃ F^!≃ S_X[-d]∘ F^* but we cannot a priori show T≃ S_[-d]. We note a condition that is often easier to check than (<ref>) above Let F:→ be a spherical functor, such that has a Calabi-Yau structure ϕ of dimension d, and has a Serre functor S_. Let α:T≃ S_[-d] be an isomorphism. Suppose that αρ=(S_[-d]ϵ)∘(κ F), where ϵ is the counit of F^*F. F^!F[r]^ρ[d]^κ F T[d]^α S_[-d]F^*F[r]^S_[-d]ϵ S_[-d] Then α is a compatible spherical structure for F. It remains to show that the diagram (<ref>) commutes. We have a diagram F^![r][d]^κ F^!FF^*[r][d]^κ FF^* TF^*[d]^α F^* S_[-d]F^*[r] S_[-d]F^*FF^*[r] S_[-d]F^* The left square clearly commutes and the right square commutes by our assumption. The morphism S_[-d]F^*→ S_[-d]F^* is the identity by the unit-counit relations, so taking the bottom path yields the map κ:F^!→ S_[-d]F^*. But the top map is precisely τ, so we get (α F^*)∘τ∼κ, as desired. We have three natural families of compatible spherical functors : * Let Y be a smooth Calabi-Yau variety of dimension d and i:D→ Y a smooth divisor. Then F=i_*:(D)→(Y) is spherical with F^*=i^* and F^!=i^!. For E∈(D), we have i^!i_*E≃ E⊕ E(D)[-1], where E(D)=E⊗_Y(D)|_D. The unit E→ i^!i_*E is inclusion into the first factor, so that TE=(E→ i^!i_*E)≃ E(D)[-1]≃ (E⊗ω_D[d-1])[-d]≃ S_X(E)[-d], where _X(D)≃ω_D by the adjunction formula. This gives an isomorphism T≃ S_X[-d]. Note that ρ:i^!i_*E≃ E⊕ E(D)[-1]→ TE is just the projection onto the second factor. We also have i^*i_*E≃ E(-D)[1]⊕ E, and the counit i^*i_*E→ E is also projection onto the second factor. Thus the diagram i^!i_*[r][d] (-)⊗ω_D[-1][d] S_X[-d]i^*i_*[r] S_X[-d] commutes and we have a compatible spherical structure. * Let X be a Fano variety of dimension d+1 and a:Y→ X a smooth anticanonical divisor. Then F=a^*:(X)→(Y) is spherical with F^*=a_!=a_*(-)(Y)[-1] and F^!=a_*. For E∈(X), a_*a^*E≃(E(-Y)→ E), where E(-Y)→ E is the defining section of Y, and the map E→ a_*a^*E is given by the diagram 0[r][d] E[d]^id E(-Y)[r] E . Then TE=(E→ a_*Ea^*E)≃ E(-Y)[1]≃ (E⊗ω_X[d+1])[-d]≃ S_X[-d](E), where we have again used the adjunction formula. Thus we have an isomorphism T≃ S_X[-d]. Note that the map a_*a^*E→ TE projects onto the E(-Y) term, sending the E term to 0. We also check that a_!a^*E≃(E→ E(Y)). Again, the map a_!a^*E→ E projects onto the E term, sending E(Y) to 0. Thus we have a commutative diagram a_*a^*[r][d] (-)⊗ω_X[1][d] S_X[-d]a_!a^*[r] S_X[-d] commutes and we have a compatible spherical structure. * Let w: X→ be a Lefschetz fibration with smooth fiber Y. Then F=∩:(X,w)→(Y) is spherical with F^!=∪ and F^*=∪∘ S^-1[1] <cit.>. We expect this is a compatible spherical functor. Let and be proper categories, let F:→ be a functor, and let ϕ̃:_*()_S^1→ k[-d] be a right Calabi-Yau structure. Recall (Definition <ref>) that a right relative Calabi-Yau structure on F is a homotopy F^*ϕ̃∼ 0 such that for all x,y∈, the sequence _(x,y)→_(Fx, Fy)≃_(Fy, Fx)^∨[-d]→_(y,x)^∨[-d] is a fiber sequence. Here the middle equivalence is from the equivalence _Δ(Fx, Fy)≃^∨(Fx, Fy)[-d]. Similarly, if ϕ:_*()→ k[-d] is a weak right Calabi-Yau structure, then a weak right relative Calabi-Yau structure on F is a homotopy F^*ϕ∼0 such that the above sequence is a fiber sequence. Here is what we can say about the existence of relative Calabi-Yau structures for the three families of functors described in Example <ref>: * Let Y be a smooth Calabi-Yau variety and i:D→ Y a smooth divisor. Then the map i_*:(D)→(Y) on derived stacks induced by the functor i_*:(D)→(Y) carries a Lagrangian structure. This suggests that the functor i_*:(D)→(Y) should carry a relative Calabi-Yau structure. * Let X be a Fano variety and a:Y→ X a smooth anticanonical divisor. Then the functor a^*:(X)→(Y) has a relative Calabi-Yau structure. <cit.> * Let w:X→ be a Lefschetz fibration with smooth fiber Y. Then the functor ∩:(X,w)→(Y) has a weak relative Calabi-Yau structure.By mirror symmetry (this example is mirror to the previous one), we expect this functor to carry a relative Calabi-Yau structure. At this point it is natural to conjecture that a compatible spherical functor will always carry a relative Calabi-Yau structure. We will show weaker results: Let and be smooth and proper categories and suppose F: → has a weak relative right Calabi-Yau structure. Suppose that F admits left and right adjoints. Then F is a compatible spherical functor Let F^* and F^! be left and right adjoints of F, respectively. Letting T=(𝕀_→ F^!∘ F), we give an isomorphism T≃ S_[-d]. For all x,y∈, we have (x,y)[r]@=[d] (x,F^!Fy)[r][d]^[@]∼ (x,Ty)[d] (x,y)[r] (Fx,Fy)[r] _(x,S_[-d]y), where the bottom row is the fiber sequence given by the relative Calabi-Yau structure on F. Since the first two vertical maps are isomorphisms, so is the third; this gives T≃ S_[-d]. For the second condition, we have a diagram (x,F^!y)[d][r]^∼ (Fx,y)[d][r]^∼ (y,Fx)^∨[-d][d][dr]^[@]∼ (x,F^!FF^*y)[r]^∼ (Fx,FF^*y)[r]^∼ (FF^*y,Fx)^∨[-d][r] (F^*y,x)^∨[-d][d]^[@]∼ (x,S_[-d]F^*y) Here the squares commute by naturality of adjunction and functoriality of the Serre functor, respectively, and the triangle commutes because F^*→ F^*FF^*→ F^* is the identity. Note that the map (x,F^!FF^*y)→(x,S_[-d]F^*y) of the bottom row is the one obtained from the composition F^!F→ T→ S_[-d] described above. Thus taking the bottom path is precisely the composition F^!→ F^!FF^*→ TF^* we want. The top path is an isomorphism, and in particular, is exactly the map κ. Thus our result is proven. We now turn to the converse: Let and be smooth and proper categories and suppose has a weak right Calabi-Yau structure ϕ:_*()→ k[-d]. Let F:→ be a compatible spherical functor. Then F has a weak relative Calabi-Yau structure. We have isomorphisms _*()^∨≃_^e(,^∨)≃__(𝕀_, S_) (<cit.> 2.8, 2.12). Under this identification the pullback F^*:_*()^∨→_*()^∨ sends α∈(, ^∨) to the composition → (F^e)^*_Δ (F^e)^*^∨≃ ((F^e)^*_Δ)^*→^∨ where F^e: ^e→^e (see Notation <ref>) is the functor induced by F. Identifying an endofunctor T:→ with the bimodule M_T(x,y)=_(x,Ty), this becomes the composition 𝕀_→ F^!F→ F^!S_ F≃ S_ F^*F→ S_. In our case, we have S_≃𝕀_[d] via ϕ, under which identification α=ϕ=𝕀_𝕀_ in degree -d. Furthermore the structure map F^!F→(𝕀_→ F^!F)≃ S_ is given by F^!F≃ S_ F^*F→ S_. (<cit.>, proof of Theorem 2.13). Thus the pullback F^*ϕ is given by 𝕀_→ F^!F→(𝕀_→ F^!F), which has a canonical homotopy to 0. For nondegeneracy, we have a diagram _(x,y)[r]@=[d] _(x,F^!Fy)[r][d]^(0.45)[@]∼ _(x,S_[-d]y)[d]^(0.45)[@]∼ _(x,y)[r] _(Fx,Fy)[r] _(y,x)^∨[-d]. Here the first square clearly commutes; it remains to show commutativity of the second. Expanding this square a little, we have: (x,F^!Fy)[r]^(0.4)[@]∼[dd]^(0.45)[@]∼ (x,S_[-d]F^*Fy)[r][d]^(0.45)[@]∼ (x,S_[-d]y)[d]^(0.45)[@]∼ (F^*Fy,x)^∨[-d][r][d]^(0.45)[@]∼ (y,x)^∨[-d]@=[d] (Fx,Fy)[r]^(0.45)[@]∼ (Fy,Fx)^∨[-d][r] (y,x)^∨[-d]. For the rectangle on the left, note that the top map is induced by κ by the compatibility condition, and going down, right, and back up is exactly the definition of κ. The two squares on the right are clear, so we have our commutativity and thus our result. As mentioned in <cit.>, the information of a (compatible) spherical functor is roughly analogous to the naive definition of a Calabi-Yau category S_≃𝕀_[d] and likely needs to be supplemented with higher homotopical data. How can we naturally describe the higher-homotopical data on a compatible spherical functor that is needed to promote it to a relative Calabi-Yau structure? Can every compatible spherical functor be promoted in this way? § CALABI-YAU STRUCTURES ON PERVERSE SCHOBERS The main purpose of this section is to prove Theorem <ref>, our main gluing theorem for Calabi-Yau structures, and to discuss its ramifications. The section is organized as follows. <ref> is entirely devoted to the statement and proof of Theorem <ref>. In <ref>, the we discuss the implications of this theorem for the study of Calabi-Yau structures on Fukaya categories with coefficients in a perverse Schober. We do not develop a general theory of CY-structures on perverse Schobers here; that will appear elsewhere. Instead, we focus on some examples to illustrate the geometric content of the gluing theorem. By reformulating the gluing construction in terms of spherical functors using Theorem <ref>, we have attempted to bring out the relationship of the result of <ref> to the monodromy of Lefschetz fibrations. Furthermore, we study the effect of modifying the spherical functors defining a Schober (“categorical surgery”), and show how to obtain Kronecker quivers by performing categorical surgery on the LG-mirror of ℂℙ^2 (Example <ref>). §.§ Calabi-Yau spans and the main gluing theorem In order to formulate the main gluing theorem for Calabi-Yau structures, we first introduce the categorical analogue of a cobordism between oriented manifolds, and some relevant notation. Let be a k-linear stable ∞-category equipped with a strong (resp. weak) d-Calabi-Yau structure (Definition <ref>) ϕ̃_̃: _*() → k[-d] (resp. ϕ_: _*() → k[-d]). When clear from the context, we will omit ϕ_ from the notation, and simply write for the Calabi-Yau category (, ϕ̃_̃) (resp. the weak Calabi-Yau category (, ϕ_)). We will write to denote the Calabi-Yau category (, - ϕ̃_̃) (resp. the weak Calabi-Yau category (, -ϕ_)). Here -ϕ_ is determined up to contractible ambiguity, and therefore, so is . Recall the definition of a relative Calabi-Yau structure (Definition <ref>), and the notation introduced in <ref>. An oriented cobordism from a d-Calabi-Yau category to a d-Calabi-Yau category is a functor →× equipped with a relative Calabi-Yau structure. Weak oriented cobordisms are defined similarly, by simply replacing all the Calabi-Yau structures by weak Calabi-Yau structures. The following theorem, which should be compared to Theorem 6.2 of <cit.> and Theorem 2.9 of <cit.>, states the oriented cobordisms can be glued together in a natural way. Let , and be k-linear stable ∞-categories equipped with right d-Calabi-Yau structures ϕ_, ϕ_ and ϕ_, respectively. Let →× and →× be oriented cobordisms (resp. weak oriented cobordisms) in the sense of Definition <ref>. Then the natural functor := ×_→× is equipped in a canonical way with the structure of an oriented cobordism (resp. weak oriented cobordism). The proof of this theorem is given later in the section, immediately after Construction <ref>, which gives a construction of isotropy data on →×. We begin first with some elementary lemmas that will be needed in the proof. The reader may wish to jump ahead to Construction <ref>, referring back to these lemmas when necessary. Let [r]^H[d]_I [d]^G [r]_F be a pullback square of ∞-categories. Then for each u, v in , there is a pullback square of spaces (u,v) [r] [d] (Hu, Hv) [d] (Iu, Iv) [r] (FIu, GHv) First, note that the commutativity of the first square is given by an equivalence G ∘ H ≃ F ∘ I, which in turn gives equivalences (FIu, FIv) ≃(FIu, GHv) ≃(GHu, GHv). The lower right map in the second square is given by the composite of the map (Iu, Iv) →(FIu, FIv) with the equivalence (FIu, FIv) →(FIu, GHv), and a similar construction gives the right vertical map. Let Δ^n denote the n-simplex, thought of as an ∞-category in the quasicategory model. Let be an ∞-category, and let c,d be objects in . Then there is a natural pullback square of ∞-categories _(c,d) [r] [d] (Δ^1, ) [d] Δ^0[r]^(c,d) × where the right vertical map is obtained by applying (-, ) to the natural map Δ^0∐Δ^0→Δ^1 of simplicial sets. Applying (Δ^1, -) and (Δ^0 ∐Δ^0, -) to our original pullback square of categories, we obtain two pullback squares of ∞-categories (Δ^1, ) [r] [d] (Δ^1, ) [d] (Δ^1, ) [r] (Δ^1, ) and ×[r] [d] ×[d] ×[r] × respectively. Furthermore, pulling back along the map Δ^0∐Δ^0→Δ^1 gives a map from the first square to the second. By the previous paragraph, the homotopy fiber of this map is equivalent to the square of mapping spaces in the statement of the lemma. Since limits commute with limits, this square is a pullback square. A stable ∞-category is locally bounded below if for each c, d in there exists n ∈ℤ such that (c,d) is n-connective, i.e., such that π_k (c,d) = 0 for k < n. Let [r]^H[d]_I [d]^G [r]_F be a pullback square in the ∞-category of stable ∞-categories and exact functors. Suppose each of the categories is locally bounded below (Definition <ref>). Then for each u, v in , there is a pullback square of spectra (u,v) [r] [d] (Hu, Hv) [d] (Iu, Iv) [r] (FIu, GHv) The idea is to reduce the statement to Lemma <ref> using the connectivity hypothesis. Since the categories are locally bounded below, there exists n ∈ℤ such that the n-fold suspensions of the spectra appearing in the statement of the lemma are all 1-connective. Since Σ^n (-,-) ≃(-, Σ^n - ), by replacing v with Σ^n v = v[n] we may assume that the morphism spectra appearing in the statement of the lemma are all 1-connective. The commutativity of this diagram of spectra is clear from functoriality. Since the category of spectra admits pullbacks, there is a pullback square ♮ K [r] [d] (Hu, Hv) [d] (Iu, Iv) [r] (FIu, GHv) Applying Lemma <ref> to the square (<ref>), and using the long exact sequence of on homotopy groups, we deduce that K is 0-connective. By the universal property of pullbacks, there is a natural morphism (u,v) → K of connective spectra. Our goal is to show that this map is an equivalence. Since Ω^∞ is conservative when restricted to connective spectra, it suffices to show that Ω^∞(u,v) →Ω^∞ K is an equivalence. The functor Ω^∞ from spectra to spaces is a right adjoint, and therefore preserves all limits. Applying Ω^∞ to the square (<ref>), we obtain a pullback square in the ∞-category of spaces Ω^∞ K [r] [d] (Hu, Hv) [d] (Iu, Iv) [r] (FIu, GHv) By Lemma <ref>, (u,v) ≃Ω^∞(u,v) is also characterized as a pullback of the same maps. By the universal property of pullbacks, it follows that the map Ω^∞(u,v) →Ω^∞ K is an equivalence, completing the proof. It is well known that the data of a pushout-pullback square in an abelian category is equivalent to the data of an exact sequence. The following lemma is a homotopical analogue of this fact, with abelian categories being replaced by stable ∞-categories, and exact sequences by fiber sequences. Let A, B, C and D be objects in a stable ∞-category , and let B ⊕ C be a biproduct (product and coproduct) of B and C, which exists since is stable. Let f ∈(B,D), g ∈(C,D), h ∈(A,B), i ∈(A,C), k ∈(B ⊕ C, D) and l ∈(A, B ⊕ C). Suppose k maps to the connected component of (f,-g) under the equivalence (B ⊕ C, D) ≃(B,D) ×(C,D) induced by the universal property of B ⊕ C, and l maps to the component of (h,i) under the equivalence (A, B ⊕ C) ≃(A,B) ×(A,C) . * The following spaces are homotopy equivalent: * The space of paths from f ∘ h to g ∘ i in (A,D), i.e., the space of 2-cells witnessing the commutativity of the square ♭ A [r]^h[d]_i B [d]^f C [r]_g D * The space of paths from k ∘ l to 0 in (A,D), i.e., the space of 2-cells witnessing the commutativity of the diagram ♭♭ A[r]^l[d] B ⊕ C [d]^k 0 [r] D * A commutative square of the form (<ref>) is bicartesian if and only if its image of the form (<ref>) under the homotopy equivalence of (1) is bicartesian. Since is stable, it admits a natural spectral enrichment. In particular, the mapping spaces are grouplike infinite loop spaces; for each pair of objects E, F in , there is a mapping spectrum (E,F), and (E,F) ≃Ω^∞(E,F). Thus, for any e ∈(E,F), there is a map _e: (E,F) →(E,F), well defined up to homotopy, such that [_e(e')] = [e] + [e'] in π_0(E,F); and furthermore, _e is a homotopy equivalence with a homotopy inverse given by _-e, where -e ∈(E,F) is an element such that _e(-e) is a zero map. Note that the statement of (1) is independent of the choice of composites f ∘ h and g ∘ i, since it depends only on the connected components [f ∘ h] and [g ∘ i], and composition is well defined up to homotopy. Now we turn to the proof of (1). Let -g be an additive inverse for g and consider the map _(-g) ∘ i : (A,D) →(A,D). By the universal property of the biproduct B ⊕ C, we have [_(-g) ∘ i(f ∘ h)] = [k ∘ l] for any composite k ∘ l of k and l. One the other hand, using the fact that composition of morphisms extends to a map of spectra (A,C) ⊗(C,D) →(A,D), we see that [_(-g) ∘ i(g ∘ i)] = [_-(g ∘ i)(g ∘ i)] = [0]. This, _-g ∘ i: (A,D) →(A,D) is a homotopy equivalence that carries the connected component of g ∘ i to the connected component of zero maps, and carries the connected component of f ∘ h to the connected component of k ∘ l. Passing to path spaces, this proves (1). Part (2) of the lemma follows immediately from part (1), and the universal property of bicartesian squares. Let , and be locally proper k-linear stable ∞-categories, and let F = (F_, F_) : →× be a functor. Let d ∈ℕ and let ϕ_: _*() → k[-d] and ϕ_: _*() → k[-d] be weak right Calabi-Yau structures (Definition <ref>). Then there is a natural homotopy equivalence between the following spaces: * The space of weak isotropy structures (see Definition <ref>) on F with respect to the weak right Calabi-Yau structure (- ϕ_, ϕ_) on ×. * The space of paths from ϕ_∘_*(F_) to ϕ_∘_*(F_) in _(k)(_*() , k[-d]), i.e., the space of 2-cells witnessing the commutativity of the following square 5pc_*() [r]^_*(F_)[d]__*(F_) _*() [d]^ϕ_ _*() [r]_ϕ_ k[-d] The lemma follows immediately from part (1) of Lemma <ref>. Let , and be locally proper k-linear stable ∞-categories, and let F = (F_, F_) : →× be a functor. Let d ∈ℕ and let ϕ_: _*() → k[-d] and ϕ_: _*() → k[-d] be weak right Calabi-Yau structures (Definition <ref>). Then there is a natural homotopy equivalence between the following spaces: * The space of isotropy structures (Definition <ref>) on F with respect to the right Calabi-Yau structure (- ϕ̃_̃, ϕ̃_̃) on ×. * The space of paths from ϕ̃_̃∘_*(F_) to ϕ̃_̃∘_*(F_) in _(k)(_*() , k[-d]), i.e., the space of 2-cells witnessing the commutativity of the following square 5pc_*() [r]^_*(F_)[d]__*(F_) _*() [d]^ϕ_ _*() [r]_ϕ_ k[-d] The lemma follows immediately from part (1) of Lemma <ref>. All the categories appearing in the following construction are k-linear stable ∞-categories. Let d ∈ℕ, and suppose that the categories , and are equipped with weak right d-Calabi-Yau structures ϕ_, ϕ_ and ϕ_. Let F = (F', F”): →× and G = (G', G”): →× be functors equipped with weak isotropy data with respect to the Calabi-Yau structures (-ϕ_, ϕ_) on × and (-ϕ_, ϕ_) on ×. Let := ×_, and consider the diagram: ♯_*() [r] [d] _*() [r] [d] _*() [d]^ϕ_ _*() [r] [d] _*() [r]_ϕ_[d]^ϕ_ k[-d] [d]^𝕀 _*() [r]_ϕ_ k[-d] [r]_𝕀 k[-d] All the unmarked arrows in this diagram are given by applying the functor _* to the natural diagram of categories [r]^H”[d]_H' [r]^G”[d]^G' [r]_F”[d]_F' The upper left square in (<ref>) is equipped with the structure of a homotopy square by virtue of the functoriality of _*. The lower right square is equipped with the trivial homotopy commutative structure. By virtue of Lemma <ref>, the lower left square and the upper right square are equipped with commutative structures induced by the isotropy data on the functors F: →× and G: →×, respectively. Since each of the four adjacent squares in the diagram (<ref>) is equipped with a homotopy commutative structure, it follows that the outer square is equipped with a homotopy commutative structure. Applying Lemma <ref> again, we deduce that commutativity data for the outer square in (<ref>) equips the natural functor →× with a weak isotropy structure, with respect to the Calabi-Yau structure (-ϕ_, ϕ_). If each of the functors F: →× and G: →× is equipped with a strong isotropy structure, then the argument of the paragraph above, applied to the analogue of the diagram (<ref>) with _* replaced by _*, constructs a strong isotropy structure on the functor →×. Let →× and →× be as in the statement of the theorem, and let ≃×_. Construction <ref> endows the natural functor (see diagram (<ref>)) →× with isotropy data. It remains to show that this isotropy data satisfies the nondegeneracy condition of Definition <ref>. To this end, consider the following diagram ♮♮_(u,v) [r] [d] _(H”u, H”v) [r] [d] _(G”H”u, G”H”v) [d] _(H'u, H'v) [r] [d] _(F”H'u, G'H”v) [r] [d] _(H”v, H”u)^∨[d] [d] _(F'H'u, F'H'v) [r] _(H'v, H'u)^∨[d] [r] _(v,u)^∨[d] The upper left square of this diagram is the commutative square of Lemma <ref> (see also Lemma <ref>). Recall that the central term in this diagram can be described in several ways: we have natural equivalences _(G'H”u, G'H”v) ≃_(F”H'u, G'H”v) ≃_(F”H'u, F”H'v). The lower vertical map in the central column is the composite (F”)^∨[d] ∘Ξ(ϕ_) of the equivalence Ξ(ϕ_): _(F”H'u, F”H'v) →_(F”H'v, F”H'u)^∨[d] induced by the Calabi-Yau structure on with the map (F”)^∨[d]: _(F”H'v, F”H'u)^∨[d] →_(H'v, H'u)^∨[d]. Similar remarks apply for the right hand map in the central row, the left hand map in the bottom row, and the upper map in the rightmost column. The reader is referred to Construction <ref>, and Diagram (<ref>) therein, for a discussion of this construction. Our categories are all locally proper, and therefore locally bounded below. Since ≃×_, the hypotheses of Lemma <ref> are satisfied, and we conclude that the square in the upper left hand corner is a bicartesian square of k-module spectra. Using the identification _(F”H'u, F”H'v) ≃_(F”H'v, F”H'u)^∨[d] induced by the Calabi-Yau structure on , the square in the lower right hand corner is identified with the k-linear dual, shifted by d, of the square in the upper left corner. Since (-)^∨[d] is an exact functor, it follows that the lower right square is a pullback square as well. Now let us consider the square in the lower left corner. Since the functor F: →× is equipped with a relative Calabi-Yau structure, the underlying isotropy structure gives rise to a commutative diagram _(H'u, H'v) [r] [d] _(F'H'u, F'H'v) ⊕_(F”H'u, F”H'v) [d] 0 [r] _(H'v, H'u)^∨[d] by Construction <ref>. By Lemma <ref>, this implies that the lower left square in Diagram (<ref>) is homotopy commutative. Since the isotropy structure on F defines a Calabi-Yau structure, it is non-degenerate (Definition <ref>), which means that (<ref>) is in fact a pullback square. Applying Lemma <ref> once again, we conclude that the lower left square in (<ref>) is a pullback square. The same argument shows that the square in the upper right corner of Diagram (<ref>) is a pullback square. Thus we have shown that all the four adjacent squares in Diagram (<ref>) are pullback-pushout squares of k-module spectra. It follows that the outer square of this diagram is also a pullback-pushout square. Applying Lemma <ref> to the outer square, we conclude that the corresponding diagram _(H'u, H'v) [r] [d] _(F'H'u, F'H'v) ⊕_(G”H”u, G”H”v) [d] 0 [r] _(v, u)^∨[d] is a pullback square. An elementary, albeit somewhat tedious, diagram-chase (whose details we leave to the reader) shows that this square is in fact the square obtained by applying Construction <ref> to the “glued” isotropy structure on →× obtained by Construction <ref>. Thus we have proven that the glued isotropy structure on →× is non-degenerate, which is what we set out to do. §.§ CY-structures and surgery on Schobers Theorem <ref> has the following familiar analogue in topology. Let M_1,M_2 be oriented manifolds with boundary, and partition the boundary of each one as M_1= N_1∐ N'_12 and M_2=N_12∐ N_2. Giving each boundary component the induced orientation, let us further suppose that N'_12≃ N_12 via an orientation-reversing homeomorphism; we write M_1=N_1∐N_12. Then we can glue M_1 and M_2 along N_12 to get a new manifold M=M_1∐_N_12M_2 with boundary N_1∐ N_2 (see Figure <ref>). We can relate this to the categorical case as follows. Letting (M_i) be the category of dg-local systems on M_i, we expect a relative Calabi-Yau structure on the pullback (M_i)→( M_i). Furthermore, we have isomorphisms ( M_1)≃(N_1∐N_12)≃(N_1)×(N_12) and similarly ( M2)≃(N_12)×(N_2). We also have (M)≃(M_1)×_(N_12)(M_2). Then the theorem gives us that (M)→(N_1)×(N_2) is compatible spherical, which corresponds to the fact that M has boundary N_1∐ N_2. Now we consider the categorical generalization of the following situation from symplectic geometry. Let w:X→ be a Landau-Ginzburg model with smooth fiber Y and compact critical locus. Let U_1, U_2 be bounded open sets of such that U_1∪ U_2 contains all critical values of w and U_1∩ U_2=∅. Further assume (for i=1,2) that the maps w|_U_i:X|_U_i→ U_i can be extended to w_i:X_i→ for some space X_i and map w_i such that w_i is a fibre bundle above an open set containing -U_i. For example, if each U_i is a convex region this latter condition is certainly possible, and in particular if w is a Picard-Lefschetz fibration we can partition the (isolated) critical values between two such regions. In this situation we expect the structure of X to be related to the structure of the U_i, and on a categorical level we expect to obtain (X,w) and ∩:(X,w)⇄(Y):∪ from (X_i,w_i) and ∩_i:(X_i,w_i)⇄(Y):∪_i. More generally, suppose has a weak right d-Calabi-Yau structure and F_i:_i→ is a spherical functor for i=1,2. Viewing the case of Fukaya-Seidel categories as the prototypical spherical functor, we seek to glue _1 and _2 in a similar way to obtain a composite and spherical functor F:→. We will represent (X_i,w_i) or its objects by the diagram [dashed] (0,0) circle [radius=0.5]; (0.5,0) – (2.5,0); at (0,0) U_i; . Here we think of the exiting line as having a fiber (Y), and the application of ∩_i on an object L∈(X_i,w_i) as looking at the intersection of a L with a fiber above the line. In general, for each outgoing path, we can construct a functor in this way. Analogously we will represent _i by [dashed] (0,0) circle [radius=0.5]; (0.5,0) – (2.5,0); at (0,0) _i; . In the symplectic case, ∩_i has a right (left) adjoint ∪_i (∪_i^L) given by parallel transport of a Lagrangian on a loop from +∞ to +∞ clockwise (counter-clockwise) around U_i. We will represent these by the pictures [dashed] (0,0) circle [radius=0.5]; [->] (2.5,.1) to (1,.1) to [out=180,in=0] (0,.6); (0,-.6) to [out=0,in=180] (1,-.1) to (2.5,-.1); [->] (0,.6) arc (90:270:.6); [fill] (2,.1) circle [radius=0.05]; at (0,0) U_i; and [dashed] (0,0) circle [radius=0.5]; (2.5,.1) to (1,.1) to [out=180,in=0] (0,.6); [<-] (0,-.6) to [out=0,in=180] (1,-.1) to (2.5,-.1); [<-] (0,.6) arc (90:270:.6); [fill] (2,-.1) circle [radius=0.05]; at (0,0) U_i; respectively, and we will draw similar pictures for the adjoints F^!, F^*:_i→_i. Using Theorem <ref>, the functor _1×__2→ 0 is compatible spherical. In the Landau-Ginzburg situation above, this corresponds to the weak Calabi-Yau structure on (X), the Fukaya category of compact Lagrangians, which is not exactly what we want (Figure <ref>). Instead, we model (X,w) with the following construction. Let _±(X_2,w_2) be the two-sided Fukaya-Seidel category of (X_2,w_2) consisting of Lagrangians in X_2 which, outside of some compact set K⊂, consist of fibrewise Lagrangians parallel transported along rays to +∞ or -∞. In this scenario, we have two restriction functors ∩_-, ∩_+:_±(X_2,w_2)→(Y), which are the fibers at -∞ and +∞. We draw this as in Figure <ref>, with ∩_- and ∩_+ corresponding to the left and right path, respectively. Then we can construct the fiber product (X,U_1,U_2,w)=(X_1,w_1)×_∩_1,(Y),∩_-_±(X_2,w_2), which we expect to coincide with (X,w). Furthermore, we have a map ∩':(X,U_1,U_2,w)→_±(X_2,w_2)(Y) which we expect to agree with ∩:(X,w)→(Y). See Figure <ref>; the outgoing path is ∩'. Whether we actually have an equivalence (X,U_1,U_2,w)≃(X,w) seems to depend on a Mayer-Vietoris type result for Fukaya-Seidel categories. This should be more straightforward to verify if (X,w) is generated by thimbles. In the case of general spherical functors, we wish to construct a category _2,± and two functors F_±:_2,±→ as above. To do this, we note that Figure <ref> is equivalent “up to homotopy” to Figure <ref>, and the latter has an existing categorical interpretation, which we will take as our _2,±. Consider (A_2)⊗. There are three natural functors P_1,P_2,P_3:(A_2)→(A_1)≃ D^b(k) sending a representation E_1→ E_2 to E_1, E_2, and (E_1→ E_2) respectively. These induce functors P_i⊗𝕀:(A_2)⊗→(A_1)⊗≃. We represent this as in Figure <ref>; the left, top, and right paths correspond to P_1,P_2, and P_3 respectively. Then we set _2,±=_2×_F_2, , P_2⊗𝕀(A_2)⊗ and maps F_-,F_+ by compositions F_- :_2,±→(A_2)⊗_2 F_+ :_2,±→(A_2)⊗_2. See Figure <ref> again. The left and right paths out are the functors F_- and F_+, respectively. Now we wish to show that F_±=(F_+,F_-):_2,±→× is spherical. The functor P=(P_1,P_2,P_3):(A_2)→(A_1)^×3 is compatible spherical. For i=1,2,3 define Q_i:(A_1)→(A_2) by Q_1x =(x→ 0) Q_2x =(x x) Q_3x =(0→ x). It is easy to check that P_i⊣ Q_i for all i, Q_i+1⊣ P_i for i=1,2, and Q_1[-1]⊣ P_3. Then we have P^!(x,y,z)=Q_1x⊕ Q_2y⊕ Q_3z and P^*(x,y,z)=Q_2x⊕ Q_3 y⊕ Q_1z[-1]. Furthermore, for (xy)∈(A_2), we have P^!P(x→ y)=(x⊕ y→ y⊕(f)) and (𝕀→ P^!P)(x→ y)=(y→(f))=S_(A_2)(x→ y). Let have a weak right d-Calabi-Yau structure and a weak right d'-Calabi-Yau structure. Let F:→ be a compatible spherical functor. Then F⊗𝕀_:⊗→⊗ is compatible spherical. First, we check that ⊗ has a weak right (d+d')-Calabi-Yau structure. Using the isomorphisms 𝕀_[d]≃ S_ and 𝕀_[d']≃ S_ we have, for y⊗ z,y'⊗ z'∈⊗ (y⊗ z,y'⊗ z'[d+d']) ≃(y,y'[d])⊗(z,z'[d']) ≃(y',y)^∨⊗(z',z)^∨ ≃(y'⊗ z',y⊗ z)^∨, yielding an isomorphism 𝕀_⊗[d+d']≃ S_⊗. This is equivalent to the desired weak right (d+d')-Calabi-Yau structure. Similarly it is easy to check that S_⊗≃ S_⊗𝕀_[d']. Now, the left and right adjoints of F⊗𝕀 are F^*⊗𝕀 and F^!⊗𝕀 respectively. We have (𝕀_⊗𝕀_→ (F^!⊗𝕀_)∘(F⊗𝕀_)) ≃(𝕀_→ F^!F)⊗𝕀_ ≃ S_⊗[-(d+d')]. It is easy to check that under this identification the map F^!⊗𝕀_→ (F^!⊗𝕀_)∘ (F⊗𝕀_)∘ (F^*⊗𝕀_)→ S_⊗[-(d+d')]∘ (F^*⊗𝕀_) coincides with κ_F⊗𝕀_≃κ_F⊗𝕀_𝕀_. The functor F_±=(F_+,F_-):_2,±→× is compatible spherical. By the previous two lemmas, the functor (P_1,P_2,P_3)⊗𝕀_:(A_2)⊗→^×3 is compatible spherical. By assumption, F_2:_2→ is compatible spherical. Applying Theorem <ref>, we see that F_± is compatible spherical. 5pc_2,±[r][d] (A_2)⊗[r]^-(P_1,P_3)⊗𝕀_[d]^P_2⊗𝕀_ × _2[r][d] 0 With this in hand, we can construct _=_1×_F_1,,F_-_2,±. Applying Theorem <ref> once more, we see that the composition F_:_→_2,± is compatible spherical, assuming it has both adjoints. These adjoints can be shown to exist by general results, but we will construct them explicitly shortly. Before this, we note that _=_1×_F_1,,P_1⊗𝕀_(A_2)⊗×_P_1⊗𝕀_,,F_2_2 and can be equivalently described as follows: * The objects of _ are triples (x,y,f) with x∈_1, y∈_2, and f:F_1x→ F_2y. * For (x,y,f), (x',y',f'), ((x,y,f),(x',y',f')) is the space of triples (g,h,α), with g:x→ x', h:y→ y', and α:f'∘ F_1g∼ F_2h∘ f. With this description, F_ sends (x,y,f) to (f). Pictures of _ are shown in Figure <ref>; the left diagram is in line with previous ones, but the (equivalent) right diagram stresses the symmetry between _1 and _2 and we will generally prefer it. As a bit of notation, for i=1,2, we let T_i'=(F_iF_i^!→𝕀_); in the case of the Picard-Lefschetz fibration T_i'[1] is the counterclockwise monodromy on (Y) around U_i. Note that T_i' is an autoequivalence, and there is a natural choice of inverse (T_1'x)^-1=(𝕀_→ F_iF_i^*); in particular, we have a morphism F_iF_i^*→ (T_1'x)^-1 (<cit.>, Theorem 1.1). The right adjoint F_^! of F_ is given by F_^!x=(F_1^!T_2'x,F_2^!x,g_x), where g_x=ξ∘η_1,T_2'x is the morphism F_1F_1^!T_2'x T_2'x F_2F_2^!x, where the η_1,T_2'x is the counit of F_1⊣ F_1^! and ξ comes from the fiber sequence T_2'→ F_2F_2^!→𝕀_. Similarly, the left adjoint F_^* is given by F_^*x=(F_1^*x,F_2^*(T_1')^-1x, h_x), where h_x is the morphism F_1F_1^*x→ (T_1')^-1x→ F_2F_2^*(T_1')^-1x; here the first map is the previously mentioned F_iF_i^*→ (T_1'x)^-1 and the second is the unit of F_2^*⊣ F_2. We show the case of F_^!; the proof for F_^* is similar. Fix x∈ and (y,z,f)∈_. We have _(F_(y,z,f),x)≃_((f),x). The latter of these is equivalent to the space of (h,α) in the diagram F_1y[r]^f[rd]_0^<-2>α F_2z[d]^h x That is, h:F_2z→ x and α:h∘ f∼0. Using the adjunction (F_2z,x)≃(z,F_2^!x), the former is equivalent to the choice of h̃:z→ F_2^!x. Further, we have a natural homotopy F_2z[dl]_F_2h̃[d]^h^<-2> F_2F_2^!x[r] x . We then have a diagram F_1y[r][d]_F_2h̃∘ f[dr]^(.7)0 F_2z[d][dl]| F_2F_2^!x[r]_η_2,x x ; filling in the upper left triangle with a 2-morphism witnessing the composition F_2h̃∘ f, we are left with a horn we can fill in with a 3-simplex, and in particular we get a homotopy β:η_x∘ F_2h∘ f∼ 0; further, it is clear that given h, the choice of α and β are equivalent. Finally, the choice of such a β is equivalent to a map k:F_1y→(η_2,x)=T_2'x plus a homotopy γ:ξ∘ k∼ F_2h̃∘ f, where ξ:T_2'x→ F_2F_2'x is the natural map. This k and γ are then equivalent to k̃:y→ F_1^!T_2'x and a homotopy γ:ξ∘η_1,T_2'x∘k̃∼ F_2h̃∘ f F_1F_1^!T_2'x[d]_η_1,T_2'x F_1y[d]^F_2h̃∘ f[l]_k̃^<6>γ T_2'x[r]^ξ F_2F_2'x[r] x Taking g_x=ξ∘η_1,T_2'x, this γ is the same as commutativity data for the square F_1y[r]^f[d]^k̃ F_2z[d]^h̃ F_1F_1^!T_2'x[r]^g_x F_2F_2^!x . Thus we have _(F_(y,z,f),x)≃__((y,z,f),F_^!x). The functor F_:_→ is compatible spherical. We may represent the situation of Lemma <ref> with Figure <ref>. The action of F_^! transports an object x counter-clockwise around both _1 and _2. The counter-clockwise loop around _2 gives us the F_2^!x∈_2 and leaves us with T_2'x coming out the other end. Transporting T_2'x around _1 gives us the F_1^!T_2'x of the statement. Further, the outgoing line of this loop is then T_1'T_2'x, so that the total twist is given by T_1'T_2' (see Lemma <ref> below). With these in hand, we can compute the total twist T'_ and Serre functor S_=(𝕀__→ F_^!F_)[d]. For the former, we have T'_=T'_1T'_2[1] Consider the diagram 5pc F_1F_1^!T_2'[r][d] T_2'[r][d] T_1'T_2'[1][d] F_1F_1^!T_2'[r][d] F_2F_2^![r][d] F_F_^![d] 0[r] 𝕀_[r] 𝕀_ All rows and the two left columns are fiber sequences, so the right column is as well. Let us rewrite this result as T'_[1]=(T'_1[1])(T'_2[1]) and recall in the case of a Landau-Ginzburg model that T'_i[1] is the monodromy around U_i. This result then says that the total monodromy is obtained by composing the monodromy around each U_i. The Serre functor cannot be written in such a simple manner, but we can simplify it a little. Let S_i denote the Serre functor of _i. Let (x,y,f)∈_. Then S_(x,y,f)≃(c_1,c_2,k)[d], where c_1=(x→ F_1^!T_2'(f)), c_2=(y,F_2^!(f)), and k is induced by f and g_(f). x[r][d]^f F_1^!T_2'(f)[r][d]^g_(f) c_1@–>[d]^k y[r] F_2^!(f)[r] c_2 Furthermore, we have c_1≃(F_1^!F_2c_2[-1]→ S_1x[-d]) and c_2≃(F_2^!F_1x→ S_2y[-d]), where the maps will be defined below. As mentioned above, since F_ is compatible spherical, we have S_≃(𝕀__→ F_^!F_)[d], from which the first statement follows immediately. For the second statement, consider the diagram 0[r][d] F_2^!F_1x@=[r][d] F_2^!F_1x[d] y[r]@=[d] F_2^!F_2y[r][d] S_2[-d]y[d] y[r] F_2^!(f)[r] c_2 . All three rows and the first two columns are fiber sequences, so the last column is as well. Thus we have c_2≃(F_2^!F_1x→ S_2y[-d]). For c_1, we consider F_1x[r][d] T'_2(f)[r][d] F_2c_2@=[d] F_2y[r][d] F_2F^!_2(f)[r][d] F_2c_2[d] (f)@=[r] (f)[r] 0. Since the columns and bottom two rows are fiber sequences, we get a fiber sequence F_1x→ T'_2(f)→ F_2c_2. Then consider x[r]@=[d] F_1^!F_1x[r][d] S_1[-d]x[d] x[r][d] F_1^!T'_2(f)[r][d] c_1[d] 0[r] F_1^!F_2c_2@=[r] F_1^!F_2c_2. Once again, the rows and first two columns are fiber sequences, so the third column is as well; rotating this gives the isomorphism c_1≃(F_2^!F_2c_2[-1]→ S_1x[-d]). The above results may be generalized as follows. We note that (A_n)≃(A_n-1)×_(A_1)(A_2) By inductively applying Theorem <ref> we may construct a map (A_n)→(A_1)^×(n+1) with a compatible spherical structure. Furthermore, if has a weak Calabi-Yau structure, then applying Lemma <ref>, we have a compatible spherical structure on (A_n)⊗→^×(n+1) Now fix a ribbon graph and let us denote some number e of leaves (1-valent vertices) as belonging to outgoing edges; the other vertices will be internal vertices. Assume that any edge is incident to at least one internal vertex, and that there are no loops. Let us fix a “generic” category , and, for each internal leaf, a category _i equipped with a spherical functor F_i:_i→ defining a perverse Schober 𝔛 on a thickening of . Then the construction of <cit.> allows us to associate to a sheaf of categories 𝔛_ on (see also the introduction of this paper). Considering Γ(,𝔛_), we have a structure map Γ(,𝔛_)→ for each outgoing edge of . We then obtain a map F_:Γ(,𝔛_)→^× e which we may explicitly describe as follows: * If has a single internal vertex i, and this is a leaf corresponding to a spherical functor F_i:_i →, then e=1, Γ(,𝔛_G)≃, and F_= F_i. * If has a single internal vertex, and this is an n-valent vertex with n>1, then Γ(,𝔛_)≃(A_n-1)⊗ and F_:(A_n-1)⊗→^× n is the map described above. * If has more than one internal vertex, fix some internal n-valent vertex v and construct a graph ' by removing v and any outgoing edge incident to v; for any non-outgoing edge incident to v and some v' we add a new outgoing edge between v' and a new leaf. Let r be the number of such edges. Let ” be the graph with a single n-valent vertex; if n=1 we assign to this vertex the functor F_v assigned to v. Then Γ(,𝔛_)≃Γ(',𝔛_')×_^rΓ(”,𝔛_”). We inductively have maps F_':Γ(',𝔛_')→^e' and F_”:Γ(”,𝔛_”)→^e”. Analogously to the construction of the map _→ we may then construct F_:Γ(,𝔛_)→^e. If, furthermore, the category is equipped with a weak right Calabi-Yau structure and the functors F_i: _i→ are equipped with weak (resp. strong) relative Calabi-Yau structures, then the functor F_ also has a weak (resp. strong) relative Calabi-Yau structure. In the first case, this is by assumption. In the second case, this is the structure on (A_n-1)⊗→^× n described above. In the third, we may apply Theorem <ref>. We further expect that one can find formulas for adjoints similar to those of Lemma <ref>, so that F_ has a compatible spherical structure. We do not show this here, although we note that in the simple case that is a binary tree (i.e. there are no cycles, exactly one outgoing edge, and all internal vertices are uni- or trivalent) we may simply inductively apply Lemma <ref>. Let us describe a simple representation of the category _. For i=1,2 we define maps J_i:_i→_ by J_1x =(x[-1],0,0) J_2y =(0,y,0). In the case of a Picard-Lefschetz fibration, these correspond to the inclusions (X_i,w_i)→(X,w) of thimbles ending in critical points of U_i. It is easy to check the following: With _i,,_,F_i,F_,J_i as above, we have * F_J_i=F_i. * For i=1,2, J_i is full and faithful. * For x∈_1, y∈_2, we have __(J_1x,J_2y)≃_(F_1x,F_2y). * _ has a semi-orthogonal decomposition _=⟨ J_1_1,J_2_2⟩. We will now construct Kronecker quivers as Fukaya-Seidel categories using this method. Let w':X'→ be a mirror to ^2, so that w' has three critical points corresponding to ,(1),(2)∈ D^b(^2), and let E be a smooth fiber. Take an open disc U containing two of these points, z_1 and z_2, and extend X'|_U to a Picard-Lefschetz fibration w:X→ so that its only critical points are z_1 and z_2. Then (X,w) is K_3, the Kronecker quiver with three arrows. More specifically, (X,w) is generated by thimbles L_i ending at z_i such that ∩ L_1 and ∩ L_2 are Lagrangians in E intersecting in three points. Let U_i be a small disc around z_i, and let (X_i,w_i) be extensions of the fibration over U_i to all of . Then we have (X_i,w_i)=(A_1)=D^b(k), with ∩_i:(X_i,w_i)→(E) sending the generator to ∩ L_i. Taking _i=(X_i,w_i) and F_i=∩_i, it is clear from Lemma <ref> that _ is again K_3. We can alter this example with a surgery. Fix a symplectomorphism R of E, say a Dehn twist, and a cover U=V_1∪ V_2 with z_i∈ V_i, z_i∉ V_j for i≠ j. Let X_i=X|_V_i. Then we have X|_U≃ X_1∐ X_2/g, where g is an isomorphism identifying X_1|_V_1∩ V_2≃ X_2|_V_1∩ V_2. Choosing a trivialization of X|_V_1∩ V_2, and let R̃ be the fibrewise action of R on X|_V_1∩ V_2. Then we may glue V_1 and V_2 back together via R by setting X_R=X_1∐ X_2/gR̃. Let (X_R,w_R) be the resulting Landau-Ginzburg model. Fixing a fiber E in the same component as z_2, this has the effect of replacing ∩ L_1 with R∩ L_1; if HF(R∩ L_1,∩ L_2)≃ k^n, we will have (X_R,w_R)≃ K_n, the Kronecker quiver with n arrows. Let R also denote the induced autoequivalence of (E). Then the surgery has the effect that ∩_1 will be replaced by R∩_1. Thus we construct the Kronecker quiver by the diagram K_n[r][d] _2,±[r]^∩_+[d]^∩_- (E) _1[r]^R∩_1 (E) Now let us look more at K_n from the point of view of the gluing. We have _1≃_2≃ D^b(k), so that the functors F_i:_i→ are determined by the objects y_i=F_ik. Furthermore, it is easy to check that F_i^!z=_(y_i,z) and F_i^*z=_(z,y_i)^∨. Let us calculate the Serre functor S_. By Lemma <ref>, _ has a generating exceptional collection ⟨ E_1,E_2⟩, where E_1=J_1k=(k[-1],0,0) and E_2=J_2k=(0,k,0). Therefore we will determine S_E_i. For any z∈, Lemma <ref> gives us F_^!z=((_(y_1,y_2)⊗_(y_2,z)→_(y_1,z)),_(y_2,z),g_z) Here g_z is the composition (_(y_1,y_2)⊗_(y_2,z)→_(y_1,z))⊗ y_1 →_(y_1,y_2)⊗_(y_2,z)⊗ y_1 →_(y_2,z)⊗ y_2 with the second map coming from evaluation (y_1,y_2)⊗ y_1→ y_2. Let us set V=_(y_1,y_2) and recall that the Calabi-Yau structure on gives us _(y_2,y_1)≃ V^∨[-1]. Furthermore, _(y_i,y_i)≃ C^*(S^1)≃ k⊕ k[-1]. Then F_^!F_E_1≃((V⊗ V^∨[-1]→ k⊕ k[-1]),V^∨[-1],g_y_1) and F_^!F_E_2 ≃((V⊗ (k⊕ k[-1])→ V),k⊕ k[-1],g_y_2) ≃(V[-1], k⊕ k[-1], g_y_2). Then using S_E_i≃(E_i→ F_^!F_E_i), we have S_E_1 ≃((V⊗ V^∨[-1]→ k[-1]),V^∨[-1],g_y_1) ≃(sl(V)[-1],V^∨[-1],g_y_1) and S_E_2≃ (V[-1], k[-1],g_y_2). § AN A-MODEL RELATIVE CALABI-YAU STRUCTURE Let w:X→ be a Landau-Ginzburg model with smooth and compact fiber Y of dimension d. Then we have a functor ∩:(X,w)→(Y) which is restriction to the fiber at infinity, and the Orlov functor ∪:(Y)→(X,w) which is a Hamiltonian flow of a Lagrangian along an arc from +∞ to itself going around all the critical values. Furthermore, ∩ is a left adjoint of ∪. This pair of adjoint functors is constructed in the paper <cit.>, which is currently in preparation. Abouzaid has announced <cit.> a proof of the fact that ∩ is spherical. The mirror of (X,w) is a Fano variety X^∨. Here we use the term “mirror” in the sense of homological mirror symmetry, so we have an equivalence (X,w) ≃(X^∨). By Homological Mirror Symmetry, since (Y) has a Calabi-Yau structure, we expect its mirror (Y) to have one as well. When Y is compact, Ganatra <cit.> has shown that (Y) does indeed carry a right Calabi-Yau structure. Furthermore, the functor ∩:(X,w)→(Y) is mirror to a^*:(X^∨)→(Y^∨), the restriction functor from X^∨ to an anticanonical divisor Y^∨↪ X^∨. The latter functor is shown to carry a relative Calabi-Yau structure in <cit.>; this is a noncommutative/categorical version of a result of Calaque <cit.> stating that the induced map on moduli spaces of perfect complexes carries a Lagrangian structure. By mirror symmetry, we should expect ∩:(X,w)→(Y) to have a relative Calabi-Yau structure. This section is devoted to outlining an argument demonstrating the existence of this structure. The main result of this section, Theorem <ref>, is essentially a formulation of the statement that ∩:(X,w)→(Y) carries a relative CY-structure. Our formulation axiomatizes the inputs from symplectic geometry that are required in the proof. The reasons for treating these inputs as a black-box are twofold. First, proofs for many of the facts that we need from symplectic geometry are not yet available, and are the subject of works by experts in symplectic geometry that are currently in progress <cit.>. Second, we wish to make manifest the extent to which our argument is robust, and independent of specific features of the symplectic setup. Outline. Here is an outline of the argument: - In Definition <ref>, we introduce extra structure on the functor that allows us to reduce the construction of isotropy data to elementary topology, as described in Construction <ref>. - Definition <ref> axiomatizes various formal features of explicit chain models for mapping spaces in (X,w) and (Y), including compatibility conditions with ∩ and the Calabi-Yau structure on (Y), that enter into the proof of the non-degeneracy of the isotropy structure given by Construction <ref>. In this section, by a Landau-Ginzburg model we mean a symplectic manifold (X,ω) equipped with a smooth morphism w: X →ℂ with the following properties: - there is a finite collection of points {p_1,...,p_n} in ℂ such that w defines a locally trivial fibration on ℂ -{p_1,...,p_n} whose generic fiber is a symplectic manifold (Y, ω_|Y). - the structure group of this locally trivial fibration ℂ -{p_1,...,p_n} is contained in the symplectomorphism group Symp(Y, ω_|Y) of the fiber. Given a Landau-Ginzburg model (X,w), let X_∞ be its fiber at infinity. We are concerned only with its homotopy type. For instance, let U be the intersection of a sector in that contains the positive real axis with the complement of a large compact set containing the critical values of w. Then we can model X_∞ by w^-1(U) ⊂ X. Then clearly we have a (canonical up to contractible ambiguity) homotopy equivalence X_∞≃ Y_t for any fiber Y_t := w^-1(t) with t ∈ U. If Y = Y_s is any smooth fiber of w, then we still have an non-canonical homotopy equivalence X_∞≃ Y_s. In the discussions that follow, the fiber Y will always be fixed, and we will assume that we have chosen such a homotopy equivalence. Composing the boundary map ∂ in relative homology with this map induced by this homotopy equivalence, we obtain a map ∂: C_*(X,X_∞,k) → C_*(Y,k) which we continue to denote by ∂, abusing notation. Let and be locally proper k-linear stable ∞-categories, let F: → be an exact functor, and let ϕ_: _*() → k[-d] be a weak Calabi-Yau structure on . A weak pre-isotropy structure on F with respect to w consists of the following data: * a homotopy commutative diagram as follows in the ∞-category of k-modules: _*() [r] [d]_θ_ _*() [d]^θ_ _*(X,X_∞)[-d-1] [r]^(0.6)∂ _*(Y)[-d] Here Y is some fixed smooth fiber of w and _*(Y) = _*(Y;k) is a chain complex computing the homology _*(Y;k) of the topological space Y with coefficients in k, and X_∞ is the fiber at infinity, so that _*(X; X_∞) can be identified with the chain complex of vanishing cycles of the fibration defined by w. * a homotopy commutative diagram of k-modules: _*() [r]^ϕ_[d]_θ_ k[-d] _*(Y)[-d] [r]__0[-d] _0(Y)[-d] [u]_[-d] Here _0(Y) ≃τ_≤ 0_*(Y) since _*(Y) is connective, and _0: _*(Y) →τ_≤ 0_*(Y) is simply the unit of the natural adjunction between spectra and connective spectra. The map is the linear extension of the map that sends the class of any point in Y to the element 1 ∈ k. Equip the chain complex _*(-) with a homotopically trivial S^1-equivariant structure, and suppose that is equipped with a strong Calabi-Yau structure ϕ̃_: _*() → k[-d]. Then a strong pre-isotropy structure on F with respect to w is the data of commutative diagrams as in (1) and (2) above, taking values in the ∞-category of S^1-equivariant k-modules. Let X be an exact symplectic manifold, and let w: X → be a Lefschetz fibration satisfying the hypotheses in <cit.>, with exact generic fiber Y. Then the Fukaya-Seidel category (X,w) and the Fukaya category (Y) are defined over k =. By <cit.>, we have _*((Y))≃^*-d(Y), the symplectic cohomology of F. We may take a model of ^*-d(Y) whose generators are constant loops so that (as vector spaces) we have ^*-d(Y)≃_d-*(Y, ). For (X,w) we have _*((X,w))≃_d+1-*(X, X_∞, ), the vanishing cycles of X, with the trivial S^1 action <cit.>. Furthermore, in the forthcoming paper <cit.> a functor ∩: (X,w) →(Y) is constructed, which acts on the support of an A-brane by simply intersecting it with a generic smooth fiber. We expect that the results proven therein will imply that the diagram _*((X,w)) [r] [d] _*((Y)) [d] _*(X,X_∞)[-d-1] [r] _*(Y)[-d] commutes in S^1-modules, thus equipping the functor ∩ with a pre-isotropy structure with respect to w. In the diagram above, the right vertical equivalence is the composite of the open-closed map _*((Y) →^*-d(Y) with the equivalence ^*-d(Y)≃_d-*(Y) mentioned earlier. Let X be an symplectic manifold, and let w: X → be a symplectic fibration, with compact generic fiber Y. In this situation one expects to be able to define categories (X,w) and (Y) that are linear over the Novikov field Λ := ((t^)), and a functor ∩: (X,w) →(Y) as in Remark <ref>. The properness of the fiber Y will be reflected in the properness of these categories. It is expected <cit.> (see also <cit.>) that _*((Y)) is equivalent to the quantum cohomology QH^*(Y) via the open-closed map. The underlying chain complex QH^*(Y) is the cochain complex ^*(Y, Λ), which via Poincare duality is identified with _*(Y, Λ)[-d]. Similarly, one expects that _*((X,w)) ≃_*(X, X_∞, Λ) as S^1-equivariant chain complexes. As in Remark <ref>, the functor ∩ should be equipped with a pre-isotropy structure with respect to w; the main difference is that all the structures are defined over Λ instead of . Suppose that we are given a functor F: → that is equipped with a weak (resp. strong) pre-isotropy structure with respect to a Landau-Ginzburg model w: X → as in Definition <ref>. We are going to construct a weak (resp. strong) isotropy structure on F (Definition <ref>) from this data. Consider the diagram: _*() [r]^_*(F)[d]_θ_ _*() [d]^θ_ _*(X,X_∞)[-d-1] [r]^(0.6)∂[d]__1[-d] _*(Y)[-d] [d]^_0[-d] _1(X,X_∞)[-d] [r]^(0.6)∂[d] _0(Y)[-d] [d]^[-d] 0 [r] k[-d] Observe that _*(X,X_∞) is 1-connective and so _1(X,X_∞) ≃τ_≤ 1_*(X, X_∞); the map _1: _*(X,X_∞) →τ_≤ 1_*(X,X_∞) is simply the unit of the natural adjunction between spectra and 1-connective spectra. The two lower squares are manifestly commutative as diagrams in the ∞-category of S^1-equivariant k-modules. In particular, they are also commutative as diagrams in the ∞-category _k of k-modules. Thus, * If F is equipped with a weak pre-isotropy structure with respect to w, then the upper square defines a homotopy commutative square in _k. Therefore the outer square is also commutative in _k, and defines a weak isotropy structure on F. * If F is equipped with a strong pre-isotropy structure with respect to w, then the upper square defines a homotopy commutative square of S^1-equivariant k-modules. Therefore the outer square is also equipped with the structure of a commutative square of S^1-equivariant k-modules, and thus defines a strong isotropy structure on F. Having discussed the geometric structures required to produce an isotropy structure on F, we now turn our attention to non-degeneracy. The following lemma will allow us to reduce the question of non-degeneracy to a generator set of objects that have geometric representatives: Let F: → be a functor of locally proper k-linear stable ∞-categories equipped with an isotropy structure with respect to a Calabi-Yau structure ϕ_ on , and let _⊂_ be a set of objects that generates under finite colimits. For x, y ∈_, consider the homotopy commutative square: _(x,y) [r] [d] _(Fx, Fy) [d] 0 [r] _(y,x)^∨[-d] obtained by applying Construction <ref>. Assume that (<ref>) is a pullback square for all x,y in _. Then (<ref>) is a pullback square for x, y in _. Let x ∈_. Define ^r(x):= {y ∈_| Diagram (<ref>) is a pullback square}. Let y', y ∈^r(x), and suppose y' → y → y” is a fiber-cofiber sequence in the stable ∞-category . Consider the diagram _(x,y') [r] [d] _(Fx, Fy') [r] [d] _(y',x)^∨[-d] [d] _(x,y) [r] [d] _(Fx, Fy) [r] [d] _(y,x)^∨[-d] [d] _(x,y”) [r] _(Fx, Fy”) [r] _(y”,x)^∨[-d] The commutativity of all the squares in this diagram derives from the functoriality of (-,-), and the fact that the collection of morphisms Ξ(ϕ): _(Fx,Fy) →_(Fy, Fx)^∨[-d] underlies a natural transformation of functors. For each row in this diagram, the composite of the morphisms in that row is equipped with a nullhomotopy given by the isotropy structure on F. All of this information fits together to give a functor Δ^2 ×Δ^1 ×Δ^1 →_k. The columns of the diagram above are fiber sequences since _(x,-) and F are exact functors, and y' → y → y” is a fiber sequence. Thus the bottom row is the cofiber of the map from the top row to the middle row. Furthermore, the upper two rows of this diagram are fiber sequences, since y', y ∈^r(x) by assumption. Since cofiber of a map between fiber sequences is a fiber sequence, the bottom row is also a fiber sequence. We conclude that y”∈^r(x), and hence that ^r(x) is stable under taking cones. Clearly 0 ∈^r(x). Furthermore, by the hypothesis in the statement of the lemma, _⊂^r(x). Since _ generates under finite colimits, we deduce from the conclusion of the previous paragraph (and the fact that is stable) that ^r(x) = _. Now let y ∈_ be an arbitrary object, and let ^l(y) := {x ∈_| Diagram (<ref>) is a pullback square}. Then clearly 0 ∈^l(y), and _⊂^l(y) by the conclusion of the previous paragraph. By applying the same argument as above to a fiber sequence x' → x → x”, we see that if x', x ∈^l(y), then x”∈^l(y). Since is generated under finite colimits by _, and _∪{0}⊂^l(y), we conclude that ^l(y) = _ for arbitrary y ∈_. This is exactly what we set out to prove. In order to prove that the isotropy structure that we have produced is non-degenerate, we will Let (X,ω) be a symplectic manifold, and let - H ∈ C^∞(X, ℝ) - ⊂ C^∞(X, ℝ) - L, L' ∈Lag(X), the set of smooth Lagrangian submanifolds of X. Then we will use the following notation: * _H denotes the set of time 1 H-Hamiltonian trajectories; i.e., _H := {γ:[0,1] → X |γ is smooth and γ̇(t) = 𝔛_H(γ(t))}. Here 𝔛_H is the vector field on X characterized by ω(𝔛_H,-) = -dH. * _ : = ∪_H ∈_H. * _(L,L') := {γ∈_|γ(0) ∈ L and γ(1) ∈ L'}. For any ⊂ C^∞(X), and for any Lagrangians L,L', there is a natural bijection _: _(L,L') [r]^∼ _- (L',L) obtained by sending γ∈_(L,L') to γ∘δ where δ: [0,1] → [0,1] is given by δ(t) = 1-t. Some notation for linear algebra constructions: * We will write (-)^♮: Chain(k) →Vect for the forgetful functor from the 1-category of chain complexes of vector spaces over k to the 1-category of vector spaces. * _k: Sets→Vect_k is the left adjoint to the forgetful functor from vector spaces over k to sets. * Let be a basis for a graded vector space V. We will write _d^* for the dual basis of V^∨[d] = (V,k[d]). For γ∈, γ_d^* ∈_d^* will denote the dual element, so that γ^*_d(γ) = 1 and γ^*_d(γ') = 0 for all γ' ≠γ. When d is clear from the context we will suppress d in the notation. Let w: X → be a Landau-Ginzburg model as in Definition <ref>, and let k be a field. Suppose that the generic fiber Y is of real dimension 2d. A weak (resp. strong) admissible categorical LG-formalism for (X,w) consists of (D1) A smooth function H: X →ℝ. Define := {H}, := - and := ∪. (D2) An exact functor F: → of locally proper k-linear stable infinity categories equipped with a weak (resp. strong) pre-isotropy structure (Definition <ref>) for a given weak (resp. strong) d-Calabi-Yau structure ϕ on . (D3) A set _⊂_ that generates under finite colimits, a map (Notation <ref>) _×_→Lag(X) ×Lag(X) (x,y) ↦ (L_x,y, L_x,y') and isomorphisms of sets σ_x,y: _(L_x,y, L'_x,y) →_(L_y,x', L_y,x) (see Remark <ref>). (D4) For all (x,y) ∈^× 2_ a model for _(x,y) as a chain complex _(x,y) whose underlying vector space has as basis the set _x,y := _(L_x,y,L_x,y'). In particular, we have an isomorphism of vector spaces μ_x,y: _k(_x,y) →_(x,y)^♮ (Notation <ref>). (D5) For all (x,y) ∈^× 2_ a model for _(Fx,Fy) as a chain complex _(Fx,Fy) whose underlying vector space has as basis the set _x,y := _(L_x,y,L_x,y'). In particular, we have an isomorphism of vector spaces ν_x,y: _k(_x,y) →_(Fx,Fy)^♮. satisfying the following conditions: (A1) For all (x,y) ∈_^× 2, we have L_x,y∩ L'_x,y = ∅, so that, in particular, _(L_x,y, L_x,y') ∩_(L_x,y, L_x,y') = ∅ and consequently _x,y = _x,y∐_x,y is a disjoint union. (A2) For each x,y ∈, the following diagram in the 1-category of graded vector spaces commutes (see Notation <ref>): _k(_x,y) [r]^_k(ι)[d]^(0.45)[@]∼_μ_x,y _k(_x,y) [d]^ν_x,y_(0.45)[@]∼ _(x,y)^♮[r]_F^♮ _(Fx, Fy)^♮ Here the upper horizontal map is induced by the inclusion ι: _x,y→_x,y. (A3) For each x,y ∈, the following diagram in the 1-category of vector spaces commutes : _k( _x,y) [r]^_k(η)[d]^(0.45)[@]∼ _k(^*_y,x) [d]^(0.45)[@]∼ _(Fx,Fy)^♮[r]_Ξ(ϕ)^♮ _(Fy,Fx)^∨[-d]^♮ Here - The vertical maps are given by (D5). - The lower horizontal map is induced by the Calabi-Yau structure on (Definition <ref>). - The map η defining the upper horizontal arrow arises as follows. Composing the map _ of Remark <ref> with the map σ_x,y from item (D3) gives an isomorphism θ: _∘σ_x,y: _(L_x,y, L'_x,y) →_(L_y,x,L'_y,x). Similarly, θ':= _∘σ_y,x is an isomorphism. Let τ: _y,x→^*_y,x;d be the tautological isomorphism. Then η := τ∘ (θ⊔ (θ')^-1) : _x,y→^*_y,x is the isomorphism of the upper horizontal row. A Landau-Ginzburg model is admissible if the functor ∩: (X,w) →(Y) is defined, and underlies an admissible categorical LG-formalism for (X,w). For our purposes, two admissible categorical formalisms will be equivalent if the underlying functors with pre-isotropy data are equivalent in the obvious sense. So, without loss of generality, we may choose (D3) in Definition <ref> so that for all x ≠ y, L_x,y = L'_y,x and σ_x,y = 𝕀. Based on ongoing work of Abouzaid and Ganatra <cit.>, we expect that the general Landau-Ginzburg model is in fact admissible in the sense of Definition <ref>. Remarks <ref> and <ref> discussed pre-isotropy data on ∩. Specific details pertaining to the other items in <ref> will depend on the particular model we choose for (X,w). Suppose that (X,w) is generated by noncompact (decorated) Lagrangians L ⊂ X whose image under w is required to intersect the complement of some fixed compact set K ⊂ in a finite union of rays parallel to the positive real axis. In this situation, the Hamiltonian H: X →ℝ from Definition <ref>, (D1), can be taken to be the pullback of a smooth function on H^♮: →ℝ with dH^♮ supported near infinity in some sector containing the positive real axis and generating a counterclockwise flow on satisfying some technical hypotheses. Let x,y ∈(X,w), and suppose that we can choose Lagrangians L,L' representing x, y respectively. Then _(X,w)(x,y) is computed by the Floer complex CF^*(Φ_H^1(L), L'), where Φ_H^t is the flow generated by the Hamiltonian H. A basis for this vector space is given by the set of intersection points of Φ_H^1(L) and L', which in turn can be identified with the set of time 1 Hamiltonian trajectories starting on L and ending on L'. This motivates the item (D4) in Definition <ref>. At a heuristic level, the functor ∩ is given by intersecting a Lagrangian L as above with a smooth fiber. Fix a smooth fiber Y := w^-1(t_0) for some t_0 near infinity along the positive real axis. Suppose that we are given an object x in (X,w) that is represented by a Lagrangian L that, outside some compact set, projects to a finite union l := ⊔_i l_i of rays parallel to the positive real axis in . For t ∈ℝ, let S_t := {Re(z) = t}∩ l =: {t_1,...,t_n} be the intersection of the vertical line through t with our family of horizontal rays. For each t_i ∈ S_t, let γ_i be a path in connecting t_i to t_0. Let L_i ⊂ Y be the Lagrangian obtained by applying symplectic parallel transport along γ_i to L ∩ w^-1(t_i). Then, roughly speaking, ∩ x can be represented by the Lagrangian ∪ L_i. Now, if y is another object in (X,w), represented by a Lagrangian L', then the mapping space _(Y)(∩ x, ∩ y) is computed by the Floer complex CF^*(∪_i L_i, ∪_j L'_j), which has as vector space basis the set of intersection points of ∪_i L_i and ∪_j L'_j. If L and L' are disjoint, one can argue that this set coincides with the set _(L,L') of time 1 Hamiltonian trajectories starting on one of the Lagrangians and ending on the other. This motivates (D5) and (A1) in Definition <ref>. Let w:X→ be a Landau-Ginzburg model equipped with a strong (resp. weak) admissible categorical LG-formalism (Definition <ref>) with underlying functor F: →. Then the functor F carries a strong (resp. weak) relative Calabi-Yau structure (Definition <ref>). By Definition <ref>, (D2), F is equipped with a strong (resp. weak) pre-isotropy structure, as defined in Definition <ref>. Applying Construction <ref>, we obtain a strong (resp. weak) isotropy structure on F. In order to prove that this isotropy structure defines a relative Calabi-Yau structure, it remains only to verify that for all x, y ∈_, the square _(x,y) [r] [d] _(Fx, Fy) [d] 0 [r] _(y,x)^∨[-d] obtained by applying Construction <ref> to this isotropy structure is a pullback square of spectra. By Lemma <ref>, it suffices to check that this diagram is a pullback square for all x,y in the set of generators _ given by item (D3) of Definition <ref>. To this end, for x,y ∈_, consider the diagram of vector spaces given by (D4) and (D5): _k(_x,y) [r] [d]_μ_x,y^(0.45)[@]∼ _k(_x,y∐_x,y) [r]^(0.45)∼[d]_ν_x,y^(0.45)[@]∼ _k((_y,x∐_y,x)^*) [r] [d]^(0.45)[@]∼ _k((_y,x)^*) [d]^(0.45)[@]∼ _(x,y)^♮[r] _(Fx,Fy)^♮[r]^(0.4)∼ (_(Fy,Fx)^∨[-d])^♮[r] (_(y,x)^∨[-d])^♮ The left hand square is the commutative square of (A2) of Definition <ref>. The middle square is the commutative square of (A3) from the same definition. The right hand square is the k-linear dual (shifted by [-d]) of the commutative diagram of (A2), with the roles of x and y interchanged. Thus the entire diagram commutes. Let ι^*: _y,x: →_x,y∐_x,y be the composite of the natural inclusion _y,x⊂_y,x∐_y,x and the isomorphism η^-1 from (A3) of <ref>. Then ι∐ι^*: _x,y∐_y,x→_x,y realizes _x,y as a coproduct. Since the free vector space functor carries coproducts to direct sums, we conclude that the sequence 0 →_k(_x,y) →_k(_x,y∐_x,y) →_k((_y,x)^*) → 0 from the top row of the previous diagram is an exact sequence of vector spaces. The vertical arrows in said diagram are all isomorphisms by hypothesis (see (D4) and D5)), so the lower row allows defines an exact sequence. Since the forgetful functor from the abelian 1-category of chain complexes to the category of vector spaces reflects exact sequences, we conclude that the sequence 0 →_(x,y) →_(Fx,Fy) →_(y,x)^∨[-d] → 0 from the bottom row is an exact sequence of chain complexes. Passing from the abelian category of chain complexes to the stable ∞-category _k of k-module spectra, this gives rise to a pullback-pushout square that agrees with the diagram <ref> on the outer 1-simplices. To complete the proof, we need to show that the homotopies witnessing the commutativity of the two diagrams are essentially the same; more precisely, we need to show that the pullback square that we have just constructed is actually equivalent to the square constructed from isotropy data in Diagram <ref> as a homotopy coherent diagram. We expect that the techniques to be developed in <cit.> will allow us to verify this. § SHIFTED SYMPLECTIC STRUCTURES This section consists of two logically independent subsections: * <ref> is devoted to the proof of Theorem <ref>, which states that the moduli of compactly supported perfect complexes on certain open Calabi-Yau varieties carries a natural shifted symplectic structure. * <ref> is devoted to the proof of Theorem <ref>, which states that the map on moduli spaces of objects induced by the pushforward functor from perfect complexes on a smooth divisor to perfect complexes on an ambient smooth and proper Calabi-Yau variety carries a natural Lagrangian structure. These theorems were motivated and placed within the larger context of this paper in <ref> of the introduction. Here, we begin by cursorily recalling the relevant definitions and results regarding shifted symplectic and Lagrangian structures, referring the reader to <cit.> for a more detailed and precise discussion. In this section, k will be a fixed base field, of characteristic 0. Let X be a derived Artin stack with cotangent complex Ω_X. We can form the de Rham algebra Ω^*_X=^*__X(Ω_X[1]). This is a weighted sheaf whose weight p piece is Ω^p_X=^p__X(Ω_X[1])=∧^pΩ_X[p]. The space of p-forms of degree n on X is ^p(X,n)=Ω^∞_(X)(_X,∧^pΩ_X[n]) ≃_(X)(_X,∧^pΩ_X[n]) We also construct the weighted negative cyclic chain complex NC^w, whose degree n, weight p part is NC^w,n(Ω_X)(p)=(⊕_i≥0∧^p+iΩ_X[n-i],d_Ω_X+d_dR). The space of closed p-forms of degree n is ^p,cl(X,n)=τ_≤ 0_(X)(_X,NC^n(Ω_X)(p)). There is a natural “underlying form” map ^p,cl(X,n)→^p(X,n) corresponding to the projection ⊕_i≥0∧^p+iΩ_X[n-i]→∧^pΩ_X[n]. A 2-form ω:_X→∧^2Ω_X[n] of degree n is nondegenerate if the adjoint map _X→Ω_X[n] is a quasi-isomorphism. An n-shifted symplectic form on X is a closed 2-form whose underlying form is nondegenerate. Let X be a derived Artin stack with an n-shifted symplectic form ω and let f:Y→ X be a morphism. An isotropic structure on f is a homotopy h:0∼ f^*ω in the space of closed forms on Y. An isotropic structure on f defines a map Θ_h:_f→Ω_Y[n-1]. We say h is Lagrangian if Θ_h is a quasi-isomorphism of complexes. §.§ Perfect Complexes on Open Varieties Let U=X\ D be an open variety. In general, it is unrealistic to hope for a symplectic structure on (U,Y), because we need to integrate on U, which is noncompact. But in the particular case of Y=, we can consider the space _c(U) of compactly supported perfect complexes on U. We can express this via a pullback square _c(U)[r][d] (X)[d] ∙[r]^0 (D), with ∙→(D) corresponding to the 0 complex. Thus _c(U) is a geometric stack, and, in particular, an open substack of (X). In general we will not distinguish between a compactly supported complex on U and its extension by 0 to X. We claim that _c(U) carries a symplectic structure: Let X be a smooth d-dimensional variety, D⊂ X a divisor, and U=X\ D. Let α be a meromorphic section of _X(K_X) which is holomorphic nonvanishing on U. Then α induces a (2-d)-shifted symplectic structure on _c(U). The construction of the form closely mimics that of Theorem <ref>. Let V(α)=D_+-D_- with D_+ and D_- effective; note that D_+∪ D_-⊆ D. We can consider _c(U) a substack of (X). Thus we have an evaluation :_c(U)× X→. Now, ^*_ has the following description: for g: A→_c(U)× X corresponding to a perfect complex E on A× U and g: A→ X, we have (^*_)_g≃((𝕀_A× g)^*E,(𝕀_A× g)^*E)[1]. If g factors through _c(U)× D_-, then this vanishes, as E is supported away from D. Then for p≥ 1, (^*∧^pΩ_)_D-∼0, and in particular, if ω is a p-form on , ^*ω vanishes on _c(U)× D_- as well. Similarly to the proof of Theorem <ref> (<cit.>) we then get a map DR(_c(U)× X)(-_c(U)× D_-)→ DR(_c(U))⊗Γ(X,(-D_-)). Further, we have an orientation map Γ(X,(-D_-)) →Γ(X,(D_+-D_-)) Γ(X,K_X) → k[-d], where the last map is projection onto H^d,d. Combining these yields an integration map ∫_α: DR(_c(U)× X)(-_c(U)× D_-)→ DR(_c(U))[-d] and similarly on the level of negative cyclic complexes ∫_α: NC^w(_c(U)× X)(-_c(U)× D_-)→ NC^w(_c(U))[-d]. Then if ω is the symplectic form on , we get a closed 2-form ∫_α^*ω on _c(U). As in the proof of Theorem <ref>, we can describe the pairing on _c(U) as follows. For g: A→_c(U) corresponding to a perfect complex E on U× A compactly supported over A, the pairing (E,E)[1]∧(E,E)[1]→ A[2-d] is given by cup product, followed by trace and integration multiplied by α. This is nondegenerate because α is nonvanishing on the support of E. Note from the description at the end of the proof that the symplectic structure on _c(U) depends only on α|_U, and not on what the compactification X is. In the special case that X is Fano and D is a smooth effective anticanonical divisor, we can write _c(U)=(X)×_(D) k, where k→(D) corresponds to the zero complex. This is a Lagrangian intersection over a (3-d)-shifted symplectic derived stack. We note that Anatoly Preygel has obtained similar results in <cit.> using different methods. §.§ Pushforwards for Perfect Complexes For maps i:X→ Y there is an induced map i^*:(Y,Z)→(X,Z). In the particular case that Z= is the derived stack of perfect complexes, we get a map i_*:(X)→(Y) going the other way as well. If Y is a smooth Calabi-Yau variety, (Y) will be symplectic and we can investigate the properties of this map. Let Y be a smooth Calabi-Yau variety and i:D→ Y a smooth divisor. Then the map i_*:(D)→(Y), induced by the functor i_*: (D) →(Y), carries a natural Lagrangian structure. Consider the diagram 5pc5pc D×(D)[d]^𝕀_d× i_*[rrd]^_D D×(Y)[r]^i×𝕀_(Y) Y×(Y)[r]^_Y The symplectic form ω_(Y) on (Y) is given by ∫_[Y]^*(), where is the universal perfect complex on . Write _Y=_Y^* for the universal complex on Y×(Y), and similarly _D=_D^*. Integration clearly commutes with pullback by i_*, so we have (i_*)^*∫_[Y](_Y)=∫_[Y]((𝕀_Y× i_*)^*_Y). Now, for any point p: A→(Y) corresponding to a perfect complex E∈(Y× A), we have (𝕀_D× p)^*(𝕀_Y× i_*)^*_Y ≃ (i×𝕀_A)^*E ≃ (𝕀_D× p)^*(i×𝕀_(Y))_*_D as sheaves on D× A. Thus we have an isomorphism (𝕀_Y× i_*)^*_Y≃ (i×𝕀_(Y))_*_D. Thus we have (i_*)^*ω_(Y)≃∫_[Y]((i×𝕀_(Y))_*_D). Now, ((i×𝕀_(Y))_*_D=c_1(D)ω' for some form ω'. For the integration map, recall that we use the Künneth formula DR(Y×(D))≃ DR(Y)⊗ DR((D)) followed by the projection DR(Y)→_Y. But then c_1(D)ω' will decompose as some sum of terms c_1α⊗β, and c_1(D)α will project to 0 in _Y because c_1(D) is a (1,1) form. This gives our isotropic structure. For Lagrangianness, consider an A-point g: A→_c(D), corresponding to a perfect complex E on D× A=D_A. Then _(D),g≃_D_A(E,E)[1], and (i_*)^*_(Y),(i_*)^g≃_Y_A(i_*E,i_*E)[1]. The symplectic structure ω on (U) at some F is given by ∧^2_U_A(F,F)[1] _U_A(F,F)[2] Γ(U_A,_U_A)[2] → H^d(U_A,_U_A)[2-d] A[2-d]. We claim that _Y_A(i_*E,i_*E)≃_D_A(E,E)⊕_D_A(E,E⊗ K_D)[-1]. * To see this, we note that _Y_A(i_*E,i_*E)≃_D_A(i^*i_*E,E), and i^*i_*E≃_D_A⊗_i^-1_Y_AE≃{E(-D_A)→ E}, where the map {E(-D_A)→ E} is multiplication by the defining section of D_A. Since E is supported on D_A, this is 0, so i^*i_*E≃ E⊕ E(-D_A)[1]. Then we have _Y_A(i_*E,i_*E) ≃_D_A(E⊕ E(-D_A)[1],E) ≃_D_A(E,E)⊕_D_A(E,E(D_A[-1]) ≃_D_A(E,E)⊕_D_A(E,E⊗ K_D)[-1]. Furthermore, the map _D_A(E,E)→_D_A(E,E)⊕_D_A(E,E⊗ K_D)[-1] ** comes from the counit E⊕ E(-D_A)[1]≃ i^*i_*E→ E which is the projection, so (<ref>) is the obvious inclusion. In the decomposition (<ref>), the multiplication structure of the right hand side is the obvious square zero extension of _D_A(E,E), where _D_A(E,E⊗ K_D)[-1] has the obvious (left and right) module structure. Furthermore, trace map is given by the composition _D_A(E,E)⊕_D_A(E,E⊗ K_D)[-1]_D_A(E,E⊗ K_D)[-1]_D_A(-D_A)→_Y_A. Then the pairing of _(D),g≃_D_A(E,E)[1] with _i_*, g[d-1]≃_D_A(E,E⊗ K_A)[d-1] is the nondegenerate pairing of Serre duality, and we have nondegeneracy. Theorem <ref> should be contrasted with the result of <cit.> saying that if X is a Fano variety and a:Y→ X a smooth anticanonical divisor, then a^*:(X)→(Y) has a Lagrangian structure (<cit.> Theorem 2.10). The functor i_*: (D) →(Y) is one of the compatible spherical functor of Example <ref>. By Theorem <ref> this functor carries a weak relative Calabi-Yau structure. We have conjectured that this can be promoted to a strong Calabi-Yau structure. Theorem <ref> can be viewed as giving evidence for this statement, because the map on moduli spaces of objects induced by a relative Calabi-Yau functor is expected to carry a Lagrangian structure. § FURTHER DIRECTIONS In the introduction, we described some of the geometric background from symplectic topology that led us to the results presented in this paper. In this section, we point to potential ramifications of these results in a different direction: we outline a plan to introduce and develop a theory of derived n-shifted hyperkähler stacks, and noncommutative hyperkähler spaces. Hyperkähler manifolds were first defined by Calabi in <cit.>. The initial development of this subject was pioneered by Bogomolov, Beauville and Hitchin. In recent years there has been a tremendous revival of interest in this subject, with several splendid results being obtained by Verbitsky, Kamenova, Voisin, Kaledin, Huybrechts, Kollar, Laza, Saccà and others (see, e.g., <cit.>, <cit.>, <cit.>, <cit.>). Recall that a hyperkähler manifold is a real C^∞ Riemannian manifold of dimension 4n whose holonomy is contained in Sp(n) = O(4n) ∩ GL_n(ℍ). More explicitly, a hyperkähler manifold is a Riemannian manifold (X,g) whose tangent bundle is equipped with covariantly constant endomorphisms I, J, K satisfying the quaternionic identities: I^2 = J^2 = K^2 = IJK = -1. Define symplectic forms ω_1(v,w) := g(Iv,w), ω_2(v,w) := g(Jv,w), and ω_3(v,w) := g(Kv,w). It is straightforward to check that the form ω_+ := ω_2 + √(-1)ω_3 is holomorphic with respect to the complex structure I. Thus, underlying every hyperkähler manifold is a holomorphic symplectic manifold. Conversely, if X is a compact holomorphic symplectic manifold, then by an application of Yau's celebrated theorem on the existence of Ricci flat metrics and an earlier theorem of Bochner, on can show that every Kähler class on X contains a hyperkähler metric. It is this close relationship between holomorphic symplectic geometry and hyperkähler geometry that serves as the starting point for our proposed generalization of hyperkähler geometry to derived stacks: the basic idea is to replace the holomorphic symplectic form by an n-shifted symplectic structure on a derived stack. In fact, even without appealing to Yau's powerful result, it turns out that it is possible to reformulate the notion of a hyperkähler metric in purely holomorphic terms using Penrose's twistor geometry. It is easy to see that each imaginary unit quaternion 𝐮 defines an almost complex structure I_𝐮 on the tangent bundle of X. Identify the imaginary unit quaternions with ℂℙ^1 ≃ S^2. Then the C^∞-manifold X × S^2 can be equipped with a unique almost complex structure that restricts to I_𝐮 on X ×{𝐮}, and is compatible with the projection to S^2 ≃ℂℙ^1. It turns out that this almost complex structure is integrable, and defining complex manifold Z with a projection π: Z →ℂℙ^1 called the twistor family of X. <cit.> The twistor space Z constructed above admits the following structures * π: Z →ℂℙ^1 is a holomorphic fiber bundle. * There is a real structure τ on Z covering the antipodal map on the projective line. * There is a holomorphic symplectic form ω_rel on the fibers of π, which is real with respect to τ and given by a section of ∧^2 T_π^∨⊗π^* 𝒪_ℙ^1(2). * There is a family of global holomorphic sections of π, real with respect to τ, whose normal bundles are given by ℂ^2n⊗_ℂπ^* 𝒪_ℙ^1(1). Moreover, the twistor family completely characterizes the hyperkähler manifold: <cit.> Let Z be a complex manifold of complex dimension 2n+1 equipped with the structures of Theorem <ref>. Then the space of real sections (4) of the fibration π: Z →ℂℙ^1 can be equipped with a natural Riemannian metric that turns it into a hyperkähler manifold of real dimension 4n whose twistor family is π: Z →ℂℙ^1. Theorems <ref> and <ref> motivate the following tentative definition: A d-shifted derived twistor family of hyperkähler type is given by a locally geometric derived analytic ∞-stack Z over ℂ equipped with the following structures: * A map π: Z →ℙ_ℂ^1 of derived stacks. * A real structure τ on Z covering the antipodal map on the projective line. * A d-shifted symplectic structure ω_rel on the fibers of π, which is real with respect to τ and whose underlying 2-form is given by a section of degree d of ∧^2𝕋_π^∨⊗π^* 𝒪_ℙ^1(2). Here 𝕋_π is the homotopy fiber of the natural map of tangent complexes 𝕋_Z→π^* 𝕋_ℙ^1. More precisely, the closed form ω_rel itself is a section of HC^-,w(Z/ℙ^1) ⊗𝒪_ℙ^1(2), the relative weighted negative cyclic complex of Z over ℙ^1, twisted by 𝒪_ℙ^1(2). * a connected component X of the homotopy fixed points _ℙ^1(ℙ^1, Z)^τ of the induced action of τ on the derived mapping stack of analytic sections of π, such that the natural map X ×ℙ^1→ Z is an equivalence of C^∞ derived stacks. It will be an interesting question whether it is possible to define an analogue of the notion of a “derived hyperkähler metric” on a real C^∞ derived stack, so that the derived analogues of Theorems <ref> and <ref>. In the absence of such a definition, we propose to treat the derived twistor families of <ref> as a proxy for the notion of a d-shifted derived hyperkähler stack. Hyperkähler manifolds are difficult to come by, and thus one of the central problems in hyperkähler geometry is the problem of constructing hyperkähler manifolds. Therefore, the first order of business will be to address the following: Formulate and prove twistor family versions of * the result stating that the mapping stack from d-oriented derived stack into an n-shifted symplectic stack admits a n-d shifted symplectic structure, and the relative version of the same (Theorem <ref> and Theorem <ref>). * the theorem stating that Lagrangian intersections are symplectic (Theorem <ref>). An important technique for constructing new hyperkähler manifolds from old ones is hyperkähler reduction. Indeed, many of the interesting examples of hyperkähler manifolds arise from solutions to the anti-self-dual Yang-Mills equations, and thus can be viewed as arising from infinite-dimensional hyperkähler reduction. Safronov has explained <cit.> how to interpret symplectic reduction as a Lagrangian intersection in derived algebraic geometry. By implementing his construction in twistor families using the solution to Problem <ref>, it should be possible to address the following: Introduce a derived version of hyperkähler reduction, using the methods of <cit.>, and use this to construct new derived hyperkähler stacks. In order to successfully address these problems, it will be necessary to develop a robust theory of shifted symplectic structures in families and on derived analytic stacks. This theory should be of independent interest. A successful solution to Problems <ref> and <ref> should lead to a new conceptual understanding of classical hyperkähler spaces, such as moduli spaces of sheaves on K3 surfaces, and instanton moduli spaces such as bow varieties. The first examples where we hope to obtain hyperkähler structures with a non-trivial and interesting derived/stacky structure are singular moduli spaces of sheaves on K3 surfaces and coadjoint orbits. One of our motivations for studying derived hyperkähler geometry comes from nonabelian Hodge theory <cit.>. Let Y be a smooth projective variety over ℂ of dimension d, and let G be a reductive algebraic group over ℂ. Then, by results of Hitchin, Simpson, Fujiki, etc <cit.> the moduli space ℳ_Harm(Y,G) of harmonic G-bundles (solutions to Hitchin's equations) on the Kähler manifold Y(ℂ) carries a natural hyperkähler metric. The associated twistor family (ℳ_Harm(Y,G)) is the Deligne-Simpson space π: ℳ^ss_Del(Y,G) →ℙ^1. The fiber over λ∈𝔸^1 ⊂ℙ^1 is the space of semistable λ-connections on Y. For λ = 0 this is the moduli space of semistable Higgs bundle ℳ^ss_Dol(Y,G), ad for λ = 1 this is the de Rham moduli space ℳ^ss_DR(Y,G). It follows that ℳ^ss_Dol(Y,G) and ℳ^ss_DR(Y,G) carries natural holomorphic (0-shifted) symplectic structures. On the other hand, <cit.> construct natural 2(1-d)-shifted structures on the derived stacks ℳ_Dol(Y,G):= (Y_Dol, BG) and ℳ_DR(Y,G):= (Y_DR, BG). This leads to the following question, which we plan to investigate: * When d= 0 we have two natural 0-shifted holomorphic symplectic forms on the moduli space ℳ^ss_Dol(Y,G) (resp. ℳ^ss_DR(Y,G) ), one coming from its realization as a fiber of the twistor family of the Hitchin's hyperkähler manifold of harmonic bundles, and the other coming from the <cit.> mapping space construction. Are these two symplectic structures relates in any way? * When d > 0, do the 2(1-d)-shifted <cit.> symplectic structures on these moduli spaces have any relation to the other structures associated with the Hitchin-Deligne-Simspson twistor family? Do these 2(1-d)-shifted symplectic structures give rise to an n-shifted derived twistor family of hyperkähler type (Definition <ref>) for which the fiber over λ∈𝔸^1 is a suitably rigidified version of the derived moduli stack of λ-connections? Next, we turn to the problem of studying (derived) hyperkähler geometry through the lens of categorical noncommutative geometry. The functor from commutative spaces to noncommutative spaces has an adjoint, namely the functor ↦Moduli_ <cit.> carrying a category to the moduli space of compact objects in it. Abuaf <cit.> has introduced the notion of a hyperkähler category that takes as the starting point, in the following sense: his definition is designed so that, when X is an ordinary variety, the category (X) is hyperkähler if and only if X is a hyperkähler manifold. For our purposes, the dual point of view is more natural; thus, the notion of -hyperkähler space should have the property that the functor Moduli carries an -hyperkähler space of dimension d to a (2-d)-shifted derived hyperkähler stack in the sense of Definition <ref> and Remark <ref>. The functor Moduli carries d-Calabi-Yau structures on categories to (2-d)-shifted symplectic structures on derived stacks. Therefore, the noncommutative analogue of Definition <ref>, should, very roughly speaking, incorporate the following elements: - a quasi-coherent sheaf of proper ℂ-linear ∞-categories on ℙ^1 - an S^1-equivariant morphism _*() →𝒪_ℙ^1(2)[-d] in (ℙ^1), which defines Calabi-Yau structures on the stalks of . - a real structure τ: σ^* covering the antipodal map σ. - a family of preferred real global sections of with deformation-theoretic properties analogous to Theorem <ref> (4). Give a precise definition of -hyperkähler spaces incorporating the elements described in Remark <ref>. Prove a twistor family analogue of Theorem <ref> that allows one to construct new -hyperkähler spaces by gluing. Such a noncommutative set-up opens the door to many new avenues for studying hyperkähler geometry. For instance, if the general fiber _x of the sheaf in Remark <ref> arises as the global sections of a perverse Schober, then we can use the techniques of symplectic topology and the methods developed in this paper to decompose and study _x in terms of simpler constituents. In addition to being of interest in its own right, the added flexibility afforded by the noncommutative framework could lead to new insights about classical hyperkähler geometry. amsalpha Ludmil Katzarkov Universität Wien, Fakultät für Mathematik, 1090 Wien, Österreich National Research University, Higher School of Economics, Russian Federation Email: lkatzarkov@gmail.com Pranav Pandit Universität Wien, Fakultät für Mathematik, 1090 Wien, Österreich Email: pranav.pandit@univie.ac.at Ted Spaide Universität Wien, Fakultät für Mathematik, 1090 Wien, Österreich Email: theodore.spaide@univie.ac.at

http://arxiv.org/abs/1701.07591v2

20170126070414

Is BaCr$_2$As$_2$ symmetrical to BaFe$_2$As$_2$ with respect to half $3d$ shell filling?

cond-mat.supr-con

p.richard@iphy.ac.cn Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing, China CEA, LITEN, 17 Rue des Martyrs, 38054 Grenoble, France Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing, China Diamond Light Source, Harwell Campus, Didcot, OX11 0DE, United Kingdom Diamond Light Source, Harwell Campus, Didcot, OX11 0DE, United Kingdom Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing, China Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6114, USA Centre de Physique Théorique, Ecole Polytechnique, CNRS UMR 7644, Université Paris-Saclay, 91128 Palaiseau, France Collège de France, 11 place Marcelin Berthelot, 75005 Paris, France European Theoretical Synchrotron Facility, Europe Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China Collaborative Innovation Center of Quantum Matter, Beijing, China We have performed an angle-resolved photoemission spectroscopy study of BaCr_2As_2, which has the same crystal structure as BaFe_2As_2, a parent compound of Fe-based superconductors. We determine the Fermi surface of this material and its band dispersion over 5 eV of binding energy. Very moderate band renormalization (1.35) is observed for only two bands. We attribute this small renormalization to enhanced direct exchange as compared to Fe in BaFe_2As_2, and to a larger contribution of the e_g orbitals in the composition of the bands forming the Fermi surface, leading to an effective valence count that is reduced by Fe d - As p hybridization. Is BaCr_2As_2 symmetrical to BaFe_2As_2 with respect to half 3d shell filling? H. Ding December 30, 2023 ============================================================================== § INTRODUCTION The electronic correlations are widely believed to play a major role for unconventional superconductivity in the Fe-based superconductors <cit.>. It was established theoretically that the strength of the electronic correlations in these materials is tuned by the filling of the 3d shell in presence of a large Hund's coupling <cit.>. Accordingly, angle-resolved photoemission spectroscopy (ARPES) studies report that the band renormalization factor 1/Z decreases monotonically upon filling the 3d shell: 1/Z=3 in BaFe_2As_2 (d^6) <cit.> (consistent with density functional theory (DFT) + dynamic mean-field theory (DMFT) calculations <cit.>), 1/Z=1.4 in BaCo_2As_2 (d^7) <cit.>, 1/Z=1.1 in SrNi_2As_2 (d^8) <cit.>, and 1/Z=1 in BaCu_2As_2 (d^10) <cit.>. Within this framework, the electronic correlations should evolve similarly as a function of band filling with respect to the half 3d shell case (d^5). Therefore, the d^4 case of Cr^2+, the symmetric counterpart of the d^6 filling of Fe^2+, raises the possibility of unconventional superconductivity for compositions in proximity of BaCr_2As_2. As with BaFe_2As_2, BaCr_2As_2 is an antiferromagnetic metal, and a sizable renormalization factor of 2 was derived from specific heat measurements <cit.>. A recent DMFT study also suggests mass enhancement by a factor of 2 <cit.>, thus reinforcing the view that BaCr_2As_2 can be regarded as the symmetrical analog of BaFe_2As_2. A possible pairing instability is even proposed upon negative pressure or electron doping <cit.>. However, there is to date no direct experimental characterization of the electronic band dispersion of BaCr_2As_2 [After completion of our work, we have been aware of another ARPES study of BaCr_2As_2 showing results similar to ours <cit.>], which is essential to answer the question: Is BaCr_2As_2 symmetrical to BaFe_2As_2 with respect to half 3d shell filling? Here we present an ARPES study of BaCr_2As_2. Despite evidence for a surface state, the Fermi surface (FS) is very similar to the one expected from local density approximation (LDA) calculations. We find that while most bands are not renormalized, two bands near the Fermi energy (E_F) are renormalized by a factor 1.35, which is much smaller than reported theoretically. We show that unlike in BaFe_2As_2, the spectral weight at E_F in BaCr_2As_2 is significantly contaminated by e_g orbitals and that direct exchange between Fe neighboring atoms play a more significant role. From our experimental results and analysis, we conclude that although there are obvious similarities between BaCr_2As_2 and BaFe_2As_2, the analogy between these two compounds cannot be pushed too far. § METHODS High-quality single crystals of BaCr_2As_2 were grown by the self-flux method <cit.>. ARPES measurements were performed using photon energies (hν) of 56 eV and 73 eV at Beamline I05 of Diamond Light Source equipped with a VG-Scienta R4000 analyzer. The energy and angular resolutions were set at 12 - 15 meV and 0.2^o, respectively. Additional measurements in the 22 - 80 eV hν range have been recorded at the PGM beamline of the Synchrotron Radiation Center equipped with a VG-Scienta R4000 analyzer, with the energy and angular resolutions set at 20 - 30 meV and 0.2^o, respectively. All samples were cleaved in situ and measured at 20 K in a vacuum better than 5× 10^-11 Torr. Throughout the paper, we label the momentum values with respect to the 1 Cr/unit-cell Brillouin zone (BZ), and use c^' = c/2 as the distance between two Cr planes. We have performed DFT calculations within the LDA and in the G-type antiferromagnetic order, as implemented in the Wien2k code <cit.>. The lattice parameters were taken from experiment (a = 3.963 Å, c= 13.600 Å) <cit.> and we chose z_As = 0.3572, similar to what was found in the calculations of Singh et al. <cit.>. § RESULTS AND DISCUSSION In Figs. <ref>(a) and <ref>(b), we display our calculations of the FS of BaCr_2As_2 in the antiferromagnetic state for k_z=0 and k_z=π, respectively. Our results are consistent with previous calculations <cit.> and show that the FS differs substantially from that of the Fe-based superconductors. The calculations indicate the existence of 3 hole FS pockets centered at the Γ point (0,0,0), and 2 hole FS pockets centered at the Z point (π,π,0)=(0,0,π). Here we call α the inner FS pocket, which is 3D and does not appear around the Z point. The β band, which forms the β FS in the k_z=0 plane, disperses only slightly along k_z and gives the β' FS pocket around the Z point. In contrast, the γ band varies more strongly along k_z to give a γ FS pocket in the k_z=0 plane and a γ' FS pocket in the k_z=π plane that have different shapes and sizes. The orbital projections of the calculated band dispersions are shown in Fig. <ref>, with the weight of the majority spin (Cr1) and minority spin (Cr2) plotted separately. Our calculations indicate that both the α and β FSs derive mainly from Cr1 d_xz+d_yz, but that their orbital composition is not pure and includes other Cr1 and Cr2 d orbitals as well. In contrast, the γ band derives almost only from Cr1 d_xy. Although most of the As p states locate below E-E_F=-1.5 eV, our calculations also predict some weight for As p_z at E_F. The experimental sections of FS displayed in Figs. <ref>(c) and <ref>(e) for k_z≈ 0, and in Figs. <ref>(d) and <ref>(f) for k_z≈π, are very similar to the calculations. This is notably true for the γ and γ' FS pockets. Small discrepancies need to be reported though. For example, while the calculations predict a slightly octogonalish shape for the section of the β FS around Γ, what is observed experimentally is more squarish. The α FS in Figs. <ref>(c) and <ref>(e) is also circular, in contrast to the rather squarish calculated one. More importantly, a replica of that FS pocket is detected at the Z point, suggesting a band folding or a surface state. Our analysis shows that the α, β and γ sections of FS cover respectively 7.4 %, 34.5% and 63.3% of the 2Cr/unit-cell BZ (dashed red squares in Fig. <ref>). At k_z≈π, the β' and γ' sections of FS cover 50.2 % and 35.1 % of the 2Cr/unit-cell BZ. After averaging the total areas at k_z≈ 0 and k_z≈π, we find 95.3 % of the 2Cr/unit-cell BZ. Considering spin degeneracy, this coincides to slightly less than the expected d^4 filling of Cr^2+. Our approximation is justified by the k_y-k_z mapping showed in Fig. <ref>(m), obtained by scanning hν from 22 to 80 eV. The conversion from hν to k_z is done using the three-step model with the sudden approximation and the free-electron final state, and we used an inner potential of 13 eV <cit.>. The α FS, small compared to the others, is dispersionless, but has stronger intensity around the Γ point, giving further support for the existence of a surface state. While the β band is nearly dispersionless along k_z, the γ band shows sizable dispersion along k_z. In Figs. <ref>(a) and <ref>(b) we show the electronic band dispersion observed experimentally from ARPES intensity cuts along Γ-M (k_x=0) recorded using 73 eV photons (k_z≈ 0) in the s and p polarization configurations, respectively. In a similar fashion, we display in Figs. <ref>(g) and <ref>(h) the ARPES intensity cuts recorded with 56 eV photons (k_z≈π), corresponding to the Z-A high-symmetry line. The LDA calculations are superimposed for comparison. The curvature intensity plots <cit.> and the energy distribution curves (EDCs) corresponding to the ARPES cuts are displayed on their right side. The agreement between experiments and calculations is rather good over 4.5 eV and most bands would need only small energy shifts to match the ARPES intensity plot. While the s polarization configuration favors the observation of states with odd symmetry or with z-oriented orbitals such as d_z^2 or p_z, the p polarization configuration favors even symmetry states. Accordingly, the γ (d_xy) band is more intense under s polarization. The experimental results also indicate that the β and α bands have dominant odd and even symmetries, respectively. The most obvious difference between the calculations and the experimental results is the presence of the band labeled ζ_2 in Fig. <ref>, which is not predicted. As with the ζ_1 band associated to the Cr1 d_z^2 orbital, it appears clearly only under s polarization. Interestingly, the dispersions of the ζ_1 and ζ_2 bands are identical within uncertainties, and the location of the bottom of the ζ_2 band is nearly the same under both 56 eV and 73 eV photon excitations. Consequently, we conjecture that the ζ_2 band is a surface state with d_z^2 orbital character. As mentioned above, the unrenormalized LDA calculations are in good agreement with the experimental data. Even the α band, near E_F, is well reproduced by the unrenormalized LDA calculations. However, the situation is not as good for the β and γ bands. As we show in Fig. <ref> by zooming on the band dispersion near E_F, these two bands fit the calculations better after renormalizing the LDA bands by a factor of 1.35. This factor is far from that observed in the Fe-based superconductors and more consistent with the renormalization found in BaCo_2As_2 <cit.>, which corresponds to a d^7 filling. From that perspective, the case of BaCr_2As_2 is thus not symmetrical to BaFe_2As_2. A similar asymmetry was found theoretically <cit.> in the 1111 phase (corresponding to LaFeAsO and LaCrAsO). There, it was attributed to the orbital-dependence of the electronic filling <cit.>. Indeed, the t_2g orbitals (d_xz/d_yz and d_xy) are closer to half-filling for the d^6 configuration (Fe) than for a d^4 filling (Cr), whereas the less correlated e_2g orbitals (d_z^2 and d_x^2-y^2) are the ones closest to half-filling for the d^4 configuration <cit.>. In that picture, recovering a similar electronic mass enhancement as in LaFeAsO would require a d^n filling of around n=4.6 to 4.8. On the other hand, the situation might not be only determined by the respective orbital fillings. In fact, another – possibly more important – factor is the orbital-resolved kinetic energy that is larger for orbitals having a low density-of-states at the Fermi level. As discussed in Ref. <cit.> for FeSe, the orbitals that exhibit a single-particle pseudogap are less correlated. This can be rationalized by the presence of more possibilities for fluctuations around the Fermi level in the other three orbitals. A similar mechanism could be at work in BaCr_2As_2. We now try to understand why the 122 structure leads to superconductivity with a high superconducting transition temperature (T_c) for Fe-based compounds, whereas no superconductivity is reported for their Cr-based counterparts. The strength of the electronic correlations, which we show in this work to be very different in these two systems, may play an important role. Possible superconductivity was predicted for 3d^n fillings somewhere between n=4 and n=5 <cit.>, where the strength of the electronic correlations should be comparable to that in Fe-based superconductors. Yet, additional issues need to be addressed. For instance, the magnetic ordering is different in the two systems. BaCr_2As_2 shows a G-type ordering, and a Néel temperature of 580 K has been reported <cit.>, with Cr spins pointing along the c-axis in EuCr_2As_2 <cit.>. In contrast, BaFe_2As_2 orders antiferomagnetically around 140 K with spins aligned inside the Fe plane in a bi-collinear structure <cit.>. The antiferromagnetic wave vector in the Fe-based superconductors connects hole and electron pockets FS pockets, giving rise to the quasi-nesting scenario suggesting that low-energy spin fluctuations may contribute to the superconducting pairing <cit.>. Although the quasi-nesting scenario is now seriously challenged <cit.>, in particular due to the existence of high-T_c superconductivity without hole FS pocket in A_xFe_2-xSe_2 <cit.> and FeSe monolayers <cit.>, the Γ-M wave vectors connects the top of the holelike bands and the bottom of the electron bands within an energy range smaller than a few hundreds of meV in all Fe-based superconductors, suggesting that high-energy spin fluctuations (local moments) may play a role. There is no such electron-hole connection with the antiferromagnetic wave vector in BaCr_2As_2. The difference in the magnetic ordering of BaFe_2As_2 and BaCr_2As_2 hides another important property of their respective electronic structures. Although it is known that the exchange interactions between next nearest Fe neighbors (super-exchange J_2) is determinant in shaping the magnetic structure of the Fe-based superconductors <cit.>, G-type antiferromagnetism is favored by interactions between the nearest Cr neighbors (direct exchange J_1). This suggests stronger d orbital mixing in BaCr_2As_2, which is compatible with our LDA calculations, as mentioned above. It was argued recently that the mixing of the t_2g and e_g orbitals (which implies enhanced direct exchange) was detrimental to superconductivity with high T_c, and that a suitable electronic configuration that leaves pure and half-filled t_2g orbitals at E_F in structures with tetrahedrally-coordinated transition metals was found only for the d^6 filling of Fe^2+ <cit.>. With that respect, BaFe_2As_2 and BaCr_2As_2 are very different. Finally, one might speculate that the enhanced d-p-hybridization leads to an effectively reduced d-electron count in BaCr_2As_2, which would then be consistent with a correlation strength comparable to the d^7 compound BaCo_2As_2. A similar mechanism was indeed discussed in Ref. <cit.> where it was highlighted that the correlation strength depends on the effectively available charge rather than on the nominal d-electron count. Finally, one might also speculate that the increased hybridization also influences the effective crystal field splittings (through ligand field effects) thus modifying more profoundly the scenario of the Hund's metal phase appearing around the d^6 filling. § SUMMARY In summary, we performed an ARPES study of BaCr_2As_2 that indicates good consistency with LDA calculations in the antiferromagnetic state. A renormalization factor of 1.35 is observed for only two bands, indicating that BaCr_2As_2 is much less correlated than BaFe_2As_2. Our analysis suggests a stronger weight of the e_g orbitals at E_F, which may be responsible for the observed asymmetry with respect to half 3d shell filling. § ACKNOWLEDGEMENT We thank E. Bascones for useful discussions. This work was supported by grants from MOST (Grants Nos. 2015CB921301, 2016YFA0300300 and 2016YFA0401000) and NSFC (Grants Nos. 11274362 and 11674371) from China, a Consolidator Grant of the European Research Council (Project No. 617196) and supercomputing time at IDRIS/GENCI Orsay (Project No. t2016091393). We acknowledge Diamond Light Source for time on beamline I05 under proposal SI9469, which contributed to the results presented here. This work is based in part on research conducted at the Synchrotron Radiation Center, which was primarily funded by the University of Wisconsin-Madison with supplemental support from facility users and the University of Wisconsin-Milwaukee. The work at ORNL was supported by the Department of Energy, Basic Energy Sciences, Materials Sciences and Engineering Division.

http://arxiv.org/abs/1701.07843v1

20170126190707

Galactic Winds with MUSE: A Direct Detection of FeII* Emission from a z = 1.29 Galaxy

astro-ph.GA

Université de Toulouse, UPS-OMP, 31400 Toulouse, France hayley.finley@irap.omp.eu IRAP, Institut de Recherche en Astrophysique et Planétologie, CNRS, 14 avenue Édouard Belin, 31400 Toulouse, France IRAP, Institut de Recherche en Astrophysique et Planétologie, CNRS, 9 avenue Colonel Roche, 31400 Toulouse, France Aix Marseille Univ, CNRS, LAM, Laboratoire d'Astrophysique de Marseille, Marseille, France CRAL, Observatoire de Lyon, CNRS, Université Lyon 1, 9 Avenue Ch. André, F-69561 Saint Genis Laval Cedex, France Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, PT4150-762 Porto, Portugal ETH Zurich, Institute of Astronomy, Wolfgang-Pauli-Str. 27, CH-8093 Zürich, Switzerland Observatoire de Genève, Université de Genève, 51 Ch. des Maillettes, 1290 Versoix, Switzerland Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, D-14482 Potsdam, Germany Institut für Physik und Astronomie, Universität Potsdam,Karl-Liebknecht-Str. 24/25, 14476 Golm, Germany Emission signatures from galactic winds provide an opportunity to directly map the outflowing gas, but this is traditionally challenging because of the low surface brightness. Using very deep observations (27 hours) of the Hubble Deep Field South with the Multi Unit Spectroscopic Explorer (MUSE) instrument, we identify signatures of an outflow in both emission and absorption from a spatially resolved galaxy at z=1.29 with a stellar mass , star formation rate , and star formation rate surface brightness within the λλ3727,3729 half-light radius R_1/2, = kpc. From a component of the strong resonant MgII and FeII absorptions at -350 km s^-1, we infer a mass outflow rate that is comparable to the star formation rate. We detect non-resonant FeII* emission, at λ2626, λ2612, λ2396, and λ2365, at 1.2 - 2.4 - 1.5 - 2.7 × 10^-18 respectively. These flux ratios are consistent with the expectations for optically thick gas. By combining the four non-resonant emission lines, we spatially map the FeII* emission from an individual galaxy for the first time. The emission has an elliptical morphology that is roughly aligned with the galaxy minor kinematic axis, and its integrated half-light radius, R_1/2, =  kpc, is 50% larger than the stellar continuum (R_1/2,⋆≃ ) or the nebular line. Moreover, the FeII* emission shows a blue wing extending up to -400 km s^-1, which is more pronounced along the galaxy minor kinematic axis and reveals a C-shaped pattern in a p-v diagram along that axis. These features are consistent with a bi-conical outflow. Galactic Winds with MUSE: A Direct Detection of FeII* Emission from a z=1.29 Galaxy Based on observations of the Hubble Deep Field South made with ESO telescopes at the La Silla Paranal Observatory under program ID 60.A-9100(C). Advanced data products are available at http://muse-vlt.eu/ science. Hayley Finley1,2 Nicolas Bouché3 Thierry Contini1,2 Benoît Epinat 1,2,4 Roland Bacon 5 Jarle Brinchmann 6,7 Sebastiano Cantalupo 8 Santiago Erroz-Ferrer 8 Raffaella Anna Marino 8 Michael Maseda 6 Johan Richard 5 Anne Verhamme 5,9 Peter M. Weilbacher 10 Martin Wendt 10,11 Lutz Wisotzki 10 ===================================================================================================================================================================================================================================================================================================================================================================== § INTRODUCTION Galactic winds, driven by the collective effect of hot stars and supernovae explosions, play a major role in regulating galaxy evolution. By expelling enriched matter beyond the halo, galactic winds can address discrepancies between observations and ΛCDM models that over-predict the number of low-mass galaxies <cit.> and enrich the intergalactic medium <cit.>. Likewise, galactic winds may play a major role in regulating the mass-metallicity relation <cit.>. Therefore, quantifying the mass fluxes of galactic outflows (and their extents) is necessary to gain a complete understanding of galaxy evolution. However, while galactic winds appear ubiquitous <cit.>, observational constraints for the physical properties of galactic outflows, including their extents and mass outflow rates, are sparse. Traditional "down the barrel" 1D galaxy spectroscopy provides direct constraints on the wind speed from the blue-shifted absorption lines but cannot constrain the physical extent of outflows, leading to large uncertainties in outflow rates. Techniques that use a background source can address this question. For instance, the background quasar technique provides constraints on the physical extent of gas flows from the impact parameter between the galaxy and the absorbing gas <cit.>. These recent studies have made progress investigating the kinematics, orientation, and extent of gas flows around star forming galaxies. As a variation on this technique, spectroscopy against a background galaxy probes absorption from the foreground galaxy halo over a larger solid angle <cit.>. However, these constraints on the physical extent of outflows are usually limited due to their one-dimensional nature, except for <cit.>. Mapping the extent of gas flows in two dimensions is critical to better constrain mass outflow rates. Mapping outflows in emission, such as for M82 <cit.> and other nearby galaxies <cit.>, is difficult at high redshift, because the emitting gas inherently has a very low surface brightness. Beyond the local universe, galaxies with emission signatures from outflows are beginning to be detected. Currently, rest-frame UV and optical spectroscopy use three types of emission signatures to map the extent of outflows: the nebular, resonant, and non-resonant emission lines. The most common nebular emission lines seen in HII regions are hydrogen recombination and forbidden lines, such as [OII] λλ3727,3729. A transition is resonant when a photon can be absorbed from the ground state and re-emitted to the same lowest level of the ground state, as for Lyman-alpha and the MgII λλ2796,2803 transitions. A transition is non-resonant when the photon can be re-emitted to an excited level of a ground state that has multiple levels due to fine structure splitting. Non-resonant transitions are commonly denoted with a *, like FeII*. Due to the slight energy difference between the ground and excited states, photons from non-resonant emission no longer have the correct wavelength to be re-absorbed through a resonant transition and instead escape. In other words, the gas is optically thin to photons that are emitted through a non-resonant transition. The first type of emission signature (nebular lines) from outflows can appear as a broad component in nebular emission lines such as . Such broad component is regularly seen in local Ultra-Luminous Infra-Red Galaxies <cit.> and more recently in normal star-forming galaxies <cit.>. At high redshifts, <cit.> detected a broad component in composite spectra of z ∼ 2 star-forming galaxies and <cit.> observed this broad component in a few individual galaxies. <cit.> found that the broad emission is spatially extended beyond the half-light radius, R_1/2. The second possible emission signature of outflows comes from resonant transitions such as , a line which is often more extended than the stellar continuum <cit.> but might be strongly affected by dust absorption. Emission from resonant metal lines, such as Siii, Feii, or Mgii, is less affected by dust and may be observed as P-cygni profiles. The relative strength between the (mostly) blueshifted absorption and (mostly) redshifted emission dictates whether the signature appears as a traditional P-cygni profile or as emission `infilling'. Contrary to the resonant FeII lines observed across a similar wavelength range (FeII λ2344, λλ 2374,2382, and λλ 2586,2600), the MgII doublet is particularly sensitive to emission infilling, since its lower energy level does not have fine structure splitting. As a result of the different possible relative strengths of the emission and absorption components, observed profiles for the resonant MgII λλ 2796,2803 transitions vary greatly for different star-forming galaxies <cit.>. The third possible signature of outflows in emission is from non-resonant transitions such as CII*, SiII* <cit.> or FeII* <cit.>. Detecting non-resonant emission typically requires stacking hundreds of galaxy spectra. Using more than 800 Lyman break galaxies (LBGs) at z > 2, <cit.> first detected SiII* in the composite spectrum, and <cit.> more recently detected CII* and SiII* in the composite spectrum of 59 LBGs. Since the non-resonant FeII* lines are at redder wavelengths than CII* and SiII*, they are practical for investigating outflows at lower redshifts, like z∼1. Based on comparing composite spectra from samples of ∼ 100 or more star-forming galaxies at z ∼ 1-2 <cit.>, FeII* emission may vary with galaxy properties, such as galaxy mass and dust attenuation. <cit.> present individual spectra with different combinations of blue-shifted absorption, resonant MgII emission, and non-resonant FeII*. In two notable direct detections of FeII* emission from galaxies at z = 0.694 and z = 0.9392 <cit.>, the non-resonant emission is observed along with blue-shifted absorption lines and resonant MgII emission, allowing the authors to constrain and model the outflows. Similarly, <cit.> use non-resonant CII* and SiII* emission in UV spectra of four green pea galaxies at z ∼ 0.14 - 0.2 to infer the geometry of their outflows. These studies provide information about outflow properties on galactic scales, but it is also possible to characterize outflows from individual star-forming regions across z > 1 galaxies thanks to adaptive optics or gravitational lensing <cit.>. Using adaptive optics, <cit.> identify star-forming regions in five z > 2 galaxies and argue that bright regions (or clumps) with a broad component in the nebular emission are the launch sites for massive galactic winds. With the benefit of gravitational lensing, <cit.> characterize MgII emission, FeII* λλ2612, 2626 emission, and FeII absorption from multiple star-forming regions across a supernova host galaxy at z = 1.49 at locations both associated with and independent of the supernovae explosion. <cit.> likewise detect blueshifted FeII and MgII absorptions, redshifted MgII emission, and non-resonant FeII* λλ2612, 2626 emission in four star-forming regions of a gravitationally lensed galaxy at z = 1.70, but find that the outflow properties vary from region to region. Spatially resolved observations suggest that outflow properties could be localized and strongly influenced by the nearest star-forming clump. Despite advances from these diverse studies, we have not yet been able to map the morphology and extent of outflows from individual galaxies beyond the local universe. The new generation of integral field spectrographs, the Multi Unit Spectroscopic Explorer <cit.> on the VLT and the Keck Cosmic Web Imager <cit.>, are well-suited for studying galactic winds in emission and tackling this challenge. While slit spectroscopy can inadvertently miss scattered emission if the aperture does not cover the full extent of the outflowing envelope <cit.>, integral field observations eliminate aperture effects for distant galaxies, making emission signatures easier to detect. The combined spatial and spectroscopic data facilitate characterizing the morphology and kinematics of both star-forming galaxies and the outflows they produce. In this paper, we analyze galactic wind signatures from a spatially resolved star-forming galaxy at z = 1.2902 observed with MUSE. We present the observations in Section <ref> and summarize the galaxy properties in Section <ref>. With the integrated 1D MUSE galaxy spectrum, we characterize outflow signatures from FeII, MgII, and MgI transitions in absorption and FeII* transitions in emission in Section <ref>. We then investigate the spatial extent and the kinematic properties of the FeII* emission in Sections <ref> and <ref>, respectively. In Section <ref>, we compare our observations with radiative transfer wind models and estimate the mass outflow rate. We review our findings in Section <ref>. Throughout the paper, we assume a ΛCDM cosmology with Ω_ m = 0.3, Ω_Λ = 0.7, and H_0 = 70 km s^-1 Mpc^-1. With this cosmology, 1 arcsec corresponds to 8.37 kpc at the redshift of the galaxy. § DATA MUSE fully covers the wavelength range 4650-9300 Å with 1.25 Å per spectral pixel. The field of view spans 1× 1 with a pixel size of 0.2. The instrument is notable both for its high throughput, which reaches 35% at 7000 Å (end-to-end including the telescope), and its excellent image quality sampled at 0.2 per spaxel. While MUSE opens new avenues to address a wide variety of scientific questions, these two characteristics make the instrument optimal for deep field observations. As part of commissioning data taken during July and August 2014, MUSE observed a 1× 1 field of view in the Hubble Deep Field South (HDFS) for a total integration time of 27 hours. The final data cube is a 5σ-clipped mean of 54 individual exposures that were taken in dark time under good seeing conditions (0.5-0.9). The 1σ emission-line surface brightness limit for this cube is 1 × 10^-19 erg s^-1 cm^-2 arcsec^-2. The MUSE observations provided spectroscopic redshifts for 189 sources with magnitude I_814≤ 29.5 (8 stars and 181 galaxies), a factor-of-ten increase over the 18 previously-measured spectroscopic redshifts in this field. A catalogue of sources in the MUSE HDFS field includes the redshifts, emission-line fluxes, and 1D spectra. The observations, the data cube, and an overview of scientific exploitations are fully described in <cit.>. Both the data cube and the catalogue of sources are publicly available.[<http://muse-vlt.eu/science/hdfs-v1-0/>] The deep IFU observations reveal emission from FeII* transitions directly detected from one galaxy in the MUSE HDFS. The galaxy has ID #13 in the MUSE catalogue, with coordinates α = 22h 32m 52.16s, δ = -60^∘ 33 23.92 (J2000) and magnitude I_814 = 22.83 ± 0.005. It is part of a 9-member group at z ≃ 1.284, discussed in <cit.>, that also includes two AGN and an interacting system with tidal tails. This direct detection of a galaxy with FeII* emission offers a new opportunity to characterize galactic winds. § GALAXY PROPERTIES Galaxy ID#13 is part of a sample of 28 spatially resolved galaxies that <cit.> selected from the MUSE HDFS according to the criterion that the brightest emission line covers at least 20 spatial pixels with a signal-to-noise ratio (S/N) higher than 15. For this galaxy, emission from the [OII] λλ3727,3729 doublet is the dominant feature in the MUSE spectrum. We determined the galaxy systemic redshift from a p-v diagram extracted from the MUSE data cube along the galaxy kinematic major axis by fitting a double Gaussian profile to the [OII] λλ3727,3729 emission at each position along the slit. The systemic redshift of z = 1.29018 ± 0.00006 is the mean value between the two asymptotes of the rotation curve. <cit.> investigated the morphological and kinematic properties of the galaxy ID#13, as part of the MUSE HDFS spatially resolved galaxy sample. They constrained the morphology from HST images in the F814W band by modelling the galaxy with Galfit <cit.> as a bulge plus an exponential disk. <cit.> then performed the kinematic analysis with two different techniques: a traditional 2D line-fitting method with the Camel algorithm <cit.> combined with a 2D rotating disk model, which requires prior knowledge of the galaxy inclination, and a 3D fitting algorithm, GalPaK^3D <cit.>, which simultaneously fits the morphological and kinematic parameters directly from the MUSE data cube. The parameters from the 2D and 3D models are in good agreement overall (see Table <ref>). From the morphological analysis on the HST images, galaxy ID#13 is compact with a disk scale length of R_d=1.25 kpc (correspondingly R_1/2=2.1 kpc) and has a low inclination angle of i=33^∘. The inclination from 3D fitting yields a lower value of ∼ 20^∘. The disagreement likely arises from an asymmetric morphology seen in the HST images, since statistically the two techniques measure inclinations that are in good agreement <cit.>. The galaxy also shows a misalignment between the morphological position angle measured from the HST image, -46^∘, and the MUSE kinematic position angle, -13^∘, again likely due to the asymmetric light distribution that only appears at higher spatial resolution. Regardless, the galaxy has a low inclination with i∼ 20^∘ - 30^∘. From the kinematic analysis on the MUSE data, the velocity field has a low gradient, ± 10 km s^-1, a low maximum velocity, 24 km s^-1, and a velocity dispersion of 45-50 km s^-1. Therefore, non-circular motions dominate the gas dynamics within the disk, with V/σ≈ 0.5, i.e. below the commonly-used V/σ≤ 1 threshold for identifying dispersion-dominated galaxies. Note that the different maximum velocities from the 2D and 3D methods are entirely due to the different inclination values (Table <ref>). Nonetheless, the ratio remains V/σ≲ 1 for the range of possible inclinations, 17^∘-33^∘. <cit.> estimated the visual extinction, A_V = 1.20 mag, stellar mass, , and star formation rate , from Stellar Population Synthesis using broad-band visible and near infra-red photometry [The [OII]-derived SFR for a <cit.> IMF is 65 M_⊙ yr^-1 using the <cit.> calibration, which also yields an extinction of A_V=1.5 in the gas.]. The galaxy ID#13 is one of the most massive of the 28 spatially-resolved galaxies in the MUSE HDFS sample and also has the highest star formation rate (SFR). This SFR places galaxy ID#13 above the main sequence <cit.> by almost 1 dex, indicating that this galaxy is undergoing a starburst with a high specific SFR of sSFR =10 Gyr^-1. The starburst phase of galaxy evolution can produce large-scale outflows when many short-lived massive stars explode as supernovae. The properties of this galaxy are conducive to detecting signatures from galactic winds. The low inclination angle favors observing blue-shifted absorptions, given that this signature increases substantially towards face-on galaxies <cit.>. The [OII] luminosity (∼ 10^43 erg s^-1) and rest-frame equivalent width (∼ 50 Å, see Table <ref>) indicate that the galaxy ID#13 is also well-suited for investigating winds in emission, since FeII* and MgII emission correlate with L_OII or [OII] rest-frame equivalent width <cit.>. § ABSORPTION AND EMISSION PROFILES FROM THE 1D SPECTRUM In this section, we analyze the galaxy ID#13 1D spectrum extracted from the MUSE data using a white-light weighting scheme. The 1D MUSE spectrum (Figure <ref>) reveals resonant FeII, MgII, and MgI self-absorption, non-resonant FeII* emission, and CII] and [OII] nebular emission lines. The transitions occur in three multiplets[See <cit.> or <cit.> for energy level diagrams.]. In the FeII UV1, UV2, and UV3 multiplets, a photon can be re-emitted either through a resonant transition to the ground state, which produces emission infilling, or through a non-resonant transition to an excited state in the lower level, in which case the emission occurs at a slightly different wavelength. We investigate the integrated absorption and emission profiles, focusing first on the resonant absorption and emission properties (Section <ref>), then on the non-resonant emission properties (Section <ref>). §.§ Resonant Fe and Mg profiles Figure <ref> compares the velocity profiles of each of the individual FeII, MgII, and MgI transitions relative to the galaxy systemic redshift. The self-absorption profiles are asymmetric, with the strongest component centered on the galaxy systemic redshift, and a significant blue wing extending to -800 km s^-1. We fit these profiles simultaneously with VPFIT[<http://www.ast.cam.ac.uk/ rfc/vpfit.html>] v10, using several components and requiring each to have the same redshift and Doppler parameter across the different transitions. The absorptions are well-fitted with three components at redshifts 1.28514 ± 0.00021, 1.28752 ± 0.00009, and 1.29024 ± 0.00006, corresponding to shifts of -660 ± 28 km s^-1, -349 ± 12 km s^-1 and +8.5±6.5 km s^-1 relative to the galaxy systemic velocity. Table <ref> summarizes the total rest-frame equivalent widths for each transition, calculated both from the fit and directly from the flux. Globally, the resonant transitions in Figure <ref> reveal several key features: (1) the profiles are very similar to one another, and (2) the strongest component is roughly centered at the galaxy systemic redshift. As <cit.> first demonstrated, emission infilling in resonant absorption lines can alter doublet ratios and mimic partial coverage. However, here we find that emission infilling does not play a significant role in this galaxy for the following two qualitative arguments. First, while strong emission infilling would produce clear P-cygni profiles (which are not observed), moderate amounts of emission infilling would cause a blue-shift to the centroid of the absorption, an effect commonly seen in stacked spectra <cit.> or individual cases <cit.>. None of the absorptions in the galaxy ID#13 spectrum (Figure <ref>) have blue-shifted centroids. Second, because has multiple channels to re-emit the photons (through resonant and non-resonant transitions), the degree of infilling for a particular absorption line depends on the likelihood of re-emission through the different channels within a multiplet. Purely resonant transitions, such as MgII and λ2383, are the most sensitive to emission infilling. <cit.> demonstrated that the resonant absorptions that are the least (most) affected by emission infilling are λ2374 ( λ2600 and λ2383) respectively. Figure <ref> shows that the λ2374, λ2600 and λ2383 absorption profiles are all very similar for the galaxy ID#13. The lack of blue-shifted centroids and the consistent absorption profiles argue strongly against the presence of detectable emission infilling in this galaxy. We quantify (and put a limit on) the global amount of infilling using the method proposed by <cit.>, which consists of comparing the observed rest-frame equivalent widths of the resonant lines to those seen in intervening quasar spectra (see their Figure 12). Using the averaged rest-frame equivalent widths of resonant and MgII absorptions from a stacked spectrum of ∼ 30 strong MgII absorber galaxies at 0.5<z<1.5 from <cit.>, we find that our data is consistent with no emission infilling. Our data could allow for at most <0.8 Å (<1.8 Å) of infilling for λ2600 ( λ2383), the two transitions most susceptible to infilling <cit.>. This means that at most 22% (55%) of these absorptions could be affected by infilling and that the impact on the other absorptions is even smaller. Similarly, we separately estimate the amount of infilling for each of the three sub-components shown in Figure <ref> (Table <ref>). We are unable to put constraints on the weak component `A', but the blue-shifted component `B' at -350 km s^-1 does not allow for emission infilling that would increase the λ2383 equivalent width by more than 10%. The component `C' at the galaxy systemic redshift allows for the largest amount of emission infilling with 60% corrections for λ2600 and λ2383, 40% for λ2344 and 20% for λ2586. As we discuss later in Section <ref>, the blue-shifted galactic wind component (`B') appears to be less affected by emission infilling than the systemic component associated with the galaxy ISM (`C'). We end this section by mentioning that, as we will argue in section <ref>, the and MgII gas is likely optically thick. The absorptions ought to be saturated, and the reason we do not observe fully absorbed profiles is either due to a partial covering fraction (rather than emission infilling) or more likely to the low spectral resolution. As we will show in the next section, the non-resonant emission pattern is also consistent with optically thick gas. §.§ Non-resonant emission Figure <ref> shows the non-resonant transitions FeII* λ2365, λ2396, λ2612, and λ2626 that we detect in the MUSE HDFS galaxy ID#13 1D spectrum at 2.5σ - 6σ significance. No FeII* λ2632 emission is detected (Figure <ref>). The fluxes in the non-resonant transitions FeII* λ2365, λ2396, λ2612, λ2626 transitions are 1.2 - 2.4 - 1.5 - 2.7 ×10^-18 , respectively. Table <ref> gives the emission peak fluxes and rest-frame equivalent widths measured for all of the FeII* transitions. These flux ratios of 0.5:1.0:0.6:1.0 are consistent with the expectation (0.66:1.0:0.66:1.0) for optically thick gas discussed in <cit.>. In the optically thin regime, the flux ratios should be on the order of ∼ 1. Regarding the non-detection of λ2632, we note that this transition is usually not detected in stacked spectra <cit.>, except for in the <cit.> stacked spectrum, but that it is observed in the other individual cases <cit.>. <cit.> explore whether underlying stellar absorption suppresses the FeII* λ2632 emission in their stacked spectra. However, for this starburst galaxy, the F- and G-type stars that produce the underlying absorption are unlikely to significantly contribute to the stellar continuum. We perform a joint Gaussian fit to the four non-resonant emission peaks and find that they appear symmetric and centered on the galaxy systemic redshift measured from [OII] λλ3727,3729 (Figure <ref>). This is in contrast to <cit.>, who found that the emission from their stacked spectrum of 8,600 galaxies is slightly asymmetric, and in contrast to <cit.>, who observed FeII* emission peaks that are slightly (∼ 30 km s^-1) redshifted relative to the nebular emission lines. § MORPHOLOGY OF THE FEII* EMISSION In this section, we investigate whether the FeII* emission has a similar spatial extent and morphology as the stellar continuum and the [OII] λλ3727,3729 emission. For the FeII* emission, first we produce a sub-cube of size 1.5× 1.5 for each of the four emission lines and transform the wavelength axis to velocity space. We interpolate each sub-cube to the same velocity scale with pixels of 30 km s^-1 that span ±930 km s^-1 and zero velocity at the galaxy systemic redshift, z = 1.2902. We subtract the continuum and combine the four sub-cubes. To estimate the stellar continuum, we use the mean value from two regions redwards of the FeII* emission peaks at ∼λ2425 and ∼λ2700 Å that span 115 Å and 300 Å respectively. The continuum pseudo narrow band image shown in Figure <ref> (middle left) is from the mean of these two continuum regions, which have a flat slope. From the combined FeII* emission velocity cube, we then extract a narrow band (NB) image by summing 13 pixels (± 390 km s^-1). The top left panel of Figure <ref> shows the pseudo-narrow band FeII* image with 2 × 2 smoothing, and we use this image for the analysis. For comparison, we also tested an automated extraction with the CubExtractor software (Cantalupo et al. in prep.), which selects connected volume pixels (voxels) that are above a specified SNR threshold (2.7 was optimal in our case) to produce optimally extracted images, as in <cit.>. Our morphological results are independent of the method used to produce the FeII* NB image. Similarly, we create the [OII] pseudo-narrow band image from a 30 × 30 pixel (1.5× 1.5) sub-cube that spans 18 spectral pixels (22.5 Å) to cover the λλ 3727,3729 doublet. Again, we subtract the continuum estimated between ∼3550 - 3600 Å to obtain the [OII] surface brightness map shown in the bottom left panel of Figure <ref>. The FeII* map in Figure <ref> is the first two-dimensional spatial map of the non-resonant emission in a individual galaxy at intermediate redshift. Previous studies have looked for signatures of extended FeII* emission in stacked spectra <cit.>. In a stacked spectrum from 95 star-forming galaxies at 1 < z < 2, <cit.> found that the FeII* λ2626 emission line is slightly more spatially extended that the stellar continuum. <cit.> performed a similar analysis with 97 star-forming galaxies at 1 ≲ z ≲ 2.6, but were not able to spatially resolve the FeII* emission. Thanks to the sensitivity of MUSE, we are able to address whether FeII* is more extended than the continuum and to characterize the FeII* emission morphology for the first time. The top left panel of Figure <ref>) shows that the extended FeII* emission (solid contour) appears to be more extended than the continuum (dashed contour) and has a privileged direction. Comparing the FeII* emission position angle with the kinematic axis of the galaxy, indicated with crosses, shows that the FeII* is more extended along the minor kinematic axis of the galaxy. To quantify the extent of the FeII*, stellar continuum, and [OII] λλ3727,3729 emission, we use a custom Python MCMC algorithm to fit each of the surface brightness maps in the left column of Figure <ref> with a Sersic profile. The fit provides us with intrinsic parameters and with an intrinsic model of the emitting region, i.e. deconvolved from the seeing, because we convolve the Sersic profile with the actual PSF taken from the brightest star in the same data cube, MUSE HDFS ID#1 <cit.>, across wavelengths corresponding to the galaxy emission lines [The PSF can be approximately described by a Moffat profile with FWHM 0.70 (0.63) at the FeII* and stellar continuum emission ([OII] emission) wavelengths, which corresponds to a half-light radius of 0.50 (0.44).]. In practice, we fix the Sersic index n to n=1 or n=0.5 because the Sersic index n is unconstrained [The Sersic n index is unconstrained because the seeing radius is much larger than the emission. Indeed, the seeing radius is FWHM/2=0.35, corresponding to R_1/2≈0.5 for a Moffat profile, whereas the galaxy's intrinsic half-light radius R_1/2 is only ≈0.3.]. The size estimate, R_1/2, is nonetheless robust and independent of the Sersic index n, since it is determined empirically from the flux growth curve, an integrated quantity. Table <ref> summarizes the results from this analysis and Figure <ref> (middle column) shows the modeled profiles for n=1 for the FeII*, stellar continuum, and [OII] emission. The right column of Figure <ref> gives the residual maps, which are the difference between the observed data and the intrinsic model convolved with the seeing. The stellar continuum emission (Figure <ref>, middle row) appears round and compact. The intrinsic emission from the exponential disk fit yields an inclination of 28 ± 3^∘ and a half-light radius, R_1/2, of around ( kpc). These continuum emission properties from MUSE are comparable to the measurements from HST images discussed in Section <ref> and shown in Table <ref>. The [OII] λλ3727,3729 emitting region has the same morphology but is slightly more extended than the stellar continuum with R_1/2, = ( kpc). The corresponding star formation rate surface density is . The FeII* emission has a morphology and physical extent that are different from the stellar continuum and [OII] emission. The intrinsic FeII* emission is more elliptical with an axis ratio of b/a=0.57, compared to the rounder continuum and [Oii] emission, which both have b/a≃0.9. The FeII* emission is elongated along the direction (PA≈ +60^∘) that roughly corresponds to the galaxy minor kinematic axis (PA≈ +75^∘, Table <ref>). Moreover, the intrinsic half-light radius of the FeII* emission is R_1/2, = , i.e. about 50% larger than that of the stellar continuum. In other words, the FeII* half-light radius, R_1/2, =  kpc, extends ≳ 1.5 kpc beyond the stellar continuum and the [OII] emission, which both have R_1/2≈ 2.5 kpc (Table <ref>). This is apparent from comparing the extent of the FeII* emission (solid contour) to the continuum emission (dashed contour) in the top left panel of Figure <ref>. See Table <ref> for the emission properties. § KINEMATICS OF THE FEII* EMISSION In this section, we investigate whether it is possible to trace the kinematics of the FeII* emission. To do so, we visually inspected the velocity cube produced in the previous section and found that the kinematic major axis from the FeII* emission follows a PA of about 70 deg, which happens to correspond roughly to the galaxy minor kinematic axis. Figure <ref> shows p-v diagrams for this 70 deg slit orientation (bottom row) and for a slit oriented at -20 deg (top row). In both cases, the slit width is 1. Following the peak of the emission, we see that the FeII* emission has a velocity gradient along the galaxy minor kinematic axis. The white solid line in the bottom p-v diagram guides the eye along this velocity gradient. Black contours trace the FeII* emission, and the red shaded area indicates the continuum. Figure <ref> reveals two additional results. First, the FeII* emission shows an extended blue-wing, which is more pronounced in the PA = +70 profile. Secondly, the blue-side of the FeII* emission contours in the bottom panel, with the slit oriented at +70 deg, shows a C-shaped pattern. The contours extend to -400 km s^-1 near +1 and -1.5, but decrease to -200 km s^-1 in between. This C-shape pattern is characteristic of a hollow conical emission, as could be expected from an outflow. § DISCUSSION From deep MUSE observations of the HDFS, we identify a spatially-resolved galaxy (ID#13) at z=1.2902 that has a low inclination (i = 33^∘), an orientation that may favor detecting galactic outflows in emission <cit.>. This galaxy has a star formation rate of , which places it in the starburst category <cit.>. Its star formation rate surface density is , well above the threshold for galactic winds <cit.>. The star formation rate and stellar mass are nearly identical to those of two galaxies at z = 0.694 <cit.> and z = 0.9392 <cit.> that both show direct evidence of galactic winds from blueshifted absorptions, redshifted MgII emission, and non-resonant FeII* emission. In the integrated 1D MUSE spectrum, we detect non-resonant FeII* emission but no apparent resonant FeII and MgII emission, differentiating this galaxy somewhat from the two previous examples. We also obtain the first direct detection of spatially extended FeII* emission (Section <ref>) from an individual galaxy by stacking the FeII* transitions (Figure <ref>). We discuss here (in Section <ref>) the implications for these results in the context of radiative transfer wind models, since the emitting gas is likely entrained in a galactic-scale outflow. The strong, asymmetric FeII and MgII absorptions in the 1D galaxy spectrum, which have blueshifted components at -660 km s^-1 and -350 km s^-1 relative to the systemic redshift (Figure <ref>), are a clear signature of outflowing gas. In Section <ref>, we will estimate the mass outflow rate and compare it to the galaxy SFR. §.§ Implications for outflow models The MUSE surface brightness maps (Figure <ref>) reveal that the FeII* emission has a more elliptical shape than the stellar continuum and the [OII] emission. Detecting FeII* emission that is more extended along one axis suggests that the outflow is not isotropic. Isotropic outflows are, however, the fiducial geometry for radiative transfer and semi-analytic wind models <cit.>. Comparing these models and their variations with direct detections, such as the MUSE HDFS galaxy ID#13, can help to interpret the observations while motivating additional refinements to the models. The radiative transfer and semi-analytic models of galactic outflows from <cit.> and <cit.> both predict concurrent resonant and non-resonant emission, because absorbed photons can be re-emitted via either non-resonant or resonant transitions for multiplets <cit.>. The spectrum of the HDFS galaxy ID#13 (Figures <ref> and <ref>) shows strong and MgII absorptions (with total rest-frame equivalent widths from 3 to 5 Å), but no evidence for P-cygni profiles and globally small amounts of possible emission infilling as discussed in Section <ref>. We next discuss whether this apparent lack of resonant emission occurs in other galaxies, and how to potentially reconcile the models with such data. Like the HDFS ID#13 galaxy, the two previously published direct detections of FeII* emission <cit.> do not have P-cygni profiles, although they might have moderate resonant emission infilling. Unlike the HDFS ID#13 galaxy, these galaxies have strong P-cygni MgII profiles. The published composite spectra <cit.> show non-resonant FeII* emission without obvious resonant FeII emission as P-cygni profiles. Among these composite spectra, only <cit.> and <cit.> reveal MgII P-cygni profiles. In order to investigate the origin of the apparent lack of resonant emission, in Figure <ref> we compare rest-frame equivalent width measurements from the MUSE HDFS ID#13 galaxy with predictions from the <cit.> radiative transfer models, following <cit.>. In Figure <ref>, we also include the two previously published direct detections of emission from <cit.> and <cit.>. Each panel pairs a non-resonant FeII* emission transition with its corresponding resonant FeII absorption transition from within a multiplet. <cit.> produce models for the FeII UV1 multiplet and MgII λλ2797,2803 doublet to explore how varying model geometries and physical assumptions about the dust content, ISM contribution, gas density, and wind speeds impact the line profiles from the resonant and non-resonant transitions. In nearly all of the tested models, the resonant transitions produce P-cygni profiles with blueshifted absorption and redshifted emission. Varying each of the physical properties individually from the fiducial model is not sufficient to suppress the resonant FeII and MgII P-cygni profiles. Indeed, the only model that substantially suppresses the resonant emission combines an ISM component with dust extinction. In order to reproduce the observed profiles of the MUSE HDFS galaxy ID#13, the models will need to simultaneously incorporate more properties. To gain physical intuition for the impact of the individual properties, we now discuss varying the outflow geometry, dust extinction and ISM component affects the profiles. In contrast to their fiducial model (black point in Figure <ref>), which assumes angular isotropy, <cit.> also modeled bi-conical outflows (purple points), where the wind fills an opening angle into and out of the plane of the sky along the line of the sight to the galaxy. Collimating the outflow suppresses both the resonant and non-resonant emission, and for highly collimated outflows, absorption dominates the profile. The lack of resonant emission in the HDFS galaxy ID#13 suggests that the outflow could be bi-conical and collimated. However, since this geometry also suppresses the non-resonant FeII* emission, highly collimated wind models create a double-peaked FeII* profile that is not observed. The <cit.> fiducial model assumes no dust extinction, but they show that dust absorption (pink points in Figure <ref>) can have a strong impact on the line profiles. Increasing the amount of dust extinction suppresses the resonant emission slightly more than the non-resonant emission. Adding τ_ dust=3 to the fiducial model leaves a weak P-cygni profile, whereas the model with τ_dust = 10 suppresses the resonant emission while leaving weak non-resonant FeII* emission. However, such copious dust extinction would extinguish the source by 15 magnitudes and make it unobservable. The moderate visual extinction for the galaxy ID#13, A_V = 1.20^+0.59_-0.26 mag, suggests that dust extinction alone is not sufficient to explain the diminished resonant emission. Adapting the outflow model to include gas that represents the ISM of the galaxy (red circular point in Figure <ref>), i.e., gas that is centralized and lacks a significant radial velocity, also produces line profiles with similarities to the galaxy ID#13. Adding the ISM component increases the absorption around zero systemic velocity and suppresses the resonant emission. Moreover, the ISM component can boost the emission. For FeII, more resonant absorption due to the ISM component allows more photons to escape through non-resonant re-emission. Indeed, the non-resonant FeII* emission becomes ∼ 10 times stronger than in the fiducial model. With suppressed emission but increased emission, an ISM component is essential to re-creating the observations from the galaxy ID#13. Finally, <cit.> include dust extinction in the ISM component (orange point in Figure <ref>). Compared to the dusty wind model discussed earlier, this model suffers much more from dust extinction because the simple kinematic structure of the ISM allows multiple scattering events. For MgII, the photons are resonantly trapped in the dusty ISM. For , all of the emission lines diminish compared to the same model without dust, but the ratio between the FeII* emission to emission remains stronger than in the fiducial model. This model best describes the ID#13 galaxy. Further exploring the same physical conditions with a bi-conical outflow may highlight subtleties that could indicate preferring one geometry over another. To summarize, we suggest that a model combining a dusty ISM with a bi-conical outflow that has a moderate amount of dust opacity in the wind would be able to match the data for the HDFS galaxy ID#13. §.§ Mass Outflow Rate Estimation To estimate the mass entrained in the outflow, we consider only the absorption components that are not affected by the ISM. Consequently, we exclude the component `C' at the systemic velocity (Figure <ref>, Table <ref>). The other two components are blueshifted by -660 km s^-1 (`A') and -350 km s^-1 (`B'), respectively. The wind component `B' at -350 km s^-1 dominates the bulk of the mass flux given the equivalent width ratios between components `A' and `B.' As discussed in Section <ref>, the wind component `B' is the least affected by emission infilling (at or below the 10% level), whereas the ISM component `C' is the most affected by emission infilling. Hence, emission infilling does not affect our estimate of the mass outflow rate from the wind component `B.' Similar to <cit.>, we estimate the mass outflow rate from: dMdt≈ 1  M_⊙  yr^-1 C_fN_ flow( H) 10^20 cm^-2A_ flow45  kpc^2v 300 km s^-1 5 kpcD where C_f is the covering fraction of the outflowing gas, N_ flow( H) is the column density of hydrogen associated with the outflow, A_ flow is the projected surface area of the outflow, v is the outflow velocity, and D is the physical distance the outflow extends from the galaxy center. We estimate N_ flow( H) from the metal column densities N(Fe) and N(Mg). Because VPFIT column densities and Doppler b parameters are degenerate for optically thick lines, we determine the metal column densities N(Fe) and N(Mg) from the equivalent width, W_0, following <cit.>, with an additional term for the covering fraction: log N = logW_0λ - log2F(τ_0)π^1/2τ_0 - logλ f - log C_f + 20.053 where τ_0 is the optical depth at line center, λ is the transition wavelength in Å, and f is the oscillator strength. The optical depth τ_0 is determined from the ratio of equivalent widths from two lines within the same multiplet, as in <cit.>, which is referred to as the `doublet ratio' method. For MgII, the oscillator strengths indicate that the equivalent width ratio is 2:1 in the optically thin case. The MgII equivalent width ratio follows F(2τ_0)/F(τ_0) for the transmission integral: F(τ_0) = ∫_0^+∞ (1 - e^-τ_0 exp(-x^2))dx From our measured MgII equivalent width ratio, 1.06 ± 0.13, we numerically solve for τ_0, 2803≈ 240. Both the equivalent width ratio and the high optical depth value indicate that MgII is saturated. For FeII, we can calculate the optical depth for two different sets of transitions: τ_0, 2586 from W_0, 2600/W_0, 2586 and τ_0, 2374 from W_0, 2382/W_0, 2374. The optical depth ratios are 3.46:1 and 10.22:1 respectively, using the oscillator strength values from <cit.>. After again solving numerically, the optical depth values are τ_0, 2586 = 3.53 and τ_0, 2374 = 1.55. We can therefore use these optical depth and equivalent width values to obtain a good estimate of the FeII column density, since <cit.> find accurate column densities even for blended components that result from multiple clouds, as long as the optical depth in the weaker transition is τ_0 < 5. With knowledge of the optical depth, we can determine the covering fraction from the residual intensities between the zero level and the doublet lines <cit.>. Using Equation 5 from <cit.> with MgII, we find a covering fraction of at least 0.4. However, this formula ignores the instrument resolution, which could lead to a much higher covering fraction. To estimate a lower limit on the column density, we take C_f=1, as in <cit.>. Applying Equation <ref>, the column density measurements are N(MgII λ2803) = 15.89, N(FeII λ2586) = 14.74, and N(FeII λ2374) = 14.76. These measurements are in good agreement with the values from vpFit, N(MgII) = 15.87 ± 0.68 and N(FeII) = 14.75 ± 0.16. From the metal column densities, in order to estimate the gas flow column density N_ flow(H), we use solar abundances <cit.> and a dust depletion correction but no ionization correction, as in <cit.>. To estimate the dust depletion factor, we use the <cit.> method to simultaneously fit for the depletion level using the column densities of these two elements (Mg, Fe). The fit yields a global depletion factor of F_⋆ = 1.25 ± 0.39, corresponding to δFe of -2.60 dex and δMg of -1.50 dex. With these depletion corrections, the total gas column density is thus at least log N(H) ≥ 21.76 ± 0.48 - log Z/Z_⊙, given that we used solar abundances[Ionization corrections would further increase the column density, but they are small at this level.]. We can estimate the projected area of the ouflow A_ flow from the size of the stellar continuum, since we detect MgII and FeII in absorption against the continuum. The MUSE stellar continuum (Section <ref>) has an intrinsic half-light radius of  kpc. Because the spectrum is optimally extracted with a white-light image weighting scheme, the effective half-light radius of the extracted 1D spectrum is R_1/2, eff=√(2)× R_1/2,⋆ or 3.3 kpc. The stellar continuum therefore covers a surface area of A_ flow = π R_1/2,⋆^2 b/a =30 kpc^2. Finally, we must assume an effective or characteristic distance for the gas at -350 km/s with a total column of log N_ flow(H)>21.80. For a mass-conserving flow, the gas closer to the galaxy will dominate the column density. However, outflowing gas moving at -350 km s^-1 needs a few kpc (1-5) to accelerate to that speed <cit.>. Hence, we conservatively use an upper limit of D<5 kpc, as in <cit.>, which leads to an outflow rate of >45 M_⊙ yr^-1. For plausible values of 2-3 kpc, the outflow rate would be 75 -110 M_⊙ yr^-1. In comparison, the * emission has a characteristic size of ∼ 4 kpc. Overall, the outflow rate is comparable to the star formation rate of 78 M_⊙ yr^-1. § CONCLUSIONS The direct detection of FeII* emission from the spatially-resolved MUSE HDFS galaxy ID#13 at z = 1.29 opens a new avenue for studying galactic outflows in emission. From an analysis of the deepest MUSE field so far (27 hours), the properties of this individual galaxy, including the inclination, stellar mass, star formation rate, and gas kinematics, are well characterized (Table <ref>). This galaxy has a low inclination (i∼33 deg), , and . Using the 1D integrated spectrum and 2D pseudo-narrow band images, we identify signatures of winds in emission from the FeII*, FeII, and MgII transitions and investigate the wind morphology and extent. Specifically, we find: 2pt * The star formation rate surface density from [OII] λλ3727,3729 is , well above the threshold for galactic winds <cit.>. * Asymmetric FeII, MgII, and MgI self-absorptions in the MUSE 1D spectrum have a strong blue wing that extends beyond ∼ -700 km s^-1. The profiles are well-fitted with three components at -660 km s^-1, -350 km s^-1, and +9 km s^-1 (Figure <ref>). These blue-shifted absorptions indicate outflowing material along the line of sight, and we estimate a mass outflow rate in the range of 45 - 110 M_⊙ yr^-1. * Emission infilling does not appear to be present because (i) all absorptions have very similar shapes, whereas infilling is varies significantly for different transitions <cit.> and (ii) the strongest component for all absorptions (including λ2600 and λ2382) is close to the galaxy systemic redshift (Figure <ref>). A quantitative analysis following the <cit.> empirical method shows that emission infilling could impact the λ2600 (λ2383) rest-frame equivalent width by at most 10% (25%), respectively, and less for the other transitions. * Non-resonant FeII* emission from the λ2365, λ2396, λ2612, and λ2626 transitions have fluxes of 1.2 - 2.4 - 1.5 - 2.7 ×10^18 , respectively, and flux ratios that are consistent with optically thick gas <cit.>. The FeII* λ2632 transition has a 1-σ flux limit of < 8 × 10^-19 . Contrary to stacked spectra <cit.>, the emission in this galaxy appear to be symmetric and well-centered on the galaxy systemic redshift (Figure <ref>). * After stacking the four non-resonant FeII* emission lines, we obtain the first spatially-resolved 2D map of this non-resonant emission from a z∼ 1 galaxy (Figure  <ref>). The FeII* emission is more extended than the stellar continuum or [OII] emission. The FeII* emission half-light radius is R_1/2, =  kpc, about 50% larger than that of the continuum which has R_1/2,⋆=  kpc. The emission has a different morphology; it is more elongated in the direction that roughly corresponds to the galaxy minor kinematic axis. * The FeII* emission displays a velocity gradient along the kinematic minor axis, and the blue wing of the emission contours reveals a C-shape pattern in a p-v diagram from a pseudo-slit extracted along this axis (Figure <ref>). These features are consistent with a conical outflow. * Comparing the observed emission and absorption properties with predictions (Figure <ref>) from the radiative transfer models of <cit.> suggests that the isotropic fiducial wind model fails, but that a biconical wind model including a dusty ISM component could more likely reproduce the observations from galaxy ID#13. This geometry agrees with a growing body of models and observations that suggest outflowing gas driven by supernovae explosions escapes the disk preferentially along the galaxy minor axis in a bi-conical flow <cit.>. FeII* emission from the MUSE HDFS galaxy ID#13 was identified serendipitously, but by systematically searching through field galaxies in similar IFU data sets it will be possible to construct samples of z ∼ 1 galaxies that each show evidence of outflows in emission. Observational constraints from these samples can then drive improvements to models of galactic-scale outflows. This work has been carried out thanks to the support of the ANR FOGHAR (ANR-13-BS05-0010-02), the OCEVU Labex (ANR-11-LABX-0060), and the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the “Investissements d'avenir” French government program. NB acknowledges support from a Career Integration Grant (CIG) (PCIG11-GA-2012-321702) within the 7th European Community Framework Program. RB acknowledges support from the ERC advanced grant 339659-MUSICOS. JB is supported by FCT through Investigador FCT contract IF/01654/2014/CP1215/CT0003, by Fundação para a Ciência e a Tecnologia (FCT) through national funds (UID/FIS/04434/2013), and by FEDER through COMPETE2020 (POCI-01-0145-FEDER-007672). BE acknowledges support from the “Programme National de Cosmologie and Galaxies” (PNCG) of CNRS/INSU, France. RAM acknowledges support by the Swiss National Science Foundation. JR acknowledges support from the ERC starting grant 336736-CALENDS. aa

http://arxiv.org/abs/1701.07935v3

20170127034434

Cutoff-free Circuit Quantum Electrodynamics

quant-ph

Department of Electrical Engineering, Princeton University, Princeton, New Jersey, 08544 Any quantum-confined electronic system coupled to the electromagnetic continuum is subject to radiative decay and renormalization of its energy levels. When coupled to a cavity, these quantities can be strongly modified with respect to their values in vacuum. Generally, this modification can be accurately captured by including only the closest resonant mode of the cavity. In the circuit quantum electrodynamics architecture, it is however found that the radiative decay rates are strongly influenced by far off-resonant modes. A multimode calculation accounting for the infinite set of cavity modes leads to divergences unless a cutoff is imposed. It has so far not been identified what the source of divergence is. We show here that unless gauge invariance is respected, any attempt at the calculation of circuit QED quantities is bound to diverge. We then present a theoretical approach to the calculation of a finite spontaneous emission rate and the Lamb shift that is free of cutoff. Cutoff-free Circuit Quantum Electrodynamics Hakan E. Türeci December 30, 2023 =========================================== Introduction. An atom-like degree of freedom coupled to continuum of electromagnetic (EM) modes spontaneously decays. When the atom is confined in a resonator, the emission rate can be modified compared with its value in free space, depending on the EM local density of states at the atomic position <cit.>, which is called the Purcell effect <cit.>. An accompanying effect is the Lamb shift, a radiative level shift first observed in the microwave spectroscopy of the hydrogen ^2P_1/2- ^2S_1/2 transition <cit.>. These quantities have been experimentally accurately characterized for superconducting Josephson junction (JJ) based qubits coupled to coplanar transmission lines <cit.> and three-dimensional resonators <cit.>. In the dispersive regime where a qubit with transition frequency ω_j is far-detuned from the nearest resonant cavity mode (frequency ν_r, loss κ_r), single mode expressions exist for the Purcell decay rate, γ_P = (g/δ)^2 κ_r and the Lamb shift, Δ_L = g^2 / δ. Here g denotes the coupling between the qubit and the cavity mode and δ=ω_j - ν_r denotes their detuning <cit.>. However, for large couplings accessible in circuit QED, the single mode approximation is often inaccurate <cit.>. In addition, due to particular boundary conditions imposed by the capacitive coupling of a resonator to external waveguides, the qubit relaxation time is limited by the EM modes that are far-detuned from the qubit frequency <cit.>. Similarly the measured Lamb shift in the dispersive regime can only be accurately fit with an extended Jaynes-Cummings (JC) model including several modes and qubit levels <cit.>. The Purcell rate has been generalized to account for all modes γ_P=∑_n(g_n/δ_n)^2 κ_n, where g_n and δ_n =ω_j - ν_n are coupling to and detuning from resonator mode n with frequency ν_n and decay rate κ_n. Expression (<ref>) is divergent without imposing a high-frequency cutoff <cit.>. Divergences appear as well in the Lamb shift and other vacuum-induced phenomena, e.g. photon-mediated qubit-qubit interactions <cit.>. These divergences are neither specific to the dispersive limit nor to the calculational scheme used to compute QED quantities. This issue is well-known for the Lamb shift <cit.>, but less noted for the spontaneous emission rate. Indeed, free space spontaneous emission rate diverges as well, as we show in <cit.>. The finite result by Wigner and Weisskopf <cit.> is due to Markov approximation which filters out the ultraviolet divergence. Recent generalizations of the Wigner-Weisskopf approach impose an artificial cut-off to obtain a finite result <cit.>. So far, no satisfactory theoretical explanation has been given for these divergences. Here we address this issue within the framework of circuit quantum electrodynamics <cit.> (QED) and show that finite expressions can be obtained when gauge invariance is respected. We focus here on a superconducting artificial atom coupled to an open transmission-line resonator, but our results should be valid for other types of one-dimensional open EM environments as well. Gauge invariance in circuit QED. The role of gauge invariance in accounting for light-matter interaction has been a vexing question since the beginnings of QED (see Ref. <cit.>, and references therein). Hence, we first discuss gauge invariance in superconducting electrical circuits, and its impact on QED observables. We consider a weakly nonlinear charge qubit (e.g. transmon <cit.><cit.>) capacitively coupled to a transmission-line resonator that in turn is coupled at both ends to semi-infinite waveguides (Fig:cQED-open). We assign flux variables to nodes, Φ_n (t) = ∫^t dτ V_n(τ), with V_n(t) being the instantaneous voltage at node n with respect to the ground node <cit.>. Fixing the ground amounts to a particular gauge choice <cit.>. For the connection geometry in Fig <ref>, the light-matter interaction derives from the energy on the coupling capacitor in the dipole approximation, T_int = 1/2 C_g [Φ̇(x_0) - Φ̇_j]^2 <cit.>, with x_0 the qubit position. If from the three terms in its expansion, T_EM = 1/2 C_g Φ̇(x_0)^2, T_ EM-JJ = - C_g Φ̇(x_0) ·Φ̇_j and T_JJ = 1/2 C_g Φ̇_j^2, only the direct interaction T_ EM-JJ is kept, a multimode JC model in terms of circuit parameters can be derived [See also the Supplementary Material for a brief derivation of the Heisenberg-Langevin equation of motion and a discussion of the multimode convergence of its characteristic function, which includes Refs. <cit.>], but gives rise to a diverging Purcell rate using Eq. (<ref>). This open JC Model involves a two level approximation (TLA) of the JJ Hilbert space, the rotating wave approximation (RWA) to drop nonresonant contributions, and the Born and Markov approximations leading to a Master equation accounting for losses due to resonator-waveguide coupling. It is unclear which approximation underlies the divergence, or whether the divergence can be resolved within the effective subgap circuit QED field theory. We first note that keeping only the direct interaction T_ EM-JJ violates gauge invariance. We find that inclusion of all terms, in particular T_EM, equivalent to the diamagnetic A^2 term in the minimal coupling Hamiltonian (p - e A)^2/2m <cit.>, is essential to make all studied QED observables finite. The A^2-term is thought to have no impact on transition frequencies in vacuum-induced effects such as the Lamb shift. Because it does not involve atomic operators, it is expected to make the same perturbative contribution to every atomic energy level, precluding observable shifts in transition frequencies <cit.>. This argument relies on perturbation theory in the A^2-term. We show that the diamagnetic term does have an impact when accounted for exactly to all orders. Heisenberg equations of motion describing the infinite network in Fig:cQED-open, extending from x=-∞ to x=∞, are <cit.> φ̂̈̂_j(t)+ ( 1- γ) ω_j^2sin[φ̂_j(t)] = γ∂_t^2φ̂(x_0,t), [∂_x^2-χ(x,x_0)∂_t^2]φ̂(x,t) = χ_sω_j^2 sin[φ̂_j(t)]δ(x-x_0), Here φ̂_j(t) and φ̂(x,t) are dimensionless flux operators for the JJ and the resonator-waveguide system, respectively, γ≡ C_g/(C_g+C_j) is a capacitive ratio, χ_s = γ C_j / cL is the dimensionless series capacitance of C_g and C_j, ω_j is the dimensionless transmon frequency, and χ_i≡ C_i/(cL) for i=g,j,R,L <cit.>. These two inhomogeneous equations show that the flux field at x_0 drives the dynamics of the JJ [eqn:TransDyn], while the JJ acts as a source driving the EM fields [eqn:ResDyn]. In addition, the fields are subject to continuity conditions at the ends of the resonator x=0,1 (in units of L). It is instructive to trace the individual terms of T_int in Eqs. (<ref>-<ref>). T_JJ modifies the qubit frequency, renormalizing γ from C_g/C_j to C_g/(C_g+C_j), while the direct interaction term T_ EM-JJ gives source terms in both equations. Most importantly, T_EM introduces an effective scattering term in the wave equation describing the fields in the transmission line, by modifying the unitless capacitance per length from 1 to χ(x,x_0) = 1 + χ_s δ(x-x_0). Consequently, these equations are consistent <cit.> with Kirchhoff's law of current conservation. In particular, at x=x_0, Eq. (<ref>) yields .∂_xφ̂(x,t)]_x_0^-^x_0^+=χ_s∂_t^2φ̂(x_0,t)+χ_sω_j^2sin[φ̂_j(t)], where the discontinuity in the resonator current is equal to the total current through the capacitive and Josephson branches of the transmon. Similar modification of resonator dynamics has been pointed out before for JJ-based qubits <cit.>. Equation <ref> can be solved in the Fourier domain, where φ̂̃̂(x,ω) = ∫_-∞^∞ dt φ̂(x,t) e^-iω t can be expanded in the basis φ̃_n (x,ω) that solves the generalized eigenvalue problem [∂_x^2 + χ(x,x_0) ω^2 ] φ̃_n (x,ω)=0, subject to continuity conditions at the ends of the resonator, i.e. ∂_x φ̃_n (1^-,ω)= χ_Rω^2[ φ̃_n (1^-,ω)-φ̃_n (1^+,ω)] and ∂_x φ̃_n (0^+,ω)= χ_Lω^2 [φ̃_n (0^-,ω)-φ̃_n (0^+,ω)], which models the coupling to the waveguides and associated loss. The Dirac δ-function in χ(x,x_0) leads to the discontinuity -.∂_x φ̃_n(x)]_x_0^-^x_0^+ = χ_s ω_n^2 φ̃_n(x_0), resulting in a modified current-conserving (CC) basis <cit.>. These modifications in the spectrum of the transmission line resonator impact the qubit dynamics that is driven by resonator fluctuations. The role of modal modification in eqn:Current-Conservation can be illustrated with a phenomenological model. Previously, the Purcell rate and the Lamb shift have been calculated using the Lindblad formalism in the dispersive limit <cit.>. An effective multimode JC model ℋ̂_JC = ω_j/2σ̂_z + ∑_n ν_n â_n^†â_n + ∑_n g_n (σ̂^+â_n+σ̂^- â_n^†) can be obtained from our first principles model <cit.>, which incorporates the modifications to the resonator modes and the qubit dynamics. Resonator losses are included through a Bloch–Redfield equivalent zero-temperature master equation for the reduced density matrix of the resonator and qubit ρ̂̇̂ = -i [ ℋ̂_JC, ρ̂ ] + κ_n ( 2 â_n ρ̂â_n^† - {ρ̂, â_n^†â_n }). The expressions of cavity frequencies ν_n, associated losses κ_n and modal interaction strengths g_n are given in the Supplementary Material <cit.>. All these quantities are functions of χ_s, the strength of the modification of the capacitance per unit length. In particular, the light-matter coupling is found as g_n=1/2γ√(χ_j)√(ω_jν_n)φ̃_n(x_0). We show in Fig. <ref> that g_n is non-monotonic <cit.> for any χ_s≠ 0, first increasing, then turning over at a critical χ_s-dependent mode n, decreasing as g_n ∼ 1/√(n) in the large-n limit <cit.>. This high frequency behavior of g_n renders the multimode Purcell rate finite, without an imposed cutoff [We note that this result is valid in the dispersive limit i.e. away from cavity resonances. In that limit, we expect this result to be fairly accurate when compared to the rate extracted from the exact time evolution of the Master equation for the multimode JC model.]. This phenomenon is not specific to the resonator geometry in Fig:cQED-open. The underlying physics is the conservation of current at the position x_0 of the qubit. At high frequency, the series capacitance χ_s becomes a short-circuit to ground, acting as a low-pass filter and suppressing mode amplitude at x_0. This is the cause of the power law drop of g_n as n →∞ (subfig:gXrXl1Em3). Moreover, eliminating the continuum degrees of freedom of the waveguides gives an effective decay rate for each mode, κ_n, which increases monotonically as κ_n ∼ n^0.3 (subfig:KappaXrXl1Em3). In the Supplementary Material, we show that for χ_s=0 the resulting series eqn:Multimode Purcell Rate diverges <cit.>, as pointed out in previous studies <cit.>. For any nonzero χ_s, individual terms in the sum (<ref>) display a universal power law ∼ n^-2.7 (subfig:LogLogSpEmRateXrXl1Em3), which guarantees convergence [The power law dependence of κ_n and g_n, though universal with respect to χ_s, are specific to the chosen circuit topology.]. Solution of the Heisenberg-Langevin equations. Although we showed that the expression (<ref>) for the Purcell decay rate converges, it is only valid in the dispersive regime g_n ≪δ_n. This estimate for the Purcell decay rate and the Lamb shift will deviate substantially from the exact result for a range of order g_n around each cavity resonance, diverging as the qubit frequency approaches the resonance (see Fig. <ref>). This fictitious divergence can in principle be cured by solving the full multimode Master equation. Even if computational challenges relating to the long-time dynamics in such a large Hilbert space can be addressed, the resulting rate would still be subject to the TLA, RWA, Born and Markov approximations, casting a priori an uncertainty on its reliability. An improved analytic result that is uniformly valid in the transmon frequency, and is not limited by the aforementioned approximations can be found by solving eqn:TransDyneqn:ResDyn perturbatively in the transmon's weak nonlinearity. EM degrees of freedom can be integrated out by solving eqn:ResDyn exactly, plugging into eqn:TransDyn and tracing over the photonic Hilbert space. To lowest order in the transmon nonlinearity ϵ = (E_c/E_j)^1/2, where E_c and E_j are the charging and Josephson energy, respectively, the effective equation for the qubit is <cit.> Ẍ̂_j(t)+ω_j^2[1-γ+i𝒦_1(0)]X̂_j(t) =- ω_j^2 ∫_0^tdt' 𝒦_2(t-t') X̂_j(t'), where X̂_j (t) = _ph{ρ̂_ph(0)φ̂_j(t)}/ϕ_zpf is the reduced flux operator traced over the photonic degrees of freedom and ϕ_zpf≡(√(2)ϵ)^1/2 is the magnitude of the zero-point phase fluctuations. This delay equation features the memory kernels 𝒦_n(τ)≡γχ_s∫_-∞^+∞dω/2π ω^n G (x_0,x_0,ω)e^-iωτ, where G(x,x',ω) is the classical EM Green's function defined by [∂_x^2-χ(x,x_0)∂_t^2] G(x,x',ω) e^-i ω t = e^-i ω tδ(x-x') implying that G (x,x',ω) is the amplitude of the flux field created at x by a transmon oscillating with a frequency ω at x' <cit.>. The term on the right hand side of <ref> is therefore proportional to the fluctuating current driving the qubit at time t, that was excited by itself at an earlier time t'. This Green's function correctly encodes the modification of the capacitance per length. Equation (<ref>) can be solved exactly in the Laplace domain X̂̃̂_j(s)=sX̂_j(0)+ Ẋ̂_j(0)/D_j(s), where h̃(s)≡∫_0^∞ dt h(t) e^-st, with D_j(s) defined as <cit.> D_j(s)≡ s^2+ω_j^2[1-γ+i𝒦_1(0)+𝒦̃_2(s)]. We express the characteristic function D_j(s) in meromorphic form D_j(s)=(s-p_j)(s-p_j^*)∏_m(s-p_m)(s-p_m^*)/(s-z_m)(s-z_m^*). The poles of 1/D_j(s) are the hybridized qubit-like and resonator-like complex-valued excitation frequencies, p_j≡ -α_j-iβ_j and p_n≡ -α_n-iβ_n, respectively, of the qubit-resonator system, while its zeroes z_n ≡ -iω_n=-κ_n-iν_n correspond to bare non-Hermitian <cit.> cavity resonances. The real part of the qubit-like pole, α_j, is the Purcell loss rate, while β_j - ω_j is the Lamb shift, as shown in Fig:poles_sf_labeled. In the Supplementary Material, we show that D_j(s) is convergent, and hence so are all hybridized frequencies, for any nonzero χ_s. The A^2-term kept in our calculation to enforce gauge invariance plays the role of the “counterterm" discussed by Caldeira and Leggett to cancel infinite frequency renormalization <cit.>. This problem has also been discussed in the context of the quantum theory of laser radiation <cit.>. Perturbative corrections. The transmon nonlinearity neglected in Eq. (<ref>) can be reintroduced as a weak perturbation. The leading order correction to the hybridized resonances amounts to self- and cross-Kerr interactions <cit.>. Using multi-scale perturbation theory <cit.>, the correction to the transmon qubit-like resonance β_j is given by β̂_j=β_j-√(2)ϵ/4ω_j[u_j^4ℋ̂_j(0)+∑_n 2u_j^2u_n^2ℋ̂_n(0)] where the coefficients u_j,n define the transformation from the hybridized to the unhybridized modes and ℋ̂_j,n(0) are the free Hamiltonians of the transmon and mode n, respectively. For χ_g→0, we find u_j→1, u_n=0 and β_j→ω_j such that we recover the frequency correction of free quantum Duffing oscillator ω̂̅̂_j=ω_j[1-√(2)ϵ/4ℋ̂_j(0)] <cit.>. We note three features of this result. Firstly, the correction is an operator and that expresses the fact that transmon levels are anharmonic. The anharmonicity can be calculated from the expectation value of a corrected quadrature operator <cit.>. Secondly, by virtue of the lowest order result being convergent without a cutoff, the perturbative corrections are also convergent in the number of modes included. Finally, this result is not limited by the qubit-resonator coupling strength or the openness of the cavity. The final result is finite for all qubit frequencies, as opposed to the dispersive-limit result. The correction to the Purcell decay is higher order and forms the subject of future work. We compared the spontaneous decay from the linear theory (blue solid) to the dispersive limit estimate γ_P in Eq. (<ref>) (black dashed) as the transmon frequency is tuned across the fundamental mode in Figs. <ref>-<ref>. First, the spontaneous decay is asymmetric, since there are (in)finitely many modes with frequency (larger) smaller than ω_j. This feature is captured by both theories. Second, the spontaneous decay is enhanced as the qubit frequency approaches the fundamental resonator frequency. However, the dispersive limit estimate is perturbative in g_n/δ_n and hence yields a divergent result (fake kink) on resonance regardless of coupling constant, contrary to our result <ref> which predicts a finite value even at ultrastrong coupling (Fig. <ref> and caption). In Figs. <ref>-<ref> we compare the Lamb shift from the linear theory (blue solid) and the leading order perturbation theory (red dotted) to the dispersive multimode estimate (black dashed) ∑_n g_n^2/δ_n <cit.>. Below the fundamental mode, the Lamb shift is negative due to the collective influence of all higher modes that redshifts the qubit frequency. Above the fundamental mode, there appears a competition between the hybridization with the fundamental mode and all higher modes. Close enough to the fundamental mode, the Lamb shift is positive until it changes sign, as predicted by all three curves. Conclusion. We have presented a framework to calculate the spontaneous decay and the Lamb shift of a transmon qubit, convergent in the number of resonator modes without the need for rotating-wave, two-level, Born or Markov approximations, or a high frequency cutoff. This is achieved by an ab initio treatment of the quantum circuit equations of motion containing the A^2-term to enforce gauge invariance. Therefore, the modes of the resonator are modified such that the light-matter coupling is suppressed at high frequencies. Formulating the cavity resonances in terms of non-Hermitian modes provides access to the spontaneous decay, the Lamb shift, and any other QED observables in a unified way. Acknowledgements. We acknowledge helpful discussions with Zlatko Minev and S. M. Girvin. This work was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Award No. DE-SC0016011. Note. While finishing this manuscript we became aware of Ref. Gely_Convergence_2017, which arrives at a similar conclusion for the Lamb shift in the dispersive regime through a different approach. Supplementary Material: Cutoff-free Circuit Quantum Electrodynamics § HEISENBERG EQUATIONS OF MOTION In this section, we present the Heisenberg equations of motion in terms of flux variables <cit.>. These equations were derived before by the authors <cit.> (see App. A), but the main steps are summarized below for clarity. The flux variable is defined at any node n in terms of the voltage at that node with respect to a fixed ground node Φ_n(t)≡∫_0^tdt' V_n(t'). The classical Lagrangian is the sum of the Lagrangians for the Josephson junction, resonator, right and left waveguides, capacitive coupling between the resonator and the waveguides and the transmon-resonator capacitive coupling, respectively (let U_j(Φ_j) be the nonlinear Josephson potential): ℒ =1/2C_jΦ̇_j(t)^2-U_j(Φ_j(t))_ℒ_j +∫_0^+^L^- dx[1/2 c(∂_tΦ)^2-1/2l(∂_tΦ)^2]_ℒ_Res +∫_L^+^∞ dx[1/2 c(∂_tΦ_R)^2-1/2l(∂_xΦ_R)^2]_ℒ_RW +∫_-∞^0^- dx[1/2 c(∂_tΦ_L)^2-1/2l(∂_xΦ_L)^2]_ℒ_LW +1/2C_L[Φ̇_L(0^-,t)-Φ̇(0^+,t)]^2_ℒ_C_L +1/2C_R[Φ̇_R(L^+,t)-Φ̇(L^-,t)]^2_ℒ_C_R +1/2C_g[Φ̇_j(t)-Φ̇(x_0,t)]^2_ℒ_C_g, From Eq. (<ref>) one can derive, via a Legendre transformation followed by quantization <cit.>, the Hamiltonian operator associated with the quantum circuit. The quantum Hamiltonian for C_R,L→ 0 is in Ref. Malekakhlagh_Origin_2016. C_R,L≠ 0 leave equations of motion unchanged, but change boundary conditions (BCs) at x=0,L. Importantly, Heisenberg equations of motion for the quantum flux operators Φ̂_j, Φ̂(x,t) and Φ̂_R,L(x,t) turn out to be formally identical to Euler-Lagrange equations for (<ref>) with classical fields promoted to operators. To express the Heisenberg equations of motion in a compact way, we introduce the following notations. Φ_0≡h/2e is the superconducting flux quantum and E_j is the Josephson energy. C_s ≡ C_gC_j/(C_g+C_j) is the series capacitance of C_j and C_g and γ≡ C_g/(C_g+C_j). There is a modified capacitance per unit length in the resonator due to the coupling to the transmon qubit at position x_0: c(x,x_0)≡ c+C_sδ(x-x_0). c and l are the capacitance and inductance per unit length in the resonator and the waveguides. We pass to unitless coordinates and operators (v_p≡ 1/√(lc)) x→x/L, t→t/L/v_p, ω→ω/v_pL, φ̂≡ 2πΦ̂/Φ_0, n̂≡Q̂/2e The newly introduced operators φ̂ and n̂ represent phase and number and are canonically conjugate: [φ̂_j,n̂_j]=i and [φ̂(x,t),n̂(x',t')]=iδ(x-x')δ(t-t'). Below we use unitless capacitances χ_i≡ C_i/(cL), i=R,L,j,g,s, and the unitless capacitance per unit length becomes χ(x,x_0)≡ 1+χ_s δ(x-x_0). In terms of the quantities introduced, the Heisenberg equations of motion for the superconducting phase operators are: φ̂̈̂_j(t)+(1-γ)ω_j^2sin[φ̂_j(t)]=γ∂_t^2φ̂(x_0,t), [∂_x^2-χ(x,x_0)∂_t^2]φ̂(x,t)=χ_sω_j^2 sin[φ_j(t)]δ(x-x_0), ∂_x^2φ̂_R,L(x,t)-∂_t^2φ̂_R,L(x,t)=0, with boundary conditions -.∂_xφ̂|_x=1^- =-.∂_xφ̂_R|_x=1^+ =χ_R∂_t^2[φ̂(1^-,t)-φ̂_R(1^+,t)], -.∂_xφ̂|_x=0^+ =-.∂_xφ̂_L|_x=0^- =χ_L∂_t^2[φ̂_L(0^-,t)-φ̂(0^+,t)], φ̂(x=x_0^-,t)=φ̂(x=x_0^+,t), . ∂_xφ̂|_x=x_0^+ - .∂_xφ̂|_x=x_0^- -χ_s∂_t^2 φ̂(x_0,t) =χ_s ω_j^2 sin[φ_j(t)]. In Eqs. (<ref>) and (<ref>), the oscillation frequency is unitless ω_j^2 = 8ℰ_cℰ_j, in terms of unitless Josephson and charging energies ℰ_j,c≡√(lc) LE_j,c/ħ, E_c≡e^2/2C_j. Equations (<ref>-<ref>) are Eqs. (2-3) in the main text. § SPECTRAL REPRESENTATION OF THE GREEN'S FUNCTION In this section we introduce a spectral representation of the Green's function. The Green's function enters the effective Heisenberg equation of motion for the superconducting phase of the transmon qubit (see Ref. Malekakhlagh_NonMarkovian_2016 for a complete derivation). The resonator Green's function appears if one follows this aim in Eqs. (<ref>) and (<ref>): one has to solve for φ̂(x,t), which is driven by the qubit in Eq. (<ref>), and substitute into (<ref>). The resonator Green's function is defined as the response of the resonator fields, described by the left hand sides of Eqs. (<ref>-<ref>), to a δ-function source in space-time [∂_x^2 -χ(x,x_0)∂_t^2] G(x,t|x_0,t_0)=δ(x-x_0)δ(t-t_0), obeying BCs (<ref>-<ref>) with φ̂(x,t) replaced by G(x,t|x_0,t_0). Introducing Fourier transforms G̃(x,x_0,ω)=∫_-∞^∞dt G(x,t|x_0,t_0) e^+iω(t-t_0), G(x,t|x_0,t_0)=∫_-∞^∞dω/2πG̃(x,x_0,ω)e^-iω(t-t_0), Equation (<ref>) becomes a Helmholtz equation [∂_x^2 +ω^2χ(x,x_0) ]G̃(x,x_0,ω)=δ(x-x_0). while the BCs (<ref>-<ref>) are transformed by replacing ∂_t → -iω to .∂_xG̃|_x=1^- =.∂_x G̃|_x=1^+ =χ_R ω^2 (.G̃|_x=1^--.G̃|_x=1^+), .∂_x G̃|_x=0^- = .∂_x G̃|_x=0^+ =χ_L ω^2 (.G̃|_x=0^--.G̃|_x=0^+). .G̃|_x=x_0^+=.G̃|_x=x_0^-, .∂_x G̃|_x=x_0^+-.∂_x G̃|_x=x_0^-+χ_sω^2.G̃|_x=x_0=1, Lastly, outgoing BCs at infinity model the baths: . ∂_x G̃(x,x_0,ω)|_x→±∞=± iωG̃(x→±∞,x_0,ω). Excitations leaving the resonator never reflect back towards it. §.§ Spectral representation of Green's function for χ_R,L = 0 Setting χ_R=χ_L=0 (amounting to a closed resonator) imposes Neumann BC ∂_x G̃|_x=0,1=0 and the problem for G̃ is Hermitian. G̃ can be expanded in terms of a discrete set of normal modes satisfying ∂_x^2φ̃_n(x)+χ(x,x_0)ω_n^2φ̃_n(x)=0, .∂_x φ̃_n(x)|_x=0,1=0. An important feature of the modes is that their derivative is discontinuous -.∂_x φ̃_n(x)|_x_0^-^x_0^+=χ_sω_n^2 φ̃_n(x_0), Physically, this is the continuity equation at x_0, or current conservation. The mode amplitude at x_0 is suppressed. These observations lead us to name this set of resonator eigenmodes the current-conserving (CC) basis. The CC basis eigenfrequencies obey a transcendental equation sin(ω_n)+χ_sω_ncos(ω_n x_0)cos[ω_n (1-x_0)]=0, while the eigenfunctions read φ̃_n(x)∝cos[ω_n (1-x_0)]cos(ω_n x), 0<x<x_0 cos(ω_n x_0)cos[ω_n (1-x)], x_0<x<1 and the basis is orthonormal over [0,1]: ∫_0^1dxχ(x,x_0)φ̃_m(x)φ̃_n(x)=δ_mn. Equation (<ref>) can be solved numerically or asymptotically as n→∞, as we do in Sec. <ref>. The spectral representation of G̃(x,x',ω) <cit.> is G̃(x,x',ω)=∑_n∈ℕφ̃_n(x)φ̃_n(x')/ω^2-ω_n^2=∑_n∈ℤ n≠ 01/2ωφ̃_n(x)φ̃_n(x')/ω-ω_n, since ω_-n=-ω_n and φ̃_-n(x)=φ̃_n(x). §.§ Spectral representation of Green's function for χ_R,L≠ 0 If the resonator is open, χ_L,R≠ 0, we resort to a spectral representation in terms of a discrete set of non-Hermitian modes <cit.> that carry constant flux away from the resonator, Constant Flux (CF) modes <cit.>. CF modes satisfy the homogeneous wave equation  ∂_x^2φ̃_n(x,ω)+χ(x,x_0)ω_n^2(ω)φ̃_n(x,ω)=0, with BCs (<ref>)-(<ref>) and (<ref>). Both the modes φ̃_n(x,ω) and their frequencies ω_n(ω) depend on the source frequency ω. An outgoing plane wave solution for the left and right waveguides obeying (<ref>), is φ̃_n(x,ω)= A_n^<e^iω_n(ω) x+B_n^< e^-iω_n(ω) x, 0<x<x_0 A_n^>e^iω_n(ω) x+B_n^> e^-iω_n(ω) x, x_0<x<1 C_ne^iω x, x>1 D_ne^-iω x, x<0 Applying BCs (<ref>-<ref>) leads to a transcendental equation analogous to the closed case which fixes the parametric dependence ω_n(ω) <cit.>. The CF modes satisfy now a biorthonormality <cit.> condition ∫_0^1dxχ(x,x_0)φ̅̃̅_m^*(x,ω)φ̃_n(x,ω)=δ_mn, where {φ̅̃̅_m(x,ω)} obey the Hermitian adjoint of (<ref>). φ̃_n(x,ω) and φ̅̃̅_n(x,ω) are right and left eigenfunctions and obey φ̅̃̅_n(x,ω)=φ̃_n^*(x,ω). The CF mode spectral representation of the Green's function of the open resonator is G̃(x,x',ω)=∑_nφ̃_n(x,ω)φ̅̃̅_n^*(x',ω)/ω^2-ω_n^2(ω). There are two sets of poles of G̃(x,x',ω) in the complex plane. When the denominator of (<ref>) vanishes, ω=ω_n(ω), which corresponds to quasi-bound eigenfrequencies that obey [e^2iω_n-(1-2iχ_Lω_n)(1-2iχ_Rω_n)] +i/2χ_sω_n[e^2iω_n x_0+(1-2iχ_Lω_n)] ×[e^2iω_n (1-x_0)+(1-2iχ_Rω_n)]=0. The solutions reside in the lower half of complex ω-plane and come in symmetric pairs with respect to the {ω} axis, i.e. if ω_n satisfies (<ref>), so does -ω_n^*. Therefore the eigenfrequencies are ω_n= -iκ_0, n=0 +ν_n-iκ_n, n∈+ℕ -ν_n-iκ_n, n∈-ℕ where ν_n > 0 and κ_n >0 are the oscillation frequency and decay rates of quasi-bound mode n, respectively. The dependence of κ_n on mode number n is plotted in Fig. 2 of the main letter. Note the existence of a pole at ω=0, which comes from the ω-dependence of CF states φ̃_n(x,ω) <cit.>. § MULTIMODE JAYNES-CUMMINGS HAMILTONIAN The classical Hamiltonian for the cQED system can be found from the circuit Lagrangian (<ref>) <cit.> ℋ_sys =4ℰ_c n_j^2(t)-ℰ_jcos[φ_j(t)] +∫_0^1dx {n^2(x,t)/2χ(x,x_0)+1/2[∂_x φ(x,t)]^2} +2πγ z n_j(t)∫_0^1dxn(x,t)/χ(x,x_0)δ(x-x_0), where z≡ Z/R_Q where Z≡√(l/c) is the characteristic impedance of the resonator and R_Q≡ h/(2e)^2 is the superconducting resistance quantum. The modification in capacitance per length originates from the system Lagrangian that contains the gauge-invariant qubit-resonator coupling χ_g[φ̇_j(t)-φ̇(x_0,t)]^2/2. In contrast, a phenomenological product coupling χ_gφ̇_j(t)φ̇(x_0,t) would yield a ℋ_sys with χ_s=0 which results in bare resonator modes. For the purpose of quantizing ℋ_sys, we find the spectrum of the resonator by solving the corresponding Helmholtz eigenvalue problem that has been discussed in Sec. (<ref>). We find ℋ̂_sys≡ω_j/4{𝒴̂_j^2-√(2)/ϵcos[(2ϵ^2)^1/4𝒳̂_j]} +∑_n{ν_n/4[𝒳̂_n^2+𝒴̂_n^2]+g_n𝒴̂_j𝒴̂_n}, where have defined the canonically conjugate variables 𝒳̂_l≡(â_l+â_l^†) and 𝒴̂_l≡ -i(â_l-â_l^†), where â_l represent the boson annihilation operator of sector l≡ j,c. Moreover, ω_j≡√(8ℰ_jℰ_c) and ϵ≡√(ℰ_c/ℰ_j) is a measure for the strength of transmon nonlinearity. For ϵ=0, we recover ω_j(𝒳̂_j^2+𝒴̂_j^2)/4, the Hamiltonian of a simple harmonic oscillator. In the transmon regime where ϵ≪ 1, the leading contribution is -√(2)ϵω_j𝒳̂_j^4/48. The coupling between qubit and the nth CC mode of the resonator is g_n=1/2γ√(χ_j)√(ω_jν_n)φ̃_n(x_0). There are typically two approaches to diagonalize Eq. (<ref>). In the first approach, assuming that the qubit nonlinearity is strong, one performs a two level reduction. Then, the multimode Rabi Hamiltonian can be derived from Eq. (<ref>) by projecting the quadratures to Pauli sigma matrices, 𝒳̂_j→σ̂^x and 𝒴̂_j→σ̂^y, which yields ℋ̂_Rabi =ω_j/2σ̂^z+∑_nν_nâ_n^†â_n -∑_ng_n(â_n-â_n^†)(σ̂^–σ̂^+). In the rotating wave approximation, Eq. (<ref>) transforms into the multimode Jaynes-Cummings Hamiltonian ℋ̂_JC =ω_j/2σ̂^z+∑_nν_nâ_n^†â_n+∑_ng_n(σ̂^+â_n+σ̂^-â_n^†) used in the main text. Analytic results can be found for the Purcell decay rate and the Lamb shift in the dispersive limit where g_n≪|ω_j-ω_n| <cit.>. In a Lindblad calculation, resonator losses are included by a Bloch-Redfield approach through the Master equation for the reduced density matrix of the resonator and qubit degrees of freedom ρ̂̇̂ = -i [ ℋ̂_JC, ρ̂ ] + κ_n/2( 2 â_n ρ̂â_n^† - {ρ̂, â_n^†â_n }), where κ_n can be replaced from the solutions to Eq. (<ref>). The second approach treats the nonlinearity as a weak perturbation and is explained in the next section. § WEAKLY NONLINEAR TRANSMON In this section we summarize the steps necessary to derive Eq. (10) of the main text. The full development of multi scale perturbation theory is in Ref. Malekakhlagh_NonMarkovian_2016. By keeping the lowest order nonlinearity (Kerr terms which are quartic in the transmon quadrature), the Hamiltonian can be rewritten in a new basis that diagonalizes the quadratic part ℋ̂_sys ≡β_j/4(𝒳̂̅̂_j^2+𝒴̂̅̂_j^2)+∑_nβ_n/4(𝒳̂̅̂_n^2+𝒴̂̅̂_n^2) -εω_j/8(u_j 𝒳̂̅̂_j+∑_nu_n 𝒳̂̅̂_n)^4, where ε≡√(2)ϵ/6, β_j,n are hybridized frequencies and u_j,n are hybridization coefficients: 𝒳̂_j=u_j𝒳̂̅̂_j+∑_n u_n𝒳̂̅̂_n. The Heisenberg equations of motion for quadratures become a set of quantum Duffing equations coupled via the quartic terms 𝒳̂̈̂̅̂̈̂_l(t)+β_l^2{𝒳̂̅̂_l(t)-ε_l[u_j𝒳̂̅̂_j(t)+∑_n u_n𝒳̂̅̂_n(t)]^3}=0, where ε_l≡ω_j/β_lu_lε for l≡ j,n. Up to lowest order in the perturbation <cit.>, we find an operator valued correction of the linear theory qubit-like frequency β_j: β̂_j=β_j-√(2)ϵ/4ω_j[u_j^4ℋ̂̅̂_j(0)+∑_n2u_j^2u_n^2ℋ̂̅̂_n(0)], and an analogous correction of the resonator like frequency β_n as β̂_n=β_n-√(2)ϵ/4ω_j [u_n^4ℋ̂̅̂_n(0)+2u_n^2u_j^2ℋ̂̅̂_j(0). +.∑_m≠ n2u_n^2u_m^2ℋ̂̅̂_m(0)], where ℋ̂̅̂_l(0)≡1/4[𝒳̂̅̂_l^2(0)+𝒴̂̅̂_l^2(0)] for l=j,n. In the main text, Eq. (10), the bar notation is dropped. The lowest order MSPT solution for the qubit quadrature becomes, in terms of renormalized frequencies β̂_j,n, <cit.> 𝒳̂_j^(0)(t) =u_jâ_j(0)e^-iβ̂_j t+e^-iβ̂_j tâ_j(0)/2cos(3ω_j/4u_j^4ε t)+H.c. +∑_n[u_nâ_n(0)e^-iβ̂_n t+e^-iβ̂_n tâ_n(0)/2cos(3ω_j/4u_n^4ε t )+H.c.]. This equation takes into account corrections up to 𝒪(ε) in frequencies. To extract these corrections, we must evaluate the expectation value of Eq. (<ref>) with respect to the initial density matrix. We chose ρ̂=|Ψ⟩_j⟨Ψ|_j⊗|0⟩_ph⟨0|_ph with |Ψ⟩_j=(|0⟩_j+|1⟩_j)/√(2). The correction to the transmon like frequency is obtained from the Fourier components of <𝒳̂_j(t)>. This is the correction plotted in Fig. 3 of the main text. § ASYMPTOTIC BEHAVIOR OF LIGHT-MATTER COUPLING In this section we find the asymptotic behavior of the eigenfrequencies ω_n and eigenmodes φ̃_n(x) of the resonator discussed in the main text. This provides an analytical understanding of the high frequency suppression in the light-matter coupling g_n. To point out the origin of the suppression that arise from a nonzero χ_s, let us consider the closed resonator (χ_R,L=0) case. Consider the special case of x_0=0^+ first. This is of experimental interest in order to achieve the maximum coupling to all modes of a resonator. Then, the transcendental Eq. (<ref>) simplifies to sin(ω_n)+χ_sω_ncos(ω_n)=0, which can be rewritten as tan(ω_n)=-χ_sω_n. The large ω_n solution for χ_s≠0 is then obtained lim_n→∞ω_n=nπ-π/2, which is independent of the value for χ_s. This implies that the effect of a nonzero χ_s on ω_n is a total shift π/2 (half of the free spectral range) in comparison with the case χ_s=0. Substituting x_0=0^+ in Eq. (<ref>), the normalization factor 𝒩_n is found via Eq. (<ref>) as ∫_0^1dxcos^2[ω_n(1-x)]+χ_scos^2(ω_n)=1/𝒩_n^2, which gives 𝒩_n=√(2)/√(1+χ_scos^2(ω_n)). Therefore the eigenmode is found as φ̃_n(x_0=0^+)=√(2)cos(ω_n)/√(1+χ_scos^2(ω_n)). Using the trigonometric identity cos^2(ω_n)=1/1+tan^2(ω_n) and Eq. (<ref>) we can rewrite Eq. (<ref>) as φ̃_n(x_0=0^+)=√(2)/√(1+χ_s+χ_s^2ω_n^2), which now provides the algebraic dependence of φ̃_n(x_0) on ω_n. According to Eq. (<ref>), for large enough ω_n (χ_sω_n ≫ 1+χ_s), we find φ̃_n(x_0)1/ω_n, where the symbol represents asymptotic equivalence. This imposes a natural cut-off on the light matter coupling for n→∞, since g_n∝√(ω_n)φ̃_n(x_0)1/√(ω_n). Next, we would like to find the asymptotic behavior of ω_n and φ̃_n(x_0) for a general x_0. In order to bring Eq. (<ref>) into a similar form to Eq. (<ref>), we first replace sin(ω_n)=sin[ω_nx_0+ω_n(1-x_0)] and then divide by cos(ω_n x_0)cos[ω_n (1-x_0)] to obtain tan(ω_n x_0)+tan[ω_n (1-x_0)]=-χ_sω_n. Next, the normalization factor 𝒩_n is found from Eq. (<ref>) as 𝒩_n=√(2)/√(x_0cos^2[ω_n(1-x_0)]+(1-x_0)cos^2(ω_n x_0)+χ_scos^2[ω_n(1-x_0)]cos^2(ω_nx_0)), Plugging this into Eq. (<ref>) we find φ̃_n(x_0)=√(2)/√(1+χ_s+x_0tan^2(ω_nx_0)+(1-x_0)tan^2[ω_n(1-x_0)]) Equations (<ref>) and (<ref>) provide the asymptotic behavior of ω_n, φ̃_n(x_0) and g_n for a general x_0. § CHARACTERISTIC FUNCTION D_J(S) AND ITS CONVERGENCE In this section we derive the expression for the characteristic function D_j(s) and compare its convergence in number of resonator modes with and without the modification we found for g_n. Consider the Heisenberg-Langevin equations of motion corresponding to Hamiltonian (<ref>) in the linear regime (ϵ=0) for 𝒳̂_j,n(t) as (d_t^2+ω_j^2)𝒳̂_j(t)=-∑_n2g_nω_n𝒳̂_n(t), (d_t^2+2κ_n d_t+ω_n^2)𝒳̂_n(t)=-2g_nω_j𝒳̂_j(t)-f̂_n(t), where κ_n and f̂_n are the decay rate and noise operator coming from coupling to the waveguide degrees of freedom <cit.>. Equations (<ref>-<ref>) are linear constant coefficient ODEs and can be solved exactly via the unilateral Laplace transform h̃(s)=∫_0^∞dt h(t)e^-st. Taking the Laplace transform of Eqs. (<ref>-<ref>) we obtain (s^2+ω_j^2)𝒳̂̃̂_j(s)+∑_n2g_nω_n 𝒳̂̃̂_n(s)= s𝒳̂_j(0)+𝒳̂̇̂_j(0), (s^2+2κ_ns+ω_n^2)𝒳̂̃̂_n(s)+2g_nω_j𝒳̂̃̂_j(s)= (s+2κ_n)𝒳̂_n(0)+𝒳̂̇̂_n(0)+f̂̃̂(s). The solution for 𝒳̂̃̂_j(s) then reads 𝒳̂̃̂_j(s)=N̂_j(s)/D_j(s), where the numerator N̂_j(s) =s𝒳̂_j(0)+𝒳̂̇̂_j(0) -∑_n2g_nω_n[(s+2κ_n)𝒳̂_n(0)+𝒳̂̇̂_n(0)-f̂̃̂_n(s)]/s^2+2κ_n s+ω_n^2, contains the operator initial conditions and the denominator D_j(s)≡ s^2+ω_j^2-∑_n4g_n^2ω_jω_n/s^2+2κ_n s+ω_n^2. is the characteristic function whose roots give the hybridized poles of the full system. Therefore, we can represent D_j(s) as D_j(s)=(s-p_j)(s-p_j^*)∏_n(s-p_n)(s-p_n^*)/(s-z_n)(s-z_n^*), where p_j,n≡ -α_j,n-iβ_j,n stand for the transmon-like and the nth resonator-like poles, respectively. Furthermore, z_n ≡-κ_n-i√(ω_n^2-κ_n^2) is the nth bare non-Hermitian resonator mode. The notation (p for poles and z for zeros) is chosen based on 1/D_j(s) that appears in the Laplace solution (<ref>). In order to compute the hybridized poles in practice, we need to truncate the number of resonator modes in D_j(s). This truncation is only justified if the function D_j(s) converges as we include more and more modes. First, note that without the correction give by χ_s this sum is divergent, since g_n√(ω)_n√(n) and for a fixed s we obtain 4g_n^2ω_jω_n/s^2+2κ_n s+ω_n^2ω_n^2/ω_n^2 1. Hence, the series in divergent. On the other hand, we found that for a non-zero χ_s, g_n 1/√(ω_n) 1/√(n). Therefore we find 4g_n^2ω_jω_n/s^2+2κ_n s+ω_n^21/ω_n^21/n^2, and the series becomes convergent. In writing Eq. (<ref>), we used the fact that ω_n n and κ_n has a sublinear asymptotic behavior found numerically. § DIVERGENCE IN THE WIGNER-WEISSKOPF THEORY OF SPONTANEOUS EMISSION Divergence of the Purcell decay rate appears in other frameworks besides the dispersive limit Jaynes-Cummings model as well. In this appendix, we show that the spontaneous decay rate of a qubit coupled to continuum of modes is also divergent, unless the gauge invariance of the interaction is incorporated as presented in this manuscript. The impression of an (erroneous) finite decay rate in free space goes back to Wigner and Weisskopf's original work on spontaneous atomic decay, which implicitly makes a Markov approximation (See Sec. 6.3 of <cit.>). We emphasize that employing the Markov approximation always yields a finite value for the decay rate regardless of the form of spectral function for electromagnetic background. To see this explicitly, we go over the Wigner-Weisskopf theory of spontaneous emission for a two-level system coupled to a continuum of modes inside an infinitely long 1D medium. In interaction picture, the Hamiltonian reads ℋ̂_I=∑_kħ[g_k^*(x_0)σ̂^+â_ke^i(ω_j-ω_k)t+H.c.], which conserves the total number of excitations N̂≡σ̂^+σ̂^-+∑_k⃗â_k⃗^†â_k⃗. As a result, a number conserving Ansatz for the wavefunction can be written as |Ψ(t)⟩=c_e(t)|e,0⟩+∑_kc_g,k(t)|g,1_k⟩, where there is either no photon in the cavity and the qubit is in excited state |e⟩, or there is a photon at frequency ω_k with qubit in the ground state |g⟩. By solving the Schrodinger equation we obtain the time evolution of the unknown probability amplitudes c_e(t) and c_g,k(t). Combining these equations yields an effective equation for c_e(t) as ċ_e(t)=-∫_0^tdt'𝒦(t-t')c_e(t'), where the memory Kernel 𝒦(τ) is given by 𝒦(τ)≡∑_k|g_k(x_0)|^2e^i(ω_j-ω_k)t. Next, we replace the expression for g_k(x_0), derived in Sec. <ref>, as |g_k(x_0)|^2=γχ_s/4ω_jω_k|φ̃_k(x_0)|^2. Note that without respecting the gauge symmetry of interaction |φ̃_k(x_0)|=𝒩(x_0) is k-independent. Moreover, the sum over k can be replaced as ∑_k→L/2π∫_0^∞dk=L/2π v_p∫_0^∞dω_k, for a continuum of modes, where v_p is the phase velocity of the medium. Inserting Eqs. (<ref>) and (<ref>) into the effective Eq. (<ref>) we obtain ċ_e(t) =-1/2πγχ_sω_j 𝒩^2(x_0) L/4v_p ×∫_0^∞dω_k ω_k ∫_0^tdt' e^i(ω_j-ω_k)(t-t')c_e(t') Importantly, the integral over ω_k in Eq. (<ref>) does not converge since the integrand grows unbounded as ω_k→∞. To resolve this, Wigner and Weisskopf assumed that the dominant contribution comes from those modes of continuum whose frequency are close to the qubit frequency. Therefore, the factor ω_k can be replaced by ω_j and by extending the lower limit of integral over ω_k to -∞ we can use the identity ∫_-∞^+∞dω_ke^i(ω_j-ω_k)(t-t')=2πδ(t-t'), to arrive at a finite value for the spontaneous decay as ċ_e(t)≈-Γ_sp/2c_e(t), Γ_sp≡γχ_sω_j^2 𝒩^2(x_0) L/2v_p. It is worth mentioning that using Markov approximation, one always obtains a finite expression for the spontaneous decay rate regardless of the form for the spectral function. This happens because instead of integrating over the entire frequency span, the Markov approximation picks a small window around qubit frequency. Next, we show how our natural high frequency cut-off for light-matter coupling resolves the divergence of Wigner-Weisskopf theory. First, note that applying Markov approximation is indeed unnecessary, since the Volterra Eq. (<ref>) with the memory kernel 𝒦(τ)=1/2πγχ_sω_jL/4v_p∫_0^∞dω_k ω_k|φ̃_k(x_0)|^2e^i(ω_j-ω_k)τ, has an exact solution in Laplace domain as c̃_e(s)=c_e(0)/s+𝒦̃(s), where 𝒦̃(s)≡∫_0^∞dτ𝒦(τ)e^-sτ is the Laplace transform and is found as 𝒦̃(s)=1/2πγχ_sω_jL/4v_p∫_0^∞dω_k ω_k|φ̃_k(x_0)|^2/s+i(ω_k-ω_j). Second, when the gauge-invariance of the interaction is incorporated, the mode amplitude is frequency dependent that experiences a high frequency suppression as |φ̃_k(x_0)|∼1/ω_k. Replacing Eq. (<ref>) into expression (<ref>) for 𝒦̃(s) we obtain 𝒦̃(s)∝∫ dω_k1/ω_k[s+i(ω_k-ω_j)]. Interestingly, with the corrected expression for the eigenmodes, the integrand behaves like 1/ω_k^2 at ω_k→∞, and as a result the integral converges. Otherwise, the integrand behaves like a constant at ω_k→∞ and the result is divergent. apsrev4-1

http://arxiv.org/abs/1701.07572v1

20170126042005

Adaptive posterior convergence rates in non-linear latent variable models

math.ST

Adaptive posterior convergence rates in non-linear latent variable models Shuang Zhou[Corresponding author: shuang.zhou@stat.fsu.edu], Debdeep Pati Department of Statistics, Florida State University Anirban Bhattacharya Department of Statistics, Texas A&M University David Dunson Department of Statistical Science, Duke University December 30, 2023 ================================================================================================================================================================================================================================================================================== Abstract Non-linear latent variable models have become increasingly popular in a variety of applications. However, there has been little study on theoretical properties of these models. In this article, we study rates of posterior contraction in univariate density estimation for a class of non-linear latent variable models where unobserved (0,1) latent variables are related to the response variables via a random non-linear regression with an additive error. Our approach relies on characterizing the space of densities induced by the above model as kernel convolutions with a general class of continuous mixing measures. The literature on posterior rates of contraction in density estimation almost entirely focuses on finite or countably infinite mixture models. We develop approximation results for our class of continuous mixing measures. Using an appropriate Gaussian process prior on the unknown regression function, we obtain the optimal frequentist rate up to a logarithmic factor under standard regularity conditions on the true density. Keywords: Bayesian nonparametrics; Density estimation; Gaussian process; One factor model; Rate of convergence § INTRODUCTION Latent variable models are popular in statistics and machine learning for dimension reduction, parsimonious dependence modeling and data visualization. Linear latent variable models, such as factor models or probabilistic principal components, assume a linear relationship between the observed and latent variables. A number of non-linear latent variable models have been proposed in the literature for structured dimension reduction and manifold learning; example include the generative tomographic mapping (GTM; <cit.>) and the Gaussian process latent variable model (GP-LVM; <cit.>). These models flexibly model the relationship between observed and latent variables, notably using Gaussian process (GP) priors. In spite of their empirical success, a general theoretical framework studying the properties of the induced density of the data after marginalizing out the latent variables seems lacking. <cit.> proposed an NL-LVM (non linear latent variable model) approach for univariate density estimation in which the response variables are modeled as unknown functions (referred to as the transfer function) of uniformly distributed latent variables with an additive Gaussian error. Operationally similar to a univariate GP-LVM model, the latent variable specification allows straightforward posterior computation via conjugate posterior updates. Since inverse c.d.f. transforms of uniform random variables can generate draws from any distribution, a prior with large support on the space of transfer functions can approximate draws from any continuous distribution function arbitrarily closely. One can also conveniently center the non-parametric model on a parametric family by centering the prior on the transfer function on a parametric class of quantile (or inverse c.d.f.) functions {F_θ^-1 : θ∈Θ}. While such centering on parametric guesses can be achieved in Dirichlet process (DP; <cit.>) mixture models by appropriate choice of the base measure G_0, posterior computation becomes complicated unless the base measure is conjugate to the kernel 𝒦. Although there is an increasingly rich literature on asymptotic properties of Bayesian density estimation, this literature mainly focuses on discrete mixture models that have a fundamentally different form from the NL-LVM models. Hence, it is unclear what types of asymptotic properties NL-LVMs have for density estimation, and technical tools developed in the existing literature cannot be fully utilized to study this problem. Our focus is on closing this gap focusing in particular on studying how the posterior measure for the unknown density concentrates around the true density f_0 as the sample size n increases. Assuming f_0 belongs to a Hölder space of univariate functions with smoothness β, it is known that the minimax optimal rate of convergence for an estimate of the density is n^-β/(2β +1). Assuming the prior for the unknown density is induced through a discrete mixture of exponential power distributions, <cit.> showed that the posterior measure for the density concentrates at the optimal rate up to a logarithmic factor. Their result shows rate adaptivity to any degree of smoothness of the true density f_0, generalizing previous results, such as posterior consistency <cit.> or optimal rates for a particular smoothness level <cit.>. We seek to obtain an adaptive rate result for a class of NL-LVM models, and in the process significantly advance technical understanding of this relatively new class of models. The main contributions of this article are as follows. We provide an accurate characterization of the prior support in terms of kernel convolution with a class of continuous mixing measures. We provide conditions for the mixing measure to admit a density with respect to Lebesgue measure and show that the prior support of the NL-LVM is at least as large as that of DP mixture models. We then develop approximation results for the above class of continuous mixing measures, and show adaptive convergence rates. This involves some novel issues and technical details relative to the existing literature. The rest of the article is organized as follows. We introduce relevant notations and terminologies in Section <ref>. To make the article self-contained, we also provide a brief background on Gaussian process priors. In Section <ref>, we formulate our assumptions on the true density f_0 and in the following section, we describe the NL-LVM model and relate it to convolutions. We state our main theorem on convergence rates in Section <ref>. Section <ref> provides auxiliary results and Section <ref> proves the main theorem of posterior concentration rate. We discuss some implications of our results and outlines possible future directions Section <ref>. § NOTATIONS Throughout the article, Y_1,…, Y_n are independent and identically distributed with density f_0 ∈ℱ, the set of all densities on ℝ absolutely continuous with respect to the Lebesgue measure λ. The supremum and _1-norm are denoted by ·_∞ and ·_1, respectively. We let ·_p, ν denote the norm of L_p(ν), the space of measurable functions with ν-integrable pth absolute power. For two density functions f, g ∈ℱ, let h denote the Hellinger distance defined as h^2(f, g) = √(f)-√(g)_2, λ=∫ (f^1/2 - g^1/2)^2 dλ, K(f,g) the Kullback-Leibler divergence given by K(f,g) = ∫log(f/g) f dλ and V(f,g)=∫log(f/g)^2 f dλ. The notation C[0, 1] is used for the space of continuous functions f : [0, 1] →ℝ endowed with the supremum norm. For β >0, we let C^β[0, 1] denote the Hölder space of order β, consisting of the functions f ∈ C[0, 1] that have ⌊β⌋ continuous derivatives with the ⌊β⌋th derivative f^⌊β⌋ being Lipschitz continuous of order β -⌊β⌋. The ϵ-covering number N(ϵ,S,d) of a semi-metric space S relative to the semi-metric d is the minimal number of balls of radius ϵ needed to cover S. The logarithm of the covering number is referred to as the entropy. By near-optimal rate of convergence we mean optimal rate of convergence slowed down by a logarithmic factor. We write “≾” for inequality up to a constant multiple. Let ϕ(x) = (2π)^-1/2exp(-x^2/2) denote the standard normal density, and let ϕ_σ(x) = (1/σ) ϕ(x/σ). Let an asterisk denote a convolution e.g., (ϕ_σ * f)(y) = ∫ϕ_σ(y - x)f(x)dx, and let ϕ_σ^(i) * f denote the i-fold convolution. The support of a density f is denoted by supp(f). We briefly recall the definition of the RKHS of a Gaussian process prior; a detailed review can be found in <cit.>. A Borel measurable random element W with values in a separable Banach space (𝔹, ·) (e.g., C[0,1]) is called Gaussian if the random variable b^*W is normally distributed for any element b^* ∈𝔹^*, the dual space of 𝔹. The reproducing kernel Hilbert space (RKHS) ℍ attached to a zero-mean Gaussian process W is defined as the completion of the linear space of functions t ↦ EW(t)H relative to the inner product ⟨ W(·)H_1; W(· )H_2⟩_ℍ = H_1H_2, where H, H_1 and H_2 are finite linear combinations of the form ∑_ia_iW(s_i) with a_i ∈ℝ and s_i in the index set of W. Let W = (W_t: t ∈ℝ) be a Gaussian process with squared exponential covariance kernel. The spectral measure m_w of W is absolutely continuous with respect to the Lebesgue measure λ on ℝ with the Radon-Nikodym derivative given by dm_w/dλ(x) = 1/2π^1/2e^-x^2/4. Define a scaled Gaussian process W^a=(W_at: t ∈ [0,1]), viewed as a map in C[0,1]. Let ℍ^a denote the RKHS of W^a, with the corresponding norm ·_ℍ^a. The unit ball in the RKHS is denoted ℍ^a_1. Throughout the paper, C, C_1, C_2, … denote global constants whose value may change one line to another. § ASSUMPTIONS ON THE TRUE DENSITY It has been widely recognized that one needs certain smoothness assumptions and tail conditions on the true density f_0 to derive posterior convergence rates. We make the following assumptions in our case: Assume logf_0∈ C^β[0, 1]. Let l_j(x)=d^j/dx^jlog f_0(x) be the derivatives for j=1, …, r with r= ⌊β⌋. For any β >0, assume that there exists a constant L > 0 such that |l_r(x) - l_r(y)| ≤ L|x-y|^β - r, for all x ≠ y. Assume f_0 is compactly supported on [a_0,b_0], for -∞ < a_0 <b_0 < ∞, and that there exists some interval [a,b] ⊂ [a_0, b_0] such that f_0 is nondecreasing on [a_0, a], bounded away from 0 on [a,b] and non-increasing on [b, b_0]. Assumption <ref> is useful in simplifying expressions for f_0 as convolutions with a given density, providing a key piece in our theoretical developments. Similar assumption on the local smoothness appeared in <cit.>, while in our case the global smoothness assumption is sufficient since f_0 is assumed to be compactly supported. Assumption <ref> guarantees that for every δ > 0, there exists a constant C > 0 such that f_0 * ϕ_σ≥ Cf_0 for every σ < δ. § THE NL-LVM MODEL Consider the nonlinear latent variable model, y_i = μ(η_i) + ϵ _i, ϵ_i ∼(0, σ^2), (i=1, …, n) μ ∼Π_μ, σ∼Π_σ, η_i ∼(0,1), where η_i's are latent variables, μ∈ C[0, 1] is a transfer function relating the latent variables to the observed variables and ϵ_i is an idiosyncratic error. The density of y conditional on the transfer function μ and scale σ is obtained on marginalizing out the latent variable as f(y; μ, σ) def= f_μ, σ(y)= ∫_0^1ϕ_σ(y-μ(x))dx. Define a map g: C[0,1] × [0,∞) →ℱ with g(μ, σ) = f_μ, σ. One can induce a prior Π on ℱ via the mapping g by placing independent priors Π_μ and Π_σ on C[0,1] and [0, ∞) respectively, with Π = (Π_μ⊗Π_σ) ∘ g^-1. <cit.> assumed a Gaussian process prior with squared exponential covariance kernel on μ and an inverse-gamma prior on σ^2. It is not immediately clear whether the class of densities f_μ, σ in the range of g encompass a large subset of the density space. The following intuition relates the above class with continuous convolutions which plays a key role in our proofs. Let f_0 be a continuous density with cumulative distribution function F_0(t) = ∫_-∞^t f_0(x) dx. Assume f_0 to be non-zero almost everywhere within its support, so that F_0 : (f_0) → [0,1] is strictly monotone and hence has an inverse F_0^-1 : [0,1] →(f_0) satisfying F_0{F_0^-1(t)} = t for all t ∈(f_0). Letting μ_0(x) = F_0^-1(x), one obtains f_μ_0, σ(y) = ∫_0^1ϕ_σ(y-F_0^-1(x))dx = ∫_-∞^∞ϕ_σ(y-t) f_0(t) dt, where the second equality follows from the change of variable theorem. Thus, f_μ_0,σ(y) = ϕ_σ*f_0, i.e., f_μ_0, σ is the convolution of f_0 with a normal density having mean 0 and standard deviation σ. It is well known that the convolution ϕ_σ*f_0 can approximate f_0 arbitrary closely as the bandwidth σ→ 0 in the sense that for f_0 ∈ L_p(λ) with p ≥1, ϕ_σ*f_0 - f_0_p, λ→ 0 as σ→ 0. For Holder-smooth functions, the order of approximation can be characterized in terms of the smoothness. If f_0 ∈ C^β[0, 1] with β≤ 2, it follows from standard Taylor series expansion that || ϕ_σ*f_0 - f_0||_∞ = O(σ^β). For β > 2, a similar Taylor series expansion yields a sub-optimal error || ϕ_σ*f_0 - f_0||_∞ = O(σ^2). In this case, we can refine the approximation by convoluting with a sequence of functions f_j constructed by the procedure, f_j+1 = f_0 - _σ f_j, _σf_j = ϕ_σ*f_j - f_j, j ≥ 1. For f_0 ∈ C^β[0,1] with β∈ (2j, 2j+2] we have ϕ_σ*f_j - f_0_∞ = O(σ^β) <cit.>. Although the f_js need not be non-negative in general, we show that they are non-negative on (f_0) when f_0 is compactly supported. It can be additionally shown that the normalizing constant is 1 + O(σ^β); let h_j denote the density obtained by normalizing f_j. We then approximate f_0 by ϕ_σ * h_β, where h_β = h_j for β∈ (2j, 2j+2]. Let λ̃ denote the Lebesgue measure on [0,1], or equivalently, the [0,1] distribution. For any measurable function μ : [0,1] →ℝ, let ν_μ denote the induced measure on (ℝ, ℬ), with ℬ denoting the Borel sigma-field on ℝ. Then, for any Borel measurable set B, ν_μ(B) = λ̃(μ^-1(B)), where μ^-1(B) = {x ∈ [0,1]   :  μ(x) ∈ B}. By the change of variable theorem for induced measures, ∫_0^1ϕ_σ(y-μ(x))dx = ∫ϕ_σ(y-t) dν_μ(t), so that f_μ, σ can be expressed as a kernel mixture form with mixing distribution ν_μ. It turns out that this mechanism of creating random distributions is very general. Depending on the choice of μ, one can create a large variety of mixing distributions based on this specification. For example, if μ is a strictly monotone function, then ν_μ is absolutely continuous with respect to the Lebesgue measure, while choosing μ to be a step function, one obtains a discrete mixing distribution. However, it is easier to place a prior on μ supported on the space of continuous functions C[0, 1] without further shape restrictions and Theorem <ref> assures us that this specification leads to large L_1 support on the space of densities. Suppose the prior Π_μ on μ has full sup-norm support on C[0,1] so that (μ - μ^*_∞ < ϵ) > 0 for any ϵ > 0 and μ^* ∈ C[0,1], and the prior Π_σ on σ has full support on [0, ∞). If f_0 is compactly supported so that the quantile function μ_0 ∈ C[0,1], then it can be shown that under mild conditions, the induced prior Π assigns positive mass to arbitrarily small L_1 neighborhoods of any density f_0. We summarize the above discussion in the following theorem, with a proof provided in the appendix. If Π_μ has full sup-norm support on C[0,1] and Π_σ has full support on [0, ∞), then the L_1 support of the induced prior Π on ℱ contains all densities f_0 which have a finite first moment and are non-zero almost everywhere on their support. The conditions of Theorem <ref> are satisfied for a wide range of Gaussian process priors on μ (for example, a GP with a squared exponential or Matérn covariance kernel). When f_0 has full support on ℝ, the quantile function μ_0 is unbounded near 0 and 1, so that μ_0_∞ = ∞. However, ∫_0^1μ_0(t) dt = ∫_ℝx f_0(x) dx, which implies that μ_0 can be identified as an element of L_1[0,1] if f_0 has finite first moment. Since C[0,1] is dense in L_1[0,1], the previous conclusion regarding L_1 support can be shown to hold in the non-compact case too. § THE MAIN THEOREM We consider the case where f_0 satisfies Assumption <ref> and Assumption <ref>. For β-Hölder density f_0, we consider density h_β as defined after expression (<ref>). Denote μ_0 the quantile function F_h_β^-1: [0,1] → [a_0, b_0], a continuous monotone function inheriting the smoothness of h_β. Note that h_β has the same smoothness of f_0 based on the construction of f_j, therefore with the fundamental theorem of calculus it is easy to see that μ_0 ∈ C^β+1[0,1]. We now mention our choices for the prior distributions Π_μ and Π_σ. We assume μ follows a centered and rescaled Gaussian process denoted by (0, c^A), where A denotes the rescaled parameter, and assume A is a density satisfying for a>0, C_1a^pexp(-D_1a log^q a)≤ g(a) ≤ C_2a^pexp(-D_2a log^q a) for positive constants C_1, C_2, D_1, D_2, nonnegative constant p and q, and every sufficiently large a>0. We assume σ∼(a_σ, b_σ). Note that contrary to the usual conjugate choice of an inverse-Gamma prior for σ^2, we have assumed an inverse-Gamma prior for σ. This enables one to have slightly more prior mass near zero compared to an inverse-Gamma prior for σ^2, leading to the optimal rate of posterior convergence. Refer also to <cit.> for a similar prior choice for the bandwidth of the kernel in discrete location-scale mixture priors for densities. We state below the main theorem of posterior convergence rates. If f_0 satisfies Assumption <ref> and the priors Π_μ and Π_σ are as in Assumptions <ref> and <ref> respectively, the best obtainable rate of posterior convergence relative to Hellinger metric h is ϵ_n = max(ϵ̃_n, ϵ̅_n), where ϵ̃_n=n^-β/2β+1(log n)^t_1, ϵ̅_n= n^-β/2β+1(log n)^t_2, with nonnegative constants t_1=β(2∨ q)/(2β +1), t_2= t_1+1. Unlike the treatment in discrete mixture models <cit.> where a compactly supported density is approximated with a discrete mixture of normals, the main trick here is to approximate the true density f_0 by the convolution ϕ_σ*f_0 and allow the prior on the transfer function to appropriately concentrate around the true quantile function μ_0 ∈ C[0,1]. § AUXILIARY RESULTS To guarantee that the above scheme leads to the optimal rate of convergence, we first derive sharp bounds for the Hellinger distance between f_μ_1, σ_1 and f_μ_2, σ_2 for μ_1, μ_2 ∈ C[0, 1] and σ_1, σ_2 > 0. We summarize the result in the following Lemma <ref>. For μ_1, μ_2 ∈ C[0, 1] and σ_1, σ_2 > 0, h^2(f_μ_1,σ_1, f_μ_2, σ_2) ≤ 1- √(2σ_1σ_2/σ_1^2 + σ_2^2)exp{-μ_1 - μ_2_∞^2/4(σ_1^2 + σ_2^2)}. Note that by Hölder's inequality, f_μ_1, σ_1(y)f_μ_2, σ_2(y) ≥{∫_0^1√(ϕ_σ_1(y - μ_1(x)))√(ϕ_σ_2(y - μ_2(x)))dx}^2. Hence, h^2(f_μ_1,σ_1, f_μ_2, σ_2) ≤ ∫[∫_0^1ϕ_σ_1(y-μ_1(x))dx + ∫_0^1ϕ_σ_2(y-μ_2(x))dx - 2∫_0^1√(ϕ_σ_1(y - μ_1(x)))√(ϕ_σ_2(y - μ_2(x)))dx]dy. By changing the order of integration (applying Fubini's theorem since the function within the integral is jointly integrable) we get, h^2(f_μ_1,σ_1, f_μ_2, σ_2) ≤ ∫_0^1h^2(f_μ_1(x),σ_1, f_μ_2(x), σ_2)dx = ∫_0^1[1- √(2σ_1σ_2/σ_1^2 + σ_2^2)exp{-(μ_1(x) - μ_2(x))^2/4(σ_1^2 + σ_2^2)}]dx ≤ 1- √(2σ_1σ_2/σ_1^2 + σ_2^2)exp{-μ_1 - μ_2_∞^2/4(σ_1^2 + σ_2^2)}. When σ_1 = σ_2 = σ, h^2(f_μ_1,σ, f_μ_2, σ) ≤ 1 - exp{μ_1 -μ_2_∞^2/ 8 σ^2}, which implies that h^2(f_μ_1,σ, f_μ_2, σ) ≾μ_1 -μ_2_∞^2/σ^2. The standard inequality h^2(f_μ_1,σ_1, f_μ_2, σ_2) ≤f_μ_1,σ_1- f_μ_2, σ_2_1 relating the Hellinger distance to the total variation distance leads to the cruder bound h^2(f_μ_1,σ_1, f_μ_2, σ_2) ≤ C_1 μ_1 -μ_2_∞/(σ_1 ∧σ_2) + C_2|σ_2 - σ_1|/(σ_1 ∧σ_2), which is linear in μ_1 -μ_2_∞. This bound is less sharp than what is obtained in Lemma <ref> and does not suffice for obtaining the optimal rate of convergence. To control the Kullback-Leibler divergence between the true density f_0 and the model f_μ, σ, we derive an upper bound for logf_0/f_μ, σ_∞ in Lemma <ref>. If f_0 satisfies Assumption <ref>, logf_0/f_μ, σ_∞≤ C + μ - μ_0_∞^2/σ^2 for some constant C > 0. Note that f_μ, σ(y) = 1/√(2π)σ∫_0^1exp{-(y-μ(x))^2/2σ^2}dx ≥ 1/√(2π)σ∫_0^1exp{-(y-μ(x))^2/σ^2}dx exp{-μ-μ_0_∞^2/σ^2} ≥ C ϕ_σ/√(2) * f_0 (y) exp{-μ-μ_0_∞^2/σ^2} ≥ C f_0(y) exp{-μ-μ_0_∞^2/σ^2}, where the last inequality follows from Lemma 6 of <cit.> since f_0 is compactly supported by Assumption <ref>. This provides the desired inequality. For β∈ (2j, 2j+2], j ≥ 0 and f_j constructed by <ref>, we have the expression f_j = ∑_i=0^j(-1)^i j+1 i+1ϕ_σ^(i) f_0. The proof can be found in Appendix <ref>. The expression of f_j as a linear combination of true density and the folded convolutions indicates that f_j is as smooth as f_0. One can get infinitely differentiable function by convoluting with the kernel, so for the true density with higher regularity degree, we need add the "smoother" function into the approximation f_j to ensure the approximation error remains of order O(σ^β). For any β > 0, let f_0 satisfy Assumption <ref> and <ref>, integer j be such that for β∈ (2j, 2j+2], f_j constructed by (<ref>). For any constant L and all x ∈ [a_0, b_0], we have ϕ_σ * f_β (x) = f_0(x) (1 + O(D(x)σ^β)), where D(x) = ∑_i=1^r c_i |l_j(x)|^β/i + c_r+1, for nonnegative constants c_i, i = 1, …, r, and c_r+1 a multiple of L. Following the proof of Lemma 1 in <cit.>, for any x, y ∈ [a_0, b_0], logf_0(y)≤logf_0(x) + ∑_i=1^rl_j(x)/j!(y-x)^j + L|y-x|^β, logf_0(y)≥logf_0(x) + ∑_i=1^rl_j(x)/j!(y-x)^j - L|y-x|^β. Define B^u_f_0,r(x,y) = ∑_i=1^rl_j(x)/j!(y-x)^j + L|y-x|^β, B^l_f_0,r(x,y) = ∑_i=1^rl_j(x)/j!(y-x)^j - L|y-x|^β. Then we have e^B^u_f_0,r≤ 1 + B^u_f_0,r + 1/2!(B^u_f_0,r)^2 + … + M |B^u_f_0,r|^r+1, e^B^l_f_0,r≥ 1 + B^l_f_0,r + 1/2!(B^l_f_0,r)^2 + … - M |B^l_f_0,r|^r+1. where M = 1/(r+1)!exp{sup_x, y ∈ [a_0, b_0], x ≠ y (|∑_j=1^rl_j(x)/j! (y-x)^j| + L|y-x|^β)}. Note that f_0 is bounded on [a_0, b_0], we consider the convolution on the whole real line by extending f_0 analytically outside [a_0, b_0]. For β∈ (1, 2], r = 1 and x ∈ (a_0, b_0), ϕ_σ* f_0(x) ≤ f_0(x) ∫ e^B^u_f_0,r(x, y)ϕ_σ(y-x) dy ≤ f_0(x) ∫_ℝϕ_σ(y-x) [ 1 + L|y-x|^β + M ( l^2_1(x)(y-x)^2 + L^2|y-x|^2β) ] dy. Since l_j(x)'s are all continuous on [a_0, b_0], there exist finite constants M_j such that |l_j| ≤ M_j and |y-x| ≤ |b_0 - a_0|. The integral in the last inequality can be bounded by ∫_ℝϕ_σ(y-x) [ 1 + L|y-x|^β + M ((M_1|b_0 - a_0|)^2-β |l_1(x)(y-x)|^β + (L^2|b_0 - a_0|^β )|y-x|^β) ] dy Therefore, ϕ_σ* f_0(x) ≤ f_0(x) ( 1 + (r_1|l_1(x)|^β +r_2) σ^β), where r_1 = M (M_1|b_0 - a_0|)^2-βμ_β, r_2 = (L + ML^2)μ_β. In the other direction, ϕ_σ* f_0(x) ≥ f_0(x) ∫ϕ_σ(y-x) [ 1 - L|y-x|^β - M ( l^2_1(x)(y-x)^2 + L^2|y-x|^2β) ] dy. Thus we achieve (<ref>). For any β > 2, suppose β∈ (2j, 2j +2], j > 1. First we calculate ϕ_σ* f_0, ϕ^(2)_σ* f_0, …, ϕ^(j)_σ* f_0(x), by Lemma <ref> to get ϕ_σ* f_β(x). The calculation of ϕ^(i)_σ* f_0(x) is the same as ϕ_σ* f_0(x) except taking the convolution with ϕ_√(i)σ. The terms σ^2, σ^4, …, σ^2j caused by the factors containing |y-x|^k, k < β in ϕ^(i)_σ f_0 can be canceled out by Lemma <ref>. For terms containing |y-x|^k, k ≥β, we take out |y-x|^β and bound the rest by a certain power of |l_j(x)| or some constant. Let f_0 satisfy Assumption <ref> and <ref>. With A_σ = {x: f_0(x) ≥σ^H} , we have ∫_A^c_σf_0(x)dx = O(σ^2β), ∫_A^c_σϕ_σ*f_j(x) dx = O(σ^2β), for all non-negative integer j, sufficiently small σ and sufficiently Large H. Under Assumption <ref> there exists (a, b) ⊂ [a_0, b_0] such that A^c_σ⊂ [a_0, a) ∪ (b, b_0] if we choose σ sufficiently small, so that f_0 (x) ≤σ^H for x ∈A^c_σ. Therefore, ∫_A^c_σf_0(x) ≤σ^H |b_0 - a_0| ≤ O(σ^2β) if we choose H ≥ 2β. Using Lemma <ref>, ∫_A^c_σϕ_σ* f_j(x) dx = ∫_A^c_σ f_0(x) (1 + O(D(x)σ^β)) ≤ O(σ^H), with bounded D(x) and H ≥ 2β it is easy to bound the second integral by O(σ^2β). Suppose f_0(x) satisfies Assumption <ref> and <ref>. For β >2 and j such that β∈ (2j,2j+2], we can construct the density h_β from (<ref>) and show that h_β satisfies Lemma <ref> and Lemma <ref>. To get the density function we first show that f_j is nonnegative and compute the normalizing constant ∫ f_j(x) = 1 + O(σ^β). Following the proof of Lemma 2 in <cit.>, we treat log f_0 as a function in C^2 [0,1] and obtain the same form of ϕ_σ*f_0 as (<ref>). For small enough σ we can find ρ_1 ∈ (0,1) very close to 0 such that ϕ_σ* f_0(x) = f_0(x) (1 + O(D^(2)(x)σ^2)) < f_0(x)(1 + ρ_1), where D^(2) contains |l_1(x)| and |l_2(x)| with certain power, so D^(2) is bounded. Then we have f_1(x) = 2f_0(x) - K_σ f_0(x) > 2f_0(x) - f_0(x) (1 + ρ_1) = f_0(x) (1 - ρ_1). Then we treat logf_0 as a function with β = 4, j = 1. Similarly, we can get ϕ_σ* f_1(x) = f_0(x) (1 + O(D^(4)(x)σ^4)), where D^(4) contains |l_1(x)|, …. |l_4(x)|. We can find 0 < ρ_2 < ρ_1 such that ϕ_σ* f_1(x) < f_0(x) (1 + ρ_2), then can get f_2 (x) = f_0(x) - (ϕ_σ* f_1(x) - f_1(x)) > f_0(x)(1 - ρ_1 - ρ_2) > f_0(x)(1 - 2ρ_1). Continuing this procedure, we can get f_j(x) > f_0(x) (1 - jρ_1), with sufficiently small σ, and 1 - jρ_1 ∈ (0,1) but very close to 1. Obviously f_j is nonnegative. Now we calculate the normalizing constant for f_j. When β < 2, with Lemma <ref>, ∫ f_1(x) = ∫ f_0 - (ϕ_σ* f_0 - f_0) ≤∫ f_0 + |∫ (ϕ_σ*f_0 - f_0)| ≤ 1 + O(σ^β). For β∈ (2,4], ∫ f_2(x) = ∫ f_0 - (ϕ_σ* f_1 - f_1) ≤∫ f_1 + |∫ (ϕ_σ*f_1 - f_0)| ≤ 1 + O(σ^β). Then by induction, we have ∫ f_j = 1 + O(σ^β), so that we have the density h_β= f_j/1 + O(σ^β), β∈ (2j, 2j+2]. Now to show h_β satisfying (<ref>), note that ϕ_σ* h_β = ϕ_σ* f_j/∫ f_j = f_0(x)(1+ O(D(x)σ^β))/1+ O(σ^β). Since D(x) is bounded and for sufficiently small σ we consider term (1+ O(D(x)σ^β))/1+ O(σ^β)≾(1+ O((D(x)+1)σ^β)+ O(D(x)σ^2β))/1+ O(σ^β) = 1+ O(D(x)σ^β), which directly leads to the same form as (<ref>), and obviously Lemma <ref> is satisfied. Let f_0 satisfy Assumption <ref> and <ref>, and integer j be such that β∈ (2j, 2j+2]. Then we can show that the density h_β defined by (<ref>) satisfies, ∫ f_0(x) logf_0(x)/ϕ_σ* h_β(x) = O(σ^2β), for sufficiently small σ and all x ∈ [a_0, b_0]. Again consider the set A_σ = {x: f_0(x) ≥σ^H} with arbitrarily large H. We separate the Kullback-Leibler divergence into ∫_[a_0, b_0] f_0 logf_0/ϕ_σ* h_β≤∫_A_σ(f_0 - ϕ_σ*h_β)^2/ϕ_σ* h_β + ∫_A^c_σ f_0 logf_0/ϕ_σ* h_β + ∫_A^c_σ(ϕ_σ*h_β - f_0). Under Assumption <ref> and by Remark 3 in <cit.>, for small enough σ there exists constant C such that for all x ∈ [a_0, b_0], ϕ_σ*f_0 ≥ Cf_0, especially on set A^c_σ f_0 satisfies ϕ_σ*f_0 ≥ f_0/3. Also in the proof of Lemma <ref> we can find ρ∈ (0,1) such that f_β > ρ f_0. Then we have on set A_σ with sufficiently small σ ϕ_σ*h_β = ϕ_σ*f_β/1+O(σ^β)≥ρϕ_σ*f_0/1 + O(σ^β)≥ K f_0, for some positive and finite constant K. Applying Lemma <ref>, the first integral on the right side of <ref> can be bounded by ∫_A_σ(f_0 - ϕ_σ*h_β)^2/ϕ_σ* h_β ≤∫_A_σ[f_0(x) - f_0(x)(1 + O(D(x)σ^β))]^2/K_1 f_0(x) ≾∫_A_σ f_0(x) O(D^2(x)σ^2β) = O(σ^2β). To bound the second integral of r.h.s again by Remark 3 in <cit.> we get ϕ_σ*h_β≥ρ/3(1+ O(σ^β))f_0, so easily we can find a constant C < 1 such that ϕ_σ*h_β≥ C f_0. With Lemma <ref> clearly the second and third term can be bounded by O(σ^2β). Let ℍ_1^a denote the unit ball of RKHS of the Gaussian process with rescaled parameter a and 𝔹_1 be the unit ball of C[0,1]. For r >1, there exists a constant K, such that for ϵ < 1/2, log N(ϵ, ∪_a ∈ [0,r]ℍ_1^a, ·_∞) ≤ Kr( log1/ϵ)^2. Since we can write any element of ℍ_1^a as a function of Re(z) by <cit.> which can be analytically extended to some interval containing Ω^a = {z∈ℂ: |Re(z)| ≤ R} with R = δ/(6max(a,1)), so for any h ∈Ω^a, |2aRe(z)| ≤ |δ/6max(a,1)· 2a| = δ/3. Consider any b with |b-a|≤1, we can show that any element of ℍ_1^b can be extended analytically to Ω^a noting that for z ∈Ω^a related to the maximum norm, 2bRe(z)≤2aRe(z) + 2(b-a)Re(z)≤δ/3+2Re(z)≤2/3δ. Therefore, ℱ^a forms one ϵ-net over ℍ_1^b. We find one set Γ = {a_i, i = 1, …, k} with k = ⌊ r⌋ +1 and a_k = r, such that for any b∈ [0,r] there exists some a_i satisfying |b-a_i|≤ 1, so that ∪_i≤ kℱ^a_i forms an ϵ-net over ∪_a ≤ rℍ_1^a. Since the covering number of ∪_i≤ kℱ^a_i is bounded by summation of covering number of ℱ^a_i, we obtain log N(ϵ, ∪_a ∈ [0,r]ℍ_1^a, ·_∞) ≤log(∑_i=1^k #(ℱ^a_i)) ≤log(k ·#(ℱ^r)) ≤ Kr( log1/ϵ)^2. To complete the proof, note that the piecewise polynomials are constructed on the partition of [0,1], ∪_i≤ mB_i, where B_i's are disjoint interval with length shorter than R= δ/(6max(a,1)), so the total number of polynomials is a non-decreasing in a. Also we find that when building the mesh grid of the coefficients of polynomials in each B_i, both the approximation error and tail estimate are invariant to interval length R, therefore we have #(ℱ^a) ≤#(ℱ^b) if a ≤ b, for a,b ∈ [0,r]. With larger a we need a finer partition on [0,1] while the grid of coefficients of piecewises polynomial remains the same except the range and the meshwidth will change together along with a. Since we can see the element h of RKHS ball as a function of it and with Cauchy formula we can bound the derivatives of h by C/R^n, where |h|^2 ≤ C^2. § PROOF OF THE MAIN THEOREM Proof of Theorem <ref>: Following <cit.>, we need to find sequences ϵ̅_n,ϵ̃_n → 0 with nmin{ϵ̅_n^2,ϵ̃_n^2}→∞ such that there exist constants C_1, C_2, C_3, C_4> 0 and sets ℱ_n ⊂ℱ so that, log N(ϵ_n, ℱ_n, d) ≤ C_1nϵ̅_n^2 Π(ℱ_n^c) ≤ C_3exp{-nϵ̃_n^2(C_2+4)} Π( f_μ, σ: ∫ f_0 logf_0/f_μ, σ≤ϵ̃_n^2, ∫ f_0 log(f_0/f_μ, σ)^2 ≤ϵ̃_n^2 ) ≥ C_4exp{-C_2nϵ̃_n^2}. Then we can conclude that for ϵ_n = max{ϵ̅_n, ϵ̃_n} and sufficiently large M > 0, the posterior probability Π_n(f_μ, σ: d(f_μ, σ, f_0) > Mϵ_n | Y_1, …, Y_n) → 0 a.s. P_f_0. We consider the Gaussian process μ∼ W^A given A, with A satisfying Assumption <ref>. We will first verify (<ref>) along the lines of <cit.>. Note that h^2(f_0, f_μ, σ) ≾ h^2(f_0, f_μ_0, σ) + h^2(f_μ_0, σ, f_μ, σ). Since f_μ_0, σ = ϕ_σ*h_β, by Lemma <ref>, one obtains under Assumptions <ref> and <ref>, h^2(f_0, f_μ_0, σ) ≤∫ f_0 log(f_0/f_μ_0, σ) ≾ O(σ^2β). From Lemma <ref> and the following remark, we obtain h^2(f_μ_0, σ, f_μ, σ) ≾μ- μ_0_∞^2/σ^2. From Lemma 8 of <cit.>, one has ∫ f_0 log(f_0/f_μ, σ)^i≤ h^2(f_0, f_μ,σ)(1 + logf_0/f_μ, σ_∞)^i for i=1,2. From (<ref>)-(<ref>), for any b ≥ 1 and ϵ̃_n^2 = σ_n^2β, {σ∈ [σ_n, σ_n + σ_n^b], μ - μ_0_∞≾σ_n^β+1}⊂ {∫ f_0 logf_0/f_μ, σ≾σ_n^2β, ∫ f_0 log(f_0/f_μ, σ)^2 ≾σ_n^2β}. Since μ_0 ∈ C^β+1[0,1], from Section 5.1 of <cit.>, (μ - μ_0_∞≤ 2δ_n) ≥ C_4exp{-C_5(1/δ_n)^1/β+1log(1/δ_n)^2∨ q}(C_6/δ_n)^p+1/β+1, for δ_n → 0 and constants C_4, C_5, C_6 > 0. Letting δ_n = σ_n^3, we obtain (μ - μ_0_∞≤ 2δ_n) ≥exp{-C_7(1/σ_n)log^2∨ q(1/σ_n^β+1)}, for some constant C_7 > 0. Since σ∼ IG(a_σ, b_σ), we have (σ∈ [σ_n, 2σ_n ]) = b_σ^a_σ/Γ(a_σ)∫_σ_n^2σ_nx^-(a_σ+1) e^-b_σ/xdx ≥ b_σ^a_σ/Γ(a_σ)∫_σ_n^2σ_n e^-2b_σ/xdx ≥ b_σ^a_σ/Γ(a_σ)σ_nexp{-b_σ/σ_n } ≥ exp{-C_8/σ_n }, for some constant C_8> 0. Hence {σ∈ [σ_n, 2σ_n ], μ - μ_0_∞≾σ_n^β+1} ≥ exp{-C_7(1/σ_n)log^2∨ q(1/σ_n^β+1)}exp{-C_8/σ_n } ≥ exp{-2C_7(1/σ_n)log^2∨ q(1/σ_n^β+1)}. Then (<ref>) will be satisfied with ϵ̃_n = n^-β/2β+1log^t_1(n), where t_1=β(2∨ q)/2β+1 and some C_9 > 0. Next we construct a sequence of subsets ℱ_n such that <ref> and <ref> are satisfied with ϵ̅_n = n^-β/2β+1log ^t_2n and ϵ̃_n for some global constant t_2 > 0. Letting 𝔹_1 denote the unit ball of C[0,1] and given positive sequences M_n, r_n, define B_n = ∪_a < r_n(M_n ℍ^a_1) + δ̅_n𝔹_1 as in <cit.>, with δ̅_n =ϵ̅_nl_n/K_1, K_1 = 2(2/π)^1/2 and let ℱ_n = {f_μ, σ: μ∈ B_n, l_n < σ < h_n }. First we need to calculate N(ϵ̅_n, ℱ_n, ·_1). Observe that for σ_2 >σ_1 > σ_2/2, f_μ_1, σ_1 - f_μ_2, σ_2_1 ≤(2/π)^1/2μ_1 - μ_2_∞/σ_1 + 3(σ_2 - σ_1)/σ_1. Taking κ_n =min{ϵ̅_n/6, 1} and σ_m^n = l_n (1+ κ_n)^m, m ≥ 0, we obtain a partition of [l_n, h_n] as l_n=σ_0^n < σ_1^n < ⋯ < σ_m_n-1^n < h_n ≤σ_m_n^n with m_n= (logh_n/l_n) 1/log( 1+κ_n) +1. One can show that 3(σ_m^n -σ_m-1^n)/σ_m-1^n = 3κ_n ≤ϵ̅_n/2. Let {μ̃_k^n, k=1, …, N(δ̅_n, B_n, ·_∞)} be a δ̅_n-net of B_n. Now consider the set {(μ̃_k^n, σ_m^n): k=1, …, N(δ̅_n, B_n, ·_∞), 0≤ m≤ m_n }. Then for any f= f_μ, σ∈ℱ_n, we can find (μ̃_k^n, σ_m^n) such that μ - μ̃_k^n_∞ < δ̅_n. In addition, if one has σ∈ (σ_m-1^n, σ_m^n], then f_μ, σ - f_μ_k^n, σ^n_m_1 ≤ϵ̅_n. Hence the set in (<ref>) is an ϵ̅_n-net of ℱ_n and its covering number is given by m_nN(δ̅_n, B_n, ·_∞). From the proof of Theorem 3.1 in <cit.>, for any M_n, r_n with r_n > a_0, we obtain log N(2δ̅_n, B_n, ·_∞) ≤ K_2 r_n ( log(M_n/δ̅_n))^2. Again from the proof of Theorem 3.1 in <cit.>, for r_n > 1 and for M_n^2 > 16K_3r_n (log (r_n / δ̅_n))^2, we have (W^A ∉ B_n) ≤K_4r_n^p e^-K_5r_nlog^q r_n/K_5log^q r_n + exp{-M_n^2/8} for constants K_3, K_4, K_5 > 0. Next we calculate P(σ∉ [l_n, h_n]). Observe that (σ∉ [l_n, h_n ]) = P(σ^-1 < h_n^-1) + P(σ^-1 > l_n^-1) ≤ ∑_k=α_σ^∞e^-b_σh_n^-1(b_σh_n^-1)^k/k! + b_σ^a_σ/Γ(a_σ)∫_l_n^-1^∞ e^-b_σx/2dx ≤ e^-a_σlog(h_n) + b_σ^a_σ/Γ(a_σ)e^-b_σl_n^-1/2. Thus with h_n = O(exp{n^1/2β+1(log n)^2t_1}), l_n = O(n^-1/2β+1(log n)^-2t_1), r_n =O(n^1/2β+1(log n)^2t_1), M_n = O(n^1/2β+1(log n)^t_1+1), (<ref>) and (<ref>) implies Π(ℱ_n^c)= exp{-K_6nϵ̃^2_n} for some constant K_6 > 0 guaranteeing that (<ref>) is satisfied with ϵ̃_n = n^-β/2β+1(log n)^t_1. Also with ϵ̅_n = n^-β/2β+1(log n)^t_1+1, it follows from (<ref>) and (<ref>) that log N(ϵ̅_n, ℱ_n, ·_1) ≤ K_7 n^1/2β+1(log n)^2t_1+2 for some constant K_7 > 0. Hence max{ϵ̅_n, ϵ̃_n} = n^-β/2β+1(log n)^t_1+1. § DISCUSSION Non-linear latent variable models offer a flexible modeling framework in a broad variety of problems and improved practical performance has been demonstrated by <cit.> among others. The univariate density estimation model studied here can be extended to multivariate density estimation, latent factor models and density regression problems. When the density is compactly supported, the quantile function is a continuous function on [0, 1]. Hence one can use standard results on concentration bounds for Gaussian processes <cit.>. However, for densities supported on ℝ, the results fail as the corresponding quantile function is unbounded near zero and one. In this case, assumptions on the tail behavior of the true density as well as a careful analysis on the behavior of the corresponding quantile function near boundary are required. We propose to address this problem elsewhere. § APPENDIX §.§ Proof of Theorem <ref> Let f_0 be a density with quantile function μ_0 that satisfies the conditions of Theorem <ref>. Observe that μ_0_1 = ∫_t=0^1 μ_0(t) dt = ∫_-∞^∞z f_0(z) dz < ∞ since f_0 has a finite first moment, and thus μ_0 ∈_1[0, 1]. Fix ϵ > 0. We want to show that Π{ B_ϵ(f_0) } > 0, where B_ϵ(f_0) = {f  : f - f_0_1 < ϵ}. Note that μ_0 ∉ C[0, 1], so that ( μ - μ_0_∞ < ϵ) can be zero for small enough ϵ. The main idea is to find a continuous function μ̃_̃0̃ close to μ_0 in L_1 norm and exploit the fact that the prior on μ places positive mass to arbitrary sup-norm neighborhoods of μ̃_̃0̃. The details are provided below. Since ϕ_σ*f_0 - f_0_1 → 0 as σ→ 0, find σ_1 such that ϕ_σ*f_0 - f_0_1 < ϵ/2 for σ < σ_1. Pick any σ_0 < σ_1. Since C[0, 1] is dense in _1[0, 1], for any δ > 0, we can find a continuous function μ̃_̃0̃ such that μ_0 - μ̃_̃0̃_1 < δ. Now, f_μ, σ - f_μ̃_̃0̃, σ_1 ≤ C μ - μ̃_̃0̃_1/σ for a global constant C. Thus, for δ = ϵσ_0/4, {f_μ, σ : σ_0 < σ < σ_1, μ - μ̃_̃0̃_∞ < δ}⊂{f_μ, σ : f_0 - f_μ, σ_1 < ϵ}, since f_0 - f_μ, σ_1 < f_0 - f_μ_0, σ_1 + f_μ_0, σ - f_μ̃_̃0̃, σ_1 + f_μ̃_̃0̃, σ - f_μ, σ_1 and f_μ_0, σ = ϕ_σ*f_0. Thus, Π{ B_ϵ(f_0) } > (μ - μ̃_̃0̃_∞ < δ)  (σ_0 < σ < σ_1) > 0, since Π_μ has full sup-norm support and Π_σ has full support on [0, ∞). §.§ Proof of Lemma <ref> Consider f_j constructed by (<ref>). When i=1, f_1 = 2f_0 - ϕ_σ*f_0, so the form holds. By induction, suppose this form holds for j > 1, then f_j+1 = f_0 - (ϕ_σ*f_j - f_j) = f_0 + ∑_i=0^j(-1)^i+1j+1 i+1ϕ_σ^(i+1)* f_0 + ∑_i=0^j(-1)^i j+1 i+1ϕ_σ^(i)* f_0 = (j+2)f_0 + ∑_i=1^j+1(-1)^ij+1 i+1ϕ_σ^(i)* f_0 + ∑_i=1^j(-1)^ij+1 iϕ_σ^(i)* f_0 = (j+2)f_0 + ∑_i=1^j (-1)^i( j+1 i+1 + j+1 i) ϕ_σ^(i)*f_0 + (-1)^j+1ϕ_σ^(i +1)*f_0 = (j+2)f_0 + ∑_i=1^j(-1)^ij+2 i+1ϕ_σ^(i)* f_0 + (-1)^j+1ϕ_σ^(i +1)*f_0 = ∑_i=0^j+1(-1)^ij+2 i+1ϕ_σ^(i)* f_0. It holds for j +1, which completes the proof. plain

http://arxiv.org/abs/1701.07597v1

20170126073357

Memory effect and pseudomode amplitude in non-Markovian dynamics of a two level system

quant-ph

Graduate School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan Graduate School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan We study non-Markovian dynamics of a two level atom using pseudomode method. Because of the memory effect of non-Markovian dynamics, the atom receives back information and excited energy from the reservoir at a later time, which causes more complicated behaviors than Markovian dynamics. With pseudomode method, non-Markovian dynamics of the atom can be mapped into Markovian dynamics of the atom and pseudomode. We show that by using pseudomode method and quantum jump approach for Markovian dynamics, we get a physically intuitive insight into the memory effect of non-Markovian dynamics. It suggests a simple physical meaning of the memory time of a non-Markovian reservoir. Memory effect and pseudomode amplitude in non-Markovian dynamics of a two level system Yasuhiro Tokura December 30, 2023 ====================================================================================== § INTRODUCTION All realistic quantum systems are open quantum system; the system interacts with reservoir systems which cause decoherence and relaxation <cit.>. According to characters of the interaction and the structure of the reservoirs, the dynamics of open quantum systems can be classified into Markovian dynamics with no memory effect and non-Markovian one with memory effect. In Markovian open system, the reservoir acts as a sink for the system information; the information that the system of interest loses into the reservoir does not play any further role in the system dynamics. However, in non-Markovian case, this lost information is temporarily stored in the reservoir and comes back at a later time to influence the system <cit.>. This is the memory effect of non-Markovian dynamics and causes more complicated behaviors than Markovian dynamics. There are stochastic approach for Markovian dynamics <cit.>; quantum jump, Monte Carlo wave function and quantum trajectory. In these methods, the dissipation caused by the interaction with the reservoir is interpreted as an incoherent jump between two states and the state of the system is described by the sum of the ensembles which are identified by the jump times. Therefore, we can get the intuitive understanding of the dynamics. Recently, non-Markovian dynamics has been investigated <cit.>. In these papers, non-Markovianity of quantum processes is discussed. The measures for the degree of non-Markovianity are based on the distinguishability of quantum states, which focuses on the dynamics of the system of interest. In this paper, we use pseudomode method <cit.>. With pseudomode method, non-Markovian dynamics of the system of interest can be mapped into Markovian dynamics of a combined system of the system of interest and pseudomode. Therefore, the dynamics of the extended system can be discussed. Non-Markovian quantum jump has been also investigated <cit.>. Because of the memory effect and back flow from the reservoir into the system, the description is more complicated than Markovian case and pure state quantum trajectories for general non-Markovian systems do not exist <cit.>. By connecting this method and pseudomode method, we get a simple intuitive physical picture of the memory of a non-Markovian reservoir and of how such memory allows to partly restore some of the coherence lost to the environment <cit.>. This result also suggests that pseudomode could be seen as an effective description of the reservoir memory. The purpose of this paper is to get a more physically intuitive insight into the memory effect of non-Markovian dynamics. For this purpose, we use pseudomode method and quantum jump approach. With pseudomode method, non-Markovian dynamics of the system is described by Markovian dynamics of a combined system, so that we can apply quantum jump approach for Markovian dynamics to the combined system. The result gives us a simple physical meaning of the memory time of a non-Markovian reservoir. The paper is organized as follows. In Sec. <ref>, we persent the model discussed in the paper. In Sec. <ref>, the property of the model we have presented in Sec. <ref> is evaluated using quantum jump approach for Markovian dynamics and, in Sec. <ref>, we study the dynamics of the damped Jaynes Cummings model, which is a typical example of non-Markovian system. Finally, we conclude the paper in Sec. <ref>. § MODEL Non-Markovian systems appear in many branches of physics. Here we consider a two level atom interacting with a structured electromagnetic reservoir which is described by the Jaynes-Cummings model with rotating approximation <cit.>. The Hamiltonian for the total system is H=ħω_02σ_z +∑_k ħω_k b_k^† b_k+∑_k ħ g_k (σ_+ b_k +σ_-b_k^†), where σ_z = |eł̊e| - |gł̊g|, σ_+ = (σ_-)^† = |eł̊g|, b_k^† and b_k are the bosonic creation and annihilation operators for the reservoir mode k with frequency ω_k ≥ 0 and g_k is the coupling between the two level atom and the reservoir mode k in the reservoir. |e$̊ and|g$̊ are excited and the ground states of the two level atom, respectively. Total excitation number is a conserved quantity in this model. Let the atom be in an arbitrary superposition and the reservoir be in vacuum state at t=0, therefore the initial state is given by |Ψ(0)=̊ (α|e+̊β |g)̊⊗ |0,̊ where |0$̊ denotes the vacuum state of the reservoir, andαandβsatisfy the normalization condition|α|^2 + |β|^2=1. In the interaction picture, the total state att≥0can be expanded as |Ψ(t)_̊I = α(a_0(t)|e,0+̊∑_k a_k(t)|g,1_k)+β|g,0.̊ where|1_k=̊ b_k^†|0$̊ and the coefficient of |g,0$̊ is independent of time. Inserting this state into Schrödinger equation, we get the integro-differential equation for atomic amplitudea_0(t), ddta_0(t)= - ∫_0^t dt' f(t-t') a_0(t'), wheref(t) ≡∑_k g_k^2 e^-i (ω_k-ω_0)t is a correlation function. Here we assume that the couplingg_kdepends only on the frequencyω_k. In a continuous distribution limit, the sum on the reservoir modekis replaced by an integral byωas follows, ∑_k g_k^2 ≃∫_-∞^∞ dωρ(ω) g^2(ω) = 12π∫_-∞^∞ dω D(ω), whereρ(ω)is the density of states of the reservoir. The structure of the reservoir is characterized by the positive definite functionD(ω). Because we have extended the integral to-∞, this function should be vanished in the negativeωregion. With these equations, the correlation function becomes f(t) = 12π∫_-∞^∞ dω D(ω)e^-i (ω-ω_0)t. If the reservoir has no structure,D(ω) does not depend onω. In this case, the correlation function is proportional to the delta function, so that the dynamics of the system is Markovian dynamics. In the following, we restrict thatD(ω)can be approximated by a sum of Lorentz functions. This is not the necessary condition for using pseudomode method but an assumption for simplicity. We set the explicit form ofD(ω)as D(ω) ≃∑_l =1^L γ_l λ_l^2(ω -ω_l)^2 + λ_l^2, whereγ_lis the coupling strength andλ_l^-1is the reservoir's correlation time. SinceD(ω)is vanished in the negativeωregion, the resonant frequencyω_lshould be much lager than the widthλ_l. From the residue theorem andt-t'≥0, only the poles in the lower half plane have contribution for the dynamics ofa_0(t). So we define thatλ_lis positive for anyl. SinceD(ω)should be non-negative for anyω, we considerγ_l >0for anyl, which is also not the necessary condition but an assumption for simplicity. Integral of Lorentz function is ∫_-∞^∞ dωλ_l^2 (ω -ω_l)^2 + λ_l^2 =πλ_l. Using this assumption, we get the integro-differential equation ddta_0(t) =- ∑_l=1^L γ_l λ_l2∫_0^t dt' e^-i Δ_l (t-t')e^- λ_l(t-t') a_0(t'), whereΔ_l = ω_l-ω_0. From this equation, we can see that the parameterλ_lrepresents how long past state affects the present dynamics. If there exists finiteλ_l, the present dynamics depends on the past dynamics. Therefore, the system interacts with non-Markovian reservoir and its non-Markovianity is characterized byλ_l. With the pseudomode method <cit.>, the dynamics of this system can be mapped into Markovian dynamics of a combined system of the two level system andLpseudomodes system. For the present model, pseudomode method leads to the following Markovian master equation ddtρ_SP^I(t) = 1i ħ[H_SP,ρ_SP^I(t)] + ∑_l=1^L 2 λ_l 𝒟[c_l]ρ_SP^I(t), whereρ_SP^I(t)is the density operator of the combined system,𝒟[·]is a superoperator 𝒟[A]ρ = Aρ A^† - 12(A^† A ρ- ρ A^† A ), which describes the dissipation andH_SPis a combined system Hamiltonian H_SP = ∑_l=1^L ħΔ_l c_l^† c_l + ∑_l=1^L ħ√(γ_lλ_l/2)(σ_+ c_l+σ_-c_l^†), wherec_l^†andc_lare the creation and annihilation operators for the pseudomode labeled byl. From the definition of the pseudomdoes, the initial state of the pseudomodes is the vacuum (see Refs. <cit.> for details). From Eq. (<ref>), we can see that the system coherently interacts with pseudomodes and each pseudomode dissipatively interacts with a Markovian reservoir (FIG. <ref>). The information contained in the atom first flows to pseudomodes and then from each pseudomode to its reservoir. The flow from each pseudomode to its reservoir is one-way, but the flow between the atom and pseudomodes is two-way. In non-Markovian dynamics, the atom recieves back information and excitation energy from the reservoir due to memory effect. Therefore pseudomodes could be seen as an effective description of the reservoir memory <cit.>. To get the state of the system, we should trace out the pseudomodes, ρ_S^I(t) = Tr_Pρ_SP^I(t). § STOCHASTIC APPROACH With the pseudomode method, the dynamics of this system is effectively described by the Markovian master equation. So we can use stochastic approach for Markovian dynamics. Here we define a non-Hermitian Hamiltonian, H_ eff^I = H_SP - i∑_l=1^L ħλ_l c_l^† c_l . In this Hamiltonian, non-Hermitian term represents dissipation into the Marovian resevoir. Using Eq. (<ref>), we can rewrite Eq. (<ref>) as, ddtρ_SP^I(t) = 1i ħ[H_ eff^I ρ_SP^I(t)-ρ_SP^I(t)(H_ eff^I)^†] + ∑_l=1^L2λ_l c_lρ_SP^I(t)c_l^†. The first term of the right-hand side represents the continuous dynamics governed by the non-Hermitian HamiltonianH_eff^I. The second term represents jump process which is the loss of an excitation energy from pseudomodes. Using an unnormalized state vector|Ψ̃(t)$̊ which satisfies Schrödinger equation, i ħddt|Ψ̃(t)=̊ H_ eff^I|Ψ̃(t),̊ we can divide ρ_SP^I(t) into two terms as follows, ρ_SP^I(t) = |Ψ̃(t)ł̊Ψ̃(t)|+ Π_p(t) |g,0_Pł̊g,0_P|, where |0_P$̊ is the vacuum state of pseudomodes. The trace ofρ_SP^I(t)is conserved to 1, so that the coefficient of the second termΠ_p(t)is defined byΠ_p(t)= 1 - łΨ̃(t)|Ψ̃(t)$̊ of time. Because λ_l is defined as positive, the inner product of |Ψ̃(t)$̊ is a monotonic decreasing function of time andΠ_p(t)is a monotonic increasing function. From the quantum trajectory approach <cit.>, the unnormalized state vector|Ψ̃(t)$̊ is a trajectory under no jumps. Since the system is two level atom and the initial state of the reservoir is vacuum, the jumped part (= the second term of Eq. (<ref>)) is the ground state. The state of the two level atom is given by ρ_S^I(t) = Tr_P |Ψ̃(t)ł̊Ψ̃(t)| + Π_p(t) |gł̊g|. The probability that there is no jump until time t (= the survival probability) is given by the inner product of |Ψ̃(t)$̊, P_0(t) = łΨ̃(t)|Ψ̃(t).̊ Because we can regard the pseudomodes as memory part of the reservoir <cit.>, the survival probabilityP_0(t)can be regarded as the probability that the system interacts with its reservoir coherently until timet. The jump rate to the ground state of the combined system, which is given by the damp rate ofP_0(t), represents a memory loss rate. So the probability density of jump is given by p(t) = -ddt P_0(t). This probability density represents the information flux from pseudomode into Markovian reservoir. For the particular model, the relationship between the oscillation ofp(t)and the measure of non-Markovianity had been discussed <cit.>. Since the initial state of the atom is|ψ(0)=̊ α|e+̊ β|g$̊, the state of the combined system can be expanded as |Ψ̃(t)=̊α(a_0(t)|e,0_P+̊∑_l=1^L q_l(t)|g,1_l)̊+β|g, 0_P,̊ where |1_l=̊ c_l^† |0_P$̊,a_0(0)=1andq_l(0)=0. Thisa_0(t)is the same as the amplitudea_0(t)in Eq. (<ref>). Using Eq. (<ref>), the survival probabilityP_0(t)and the probability densityp(t)are given by P_0(t) = | α |^2( |a_0(t)|^2 +∑ _l=1^L|q_l(t)|^2) +|β|^2, p(t) = ∑_l=1^L 2λ_l|α q_l(t)|^2. The probability densityp(t)is the2λ_l|αq_l(t)|^2represents the energy flow from pseudomodelto its reservoir. If the jump to the ground state occurs during the measurement timeT, the expectation value of the jump time is given by ł t _̊T = ∫_0^T t p(t) dt∫_0^T p(t)dt, and we define the expectation valuełt $̊ as the long measurement time limit of ł t _̊T, ł t ≡̊ lim_T→∞ł t _̊T = ∫_0^∞ (| a_0(t)|^2 +∑ _l=1^L| q_l(t)|^2)dt. Moreover we define that ł t_S =̊ ∫_0^∞ | a_0(t)|^2dt , ł t_l =̊ ∫_0^∞ | q_l(t)|^2dt, and then we get ł t =̊ł t_S +̊∑_l ł t_l $̊, where łt_S $̊ is the expected time length that the two level system is in the excited state |e$̊ andłt_l $̊ is one that a pseudomode l is in the excited state |1_l$̊. Since we can regard the pseudmodes as the degree of freedom of the reservoir that interact with the system of interest coherently <cit.>,∑_l łt_l $̊ can be regarded as the expectation value of memory time of the non-Markovian reservoir and reflects non-Markovianity of the system dynamics. We consider the Markovian limit (λ_l→∞). When λ_l ≫Δ_l ,γ_l, the unnormalized state vector is approximated as |Ψ̃(t)≃̊(α e^-1/2∑_l γ_l t |e+̊β |g) |0_P.̊ Therefore, we can see that q_l(t)=0 for any t>0 and ł t_l =̊ 0 in the Markovian limit . Because time t is positive, ł t_l =̊0 means that there is no time that pseudomodes are in their excited states. The Markovian limit is the limit the reservoir has no memory. This is consistent and intuitive with the result we have got here; pseudomodes are vanished and ∑_l ł t_l $̊ converges to0in the Markovian limit, so that pseudomode is a memory part of the reservoir and∑_l łt_l $̊ is an expectation value of memory time of the reservoir. This result also suggests the following criterion. * When ∑_l ł t_l =̊ 0, its dynamics is Markovian. * When ∑_l ł t_l ≠̊0, its dynamics is non-Markovian. § DAMPED JAYNES-CUMMINGS MODEL In this section, we discuss the dynamics of a two level atom in a lossy cavity <cit.>. The reservoir is electromagnetic field inside and outside the cavity and its density of state has a peak at the cavity resonant frequency. Therefore, we can assume that the structure is a single Lorentz function, D(ω) = γλ^2(ω -ω_c)^2 + λ^2, where ω_c is the resonant frequency of the cavity. This is called the damped Jaynes-Cummings model, which is a typical example of non-Markovian system and L=1 case of what we have discussed. Therefore, we can use pseudomode method and the result we have got. The effective Hamiltonian H_ eff^I is H_ eff^I = H_SP - iħλ c_p^† c_p = ħ(Δ- i λ) c_p^† c_p +ħ√(γλ2)(σ_+ c_p+σ_-c_p^†), and the unnormalized state |Ψ̃(t)$̊ is |Ψ̃(t)=̊α(a_0(t)|e, 0_P+̊ q(t)|g,1_P)̊+β|g, 0_P,̊ whereΔ=ω_c-ω_0is the detuning between the two level system and the pseudomode,c_p^†andc_pare creation and annihilation operators for the pseudomode and|1_P=̊ c_p^†|0_P$̊. Inserting the effective Hamiltonian H_ eff^I and the unnormalized state |Ψ̃(t)$̊ into Schrödinger equation, we get two simultaneous differential equations, {[ iddta_0(t) =√(γλ2)q(t); iddtq(t) = (- iλ+Δ) q(t) +√(γλ2)a_0(t).; ]. Eigenvalues of these equations are -(λ + i Δ) ±√((λ + i Δ)^2 -2γλ)2 . Under the initial condition,a_0(0)=1andq(0)=0, the solution is {[ a_0(t) = e^-λ/2te^-iΔ/2t[cosh(dt2)+λ+iΔdsinh(dt2)],; q(t) = - i √(2γλ)/d e^-λ/2t e^-iΔ/2tsinh(dt2) , ]. where we defined =√( (λ+ i Δ)^2 - 2γλ). As a result, we get the probability density, p(t) = 2λ |α q(t)|^2 = 2 |α|^2 γλ ^2|d|^2 e^-λ t(cosh( Re[d] t)-(cos( Im[d] t)) . FIG. <ref> shows the dynamics of the populations as a function of time. In FIG. <ref>,|a_0(t)|^2and|q(t)|^2oscillate andΠ_p(t)monotonically increases. The correlated oscillations of|a_0(t)|^2and|q(t)|^2are caused by non-Markovianity. There is no energy flow from Markovian reservoir into the combined system, so thatΠ_p(t)monotonically increases. FIG. <ref> shows the dynamics of the probability density of jump as a function of time. From FIG. <ref>, we can see thatp(t)is positive except fort=0andt→∞, becausecoshx >1for∀x>0. When there is no detuningΔ=0,d=√(λ^2-2γλ), so thatdis real or pure imaginary. When2γ<λ,dis real so thatp(t)≠0fort>0. On the other hand, whenλ<2γ,dis pure imaginary so that p(t) = 2|α|^2γλ2γ-λ e^-λ t(1-cos(√(2γλ-λ^2) t)). Therefore when the time satisfies t= 2πn(2γ-λ)^-1/2 forn ∈{0,𝐍}, the pribability densityp(t)is0. Here the structure of the reservoir is a single Lorentz function so that there is only a single pseudomode and the expectation value of jump time is given by ł t = ł t_S +̊ł t_P ,̊ ł t_S =̊∫_0^∞ | a_0(t)|^2dt, ł t_P =̊∫_0^∞ | q(t)|^2dt. Using the result we calculated above, we get {[ ł t_S =̊1γ(1+ (Δλ)^2)+ 12λ; ł t_P =̊12λ. ]. As noted at Eq. (<ref>), non-Markovianity is characterized byλ. The decrease ofλmeans an increase of the reservoir correlation time, hence the non-Markovianity becomes stronger. In Eq. (<ref>), the expectation valuesłt_S $̊ and ł t_P$̊ are monotonically decreasing functions forλ. Therefore, the result shows that non-Markovianity of the system dynamics is reflected to the delay of the expectation value of the jump timełt $̊. From Eq. (<ref>), we see that ł t_S $̊ depends on the detuningΔandłt_P$̊ do not depend on it. This can be understood as follows. In the detuned Rabi oscillation, the oscillating amplitude is smaller than unity, which depends on the value of the detuning. The damp rate of the system excited state population and the maximum value of the pseudomode excited state population are suppressed by increasing the detuning, which are shown in FIG. <ref>. Therefore, the damp of P(t) is slower than resonance case and the expected time length that the atom is in the exited state increases as the detuning increases. However, the leak rate from the pseudomode into the Markovian reservoir is 2λ, which does not depend on the detuning. This is the reason why the expected time length the pseudomode is in the exited state is the invariant value for the detuning. As shown in FIG. <ref>, instead of the suppression of the maximum value, the population in the later time increases. We also calculate the generating function χ(ω) ≡∫_0^∞ p(t) e^iω t dt. Using Eq. (<ref>), we get explicit form of the generation function χ(ω) = 2γλ^2(λ-iω)(λ-iω)^4-(λ^2-Δ^2-2γλ)(λ-iω)^2-(Δλ)^2. From this function, we can get the expectation value ł t =̊ . dlnχ(ω)d (iω)|_ω=0 = 1γ(1+(Δλ)^2)+1λ, and the variance ł (δ t)^2 =̊ . d^2lnχ(ω)d (iω)^2|_ω=0 = 1γ^2(1+(Δλ)^2)^2-1γλ(1-3(Δλ)^2)+1λ^2. In order to understand the relationship between these values, we define the function Λ_ł t≡ ł (δ t)^2 -̊(ł t )̊^2(ł t )̊^2 = -(3λ^2-Δ^2)γλ(λ^2+γλ+Δ^2)^2. This function can be divided into 3 cases as follows Λ_ł t= {[ >0 ⋯ √(3)λ < |Δ|; =0 ⋯ √(3)λ = |Δ|; <0 ⋯ √(3)λ > |Δ| ]. The sign of Λ_ł t changes at λ_0=|Δ|/√(3). In the Markovian limit (λ→∞), the expectation value and the variance converge to ł t →̊γ^-1 and ł (δ t)^2 →̊γ^-2, respectivelity. Thus Λ_ł t converges to 0 in the Markovian limit. Because the correlation time λ_0^-1=√(3)/|Δ| is small for large detunig |Δ|, the function Λ_ł t is positive for relatively large λ for large detuning. When it is negative, the variance is relatively smaller than one of Markovian dynamics. ł t $̊ is the expected time length that the system and the reservoir can interact with each other coherently. Therefore, negativeΛ_łtmeans that the memory is lost at more definite time compared with Markovian dynamics. § CONCLUTIONS We have studied the non-Markovian dynamics of a two level atom, using pseudomode method and the stochastic approach for Markovian dynamics. In this paper, we have assumed that the structure of the reservoir is given by a sum of Lorentz functions. With pseudomode method, the non-Markovian dynamics of a two level atom can be mapped to Markovian dynamics of a combined system of the system and pseudomodes whose number is the same as that of the Lorentz functions. The expectation value of jump time to the ground state of the combined system is given by the sum of the expected time length that the two level system is in the exited state and one that each pseudomode is in its excited state. The later time length represents the memory time of a non-Markovian reservoir. In the Markovian limit, we get the result that the probability that pseudomodes are in their exited states is 0. Then the expected time length that pseudomodes in their excited state also converges to 0. In particular, we have discussed the damped Jaynes-Cummings model, which is a model of a two level atom in a lossy cavity. This is analytically solvable so that we can get an exact solution of the dynamics and the expectation values, explicitly. As a result, we have found that the expected time length that the system and the pseudomode are in their excited state took the value reflecting non-Markovianity. Since Markovian approximation is the approximation that the reservoir has no memory, we can say that our result suggest the pseudomode is the degree of the freedom characterizing the memory of the reservoir. § ACKOWLEDGEMENTS The authors are grateful to C. Uchiyama for thoughtful comments and suggestions. This work was supported by CREST, JST.

http://arxiv.org/abs/1701.08129v3

20170127174506

Extension and restriction principles for the HRT conjecture

math.CA

Kasso A. Okoudjou Department of Mathematics & Norbert Wiener Center University of Maryland College Park, MD, 20742 USA kasso@math.umd.edu This work was partially supported by a grant from the Simons Foundation # 319197, and ARO grant W911NF1610008. Part of this material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester. [2000]Primary 42C15; Secondary 42C40 The HRT (Heil-Ramanathan-Topiwala) conjecture asks whether a finite collection of time-frequency shifts of a non-zero square integrable function on is linearly independent. This longstanding conjecture remains largely open even in the case when the function is assumed to be smooth. Nonetheless, the conjecture has been proved for some special families of functions and/or special sets of points. The main contribution of this paper is an inductive approach to investigate the HRT conjecture based on the following. Suppose that the HRT is true for a given set of N points and a given function. We identify the set of all new points such that the conjecture remains true for the same function and the set of N+1 points obtained by adding one of these new points to the original set. To achieve this we introduce a real-valued function whose global maximizers describe when the HRT is true. To motivate this new approach we re-derive a special case of the HRT for sets of 3 points. Subsequently, we establish new results for points in (1,n) configurations, and for a family of symmetric (2,3) configurations. Furthermore, we use these results and the refinements of other known ones to prove that the HRT holds for certain families of 4 points. Extension and restriction principles for the HRT conjecture Kasso A. Okoudjou December 30, 2023 =========================================================== myheadings plain K. A. OKOUDJOUEXTENSION PRINCIPLE FOR THE HRT § INTRODUCTION For a, b ∈ and a function g defined on , let M_bf(x)=e^2π i bxf(x) and T_af(x)=f(x-a) be respectively the modulation operator, and the translation operator. Given a function g∈ L^2() and Λ={(a_k, b_k)}_k=1^N⊂^2, we define 𝒢(g, Λ)={e^2π i b_k ·g(· - a_k)}_k=1^N. 𝒢(g, Λ) is called a (finite) Weyl-Heisenberg or Gabor system <cit.>. The HRT conjecture <cit.>, states that Given any 0≠ g ∈ L^2() and Λ={(a_k, b_k)}_k=1^N⊂^2, 𝒢(g, Λ) is a linearly independent set in L^2(). To date a definitive answer to the Conjecture has not been given even when one assumes that the function g is very smooth and decays fast, e.g., when g ∈ S(), the space of Schwartz functions on . In particular, the following (sub-conjecture) is also open Given any g ∈ S(), g≠ 0 and Λ={(a_k, b_k)}_k=1^N⊂^2, 𝒢(g, Λ) is a linearly independent set in L^2(). While the statement of the problem seems simple, a variety of sophisticated tools such as the ergodic theorems, von Neumann algebra methods, number theory arguments, random Schrödinger operators, harmonic analysis, operator theory, has been used to prove the few known results. Perhaps the lack of unifying theme in the proofs of the known results attests to the difficulty of this problem. The HRT conjecture contains two fundamental data: the function g ∈ L^2() and the set of points Λ={ (a_k, b_k)}_k=1^N ⊂^2. Most of the known results either assume g ∈ L^2 and Λ is restricted to some special family of points, or that Λ is very general and restrictions are imposed on g. We outline all the known results about the HRT conjecture of which we are aware and we refer to the surveys <cit.> for more details. The following statements hold. (i) Conjecture <ref> holds for any Λ⊂^2, when g is compactly supported, or just supported within a half-interval (-∞, a], or [a, ∞) <cit.>. (ii) Conjecture <ref> holds for any Λ⊂^2, when g(x)=p(x)e^-π x^2 where p is a polynomial <cit.>. (iii) Conjecture <ref> holds for any g ∈ L^2(), when Λ is a finite set with Λ⊂ A(^2) + z where A is a full rank 2× 2 matrix and z∈^2 <cit.>. In particular, Conjecture <ref> holds when #Λ≤ 3 for any g ∈ L^2 <cit.>. (iv) Conjecture <ref> holds for any g ∈ L^2, when #Λ =4 and two of the four points in Λ lie on a line and the remaining two points lie on a second parallel line <cit.>. (v) Conjecture <ref> holds for any g ∈ S(), when #Λ =4 and three of the four points in Λ lie on a line and the fourth point is off this line <cit.>. (vi) Conjecture <ref> holds for any Λ⊂^2, when lim_x →∞|g(x)|e^cx^2=0 for all c>0 <cit.>. (vii) Conjecture <ref> holds for any Λ⊂^2, when lim_x →∞|g(x)|e^cxlog x=0 for all c>0 <cit.>. (viii) Conjecture <ref> holds when g is ultimately positive, and Λ ={(a_k, b_k)}_k=1^N ⊂^2 is such that {b_k}_k=1^N are independent over the rationals <cit.>. (ix) Conjecture <ref> holds for any #Λ =4, when g is ultimately positive, and g(x) and g(-x) are ultimately decreasing <cit.>. (x) Conjecture <ref> holds for any g ∈ L^2(), when Λ consists of collinear points <cit.>. (xi) Conjecture <ref> holds for any g ∈ L^2(), when Λ consists of N-1 collinear and equi-spaced points, with the last point located off this line <cit.>. We note that there is some redundancy in Proposition <ref> as part (vii) implies parts (i), (ii), and (vi). Nonetheless, we include all these results to give an historical perspective on the HRT conjecture. In addition to these, perturbation arguments <cit.> have been used on either the function g or the set Λ to get related results. A spectral result related to the HRT has been presented in <cit.>, and estimates of frame bounds for Gabor systems related to the HRT conjecture have appeared in <cit.>. A connection between the HRT, the Bargmann-Fock space and the Segal-Bargmann transform was presented in <cit.>. Other results concerning the HRT can be found in <cit.>, and for an overview of the status of the conjecture we refer to <cit.>. When g ∈ L^2(^d), d≥ 2, and Λ⊂^2d not much is known about the conjecture, see <cit.>. We refer to <cit.> for a related problem for pure translation systems, and to <cit.> for some generalizations of the conjecture. A set Λ of the form given in (iv) or (v) of Proposition <ref>, is referred to as a (2,2) configuration and (1,3) configuration, respectively. More generally, An (n, m) configuration is a collection of n+m distinct points in the plane, such that there exist 2 distinct parallel lines such that one of them contains exactly n of the points and the other one contains exactly m of the points. One of the goals of this paper is to present two different approaches to investigate the HRT conjecture. On the one hand, we prove an extension principle and use it to attack the HRT conjecture. No such extension or other inductive methods related to the HRT have ever been proved. More specifically, knowing that the Conjecture holds for a given function g ∈ L^2() and a given set Λ={(a_k, b_k)}_k=1^N⊂^2, we identify the set of all (new) points (a, b) ∈^2 ∖Λ such that the conjecture remains true for the same function g and the new set Λ'=Λ∪{(a,b)}. On the other hand, we consider the related restriction principle which asks the following question; knowing that the HRT is true for a specific set of N+1 points and g, can one establish the conjecture for a family of N related points and the same function g? To answer these questions we introduce a real-valued function that is generated by the two data in the HRT conjecture, namely, the function g and the set Λ. As we shall show, this function is derived from the Gramian of 𝒢(g, Λ) and is based on a fundamental time-frequency analysis tool: the short-time Fourier transform. Using this function along with refinements of some of the techniques introduced by Demeter in <cit.> allow us to recover some known results and establish new ones. In particular, the main contributions of this paper are: ∙ a proof that HRT conjecture holds for all (1,3) configurations when g is real-valued, ∙ a proof that HRT holds for a family of symmetric (2,3) configurations, ∙ a proof that HRT holds for a large family of 4 points (not in (1,3) nor (2,2) configurations) and real-valued functions in L^2(). Furthermore, as a byproduct of our approach we obtain:∙ a new proof of HRT for collinear points, ∙ a new proof of HRT for sets of 3 unit-lattice points and real-valued functions. The rest of the paper is organized as follows. In Section <ref>, we introduce some of the technical tools needed to state our results. We then use Bochner's theorem to provide a new proof of the HRT conjecture for collinear points (Theorem <ref>). We also motivate the extension principle by offering a new proof of the HRT conjecture for 3 points on the unit lattice and real-valued functions (Proposition <ref>). In Section <ref> we introduce and collect the main properties of the extension function which is the basis of the extension principle we propose. Subsequently, we prove in Section <ref> that there exists at most one (equivalence class of) (1,n) configuration for which the HRT conjecture could fail whenever n≥ 3 (Theorem <ref>). Furthermore, when the generator is real-valued we show that the HRT holds for all (1,3) configurations (Theorem <ref>). In Section <ref> we introduce the restriction principle. For this case, we refine Demeter's “conjugate trick” argument to establish both Conjecture <ref> (Theorem <ref>) and Conjecture <ref> (Theorem <ref>) for a family of symmetric (2,3) configurations. Subsequently, we apply the restriction principle to prove Conjecture <ref> for real-valued functions and a related family of 4 points that are not (1,3) nor (2,2) configurations (Corollary <ref>). § PRELIMINARIES AND MOTIVATION In this section, we collect some properties of the Short-Time Fourier Transform (STFT) as well as some results concerning positive definite matrices, see Section <ref>. Using the Gramian of 𝒢(g, Λ) and Bochner's theorem we then give a new proof of the HRT conjecture for collinear points, see Section <ref>. Finally, in Section <ref> we revisit the HRT for 3 points and provide a new proof of the validity of the conjecture in this case. This new proof serves as a motivation for the extension principle that we propose. The methodology we develop below is fundamentally based on the analysis of the Gramian of 𝒢(g, Λ). In particular, the notions of positive definiteness of functions and matrices constitute the overarching themes of this methodology. §.§ Preliminaries Let f, g ∈ L^2(). The Short-Time Fourier Transform (STFT) of a function f with respect to a window g is V_g f(x, y)=∫_ f(t) g(t-x) e^-2π i y t dt. It is easy to prove that V_g f is a bounded uniformly continuous function on ^2, and that lim_|x|, |y|→∞V_gf(x, y)=0; see <cit.>. We will also need the following the orthogonality and covariance properties of the STFT: given g_i, f_i, ∈ L^2(), i=1, 2, we have V_g_1f_1V_g_2f_2=f_1f_2g_1g_2, and V_g(T_aM_bf)(x, y)=e^-2π i ayV_gf(x-a, y-b), see <cit.>, and <cit.>. We will also use the following formula whenever it is well defined: ℱ_2(V_g_1f_1 V_g_2f_2)(ξ, η) = (V_f_2f_1 V_g_2g_1)(-η, ξ), where ℱ_2 denotes the two dimensional Fourier transform. We refer to <cit.> for a proof of this statement. We need some facts about positive definite matrices. We refer to <cit.> for details. In particular, given N × N Hermitian matrices A, and B, we write A ≻ B if A-B is positive definite. We will repeatedly use the following theorem. <cit.> Let E be an Hermitian N × N matrix such that E=[ A B; B^* C ], where A, C are square matrices. Then, E is positive definite if and only if A is positive definite and C≻ B^*A^-1B. In our setting, E=(f_kf_ℓ)_k, ℓ=1^N will be the Gramian of a set of N functions {f_k}_k=1^N ⊂ L^2(). As a consequence, E is automatically positive semidefinite, i.e., E≽ 0. Furthermore, A and C will always be positive definite matrices. In particular, we will consider the case where A is an (N-1)× (N-1) positive definite matrix, C=1, and B=u will be a vector in ^N-1. In this case we will make repeated use of the following corollary of Theorem <ref>. With the above notations the following assertions hold: * E≽ 0 if and only if A^-1uu≤ 1. Furthermore, E≻ 0 if and only if A^-1uu< 1. Consequently, E is singular if and only if A^-1uu= 1. * E=(1-A^-1uu)A. The proof is given in <cit.>. However, we outline it for the sake of completeness. We are given that A is positive definite. Now assume that E is positive semidefinite and let X=-A^-1u. Then [ I 0; X^* 1 ][ A u; u^* 1 ][ I X; 0 1 ]=[ A 0; 0 1-A^-1uu ] is positive semidefinite. Thus 1≥A^-1uu. And the converse is trivially seen. The last two parts easily follow as well. §.§ Revisiting the HRT for collinear points To the best of our knowledge no result on the HRT conjecture has been obtained through the analysis of the Gramian of 𝒢(g, Λ)={e^2π i b_k ·g(· - a_k)}_k=1^N. Recall that the Gramian G_g of 𝒢(g, Λ)={e^2π i b_k ·g(· - a_k)}_k=1^N is the matrix given by G_g =(e^2π i b_k·g(· -a_k)e^2π i b_ℓ·g(· - a_ℓ))_k, ℓ = 1^N =(e^-2π i a_k(b_ℓ -b_k)V_gg(a_ℓ - a_k, b_ℓ-b_k))_k, ℓ=1^N. It follows that G_g is positive semidefinite matrix and that the HRT conjecture holds if and only if G_g is strictly positive definite. In this section, we motivate our approach to analyze this Gramian by offering a new proof of the HRT conjecture for collinear points. While this result is well-known <cit.>, the new proof we provide illustrates the role of positive definiteness vis-a-vis the HRT conjecture. We will need the following version of Bochner's theorem, and refer to <cit.> for more on the classical Bochner's theorem. We recall that a continuous complex-valued function f:^d → is positive definite if ∑_j=1^N∑_k=1^Nc_jc_kf(x_j-x_k)≥ 0 for any pairwise distinct points x_1, x_2, , x_N∈^d, and (c_k)_k=1^N ∈^N. The function f is said to be strictly positive if equality holds in (<ref>) only when c_k=0 for all k=1, 2, , N. <cit.> A continuous complex-valued function f is positive definite if and only if f=μ̂ where μ is a non-negative finite Borel measure on . Furthermore, f is strictly positive definite if and only if there does not exist a non-zero trigonometric polynomial m vanishing on the support of μ, i.e., such that ∫_ m dμ=0. Let Λ={(a_k, b_k)}_k=1^N ⊂^2 be a set of collinear points. Then, by rotating and translating we can assume that Λ={(a_k, 0)}_k=1^N ⊂^2 with a_1=0, <cit.>. In this case, the Gramian of 𝒢(g, Λ) takes the form G_g=(g(· -a_k)g(· - a_ℓ))_k, ℓ = 1^N= (ĝe^-2π i (a_k-a_ℓ) ·ĝ)_k, ℓ =1^N. We can now give a new proof of the HRT conjecture when the points are collinear. Let 0≠ g ∈ L^2(), and Λ={(a_k, 0)}_k=1^N ⊂^2 with a_0=0. Then 𝒢(g, Λ)={g(· - a_k)}_k=1^N is linearly independent. For g ∈ L^2() with g_2=1 let h(ξ)=|ĝ(ξ)|^2. We note that h is non-negative, non-identically 0, and h ∈ L^1(). Let μ be the finite nonnegative Borel measure whose density with respect to the Lebesgue measure is the function h. The function Φ defined by Φ(x)=ĥ(x)=μ̂(x)=∫_h(ξ) e^-2π i x ·ξdξ is continuous. Consequently, by Proposition <ref>, Φ is positive definite. It remains to show that Φ is strictly positive definite. To do this, suppose that m is a non-zero trigonometric polynomial given by m(x)=∑_k=1^Kc_ke^2π i ξ_k x where c_k are complex numbers (not all zeros), and ξ_k are pairwise distinct real numbers. It follows that ∑_j=1^K ∑_k=1^K c_j c_k Φ(ξ_j- ξ_k) =∑_j=1^K ∑_k=1^K c_j c_k ∫_ e^-2π i(ξ_j-ξ_k)x dμ(x) =∫_ ∑_j=1^Kc_j e^-2π i ξ_j x ∑_k=1^Kc_k e^2π i ξ_k x dμ(x) =∫_ |m(x)|^2 dμ(x) =∫_ |m(x)|^2 |ĝ(x)|^2 dx. This last integral vanishes only when m vanishes on the support of μ, which is the support of g. We can now conclude that Φ=ĥ=μ̂ is strictly positive definite. However, by (<ref>) we see that the Gramian of {g(· - a_k)}_k=1^N is exactly the matrix G_g=(g(· -a_k)g(· - a_ℓ))_k, ℓ = 1^N=(Φ(a_k - a_ℓ))_k, ℓ =1^N. Therefore, G_g is strictly positive definite. §.§ Motivation: The case of three points revisited We now motivate our approach using the analysis of the Gramian of 𝒢(g, Λ)={g(· - a_k) e^2π i b_k·}_k=1^3 for 3 points {(a_k, b_k)}_k=1^3 ⊂^2. Furthermore, we suppose that the function g is real-valued. We note that following <cit.>, without loss of generality any set of three distinct points can be transformed (through area preserving transformations) into {(0,0), (0, 1), (a, b)} where (a, b) ∈^2∖{(0,0), (0,1)}. We also know that the HRT conjecture is always true for any set of two distinct points. Thus, {g, M_1g} is linearly independent and our task is to show that for any other point (a, b) ∈^2∖{(0,0), (0,1)}, {g, M_1g, M_bT_ag} remains linearly independent. Observe that the Gramian G_g of {g, M_1g, M_bT_ag} can be written in the following block structure: G_g=[ A u(a,b); u(a,b)^* 1 ] where A=[ 1 α; α 1 ] and u(a,b)=[ V_gg(a, b); V_gg(a, b-1) ] with α=V_gg(0,1), and u(a,b)^* denoting the conjugate adjoint of u(a,b). Note that |α| =|gM_1g|<g_2M_1g_2=g_2^2=1 since {g(·) , e^2π i ·g(·)} is linearly independent. We know that G_g is positive semidefinite and we wish to show that it is strictly positive definite. Appealing to Corolloary <ref>, we see that 0≤ F(a,b)=A^-1u(a,b)u(a,b)≤ 1 and that 0≤ F(a,b)<1 if and only if {g, M_1g, M_bT_ag} is linearly independent. Thus, the function F: ^2 → has range in [0,1] and 1 is its maximum value. We can now prove the following result which serves both as a motivation to our approach and gives a new proof for the HRT conjecture for any 3 points on the integer lattice. Let 0≠ g ∈ L^2() be a real-valued function with g_2=1, and Λ={(0,0), (0, 1), (a,b)}, with (a, b)∈^2 ∖{(0,0), (0,1)}. Then the function F defined above achieves its global maximum value 1 only for (a, b) ∈{(0,0), (0,1)}. Consequently, Conjecture <ref> holds for Λ and g. The proof of Proposition <ref> is based Demeter's result on (2,2) configurations, as well as on a symmetry property of F. It also illustrates the restriction principle that will be introduced in Section <ref>. First we prove the following symmetry of F. Suppose that g∈ L^2() is real-valued. With the setting above we have F(a, b)=F(a, 1-b) for all (a, b)∈×, b≠ 1/2, and F(-a, 1/2)=F(a,1/2) for all a ∈. Let V=[ 0 1; 1 0 ]. Then F(a, 1-b)=A^-1u(a, 1-b)u(a, 1-b), and u(a, 1-b)=[ V_gg(a,1-b); V_gg(a, b) ]=V [ V_gg(a,b); V_gg(a, b-1) ]=Vu(a,b). Moreover, straightforward computations show that V^TA^-1V=VA^-1V=A̅^-1. Consequently, F(a, 1-b) =A^-1u(a, 1-b)u(a, 1-b)=A^-1Vu(a,b)Vu(a,b) =V^TA^-1Vu(a,b)u(a,b)=A̅^-1u(a,b)u(a,b) =A̅^-1u(a,b)u(a,b)=A^-1u(a,b)u(a,b) =F(a, b) where we have used the fact that A^-1 is a positive definite matrix. When b=1/2 and a∈, we see that F(-a, 1/2)=A^-1u(-a, 1/2)u(-a, 1/2). but u(-a, 1/2)=[ V_gg(-a, 1/2); V_gg(-a, -1/2) ]=[ e^π i aV_gg(a, 1/2); e^-π i aV_gg(a, -1/2) ]=Bu(a,b) where B=[ e^π i a 0; 0 e^-π i a ]. It follows that F(-a,1/2) =A^-1u(-a, 1/2)u(-a, 1/2)=A^-1Bu(a, 1/2)Bu(a, 1/2) =B^*A^-1Bu(a,1/2)u(a,1/2)=A^-1u(a,1/2)u(a,1/2) =F(a,1/2) where we used the fact that B^*A^-1B=A^-1. To see why this is the case we observe that a∈ and a series of computations shows that B^*A^-1B=11-|α|^2[ 1 -α e^-2π i a; -αe^2π i a 1 ]= 11-|α|^2[ 1 -α; -α 1 ]=A^-1. The following argument gives an alternate proof of Proposition <ref>. Because {g, M_1g, g} and {g, M_1g, M_1g} are both linearly dependent (repeated vectors) it follows that F(0,0)=F(0,1)=1. Assume that there exists (a_0, b_0) ∈×∖{(0,0), (0,1)} such that F(a_0, b_0)=1 with b_0≠ 1/2. In particular, by Corollary <ref> the system 𝒢(g, {(0,0), (0,1), (a_0, b_0)}) is linearly dependent, i.e., M_b_0T_a_0g belongs to the linear span of 𝒢(g, {(0,0), (0,1)}). Using Lemma <ref> we get that F(a_0, 1-b_0)=1 and therefore M_1-b_0T_a_0g belongs to the linear span of 𝒢(g, {(0,0), (0,1)}). Therefore, 𝒢(g, {(0,0), (0,1), (a_0, b_0), (a_0, 1-b_0)}) is also linearly dependent. But {(0,0), (0,1), (a_0, b_0), (a_0, 1-b_0)} is a (2,2) configuration and g ∈ L^2(), which contradicts <cit.>. The last case to consider is to assume that for some a_0≠ 0, b_0=1/2 and F(a_0, 1/2)=1. But then using Lemma <ref> again we see that F(-a_0, 1/2)=F(a_0,1/2) and therefore 𝒢(g, {(0,0), (0,1), (a_0, 1/2), (-a_0, 1/2)}) is linearly dependent. But {(0,0), (0,1), (a_0, 1/2), (-a_0, 1/2)} is also a (2,2) configuration and g ∈ L^2(), which contradicts <cit.>. § EXTENSION AND RESTRICTION PRINCIPLES TO THE HRT CONJECTURE In this section we describe in its full generality the aforementioned extension principle for the HRT conjecture. It could also be viewed as an inductive approach to attack the conjecture. More specifically, suppose that the Conjecture holds for a given function g ∈ L^2() and a given set Λ={(a_k, b_k)}_k=1^N⊂^2. We seek all the points (a, b) ∈^2 ∖Λ such that the conjecture remains true for the same function g and the new set Λ'=Λ∪{(a,b)}. We investigate this question by using Theorem <ref> and Corollary <ref> to relate the Gramians of 𝒢(g, Λ') and 𝒢(g, Λ). In Section <ref> we introduce the main technical tool to extend the HRT in the sense given above. Subsequently, in Section <ref> we apply this approach to (1, n) configurations. Finally, in Section <ref> we introduce a related restriction principle that allows us to establish the HRT conjecture for a family of 4 points and real-valued functions from knowing that the conjecture can be proved for a related family of symmetric (2,3) configurations. To establish the latter result we refine Demeter's “conjugate trick" arguments to handle this family of (2,3) configurations. §.§ The HRT extension principle Let g ∈ L^2() with g_2=1. Assume that Conjecture <ref> holds for some Λ={(a_k, b_k)}_k=1^N⊂^2 with (a_1, b_1)=(0,0). Let Λ'={(a_k, b_k)}_k=1^N∪{(a, b)} for (a, b) ∈^2. The Gramian G_g, N+1(a,b) of 𝒢(g, Λ')={e^2π i b_k ·g(· - a_k)}_k=1^N∪{e^2π i b ·g(· -a)} has the following block structure: G_N+1:=G_g, N+1(a,b)=[ G_N u_N(a,b); u(a,b)^* 1 ] where G_N:=G_g, N is the Gramian of {e^2π i b_k ·g(· - a_k)}_k=1^N, u_N(a,b) is a vector in ^N given by u_N(a,b)=[ e^2π i a_1b_1V_gg(a, b); e^-2π i a_2 (b-b_2)V_gg(a-a_2, b-b_2); e^-2π i a_3(b- b_3)V_gg(a-a_3, b-b_3); ⋮; e^-2π i a_N(b-b_N)V_gg(a-a_N, b-b_N) ]=[ V_gg(a, b); e^2π i a_2b_2V_g(T_a_2M_b_2g)(a, b); e^2π i a_3b_3V_g(T_a_3M_b_3g)(a, b); ⋮; e^2π i a_Nb_NV_g(T_a_NM_b_Ng)(a, b) ] and u_N(a,b)^* is the adjoint of u_N(a,b). Because G_N is positive definite, the function F_N+1:^2 → [0, ∞] given by F(a,b):=F_N+1(a, b)=G_N^-1u_N(a,b)u_N(a,b) is well-defined. For simplicity and when the context is clear, we write u(a,b) for u_N(a,b), and F(a, b) for F_N(a,b). Note that when N=2 the function F is simply the one introduced in the proof of Theorem <ref>. The following result summarizes the main properties of F. With the above notations assume that G_ N is a positive definite N× N matrix. Then, the following statements hold. (i) 0≤ F(a,b)≤ 1 for all (a, b) ∈^2, and moreover, F(a_k, b_k)=1 for each k=1, , N. (ii) F is uniformly continuous and lim_|(a,b)|→∞F(a,b)=0. (iii) ∬_^2F(a,b)da db=N. (iv) The Fourier transform F:^2 → of F given by F(ξ, η)=∬_^2F(a,b)e^-2π i (aξ+ bη)dadb, is strictly positive definite, and integrable. (v) G_g(a,b)=(1-F(a,b))G_N. (i) The Gramian of G_g(a,b) is positive semidefinite so by Corollary <ref> and the assumption that G_N is positive definite, we conclude that 0≤ F(a,b)≤ 1. Moreover, for (a,b)=(a_k, b_k) we know that the Gramian is positive semidefinite as the system is linearly dependent (one element is repeated twice). Thus, we get the moreover part of the result. (ii) This follows easily as each coordinate of u(a,b) is a uniformly continuous function that tends to 0 at infinity. (iii) Suppose that G_g,N^-1=(B_i,j)_i, j=1^N. We can now write F(a,b) =G_N^-1u(a,b)u(a,b) = ∑_k=1^N∑_ℓ=1^NB_k, ℓ(u(a,b))_ℓ (u(a,b))_k = ∑_k=1^N∑_ℓ=1^NB_k, ℓ e^2π i (a_ℓ b_ℓ-a_k b_k) V_g(T_a_ℓM_b_ℓg)(a,b) V_g(T_a_kM_b_kg)(a,b) . Integrating this last formula over ^2 and using the orthogonality and covariance properties of the STFT, i.e., (<ref>) and (<ref>) we have ∬_^2F(a,b) da db = ∑_k, ℓ=1^N B_k, ℓ e^2π i (a_ℓ b_ℓ-a_k b_k) ∬_^2 V_g(T_a_ℓM_b_ℓg)(a,b) V_g(T_a_kM_b_kg)(a,b) da db Evaluating the integral leads to ∬_^2 V_g(T_a_ℓM_b_ℓg)(a,b) V_g(T_a_kM_b_kg)(a,b) da db = V_g(T_a_ℓM_b_ℓg)V_g(T_a_kM_b_kg) =T_a_ℓM_b_ℓgT_a_kM_b_kg gg = T_a_ℓM_b_ℓgT_a_kM_b_kg Consequently, ∬_^2F(a,b) da db = ∑_k, ℓ=1^N B_k, ℓ e^2π i (a_ℓ b_ℓ-a_k b_k) T_a_ℓM_b_ℓgT_a_kM_b_kg = ∑_k, ℓ=1^N B_k, ℓ M_b_ℓT_a_ℓgM_b_kT_a_kg = ∑_k, ℓ=1^N B_k, ℓ (G_N)_ℓ,k =1det G_N∑_ℓ=1^N ∑_k=1^N (-1)^k+ℓ (G_N)_ℓ,k det G_N({ℓ}',{k}') = 1G_N∑_ℓ=1^N G_N = N where we use the fact that B_k,ℓ=(-1)^k+ℓdet G_Ndet G_N({k}',{ℓ}'). (iv) This part follows from the fact that F is nonnegative, not identically 0, continuous, and integrable. Using (<ref>), (<ref>), and the notations set in part (iii) we can compute F̂ explicitly F̂(ξ, η) = ∑_k=1^N∑_ℓ=1^NB_k, ℓ e^2π i (a_ℓ b_ℓ-a_k b_k)ℱ_2(V_g(T_a_ℓM_b_ℓg) V_g(T_a_kM_b_kg) )(ξ, η) =∑_k=1^N∑_ℓ=1^NB_k, ℓ e^2π i (a_ℓ b_ℓ-a_k b_k) V_T_a_kM_b_kg(T_a_ℓM_b_ℓg)(-η, ξ) V_gg (-η, ξ) =∑_k=1^N∑_ℓ=1^NB_k, ℓ e^2π i (a_ℓ b_ℓ-a_k b_k- a_ℓ b_k) e^-2π i(a_ℓξ +b_kη) V_gg(-η -a_ℓ +a_k, ξ -b_ℓ +b_k) V_gg (-η, ξ). It is clear that ∬_^2|F̂(ξ, η)|dξ dη≤∑_k=1^N∑_ℓ=1^N|B_k, ℓ| < ∞. (v) Follows from Corollary <ref>. The following result is a consequence of Theorem <ref>. Let g ∈ L^2() with g_2=1 and Λ={(a_k, b_k)}_k=1^N⊂^2. Assume that 𝒢(g, Λ) is linearly independent. Let Λ'={(a_k, b_k)}_k=1^N∪{(a, b)}. Then 𝒢(g, Λ') is linearly independent if and only if F(a,b)<1. Furthermore, there exists R:=R(Λ, g)>0 such that for all (a, b) ∈^2 with |(a, b)|> R, then 𝒢(g, Λ') is linearly independent where Λ'=Λ∪{(a, b)} The first part follows from part (1) of Corollary <ref> and part (i) of Theorem <ref>. The existence of R is guaranteed by part (ii) of Theorem <ref>. (a) By the last part of Corollary <ref>, the extension function F makes the HRT conjecture a “local problem”. In other words, once the conjecture is known to be true for a function g and a set Λ={(a_k, b_k)}_k=1^N, it is also automatically true for Λ'=Λ∪{(a, b)} whenever the new point lies outside a ball of radius R. So to establish the HRT everywhere for Λ' we must focus on the “local” properties of F, that is the restriction of F to the aforementioned ball. (b) Corollary <ref> makes it possible to explore the HRT from a numerical point of view. Indeed, Theorem <ref> and Corollary <ref> assert that the HRT for Λ'={(a_k, b_k)}_k=1^N∪{(a, b)} holds if and only if (a, b) is not a global maximizer of F. So in theory, one only needs to prove that the set of global maximizers of F is Λ. For a smooth function g, differential calculus can be used to check this. For example, for the Gaussian g(x)=2^1/4 e^-π x^2, using <cit.> we get that V_gg(a,b)=e^-π i a be^-π a^2/2e^-π b^2/2. In this case and using Λ={(0,0), (0,1)}∪{(a, b)}, F is simply F(a,b)= e^-π (a^2+b^2)1-e^-π[ 1 + e^π (2b-1) - 2 e^π(b-1)cos a ]. One can then used multivariable calculus to show that the global maximizers of F are exactly the two points (0,0) and (0,1), see Figure <ref>. More generally, one can numerically analyze the function F to determine its global maximizers. For example, suppose Λ={(0,0), (1,0), (0,1)}. For the Gaussian g(x)=2^1/4e^-π x^2, Figure <ref> displays the graph of the function F on the square [-4, 4]× [-4,4]. This graph illustrates the validity of the HRT in this case, by showing that the global maximum value of F is only achieved on the set Λ. Similarly, when g(x)=e^-|x|, Figure <ref> displays the graph of the function F on the square [-4, 4]× [-4,4]. This graph illustrates the validity of the HRT in this case, by showing that the global maximum value of F is only achieved on the set Λ. Recall that the HRT is known to be true for this function and any set of 4 points <cit.>. Finally, Figure <ref> displays the graph of the function F on the square [-4, 4]× [-4,4] when g(x)=2^-1/21+|x|. To the best of our knowledge, the HRT has not been proved for this function and any set of 4 points. Therefore, Figure <ref> offers some numerical evidence to the validity of the conjecture in this case. We also refer to Corollary <ref> for some new results in this setting. §.§ The HRT conjecture for (1, n) configurations In this section, we consider the HRT conjecture for (1,n) configurations and prove that the conjecture can only fail for at most one such configuration. The proof is elementary and based on some dimension arguments. We then focus on the case n=3 and show that when the generator is a real-valued function then Conjecture <ref> holds for all (1,3). Note that the strongest known results for these configurations assume either that the 3 collinear points are also equi-spaced, or that the generator is in S(). The motivation of the results presented in this section is <cit.> which states that the HRT conjecture holds for almost all (in the sense of Lebesgue measure) (1,3) configurations. A consequence of our result is that the HRT conjecture can only fail for at most one (1,3) configuration up to equivalence. For more on the HRT for (1,3) we refer to Demeter's results <cit.> and a recent improvement due to Liu <cit.>. Recall that by using the metaplectic transformations one can show that any (1,n) configuration has the form {(0,1)}∪{(a_k, 0)}_k=1^n where a_1=0 and the rest of the a_ks are distinct and nonzero <cit.>. Note that the set of metaplectic transformations in ^2 can be identified with the set of 2× 2 symplectic matrices, which, in turn is SL(2, ) <cit.>. We say that two (1, n) configurations Λ_1 and Λ_2 are equivalent if and only if there exists a symplectic matrix A∈ SL(2, ) such that Λ_2=AΛ_1. Let the set of distinct equivalence classes under this relation be denoted by Λ_(1,n). Without any loss of generality we can assume that Λ_(1,n)={(0,1)}∪{(a_k, 0)}_k=1^n with a_1=0 and a_k≠ 0 for all k=2, 3, , n. To prove that the HRT conjecture holds for all (1,n) configurations, it is enough to restrict to (1,n) configurations in Λ_(1,n). Let n≥ 3 and g∈ L^2() with g_2=1. Suppose that the HRT conjecture holds for g and any (1, n-1) configuration. Then there exists at most one (equivalence class of) (1, n) configuration Λ_0 ∈Λ_(1,n) such that 𝒢(g, Λ_0) is linearly dependent. Furthermore, suppose that Λ_0={(0,1)}∪{(a_k, 0)}_k=1^n ∈Λ_(1,n) is a (1,n) configuration such that 𝒢(g, Λ_0) is linearly dependent. Let a≠ a_k for k=1, , n. Fix any k_0 ∈{1, , n} and consider Λ ={(0,1)}∪{(a_1, 0), (a_2, 0), , (a_k_0-1, 0), (a, 0), (a_k_0+1, 0), , (a_n, 0)}. Then 𝒢(g, Λ) is linearly independent. Suppose by contradiction that there exist two distinct (1, n) configurations (or equivalent classes) Λ_1 and Λ_2 such that 𝒢(g, Λ_i) is linearly dependent for i=1,2. Further, suppose that Λ_1={(0,1)}∪{(a_k, 0)}_k=1^n and Λ_2={(0,1)}∪{(b_k, 0)}_k=1^n where a_1=b_1=0 and a_i_0≠ b_i_0 for some i_0∈{2, , n}. Then, one can write M_1g=∑_k=1^nc_kT_a_kg where c_k ≠ 0 for each k=1, , n. Indeed, if c_ℓ=0 for some ℓ∈{1, , n} then Λ'_1=Λ_1∖{(a_ℓ, 0)} will be a (1, n-1) configuration and (<ref>) will become M_1g=∑_k=1, k ≠ℓ^nc_kT_a_kg. That is 𝒢(g, Λ_1') will be linearly dependent contradicting one of the assumptions of the Theorem. Similarly, M_1g=∑_k=1^nd_kT_b_kg where d_k ≠ 0 for each k=1, , n. Taking the difference between (<ref>) and (<ref>) and rearranging leads to (c_1-d_1) g + c_i_0T_a_i_0g - d_i_0T_b_i_0g+∑_k=2, k≠ i_0^n c_kT_a_kg - ∑_k=2, k≠ i_0^n d_kT_b_kg =0 where i_0 was chosen above. But since c_i_0d_i_0≠ 0 and a_i_0≠ b_i_0, this last equation is equivalent to the fact that {g, T_a_kg, T_b_kg: k=2 , n} is linearly dependent. But this contradicts the fact the HRT conjecture holds for any 0≠ g∈ L^2() and the collinear points {(0,0), (a_k, 0), (b_k, 0): k=2, , n}, <cit.>. Therefore, there can exist at most one (class of equivalence) (1, n) configuration Λ_0 for which 𝒢(g, Λ_0) is linearly dependent. For the last part, suppose that Λ_0={(0,1)}∪{(a_k, 0)}_k=0^n ∈Λ_(1,n) is such that 𝒢(g, Λ_0) is linearly dependent. Write Λ_0= Λ_0'∪{(a_k_0, 0)} where Λ_0'={(0,1), (0,0), (a_2, 0), , (a_k_0-1, 0), (a_k_0+1, 0), , (a_n, 0)}. By assumption, 𝒢(g, Λ_0') is linearly independent since Λ_0' is a (1, n-1) configuration. Then by Corollary <ref>, F(a_k_0,0)=1 where F is the function obtained from the Gramian of 𝒢(g, Λ_0) according to Theorem <ref>. Now, let a∉{0, a_k: k=2, , n}. If F(a, 0)=1 then T_ag must belong to the linear span of {M_1g, T_a_kg: k=1, , n, k≠ k_0} whose dimension is n. But, T_a_k_0g also belongs to this linear span. Therefore, the n+1 functions g, T_a_kg, T_ag, k=2, , n belong to an n dimensional space. However, these functions are linearly independent (because the points are collinear). This is a contradiction, from which we conclude that F(a, 0)<1, concluding the proof. In the special case where n=3 we have the following result. Let g∈ L^2() with g_2=1. There exists at most one (equivalence class of) (1, 3) configuration Λ_0 such that 𝒢(g, Λ_0) is linearly dependent. When n=3 it is known that the HRT conjecture holds for g and every (1, 2) configuration <cit.>. Thus the assumption of Theorem <ref> is satisfied and the corollary follows. If we restrict to real-valued functions, then we can prove a stronger result by ruling out the existence of the single “bad” (equivalence class of ) (1,3) configuration given by Corollary <ref>. Let g∈ L^2(), g_2=1 be a real-valued function. Let a≠ b≠ 0 and set Λ={(0,0), (0,1), (a,0), ( b,0)} be a (1, 3) configuration. Then, Conjecture <ref> holds for Λ and g. Assume by way of contradiction that 𝒢(g, Λ) is linearly dependent. Then, there exists c_k ∈^*, k=1, 2, 3, such that c_1g + c_2M_1g + c_3T_ag = T_bg. Because, g is real-valued, we see that T_bg=T_bg= c̅_1g + c̅_2M_-1g + c̅_3T_ag. Hence, (c_1-c̅_1)g+ c_2M_1g - c̅_2M_-1g +(c_3-c̅_3)T_ag=0. Note that c_2≠ 0. Hence, this last equation is equivalent to the fact that 𝒢(g, Λ') where Λ'={(0,0), (a, 0), (0,1), (0, -1)} is linearly dependent. However, because the points (0,1), (0,0) and (0, -1) are equally spaced, Λ' is a (1, 3) configuration, for which Conjecture <ref> is known to hold <cit.>. Therefore, we arrive at a contradiction. §.§ A restriction principle for the HRT conjecture The goal of this section is to establish Conjecture <ref> for a large family of sets of cardinality 4 (that are not (1,3) nor (2,2) configurations) when g is a real-valued function. In addition, we establish similar results for Conjecture <ref>. In fact, we prove that the general case for (almost) any 4 points follows from a special family of (3,2) configurations. This is our restriction principle: proving that Conjecture <ref> or Conjecture <ref> hold for this special special family of (3,2) configurations implies its validity for a large family 4 points. The proof of the next result is an extension of Demeter's “conjugate trick” arguments <cit.>. Let g∈ L^2() with g_2=1. Suppose Λ is a (3,2) configuration given by Λ={(0,0), (0,1), (0,-1), (a, b), (a, -b)} where b≠ 0. Then, Conjecture <ref> holds for Λ and g whenever any of the following holds (i) a, b ∈. (ii) a∈ but b∉. (iii) a, b ∉ but ab ∈, and g is a real-valued function. We can trivially assume that a≠ 0. Indeed, if a=0 then the points in Λ will all lie on the y-axis, that is the points will be collinear, and HRT is known to be true in this case <cit.>. (i) Suppose that a=pq, b=mn∈. In this case, we see that Λ=AΛ', where A=[ 1q 0; 0 1n ] and Λ'={(0,0), (0, n), (0, -n), (p, m), (p, -m)}. In particular, Λ is a subset of a lattice and the result follows from <cit.>. (ii) Next assume that a ∈ and b∉. By using a scaling matrix (a metaplectic transform) we can assume that Λ has the following form: Λ={(0,0), (0,a), (0,-a), (1, b'), (1, -b')} with b'=ba ∉ <cit.>. To simplify the notations we will assume that Λ={(0,0), (0,a), (0,-a), (1, b), (1, -b)} with a ∈ and b ∉. Assume by way of contradiction that 𝒢(g, Λ) is linearly dependent. Then, there exist c_k ∈, k=1, 2, 3, and d_k∈, for k=1, 2 such that c_1g + c_2M_ag + c_3M_-ag= d_1M_-bT_1g + d_2M_bT_1g. Observe that c_k, d_k≠ 0 for each k, since the conjecture is true for all (2,2) configurations, and (1,3) configurations where the points on the line are equiangular. We may also assume that c_1∈. Consequently, we can write (<ref>) as |P(x)g(x)|=|Q(x)g(x-1)| a. e. where P(x)=c_1+c_2e^2π i ax +c_3e^-2π i ax and Q(x)=e^2π i (-bx+θ)(r_1+r_2e^2π i (2bx+θ') with r_1, r_2∈ (0, ∞) and θ, θ'∈ [0, 1). (Here we write d_1=r_1e^2π i θ and d_2=r_2e^2π i θ'.) Furthermore, because 0≠ g ∈ L^2() we have that lim_|n|→∞ n∈g(x-n)=0 a. e. and that supp (g)∩ [0,1] has a positive measure. Let S⊂suppg∩ [0,1] be such that S has positive measure, and such that S+ contains no zeros of P and Q (this is possible since the set of such zeros is at most countable). From now on, we assume that  (<ref>) and  (<ref>) hold for all x∈ S. Next, by the Birkhoff's pointwise ergodic theorem with 1_S, there exists x_0∈ S and n' ∈ such that x_1={-x_0-θ'b+ n'b}∈ S. Here and in what follows, we denote a fractional part of x∈ by {x}. Let m=-x_0-θ'b+ n'b-x_1=y-x_1. By iterating (<ref>), it follows that for all N>m {[ |g(x_0+N)| |g(x_0-1)|∏_n=0^N|Q(x_0+n)|∏_n=0^N|P(x_0+n)|; |g(x_1-N+m)| |g(x_1-1)|∏_n=-N+m+1^-1|P(x_1+n)|∏_n=-N+m+1^-1|Q(x_1+n)| ]. Next, observe that ∏_n=-N+m^m|Q(x_1+n)|=∏_n=0^N|Q(x_0+n)|. Consequently, ∏_n=-N+m^-1|Q(x_1+n)| =∏_n=-N+m^m|Q(x_1+n)| 1∏_n=0^m|Q(x_1+n)| =K ∏_n=-N+m^m|Q(x_1+n)| =K∏_n=0^N|Q(x_0+n)|, where K=1∏_n=0^m|Q(x_1+n)| is a positive finite constant that depends only on m, x_0, n', b, and θ'. Now assume that a=t/s ∈, then P is s-periodic. Let T(x)=∏_n=0^s-1|P(x+n)|, and assume first that T(x_1)≥ T(x_0). Then, ∏_n=-N+m+1^-1|P(x_1+n)|=K' ∏_n=0^N|P(x_1 -n)|≥∏_n=0^N|P(x_0+n)| for all N>m, where K'=1|P(x_1)|∏_n=-N^-N+m|P(x_1+n)| is a constant independent of N. Consequently, for N>m, |g(x_1-N+m)| =|g(x_1 -1)|∏_n=-N+m^-1|P(x_1+n)|∏_n=-N+m^-1|Q(x_1+n)| ≥ C |g(x_1 -1)| ∏_n=0^N|P(x_0+n)|∏_n=0^N|Q(x_0+n)| ≥ C |g(x_1-1)||g(x_0-1)| |g(x_0+N)|^-1 where C is a constant that depends only on x_0, m, r_1, r_2, c_1, c_2, and c_3. But this last inequality contradicts (<ref>). Now if instead, T(x_0)≥ T(x_1). We will have ∏_n=-N+m^-1|P(x_0+n)| ≥∏_n=0^N|P(x_1+n)| for all N>m, ∏_n=-N+m^-1|Q(x_0+n)| ≃∏_n=0^N|Q(x_1+n)|, where we used the notation A≃ B to denote B/c≤ A≤ cB for some constant c that depends only on x_0, m, r_1, and r_2. For N>m, |g(x_0-N+m)|≥ C |g(x_0-1)| ∏_n=0^N|P(x_1+n)|∏_n=0^N|Q(x_1+n)|, and |g(x_1+N)|=|g(x_1-1)| ∏_n=0^N|Q(x_1+n)|∏_n=0^N|P(x_1+n)|. Consequently, for N>m, |g(x_0-N+m)|≥ C |g(x_0-1)| |g(x_1-1)||g(x_1+N)|^-1, where C is a constant that depends only on x_0, m, r_1, r_2, c_1, c_2, and c_3. But this last inequality contradicts (<ref>). We conclude that (<ref>) cannot hold unless, c_k=0 for k=1, 2,3 and d_k=0 for k=1,2. (iii) Similar to case (ii), and using a metaplectic transform we can assume that Λ is of the form Λ={(0,0), (0,a), (0,-a), (1, b), (1, b)} with a∉, b∈. We now proceed as in part (ii) and assume that (<ref>) holds. Since, g is assumed to be real-valued we see by taking the complex conjugate of (<ref>) that c_1g+c_2M_-ag +c_3M_ag-d_1M_-bT_1g-d_2M_bT_1g=0. taking the difference between this last equation and (<ref>) , we obtain (c_2-c_3)M_ag + (c_3-c_2)M_-ag +(d_2-d_1)M_bT_1g +(d_1-d_2)M_-bT_1g=0. Now, the points {(0,a), (0,-a), (1, b), (1, -b)} form a (2,2) configuration and the HRT conjecture is true in this case. Therefore, c_3=c̅_2, d_2=d̅_1. Consequently, we let c_1=c∈, c_2=re^2π i θ, and d_1=r'e^2π i θ', where r, r'∈ (0, ∞) and θ, θ' ∈ [0,1). Therefore, (<ref>) holds with P(x)=c+2rcos 2π (ax +θ) and Q(x)=2r'cos 2π (bx +θ'). In particular, Q is a s-periodic function if we let b=t/s∈. Reversing the role of the polynomials P and Q in the proof of (ii) establishes the result in this last case. We note that the case a∉ and b∈ is equivalent (by a metaplectic transformation) to a, b, ab∉. This is the only case we have not been able to address. However, if we assume that g is smoother, then we can handle this case as well, see Theorem <ref> below. We can now prove the following result for a family of 4 points in ^2 and real-valued functions. This illustrates the restriction principle we announced in the introduction. Indeed, to establish the HRT conjecture for the family of sets of four points we use the fact the conjecture was proved for the above family of symmetric (2,3) configurations. More specifically, the following result holds. Note that any set of four distinct points can be transformed into {(0,0), (0, 1), (s, 0), (a, b)}. Let g∈ L^2(), g_2=1 be a real-valued function. Suppose that Λ= {(0,0), (0, 1), (s, 0), (a, b)}⊂^2 be a subset of four distinct points. Conjecture <ref> holds for Λ and g, whenever any of the following holds (i) a, b∈. (ii) a∈ but b∉. (iii) a, b ∉ but ab ∈ If ab=0 then, we are done by invoking Theorem <ref>. So we assume that ab≠ 0, and suppose by contradiction that there exist nonzero coefficients c_1, c_2, c_3 such that T_sg=c_1g+c_2M_1g+c_3M_bT_ag. This implies that T_sg=c̅_1g+c̅_2M_-1g+c̅_3M_-bT_ag. Hence, (c_1-c_1)g + c_2M_1g -c_2M_-1g + c_3M_bT_ag - c̅_3M_-bT_ag=0. Consequently, 𝒢(g, Λ) with Λ={(0,0), (0,1), (0,-1), (a, b), (a, -b)} is linearly dependent, which, contradicts Theorem <ref>. We recall the following conjecture. <cit.>. Suppose Λ={(0,0), (0,1), (1, 0), (√(2), √(2))}, and if 0≠ g ∈ L^2() with g_2=1 then 𝒢(g, Λ) is linearly independent. An application of Corollary <ref> settles this conjecture in the special case where g is real-valued. Indeed, this follows from part (iii) of Corollary <ref> by taking s=1, a=b=√(2)∉. If we assume that the function g is smoother, i.e., g∈() then we can extend <cit.> from (2,2) configurations to certain symmetric (3,2) configurations. It must be noted that the arguments given below were originally introduced in <cit.>[The proof given in <cit.> contains a few inaccuracies that were fixed by C. Demeter and posted on Math Arxiv as arXiv:1006.0732.]. For the sake of completeness we give the details of the proof below. Let g∈(), g_2=1. Suppose Λ is a (3,2) configuration given by Λ={(0,0), (0,1), (0,-1), (a, b), (a, -b)} where b≠ 0. Then, Conjecture <ref> holds for Λ and g whenever any of the following holds (i) a, b ∈. (ii) a∈ but b∉. (iii) a, b, ab ∉. (iv) a, b ∉ but ab ∈, and g is a real-valued function. The proof is divided in a number of cases. (i), (ii), (iv) follow from Theorem <ref>. (iii) Suppose that a, b ∉. Furthermore, assume that ab∉. Using a metaplectic transformation, we may assume that Λ is of the form Λ={(0,0), (0,a), (0,-a), (1, b), (1, b)}, with a, b, b/a∉. The rest of the proof is an extension of <cit.>. We follow the proof of part (ii) of Theorem <ref> and argue by contradiction. In particular, we assume that (<ref>), (<ref>), and (<ref>) hold for all x∈ I, where I⊂supp(g)∩ [0,1] is a set of positive measure. Recall that P(x)=c_1+c_2e^2π i ax+c_3e^-2π i ax and Q(x)=e^2π i (-bx+θ)(r_1+r_2e^2π i(2bx+θ')) where r_1, r_2∈ (0, ∞), θ, θ' ∈ [0,1), c_1∈, c_2, c_3∈ with c_k≠ 0 for k=1,2,3. We first prove that |-c_1±√(c_1^2-4c_2c_3)|2|c_2|≠ 1. Suppose by way of contradiction that |-c_1±√(c_1^2-4c_2c_3)|2|c_2|= 1. This implies that P(x)=0 has real solutions of the form x_k=ω + ka for some ω∈ and k∈. Next we prove that Q must also have some real roots. Indeed, assume that Q(x)≠ 0 for all x∈. Since a∉ we can choose k∈ with x_k>0 and {x_k}∈ I (recall that {u} is the fractional part of u). Note that g({x_k})≠ 0. We now use (<ref>) to get 0=|P(x_k)||g(x_k)|=|Q(x_k)||g(x_k-1)| Thus g(x_k-1)=0. We can continue this iteration to show that g(x_k-n)=0 for all n>0. Consequently, g({x_k})=0 which is a contradiction. Therefore, Q has real roots of the form y_n=ω' +n2b for some ω'∈ and n∈. Furthermore, the zeros of P and Q must share a -orbit. Indeed, if this was not the case, we must have that x_k-y_n∉ for all n, k∈. However, a repeated use of (<ref>) will lead to the following contradiction. For any k∈ Z we have 0=|P(x_k)||g(x_k)|=|Q(x_k)||g(x_k-1)|. Since x_k=x_k-0 is not a root of Q we see that g(x_k-1)=0. Continuing in this fashion we see that g(x_k-n)=0 for all n>0. Which is a contradiction. In fact, there must exist n≠ n' ∈ and m, m'∈ such that x_n-y_m, x_n'-y_m'∈. By taking the difference between these two numbers we see that Na+M2b=k for some N, M, k∈. Using the fact that a, b∉ we arrive at the conclusion that all N, M satisfying this equation must be of the form N=ℓ N_0 and M=ℓ M_0 for some fixed N_0, M_0∈∖{0} and arbitrary ℓ∈. In addition, all n, m∈ such that x_n-y_m∈ must be of the form {[ n n_0+ℓ N_0; m m_0+ℓ M_0, ]. for some fixed n_0, m_0, N_0, M_0∈, N_0, M_0 ≠ 0 and arbitrary ℓ∈. We also point out that for each x_n there is at most one y_m such that x_n-y_m∈. Let x_ℓ= ω_0+ℓN_0a be a zero of P where ω_0=ω+n_0a, and y_ℓ be the zero of Q such that x_ℓ-y_ℓ∈. Note that y_ℓ= ω_0'+ℓM_02b where ω_0'=ω'+m_02b. Because N_0a≠M_02b, we can choose ℓ∈ such that one of the following three alternatives holds:∙ 0<x_ℓ<y_ℓ∙ x_ℓ<0<y_ℓ∙ x_ℓ<y_ℓ<0 If we assume that the first alternative holds, by ergodicity, we can choose ℓ∈ such that u_ℓ={x_ℓ}={y_ℓ}∈ I. Note that g(u_ℓ)≠ 0 and using the recursion (<ref>) and the fact that Q is nonzero on the orbit before y_ℓ, we see that g(u_ℓ +1)≠ 0, which implies that g(u_ℓ+2)≠ 0. We can continue all the way to g(u_ℓ+n)≠ 0 where n∈ is such that u_ℓ +n+1=x_ℓ. Applying (<ref>) one more time will give 0=|P(x_ℓ)||g(x_ℓ)|=|Q(x_ℓ)||g(u_ℓ+n)|≠ 0 It follows that inf_x∈|P(x)|>0. Similarly, we show that inf_x∈|Q(x)|>0. Consequently, ψ(x)=ln|c_1+c_2e^2π ix+c_3e^-2π i x| and ϕ(x)=ln|r_1+r_2e^2π i(2x+θ')| are well-defined and continuous on Using (<ref>) and (<ref>) we see that for each x, z∈ I lim_N→∞∑_n=1^Nϕ(bx+bn)-∑_n=1^Nψ(ax+an)=-∞ and lim_N→∞∑_n=-N^-1ϕ(bz+bn)-∑_n=-N^-1ψ(az+an)=∞ We now use the approximation of a by rational and the fact |ψ'|≳ 1 to control parts of the above sums. Let p_k, q_k relatively prime integers, q →∞ such that |a-p_kq_k|≤1q_k^2. Furthermore, |na-np_kq_k|≤1q_k, -q_k≤ n≤ q_k. By a Riemann sum approximation we see that |∑_n=1^q_kψ(ax+an)-q_k∫_0^1ψ|=O(1) and |∑_n=-q_k^-1ψ(ax+an)-q_k∫_0^1ψ|=O(1) for each x∈ [0,1]. Consequently, for each y, z∈ I we have |∑_n=1^q_kψ(ay+an)-∑_n=-q_k^-1ψ(az +an)|=O(1). Now using Birkhoff's pointwise ergodic theorem for 1_I, we can choose x∈ I, n'∈ N such that z:={-x-θ'b+n'2b}∈ I. Let y:=-x-θ'b+n'2b and m=y-z. Then ∑_n=-N+m^-1+mϕ(bz+bn)=∑_n=1^Nϕ(by-bn)=∑_n=1^Nϕ(bx+bn). Observe that for each N ∑_n=-N^-1ϕ(bz+bn) =∑_n=-N^-N+m-1ϕ(bz+bn)+ ∑_n=-N+m^-1+mϕ(bz+bn)+∑_n=m^-1ϕ(bz+bn) =∑_n=1^Nϕ(bx+bn)+ ∑_n=-N^-N+m-1ϕ(bz+bn)+∑_n=m^-1ϕ(bz+bn) Consequently, for each N |∑_n=-N^-1ϕ(bz+bn)-∑_n=1^Nϕ(bx+bn)| =|∑_n=-N^-N+m-1ϕ(bz+bn)+ ∑_n=m^-1ϕ(bz+bn)| =O(1) where we bound the last sum by a constant that depends only on m, z, and b. However, (<ref>)–(<ref>) cannot simultaneously hold. This completes the proof. Next we suppose that a, b∉ but ab∈. We note that as observed in <cit.>, rather than assuming that g∈() we could assume that g∈ L^2() is continuous and is such that lim_|n|→∞ n∈|g(x-n)|=0 for all x∈ [0,1]. Suppose that g∈() is real-valued. In addition to the cases covered by Corollary <ref>, Theorem <ref> can be used to settle Conjecture <ref> when a, b, ab∉. We summarize what is known about the HRT for 4 points. In ^2, a set Λ consisting of four distinct points in ^2 can be such that: (1) The four points are collinear, in which case their convex hull is a line segment, (2) The four points form a (1,3) configuration, in which case their convex hull is a triangle, (3) The four points form a (2,2) configuration, in which case their convex hull is a trapezoid, or (4) The four points are in none of the previous three categories, in which case their convex hull is a general quadrilateral. ∙ In the first case Conjecture <ref> holds for any g ∈ L^2() <cit.>.∙ In the second case Conjecture <ref> holds for any g ∈() <cit.>. However, if g ∈ L^2, Conjecture <ref> is true when the three collinear points are also equispaced <cit.>. More generally, <cit.> has condition under which the conjecture remains true, and, in fact, Conjecture <ref> holds for almost all (1,3) configurations. The results of this paper allow us to conclude that when g∈ L^2 is also real-valued then Conjecture <ref> for all (1,3) configurations.∙ For the third case, Conjecture <ref> holds for any g ∈ L^2() <cit.>.∙ In the last case, to the best of our knowledge both Conjecture <ref> and Conjecture <ref> remain open. However, when g(x)=e^-|x|^ϵ with ϵ>0 Conjecture <ref> holds for any set of 4 points <cit.>. In fact, when g decays faster than any exponential the HRT conjecture has been established not only in dimension one, but also in higher dimensions <cit.>. In this paper we showed that when g∈ L^2() is real-valued Conjecture <ref> holds for a family of four distinct points. But is still unclear whether the HRT holds for any 4 points and every g∈ L^2(). § ACKNOWLEDGMENT The author thanks C. Heil for introducing him to this fascinating and addictive problem, and for invaluable comments and remarks on earlier versions of this paper. He also thanks R. Balan, J. J. Benedetto, and D. Speegle for helpful discussions over the years about various versions of the results presented here. He acknowledges C. Clark's help in generating the pictures included in the paper. Finally, he thanks W. Liu for helpful discussions, and the anonymous referees for their useful and insightful comments and remarks. amsplain

http://arxiv.org/abs/1701.08008v1

20170127104821

Novel processes and metrics for a scientific evaluation rooted in the principles of science - Version 1

cs.DL

< g r a p h i c s > Essay and Opinion [0.1cm] Novel processes and metrics for a scientific evaluation rooted in the principles of science Version 1 [0.5cm] , and 1. SJS – The Self-Journals of Science 2. Department of Physics – Aristotle University of Thessaloniki 3. ManyLabs (www.manylabs.org) 4. The Future of Research, Inc. Made public on Jan, 26th 2017 under Creative Commons 4.0 Attribution License Reviewed and discussed at http://www.sjscience.org/article?id=580 http://www.sjscience.org/article?id=580 Abstract Scientific evaluation is a determinant of how scientists, institutions and funders behave, and as such is a key element in the making of science. In this article, we propose an alternative to the current norm of evaluating research with journal rank. Following a well-defined notion of scientific value, we introduce qualitative processes that can also be quantified and give rise to meaningful and easy-to-use article-level metrics. In our approach, the goal of a scientist is transformed from convincing an editorial board through a vertical process to convincing peers through an horizontal one. We argue that such an evaluation system naturally provides the incentives and logic needed to constantly promote quality, reproducibility, openness and collaboration in science. The system is legally and technically feasible and can gradually lead to the self-organized reappropriation of the scientific process by the scholarly community and its institutions. We propose an implementation of our evaluation system with the platform “www.sjscience.orgthe Self-Journals of Science” (www.sjscience.org) § INTRODUCTION: THE INHERENT SHORTCOMINGS OF AN ASYMMETRIC EVALUATION SYSTEM Criticism of the current academic evaluation system traditionally focuses on the problematic use of journal reputation as a proxy for scientific quality. However, the harm caused by the research community's dependency on academic journals is more unsettling and destabilizing than we usually think. Journal-based evaluation creates an asymmetry within the scientific community between a minority of scientists sitting in editorial boards, who have the exclusive power to give value to a scientific article by accepting it in their journal, and the vast majority of scholars who strive to convince editors that their articles are highly citable in order to secure one of the limited publication slots, vital for academic career advancement. In this system of value creation, scientific recognition is artificially turned into a resource of predetermined scarcity for which scholars have to compete. In one camp, members of the scientific community must compete for limited space in a few “top” journals, which can impede the natural unrestricted progress of science by disincentivizing open research and collaboration. In the other camp, a low number of editors must also contend with each other for exclusive content to increase the reputation of their journal, a process that can have strong negative effects on scientific output and on the research enterprise as a whole. Although many scholars wear both hats –being authors and journal editors at the same time– here we do not identify the problem in individual agents but rather in the roles themselves and the power relationship between them. Thus, we argue that it is not only the kind of value that is promoted by the current system that is questionable (journal prestige and `impact', as in impact factor): more importantly, it is the way the system produces value and how its implicit asymmetric power structure is detrimental to scientific progress. This fundamental problem must be addressed by any proposed alternative. In the rest of this introduction we highlight some of the most important consequences of this asymmetry. §.§ Peer-trials undermine scientific peer-review. Peer-review is a founding principle of science. It is the process through which the community debates over the validity of a scientific proposition. It allows science to be self-correcting and to develop beyond the prejudices of the few. In the current publishing environment, since scientists are competing for the same limited resources, relations between peers can become inherently conflictive. For instance, scientists working on the same topic may tend to avoid each other for as long as possible so as not to be scooped by a competitor, whereas collectively it is likely that they would have benefited most from mutual interaction during the early research stages. The most worrying consequence of peers' diverging interests is that debating becomes socially difficult –if not impossible– in the context of a journal. The rejection and downgrade of an article to a lower-ranked journal can be a direct consequence of a scientific disagreement that few people would openly take responsibility for, to avoid reprisals. While the reliability of science comes from its verifiability, today it is being validated by a process which lacks this very property. Journal's peer-review is not a community-wide debate but a gatekeeping process tied to the local policy of an editorial board, where a small number of people hold temporary authority over an article, and whose goal is to support a binary decision or acceptance of rejection within some deadlines. We propose to rather refer to journal's peer-review as a `peer-trial', a term that in our opinion better accounts for its goal and the social dynamics at work behind it. Since peer-trials necessarily involve a degree of confidentiality and secrecy, and since they are limited in time, many errors, biases and conflicts of interest <cit.> may arise without the possibility of correction. In that sense, peer-trials are questionable as a scientific process. The peer-trial has become the standard in scientific publishing for the past 60 years <cit.>, and is the only modus operandi that the current generation of scientists has known, entrenching the belief that passing them is equivalent to attaining scientific validity. This is not to say that they are unfair, unuseful, or without intellectual added-value when all participants are competent, genuinely committed and have the interests of scientific truth at heart. Surveys report that 90% of authors, whose article has been accepted, feel that peer-trial had improved it <cit.>. Nevertheless, this is not a guarantee of scientific validity. Limitations in time and the insufficiency of available competences mean that a perceived improvement does not necessarily achieve high scientific standards <cit.> as expressed, for instance, by the general misuse of statistics in biomedical sciences <cit.>. Moreover, improving scientific validity after a peer-trial process is quite different from evaluating scientific innovation. Articles that may best contribute to the progress of science are often unexpected or disruptive to the status quo <cit.>. It is precisely in this context that the peer-trial format is most likely to fail and go wrong <cit.>, with unverifiable shortfalls for science. While peer-trial still dominates the mainstream, there are strong signs that the scientific community is actively engaged in a more continuous process of validation. Browsing websites such as https://pubpeer.comPubPeer or http://home.publons.comPublons (where “post-print peer-review” is possible) makes it clear that, although articles are improved with respect to initial submission, the discussion process continues long after publication and that the evolution of articles is a more dynamic construct <cit.>. This is at odds with the world of undisclosed email dialogues between authors and editors, and reviewers and editors during the peer-trial process. The unaccountability of peer-trials have further systemic consequences. For instance, referees cannot be credited for their work and institutions are led to promote a one-dimensional definition of scientists' utility, that would only rely on their productivity, and whose latest avatar is the h-index <cit.>. §.§ Competition between journals excludes research with perceived low impact. Editors cannot afford to neglect the impact factor of their journal <cit.>. Instead, they are motivated to inflate it by selecting articles that they think will be highly cited in the following two years. This selection bias prioritizes research that is more likely to trend at the expense of elements that are critical in the testing of scientific ideas but not conducive to the increase of a journal's impact factor (such as high-accuracy experimental data that did not prove a “positive” effect, or replication studies). §.§ Gaming the system results in low quality. Since the value of an article at present is tied to being published in a journal, and since peer-trials are not transparent, can be biased, are few in number and highly variable in quality, scientists may be tempted to game the traditional publication process. By gaming, we mean that a scientist may inappropriately generalize from the trivial, avoid statistical rigour, mislead by an elegant narrative, exploit power in the relationship between editors, reviewers and authors <cit.>, or even use fake identities <cit.> and commit fraud[The latter can be followed on retractionwatch.comRetraction Watch, a blog which keeps an active record of retraction cases, and investigate many.] <cit.>. For instance, statistics from the Nuffield Council on Bioethics on the culture of scientific research in the UK show that 58% of survey respondents reported that they were aware of scientists feeling tempted/under pressure to compromise on research integrity and standards, and one-third of scientists under 35 reported feeling this pressure themselves <cit.>. The major consequence of gaming is loss of quality which leads to irreproducible research getting the seal of approval by the publishing system <cit.>. Once an article has been published, the editor may be reluctant to have a debate open, whose outcome may damage the reputation of the journal <cit.>. The `time-to-retraction' (i.e. the time from publication of an article to publication of retraction) averages 32.91 months <cit.>. This is a concern for debate in science, especially in the sphere of public health where clinical trials can already have reached an advanced stage by this time <cit.>. There is a fundamental contradiction between the scientific need to constantly and dynamically debate, test, refine or correct scientific claims, and the private need of third parties to deliver and sell something as a static end-product. All of this can considerably delay or hinder the self-correction of science, result in a waste of time and (public) money, and can have deleterious effects on the credibility of science and what it produces. §.§ Scientific conservatism is placing a brake on the pace of change. It is questionable whether any static subset of the scientific community could appropriately manage the direction of science. Today's highly-regarded researchers naturally tend to defend the paradigm that underpins their reputation, whilst opposing tomorrow's ideas <cit.>. For example, in “The Dynamic State of Body Constituents” <cit.>, Schoenheimer tried to introduce the concept that proteins were broken down intracellularly (a fact that we now take for granted, with the study of ubiquitin). However, it took more than 30 years for this concept to be accepted, and likely delayed the discovery of ubiquitin, in part because Nobel Laureate Jacques Monod was a proponent of the theory that proteins, once formed, were ever present <cit.>. In science, good ideas may eventually prevail, but a lot of time and effort may be mis-spent before they do. Though the beauty of the scientific method is to allow humans to go beyond their own prejudices, the traditional publishing system is prone to working in favor of the current dogma. The main mechanism of selection of editors (i.e. co-opting between reputable researchers) is creating and enforcing additional constraints and limits on the progress of science without the security of collective wisdom. To address these issues, in this article we introduce a model based on a novel, open, and community-wide evaluation system that captures a well-defined notion of scientific value and which is based on scientists' collective intelligence and judgement. We advocate that the inherent logic of our model may reverse the process of privatization and fragmentation of evaluation associated with the use of journal rankings. Moreover, the system is technically and legally feasible in the current environment. In Section <ref> we define scientific value, the methodology used to assess it and associated metrics. In Section <ref> we discuss the mechanics and merits of our proposed model. In Section <ref> we highlight some implications of this novel way of evaluating research works in the context of a global competition for money, tenure and honors. Our treatise is part of a broader vision that is also developed in <cit.> and <cit.>. § A SYMMETRIC PROCESS FOR THE CREATION OF SCIENTIFIC VALUE In this section, we present a definition of scientific value and describe the open and community-wide processes required to capture it. These processes maintain symmetry in the creation of scientific value and fulfil what we consider the minimal expectations from any desirable alternative evaluation system, which are: * to promote scientific quality. * to provide incentives to authors, reviewers and evaluators. * to promote academic collaboration instead of competition. * to be able to develop in parallel to current journal publication practices (as long these remain essential for funding and career advancement). * to propose article-level metrics that are easy to calculate and interpret. * to be verifiable and hard to game. A prototype of an evaluation system driven by these processes is implemented in “the Self-Journals of Science” (SJS, www.sjscience.org): an open, free and multidisciplinary platform that empowers scientists to achieve the creation of scientific value. SJS is a horizontal environment for scientific assessment and communication and is technically governed by an international organisation[<www.openscholar.org.uk>] of volunteer research scholars whose membership is free and open to the entire scientific community. §.§ Scientific value as validity and importance A scientific article relies on refutable statements that contribute to a body of knowledge about the object of study. In this description of science, the value of an article spans two distinct notions, that require their own assessment mechanisms: the correctness of its statements, which we call its validity, and the value of its contribution, which we call its importance. The validity of an article is established by a process of open and objective debate by the whole community. Since the contribution of a scientific article essentially relies on refutable statements, debating them in principle [that is, in the limit of infinite time and infinite resources to test the statements] can eventually converge on a consensus about whether it has reached accepted scientific standards (and what these standards should be: methodological soundness, unambiguity of of presentation, satisfaction of various protocols, inclusion of appropriate references, etc.) or whether or not it needs further revision(s). The importance of an article is the outcome of its perceived importance by each member of the research community. This perceived importance is a subjective assessment that depends on personal knowledge and understanding, intuition, and anticipation of future advances in the field. Unlike validity, the perception of importance does not rely on refutable elements that could be used to automatically resolve disagreements. Broad consensus is not expected based on the importance of an article alone; for instance, scientists might rightfully diverge in their belief that a certain path of development of their field is more valuable than another. The importance of an article is thus socially determined and field, time and culture-dependent. The two different concepts cannot be measured by a single index: a methodologically valid article may not be important just as a trending article may prove to be wrong. In the traditional system, these notions are merged and uniquely expressed by a local, opaque and one-time event: journal publication.[In our implementation, we also introduce an addition notion priority which is of practical use but not an index of quality of a scientific article. Nonetheless, we have introduced priority as a simple counter that allows scientists to mark articles prior to thorough scrutiny in the same vein as the “Like” button on Facebook. Priority is a short-term notion that does not imply a scientific judgement and the reasons for prioritizing an article can be either positive (e.g. the article looks interesting) or negative (it looks terrible and needs a quick refutation). Priority offers a instant filter to help organize scientific output on a short-time scale, before assessment of its validity and importance. The reason this is important is because it allows the system to channel misleading short-term effects (such as those of a promising abstract combined with an overstating narrative) into an index that is different from that used for scientific evaluation.] §.§ Assessing validity by open peer review We have defined scientific peer-review as the community-wide debate through which scientists aim to agree on the validity of a scientific item. In our system, peer-review is an open and horizontal (i.e. a non-authoritative and unmediated) debate between peers where “open” means transparent (i.e., signed), open access (i.e., reviewer assessments are made public), non-exclusive (i.e., open to all scholars), and open in time (i.e. immediate but also continuous). This brings a new ethic to publishing <cit.>: the goal of peer-review is not to provide a one-time certification expressed in the form of a binary decision of accept or reject as per the traditional mode of publishing, rather it is to scientifically debate the validity of an article with the aim of reaching an observable and stable degree of consensus. Here, reviews are no longer authoritative mandates to revise an article, but elements of a debate where peers are also equals. The influence of a review over an article is based on its relevance or its ability to rally collective opinion, and on an open context where authors cannot afford to let relevant criticism go unanswered. The validity of an article is captured by a transparent and community-wide vote between two options: “this article has reached scientific standards” [The notion of “scientific standard” is not enforced by any authority but is dynamically defined by the majority vote that follows a global conversation about each article.] or “this article still needs revisions”. The quantifiers of the validity of an article (i.e. its metrics) are hence the number of scientists who voted, and the fraction who validated the article. Any reader can instantly access the current state of the voting process which is displayed at the header of the article (Figure 1). In our proposed implementation of such open peer-review, we introduce other features aimed at incentivizing positive interactions between participants and the proper self-organization and self-regulation of the debate: * Articles are interactive and reviews are appropriately embedded into them. Reviews therefore benefit from the same visibility as the article. * Reviews can be individually evaluated with a +/- vote system * The vote “this article still needs revisions” must be substantiated by the writing of a review or up-voting of an existing review. * Upon revision, authors can select those reviews most useful to them and the reviewers get proper acknowledgement in all subsequent versions of the article - in their header (online) or in the cover page (PDF). * The history of the article is always accessible. In this form of self-publishing, scientific articles become dynamic <cit.>, authors become active in energizing a peer-review process <cit.> whose quality is driven by collective intelligence in a symmetric environment where the best scientific ideas are subjected to natural selection <cit.>. The process produces a text initially written by its authors but which also includes the debate it has generated within the community. §.§ Assessing importance with self-journals Importance vs impact. A fundamental difference between our understanding of importance and the notion of impact as measured through citations (e.g., the impact factor, the journal citation distribution <cit.> or the RCR <cit.>) or usage statistics (e.g. number of tweets, hits, downloads, etc), is that importance should be explicitly and directly related to a scientist's judgement. In that sense, metrics of impact do not necessarily convey useful or separable information about importance. As for citation-based metrics, we argue that a citation is not an endorsement of the article we cite. For instance, we will cite an article we may want to scientifically refute; citations may be included so as to please an anonymous referee <cit.>; we often cite review articles instead of original articles, or we simply copy and paste or incorrectly cite without reading the actual articles <cit.>. Also, the narrow context of our own publications may not be relevant to cite and give credit to articles that have, however, been important to us. Usage statistics (altmetrics) are even less reliable since they can be massively gamed and may be derived from sources unrelated to science, or even from a plethora of ghost accounts on social media. Self-Journals. In our alternative evaluation system we introduce the concept of self-journals as a way for scientists to properly express their judgement regarding an article's importance for a specific field. A self-journal is a novel means of personal scientific communication; it can be thought of as a scholarly journal attached to each individual scientist that works on the curation of any scientific item available on the Web via hyperlinks (and not on appropriation of articles following a submission process). A self-journal is released into structured issues, which are collections of articles around a certain topic. Every issue has its own title and editorial providing an introduction for the community and must contain a minimal number of articles (in our implementation, we set this minimum to 4). The curator has the possibility to provide personal comments on each article that has been curated in the issue (for concrete examples, please check the first issue of the self-journal of http://www.sjscience.org/memberPage?uId=148 jId=10#journalSanli Faez, http://www.sjscience.org/memberPage?uId=90 jId=6#journalKonrad Hinsen or http://www.sjscience.org/memberPage?uId=1 jId=14#journalMichaël Bon). The consistency of the selection of articles and the relevance of the personal comments determine the scientific added value of each self-journal issue. Every scientist can curate their own self-journal, through which they can release as many issues on as may topics as they please. Curators can take advantage of self-journals to review a field, present a promising way to develop it, offer a comprehensive collection of their own research, host the proceedings of a workshop or a journal club, or popularize scientific ideas and discoveries etc. A self-journal reflects the scientific vision of its curator and bears his or her signature. Interested readers can freely subscribe to a self-journal and get notified whenever a new issue is released. An ecosystem of self-journals offers a way to quantify the importance of an article, primarily by the number of its curators. § BENEFITS OF THIS EVALUATION SYSTEM The evaluation system we propose generates easy-to-use article-level metrics for validity (the number of scholars engaged in peer-review and the fraction of them who consider that the article is up to scientific standards) and for importance (the number of scholars who have curated the article in their self-journal). Such metrics can progress in parallel to existing publication practices and metrics. On the one hand, the assessment of importance through curation via hyperlinks can apply immediately to all scientific literature on the Web, irrespective of its legal owners. On the other hand, the assessment of validity requires only the uploading of author-copyrighted content or pre-prints – a practice which is already standard in some fields such as physics, and gaining momentum in others such as biology <cit.> (see for instance http://asapbio.orgASAPBio). Below we discuss how this evaluation system satisfies the properties that we laid out in section <ref> and the benefits it provides. §.§ Avoiding conflicts in evaluation Competition for an abundant resource promotes actively collaborative behaviour. In our evaluation system, the conflict-creating competition for rare slots in a top journal is replaced by a competition for open peer recognition which is abundant and non-exclusive. Giving recognition to a peer does not deprive somebody else of it; an important article does not rise at the expenses of any other article. In this symmetric recognition-based economy of knowledge where scholars are at the same time authors, reviewers and evaluators of science, all scholars can give value to their peers and are potential benefactors of each other. To get the attention of peers and convince them of your scientific merits, the best strategy (in the sense of game theory) is to pay attention to their work and rightfully give them the recognition they deserve in a way which lets your own expertise publicly shine. This is exactly what is achieved by self-journals and open peer review. Convergent interests in open peer-review. Peer-review is transformed from an authoritative trial into an open, transparent and unmediated scientific conversation. It does not assume which participants (either authors or reviewers) will be right in advance and does not result in a definitive and binary decision of acceptance or rejection of an article. Then, openly disagreeing on a constructive basis becomes an opportunity for both the author and the reviewer to gain recognition, both from each other and from the rest of the community. * For authors, an ongoing expert debate raises the level of awareness of their article among the community. A spontaneous, signed and relevant critique is delighting evidence that a peer cared and gave some thought to their article. The reviewer may eventually turn into a curator once appropriate answers have been given. A live process of improvement of the work under review is also the best way to convince other readers that articles and research results are valid. * For reviewers, credit is received for their public display of expertise, enhancing their reputation within the community. Their reviews become an integral part of the paper and are visible in all its instances and they directly benefit from helping raise the visibility of the article they have chosen to review. In this way, reviewers get special credit and are incentivized to be relevant, constructive and courteous. The public display of expertise by engaging authors in discussion will prove beneficial to reviewers at a later stage when they author their own papers, as their reputation within the community is enhanced also by the quality of their ability to review. No conflict in the assessment of importance. The assessment of importance is also devoid of any conflicts. A self-journal is a personal tool whose content, topics of interest and release dates are completely up to its curator and his or her private interests, policy and communication strategy. There is no submission process; articles are freely picked by the curators according to the message they would like to deliver to the community in the next issue of their self-journal. The consequence of this is that no single article can be expected to be curated by a specific self-journal. An unimportant article is simply characterized by the fact that it remains in its default state with no curator, and not by any explicit and conflictive statement that it is unimportant coming from somebody in particular. The act of assessment of importance by openly taking responsibility to include such an article in an issue of one's own self-journal, can only be positive by mutually increasing the visibility of both. §.§ Incentives for maintaining one's self-journal Incentives in the absence of official recognition by institutions and funders. Self-journals have their own rationale. Firstly, they are a means of personal scientific communication that allow their curators to elaborate on an individual vision of science with the necessary depth they see fit. A self-journal therefore provides a great scientific service to its readers by providing a level of consistency in the interpretation and analysis of scientific output. In return, this service benefits curators who increase their visibility and influence over what is disseminated to the community. Self-journals give new freedom and scope to the editing process since curation, as proposed here, applies to any research work with an Internet reference. In other words, a mechanism is provided that allows scientists to fully express an aspect of their individual worth that is absent in the current system, and build a reputation accordingly. A response is also provided to the problem of the decreasing visibility of authors, articles and reviewers as the volume of scientists and scientific works grows on the Web. Each issue of a self-journal acts as a pole of attraction that is likely to have a minimum audience: the authors whose articles have been curated can be notified about what is being said about their work, and may want to follow the curator. Moreover, on a platform like SJS where the ecosystem of self-journals is well integrated, interest for a particular article can guide readers to self-journal issues where it has been uniquely commented on and contextualized in relation to other articles. We wish to emphasize that the interest value of a particular self-journal issue does not lie so much in the intrinsic value of the articles selected, but rather in the specific comments and collective perspective that is being given to them. Consequently, if a certain article is curated by a “reputable” scientist, its other curators will not lose value or visibility - even when their issues contain exactly the same articles; different self-journal issues can always maintain independent interests. Every scientist therefore has both a short-term and long-term personal interest in maintaining a self-journal, and in reading those of their peers. It is also worth mentioning that the expected primary use of a self-journal - sharing one's vision of a field, is not time-consuming as scientists have already developed such visions in the course of their research whenever they have produced a bibliography. Maintenance of a self-journal in this case is reduced to the time it takes to select the most important articles and to write useful comments if desired. Incentives with official recognition by institutions and funders. If, as an evaluation system, self-journals attract the attention of institutions and funders, then the incentive for maintaining one's own self-journal is clear: it is an indispensable tool for self-promotion that allows everyone to have a share in the direction of evolution of their field. That power, which is today in the hands of editors in a top-down certification system, will be redistributed horizontally amongst scientists who are collectively co-responsible for validation and evaluation and are properly credited for doing it well. Growing awareness of the positive impact of ease-of-access to validated, important-ranked and credible science, as well as the increase in scientific value given to researchers and, by proxy, their institutes and funding bodies, will help generate momentum for the mass adoption of self-journals and open peer-review. In turn, mass adoption is what will bring the power and richness of this evaluation system at its peak. §.§ Promotion of scientific quality The need for quality curation. To gain followers, readers and influence, scientists should maintain a thoughtful and interesting self-journal. In other words, they are incentivized to be good evaluators, i.e. to carefully select and curate relevant collections of articles in issues and enhance them with enlightening comments. Scientists that mechanically curate all articles in their field (akin to copy-pasting entire bibliographies) without adding value by providing comments or constructive criticism, will receive less attention and influence than one who focusses on quality. Moreover, because of the need for structure and the writing of an editorial for each issue, self-journals are not automatons. They are not a time feed to which articles are appended in a single click and where the latest additions decrease the visibility of prior additions (something that can occur on very short timescales e.g. minutes), such as in other environments. This “friction” ensures that self-journal issues will be released only when a curator has a point to make, and that qualitative strategies are more efficient than quantitative ones when the aim is to get as large audience as possible within the scientific community. Thus, unlike simple popularity contests (such as the number of “Likes” on Facebook), there are a feedback loop and constraints acting on the quality of the self-journal itself which encourages its curator only to include articles that reach his or her standards. Curation recognizes works necessary to the progress of science. The value of a self-journal built on the validity and importance of its articles is driven by the scientific interest and relevance of the vision developed by its curator, and not because it has monopolized the distribution of highly-citable articles as is the current situation. Consequently, as an evaluation system, it liberates articles that are essential to scientific progress which are today penalized because they are believed to be less citable (e.g. replication studies or reports of valid results which failed to prove the author's hypothesis; often unwisely referred to as “negative” results). Such contributions are as important to the scientific method as reports of statistically-significant effects, especially in the empirical sciences. When providing their analysis of a certain topic, curators are now free to integrate items they see as important to making their point as convincing as possible. In the case of replication studies there is actually an incentive for their curation. Scholars who have already curated a particular standard are interested in knowing whether or not a certain claim has been confirmed or refuted. Thus, it is likely that the replication of an original result will benefit from the same level of curation as the original article. This also means that a laboratory which is considering replicating a result can have an a priori estimate of the publishing reward expected for investing time in performing an experiment. Results that are regarded as important and highly curated provide a stronger incentive for their replication. This provides a novel positive feedback mechanism and impetus for science to move forward confidently, since major results will be verified. §.§ No artificial constraint on time, space and format Unlike academic journals, self-journals are not expected to curate only the most recent articles. It is in a curator's best interest to provide a mix of both past and present articles when creating a deep and comprehensive vision of their field <cit.>. For an article, this implies that its evaluation span becomes time-dependent. For instance, a disruptive innovation which is gradually understood will become increasingly curated and reach high importance even if the author was the only one to understand it at the beginning of the process. Furthermore, linked data that results from hyperlinking content frees self-journals from space and format restrictions, present in the current mode of publishing. By this we mean that the content of self-journals is not constrained to any particular type of scientific item (article, thesis, conference proceeding, poster, essay, technical report etc), design format or storage requirement. This mode of evaluation then, gives the scientific community a certain freedom to evolve the format of articles according to its needs. §.§ Robustness to gaming and biases Here we highlight how the inherent logic of this evaluation system and its horizontal power structure, fight against gaming and human biases. However, it is clearly impossible for us to be sure how a culturally diverse population of millions people, subjected to different local constraints, would behave in such an environment. We are therefore looking for as many feedbacks as possible on this topic as well as practical tests. Gaming. The metrics we propose are established by open and community-wide processes which make them hard to game. For instance, unlike citations and usage statistics, two accomplices are unable to create an infinite loop that boosts the importance of each other's articles. This is because the primary quantifier is the number of curators rather than the number of self-journal issues in which an article appears (which would become analogous to journal citation). If a scientist exploits friendship bias (i.e. they curate articles of an acquaintance multiple times), they would only increase their index by +1. At the same time, their self-journal would lose credibility in the eyes of the community and damage their reputation. Gaming a truly open peer-review and evaluation system like the one proposed here, would imply a successful manipulation of a significant fraction of the scientific community in open and transparent processes or involves the (highly unlikely) joint individual misconduct of a large fraction of scientists which is easy to expose as it is large. The difficulty of these scenarios and the ease at which they can backfire strongly dissuade them. Note that this evaluation approach does not assume that the majority of members of the scientific community are virtuous either. It is peer pressure which naturally and constantly exerts itself to enforce the highest scientific standards everywhere. In such an environment, the best self-interest of each scientist is aligned with the ethical requirements of science. There is no way to get recognition other than making valuable contributions to scientific knowledge, be it in the form of articles, reviews or a self-journal. Finally, since these processes of peer-review and curation are not locked up in a proprietary database, the research community is free – and expected – to develop open source modules that can detect anomalous behaviors and properly address any gaming scenario. Such modules could be for instance integrated in search engines for signaling, or could be run independently. They will enforce further self-regulation and can demonstrate the reliability of these processes as they gain momentum as an evaluation system. Biases. Similarly, the absence of vertical relationship between scientists (being replaced with horizontal and reciprocal ones) and the full transparency and accountability of the system, combine to oppose the negative expression of human biases with respect to specific works or fellows (i.e. conflicts of interest, gender-, race-, age-, country-based biases, etc.) For instance, the fact that reviews are not authoritative and are themselves subjected to peer scrutiny (with the possibility of being evaluated with a +/- voting system and being refuted by the authors when irrelevant) strongly incentivizes reviewers to strengthen valid scientific arguments. Doing otherwise might backfire and negatively impact the reputation of the (always non-anonymous) reviewer. In the logic of our system, relevant reviews contribute to attracting the attention of the community and give life to an article. Therefore, even if a reviewer wants to criticize an article because of a personal bias against the authors, the need to do so in accordance with high scientific standards actually results in a benefit to the authors. Moreover, every scientist is generally incentivized not to express negative biases against their peers because they are the ones with the freedom and power to endow value to their works. Thus, failing to reach a subgroup of peers for reasons other than science goes against the interests of the scientist, who may suffer from a shortfall in the evaluation of his or her works. In addition to deflecting or opposing the negative expression of a bias, the system possibly also offers a long-term opportunity for a desirable cultural evolution in relation to disfavored minority groups. The visibility of their individuals no longer depends on a self-reproducing vertical power structure which can limit their contributions (e.g. by making it harder for them to participate in the publishing process as an anonymous reviewer or editor). Rather, they can autonomously and publicly express their value in all dimensions of the scientific activity. This is an opportunity for them to take, that cannot go unseen by the community, and which will help form the mindset of future generations. §.§ No loss of information The current ubiquitous use of indices in decision-making by administrative structures and funders enforces the constraint that our evaluation system generates numbers so that it can be easily adopted and recognized. We have therefore proposed article-level metrics for validity and importance, but it is obvious that expressing something as complex as scientific value in the form of a few numbers inevitably implies a loss of information. The quantifiers proposed here are mere tools to provide a first sound and reliable picture of what is valid and important in science, and we argue that they are in many respects preferable to journal rankings and impact factors in capturing scientific value. However, in our evaluation system, the qualitative processes underlying the computation of our metrics remain fully accessible and they offer much richer and accurate information that can and should be made use of. An ecosystem of self-journals provides a context in which an article can be appreciated via its relationship to the other articles of the issues in which it has been curated. The perspective of curators is explicitly expressed in narrative form via editorials and individual comments. Science can be followed in many additional ways: by following the activity of specific scientists (i.e. what they publish, review and curate) that one considers to have better skills than the rest, by following the activity and connections developing around specific articles, by reading self-journals. Networks of co-curated articles, of authors, of reviewers and curators can be easily studied in a systematic and automatable fashion. The collective intelligence of the scientific community can be dug as deep as necessary by all users, and according to their needs. This complete and easy-to-process information displays a diversity of approaches that opposes an evolution of our evaluation system into another “tyranny of the metrics”, where the only visible articles would be those which rank at the top according to our quantifiers. §.§ Acceleration of the adoption or debunking of novel ideas The traditional publishing system has an inherent conservatism but also the structural means to slow down the adoption of disruptive ideas if not “believed” by dominant voices. Such mechanisms exploit the asymmetry of the evaluation system and are absent from the self-journal system. However, self-journals alone are not sufficient to overcome the natural reluctance of a community to adopt novel ideas. There is inertia. Ideas may be expressed by authors' articles and self-journals but lack of peer commentary by the community is a challenge also for the use of collective intelligence for building scientific value. Our argument is that rewardable open peer-review is the motivator and accelerator of such a process. Indeed, the author of a disruptive idea can contribute as a reviewer of peer articles and accordingly point out their possible shortcomings; the community will eventually be triggered into noticing the novel idea and responding - leading either to its more rapid adoption or debunking. §.§ Full achievement of Open Science The way that self-journals create scientific value produces an additional mechanism that triggers openness. When the goal of a scientist is shifted to convincing as large a section of the community as possible (and not simply matching the minimal standards of a journal policy), those who practice open science have an advantage over those who do not. Ensuring that the full text of article, data and code are all freely and openly available, is the vehicle for maximizing the potential for getting positive feedback from the community. If, for instance, data is not made available, members of the community are likely to conclude that the article cannot be validated (since it cannot be tested), and the authors will be penalized. With the evaluation system we propose, total openness is in the self-interest of every scientist and occurs naturally without the need for top-down mandates or bureaucracy. We believe that the correct battle for Open Science is the one for the metrics that reward it. Community-based metrics re-empower the average scientist, while at the same time providing incentives to energize and reward collective participation. They will strengthen and reach beyond open access. Paradoxically this battle is also much easier than that for open access since it only depends on scientists and not on what legacy publishers can think or do. Moreover, since the assessment of validity and importance of articles takes place in a self-organized and self-regulated way, there are no intermediaries between scientists and the direct costs of scholarly communication are reduced to the cost of storage. This is presently tiny in comparison with what is currently being paid by institutions to legacy publishers whether in the form of journal subscriptions or article processing and open access charges. It will fulfil the early promise that open access would decrease costs, which is unlikely to be the case following the current policy to expect journals to flip their business model to “gold” publishing (a model which becomes prevalent in the UK and the EU following legal requirements). § IMPLICATIONS FOR THE EVALUATION OF SCIENTISTS The competition between scientists for money, tenure and honor is necessarily competitive since these resources are scarce. We believe that this inevitable competition does not de facto negatively affect the quality of science. However, the current terms of this competition definitely do (i.e. striving to secure rare publications slots in top journals as a stamp of approval). Instead, we propose a community evaluation system that removes what is an artificial construct - the rarity of scientific recognition. In our model, the quest for individual recognition does not succeed at the expenses of peers; rather it originates from open and fruitful interactions with peers. In the terms of our model, the competition for money, tenure and individual honor drives scientists to adopt a collaborative behaviour. This apparent paradox is not a contradiction because our processes of evaluation are global, while money, positions and prestige are delivered through local processes that involve a minority of the scientific community. For instance, it is clear that two scientists competing for the same position will hardly sing each other's praises. However, the evaluation of their work will depend mostly on what the rest of the community thinks, and in their attempt to get this positive feedback, the contenders will have had to perform openly valid and important research, write thoughtful reviews, and maintain an enlightening self-journal – all things good for the progress of science. In the current system, even if they do not apply to the same grants, scientists working on the same topic are always a threat to each other - because the one who will publish first will mechanically shrink the value of the other[For instance, when two different labs conduct similar research and submit similar results, editors will tend to publish the article that is first submitted to them because journal prestige relies on citation counts, and science usage is such that due to anteriority it will be the one that accrues the most citations. Conversely, in the self-journal system, articles with similar results are likely be curated in pair, and therefore have similar levels of importance since the point that the curator may want to make based on them will be strengthened by the inclusion of two independent and convergent studies.]. Therefore, we argue that the gradual adoption of the quantifiers we introduce here by institutions and funders will inject positive incentives for both quality and openness into the scientific community. The precise way that such quantifiers will be assimilated in internal processes is a choice that is political in nature an is the responsibility of those who want to evaluate researchers. It will depend on their specific goals, vision and the particularities of how best to fund science and honour scientists at the local level. If an institution can adopt the h-index, it can devise a similar index based on our quantifiers that will have a sounder scientific basis and produce positive systemic implications. Also, the modus operandi of evaluation commitees can evolve because their members can now have access to the collective intelligence of scientists who have already thought about and contextualized the articles upon which funding decisions and assignment of prestige will be based. It will be much easier and intellectually satisfying for grant reviewers who will now have access to a whole range of explicit judgements already expressed by peers. A major incentive is that, for members of evaluation committees who cannot read full articles, reading such judgements is less time-consuming, while providing a scientific safeguard and diversity they may have lacked otherwise. Finally, our evaluation model creates the potential for also taking into account the reviewing and curation activity of a scientist. Since these activities have an accountable influence on the course of science, they provide an impetus for many institutions (especially those struggling to compete with “centres of excellence”) to push for a multi-dimensional approach to assessing a scientist's value, and develop policy accordingly. Just as journal-based evaluation has brought the publish-or-perish culture, our metrics can help recreate the sense of community now lacking in science. When such a different mindset is generalized, novel models of funding can also become possible such as the decentralized approach proposed in <cit.>. § CONCLUSION: TOWARDS A PRACTICAL CHANGE We have proposed and described an evaluation system that promotes a well-defined notion of scientific value in which objective and subjective aspects are disentangled as validity and importance. The system restores a global conversation between scientists and gives control of the evolution of science back to the scientific community through open and horizontal processes, while at the same time incentivizing fruitful interactions between peers. The system also generates novel metrics that are as easy to use as the impact factor in the context of institutional evaluation or grant reviewing. It can progress in a bottom-up fashion without conflicting with current practices of publication. Since this system gives every scientist autonomous means to review and evaluate all scientific items, we consider it to be of great appeal especially to those who lack such means at present – the junior scientists. We particularly encourage them to realize that it is their self-interest to nourish such an alternative evaluation system and reshape the power structure of tomorrow's science. The evolution we are looking forward to is not an utopia. The science world already has a shining example of a bottom-up achievement initially powered by junior scientists: the pre-print server arXiv.org. Built in the 1990s by Paul Ginsparg, it offered a service of objective scientific value and rose out of the concrete needs of a community which supported and maintained it, despite not offering any reward in terms of institutional evaluation. Today it has become the main portal where physicists, mathematicians and computer scientists share their work in the form of preprints which are often cited more than analogous journal publications. This practice is now self-evident for these communities. We further propose an implementation of the ideas developed in this article in the form of SJS, an open platform for curation and peer-review, governed by an open-membership organisation of volunteer scholars[<www.openscholar.org.uk>]. We believe SJS could become a new multidisciplinary agora where scholars can co-validate and co-evaluate their research products while receiving, at the same time, proper recognition that can translate into both direct and immediate career benefits. Maintaining a self-journal is a rewarding, simple and risk-free activity for scientists, and debating with peers is natural in our community. We look forward to watching this academic ecosystem grow, self-organize and evolve in the years ahead. unsrt 10 randomPR Lutz Bornmann, Rüdiger Mutz, and Hans-Dieter Daniel. http://dx.plos.org/10.1371/journal.pone.0014331A reliability-generalization study of journal peer reviews: a multilevel meta-analysis of inter-rater reliability and its determinants. PloS one, 5(12):e14331, 2010. resistance Juan Miguel Campanario and Erika Acedo. http://onlinelibrary.wiley.com/wol1/doi/10.1002/asi.20556/fullRejecting highly cited papers: The views of scientists who encounter resistance to their discoveries from other scientists. Journal of the American Society for Information Science and Technology, 58(5):734–743, 2007. ceci Douglas P Peters and Stephen J Ceci. https://www.cambridge.org/core/journals/behavioral-and-brain-sciences/article/peer-review-practices-of-psychological-journals-the-fate-of-published-articles-submitted-again/AFE650EB49A6B17992493DE5E49E4431Peer-review practices of psychological journals: The fate of published articles, submitted again. Behavioral and Brain Sciences, 5(02):187–195, 1982. PR_female Amber E Budden, Tom Tregenza, Lonnie W Aarssen, Julia Koricheva, Roosa Leimu, and Christopher J Lortie. Double-blind review favours increased representation of female authors. Trends in ecology & evolution, 23(1):4–6, 2008. mnielsen Michael Nielsen. http://michaelnielsen.org/blog/three-myths-about-scientific-peer-review/Three myths about scientific peer review. Michael's main blog, Jan 2009. PR_history Ray Spier. http://www.cell.com/trends/biotechnology/fulltext/S0167-7799(02)01985-6The history of the peer-review process. TRENDS in Biotechnology, 20(8):357–358, 2002. PRsurvey Adrian Mulligan, Louise Hall, and Ellen Raphael. http://onlinelibrary.wiley.com/doi/10.1002/asi.22798/fullPeer review in a changing world: An international study measuring the attitudes of researchers. Journal of the American Society for Information Science and Technology, 64(1):132–161, 2013. PRsurvey2 Mark Ware. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.214.9676 rep=rep1 type=pdfPeer review: benefits, perceptions and alternatives. Publishing Research Consortium, 4, 2008. error Sara Schroter, Nick Black, Stephen Evans, Fiona Godlee, Lyda Osorio, and Richard Smith. http://www.bmj.com/sites/default/files/attachments/resources/2011/07/what-errors-do-peer-reviewers-detect.pdfWhat errors do peer reviewers detect, and does training improve their ability to detect them? Journal of the Royal Society of Medicine, 101(10):507–514, 2008. false John PA Ioannidis. http://journals.plos.org/plosmedicine/article?id=10.1371/journal.pmed.0020124Why most published research findings are false. PLoS Med, 2(8):e124, 2005. pvalue Lewis G Halsey, Douglas Curran-Everett, Sarah L Vowler, and Gordon B Drummond. http://www.nature.com/nmeth/journal/v12/n3/full/nmeth.3288.htmlThe fickle P value generates irreproducible results. Nature methods, 12(3):179–185, 2015. sci_rev Thomas S Kuhn. The structure of scientific revolutions. University of Chicago press, 2012. horrobin David F Horrobin. http://jama.jamanetwork.com/article.aspx?articleid=380984The philosophical basis of peer review and the suppression of innovation. Jama, 263(10):1438–1441, 1990. dynamic_article Pandelis Perakakis, Michael Taylor, Marco G Mazza, and Varvara Trachana. http://www.akademiai.com/doi/abs/10.1007/s11192-011-0402-1Understanding the role of open peer review and dynamic academic articles. Scientometrics, 88(2):669–673, 2011. hindex Lutz Bornmann and Hans-Dieter Daniel. https://pdfs.semanticscholar.org/c8ee/814fa8027b2f8230d4654441df57aee73357.pdfWhat do we know about the h index? Journal of the American Society for Information Science and technology, 58(9):1381–1385, 2007. impact_game PLoS Medicine Editors et al. http://journals.plos.org/plosmedicine/article?id=10.1371/journal.pmed.0030291The impact factor game. PLoS Med, 3(6):e291, 2006. politics Peter A Lawrence. http://www.planetkc.sportsci.org/jour/03/Politics_of_Publication.pdfThe politics of publication. Nature, 422(6929):259–261, 2003. fake_ID Charlotte J Haug. http://www.nejm.org/doi/full/10.1056/NEJMp1512330#t=articlePeer-review fraud—hacking the scientific publication process. New England Journal of Medicine, 373(25):2393–2395, 2015. fraud Daniele Fanelli. http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0005738How many scientists fabricate and falsify research? A systematic review and meta-analysis of survey data. PloS one, 4(5):e5738, 2009. fraudculture Nuffield Council of Bioethics. http://nuffieldbioethics.org/wp-content/uploads/Nuffield_research_culture_full_report_web.pdfThe culture of scientific research in the UK, Dec 2014. irr_cancer Leonard P Freedman, Iain M Cockburn, and Timothy S Simcoe. http://journals.plos.org/plosbiology/article?id=10.1371/journal.pbio.1002165The Economics of Reproducibility in Preclinical Research. PLoS Biol, 13(6):e1002165, 2015. irr_psy Open Science Collaboration. http://www.sciencemag.org/content/349/6251/aac4716.shortEstimating the reproducibility of psychological science. Science, 349(6251), 2015. irr_eco Andrew C Chang and Phillip Li. https://www.federalreserve.gov/econresdata/feds/2015/files/2015083pap.pdfIs Economics Research Replicable? Sixty Published Papers from Thirteen Journals Say “Usually Not”. Finance and Economics Discussion Series 2015-083, 2015. raphaPNAS Raphaël Levy. https://raphazlab.wordpress.com/2015/11/16/pnas-your-letter-does-not-contribute-significantly-to-the-discussion-of-this-paper/PNAS: your letter does not contribute significantly to the discussion of this paper, Nov 2016. noopenreview Quirin Schiermeier. http://doi.org/10.1038/nature.2017.21342`You never said my peer review was confidential' – scientist challenges publisher. Nature, 541(446), 26 Jan 2017. retraction_time R Grant Steen, Arturo Casadevall, and Ferric C Fang. http://dx.doi.org/10.1371/journal.pone.0068397Why has the number of scientific retractions increased? PLoS One, 8(7):e68397, 2013. trial http://www.nytimes.com/2011/07/08/health/research/08genes.htmlHow Bright Promise in Cancer Testing Fell Apart, 07 Jul 2011. funeral Pierre Azoulay, Christian Fons-Rosen, and Joshua S Graff Zivin. http://www.funginstitute.berkeley.edu/sites/default/files/planck.pdfDoes Science Advance One Funeral at a Time? 2014. schoenheimer Rudolf Schoenheimer. The dynamic state of body constituents. Harvard Univ. Press, Cambridge, Mass., 1946. ubiquitin Avram Hershko. Brief history of protein degradation and the ubiquitin system. In John Mayer, Aaron Ciechanover, and Martin Rechsteiner, editors, https://books.google.fr/books?id=6ILfyx_FsIAC printsec=frontcover source=gbs_ge_summary_r cad=0#v=onepage q f=falseProtein Degradation: Ubiquitin and the Chemistry of Life. Wiley, 2005. Moi Systemic studies of scientific edition. Michaël Bon's self-journal, 1, 2015. <http://www.sjscience.org/memberPage?uId=1 jId=14#journal>. interview Richard Poynder. http://poynder.blogspot.fr/2016/05/the-oa-interviews-michael-bon-founder.htmlThe OA Interviews: Michaël Bon, Founder of the Self-Journal of Science, Jul 2016. selfpub Pandelis Perakakis and Michael Taylor. http://www.tandfonline.com/doi/abs/10.1080/08109028.2014.891712Academic self-publishing: a not-so-distant future. Prometheus, 31(3):257–263, 2013. naturalselection Pandelis Perakakis, Michael Taylor, Marco Mazza, and Varvara Trachana. http://www.openscholar.org.uk/wp-content/uploads/2015/08/nsap_perakakis_2010.pdfNatural selection of academic papers. Scientometrics, 85(2):553–559, 2010. cite_dist Vincent Lariviere, Veronique Kiermer, Catriona J MacCallum, Marcia McNutt, Mark Patterson, Bernd Pulverer, Sowmya Swaminathan, Stuart Taylor, and Stephen Curry. http://biorxiv.org/content/early/2016/07/05/062109A simple proposal for the publication of journal citation distributions. bioRxiv, page 062109, 2016. RCR Bruce Ian Hutchins, Xin Yuan, James M Anderson, and George M Santangelo. http://www.biorxiv.org/content/early/2015/10/22/029629.abstractRelative Citation Ratio (RCR): A new metric that uses citation rates to measure influence at the article level. bioRxiv, page 029629, 2015. cite_reviewer Brett D Thombs and Ilya Razykov. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3494343/A solution to inappropriate self-citation via peer review. Canadian Medical Association Journal, 184(16):1864–1864, 2012. cite_without_read Mikhail V Simkin and Vwani P Roychowdhury. https://arxiv.org/abs/cond-mat/0212043Read before you cite! arXiv preprint cond-mat/0212043, 2002. bio_preprint Ronald D Vale. http://www.pnas.org/content/112/44/13439.shortAccelerating scientific publication in biology. Proceedings of the National Academy of Sciences, 112(44):13439–13446, 2015. Sanli Single particle charge tracking. Sanli Faez's self-journal, 1, 2015. <http://www.sjscience.org/memberPage?uId=148 jId=10#journal>. funding Johan Bollen, David Crandall, Damion Junk, Ying Ding, and Katy Börner. https://pdfs.semanticscholar.org/ddd8/3eda0b304700b80210228758e266bb44f742.pdfFrom funding agencies to scientific agency. EMBO reports, 2014.

http://arxiv.org/abs/1701.07842v3

20170126190645

DroidStar: Callback Typestates for Android Classes

cs.LO

2018 2018 acmlicensed [ICSE '18]ICSE '18: 40th International Conference on Software Engineering May 27-June 3, 2018Gothenburg, Sweden ICSE '18: ICSE '18: 40th International Conference on Software Engineering , May 27-June 3, 2018, Gothenburg, Sweden 15.00 10.1145/3180155.3180232 978-1-4503-5638-1/18/05 This work was done while Arjun Radhakrishna was employed at the University of Pennsylvania. Microsoft University of Colorado Boulder University of Colorado Boulder University of Colorado Boulder University of Colorado Boulder Max Planck Institute for Software Systems University of Colorado Boulder University of Colorado Boulder Event-driven programming frameworks, such as Android, are based on components with asynchronous interfaces. The protocols for interacting with these components can often be described by finite-state machines we dub callback typestates. Callback typestates are akin to classical typestates, with the difference that their outputs (callbacks) are produced asynchronously. While useful, these specifications are not commonly available, because writing them is difficult and error-prone. Our goal is to make the task of producing callback typestates significantly easier. We present a callback typestate assistant tool, , that requires only limited user interaction to produce a callback typestate. Our approach is based on an active learning algorithm, . We improved the scalability of equivalence queries (a key component of ), thus making active learning tractable on the Android system. We use to learn callback typestates for Android classes both for cases where one is already provided by the documentation, and for cases where the documentation is unclear. The results show that learns callback typestates accurately and efficiently. Moreover, in several cases, the synthesized callback typestates uncovered surprising and undocumented behaviors. : Callback Typestates for Android Classes Pavol Černý December 30, 2023 ========================================= § INTRODUCTION Event-driven programming frameworks interact with client code using callins and callbacks. Callins are framework methods that the client invokes and callbacks are client methods that the framework invokes. The client-framework interaction is often governed by a protocol that can be described by a finite-state machine we call callback typestate. Callback typestates are akin to classical typestates <cit.>, with the key difference that their outputs (callbacks) are produced asynchronously. Our goal is to make the task of producing callback typestates significantly easier for developers. As an example of a callback typestate, consider a typical interaction between a client application and the framework when the client wants to use a particular service. The client asks for the service to be started by invoking an startService callin. After the framework receives the callin, it asynchronously starts initializing the service. When the service is started and ready to be used, the framework notifies the client by invoking a onServiceStarted callback. The client can then use the service. After the client finishes using the service, it invokes a shutdownService callin to ask the framework to stop the service. Callback typestates. Callback typestates are useful in a number of ways, but they are notoriously hard to produce. First, callback typestates are a form of documentation. They tell client application programmers in what order to invoke callins and which callback to expect. Android framework documentation for some classes already uses pictures very similar to callback typestates (Figure <ref>). Second, callback typestates are useful in verification of client code. They enable checking that a client uses the framework correctly. Third, even though we infer the callback typestates from framework code, they can be used for certain forms of framework verification. For instance, one can infer typestates for different versions of the framework, and check if the interface has changed. Callback typestates are very hard to produce manually. On one hand, inspecting code to see in what situation a callback arrives, and what callins are enabled after that is error-prone. Even developers familiar with the framework often miss corner-case behaviors. On the other hand, obtaining the callback typestate with manual testing is hard. One would need to run all sequences of callins, mixed in sequence with the callbacks they produce. We systematize this testing approach using an active learning algorithm. Callback typestate assistant . We present a tool that makes producing callback typestates significantly easier. Our target user is a developer who wrote an Android class that interacts asynchronously using callbacks with client code. is a comprehensive framework for semi-automatically inferring callback typestates. The required user interaction happens in multiple steps. In the first step, the user provides code snippets to perform local tasks, such as code for class initialization and code for invoking each callin (similarly as in unit tests). This is sufficient as long as certain widely applicable assumptions hold. First, we assume that each sequence of callins produces a sequence of callbacks deterministically (this assumption fails when for instance a callback has a parameter that is ignored at first by but that influences the typestate). Second, we assume that the resulting typestate is finite. If these assumptions fail, in the following steps, asks the user for a solution to the problem. For instance, one way to remove non-determinism is to refine one callback into two separate logical callbacks, based on the parameter values. This design allows to offer the user control over the final result while requiring only limited, local, insight from the user. is available for download at Approach. We present a method for inferring typestates for Android classes. However, our method is equally applicable in other contexts. The core algorithm is based on Angluin's algorithm <cit.> adapted to Mealy machines <cit.>. In this algorithm, a learner tries to learn a finite-state machine — in our case a callback typestate — by asking a teacher membership and equivalence queries. Intuitively, a membership query asks for outputs corresponding to a sequence of input callins, and the equivalence query asks if the learned typestate is correct. We note that the teacher does not need to know the solution, but only needs to know how to answer the queries. The key question we answer is how to implement oracles for the membership and equivalence queries. We show how to implement membership queries on Android classes using black-box testing. Our main contribution here is an efficient algorithm for implementing the equivalence query using membership query. The insight here is that the number of membership queries can be bounded by a function of a new bound we call the distinguisher bound. We empirically confirmed that for Android classes, the distinguisher bound is significantly smaller than the state bound used in previous work <cit.>. Given that the number of required membership queries depends exponentially on the distinguisher bound, the novel bound is what enables our tool to scale to Android classes. Results. We use to synthesize callback typestates for 16 Android framework classes and classes from Android libraries. The results show that learns callback typestates accurately and efficiently. This is confirmed by documentation, code inspection, and manual comparison to simple Android applications. The running time of on these benchmarks ranged between 43 seconds and 72 minutes, with only 3 benchmarks taking more than 10 minutes. The usefulness of the distinguisher bound was also confirmed. Concretely, using previously known bounds, learning the callback typestate for one of our examples (MediaPlayer) would take more than a year, whereas with the distinguisher bound, this example takes around 72 minutes. Furthermore, by inspecting our typestates, we uncovered corner cases with surprising behavior that are undocumented and might even be considered as bugs in some cases. For instance, for the commonly used AsyncTask class, if execute is called after cancel but before the onCancelled callback is received, it will not throw an exception but will never cause the asynchronous task to be run. Section <ref> presents our results in more detail. Contributions. The contributions of this paper are: [(a)] * We introduce the notion of callback typestates and develop an approach, based on the algorithm, to infer them. * We show how to implement efficiently membership and equivalence oracles required by the algorithm. * We evaluate our approach on examples from the Android framework, and show its accuracy and effectiveness. § WORKFLOW AND ILLUSTRATIVE EXAMPLE We use the Android Framework's MediaPlayer class to explain the standard workflow for inferring callback typestate using . This class is highly stateful—its interface includes many methods that are only meaningful or enabled in one or two particular player states—and makes extensive use of callins and callbacks to handle the delays of loading and manipulating large media files. These properties make callback typestate a perfect fit; in fact, MediaPlayer has one of the very few examples where we found a complete callback typestate specification in the Android libraries documentation. This callback typestate is shown in Figure <ref>. In Figure <ref>, callins are represented by single arrows and callbacks by double arrows. Let us look at one part of the protocol that governs the client-framework interaction. The client first invokes the callin setDataSource, and the protocol transitions to the Initialized state. In this state, the client can invoke the callin prepareAsync, and the protocol transitions to the Preparing state. In the Preparing state, the client cannot invoke any callins, but the framework can invoke the onPrepared callback, and then the protocol transitions to the Prepared state. At this point, the client can invoke the start callin, and the media starts playing. Our goal is to semi-automatically infer the callback typestate from the figure using the tool . The developer interacts with in several steps, which we describe now. §.§ Developer-Provided Snippets To apply to the MediaPlayer class, the developer provides a number of code snippets detailed below that act as an interface through which the tool can examine MediaPlayer instances. Test object and environment instantiation. The main callback typestate inference algorithm of works roughly by repeatedly performing tests in the form of sequences of method calls on an object of the given class, i.e., the MediaPlayer. Each test must begin with an identical, isolated, class object, and if necessary, a standard environment. In the first step, the developer provides a snippet to initialize such an object and environment. In the case of MediaPlayer, this snippet is as simple as discarding the previous instance, creating a new one with new MediaPlayer(), and registering the necessary callback listeners (explained in the Callback instrumentation paragraph below). In some cases this snippet is more complex. As an example, we cannot create new instances of the BluetoothAdapter class, so for that class this snippet would need to bring the existing instance back to a uniform initial state. Callin declaration. The next step is to declare the alphabet of “input symbols” that represent the callins in the interface of our class—the final callback typestate will be written using these symbols—and map each symbol to the concrete code snippet it represents. In most cases, there is a one-to-one correspondence between input symbols and callin methods. For example, the code snippets associated with the input symbols prepare, prepareAsync, and start are prepare();, prepareAsync();, and start();, respectively. In some cases, such as when a callin takes a parameter, the developer may instead map a symbol to a set of code snippets representing alternative forms of the input which are suspected to have different behavior. In the MediaPlayer class, the setDataSource callin method takes a URL argument. The developer might (rightly) believe that depending on the validity and reachability of the given URL, the behavior of the callin in the typestate may differ. In this case, the developer may provide the two snippets setDataSource(goodURL); and setDataSource(badURL); for the same callin. will consider both snippets for generating tests, and further, it will indicate if they behave differently with respect to the typestate. In case a difference is detected, the “non-determinism” is handled as explained later in this section. The complete set of input symbols which would be declared and mapped for the MediaPlayer class are setDataSource, prepare, prepareAsync, start, stop, reset, release, and pause. Callback instrumentation. As for the callin methods, which act as the input symbols in the callback typestate, the callback methods act as the output symbols in the callback typestate. The developer specifies the set of output callback symbols and associated snippets to detect when callbacks occur. In most cases, this involved adding the listeners for the callbacks in the initialization snippet as mentioned above. In the MediaPlayer class, the output symbols are onCompleted and onPrepared. §.§ Automated Callback-Typestate Inference Once the developer provides the input and output symbols and the associated snippets, attempts to automatically learn the callback typestate following the framework of the algorithm. inference. In , the learner tests sequences of inputs until she can form a consistent hypothesis automaton. Each such test (or sequence of inputs) is called a membership query. Once a hypothesis automaton is produced, an equivalence query is performed; i.e., the hypothesis automaton is checked for equivalence with the true callback typestate. If the two are equivalent, we are done; otherwise, a counter-example test is returned from which the tool learns. This process repeats until the produced hypothesis automaton is correct. For MediaPlayer, the first set of membership queries each consist of a single different callin. Of these, only the query containing setDataSource succeeds. The learner continues with longer membership queries while building the hypothesis automaton. For instance, it learns that prepareAsync and prepare do not lead to the same state: it is possible to invoke the start after prepare, but not after prepareAsync. Once the client receives the callback onPrepared, start may be called. The learner thus hypothesizes a transition from the Preparing to the Prepared on onPrepared. Once the hypothesis is complete, the learner asks the equivalence query. Initially, a counter-example to equivalence is returned using which the learner refines its hypothesis. The final solution is found after 5 equivalence queries. Answering Equivalence Queries. The equivalence query, i.e., checking if a learned callback typestate is in fact the true callback typestate is undecidable in general. However, assuming a bound on the size of the typestate, the equivalence query can be implemented using further testing. However, equivalence queries are still expensive and to make them practical we present an new optimization based on a distinguisher bound. We can observe in Figure <ref> that for any pair of states there is a transition in one state which leads to an error in the other. This corresponds to a distinguisher bound of 1. Small distinguisher bounds arise because typestates are not random automata but part of an API designed for ease of use and robustness. Such APIs are coded defensively and are fail-fast <cit.>, i.e., errors are not buffered but reported immediately. Each state in the typestate has a specific function and an associated set of callins and callbacks. In automata terms, the alphabet is roughly the same size as the number of states and each state has only a few transitions, making any two states easy to distinguish. In Section <ref>, we explain how to use the distinguisher bound to implement equivalence queries and discuss why distinguisher bounds are small in practice. §.§ Obstacles to Inference and Solutions The based callback typestate inference algorithm makes several assumptions about the behavior of the class that do not always hold. is designed to detect these violations of assumptions and notify the developer. Here, we discuss two such assumptions, the exceptional situations that arise when the assumptions are violated, and the additional developer intervention needed to handle such cases. Non-determinism. In input-output automata learning theory, non-determinism makes learning impossible. Non-determinism is the possibility of the same sequence of input callins producing different sequences of output callbacks across tests. Non-determinism may be due to various controllable and non-controllable factors. Controllable factors include cases where behavior depends on if a file exists, if a URL is reachable, etc. On the other hand, non-controllable factors include random number generators, device sensors, etc. In practice, most of the non-determinism was controllable. The main technique for handling non-determinism is via refinement of input or output alphabets. Here, a single callin or callback is split into multiple "logical" inputs or outputs. (a) Controllable non-determinism can be eliminated by incorporating the controlling factor into the inputs. For example, in the SQLiteOpenHelper class, the behavior of the constructor callin changes depending on if a file exists. However, after splitting the callin into two separate callins constructor/fileExists and constructor/noFileExists, the behavior of each of each callin becomes deterministic with respect to these callins. (b) Another source of non-determinism is when the same callback is used to notify logically different events. For example, a class may use a generic onComplete callback which is passed a status parameter that can have the values “Success” and “Failure”. Based on this value, different further callins are enabled, leading to non-determinism. Here, the developer may manually refine the callback into two output symbols onEvent/Success and onEvent/Failure, and the behavior is deterministic with respect to these. In summary, for controllable non-determinism, the onus is on the developer to identify the source of the detected non-determinism and provide a refinement of the input or output alphabet and corresponding code snippets to control the source. No general technique exists to handle non-controllable non-determinism, but specific cases can be handled using techniques shown in Section <ref>. Non-regularity. Another basic assumption that based inference algorithm makes is that the callback typestate under consideration is regular. This assumption is commonly violated in request-response style behavior of classes where the number of responses (output callbacks) invoked is exactly equal to the number of requests (input callins). Our solution to this problem is to restrict the learning to a subset of the class behavior, such as inputs with at most one pending request callin using a learning purpose <cit.>. These restrictions makes the behavior regular and amenable to learning. § THE CALLBACK TYPESTATE LEARNING PROBLEM We introduce formal models of interfaces, define the callback typestate learning problem, and present an impossibility result about learning typestates. Callback typestates have both inputs (corresponding to callins) and outputs (corresponding to callbacks). In automata theory, callback typestates can be seen as interface automata. Interface automata <cit.> are a well-studied model of automata that can produce outputs asynchronously w.r.t. inputs. We use the name callback typestates to emphasize that they are a generalization of typestates as used in the programming languages literature. §.§ Definitions and Problem Statement Asynchronous interfaces. Let and be the set of callins and callbacks of an asynchronous interface. We abstract away parameter and return values of callins and callbacks, and model a behavior of the interface as a trace = _0 …_n ∈ (∪)^*. The interface is given by , , where ⊆{∪}^* is the prefix-closed set of all feasible traces of the interface. In the android media-player example, one example of a trace is setDataSource() ·prepareAsync() ·onPrepared ·start() ·stop(). Interface automata. We use interface automata <cit.> to represent asynchronous interfaces. An interface automaton is given by , , , , _ where: [(a)] * is a finite set of states, * ∈ is the initial state, * and are finite sets of input and output symbols, and * _⊆×{∪}× are a set of transitions. A trace of is given by _0 …_n if ∃_0 …_n+1: _0 = ∧∀ i.( _i, _i, _i+1) ∈_. is the set of all traces of . We say that models the typestate of = , , if each trace of is a trace of , and vice versa. Problem statement. Given an interface = , ,, the callback typestate learning problem is to learn an interface automaton such that =. We allow the learner to ask a membership oracle [] membership queries. For a membership query, the learner picks = _0 _1 …_n ∈^* and the membership oracle [] returns either: [(a)] * a trace ∈ whose sequence of callins is exactly , or * if no such trace exists. §.§ The Theory and Practice of Learning Typestates In general, it is impossible to learn callback typestates using only membership queries; no finite set of membership queries fixes a unique interface automaton. However, callback typestates can be effectively learned given extra assumptions. We now analyze the causes behind the impossibility and highlight the assumptions necessary to overcome it. Unbounded asynchrony. Membership queries alone do not tell us if the interface will emit more outputs (callbacks) at any point in time. Hence, we assume: Assumption 1: Quiescence is observable. This assumption is commonly used in ioco-testing frameworks <cit.>. In our setting, we add an input and an output , where is returned after a only if there are no other pending callbacks. In practice, can be implemented using timeouts, i.e., pending callbacks are assumed to arrive within a fixed amount of time. If no callbacks are seen within the timeout, is output. Using and , in the MediaPlayer example, we have that setDataSource() ·prepareAsync() ·onPrepared ·wait ·quiet is a valid trace, but setDataSource() ·prepareAsync() ·wait ·quiet is not. Behavior unboundedness. For any set of membership queries, let k be the length of the longest query. It is not possible to find out if the interface exhibits different behavior for queries much longer than k. This is a theoretical limitation, but is not a problem in practice <cit.>; most callback typestates are rather small (≤ 10 states). Assumption 2: An upper bound on the size of the typestate being learned is known. Non-determinism. We need to be able to observe the systems' behaviors to learn them and non-determinism can prevent that. Therefore, we assume: Assumption 3: The interface is deterministic. We assume that for every trace of the interface, there is at most one output ∈ such that ·∈. In practice, the non-determinism problem is somewhat alleviated due to the nature of callback typestates (see Section <ref>). See <cit.> for a detailed theoretical discussion of how non-determinism affects learnability. Consider an interface with traces given by (𝚒𝚗𝚙𝚞𝚝· ( 𝚘𝚞𝚝1|𝚘𝚞𝚝2 ))^*. All membership queries are a sequence of input's; however, it is possible that the membership oracle never returns any trace containing out2. In that case, no learner will be able to learn the interface exactly. § LEARNING CALLBACK TYPESTATES USING Given Assumption 1 and Assumption 3, we first build a “synchronous closure” of an asynchronous interface (Section <ref>). Then, we show how to learn the synchronous closure effectively given Assumption 2 (Section <ref> and <ref>). §.§ From Asynchronous to Synchronous Interfaces Using Assumption 1 and 3, we build a synchronous version of an interface in which inputs and outputs strictly alternate following <cit.>. For synchronous interfaces, we can draw learning techniques from existing work <cit.>. Define = ∪{} and = ∪{, , }. The purpose of the extra inputs and outputs is discussed below. For any ∈ (·)^*, we define () = ∈ (∪)^* where is had from by erasing all occurrences of , , , and . Synchronous closures. The synchronous closure of an asynchronous interface = , , is given by , , where and are as above, and ⊆ (·)^* is defined as the smallest set satisfying the following: [ ϵ∈ ; ∈()·∈ ··∈; ∈()·∈ ··∈; ∈()·∉ ··∈; ∈∈()·∉ ··∈; ∈ ends in ··∈ ] Informally, in : [(a)] * Each input is immediately followed by a dummy output ; * Each output is immediately preceded by a wait input ; * Any call to an input disabled in is immediately followed by an . Further, all outputs after an are 's. * Any call to in a quiescent state is followed by . Given [] and Assumption 1, it is easy to construct the membership []. Note that due to Assumption 3, there is exactly one possible reply []() for each query . Further, by the construction of the synchronous closure, the inputs and outputs in []() alternate. Mealy machines. We model synchronous interfaces using the simpler formalism of Mealy machines rather than interface automata. A Mealy machine is a tuple , , , , , where: [(a)] * , , , and are states, initial state, inputs and outputs, respectively, * : ×→ is a transition function, and * : ×→ is an output function. We abuse notation and write (, _0…_n) = _1…_n and (, _0…_n) = ' if ∃_0,…, _n+1 : _0 = ∧_n+1 = ' ∧∀ 0 ≤ i ≤ n : (_i, _i) = _i+1∧(_i, _i) = _i. A sequence _0_0…_n_n ∈ ( ·)^* is a trace of if (, _0…_n) = _0…_n. We often abuse notation and write (_0…_n) instead of (, _0…_n). We denote by the set of all traces of . §.§ : Learning Mealy Machines For the sake of completeness, we describe the classical learning algorithm by Angluin <cit.> as adapted to Mealy machines in <cit.>. A reader familiar with the literature on inference of finite-state machines may safely skip this subsection. Fix an asynchronous interface and its synchronous closure . In , in addition to a membership oracle [], the learner has access to an equivalence oracle []. For an equivalence query, the learner passes a Mealy machine to [], and is in turn returned: [(a)] * A counterexample input = _0…_n such that = _0 …_n and []() ≠_0_0 …_n_n, or * if no such exists. The full algorithm is in Algorithm <ref>. In Algorithm <ref>, the learner maintains: [(a)] * a set ⊆^* of state-representatives (initially set to {ϵ}), * a set ⊆^* of experiments (initially set to ), and * an observation table : (∪·) → (→^*). The observation table maps each prefix and suffix to ()(), where ()() is the suffix of the output sequence of (·) of length ||. The entries are computed by the sub-procedure . Intuitively, represent Myhill-Nerode equivalence classes of the Mealy machine the learner is constructing, and distinguish between the different classes. For to form valid set of Myhill-Nerode classes, each state representative extended with an input, should be equivalent to some state representative. Hence, the algorithm checks if each ·∈· is equivalent to some ' ∈ (line <ref>) under , and if not, adds · to . If no such · exists, the learner constructs a Mealy machine using the Myhill-Nerode equivalence classes, and queries the equivalence oracle (line <ref>). If the equivalence oracle returns a counterexample, the learner adds a suffix of the counterexample to ; otherwise, it returns . For the full description of the choice of suffix, see <cit.>. Let there exist a Mealy machine with n states such that is the set of traces of . Then, given [] and [], Algorithm <ref> returns making at most ||^2 n + || n^2 m membership and n equivalence queries, where m is the maximum length of counterexamples returned by []. If [] returns minimal counterexamples, m ≤ O(n). §.§ An Equivalence Oracle Using Membership Queries Given a black-box interface in practice, it is not feasible to directly implement the equivalence oracle required for the algorithm. Here, we demonstrate a method of implementing an equivalence oracle using the membership oracle using the boundedness assumption (Assumption 2). As before fix an asynchronous interface and its synchronous closure . Further, fix a target minimal Mealy machine ^* such that ^* is the set of traces of . State bounds. A state bound of implies that the target Mealy machine ^* has at most states. Given a state bound, we can replace an equivalence check with a number of membership queries using the following theorem. Let and ' be Mealy machines having k and k' states, respectively, such that ∃ w_i ∈^* : w_i≠'w_i'. Then, there exists an input word w_i' of length at most k + k' - 1 such that w_i'≠'w_i'. The proof is similar to the proof of the bound k + k' - 2 for finite automata (see <cit.>). We can check equivalence of ^* and any given by testing that they have equal outputs on all inputs of length at most k_ + - 1, i.e., using O(||^ + k - 1) membership queries. Fix a state bound of for the target Mealy machine ^*. Given a membership oracle [] and a Mealy machine with k states, the equivalence query can be answered using at most ||^ + k - 1 membership queries. While this simple algorithm is easy to implement, it is inefficient and the number of required membership queries make it infeasible to implement in practice. Other algorithms based on state bounds have a similar problems with efficiency (see Remark in Section <ref>). Further, the algorithm does not take advantage of the structure of . The following discussion and algorithm rectifies these short-comings. Distinguisher bounds. A distinguisher bound of ∈ implies that for each pair of states _1^*, _2^* in the target Mealy machine ^* can be distinguished by an input word w_i of length at most , i.e., ^*(_1^*, w_i) ≠^*(_2^*, w_i). Intuitively, a small distinguisher bound implies that each state is “locally” different, i.e., can be distinguished from others using small length input sequences. The following theorem shows that a state bound implies a comparable distinguisher bound. State bound k implies distinguisher bound k - 1. Small distinguisher bound. In practice, distinguishers are much smaller than the bound implied by the state bound. For the media-player, the number of states is 10, but only distinguishers of length 1 are required. This pattern tends to hold in general due to the following principles of good interface design: * Clear separation of the interface functions. Each state in the interface has a specific function and a specific set of callins and callbacks. There is little reuse of names across state. The typestate's alphabet is roughly the same size as the number of states. * Fail-fast. Incorrect usage of the interface is not silently ignored but reported as soon as possible. This makes it easier to distinguish states as disabled callins lead directly to errors. * No buffering. More than just fail-fast, a good interface is interactive and the effect of callins must be immediately visible rather than hidden. A good interface is not a combination lock that requires many inputs that are silently stored and only acknowledged at the very end. This observation also is not specific to callbacks typestates and it has been already observed for libraries <cit.>. Equivalence algorithm. Algorithm <ref> is an equivalence oracle for Mealy machines using the membership oracle, given a distinguisher bound. First, it computes state representatives : →^*: for each ∈, (, ()) = (line <ref>). Then, for each transition in , the algorithm first checks whether the output symbol is correct (line <ref>). Then, the algorithm checks the “fidelity” of the transition up to the distinguisher bound, i.e., whether the representative of the previous state followed by the transition input, and the representative of the next state can be distinguished using a suffix of length at most . If so, the algorithm returns a counterexample. If no transition shows a different result, the algorithm returns . Two optimizations further reduce the number of membership queries: [(a)] * Quiescence transitions. Transitions with input and output need not be checked at line <ref>; it is a no-op at the interface level. * Error transitions. Similarly, transition with the output need not be checked as any extension of an error trace can only have error outputs. * Representative transitions. If ()· = (') at line <ref>, we once again need not check the transition. Using state representatives from instead of computing them at line <ref> usually leads to a larger number of such cases. Note that if Algorithm <ref> is being called from Algorithm <ref>, the state representatives from can be used instead of recomputing R in line <ref>. Similarly, the counterexample analysis stage can be skipped in the algorithm, and the relevant suffix can be directly returned ( in lines <ref> and <ref>; and in line <ref>). Assuming the distinguisher bound of for the target Mealy machine ^*, either [(a)] * Algorithm <ref> returns and ∀ w_i ∈^*: (w_i) = ^*(w_i), or * Algorithm <ref> returns a counterexample and ≠^*. Further, it performs at most ||·||^ + 1 membership queries. Algorithm <ref> and Theorem <ref> give us improved complexity for state bounds. Assuming a state bound , equivalence checking for a Mealy machine can be implemented using at most ||·||^ membership queries. Note that the problem being addressed here, i.e., testing the equivalence of a given finite-state machine and a system whose behavior can be observed, is equivalent to the conformance testing problem from the model-based testing literature. However, several points make the existing conformance testing algorithms unsuitable in our setting. Popular conformance testing algorithms, like the W-method <cit.> and the W_p-method <cit.>, are based on state bounds and have an unavoidable O(||^) factor in the complexity. In our experiments, the largest typestate had 10 states and 7 inputs. The O(||^) factor leads to an infeasible (i.e., > 10^8) number of membership queries. However, since distinguisher bounds are often much smaller than state bounds, O(||^) membership queries are feasible (i.e., 10^3). The W- and W_p-methods cannot directly use distinguisher bounds. The other common algorithm, the D-method <cit.>, does not apply in our setting either. The D-method is based on building a distinguishing sequence, i.e., an input sequence which produces a different sequence of outputs from every single state in the machine. However, for callback typestates, such single distinguishing sequences do not exist in practice. For similar reasons, conformance testing algorithms such as the UIO-method <cit.> do not apply either. In this light, we believe that Algorithm <ref> is a novel conformance testing algorithm useful in specific settings where resets are inexpensive and systems are designed to have small distinguisher bounds. §.§ Putting It All Together We now present the full callback typestate learning solution. Given a deterministic interface with observable quiescence and the membership oracle []. Assume there exists an interface automaton with n states with distinguisher bound modeling the typestate of . Interface automaton can be learned with O(||· n^3 + n ·||^) membership queries. Proof sketch. Starting with an asynchronous interface and a membership oracle [], using Assumption 1 and Assumption 3 we can construct the membership oracle [] for the synchronous closure of . Given the distinguisher bound (or a state bound using Assumption 2 and Theorem <ref>), we can construct an equivalence oracle [] using Algorithm <ref>. Oracles [] and [] can then be used to learn a Mealy machine with the same set of traces as . This Mealy machine can be converted into the interface automata representing the callback typestate of by: [(a)] * Deleting all transitions with output and all self-loop transitions with output , and * Replacing all transitions with input with the output of the transition. § ACTIVE LEARNING FOR ANDROID We implemented our method in a tool called . In this section we describe how it works, the practical challenges we faced when working with Android, and our solutions to overcome them. is implemented as an Android application and learns callback typestates from within a live Android system. §.§ Designing an Experiment To learn a typestate, a user creates a test configuration (an extension of the LearningPurpose class) providing necessary information about a Java class under study. If known, the distinguisher bound can be provided here directly; otherwise, it can be obtained from Assumption 2 by Theorem <ref>. The instrumented alphabet, also defined here, specifies an abstract alphabet for the learning algorithm and translation between the abstract alphabet and concrete callins/callbacks of the class under study. Several other options are available for adjusting the learning, the most important being the quiescence timeout which determines Assumption 1. §.§ Observing Asynchronous Callbacks In our approach we assume bounded asynchrony (Assumption 1) and, therefore, we can observe when the interface does not produce any new output (quiescence). We enforce this assumption on a real system with timeouts: the membership query algorithm waits for a new output for a fixed amount of time , assuming that quiescence is reached when this time is elapsed. However, Android does not provide any worst case execution time for the asynchronous operations and we rely on the user to choose a large enough . The membership query also assumes the existence of a minimum time before a callback occurs. This ensures that we can issue a membership query with two consecutive callins (so, without a input in between), i.e., we have the time to execute the second callin before the output of the first callin. Consider the MediaPlayer example from Section <ref>. The membership query setDataSource(URL) · ·prepareAsync · may not return the onPrepared if is violated, i.e., if the callback does not arrive before the timeout, and while testing it is possible that the prepareAsync ·start might not return an error as expected if the lower bound is violated. To avoid such issues we try to control the execution environment and parameters to ensure that callbacks occurred between and . In the MediaPlayer case, we must pick the right media source file. §.§ Checking and Enforcing Our Assumptions The simplest experiment to learn a class's callback typestate ties a single input symbol to each of its callins and a single output symbol to each of its callbacks. However, many Android classes have behaviors which cause this simple experiment to fail and require more detailed experiments to succeed. The main challenges when designing an experiment are [(a)] * Non-deterministic behaviors, i.e., the state of the device and external events may influence an application. These elements are inherently non-deterministic; however, non-determinism violates Assumption 3. * The parameter space required to drive concrete test cases to witness a membership query is potentially infinite. Though we have ignored callin parameters till now, they are a crucial issue for testing. * The protocol we are learning may not be a regular language. Note that this is a violation of Assumption 2. Non-Deterministic Behavior. Non-deterministic behavior is disallowed by our Assumption 3. However, to make this assumption reasonable we must make non-determinism straightforward to eliminate when it arises. We explain two primary classes of non-deterministic behaviors and strategies to eliminate these behaviors. The first class is related to controllable inputs and the second to uncontrollable ones (such as inputs from the device sensors). Because the learning algorithm cannot learn from non-deterministic systems, will terminate if such behavior is detected. To assist in this process, will report a non-deterministic behavior is detected and display the disagreeing sequences to the user. It detects this by caching all membership queries as input/output sequence pairs. When a new trace is explored, checks that the trace prefixes are compatible with the previously seen traces. In the first case, a hidden (not modeled) controllable input influences the typestate. We resolve this non-determinism by manually adding the input value and create a finer alphabet that explicate the previously hidden state of the environment. For example, in the class SQLiteOpenHelper, the getReadableDatabase() may either trigger a onCreate callback or not, depending on the parameter value to a previous callin (constructor)was the name of an existing database file. Hence, the behavior of the callin is non-deterministic, depending on the status of the database on disk. In the SQLiteOpenHelper example, we split the constructor callin into constructor/fileExists and constructor/noFileExists and pass the right parameter values in each case. With this extra modeling we can learn the interface automaton, since the execution getReadableDatabase() ends in two different states of the automaton (see Figure <ref>). The second class is the effect of the uncontrollable inputs on a typestate. Such effects, by definition, cannot be controlled or made explicit prior to the call. We can sometimes to remove this non-determinism by merging different outputs, considering them to be the same. This is the dual of the previous solution. An example is the SpeechRecognizer, for which calling startListening produces different callbacks depending on the environment. As the environment cannot be reasonably controlled, we merge outputs to go to the same state. If outputs are erroneously merged, the non-determinism will propagate and continue to manifest. Thus there is no risk of unsound results. Handling Callin Parameters. While parameter-less callins such as start and stop are common in Android classes, many parameterized callins exist. Because input symbols need to be listed in the experiment definition, the full range of parameter values cannot be explored. In practice, we found that parameters often have little effect on the typestate automaton. In cases where they do affect the automaton, multiple input symbols can be defined to represent the same method called with several different parameters. This solution is similar to splitting on environmental effects when dealing with non-determinism. Learning from Non-Regular Languages. An intrinsic limitation of is that it learns only regular languages. However, some classes expose non-regular protocols. Common cases include situations where a request callin invoked n times trigger exactly n response callbacks. In the SpellCheckerSession class, callin getSuggestion and callback onGetSuggestions follow this pattern. However, even in such cases, it can be useful to build a regular approximation of the typestate. For example, restricting the typestate to behaviors where there is at most one pending request (a regular subset) provides all the information a programmer would need. Hence, in such cases, we use the technique of learning purposes <cit.> to learn a regular approximations of the infinite typestate. § EMPIRICAL EVALUATION We evaluated our interface-learning technique, as implemented in , by using it to generate callback typestates for 16 classes, sampled from the Android Framework and popular third-party libraries. is available at . For these experiments, was run on an LG Nexus 5 with Android framework version 23. Our evaluation was designed to answer the following questions: * Does our technique learn typestates efficiently? * What size distinguisher bounds occur in practice? Do they support the small distinguisher bound hypothesis? * Do the callback typestates we learn reveal interesting or unintended behavior in the interfaces? Methodology. For each experimental class, we manually identified a reduced alphabet of relevant callins and callbacks and provided them (along with other necessary information as explained in Section <ref>) to through instances of the LearningPurpose. Relevant callins and callbacks for these experiments were those which, according to the available documentation, appeared to trigger or depend on typestate changes (enabling or disabling of parts of the interface). Each instance consisted of 50-200 lines of, mostly boiler-plate, Java or Scala code. To evaluate efficiency, we measured the overall time taken for learning, as well as the number of membership (MQ) and equivalence queries (EQ). The number of queries is likely a better measure of performance than running time: the running time depends on external factors. For example, in the media player the running time depends on play-length of the media file chosen during testing. We validated the accuracy of learned callback typestates using two approaches. First, for classes whose documentation contains a picture or a description of what effectively is an callback typestate, we compared our result to the documentation. Second, for all other classes we performed manual code inspection and ran test apps to evaluate correctness of the produced typestates. We used a distinguisher bound of 2 for our experiments; further, we manually examined the learned typestate and recorded the actual distinguisher bound. For our third question, i.e., does the learned callback typestate reveal interesting behaviors, we manually examined the learned typestate, compared it against the official Android documentation, and recorded discrepancies. §.§ Results We discuss the results (in Table <ref>) and our three questions. Question 1: Efficiency. The table shows that our technique is reasonably fast: most typestates learned within a few minutes. The longest one takes 71 minutes, still applicable to nightly testing. The numbers for membership queries are reported as X (Y)—X is the number of membership queries asked by the algorithm, while Y is the number actually executed by the membership oracle. This number is lower as the same query may be asked multiple times, but is executed only once and the result is cached. For each benchmark, the accuracy validation showed that the produced typestate matched the actual behavior. Question 2: Distinguisher Bounds. As mentioned before, we used a distinguisher bound of 2 for all experiments. However, a manual examination of the learned callback typestates showed that a bound of 1 would be sufficient in all cases except the SQLiteOpenHelper and the OkHttpCall where bounds of 2 are necessary. This supports our conjecture that, in practice, interfaces are designed with each state having a unique functionality (see Section <ref>). Question 3: Interesting Learned Behavior. Of the three questions, our experiments to examine the learned callback typestate for interesting behavior turned out to be the most fruitful, uncovering several discrepancies, including corner cases, unintended behavior and likely bugs, in the Android framework. These results reaffirm the utility of our main goal of automatically learning callback typestate, and suggest that learning typestate can serve valuable roles in documentation and validation of callback interfaces. In 2 cases, the learned typestate and documented behavior differed in certain corner cases. We carefully examined the differences, by framework source examination and manually writing test applications, and found that the learned typestate was correct and the documentation was faulty. In 5 other cases, we believe the implemented behavior is not the intended behavior, i.e., these are likely bugs in the Android implementation. These discrepancies mostly fall into two separate categories: Incorrect documentation. In such cases, it turned out that the discrepancy is minor and unlikely to produce bugs in client programs. Race conditions. Several likely bugs were due to a specific category of race conditions. These interfaces have [(a)] * a callin to start an action and a corresponding callback which is invoked when the action is successfully completed; * a callin to cancel an already started action and a corresponding callback which is invoked if the action is successfully cancelled. When the start action and cancel action callins are called in sequence, the expectation is that exactly one of the two callbacks are called. However, when the time between the two callins is small, we were able to observe unexpected behaviors, including neither or both callbacks being invoked. §.§ Selected Experiments Of our 16 benchmarks, we briefly explain 5 here. The remaining experiments are discussed in the technical report <cit.>[<http://arxiv.org/abs/1701.07842>]. MediaPlayer This is the class from the example in Section <ref>. The learned typestate differs from the existing documentation. The learned typestate: [(a)] * has the pause callin enabled in the “playback completed” state, and * shows that onPrepared is invoked even after the synchronous callin prepare. Though undocumented, these behaviors are unlikely to cause any issues. AsyncTask The AsyncTask class turns arbitrary computations into callback operations with progress tracking and results are delivered via callbacks. For our experiment, the computation is a simple timer. A constructed AsyncTask object performs its task when it receives the execute callin, and then either returns the results with the onPostExecute callback, or returns an onCancelled if cancel is called first. The object is single-use; after it has returned a callback it will accept no further execute commands. Our experiment revealed an unexpected edge-case: if execute is after cancel but before the onCancelled callback is received, it will not throw an exception but will never cause the callback task to be run. The learned interface is in Figure <ref>. SpeechRecognizer This class provides an example of uncontrollable environmental non-determinism. The particular callback that signals the end of the speech session—either an onResults or an onError—is determined by the environment (in particular, the sound around the phone during the test). In this case, to reduce the system to a deterministic one we can learn, we supposed that the state after an onResults or onError is the same and merged the two callbacks into a single onFinished symbol. Our results revealed two interesting corner cases for the ordering of inputs. First, if an app calls cancel between calling startListening and receiving the onReadyForSpeech callback (represented by our “starting” output symbol), calling startListening again will have no effect until after a certain amount of time, as shown by the wait transition from state “Cancelling” to “Finished”. Delays in readiness like this can be generally considered bugs; if a system will not be ready immediately for inputs it should provide a callback to announce when the preparations are complete, so as not to invite race conditions. Our second corner case is where the app calls stopListening as the very first input on a fresh SpeechRecognizer. This will not throw an exception, but calling startListening at any point after will fail, making the object effectively dead. SQLiteOpenHelper This class provides a more structured interface for apps to open and set up SQLite databases. It has callbacks for different stages of database initialization, allowing apps to perform setup operations only as they are needed. When a database is opened with getWritableDatabase, a callback onConfigure is called, followed by an onCreate if the database didn't exist yet or an onUpgrade if the database had a lower version number than was passed to the SQLiteOpenHelper constructor, all followed finally by an onOpen when the database is ready for reading. The database can then be closed with a close. Our experiment observed the callbacks when opening databases in different states (normal, non-existent, and out of date) and performing the operation at different points in the sequence. We found that once the method is called, calling will not prevent the callbacks from being run. VelocityTracker This class was a special case with no asynchronous behavior; it was a test of our tool's ability to infer traditional, synchronous typestates. The class has a recycle method that we expected to disable the rest of the interface, but our tool found (and manual tests confirmed) that the other methods can still be called after recycling. The documentation's warning that “You must not touch the object after calling [recycle]” is thus not enforced. § RELATED WORK Works which automatically synthesize specifications of the valid sequences of method calls (e.g. <cit.>) typically ignore the asynchronous callbacks. Static analysis has been successfully used to infer typestates specifications (importantly, without callbacks) <cit.>. The work in <cit.> infers classical typestates for Java classes using . In contrast, our approach is based on testing. Therefore, we avoid the practical problem of abstracting the framework code. On the other hand, the use of testing makes our oracles sound only under assumptions. Similarly,  <cit.> uses to infer classical typestates, including ranges of input parameters that affect behavior. However, their tool is based on symbolic execution, and thus would not scale to systems as large and complex as the Android Framework. Inferring interfaces using execution traces of client programs using the framework is another common approach <cit.>. In contrast to dynamic mining, we do not rely on the availability of client applications or a set of execution traces. The algorithm drives the testing. The analysis of event-driven programming frameworks has recently gained a lot of attention (e.g.  <cit.>). However, none of the existing works provide an automatic approach to synthesize interface specifications. Analyses of Android applications mostly focus on either statically proving program correctness or security properties <cit.> or dynamically detecting race conditions <cit.>. These approaches manually hard-code the behavior of the framework to increase the precision of the analysis. The callback typestate specifications that we synthesize can be used here, avoiding the manual specification process. Our work builds on the seminal paper of Angluin <cit.> and the subsequent extensions and optimizations. In particular, we build on for I/O automata <cit.>. The optimizations we use include the counterexample suffix analysis from <cit.> and the optimizations for prefix-closed languages from <cit.>. The relation to conformance testing methods <cit.> has been discussed in Section <ref>. § CONCLUSION We have shown how to use active learning to infer callback typestates. We introduce the notion of distinguisher bound which take advantage of good software engineering practices to make active learning tractable on the Android system. Our method is implemented in the freely available tool called . This paper enables several new research directions. We plan to investigate mining parameters of callins from instrumented trace from real user interactions, as well as the inference of structured typestates (for instance, learning a typestate as a product of simpler typestates). This research was supported in part by DARPA under agreement FA8750-14-2-0263. Damien Zufferey was supported in part by the European Research Council Grant Agreement No. 610150 (ERC Synergy Grant ImPACT (<http://www.impact-erc.eu/>)). plain

http://arxiv.org/abs/1701.07986v1

20170127094611

Optimality of codes with respect to error probability in Gaussian noise

math.MG

alexey_m39@mail.ru r_n_karasev@mail.ru http://www.rkarasev.ru/en/ sasha-cigler@mail.ru ^^^Supported by the Russian Foundation for Basic Research grant 15-31-20403 (mol_a_ved) ^^^Supported by the Russian Foundation for Basic Research grant 15-01-99563 A ^^^Moscow Institute of Physics and Technology, Institutskiy per. 9, Dolgoprudny, Russia 141700 ^^^Institute for Information Transmission Problems RAS, Bolshoy Karetny per. 19, Moscow, Russia 127994 We consider geometrical optimization problems related to optimizing the error probability in the presence of a Gaussian noise. One famous questions in the field is the “weak simplex conjecture”. We discuss possible approaches to it, and state related conjectures about the Gaussian measure, in particular, the conjecture about minimizing of the Gaussian measure of a simplex. We also consider antipodal codes, apply the Šidák inequality and establish some theoretical and some numerical results about their optimality. Optimality of codes with respect to error probability in Gaussian noise Alexander Tsigler^ Jan 15 2017 ======================================================================= § INTRODUCTION Assume a code is represented by a finite set of vectors {v_i} in ℝ^n and the decoding procedure is by taking the (Euclidean distance) closest point of {v_i} (which is optimal subject to likelihood maximization). If we want to calculate the probability that a vector is transmitted correctly in the presence of a normalized Gaussian noise then we obtain a value proportional to P(v_1,…, v_N) = ∫_ℝ^nmax_i e^-|x - v_i|^2 dx. If the probability to choose any of v_i in the code is the same then the actual probability of transmitting the signal correctly is the above value P multiplied by a constant and divided by N. In most if this text the constant in front of the integral is not relevant, but if someone want to interpret the practical meaning of the data given in Section <ref> (where we allow N to vary) then this factor has to be taken into account. Here we normalize the exponent as e^-x^2 and do not use the leading factor (π)^-n/2 to shorten the formulas. Again, in the numerical results of Section <ref> we will use the more common normalization with density (2π)^-n/2 e^-|x|^2/2. It was conjectured that in the case N=n+1 with fixed total energy |v_1|^2+… + |v_N|^2 the maximum of this functional is attained at regular simplices centered at the origin, see <cit.>. In <cit.> it was shown that the regular simplex is optimal for energy tending to infinity and locally optimal for every energy. Eventually, this conjecture turned out to be false; in <cit.> it was shown that for m≥ 7 the configuration with m-2 zero vectors and 2 antipodal vectors is better than the regular simplex with m vertices and the same energy. Now it is conjectured that For N=n+1 and fixed |v_1|=… = |v_N|=r the maximum of P(v_1,…, v_N) is attained at any configuration forming a regular simplex inscribed into the ball of radius r. The case of energy tending to zero or to infinity in this conjecture was considered in <cit.>; its validity for n=3 was established in <cit.>. Our plan is as follows: In Sections <ref> and <ref> we explain how the problems of maximizing P may be reduced to problems of optimal covering of a sphere by caps and to minimizing Gaussian measure of an outscribed simplex. In Theorem <ref> of Section <ref> and in Section <ref> we prove some optimality results for antipodal configurations, inspired by such a configuration in the example of Steiner <cit.>. In Section <ref> we give some numerical results for antipodal configurations. The paper is organized as follows: In Section <ref> we overview the techniques that prove Conjecture <ref> in dimensions ≤ 3. In Section <ref> we provide another approach that would reduce the problem to another conjecture about the Gaussian measure of a (generalized) simplex (Conjecture <ref>) and we show that this approach does give some information for “antipodal” configurations of points, where, for every point x present in the configuration, the point -x is also present, here we recall and use Šidák's lemma about the Gaussian measure. In Section <ref> we study the optimality of antipodal configurations with varying lengths of vectors keeping the total energy, establish the optimality of the equal length configuration for 4-point antipodal configurations, and make numeric tests showing that for larger number of vectors the optimal lengths have more complex behavior. §.§ Acknowledgments. The authors thank Grigori Kabatianski for explaining this problem to us. § SLICING WITH THE UNIFORM MEASURE The first thing that comes to mind is to represent the integral as “the volume under the graph”, that is P(v_1,…, v_N) = {(x, y)∈ℝ^n+1 : ∃ i 0≤ y ≤ e^-|x-v_i|^2}. Then we can fix a value y∈ [0,1] and try to maximize the n-dimensional volume of the corresponding slice of the set in the right hand side of (<ref>). If the maximum of the volume of the section will be obtained at the same configuration for every value of y then the maximum of the total volume P(v_1,…, v_N) will also be there. The corresponding slice is the set ⋃_i { |x - v_i|^2 ≤ -ln y }, that is a union of balls of the same radius. So Conjecture <ref> would follow from the following stronger For N=n+1 and fixed |v_1|=… = |v_N|=r and R>0 the maximal volume of the union of balls ⋃_i B_v_i (R) is attained at any configuration forming a regular simplex inscribed into the ball of radius r. By further slicing with the distance to the origin this conjecture would follow from an even stronger For N=n+1 and R>0 the maximal area of the union of N spherical caps of radius R in the unit sphere 𝕊^n-1 is attained when the centers of the caps form a regular simplex inscribed into the unit sphere. In fact, the case n=3 of the latter conjecture (when the sphere is 2-dimensional) was resolved positively in <cit.>, this was noted in <cit.> and resulted in Conjecture <ref> holds true for n=3 and N=4. Moreover, in <cit.> other regular configurations (corresponding to the vertices of a regular solid body) were proved to maximize the area of the union of equal caps, resulting in optimality of the corresponding spherical codes. This was also noted in <cit.>. After that, in <cit.> two analytical-geometrical lemmas about the caps on a two-dimensional sphere were shown to hold in larger dimensions and it was concluded that Conjecture <ref> was therefore established for arbitrary n. However, the proof of Conjecture <ref> for n=3 in <cit.> does not generalize to larger dimensions because the argument only work for the case when in the presumably optimal configuration the caps only intersect pairwise and no point is covered by three of them. This is not a problem in 𝕊^2, since when three caps intersect in the regular configuration then those caps cover the whole 𝕊^2 and the assertion holds trivially. In the thesis <cit.> we see that this problem in the argument of <cit.> was evident to the experts. § SLICING WITH THE GAUSSIAN MEASURE Here we propose a different approach reducing the problem to estimates for Gaussian measures instead of spherical measures. Let us rewrite the value to optimize differently: P(v_1,…, v_N) = ∫_ℝ^nmax_i e^2 x· v_i - |v_i|^2 e^-|x|^2dx. = μ̅{(x, y)∈ℝ^n+1 : ∃ i 0≤ y ≤ e^2 x· v_i - |v_i|^2}, here μ̅ is the measure with density e^-|x|^2 dx dy. Again, we can fix y now and maximize the measure μ of any section, where μ is the Gaussian measure with density e^-|x|^2dx. The set whose measure is maximized will be a union of halfspaces: C_y(v_1,…, v_N) = ⋃_i { 2 x· v_i - |v_i|^2 ≥ln y}. Taking the complement, we obtain The value P(v_1,…, v_N) is maximized at a given configuration if the Gaussian measure of S_y(v_1,…, v_N) = ⋂_i { 2 x· v_i - |v_i|^2 ≤ln y} is minimized at the same point set (v_1,…, v_N) for any value of y. An advantage of this approach is that the set S_y(v_1,…, v_N) is a (possibly unbounded) convex polyhedron. From the inequality in <cit.> we readily obtain: If we consider sets of 2N points (N≤ n) in ℝ^n of the form {v_i}_i=1^N with prescribed |v_i| =r_i then P(v_1, - v_1, …, v_N, - v_N) is maximized when all the vectors v_i are orthogonal to each other. In this case, the set S_y is an intersection of several symmetric planks P_i = {|(x, v_i)| ≤|v_i|^2 - ln y/2}, and the Gaussian measure of this intersection is minimized when all the stripes are perpendicular. This follows from the Šidák inequality <cit.> μ(P_1∩…∩ P_N) ≥μ(P_1)·…·μ(P_N), which becomes an equality in case when all the planks are perpendicular to each other. This perpendicularity is only possible when N≤ n. We continue the discussion of such antipodal configurations in Section <ref>. Similarly, Conjecture <ref> is therefore reduced to: The Gaussian measure of a simplex S containing a given ball B_0(r) is minimized at the regular simplex with inscribed ball B_0(r). Of course, by slicing and using the result <cit.> about spherical caps we conclude that this conjecture holds true for n=3. In order to make such a reduction we have to establish that unbounded generalized simplices, that is sets determined by n+1 linear equations in ℝ^n, are ruled out. Call a generalized simplex essentially unbounded if it contains an open cone. Equivalently, its outer normals of facets do not contain the origin in their convex hull. In Conjecture <ref> this corresponds to the case when the convex hull of {v_i} does not contain the origin. Let p be the closest to the origin point in this convex hull. Assume that p points to the “north” and let E be the corresponding “equator” of 𝕊^n-1. The point p is a convex combination of some of v_i's, without loss of generality let them be v_1, …, v_k. Note that these v_1,…, v_k are at the same distance from E and if we move them uniformly to E (and keep other v_i's fixed) then the pairwise distances between them increase. Moreover, any distance |v_i - v_j| for i≤ k and j > k also increases, in order to see this it is sufficient to consider the three-dimensional space spanned by v_i, p, v_j and apply the elementary geometry. Now we use the reduction of Conjecture <ref> to Conjecture <ref> and analyze the volume of the union ⋃_i=1^n+1 B_v_i (r) for every radius r>0. The continuous case of the Kneser–Poulsen conjecture established in <cit.> asserts that for every r>0 the volume of such a union does not decrease when we move v_1,…, v_k to the equator. Hence the total value P(v_1,…, v_n+1) does not decrease either. Now observe that at the end the origin will be in the convex hull of v_i's. Call a generalized simplex degenerate if it is not essentially unbounded, but is still unbounded. Every degenerate simplex is a limit (in the topology given by the family of metrics _R(X, Y) = _Haus( X∩ B_0(R), Y∩ B_0(R) ), (R>0) of honest simplices; and it is easy to see that the Gaussian measure of a degenerate simplex will be the limit of the Gaussian measures of those honest simplices. So the inequality would follow, since we do not want it to be strict. After this, one may try to establish Conjecture <ref> by taking the minimal example and studying its structure. There may be some difficulty: this minimal example may turn out to be degenerate. This could be avoided if we manage to prove the stronger version of Conjecture <ref>: Let μ be a radially symmetric measure with monotone decreasing positive density ρ(r). The value μ(S) over all simplices S containing a given ball B_0(r) is minimized at the regular simplex with inscribed ball B_0(r). This conjecture can be attacked by the analysis of the minimizer because of If the integral ∫_0^+∞ρ(r) dr diverges then the minimum in Conjecture <ref> is attained at an honest simplex. Obviously, degenerate simplices have infinite measure in this case. Let S_0 be the regular simplex outscribed about B_0(r). If for any measure μ, satisfying the assumptions of Conjecture <ref>, and any non-degenerate local minimizer (among honest simplices) S of μ(S) under the constraint S⊃ B_0(r) we have μ(S) ≥μ(S_0) then Conjecture <ref> holds. First, the assertion follows from Lemma <ref> if the integral ∫_0^+∞ρ(r) dr diverges. Let us consider the general case. Assume the contrary: suppose that μ(S) is minimized (over S⊃ B_0(r), of course) at a degenerate simplex S. If we have μ(S) < μ(S_0), then we can approximate S by an honest simplex S' and still have μ(S') < μ(S_0). Now we can change the density of μ so that it remains the same around S_0 and S', and the integral ∫_0^+∞ρ(r) dr diverges. But for the modified measure we have already shown that μ(S') ≥μ(S_0). § ANTIPODAL CONFIGURATIONS IN THE PLANE Let us focus on the case when the configuration is antipodal, that is containing the vector -v for every its vector v. Theorem <ref> thus asserts that such a configuration becomes better if we make all the pairs ± v_i in it orthogonal to each other keeping their lengths. But what about the lenghts? Let us analyze how the value P(± v_1, …, ± v_N) behaves when the vectors v_i are kept orthogonal to each other, but their lengths are allowed to vary. It would be nice if the maximization if P(± v_1, …, ± v_N) for fixed |v_1|^2+… + |v_N|^2 happened at equal lengths |v_1|= … = |v_N|; but the example in <cit.> is actually a counterexample to this naive conjecture. There only one pair of ± v_i was given the maximal possible length while all other pairs ± v_i were put to zero. Fortunately, the naive conjecture holds in dimension 2: Under fixed |v_1|^2 + |v_2|^2 the maximum of P(± v_1, ± v_2) is given at equal and orthogonal to each other v_1 and v_2. It will be clear from the proof that the conclusion remains true if consider, instead of the Gaussian measure, any measure with radially symmetric density. Of course, the picture is essentially planar. Let a and b be the lengths of v_1 and v_2 respectively, and let a≤ b without loss of generality. Consider the Voronoi regions of the the four points in the plane. Let us move every Voronoi regions so that the center of it gets to the origin, see Figure <ref>. The numbers in the picture show how many times every area gets covered after the overlap of the moved Voronoi regions. Now all the measures in each of the Voronoi regions become the same Gaussian measure centered at the origin and we count it taking the overlap into account. This turns out to be the measure of the whole plane, plus two centrally symmetric strips of width c = √(a^2 + b^2) each, plus the measure of the hexagon in the picture. Since the width of the strips c does not depend on the choice of a and b, their measure is also constant in fact. Hence, for fixed a^2 + b^2, we maximize P(± v_1, ± v_2) if and only if we maximize the Gaussian measure of the hexagon. Let us look at the hexagon closer: It is obtained from the rhombus, which is the intersection of two strips of width c, by cutting off two corners. Let us give a geometric description of the cutting: Let O be the center of the hexagon and let A, B, C (see Figure <ref>) be its vertices. Since every two points of the configuration are symmetric with respect to the wall between their respective Voronoi regions, the point N, defined as symmetric to O about BC is on the straight line AB. Let M be the base of the perpendicular from O to BC. Since AB and OC are parallel, we obtain ∠ BNM = ∠ MOC. As was mentioned, OM = MN, ∠ BMN = ∠ CMO. Hence the triangles BMN and CMO are equal. Therefore OM is the perpendicular bisector of BC and OB = CO. We conclude that the hexagon is characterized by the following properties: All its vertices are at equal distances from the origin; two antipodal pairs of its sides are at distance c/2 from the origin. In other words, the four sides touch the circle of radius c/2 centered at O at their respective midpoints. Let us fix a direction in the plane and parameterize the hexagon by six parameters: Four angles for the sides that are c/2 from the origin and two shifts along the given direction for the remaining two sides, see Figure <ref>. Let F(φ_1, φ_2, …, x_2) be the Gaussian measure of such a hexagon; the center of the measure is also O. When we change a and b keeping c = √(a^2 + b^2) the hexagon vary. We may assume that the six parameters of the hexagon are all functions of a: F(a) = F(φ_1(a), φ_2(a), …, x_2(a)). Let us find the derivative: F'_a(a) = ∑∂/∂φ_iF(φ_1, φ_2, …, x_2)φ_i'(a) + ∑∂/∂ x_jF(φ_1, φ_2, …, x_2)x_j'(a). If the parameters correspond the hexagon in question (are expressed in a) then ∀ i ∈{1,2,3,4}∀ a ∂/φ_i F(φ_1, φ_2, …, x_2) = 0. When we change the angle φ_i the corresponding side is rotated about the origin keeping in touch with the circle of radius c. Since the vertices of the hexagon are at the same distance from the origin and the Gaussian density is radially symmetric, then the mass center of the side (in this Gaussian density) is in its midpoint, which is the same as the touching with the circle. If the touching point is rotated, say, with angular velocity ω then at start the velocity at a point x of the side segment equals to ω|x| and is directed along Ox rotated by π/2. If we consider two such points symmetric to each other with respect to the midpoint of the side then we see that the densities are the same at those two points and the projections of their velocities onto the normal of the side sum to zero. Since in the linear term the measure changes by the integral over the side segment of the density multiplied by the normal component of the velocity, the total derivative of the measure with respect to the rotation turns out to be zero. Now we see that the partial derivatives of F(φ_1, φ_2, …, x_2) in the angles are zero, and its partial derivatives in x_1 and x_2 are definitely non-negative and positive for a < b. Hence the measure increases when x_1 and x_2 increase. Since x_1(a) = x_2(a) = a we have to increase a until it becomes equal to b (at this moment the picture changes). So a = b is the optimal configuration. § NUMERICAL RESULTS FOR ANTIPODAL CONFIGURATIONS §.§ Formulas for the modified example of Steiner In Steiner's example <cit.> one pair of antipodal vectors had nonzero length while the other pairs had length zero, that is all those vectors were the same at the origin. Let us generalize this as follows: k pairs of vectors have the same length, while all other vectors are in the origin and their set is not empty. Let us write down an explicit formula for the probability in this case. In order to calculate the function P for the configuration we have to take every point in the configuration and its Voronoi region, integrate the Gaussian measure centered at this point over its Voronoi region, and then sum up the results over all the points. The linear hull of our point is k-dimensional and their Voronoi regions in the ambient space are orthogonal products of k-dimensional Voronoi regions by the complementary linear subspace. The Gaussian measures also equal to the products of k-dimensional Gaussian measures by the Gaussian measure of the complement, which is 1. Therefore it is sufficient to work in the k-dimensional linear hull of the points. Now choose the coordinate frame so that our nonzero vectors are ± the basis vectors multiplied by a. Of course, it does not matter how many points of the configuration are put to the origin; so we assume there is one point at the origin. The Voronoi region of the origin is therefore the cube [-a/2, a/2]^k. Other Voronoi regions are the cones on the facets of the cube minus the cube itself. Here we start to use the standard version of the Gaussian measure with density 1/√(2π) e^-x^2/2 per dimension. This is needed to invoke the standard notation Φ(x) = 1/√(2π)∫_0^x e^-t^2/2 dt. So the cube [-a/2, a/2]^k has the Gaussian measure (Φ(a/2) - Φ(-a/2))^n = (2Φ(a/2) - 1)^k. Now consider the Voronoi region which is adjacent to the cube by its facet x_1 = a/2. When we intersect this region by the hyperplane x_1 = b, b ≥a/2, we obtain the cube [-b, b]^k-1in this hyperplane. Now we have to integrate the induced Gaussian measure of this cube from x_1=a to +∞. The induced Gaussian measure has center at the center of the cube and the additional factor 1/√(2π)e^-ρ^2/2, where ρ is the distance from the center of the original Gaussian measure to the hyperplane. Since the center is at the axis point with x_1 = a, the induced measure of the section is 1/√(2π)e^-(b - a)^2/2(2Φ(b) - 1)^k-1. Eventually, the Gaussian measure of the Voronoi region is ∫_a/2^+∞1/√(2π)e^-(b - a)^2/2(2Φ(b) - 1)^k-1 db. And the total value is P(± ae_1,…,± ae_k, 0) = 2k∫_a/2^+∞1/√(2π)e^-(b - a)^2/2(2Φ(b) - 1)^k-1 db + (2Φ(a/2) - 1)^k. This formula is not very nice, but it allows us to make some numerical experiments. §.§ Numerical experiments with the modified example of Steiner We give the table where the function P of the modified example of Steiner is calculated for different values of k (the number of nonzero pairs) and the total energy E in Table <ref>. The graph of P as a function of k and E is given in Figure <ref>. We note again that in practice one may want to vary the number N of the code vectors. First, it makes sense to take N=2k+1 putting precisely one vector to the origin. Then one has to note that the actual amount of the transmitted information multiplied by the probability of correct transmission will be our number P with the factor of log_2 N/N. §.§ Formulas for arbitrary orthogonal antipodal configuration In the more general case we have k pairs of vectors ± v_i, so that the lengths in ith pair are a_i. Again, we work in the linear hull of the configuration and consider the vectors of the configuration as proportional to the basis vectors. Consider the hyperplane of x_k=0 and move it along the kth basis vector. Let the shifted hyperplane by {x_k=t}. In the intersection with this hyperplane, the Voronoi regions of the original points are the weighted Voronoi regions of their projections. The weights are t^2 for the first k-1 pairs, and the last pair is actually represented by one of the points projected to the origin in the hyperplane with weight (a_k - t)^2. When we subtract t^2 from all the weights then 2k-2 points remain without weights, while the last one gets weight a_k^2 - 2a_kt. The latter Voronoi region is a parallelotope: ∏_1^k-1[-a_i^2 + 2a_kt-a_k^2/2a_i, a_i^2 + 2a_kt-a_k^2/2a_i], if a_i^2 + 2a_kt-a_k^2 > 0 for all i≠ k. In particular, in order to this latter Voronoi region to be nonempty, we need t > a_k^2 - min_i≠ ka_i^2/2a_k, t > 0, that is t > a_k^2 - mina_i^2/2a_k. Write down the induced measure of this parallelotope: ∏_1^k-1(2Φ(a_i^2 + 2a_kt-a_k^2/2a_i) - 1), and integrate in t to obtain the Gaussian measure of one of the points in kth pair: ∫_a_k^2 - mina_i^2/2a_k^+∞1/√(2π)e^-(t - a_k)^2/2∏_1^k-1(2Φ(a_i^2 + 2a_kt-a_k^2/2a_i) - 1) dt. The sum of all such Gaussian measures equals: P(± a_1e_1, …, ± a_ke_k) = 2∑_j = 1^k∫_a_j^2 - mina_i^2/2a_j^+∞1/√(2π)e^-(t - a_j)^2/2∏_i ≠ j(2Φ(a_i^2 + 2a_jt-a_j^2/2a_i) - 1) dt. This is the function of (a_1,…,a_k) we want to maximize. §.§ Numerical experiments for arbitrary orthogonal antipodal configuration Let us try to optimize the above function numerically. We are using the standard algorithm of Basin Hopping. The results are given in Tables <ref>, <ref>, <ref>, <ref>. It seems that for arbitrary dimension k there exists a threshold of energy E_0(k) such that for energy E > E_0(k) the optimal configuration of k antipodal pairs is the configuration with all equal lengths of the vectors. abbrv

http://arxiv.org/abs/1701.08180v2

20170127194129

Camera-trap images segmentation using multi-layer robust principal component analysis

cs.CV

Self-Organizing Systems in Planetary Physics : Harmonic Resonances of Planet and Moon Orbits Markus J. Aschwanden^1 December 30, 2023 ========================================================================================================= Camera trapping is a technique to study wildlife using automatic triggered cameras. However, camera trapping collects a lot of false positives (images without animals), which must be segmented before the classification step. This paper presents a Multi-Layer Robust Principal Component Analysis (RPCA) for camera-trap images segmentation. Our Multi-Layer RPCA uses histogram equalization and Gaussian filter as pre-processing, texture and color descriptors as features, and morphological filters with active contour as post-processing. The experiments focus on computing the sparse and low-rank matrices with different amounts of camera-trap images. We tested the Multi-Layer RPCA in our camera-trap database. To our best knowledge, this paper is the first work proposing Multi-Layer RPCA and using it for camera-trap images segmentation. Camera-trap images, Multi-Layer Robust Principal Component Analysis, background subtraction, image segmentation. § INTRODUCTION Studying and monitoring of mammals and birds species can be performed using non-invasive sampling techniques. These techniques allow us to observe animal species for conservation purposes, e.g. to estimate population sizes of endangered species. Camera trapping is a method to digitally capture wildlife images. This method facilitates the register of terrestrial vertebral species, e.g. cryptic species. Consequently, camera traps can generate large volumes of information in short periods of time. Thus, the contributions in camera trapping are important for better species conservation decisions. Camera traps are devices to capture animal images in the wild. These devices consist of a digital camera and a motion detector. They are triggered when the motion sensor detects movement and dependent on the temperature of the source in relation to the environment temperature. Biologist can monitor wildlife with camera traps for detecting rare species, delineating species distributions, monitoring animal behavior, and measuring other biological rates <cit.>. Camera traps generate large volumes of information, for example a camera trapping study can generate until 200000 images, where 1% of the information is valuable <cit.>. As a consequence, biologists have to analyze thousands of photographs in a manual way. Nowadays, software solutions cannot handle the increment of the number of images in camera trapping <cit.>. Accordingly, it is important to develop algorithms to assist the post-processing of camera-trap images. Background subtraction techniques could help to segment animals from camera-trap images. There is a significant body research of background subtraction focused in video surveillance <cit.>. Nevertheless, there are not enough methods that can handle the complexity of natural dynamic scenes <cit.>. Camera-trap images segmentation is important for animal detection and classification. Camera-traps images usually have ripping water, moving shadows, swaying trees and leaves, sun spots, scene changes, among others. Consequently, the models used to segment those types of images should have robust feature extractors. There are some segmentation methods in the literature applied to camera-trap images segmentation. Reddy and Aravind proposed a method to segment tigers on camera-trap images, using texture and color features with active contours <cit.>. They do not make an objective evaluation of their method. Ren et al. developed a method to segment images from dynamic scenes, including camera-trap images; the method uses Bag of Words (BoW), Histogram of Oriented Gradients (HOG), and graph cut energy minimization <cit.>. They do not show the results on camera-trap images. Zhang et al. developed a method to segment animals from video sequences, using camera-trap images; the method uses BoW, HOG, and graph cut energy minimization <cit.>. They obtained 0.8695 of average f-measure on their own camera-trap data set. Robust Principal Component Analysis (RPCA) is a method derived from Principal Component Analysis. RPCA assumes that a data matrix can be decomposed in a low-rank and a sparse matrix. RPCA has newly seen significant activity in many areas of computer sciences, particularly in background subtraction. As a result, there are some algorithms to solve the RPCA problem <cit.>. In this work, we proposed a Multi-Layer RPCA approach in order to segment animals from camera-trap images. Our method combines color and texture descriptors as feature extractor, and solve the RPCA problem with some state-of-the-art algorithms. To our knowledge, this paper is the first work in proposing a Multi-Layer RPCA approach and using it for camera-trap images segmentation. The paper is organized as follows. Section <ref> shows material and methods. Section <ref> describes the experimental framework. Section <ref> presents the experimental results and the discussion. Finally, Section <ref> shows the conclusions. § MATERIALS AND METHODS This section shows a brief explanation of the algorithms and metrics used in this paper. §.§ Robust Principal Component Analysis An image can be decomposed in a low-rank and sparse matrix. Equation <ref> shows the RPCA problem, where M is the data matrix, L_0 is the low-rank matrix, and S_0 is the sparse matrix. The low-rank matrix is the background and the sparse matrix is the foreground in background subtraction. M=L_0+S_0 The RPCA problem can be solved with the convex program Principal Component Pursuit (PCP). This program computes L and S, taking the objective function in the Equation <ref>, where ||L||_* denotes the nuclear norm of the low-rank matrix, ||S||_1 denotes the l_1-norm of the sparse matrix, and λ is a regularizing parameter. There are some algorithms to perform PCP such as Accelerated Proximal Gradient (APG), and Augmented Lagrange multiplier (ALM) <cit.>. minimize ||L||_* + λ ||S||_1 subject to L+S=M §.§ Multi-Layer Robust Principal Component Analysis Equation <ref> shows the data matrix M in our Multi-Layer RPCA method, where β∈ [0,1] is a weight value indicating the contribution of the texture function to the overall data matrix. Function f_t(x,y) denotes the texture descriptor extracted from each image, using the classic Local Binary Pattern (LBP) <cit.>. LBP describes the texture of an image using the neighborhood of each pixel. Function f_c(x,y) denotes the color transformation for each image, converting to gray scale in this case. Our Multi-Layer RPCA computes the L and S matrices from the M matrix in the Equation <ref>. Texture descriptors can work robustly into rich texture regions with light variation. However, they do not work in a efficient way on uniform regions such as water, the sky, and others. Color descriptors could overcome the texture descriptor limitation <cit.>. The Multi-Layer approach proposed in this paper was tested in camera traps for wildlife image segmentation. M(x,y) = β f_t(x,y) + (1-β) f_c(x,y) §.§ Evaluation Metrics The f-measure metric was chosen to evaluate the performance of the Multi-Layer RPCA. Equation <ref> shows the f-measure, where precision and recall are extracted from the confusion matrix. Precision is the proportion of predicted positives that are correctly real positives. In the same way, recall denotes the proportion of the real positives that are correctly predicted <cit.>. The confusion matrix is computed comparing the ground truth (GT) with the automatic segmented images. f-measure = 2precision*recall/precision + recall § EXPERIMENTAL FRAMEWORK This section introduces the database used in this paper, the experiments executed, and the implementation details of our Multi-Layer RPCA. §.§ Database The Alexander von Humboldt Institute realizes samplings with camera traps in different regions of the Colombian forest. We select 25 cameras from 8 regions, where each camera has a relative unalterable environment. Each camera was placed in its site between 1 to 3 months. We extract 30 days and 30 nights of images from those cameras, in daytime color and nighttime infrared formats respectively. The database consists of 1065 GT images from the 30 days and 30 nights. The images have a spatial resolution of 3264x2448 pixels. The length of each day or night data set varies from 9 to 72 images, depending on the animal activity that day or night. Figure <ref> shows an example of the GT images. §.§ Experiments The experiments computed the background models with different conditions and amount of images, observing the robustness of the Multi-Layer RPCA and the influence of pre-processing in the results. All experiments performed our method with β=[0,0.05,0.1,0.15,…,1]. Accordingly, Experiment 1 uses histogram equalization as pre-processing in the color transformed image, Figure <ref> shows the pre-processing for each raw image. The background model is computed with entire days e.g. we take all images of day 1 and solve the RPCA problem in the Experiment 1. Experiment 2 computes the background model with entire nights; Figure <ref> shows the pre-processing for each raw image. Experiment 3 takes entire days and nights e.g. we take all images of day 1 and night 1 to solve the RPCA problem. Experiment 3 uses two pre-processes, daytime images uses the pre-process in Figure <ref> and nighttime images uses the pre-process in Figure <ref>. Experiment 4 takes entire days and nights such as the Experiment 3, but it only uses the pre-processing in Figure <ref> for all images. We tested 9 algorithms to solve the RPCA problem in this paper. Active Subspace RPCA (AS-RPCA) <cit.>; Exact ALM (EALM), Inexact ALM (IALM), Partial APG (APG-PARTIAL) and APG <cit.>; Lagrangian Alternating Direction Method (LSADM) <cit.>; Non-Smooth Augmented Lagrangian v1 (NSA1) and Non-Smooth Augmented Lagrangian v2 (NSA2) <cit.>; Probabilistic Robust Matrix Factorization (PRMF) <cit.>. The foreground was obtained applying a post-process to the sparse matrix. The post-processing was the same for all experiments. This stage includes a hard threshold, morphological filters, and an active contours with a negative contraction bias <cit.>. Figure <ref> shows the post-processing used. Finally, The f-measure was computed comparing each GT with each foreground. The average f-measure was computed as the mean of all f-measures. The results are displayed as a plot of the average f-measure vs β. §.§ Implementation Details The RPCA algorithms were computed using the Sobral et al. library <cit.>. The rest of the source code was developed using the image processing toolbox of Matlab. § RESULTS This section shows the results and discussions of the experiments introduced in the Section <ref>. These results use the metrics explained in the Section <ref>. Figures <ref> and <ref> show the average f-measure vs β of the Experiments 1 and 2 for all RPCA algorithms chosen. Table <ref> shows the summary of the best results for each experiment. APG-PARTIAL was the best algorithm in the Experiments 1 and 2. Daytime images have rich texture regions. In contrast, nighttime images have uniform color. Texture representations are more important on daytime images in the Experiment 1 due to β=0.6. On the contrary, color descriptors are more important on nighttime images in the Experiment 2 due to β=0.3. Those results show the importance of combining the color and texture descriptors. Figure <ref> shows the performances of the RPCA normal algorithms when β=0. Thus, our Multi-Layer RPCA outperforms the RPCA normal methods. Figures <ref> and <ref> show the average f-measure vs β of the Experiments 3 and 4. Table <ref> shows that dividing the pre-processing per daytime or nighttime in the Experiment 3 does not make a big difference in the results, but it increases the fine-tuning parameters. Table <ref> shows that NSA2 was the best algorithm in the Experiments 3 and 4, contrary to the Experiments 1 and 2 where APG-PARTIAL was the best. NSA2 algorithm is a better choice than other RPCA algorithms, if we cannot differentiate between daytime and nighttime images, or if it is difficult to do so. On the other hand, APG-PARTIAL is better, if we have information about the infrared activation. Figure <ref> shows two visual results of the Multi-Layer RPCA. Figure <ref> shows a daytime image without any pre-processing. Figure <ref> shows an original nighttime image. Figures <ref> and <ref> show the sparse matrix after the hard threshold. Figures <ref> and <ref> show the foreground image. These color results are made with the GT images. Yellow-colored regions mean pixels that are on the GT and the automatic segmented images. Red and green regions are visual representations of the under and over segmentation. § CONCLUSIONS We proposed a Multi-Layer RPCA for camera-trap image segmentation, using texture and color descriptors. The proposed algorithm is composed of pre-processing, RPCA algorithm, and post-processing. The pre-processing uses histogram equalization, Gaussian filtering, or a combination of both. The RPCA algorithm computes the sparse and low-rank matrices for background subtraction. The post-processing computes morphological filters and an active contours with a negative contraction bias. We proved the Multi-Layer RPCA algorithm in a camera-trap images database from the Colombian forest. The database was manually segmented to extract the f-measure of each automatic segmented image. We reach 0.7539 and 0.7393 of average f-measure in daytime and nighttime images respectively. The average f-measure was computed with all GT images. To our best knowledge, this paper is the first work in proposing Multi-Layer RPCA and using it for camera-trap images segmentation. Acknowledgment. This work was supported by the Colombian National Fund for Science, Technology and Innovation, Francisco José de Caldas - COLCIENCIAS (Colombia). Project No. 111571451061. ieeetr

http://arxiv.org/abs/1701.08165v2

20170127190003

Dynamical dark energy in light of the latest observations

astro-ph.CO

firstbib naturemag et al. i.e. e.g. etc. Dynamical dark energy in light of the latest observations Gong-Bo Zhao^1,2, Marco Raveri^3,4, Levon Pogosian^5,2, Yuting Wang^1,2, Robert G. Crittenden^2, Will J. Handley^6,7, Will J. Percival^2, Florian Beutler^2, Jonathan Brinkmann^8, Chia-Hsun Chuang^9,10, Antonio J. Cuesta^11,12, Daniel J. Eisenstein^13, Francisco-Shu Kitaura^14,15, Kazuya Koyama^2, Benjamin L'Huillier^16, Robert C. Nichol^2, Matthew M. Pieri^17, Sergio Rodriguez-Torres^9,18,19, Ashley J. Ross^20,2, Graziano Rossi^21, Ariel G. Sánchez^22, Arman Shafieloo^16,23, Jeremy L. Tinker^24, Rita Tojeiro^25, Jose A. Vazquez^26& Hanyu Zhang^1 December 30, 2023 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= A flat Friedman-Roberson-Walker universe dominated by a cosmological constant (Λ) and cold dark matter (CDM) has been the working model preferred by cosmologists since the discovery of cosmic acceleration<cit.>. However, tensions of various degrees of significance are known to be present among existing datasets within the ΛCDM framework<cit.>. In particular, the Lyman-α forest measurement of the Baryon Acoustic Oscillations (BAO) by the Baryon Oscillation Spectroscopic Survey (BOSS)<cit.> prefers a smaller value of the matter density fraction Ω_ M compared to the value preferred by cosmic microwave background (CMB). Also, the recently measured value of the Hubble constant, H_0=73.24±1.74 km s^-1 Mpc^-1<cit.>, is 3.4σ higher than 66.93±0.62 km s^-1 Mpc^-1 inferred from the Planck CMB data<cit.>. In this work, we investigate if these tensions can be interpreted as evidence for a non-constant dynamical dark energy (DE). Using the Kullback-Leibler (KL) divergence<cit.> to quantify the tension between datasets, we find that the tensions are relieved by an evolving DE, with the dynamical DE model preferred at a 3.5σ significance level based on the improvement in the fit alone. While, at present, the Bayesian evidence for the dynamical DE is insufficient to favour it over ΛCDM, we show that, if the current best fit DE happened to be the true model, it would be decisively detected by the upcoming DESI survey<cit.>. The observational datasets considered in this work include the latest CMB temperature and polarisation anisotropy spectra, the supernovae (SNe) luminosity distance data, the BAO angular diameter distance data from the clustering of galaxies (gBAO) and from the Lyman-α forest (LyαFB), the measurement of H_0, H(z) measurements using the relative age of old and passively evolving galaxies (OHD), the three-dimensional galaxy power spectra, and the two-dimensional weak lensing shear angular power spectra. Further details about the datasets and associated systematic effects can be found in Methods. The KL divergence, also known as relative entropy, quantifies the proximity of two probability density functions (PDFs). Rather than focusing on particular model parameters, it is designed to compare the overall concordance of datasets within a given model. We use the difference between the actual and the expected KL divergence, called “Surprise”<cit.>, as a measure of tension between datasets. Rather than comparing the PDFs for the ΛCDM parameters for every pair of datasets, we take the combined dataset, ALL16 (see Supplementary Table 1), and find the derived PDFs for the angular diameter distance D_A(z) and the Hubble parameter H(z) at redshifts corresponding to the available data. We then compute the KL divergence between the derived PDFs and the directly observed D_A(z) and H(z) from H_0, SNe, OHD, gBAO and LyαFB, and evaluate the corresponding Surprise and the standard deviation (see Methods for details). Results are shown with cyan bars in Fig. 1a. They indicate that the H_0, LyαFB and SNe measurements are in tension with the combined dataset. Introducing Tension T as the number of standard deviations by which Surprise is greater than zero, we find values of T=4.4, 3.5, and 1.7 for the H_0, LyαFB and SNe measurements, respectively (shown in Fig. 1b), with the first two values signalling significant tension. Next, we check if the tension within the ΛCDM model can be interpreted as evidence for a dynamical DE. The dynamics of DE can be probed in terms of its equation of state w, which is equal to -1 for Λ, but is different in dynamical DE models where it will generally be a function of redshift z. Commonly considered alternatives to Λ are a model with a constant w (wCDM), and one in which w is linear function of the scale factor (w_0w_aCDM)<cit.>. We allow for a general evolution of the DE equation of state and use the correlated prior method<cit.> to perform a Bayesian non-parametric reconstruction of w(z)using the Monte Carlo Markov Chain method with other cosmological parameters marginalised over (see Methods for details). Fig. 2 presents the reconstructed w(z), along with the 68% confidence level (CL) uncertainty, shown with a light blue band, derived from the combined dataset ALL16. Table 1a shows the change in χ^2 relative to ΛCDM for each individual dataset for the best fit w(z)CDM model derived from ALL16. Overall, the χ^2 is improved by -12.3, which can be interpreted as the reconstructed dynamical DE model being preferred at 3.5σ. The reconstructed DE equation of state evolves with time and crosses the -1 boundary, which is prohibited in single field minimally coupled quintessence models<cit.>, but can be realised in models with multiple scalar fields, such as Quintom<cit.>, or if the DE field mediates a new force between matter particles<cit.>. In the latter case, which is commonly classified as Modified Gravity, it is quite generic for the effective DE equation of state to be close to -1 around z=0, but evolve towards more negative values at intermediate redshifts, before eventually approaching 0 during matter domination. Such dynamics would be consistent with our reconstruction and could be tested in the future when BAO measurements at higher redshifts become available. In addition to the reconstruction from the combined ALL16 dataset presented in Fig. 2, we present reconstructions derived from ten different data combinations in Supplementary Fig. 1. The results for tension between datasets, re-evaluated for the ALL 16 best fit w(z)CDM model, are shown with dark blue bars in Fig. 1. We find T=0.7, 1.1 and 0.7 for H_0, LyαFB and SNe, respectively, indicating that tensions that existed in the ΛCDM model are significantly released within w(z)CDM. A plot of the relevant data points along with the best fit predictions from the ΛCDM and the w(z)CDM model are provided in the Supplementary Fig. 2. With a large number of additional w-bin parameters, one may be concerned that the improvement in the fit is achieved by w(z)CDM at the cost of a huge increase of the parameter space. However, correlations between the w-bins induced by the prior constrain most of that freedom. One way to estimate the effective number of additional degrees of freedom is to perform a principal component analysis (PCA) of the posterior covariance matrix of the w-bin parameters and compare it to that of the prior. Using this method, explained in detail in Methods, we find that our w(z)CDM model effectively has only four additional degrees of freedom compared to ΛCDM. We note that the demonstration that ALL16 is capable of constraining four principal components of w(z) is one of the interesting results of this work. It is interesting to compare w(z) reconstructed from ALL16 to that obtained in Zhao  (2012)<cit.> using the same prior but a different dataset which we call ALL12 (a comparison of the ALL16 and the ALL12 datasets is provided in Supplementary Table 2). ALL16 contains about 40%new supernovae compared to ALL12, primarily provided by the SDSS-II survey. Moreover, in ALL12, the BAO measurement derived from the BOSS DR9 sample<cit.> was at a single effective redshift, while in ALL16 it is tomographic at nine redshifts from BOSS DR12<cit.>, which contains four times more galaxies than DR9. In addition, ALL16 includes a high-redshift BAO measurement from Lyman-α forest, which was not available in 2012. This helps to constrain w(z) at redshifts where the supernovae constraints are weak. The new 2016 H_0 measurement<cit.> is consistent with that in 2009<cit.>, with the error bar halved. Comparing measurements of the expansion rate and the cosmic distances in ALL12 and ALL16, we find that those in ALL16 offer information at more redshift values, and with a greater signal-to-noise (S/N) ratio (see Supplementary Figure 4 for a visual comparison). Quantitatively, the S/N in measurements of H(z), D_A(z) and d_L(z) in ALL16 is larger by 80%, 260% and 90%, respectively, compared to the ALL12 dataset. The Planck 2015 CMB data is also much more informative than the WMAP 7-year release<cit.>, thanks to a higher angular resolution of the temperature and polarisation maps, and lower levels of statistical uncertainties. Overall, ALL16 is more constraining due to a significant level of new and independent information in ALL16 compared to ALL12: the effective number of w(z) degrees of freedom constrained by ALL12 was three, compared to four constrained by ALL16. Fig. 2 compares the two results and shows that they are highly consistent. We quantify the agreement by evaluating the dot-product of the ŵ vectors from the two reconstructions (the vectors are normalised so that a dot-product is unity if they are identical) and find that ŵ_ ALL12·ŵ_ ALL16=0.94±0.02. We also evaluate the tension T between the two reconstruction results and find that T=-1.1. This indicates an excellent alignment of the two results. This agreement, and the raised significance of an evolving w(z) from 2.5σ to 3.5σ CL with more advanced observations, suggests the possibility of revealing the dynamics of DE at a much more statistically significant level in the near future, as we will present later. To check whether the improvement in the fit warrants introducing additional effective degrees of freedom, we evaluate and compare the Bayesian evidence, E ≡∫ d θ L( D|θ) P(θ), for ΛCDM and the w(z)CDM model. The Bayes' factors, which are the differences in ln E between the two models, are shown in Table 1. The Bayes' factors for both the ALL12 and the ALL16 DE models are negative, indicating that ΛCDM is favoured by this criterion. However, our forecast for a future dataset, DESI++, comprised of BAO measurements from DESI<cit.>, around 4000 supernovae luminosity distances from future surveys<cit.> and CMB (assuming the Planck sensitivity), predicts a Bayes' factor of 11.3 ± 0.3 if the ALL16 w(z)CDM reconstruction happened to be the true model, which would be decisive according to the Jeffreys scale. One may ask how much the evidence for DE depends on the particular choice of the prior parameters. In principle, the choice of the smoothing scale should be guided by theory. The value used in Zhao  (2012)<cit.> and this work, a_c=0.06, is a time-scale conservatively chosen to be sufficiently small not to bias reconstructions of w(z) expected in quintessence DE models<cit.>.For the inference to be conclusive, the evidence for a dynamical DE should be strong over a wide range of the prior parameters. Therefore, we vary the strength of our prior by adjusting σ_ D, a parameter added to the diagonal of the inverse of the prior covariance matrix, and examine how the significance of the dynamical DE detection, as well as the Bayesian evidence, change with the variation of σ_ D. As shown in Fig. 3, and with additional details given in Methods, we find that neither ALL12 nor ALL16 provide evidence for a dynamical DE over the considered wide range of prior strengths. However, the Bayes factor for ALL16 is generally much less negative than that of ALL12 for all prior strengths, , it increased from -6.7±0.3 to -3.3±0.3 for σ_ D=3, which is the prior used in this work. In fact, for ALL16, the Bayes factor remains close to zero for σ_ D≲0.4. We plot the model with σ_ D=0.4 as a light green band in Fig. 2 to demonstrate the impact of changing the prior strength, and also because it is a model that has the same Bayesian evidence as ΛCDM while deviating from Λ at a 2.7σ CL. On the other hand, our forecast for DESI++ shows that, if the w(z)CDM model was true, it would be decisively supported by Bayesian evidence over a wide range of prior strengths, as shown in Fig. 3.Various ways to relieve the tension between datasets have been proposed, including allowing for additional relativistic degrees of freedom<cit.>, massive neutrinos<cit.>, and interacting vacuum<cit.>. In addition, to relieve the tension between the ΛCDM parameters required to fit the CMB temperature anisotropy spectrum at large and small scales, the Planck team introduced<cit.> a parameter A_ Lens that rescales the amplitude of the weak lensing contribution to the temperature power spectrum. In the w(z) reconstruction discussed earlier, we fixed A_ Lens=1, assumed that neutrinos are massless and set the effective number of relativistic species at the standard ΛCDM value of N_ eff=3.04. We have checked the effect of these parameters on the reconstructed w(z) by considering model M_1, with M_ν fixed to 0.06 eV, model M_2 , with M_ν and N_ eff added as free parameters, and model M_3, with varied A_ Lens. In all these cases, we find that the shape of the reconstructed w(z) and the significance of its deviation from -1 are practically the same. The inferred values of the cosmological parameters in these models are given in Supplementary Table 3, and the corresponding reconstructed w(z) are shown in Supplementary Fig. 5. The Bayes factors for M_1, M_2 and M_3 relative to the corresponding ΛCDM models (with the same added parameters), are shown in Table 1c. We also checked that neither the constant w model (M_4), nor the linear (w_0,w_a) parametrisation of w (M_5), are capable of releasing the tensions between all datasets simultaneously (see Table 1a). Interestingly, we find that our DE model with a non-parametrically reconstructed w(z) has a larger Bayes factor compared to w_0w_aCDM despite the latter having only two parameters (see results for M_4 and M_5 in Table 1c).There is always a possibility that the tensions between datasets, quantified in terms of the KL divergence in this work, are due to yet unknown systematic effects. However, it is intriguing that they persist with improvements in the quantity and the quality of the data<cit.>. If interpreted as a manifestation of DE dynamics, they suggest a w(z) that crosses -1 and has a shape that is representative of modified gravity models. The commonly used (w_0,w_a) parametrisation would have missed this behaviour and has a lower Bayesian evidence than the reconstructed w(z) model, despite the latter having more degrees of freedom. Thus, our results demonstrate that the current data can provide non-trivial constraints on the DE dynamics. It is also intriguing that the evidence for w(z) -1, while below that of ΛCDM, has become stronger with the new independent data added since 2012, and that the ALL16 reconstruction is consistent with the ALL12 w(z). We emphasise that we have not optimised the prior to maximise either the Bayes ratio or the statistical significance of the departure from -1, as it would be contrary to the principles of Bayesian inference. Future data has the ability to conclusively confirm the DE evolution inferred in this work if it happened to be the one chosen by Nature.30 url<#>1 urlprefixURL Riess98authorRiess, A. G.et al.titleObservational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant. journalAstron. J.volume116, pages1009–1038 (year1998). Perl99authorPerlmutter, S.et al.titleMeasurements of Ω and Λ from 42 High-Redshift Supernovae. journalAstrophys. J.volume517, pages565–586 (year1999). Delubac14authorDelubac, T.et al.titleBaryon acoustic oscillations in the Lyα forest of BOSS DR11 quasars. journalAstron. & Astrophys.volume574, pagesA59 (year2015). Sahni:2014ooaauthorSahni, V., authorShafieloo, A.&authorStarobinsky, A. A.titleModel-independent Evidence for Dark Energy Evolution from Baryon Acoustic Oscillations. journalAstrophys. J. Lett.volume793, pagesL40 (year2014). Battye14authorBattye, R. A., authorCharnock, T.&authorMoss, A.titleTension between the power spectrum of density perturbations measured on large and small scales. journalPhys. Rev. Dvolume91, pages103508 (year2015). Aubourg14authorAubourg, É.et al.titleCosmological implications of baryon acoustic oscillation measurements. journalPhys. Rev. Dvolume92, pages123516 (year2015). P15authorPlanck Collaborationet al.titlePlanck 2015 results. XIII. Cosmological parameters. journalAstron. & Astrophys.volume594, pagesA13 (year2016). Raveri15authorRaveri, M.titleAre cosmological data sets consistent with each other within the Λ cold dark matter model?journalPhys. Rev. Dvolume93, pages043522 (year2016). Addison15authorAddison, G. E.et al.titleQuantifying Discordance in the 2015 Planck CMB Spectrum. journalAstrophys. J.volume818, pages132 (year2016). Bernal16authorBernal, J. L., authorVerde, L.&authorRiess, A. G.titleThe trouble with H_0. journalJ. Cosmol. Astropart. Phys.volume10, pages019 (year2016). Freedman17authorFreedman, W. L.titleCosmology at a crossroads. journalNat. Astron.volume1, pages0121 (year2017). Riess16authorRiess, A. G.et al.titleA 2.4% Determination of the Local Value of the Hubble Constant. journalAstrophys. J.volume826, pages56 (year2016). Kullback:1951vaauthorKullback, S.&authorLeibler, R. A.titleOn Information and Sufficiency. journalAnn. Math. Stat.volume22, pages79 (year1951). DESIauthorDESI Collaborationet al.titleThe DESI Experiment Part I: Science,Targeting, and Survey Design. journalArXiv e-prints (year2016). 1611.00036. Seehars:2014oraauthorSeehars, S., authorAmara, A., authorRefregier, A., authorParanjape, A.&authorAkeret, J.titleInformation gains from cosmic microwave background experiments. journalPhys. Rev. Dvolume90, pages023533 (year2014). CPLauthorLinder, E. V.titleExploring the Expansion History of the Universe. journalPhys. Rev. Lett.volume90, pages091301 (year2003). Crittenden11authorCrittenden, R. G., authorZhao, G.-B., authorPogosian, L., authorSamushia, L.&authorZhang, X.titleFables of reconstruction: controlling bias in the dark energy equation of state. journalJ. Cosmol. Astropart. Phys.volume2, pages048 (year2012). Vikman04authorVikman, A.titleCan dark energy evolve to the phantom?journalPhys. Rev. Dvolume71, pages023515 (year2005). quintomauthorFeng, B., authorWang, X.&authorZhang, X.titleDark energy constraints from the cosmic age and supernova. journalPhys. Lett. Bvolume607, pages35–41 (year2005). Das05authorDas, S., authorCorasaniti, P. S.&authorKhoury, J.titleSuperacceleration as the signature of a dark sector interaction. journalPhys. Rev. Dvolume73, pages083509 (year2006). Zhao12authorZhao, G.-B., authorCrittenden, R. G., authorPogosian, L.&authorZhang, X.titleExamining the Evidence for Dynamical Dark Energy. journalPhys. Rev. Lett.volume109, pages171301 (year2012). dr9authorReid, B. A.et al.titleThe clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: measurements of the growth of structure and expansion rate at z = 0.57 from anisotropic clustering. journalMon. Not. R. Astron. Soc.volume426, pages2719–2737 (year2012). Zhao17authorZhao, G.-B.et al.titleThe clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: tomographic BAO analysis of DR12 combined sample in Fourier space. journalMon. Not. R. Astron. Soc.volume466, pages762–779 (year2017). Riess09authorRiess, A. G.et al.titleA Redetermination of the Hubble Constant with the Hubble Space Telescope from a Differential Distance Ladder. journalAstrophys. J.volume699, pages539–563 (year2009). wmap7authorLarson, D.et al.titleSeven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Power Spectra and WMAP-derived Parameters. journalAstrophys. J. Suppl.volume192, pages16 (year2011). EuclidSNauthorAstier, P., authorGuy, J., authorPain, R.&authorBalland, C.titleDark energy constraints from a space-based supernova survey. journalAstron. & Astrophys.volume525, pagesA7 (year2011). DiValentino16authorDi Valentino, E., authorMelchiorri, A.&authorSilk, J.titleReconciling Planck with the local value of H_0 in extended parameter space. journalPhys. Lett. Bvolume761, pages242–246 (year2016). Sola16authorSolà, J., authorGómez-Valent, A.&authorde Cruz Pérez, J.titleFirst Evidence of Running Cosmic Vacuum: Challenging the Concordance Model. journalAstrophys. J.volume836, pages43 (year2017). Wang15authorWang, Y., authorZhao, G.-B., authorWands, D., authorPogosian, L.&authorCrittenden, R. G.titleReconstruction of the dark matter-vacuum energy interaction. journalPhys. Rev. Dvolume92, pages103005 (year2015). DR12LyAauthorBautista, J. E.et al.titleMeasurement of BAO correlations at z=2.3 with SDSS DR12 Lyα-Forests. journalArXiv e-prints (year2017). 1702.00176. Correspondence Correspondence and requests for materials should be addressed to G.B.Zhao (email: gbzhao@nao.cas.cn). Acknowledgements G.B.Z. is supported by NSFC Grant No. 11673025, and by a Royal Society-Newton Advanced Fellowship. G.B.Z. and Y.W. are supported by National Astronomical Observatories, Chinese Academy of Sciences and by University of Portsmouth. M.R. is supported by U.S. Dept. of Energy contract DE-FG02-13ER41958. M.R. acknowledges partial support, during the development of this work, by the Italian Space Agency through the ASI contracts Euclid-IC (I/031/10/0) and the INFN-INDARK initiative. M.R. thanks SISSA where part of this work was completed. L.P. is supported by NSERC, RC by STFC grant ST/H002774/1, and Y.W. by NSFC grant No. 11403034. G.R. acknowledges support from the National Research Foundation of Korea (NRF) through NRF-SGER 2014055950 funded by the Korean Ministry of Education, Science and Technology (MoEST), and from the faculty research fund of Sejong University in 2016. A.S. would like to acknowledge the support of the National Research Foundation of Korea (NRF - 2016R1C1B2016478). 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, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrosica 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. Author contributions G.B.Z. proposed the idea, performed the dark energy reconstruction, evidence calculation, principal component analysis and the tension calculation. M.R. and Y.W. contributed to the tension calculation. G.B.Z. and L.P. wrote the draft, and all other co-authors commented on and helped improving the manuscript, and/or contributed to the BOSS data analysis. Author Information * National Astronomy Observatories, Chinese Academy of Science, Beijing, 100012, P.R.China * Institute of Cosmology and Gravitation, University of Portsmouth, Portsmouth, PO1 3FX, UK * Kavli Institute for Cosmological Physics, Enrico Fermi Institute, The University of Chicago, Chicago, Illinois 60637, USA * Institute Lorentz, Leiden University, PO Box 9506, Leiden 2300 RA, The Netherlands * Department of Physics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada * Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK * Kavli Institute for Cosmology, Madingley Road, Cambridge, CB3 0HA, UK * Apache Point Observatory, P.O. Box 59, Sunspot, NM 88349, USA * Instituto de Física Teórica, (UAM/CSIC), Universidad Autónoma de Madrid, Cantoblanco, E-28049 Madrid, Spain * Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany * Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (IEEC- UB), Martí i Franquès 1, E-08028 Barcelona, Spain * Departamento de Física, Universidad de Córdoba, Campus de Rabanales, Edificio Albert Einstein, E-14071 Córdoba, Spain * Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA * Instituto de Astrof isica de Canarias, 38205 San Crist obal de La Laguna, Santa Cruz de Tenerife, Spain * Departamento de Astrof i sica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain * Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055, Korea * Aix Marseille Univ, CNRS, LAM, Laboratoire d'Astrophysique de Marseille, Marseille, France * Campus of International Excellence UAM+CSIC, Cantoblanco, E-28049 Madrid, Spain * Departamento de Física Teórica, Universidad Autónoma de Madrid, Cantoblanco, E-28049, Madrid, Spain * Center for Cosmology and AstroParticle Physics, The Ohio State University, Columbus, OH 43210, USA * Department of Astronomy and Space Science, Sejong University, Seoul 143-747, Korea * Max-Planck-Institut für extraterrestrische Physik, Postfach 1312, Giessenbachstr., 85741 Garching, Germany * University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Korea * Center for Cosmology and Particle Physics, Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA * School of Physics and Astronomy, University of St Andrews, North Haugh, St Andrews, KY16 9SS, UK * Brookhaven National Laboratory, Bldg 510, Upton, New York 11973, USA Competing Interests The authors declare that they have no competing financial interests. Figure 1. The tension among different datasets in ΛCDM and w(z)CDM universes. Panel a: The Surprise between the PDFs for D_A(z) and H(z) derived from the best fit model using the combined dataset of ALL16, and the directly observed D_A(z) and H(z) from H_0, JLA (the JLA sample of SNe), OHD, gBAO-9z (gBAO measurements at nine effective redshifts) and LyαFB respectively (see Methods for detailed explanation and references for data used). The cyan horizontal bars indicate the 68% confidence level (CL) range of Surprise in ΛCDM, while the dark blue bars correspond to w(z)CDM; Panel b: The corresponding values of Tension T, defined as Surprise divided by its standard deviation, shown using the same colour scheme as in Panel a. Figure 2. The reconstructed evolution history of the dark energy equation of state compared with the 2012 result and the forecasted uncertainty from future data. The mean (white solid) and the 68% confidence level (CL) uncertainty (light blue band) of the w(z) reconstructed from ALL16 compared to the ALL12 w(z) reconstructed in Zhao  (2012)<cit.> (red lines showing the mean and the 68% CL band). The red point with 68% CL error bars is the value of w(z) at z=2“predicted” by the ALL12 reconstruction. The dark blue band around the ALL16 reconstruction is the forecasted 68% CL uncertainty from DESI++. The green dashed curve and the light green band show the mean and the 68% CL of w(z) reconstructed from ALL16 using a different prior strength (σ_D=0.4) for which the Bayesian evidence is equal to that of ΛCDM. See the text for details. Figure 3. The Bayes factor and the significance level of w-1 for various correlated priors for current and future data. The Bayes factor with 68% confidence level error bars (upper panel) and the statistical significance (lower) of dynamical DE derived from the 2012 data (ALL12; red dashed)<cit.>, current data (ALL16; black solid) and future data (DESI++; blue dot-dashed)<cit.> respectively. §.§ Tension calculation The Kullback-Leibler (KL) divergence<cit.>, also known as relative entropy, has been extensively utilised as a way of quantifying the degree of tension between different datasets within the ΛCDM model<cit.>. Rather than focusing on particular model parameters, it is designed to compare the overall concordance of datasets within a given model. Alternative methods of quantification of the tension have been discussed in the literature<cit.>. The KL divergence quantifies the proximity of two probability density functions (PDFs), P_1 and P_2, of a multi-dimensional random variable θ. If both P_1 and P_2 are assumed to be Gaussian<cit.>, and data are assumed to be more informative than the priors, we can write the difference between the actual and the expected KL divergence, called the “Surprise”<cit.>, as S=1/2 ln2[(θ_1 - θ_2)^T C_1^-1(θ_1 - θ_2) - Tr( C_2 C_1^-1+𝕀)] where θ_1 and θ_2 are the best-fit parameter vectors, C_1 and C_2 are the covariance matrices for P_1 and P_2, and 𝕀 is the unity matrix. The standard deviation of the expected KL divergence is Σ=1/√(2) ln2√( Tr( C_2 C_1^-1+𝕀)^2) . We can quantify the tension between P_1 and P_2 in terms of the signal-to-noise ratio T =S/Σ. If T≲1, then P_1 and P_2 are consistent with each other<cit.>. §.§ Datasets used The datasets we consider include the Planck 2015 (P15) CMB temperature and polarization auto- and cross-angular power spectra<cit.>, the JLA supernovae<cit.> (JLA); the 6dFRS (6dF)<cit.> and SDSS main galaxy sample (MGS)<cit.> BAO measurements, the WiggleZ galaxy power spectra<cit.> in four redshift slices, containing information about the Baryon Acoustic Oscillations (BAO) and Redshift Space Distortions (RSD) (P(k)), the weak lensing shear angular power spectra in six redshift slices from CFHTLenS<cit.> (WL), the recent estimate of the Hubble constant H_0 obtained from local measurements of Cepheids<cit.> (H_0), the H(z) measurement using the relative age of old and passively evolving galaxies following a cosmic chronometer approach<cit.> (OHD), the BOSS DR12 “Consensus" BAO measurement (BAO-3z)<cit.>, the BAO and RSD measurement using the complete BOSS DR12 sample covering the redshift range of 0.2<z<0.75 at three effective redshifts, the BAO measurement using the same galaxy sample but at nine effective redshifts<cit.> (BAO-9z), and the Lyα BAO (LyαFB) measurements<cit.>. A summary of datasets and data combinations used in this work is shown in Supplementary Table 1. We account for the systematic effects in our analysis as implemented in the public likelihood codes. However, we note that there may be additional systematic effects. For example, the relative velocity between baryons and dark matter may affect the BAO distance measurements<cit.>. This effect is estimated to be at sub-percent level for the galaxy BAO measurements of BOSS<cit.>, and is currently unknown for LyαFB. For SNe, we use the conventional χ^2 statistics for the analysis, although alternative statistics may extract more information and reduce the systematic effects for the JLA sample to some extent<cit.>. §.§ Non-parametric w(z) reconstruction To start, w(z) is parameterised in terms of its values at discrete steps in z, or the scale factor a. Fitting a large number of uncorrelated bins would result in extremely large uncertainties and, in fact, would prevent the Monte Carlo Markov Chains (MCMC) from converging because of the many degenerate directions in parameter space. On the other hand, fitting only a few bins could significantly bias the result. Our approach is to introduce a prior covariance between the bins based on a specified two-point function that correlates values of w at different a, ξ_w (|a - a'|) ≡⟨ [w(a) - w^ fid(a)][w(a') - w^ fid(a')] ⟩, which can be taken to be of the form proposed in<cit.>, ξ_ CPZ(δ a) = ξ_w (0) /[1 + (δ a/a_c)^2], where a_c describes the typical smoothing scale, and ξ_w(0) is a normalisation factor set by the expected variance in the mean w, σ^2_w̅. As shown in<cit.>, results are largely independent of the choice of the correlation function. The prior covariance matrix C is obtained by projecting ξ_w (|a - a'|) onto the discrete w bins<cit.>, and the prior PDF is taken to be of Gaussian form: P_ prior( w) ∝exp[-( w- w^ fid)^T C^-1( w- w^ fid)/2], where w^ fid is the fiducial model. The reconstructed model is that which maximises the posterior probability, which by Bayes' theorem is proportional to the likelihood of the data times the prior probability, P( w| D) ∝ P( D| w) × P_ prior( w). Effectively, the prior results in a new contribution to the total χ^2 of a model, which penalises models that are less smooth. In our reconstruction of w(z), we set a_c=0.06 and σ_w̅=0.04, which is the “weak prior" used in Zhao  (2012)<cit.>. To calculate the observables, we use a version of CAMB<cit.> modified to include DE perturbations for an arbitrary w<cit.>. We use PolyChord<cit.>, a nested sampling plug-in for CosmoMC<cit.>, to sample the parameter space P≡ (ω_b, ω_c, Θ_s, τ, n_s, A_s, w_1, ..., w_30,𝒩) where ω_b and ω_c are the baryon and CDM densities, Θ_s is the angular size of the sound horizon at decoupling, τ is the optical depth, n_s and A_s are the spectral index and the amplitude of the primordial power spectrum, and w_1,...,w_30 denote the 30 w-bin parameters. The first 29 w bins are uniform in a∈[0.286,1], corresponding to z∈[0,2.5], and the last wide bin covers z∈[2.5,1100]. We marginalise over nuisance parameters such as the intrinsic SN luminosity. §.§ Principal component analysis of w(z) First, we diagonalise the posterior covariance of w-bins to find their uncorrelated linear combinations (eigenmodes), along with the eigenvalues, which quantify how well a given eigenmode is constrained<cit.>. We plot the inverse eigenvalues of the posterior covariance, ordered according to the number of nodes in the eignemodes, in Panel a of Supplementary Fig. 3. The number of nodes is representative of the smoothness in the evolution of eigenmodes, with the first four posterior eigenmodes shown in Panel b of Supplementary Fig. 3. Next, we perform a PCA of the prior covariance matrix and plot its inverse eigenvalues alongside those of the posterior. We see that the fifth and higher number eigenvalues of the two matrices coincide, which means that they are fully determined by the prior. However, the first four inverse eigenvalues of the posterior are significantly larger than that of the prior, indicating that they are constrained primarily by the data. This is precisely the intent of the correlated prior method: the smooth features in w(z) are constrained by the data, with no bias induced by the prior, while the high frequency features are constrained by the prior. Thus, our w(z)CDM model effectively has only four additional degrees of freedom compared to ΛCDM. §.§ Dependence of the result on the correlated prior To investigate the dependence of our result on the strength of the correlated prior, in Fig. 3, we plot the Bayes factor and the statistical significance of w-1 as a function of σ_ D, which is a parameter that is added to the inverse covariance matrix of the correlated prior to effectively strengthen it. We find that neither ALL12 nor ALL16 dataset provides evidence for a dynamical DE at all prior strengths, although the Bayes factor for ALL16 is generally much less negative than that of ALL12 for all priors, , it grows from -6.7±0.3 to -3.3±0.3 for the prior used in this work, which corresponds to σ_ D=3. On the other hand, the plot shows that, if the w(z)CDM model was true, DESI++ would be able to provide a decisive Bayesian evidence over a wide range of prior strengths. Data Availability The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. 31 url<#>1 urlprefixURL [31]Grandis:2015qaaauthorGrandis, S., authorSeehars, S., authorRefregier, A., authorAmara, A.&authorNicola, A.titleInformation gains from cosmological probes. journalJ. Cosmol. Astropart. Phys.volume5, pages034 (year2016). 1510.06422. [32]Seehars:2015qzaauthorSeehars, S., authorGrandis, S., authorAmara, A.&authorRefregier, A.titleQuantifying concordance in cosmology. journalPhys. Rev. Dvolume93, pages103507 (year2016). 1510.08483. [33]Raveri:2016xofauthorRaveri, M., authorMartinelli, M., authorZhao, G.&authorWang, Y.titleInformation Gain in Cosmology: From the Discovery of Expansion to Future Surveys. journalArXiv e-prints (year2016). 1606.06273. [34]Paykari:2012neauthorPaykari, P.&authorJaffe, A. H.titleSparsely sampling the sky: a Bayesian experimental design approach. journalMon. Not. R. Astron. Soc.volume433, pages3523–3533 (year2013). 1212.3194. [35]Kunz:2006mcauthorKunz, M., authorTrotta, R.&authorParkinson, D. R.titleMeasuring the effective complexity of cosmological models. journalPhys. Rev. Dvolume74, pages023503 (year2006). astro-ph/0602378. [36]Amara:2013swaauthorAmara, A.&authorRefregier, A.titleModel breaking measure for cosmological surveys. journalPhys. Rev. Dvolume89, pages083501 (year2014). 1309.5955. [37]Verde:2014qeaauthorVerde, L., authorProtopapas, P.&authorJimenez, R.titleThe expansion rate of the intermediate universe in light of Planck. journalPhys. Dark Univ.volume5, pages307–314 (year2014). 1403.2181. [38]Charnock:2017vcdauthorCharnock, T., authorBattye, R. A.&authorMoss, A.titlePlanck confronts large scale structure: methods to quantify discordance. journalArXiv e-prints (year2017). 1703.05959. [39]JLAauthorBetoule, M.et al.titleImproved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples. journalAstron. & Astrophys.volume568, pagesA22 (year2014). 1401.4064. [40]6dfauthorBeutler, F.et al.titleThe 6dF Galaxy Survey: baryon acoustic oscillations and the local Hubble constant. journalMon. Not. R. Astron. Soc.volume416, pages3017–3032 (year2011). 1106.3366. [41]MGSauthorRoss, A. J.et al.titleThe clustering of the SDSS DR7 main Galaxy sample - I. A 4 per cent distance measure at z = 0.15. journalMon. Not. R. Astron. Soc.volume449, pages835–847 (year2015). 1409.3242. [42]wigglezauthorParkinson, D.et al.titleThe WiggleZ Dark Energy Survey: Final data release and cosmological results. journalPhys. Rev. Dvolume86, pages103518 (year2012). 1210.2130. [43]Heymans:2013fyaauthorHeymans, C.et al.titleCFHTLenS tomographic weak lensing cosmological parameter constraints: Mitigating the impact of intrinsic galaxy alignments. journalMon. Not. R. Astron. Soc.volume432, pages2433–2453 (year2013). 1303.1808. [44]OHD16authorMoresco, M.et al.titleA 6% measurement of the Hubble parameter at z0.45: direct evidence of the epoch of cosmic re-acceleration. journalJ. Cosmol. Astropart. Phys.volume5, pages014 (year2016). 1601.01701. [45]AcaciaauthorAlam, S.et al.titleThe clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample. journalArXiv e-prints (year2016). 1607.03155. [46]Dalal10authorDalal, N., authorPen, U.-L.&authorSeljak, U.titleLarge-scale BAO signatures of the smallest galaxies. journalJ. Cosmol. Astropart. Phys.volume11, pages007 (year2010). 1009.4704. [47]BSVauthorBeutler, F., authorSeljak, U.&authorVlah, Z.titleConstraining the relative velocity effect using the Baryon Oscillation Spectroscopic Survey. journalArXiv e-prints (year2016). 1612.04720. [48]S3ptauthorSlepian, Z.et al.titleConstraining the Baryon-Dark Matter Relative Velocity with the Large-Scale 3-Point Correlation Function of the SDSS BOSS DR12 CMASS Galaxies. journalArXiv e-prints (year2016). 1607.06098. [49]SNsysauthorMa, C., authorCorasaniti, P.-S.&authorBassett, B. A.titleApplication of Bayesian graphs to SN Ia data analysis and compression. journalMon. Not. R. Astron. Soc.volume463, pages1651–1665 (year2016). 1603.08519. [50]SCF11authorShafieloo, A., authorClifton, T.&authorFerreira, P.titleThe crossing statistic: dealing with unknown errors in the dispersion of Type Ia supernovae. journalJ. Cosmol. Astropart. Phys.volume8, pages017 (year2011). 1006.2141. [51]Crittenden:2005wjauthorCrittenden, R. G., authorPogosian, L.&authorZhao, G.-B.titleInvestigating dark energy experiments with principal components. journalJ. Cosmol. Astropart. Phys.volume12, pages025 (year2009). astro-ph/0510293. [52]CAMBauthorLewis, A., authorChallinor, A.&authorLasenby, A.titleEfficient Computation of Cosmic Microwave Background Anisotropies in Closed Friedmann-Robertson-Walker Models. journalAstrophys. J.volume538, pages473–476 (year2000). astro-ph/9911177. [53]DEPauthorZhao, G.-B., authorXia, J.-Q., authorLi, M., authorFeng, B.&authorZhang, X.titlePerturbations of the quintom models of dark energy and the effects on observations. journalPhys. Rev. Dvolume72, pages123515 (year2005). astro-ph/0507482. [54]polychordauthorHandley, W. J., authorHobson, M. P.&authorLasenby, A. N.titlePOLYCHORD: nested sampling for cosmology. journalMon. Not. R. Astron. Soc.volume450, pagesL61–L65 (year2015). 1502.01856. [55]Lewis:2002ahauthorLewis, A.&authorBridle, S.titleCosmological parameters from CMB and other data: A Monte Carlo approach. journalPhys. Rev. Dvolume66, pages103511 (year2002). astro-ph/0205436. [56]PCAauthorHuterer, D.&authorStarkman, G.titleParametrization of Dark Energy Properties: A Principal Component Approach. journalPhys. Rev. Lett.volume90, pages031301 (year2003). astro-ph/0207517. Supplementary Information This section contains three supplementary tables and five supplementary figures. Supplementary Figure 1. The reconstructed evolution history of the dark energy equation of state using ten different data combinations. The reconstructed w(z) (white solid line) and the 68% CL uncertainty (dark blue shading) from different data combinations shown in the legend. The correlated prior parameters are a_c=0.06 and σ_ m=0.04. One can note that the dip in w(z) at z ∼ 0.4 is more pronounced for ALL16 compared to ALL16-3z, thanks to BAO-9z being more informative than BAO-3z. As we will see shortly, this makes the ALL16 result more consistent with the w(z) reconstructed in Zhao  (2012)<cit.> using a different combination of data (ALL 12). Supplementary Figure 2. A comparison between observations and theoretical predictions in ΛCDM and w(z)CDM universes. The H(z) (panels a, b) and the D_A(z) (c, d) data rescaled by the values derived from the ALL16 best fit ΛCDM (a, c) and w(z)CDM (b, d) models. Datasets are labeled by accordingly coloured text, and the shaded bands indicate the 1σ uncertainty in the rescaled H(z) and D_A. The shaded bands indicate the uncertainty in the rescaled H(z) and D_A. One can see that, in the case of w(z)CDM, the data points are much more consistent with the corresponding values derived from ALL16, demonstrating the significant reduction in tension. Supplementary Figure 3. A principal component analysis of the w(z) reconstruction result. Panel a: the inverse eigenvalues of the prior covariance matrix (black line with filled dots) and of the posterior covariance (red line with unfilled dots); Panel b: the first four posterior eigenmodes of w(z) for the ALL16 dataset combined with the correlated prior. Supplementary Figure 4. The signal-to-noise ratio of observables in ALL12 and ALL16 datasets respectively. The signal-to-noise ratio of the expansion rate of the Universe H(z) (left), angular diameter distance D_A(z) (middle) and luminosity distance D_L(z) for ALL12 (upper) and ALL16 (lower) datasets. Supplementary Figure 5. The effect of neutrinos and CMB lensing amplitude on w(z) reconstruction. The reconstructed w(z) (red dashed) with 68% CL uncertainty (red solid) in three cases. Left: the sum of neutrino masses M_ν is fixed to 0.06 eV; middle: the neutrino masses M_ν and number of relativistic species N_ eff are marginalised over; right: the CMB lensing amplitude A_ Lens is marginalised over. The white line and light blue shaded bands in each panel shows the mean and 68% CL uncertainty of our ALL16 reconstruction, which is shown in Fig. 1 of the Letter using the same colour scheme, for a comparison. 31 url<#>1 urlprefixURL si:P15authorPlanck Collaborationet al.titlePlanck 2015 results. XIII. Cosmological parameters. journalAstron. & Astrophys.volume594, pagesA13 (year2016). si:JLAauthorBetoule, M.et al.titleImproved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples. journalAstron. & Astrophys.volume568, pagesA22 (year2014). si:6dfauthorBeutler, F.et al.titleThe 6dF Galaxy Survey: baryon acoustic oscillations and the local Hubble constant. journalMon. Not. R. Astron. Soc.volume416, pages3017–3032 (year2011). si:MGSauthorRoss, A. J.et al.titleThe clustering of the SDSS DR7 main Galaxy sample - I. A 4 per cent distance measure at z = 0.15. journalMon. Not. R. Astron. Soc.volume449, pages835–847 (year2015). si:wigglezauthorParkinson, D.et al.titleThe WiggleZ Dark Energy Survey: Final data release and cosmological results. journalPhys. Rev. Dvolume86, pages103518 (year2012). si:Heymans:2013fyaauthorHeymans, C.et al.titleCFHTLenS tomographic weak lensing cosmological parameter constraints: Mitigating the impact of intrinsic galaxy alignments. journalMon. Not. R. Astron. Soc.volume432, pages2433–2453 (year2013). si:Riess16authorRiess, A. G.et al.titleA 2.4% Determination of the Local Value of the Hubble Constant. journalAstrophys. J.volume826, pages56 (year2016). si:OHD16authorMoresco, M.et al.titleA 6% measurement of the Hubble parameter at z0.45: direct evidence of the epoch of cosmic re-acceleration. journalJ. Cosmol. Astropart. Phys.volume5, pages014 (year2016). si:AcaciaauthorAlam, S.et al.titleThe clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample. journalArXiv e-prints (year2016). 1607.03155. si:Zhao17authorZhao, G. B.et al., [BOSS Collaboration], titleThe clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: tomographic BAO analysis of DR12 combined sample in Fourier space. journalMon. Not. R. Astron. Soc.volume466, pages762–779 (year2017). si:Delubac14authorDelubac, T.et al.titleBaryon acoustic oscillations in the Lyα forest of BOSS DR11 quasars. journalAstron. & Astrophys.volume574, pagesA59 (year2015). si:Zhao12authorZhao, G.-B., authorCrittenden, R. G., authorPogosian, L.&authorZhang, X.titleExamining the Evidence for Dynamical Dark Energy. journalPhys. Rev. Lett.volume109, pages171301 (year2012). si:DESIauthorDESI Collaborationet al.titleThe DESI Experiment Part I: Science,Targeting, and Survey Design. journalArXiv e-prints (year2016). 1611.00036. si:EuclidSNauthorAstier, P., authorGuy, J., authorPain, R.&authorBalland, C.titleDark energy constraints from a space-based supernova survey. journalAstron. & Astrophys.volume525, pagesA7 (year2011). si:snls3authorConley, A.et al.titleSupernova Constraints and Systematic Uncertainties from the First Three Years of the Supernova Legacy Survey. journalAstrophys. J. Suppl.volume192, pages1 (year2011). si:dr9authorReid, B. A.et al.titleThe clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: measurements of the growth of structure and expansion rate at z = 0.57 from anisotropic clustering. journalMon. Not. R. Astron. Soc.volume426, pages2719–2737 (year2012). si:Riess09authorRiess, A. G.et al.titleA Redetermination of the Hubble Constant with the Hubble Space Telescope from a Differential Distance Ladder. journalAstrophys. J.volume699, pages539–563 (year2009). si:wmap7authorLarson, D.et al.titleSeven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Power Spectra and WMAP-derived Parameters. journalAstrophys. J. Suppl.volume192, pages16 (year2011).

http://arxiv.org/abs/1701.07452v1

20170125190747

WOMBAT: A Scalable and High Performance Astrophysical MHD Code

astro-ph.IM

Cray Inc., St. Paul, MN 55101 School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 pjm@cray.com Cray Inc., St. Paul, MN 55101 nradclif@cray.com Cray Inc., St. Paul, MN 55101 kkandalla@cray.com Minnesota Supercomputing Institute for Advanced Computational Research dhp@umn.edu School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 Minnesota Supercomputing Institute for Advanced Computational Research oneill@astro.umn.edu School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 Minnesota Supercomputing Institute for Advanced Computational Research nolt0040@umn.edu Institute for Theory and Computation, Center for Astrophysics, Harvard University, Cambridge, MA 02138 pedmon@cfa.harvard.edu School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 INAF-Istituto di Radioastronomia, via. P.Gobetti 101, I-40129 Bologna Italy ERC Marie Curie Fellow donnert@ira.inaf.it School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 Minnesota Supercomputing Institute for Advanced Computational Research twj@umn.edu We present a new code for astrophysical magneto-hydrodynamics specifically designed and optimized for high performance and scaling on modern and future supercomputers. We describe a novel hybrid OpenMP/MPI programming model that emerged from a collaboration between Cray, Inc. and the University of Minnesota. This design utilizes MPI-RMA optimized for thread scaling, which allows the code to run extremely efficiently at very high thread counts ideal for the latest generation of the multi-core and many-core architectures. Such performance characteristics are needed in the era of “exascale” computing. We describe and demonstrate our high-performance design in detail with the intent that it may be used as a model for other, future astrophysical codes intended for applications demanding exceptional performance. § INTRODUCTION Magneto-hydrodynamic (MHD) simulations allow us to study the dynamics of highly conducting astrophysical fluids since many astrophysical fluids are highly conductive ionized plasmas. MHD modeling then allows us to incorporate essential consequences of magnetic fields. Even “weak” magnetic fields, whose Maxwell stresses are subdominant to inertial and to thermal pressure stresses, can have major impact on the development of turbulence and its dissipation on small scales, on momentum transport, angular momentum and energy, and on thermal conduction. If the simulations include, in addition to MHD, the transport of high energy, non-thermal “cosmic ray” particle populations, the simulations can model emission processes involving the cosmic ray interactions with the bulk fluid and its magnetic field. These include γ-ray by-products of cosmic ray proton interactions with the bulk fluid and radio to X-ray emissions from cosmic ray electrons, including synchrotron radiation.Since magnetic field properties often derive from the details of the fluid dynamics over a wide range of scales of interest, it is essential for simulations to capture the dynamics with high fidelity over this full range of scales. This is generally a very intensive and challenging computational task that, despite much progress in coding and vast improvements in computing infrastructure, has often remained beyond current capabilities. That challenge is the motivation for our efforts described here to develop an MHD code environment that can effectively utilize and adapt to the coming generations of computational infrastructure to allow solutions to these pressing astrophysical problems.Numerous codes exist for both general purpose and specific use astrophysical fluid simulations. Some examples are GADGET <cit.>, NDSPMHD <cit.>, AREPO <cit.>, ENZO <cit.>, ATHENA <cit.>, RAMSES <cit.>, CHARM <cit.>, PLUTO <cit.>, CASTRO <cit.>, and FLASH <cit.>. Codes like these have been developed over many years and often have features for adding the effects of gravity, cosmic-ray transport, non-ideal MHD, cosmic expansion, and non-adiabatic energy gains and losses, including radiative and conductive cooling and heating. “Exascale” is next next major step in the evolution of high performance computing (HPC), with systems capable of performing 10^18 floating point operations per second distributed across many levels of parallelism. Preparing applications for exascale requires a substantial investment in code re-design and optimization, to enable the community to leverage the capability of new architectures and make new scientific breakthroughs. <cit.> recently presented a survey of the challenges and potential approaches to modernizing some of the most popular community codes. The latest multi-core and many-core processors (CPUs), such as Intel Xeon and Intel Xeon Phi, feature increasing core counts per processor with decreasing clock speed along with increasing single instruction-multiple data (SIMD) vector lengths. Hence, cache blocking and vectorization are critical to obtaining good performance from modern processors. But the increasing core counts also put pressure on the traditional MPI-only (Message Passing Interface) parallelization models. Memory consumption from a large number of independent MPI processes on a node may become prohibitive. For MHD simulations that develop substantial load imbalance, possibly through the inclusion of N-body dynamics or multi-level mesh refinement, balancing work between MPI ranks is critically important. However, the process of balancing work between MPI ranks carries potentially significant overhead. This overhead is the combination of the cost of moving grid data between MPI ranks and communication of the change in decomposition to some or all MPI ranks. There are several established strategies for reducing the overhead, including decomposition meta-data replication, but these techniques come at the cost of memory and complexity <cit.>. Programming models that allow for load balancing with less explicit communication are greatly needed. One attractive approach is the hybrid OpenMP/MPI model, as discussed in <cit.>. It allows MPI ranks to hold larger portions of the world grid. In the context of mesh refinement, added work due to refinement at any single location is a lower fraction of a rank's total load. For many calculations it could also result in a more symmetric load across MPI ranks if refinement needs are not confined to a single region. Work within an MPI rank can be load balanced among threads with any form of dynamic work scheduling. Finally, on-node imbalances due to contention of shared resources, such as cache or bandwidth, also can be mitigated with attention to thread scheduling. However, typical parallel loop-based OpenMP designs have shown too little scope (amount of code effectively threaded) to scale effectively to high thread count. Modern HPC interconnects often feature low latency/high bandwidth messaging with network-offloading, which enables overlap of computation with communication. MPI-RMA (Remote Memory Access) is a feature added to the MPI standard in order to expose these capabilities to the user. It should be possible for an application to drive communication near hardware limits with a highly efficient MPI-RMA implementation. However, MPI libraries need high performance implementations for the hybrid OpenMP/MPI model to include communication parallelization. In this paper we present an application design study for a new grid-based MHD code called WOMBAT[WOMBAT is available by request or by visiting http://www.astro.umn.edu/groups/compastro]. The goal of this project is to address the optimization opportunities discussed above through a co-design process. In pursuit of this goal, we seek a base design well suited for uni-grid simulations yet formulated for complex conditions requiring load balancing. For the purpose of this paper, we review the base design for MHD uniform meshes only. WOMBAT development is a collaboration between Cray Inc. Programming Environments and the University of Minnesota. Through this collaboration we developed a design strategy (see <ref>) that adapts to architectures (CPU and interconnect) using language, OpenMP and MPI best practices. We also identified bottlenecks and optimizations for MPI (Cray MPICH) resulting in significant performance improvements. Section <ref> is a performance review of WOMBAT on three architectures that can be used as a model for assessing the quality of any similar implementation. We discuss specific implementation details in <ref>. Our design strategy is applicable to many other codes and serves as a potential path forward for exascale application readiness.In what follows “KNL” designates the Intel® Xeon® Phi many-core processor (Knights Landing), “Broadwell” a recent Intel® Xeon® multi-core processor (Broadwell) and “Interlagos” the AMD OpteronTM multi-core processor (Interlagos). In all figures, these processors are shown as red, green and blue, respectively. § SCALABLE DESIGN STRATEGY ccc 3 HPC architecture constraints and the optimization techniques and features developed in WOMBAT to address them. Hardware Constraint Optimization Approach WOMBAT Design high FLOP/Byte ratio cache blocking Patch slow scalar + wide vectors vectorization Fortran + vectorization best practices many cores thread scalability SPMD OpenMP + new Cray MPICH + Patch distributed architecture RDMA + computation/communication overlap AIO + MPI-RMA + new Cray MPICH + Patch The key design characteristic of WOMBAT is to subdivide the problem into completely independent pieces of the world grid that include their own boundary zones and necessary meta-data for updating from one time step to the next. We refer to these independent pieces as “Patches.” This design naturally accommodates any numerical method with local or semi-local communication needs. The concept is similar to data management strategies in other many MHD codes <cit.>, but our design takes a unique approach to processing and scheduling the computation and communication of Patches. A Patch is a unit of work that a thread within a WOMBAT MPI process independently operates on. No assumptions are made on the number of Patches relative to the number of threads since our design adapts to this ratio. Patch boundaries also define units of communication work done with either local (intra-process) or remote (inter-process) copies. The number of zones in each dimension of a Patch and the number of them in each dimension on a rank (and Domain, see <ref>) are input parameters. This allows us to tune them for performance on a given architecture (see <ref>). WOMBAT is written in Fortran 2008. Fortran semantics make it easy for modern compilers to identify and apply optimizations, such as vectorization, as long as developers follow simple rules (see <ref>). Code modularity, organization and maintainability benefit from the object-oriented features available in Fortran 2008. However, overuse of classes can lead to a loss of optimization opportunities for a compiler. Thus any performance critical section of WOMBAT is basic Fortran code working on arrays. The code is constructed from three main categories of classes: data managers, engines and solvers. Data managers do memory management and supporting functions. Solvers accept data managers as arguments and update their arrays following whatever numerical methods they employ. Engines orchestrate parallelism and the book-keeping and communication requirements for handing data managers to solvers. Table <ref> shows the hardware constraints on the current and next generation of HPC systems, alongside the techniques and optimizations we include in our design strategy to meet these constraints. §.§ Domain Decomposition To construct the Patches, the world grid is decomposed into N equal size sub-volumes (Domains), each assigned to an MPI rank. An MPI rank's sub-volume is further decomposed into Patches containing an equal number of zones. Patches in a Domain communicate boundary data with one another and with Patches on neighboring ranks. Figure <ref> shows a sample 2d configuration of the Patch-Domain hierarchy across an arbitrary number of MPI ranks. The figure is centered on a single MPI rank's Domain. That rank has eight neighbors labeled N0 through N7 each with their own Domain. Inside every MPI rank's Domain is a 5x5 grid of Patches, labeled P0 through P24 for the central Domain. We implement the Patch-Domain design as Fortran classes. The Domain class is responsible for tracking the MPI rank and local or remote Patches that share boundaries with it. This design can be extended to nested block static or adaptive mesh refinement (SMR or AMR). In support of that and generic load balancing between MPI ranks, Patches inside a Domain are allowed to become active or inactive, meaning member data structures can be allocated and updated in time, deallocated and not included in updates. Information about the MPI rank(s) and remote Patches sharing a boundary with a Patch can be modified over time, allowing Patches to be moved between ranks with minimal bookkeeping and communication. In particular, we do not store global data structures for tracking decomposition. §.§ Optimization and Multi-level Parallelization Strategy Three levels of parallel optimization are common to HPC systems: SIMD vectorization, intra-process threading (using OpenMP), and inter-process communication (using MPI). Cache blocking is an additional optimization that addresses memory topology on these systems. §.§.§ Cache Blocking The Patch design naturally results in cache blocking, which directly addresses bandwidth limitations. The most popular CPUs used in HPC today have FLOP/s to Byte/s ratios (ratio of floating point performance to memory bandwidth[For example, a 1.4 GHz Intel Xeon Phi processor is theoretically able to achieve ≃ 3 TFLOP/s and has a memory bandwidth of ≃ 450 GB/s to MCDRAM (90 GB/s to DDR).]) of ≃ 10, hence algorithms with similar computational intensities run most efficiently. However, computational intensities that high are difficult to achieve with stencil based numerical methods for solving MHD. Processor caches can mitigate this issue when used effectively and stencil methods provide good opportunity for reuse of loaded values between operations. Hence, good performance requires cache blocking techniques on all key loops. However, explicitly programmed cache blocking can be cumbersome, because all performance-critical nested loops over problem dimensions must be expanded into higher dimensional loops with tunable blocking parameters. Since all solvers in WOMBAT operate on a single Patch, their computationally intensive loops are all roughly the size of a Patch. The best Patch size that fits into a level of cache inherently gets reuse out of cache (typically the best size fits into level 3 (L3) cache but not entirely into level 2 (L2), see <ref>). §.§.§ SIMD Vectorization Operations on a Patch consist of floating point and data motion intensive loops. These loops are written to be good SIMD vectorization candidates following the typical rules of stride-one access, recurrence free, and limited conditional logic. To accomplish good maintainability and portability we do not explicitly program this level of parallelization and leave it to a compiler to decide if and when to use vectorization. This usually requires scalar operations be isolated to separate loops so the remaining work is available for vectorization. §.§.§ OpenMP Threading The benefits of hybrid application scaling to high thread counts was discussed in <ref>. We avoid the bottle-necks of parallel loop-based OpenMP by arranging WOMBAT such that only one OpenMP parallel region is present for the duration of execution. This design presents the threaded region as a set of completely independent processes, which mimics the parallelism of MPI. We refer to this approach as SPMD OpenMP (or single program-multiple data OpenMP) <cit.>. To illustrate this design, we show a flow diagram of the main driver in WOMBAT in Figure <ref>. A section including MPI initialization and base object construction is the only work by the main thread outside the parallel region. After that, every portion of WOMBAT is executed by all threads collaboratively. This includes array allocation/paging, computation, communication, and even I/O. §.§.§ MPI To allow fine-grained work-communication overlap, the Patch design results in a larger number of small messages, compared to traditional approaches which typically use fewer monolithic boundary exchange messages. This shifts the communication sensitivity of WOMBAT from simply bandwidth to bandwidth and message rate, depending on the number and size of Patches. A Domain decomposed into Patches will generate more MPI messages and a higher aggregate amount of data moved, especially in 3d, due to added corner and edge boundaries. This extra communication will later be leveraged for communicating load information and changes to Patch ownership. Furthermore, in the SPMD OpenMP approach every thread can participate in the MPI communication using .Most of the MPI communication in WOMBAT uses MPI-RMA (e.g., , ), because of the low overhead possible with a proper MPI-RMA implementation. MPI-RMA was added to the MPI standard primarily to give users direct access to the Remote Direct Memory Access (RDMA) features available on most HPC networks (interconnects). PUT and GET operations in MPI-RMA are inherently non-blocking, and excellent overlap of computation and communication is possible on networks that also support network-offloading. While the semantics of MPI-RMA allow for these performance characteristics, many MPI libraries today implement MPI-RMA using two-sided communication <cit.>. This adds overhead and reduces the chances for overlap, leaving MPI-RMA practically unusable for an HPC application. Recent work in MVAPICH <cit.> and OpenMPI <cit.> has corrected that issue on both Infiniband and Cray networks. In Cray MPICH, MPI-RMA is now based on the low level DMAPP library specifically designed for optimal one-sided communication on Gemini and Aries interconnects. This implementation has very low overhead, tuned to utilize the network-offload (Block Transfer Engine or BTE) capability on Cray XE/XC systems.Message rate requirements and the SPMD design make it critically important that multi-threaded MPI-RMA in a given MPI library performs well. During the design of WOMBAT we found that no MPI implementation really achieved the performance that should be possible. This is because most MPI implementations (including Cray MPICH at the time) use a global lock to provide thread safety <cit.>. This serializes all MPI calls, and more importantly, most work in the user code around those MPI calls. Through a co-design approach we have optimized Cray MPICH for high performance and thread scalable MPI-RMA communication (see <ref>). We refer to this new capability as “thread-hot RMA”. An initial version is available to Cray users starting with Cray MPICH 7.3.4 with additional enhancements from the work presented here available in an upcoming releases. Other MPI libraries are also pursuing optimizations for that will make our design performance portable beyond Cray systems <cit.>. §.§.§ Asynchronous I/O WOMBAT uses a custom asynchronous I/O (AIO) library that allows for overlap of simulation progression and data writing. If all I/O data can be buffered, data can be written out with almost no impact on execution time. Some portion of I/O work is done blocking if buffers are made smaller, which reduces overall performance. AIO is implemented as a set of specialized ranks dedicated to receiving (or sending for read operations) data from a client set of worker ranks. All threads in the worker ranks package data into I/O buffers. Non-blocking communication is used to move data to AIO server ranks, and the AIO ranks then write data out as it comes in. The full system can be tuned for I/O and overlap performance by adjusting the total number of AIO server ranks. § PERFORMANCE ccccccc 7 Test platforms used in the performance studies and their characteristics. Note: We mostly use 64 cores on KNL systems to make scaling studies simpler multiples from lower core counts. System Title Architecture Interconnect Topology CPU VL [bits] Cores per Node Blue Waters Cray XE Cray Gemini 3d torus AMD Opteron^TM 6276 “Interlagos” @ 2.3 GHz 256 16 XE_IL Cray XE Cray Gemini 3d torus AMD Opteron^TM 6281 “Interlagos” @ 2.5 GHz 256 16 XC_BDW Cray XC Cray Aries dragonfly Intel® Xeon® E5-2695 “Broadwell” @ 2.5 GHz 256 36 XC_KNL Cray XC Cray Aries dragonfly Intel® Xeon Phi^TM 7250 “KNL” @ 1.4 GHz 512 68 We measure the performance of WOMBAT for 3d MHD calculations using the directionally un-split MHDTVD solver described in <ref>. Single node tests focus on the impact of vectorization on overall execution and the parallel efficiency of the SPMD OpenMP technique. Multi-node tests at scale measure the performance of the full suite of parallelization strategies, including off-node communication, and how they interact. We stress that the problems sizes used for most experiments presented here were selected to show overheads in WOMBAT, and in particular communication. This also closely follows real-world simulations run on production systems.Table <ref> summarizes the platforms used for performance experiments. We used Blue Waters (Cray XE) at the NCSA for very large weak scaling studies. The remaining systems are internal configurations at Cray Inc. We use the Cray Compiler (CCE) in all experiments. Table <ref> shows specific test information about each processor. We used the new Cray MPICH library with the “thread-hot RMA” feature in all tests unless otherwise noted. All experiments involving KNL were run with nodes configured in so-called “quadrant” Non-Uniform Memory Access (NUMA) mode with high bandwidth memory (on package) configured as a 16 GB L3 cache. ccc 3 Processor compilation and placement notes. CPU Compilation Flags Notes Interlagos -O vector3 -h preferred_vector_width=256 Only one thread/process per floating point unit Broadwell defaults Only one thread/process per core (no hardware threads used) KNL defaults Only one thread/process per core (no hardware threads used) §.§ Single Node Performance by Architecture §.§.§ SIMD Scaling We measure the impact of increasing vector length (VL) from 64 to 512 bits on KNL[To vary the type of vectors, we use the CCE compiler flag “-h preferred_vector_width=X”, where X = {128, 256, 512}. For scalar 64 bit vectors we use “-O vector0”.]. The problem size is a 17x4^2 Domain of Patches each with 48^3 zones updated by a single MPI rank with 68 threads. Figure <ref> shows the strong scaling with increasing VL on a single node of the XC_KNL system. The time to perform a single time step update is reduced approximately by a factor of 2 going from 64 and 256 bit vectors. The final step to 512 bit vectors continues to show improved performance, but the effect has been reduced to only a ≃ 19% speedup. Factors that affect the speedup from vectorization are the amount of vector versus scalar code executed, the efficiency of the vector code generated by the compiler, and the memory bandwidth available to provide data to the cores. Overall, vectorization speeds up WOMBAT by almost a factor of 2.5X on KNL processors. The speedup is roughly consistent with Amdahl's Law, assuming the fraction of execution time benefitting from parallelization p ≈ 0.65. Broadwell has accessible hardware performance counters for floating point operations that can be measured with a number of performance tools, such as PAPI or CrayPAT. Table <ref> shows the quality of vectorization relative to a scalar build of WOMBAT. We also show the breakdown of scalar and vector operations for the Intel and GNU compilers. Vectorization with CCE reduces total double precision (DP) floating point instruction count by ∼ 71%. 79% of all floating point operations are vector. Intel produces a similar amount of vector instructions but also lower performance. The performance difference is due to much lower translation lookaside buffer (TLB) utilization despite using 2 Megabyte huge pages. We intend to file a performance bug with Intel on this issue and it will be corrected in later releases. The GNU compiler is unable to produce any vector instructions. ccccc 5 Effect of vectorization on floating point instruction count on Broadwell for a single thread running a 2^3 x 32^3 problem for 100 time steps. Compiler 10^9 Scalar Ops. 10^9 256 B SIMD Ops. sec / update % of DP Peak CCE 8.5.4 (scalar) 327 0 1.87 6.6 CCE 8.5.4 20 74 0.96 14.1 Intel 17.0.1.132 15 75 1.67 8.1 GNU 6.2.0 320 0 1.84 6.6 §.§.§ Thread Scaling ccc 3 Number of Patches in each direction of the Domain and number of zones per Patch used by each architecture in the thread strong scaling test. CPU Domain Size Patch Size Interlagos 8x7x4 40^3 Broadwell 12^2x8 32^3 KNL 8^3 48^3 We show the thread strong scaling speedup of WOMBAT on the three architectures presented here in Figure <ref> with the problem sizes given in Table <ref>. The code shows excellent speedup with threads on all the architectures. The speedup from threads on KNL is ∼ 40X at 68 threads. The “turbo” on Broadwell increases performance for small thread numbers (green line versus dashed green line). On both Interlagos and Broadwell there is notable loss of scaling once the process has threads spanning beyond a single NUMA domain. On Interlagos this is at 4 threads, because each Interlagos processor is made up of two “Bulldozer” modules for a total of four NUMA nodes on a dual socket XE node. A KNL node configured in quadrant mode has only a single NUMA node, and the deviation from ideal scaling is moved to much higher thread counts. Profiles showed that the cost of thread synchronization rising at these thread counts, but there is the additional factor of a finite amount of bandwidth available on the processor. Both of these account for most of the reduction in performance from ideal. §.§ Performance at Scale by Architecture Off-node components of an application, such as network latency and bandwidth, can modify its behavior and how it should be tuned. We present a multi-node Patch size optimization study for WOMBAT, and we also demonstrate the weak and strong scaling capabilities of WOMBAT out to large node counts. §.§.§ Patch Size Optimization In <ref> we described how Patches are logically assembled to produce any grid size per rank. We study performance with Patch size also allowing the mixture of MPI ranks to OpenMP threads to vary at a scale of 27 nodes. 27 nodes is used because any configuration of MPI ranks to threads at that scale will have unique neighbors in 3d. This ensures that the MPI work is saturated and performance is not skewed. We use “PPN” to denote the number of MPI processes per node with threads placed on all cores. The total number of zones across each Patch size was held approximately constant within a system type. We chose the problem setup so update-times were held at ≃ 10 seconds. This is sufficiently large to expect throughput values to be near their absolute peak but still include overhead sensitivity. Figure <ref> shows the throughput on each architecture given by the number of zones per second each node can update. We can identify the maximum throughput each system can achieve for these problem setups. XE_KNL nodes are able to update ≳ 6x10^6 zones/sec/node at peak compared ≳ 4.5x10^6 zones/sec/node on XC_BDW and 2x10^6 zones/sec/node on XE_IL. This demonstrates the ability of our approach to adapt to the unique many-core design of KNL. The optimal Patch size is not uniform across systems. XC_BDW has the smallest optimal Patch size of 32^3. The optimal Patch size for XE_IL is 40^3, and XC_KNL has an optimal Patch size ≃ 50^3. The performance on either side of the optimal size drops off but not by the same amount on each system. Patch sizes smaller than the optimal size have lower performance due to reduced vectorization efficiency, and larger sizes have lower performance due to spilling out of L3 on both Interlagos and Broadwell. KNL has the largest SIMD vector size, which explains why it has the largest optimal Patch size. Overheads appear to affect Interlagos more than Broadwell. This favors slightly larger Patches on Interlagos despite having the same SIMD vector length as Broadwell. The cache blocking properties of the Patch design discussed in <ref> no longer function as intended for large Patches. On KNL the performance loss is not as dramatic, with exceptions at high thread count. This is due to the 16 GB L3 cache.Both XE_IL and XC_BDW show lowest performance with only a single rank per node packed with threads. This is largely due to the NUMA issues discussed in <ref>. In both of these cases, once the number of MPI processes per node matches the number of NUMA nodes on a node there is very little spread in performance at the optimal Patch size. In the case of XC_KNL performance does not vary much until the Patch size is 50^3 or greater. While the absolute best performance on XC_KNL is with 16 PPN (∼ 6.5x10^6 zones/sec/node), there is still significant performance at 4 PPN (∼ 6x10^6 zones/sec/node). At a high level, a fixed grid calculation should not perform any different exchanging ranks for threads if both MPI and OpenMP are well implemented and hardware limitations are not present. The SPMD design in WOMBAT nearly achieves this. §.§.§ Weak Scaling ccc 3 Weak Scaling Test Setup: The number of Patches in each direction of the Domain and number of zones per Patch used for each system. System Patches per Node Patch Size XE_IL 4x4x4 40^3 XC_BDW 6x6^2 32^3 XC_KNL 8x4^2 48^3 Figure <ref> shows the weak scaling on three architectures at different values of PPN for the problem sizes given in Table <ref>. For XC_IL and XE_BDW the best performance and scaling is closely matched between a single MPI rank per NUMA node or pure MPI, which follows the conclusions in <ref>. The XC_KNL systems has best performance and scaling at 4 and 8 PPN. Relative to a single node, XE_IL has a 93% efficiency up to 150 nodes at 4 PPN, and XC_BDW at 2 PPN has a 87% efficiency up to 512 nodes. Remarkably XC_KNL has 89% efficency at 648 nodes (41,472 threads) with 4 PPN relative to a single node run. The lowest performance at scale on all systems is with a single rank per node. The increases in update times are due to increasing amounts of off-node communication a rank encounters and imperfect overlap of communication with computation. Between 3 and 27 nodes, off-node communication cost is saturated, and update times are nearly flat for larger node counts. We conclude that WOMBAT has excellent weak scaling on all systems with the optimal configuration despite the relatively small problem size chosen. We show weak scaling on Blue Waters in Figure <ref> for two values of PPN and problem sizes. We limit each run to just 10 time steps. The 1 PPN runs scale out to 16,224 nodes (259,584 threads) with a world grid containing ≃ 66 billion zones. WOMBAT scales well with 60% efficiency at 16,224 nodes for 1 PPN and 75% efficiency at 4,096 nodes for 4 PPN relative to single node runs. There are several spikes of increased update times. We hypothesize it is due to network contention with both other running applications and with WOMBAT itself on the very large 3d torus topology. Runs on the smaller dedicate XE_IL system do not show these features. Going forward, we intend to make use of the topology-aware scheduling capability provided by the NCSA, and we expect this to improve performance and reduce the contention opportunities. The right panel of Figure <ref> shows the impact of the “thread-hot RMA” capability that will be included in an upcoming release of Cray MPICH. We show weak scaling on XC_KNL to 125 nodes with 4 PPN using the same problem setup for weak scaling described above. This feature produces a 17% speedup over Cray MPICH 7.3.1. §.§.§ Strong Scaling cccc 4 Strong scaling test setup. System PPN World Grid Patches Base Patch Size XE_IL 4 14x8x12 40^3 XC_BDW 2 48x12^2 32^3 XC_KNL 2 32x16x12 40^3 The left panel of Figure <ref> shows the strong scaling of WOMBAT. We defined the problem size for each system so that the time per update is limited to ∼ 60 seconds (see Table <ref>). Performance closely follows the theoretical speedup over 2 orders of magnitude in node count, with XC_BDW showing a 236X speedup at 384 nodes. The deviations are due to overheads exposed as update times at scale fall at or below 0.3 seconds. We reduced the Patch size on XC_BDW by half and on XC_KNL by a quarter at the largest scale, for example. §.§.§ Thread Scheduling at Scale We show the impact of OpenMP scheduling in Figure <ref>. In this experiment we modified the thread scheduling for just a single loop in the Update Engine (see <ref>) that drives updates over Patches. We run a balanced (Patch count divides evenly into thread count) and an imbalanced problem with the modified code and original code. 4 MPI ranks per node on 27 nodes each with 16 threads update 40 (imbalanced) or 32 (balanced) Patches, each with 48^3 zones on the XC_KNL system. Both problems show near optimal throughput with the GUIDED schedule and lower performance with STATIC. Our SPMD OpenMP design has very few thread barriers, and using a STATIC schedule assumes threads are roughly synchronized to be efficient. § DESIGN DETAILS §.§ Update Engine In our design, the Update Engine is responsible for scheduling computation and communication across threads for any solver. It accepts a Domain and iterates over Patches, exchanging messages and partitioning the update work through the specified solver until all Patches report back as completed. To allow for iterative or sub-cycling solvers, it is not necessary that a Patch be updated completely for the current time step after only a single pass through a solver. Figure <ref> shows a schematic of the Update Engine. It is contained inside the higher level OpenMP parallel region shown in Figure <ref>, and all threads call it with the same input data and requested solver class. The outer while loop allows for iterative solvers that require any arbitrary number of passes (including messaging) to complete for a time step. The inner while loop contains the work necessary to drive both communication and updates through the requested solver. The work includes packing (unpacking) boundary data into (out of) contiguous buffers, used for either local copies within a rank or MPI transfers between ranks. There is additional work for signaling and data transfer between MPI ranks and updating Patches with resolved boundaries through the requested solver. The DomainSolver class manages all book-keeping related to marking individual boundaries for any affected Patch as resolved/unresolved. It also tracks grids within a Patch as incrementally or completely updated.An important optimization in this design relates to how boundary data is exchanged between Patches contained in the same MPI rank. An instance of the Patch class includes a buffer for incoming boundary data (there is no matching buffer for outgoing data). This buffer is only used for boundary data coming from another local Patch. With the Domain class, data destined for a local Patch is directly packed into the buffer of the destination Patch. The buffer is later unpacked, along with any non-local boundary data, into the Patch grid boundary zones. This optimization takes advantage of the shared memory aspect of OpenMP, completing local Patch boundary exchange without calling MPI or excessive buffering. Some buffering is necessary to minimize contention between threads attempting to progress the same Patch. A single node run can completely avoid calling MPI with this feature by using threads on all cores. §.§.§ MPI-RMA Engine The MPI-RMA Engine handles non-local communication between Patches. It is generic enough to manage communication of any type of data of a wide range of message lengths with memory overheads and intensity on the network that is run-time tunable. The strategy for the MPI-RMA engine was to remove all explicit synchronization between MPI ranks and utilize all threads for both message packing/unpacking and initiation of network transfers. We use a single passive exposure epoch with MPI-RMA. The passive epoch starts with ranks calling for each rank it will communicate with. The communication strategy in WOMBAT does not use protections between ranks, and the lock argument to is always set to . Locking and unlocking for RMA exposure is moved outside the time loop, which essentially removes their cost in exchange for minimal overhead introduced by a signaling scheme. Figure <ref> shows an overview of the steps in the MPI-RMA Engine. Operations from the point of view of both a source and neighbor rank are shown in time. Note that all source ranks are also a neighbor rank, meaning that the steps are symmetric. The process begins with a source rank packing some (not all) boundary data from a Patch into a buffer. The rank then sends an 8 Byte signal to the neighbor rank with indicating the size of the message that has been packed. At some point the neighbor rank starts to poll on the local address where this signal is to be deposited waiting for the value to become something other than the initial state. Reading this address must be done carefully as to not allow the compiler to cache the value in a register. We do this by performing the read on the signal address from a simple C routine, designed to prevent any register caching from the calling Fortran code. Once the signal value is modified, the value is interpreted as the message length. If it is zero there is no message to transfer, which can happen for a variety of reasons due to the generic messaging property of the MPI-RMA Engine. If the value is greater than zero the neighbor rank initiates a network transfer with an . While the network transfer is in flight, both the source and neighbor rank do other communication or computation work. At some point later the neighbor rank needs the transfer to complete and calls . The message is then unpacked, and the neighbor rank then sends a signal pack to the source rank indicating that the transfer is done and the source buffer can be freely modified. The source rank eventually polls on that signal before it can repeat the full process over again. We note that an alternative implementation of this cycle could be done entirely with . In such a design, a call would immediately move data to the destination rank completed sometime later with . Then the initial above is used to signal that data is in the destination buffer. We did not use this design because it has the potential for generating more intense many-to-one traffic patterns, which can lead to degraded performance on most HPC interconnects. The MPI-RMA Engine cycle applies to each segment of the single communication buffer that was created for the RMA window with . Multiple segments in this buffer allow for many unique messages to be exchanged with neighbor ranks. They also present potential thread parallelism for communication. The MPI-RMA Engine cycle is self-contained and can be applied to any number of independent messages to be exchanged with minimal contention or protection required between them. Multiple threads can therefore drive the engine entirely independently as long as they operate on separate buffer segments. The single buffer attached to the RMA window in WOMBAT is decomposed into multiple regions each available for communicating Patch boundary data. Figure <ref> shows the anatomy of this buffer with an example of a 2d Cartesian domain with 9 MPI ranks (similar to the domain structure in Figure <ref>). The figure begins at the top looking at the entire RMA buffer logically separated into equal size segments for each of the 8 neighbors (labeled N0 through N7) any rank might have. Note that it is possible for some of the logical neighbors to be the same MPI rank if the world grid is periodic. Each of these segments is further divided based on a run-time tunable value for the number of “mailboxes” dedicated to each neighbor rank. Increasing the number of mailboxes has the effect of putting more network transfers in-flight at any moment, which can reduce the number of iterations in the Engine. Each of these mailboxes is large enough to buffer all boundary data to and from one Patch (size is doubled for send and receive). It is not necessary or common that this data be from the same source Patch. In one of these mailboxes, there are 8 boundary segments corresponding to the 4 edges and 4 corners that will be communicated in 2d from a Patch labeled B0 through B7. A single section of a boundary segment, there are 4 distinct sections. The first two are each 8 Bytes in length and are used for the incoming and outgoing signals described above and in Figure <ref>. The next “header” section is used to encode descriptive data about the message payload. This information includes identifying information for the Patch that should receive this boundary data. The header can be leveraged for performing other communication that might be useful to exchange between rank on a regular basis, such as load imbalance statistics or changes in the ownership of a given Patch. The final section in the boundary segment is the message payload. The MPI-RMA Engine also has methods for initiating and completing non-blocking global reductions. They are used to compute time step sizes across all MPI ranks. Our implementation delays time step calculation by one step in order to overlap the collective with work. §.§.§ MPI-RMA Thread Optimization in Cray MPICH In the SPMD OpenMP model, threads do their computation, message sending, and message completion asynchronously, so contention on the interconnect resources becomes relevant to performance. On Cray XC systems the Aries interconnect provides 128 hardware “lanes” called communication domains (CDMs) for concurrent message transfers and synchronizations (although MPI does not always make use of all of them). The MPI library assigns these CDMs either statically to threads the first time a thread makes an MPI call, or dynamically each time a message is sent or completed. In SPMD OpenMP, static assignment of CDMs to threads is not feasible anymore, because it provides no means for the MPI library to dynamically minimize contention. For example, if a thread needs to complete all messages targeting a specific remote rank, it may need access to several CDMs that have been statically assigned to other threads before. Safe access to these CDMs could be handled with a mutex, but doing so can force other threads to wait for access to the CDM before sending a message. Hence, dynamic allocation of CDMs is required to minimize overhead from CDM contention and maximize performance. We have adapted Cray's MPI-RMA implementation to use lock-free dynamic allocation of CDMs. It is now designed specifically to minimize overhead due to CDM assignment and maximize performance for SPMD approaches. The library guarantees contention-free communication as long as the number of concurrent requests to send and/or complete a message does not exceed the available number CDMs. § NUMERICAL METHODS For an initial implementation in the code design discussed above we use a 2^nd order, directionally un-split version of the non-relativistic ideal MHD solver described in RJ95 and <cit.> referred to as MHDTVD. This new implementation follows the CTU+CT scheme, described in <cit.> (hereafter GS05) and <cit.> (hereafter GS08), modified for the MHDTVD solver. The algorithm outlined here solves the equations of MHD neglecting charge separation between ions and electrons, electrical resistivity, viscosity, and non-adiabatic processes, such as thermal conduction. With these assumptions the ideal MHD equations are ∂ρ/∂ t + ∇· (ρ𝐯) = 0, ∂𝐯/∂ t + 𝐯·∇ 𝐯 + 1/ρ∇P - 1/ρ(∇× 𝐁) × 𝐁 = 0, ∂ P/∂ t + 𝐯·∇P + γ P ∇·𝐯 = 0, ∂ B/∂ t - ∇×(𝐯 × 𝐁) = 0, where γ is the plasma adiabatic index. Following the convention of RJ95, we have selected our units such that 4π does not appear in these equations. For a one-dimensional flow along the X direction, Equations <ref> - <ref> can be written in the conservative form ∂𝐪/∂ t + ∂𝐅/∂ x = 0, where 𝐪 and 𝐅 are the state vector and flux vector respectively defined as 𝐪 = ( [ ρ; ρ v_x; ρ v_y; ρ v_z; B_x; B_y; B_z; E ]), 𝐅 = ( [ ρ v_x; ρ v_x^2 + P^* - B_x^2; ρ v_xv_y - B_xB_y; ρ v_xv_z - B_xB_z; 0; B_yv_x - B_xv_y; B_zv_x - B_xv_z; (E + P^*)v_x - B_x(B_xv_x + B_yv_y + B_zv_z) ]). The total pressure and total energy are given by P^* = P + 1/2(B_x^2 + B_y^2 + B_z^2) E = P/γ - 1 + ρ/2(v_x^2 + v_y^2 + v_z^2) + 1/2(B_x^2 + B_y^2 + B_z^2). A source term vector can be added to Equation <ref> to include additional physics, such as gravity, geometry corrections, cooling, and cosmic-ray feedback. This system of equations is hyperbolic under the definition that the Jacobian matrix, 𝐀 = ∂𝐅 / ∂𝐪, has all real eigenvalues and a complete set of right eigenvectors. This system is not strictly hyperbolic, however, due to conditions that can produce degenerate eigenvalues. The seven eigenvalues a_1,7 = v_x± c_f, a_2,6 = v_x± c_a, a_3,5 = v_x± c_s, and a_4 = v_x correspond to three MHD wave families and an entropy mode. The characteristic wave speeds are c_f = (1/2[a^2 + B_x^2 + B_y^2 + B_z^2/ρ + . . . . √((a^2 + B_x^2 + B_y^2 + B_z^2/ρ)^2 - 4a^2B_x^2/ρ)] )^1/2 c_s = (1/2[a^2 + B_x^2 + B_y^2 + B_z^2/ρ - . . . . √((a^2 + B_x^2 + B_y^2 + B_z^2/ρ)^2 - 4a^2B_x^2/ρ)] )^1/2 c_a = √(B_x^2/ρ), where the sound speed is defined as a = √(γ P / ρ). One of the difficulties in solving Equation <ref> is that some of the eigenvalues will coincide in limiting cases and special care must be taken to avoid singularities around points where B_x = 0 or B_y = B_z = 0 (RJ95). We summarize the one dimensional MHDTVD algorithm in <ref>. §.§ MHD in Two Dimensions The 2d directionally un-split update closely follows the steps for the CTU+CT scheme described in GS05. Our implementation utilizes 5 boundary zones, which requires only one boundary exchange per time step for both state variables and zone corner EMFs. Given a time step Δ t, the steps in the algorithm are: Step 1 Compute the directionally split fluxes in both X and Y directions using initial states 𝐪^n from Equation <ref> for a time step Δ t. Step 2 Compute a zone-centered reference EMF for use in the mid-time step constrained transport update of the face-centered magnetic field. The EMF is given by v_x × B_y + v_y × B_x with each input derived from the initial state vector 𝐪^n. Step 3 Using the upwinded algorithm in GS05, compute EMF values at zone corners using the B_y and B_x fluxes from the X and Y passes from Step 1 and the reference EMF from Step 2. Step 4 Update the face centered magnetic field 𝐛^n to 𝐛^n+1/2 from the EMFs in Step 3 over Δ t / 2. Step 5 Update the zone centered state vector from the initial states 𝐪^n to 𝐪^n+1/2_x using fluxes from the Y pass in Step 1 applied over Δ t / 2. Include the ∇· B source term vector described by GS05. Step 6 Using the preconditioned state 𝐪^n+1/2_x, compute fluxes along X from Equation <ref> for a time step Δ t. Step 7 Repeat steps 5 and 6 for the Y direction. Step 8 Compute a zone-centered reference EMF for use in the final CT update of the face-centered magnetic field. The EMF is given by v_x × B_y + v_y × B_x with v_x and v_y coming from an un-split update of 𝐪^n to 𝐪^n+1/2 using the fluxes from Steps 6 and 7. Step 9 Using the upwinded algorithm in GS05, compute EMF values at zone corners using the B_y and B_x fluxes from the X and Y passes from Steps 6 and 7 and the reference EMF from Step 8. Step 10 Use an un-split update of the state vector 𝐪^n to 𝐪^n+1 using fluxes from Steps 6 and 7 applied over Δ t. Step 11 Update the face centered magnetic field 𝐛^n to 𝐛^n+1 from the EMFs in Step 9 over Δ t. Update the zone centered magnetic field from averages of the face centered magnetic field as described in GS05. §.§ MHD in Three Dimensions The 3d un-split update is based on the 6-solve algorithm described in GS08. We again utilize 5 boundary zones as described above for 2d. The steps in the 3d algorithm are: Step 1 Compute the directionally split fluxes in the X, Y and Z directions using initial states 𝐪^n from Equation <ref> for a time step Δ t. Step 2 Compute zone-centered reference EMFs for use in the mid-time step constrained transport update of the face-centered magnetic field using inputs derived from the initial state vector 𝐪^n. Step 3 Using the upwinded algorithm in GS08, compute EMF values at zone corners using the magnetic fluxes from the directional passes from Step 1 and the reference EMF from Step 2. Step 4 Update the face centered magnetic field 𝐛^n to 𝐛^n+1/2 from the EMFs in Step 3 over Δ t / 2. Step 5 Update the zone centered state vector from the initial states 𝐪^n to 𝐪^n+1/2_x with an un-split update using fluxes from the Y and Z passes in Step 1 applied over Δ t / 2. Include the ∇· B source term vector described by GS08. Step 6 Using the preconditioned state 𝐪^n+1/2_x, compute fluxes along X from Equation <ref> for a time step Δ t. Step 7 Repeat steps 5 and 6 for the Y and Z directions using the appropriate transverse fluxes from Step 1. Step 8 Compute zone-centered reference EMFs for use in the final CT update of the face-centered magnetic field. Velocity values come from an un-split update of 𝐪^n to 𝐪^n+1/2 using the fluxes from Steps 6 and 7. Step 9 Using the upwinded algorithm in GS08, compute EMF values at zone corners using the magnetic fluxes from the directional passes from Steps 6 and 7 and the reference EMF from Step 8. Step 10 Use an un-split update of the state vector 𝐪^n to 𝐪^n+1 using fluxes from Steps 6 and 7 applied over Δ t. Step 11 Update the face centered magnetic field 𝐛^n to 𝐛^n+1 from the EMFs in Step 9 over Δ t. Update the zone centered magnetic field from averages of the face centered magnetic field as described in GS08. § TEST CALCULATIONS §.§ Linear Wave Convergence We performed linear wave convergence tests using eigenvectors of the Roe matrices for hydrodynamics and MHD following the setup used by GS05. For one-dimensional tests, we use a periodic domain L = 1 divided into N zones containing a background fluid with ρ = 1, P = 3/5, and γ = 5/3. The background is at rest for shear and entropy waves, otherwise v_x = 1. For hydrodynamic waves (sound, v_y and v_z shear, and entropy modes), B_x = B_y = B_z = 0, while the background for MHD waves (slow, Alfvén, fast, and entropy modes) has magnetic field components B_x = 1, B_y = √(2), B_z = 1/2. A sinusoidal perturbation is applied to this background state, such that the initial state vector is given by 𝐪_0 = 𝐪̅ + A_0 𝐑_k cos(2π x), where 𝐪̅ is the background state, A_0 = 10^-6 is the amplitude, and R_k is the right eigenvector for the wave mode k. Each wave is propagated for one wavelength, and then the error in the solution is computed using the L_1 error vector averaged over every zone i, defined by δ𝐪 = N^-1∑_i|𝐪_i - 𝐪_i,0|. Increasing the number of zones up to N = 1024, the solution for each wave mode in 1d converges with second-order accuracy as seen by the norm of the L_1 error vector in Figure <ref>. We also tested the convergence of MHD waves propagating oblique to a three-dimensional grid, following the setup in GS08. The wave is initialized rotated with respect to a computational grid of size (L_X, L_Y, L_Z) = (3, 3/2, 3/2) with 2NxNxN zones, such that the wave vector is k⃗ = (1/3, 2/3, 2/3). The face centered magnetic field components are initialized via a vector potential defined at the corners of the grid zones, and then zone centered magnetic field values are averaged from face centered fields. After propagating one wavelength, the L_1 error vector is computed with respect to the initial conditions. The convergence with increasing resolution is shown in Figure <ref>. §.§ RJ95 2a The next set of tests are of the shock-tube setup 2a from RJ95. The left-hand state was initialized with (ρ,v_x,v_y,v_z,B_y,B_z,P) = [1.08,1.2,0.01,3.6/(4π)^1/2,2/(4π)^1/2,0.95], and the right-hand state with [1,0,0,0,4/(4π)^1/2,2/(4π)^1/2,1]. For this test B_x = 2/(4π)^1/2. Figures <ref> and <ref> shows the evolved grid at t = 0.2 in 1d and 3d respectively. The shock normal is rotated 45^∘ out of all primary planes in 3d. §.§ RJ95 4a Figures <ref> and <ref> show the results at t = 0.15 of the 4a setup from RJ95. Figure <ref> is the 1d result, and Figure <ref> is the 3d result with the shock normal rotated 45^∘ out of all primary planes. The left-hand state was initialized with (ρ,v_x,v_y,v_z,B_y,B_z,P) = [1,0,0,0,1,0,1], and the right-hand state with [0.2,0,0,0,0,0,0.1]. For this test B_x = 1. §.§ Brio & Wu Shock-tube We performed the well-known MHD shock tube test of <cit.> on a one-dimensional domain. The left-hand state is initialized with the state vector (ρ,v_x,v_y,v_z,B_y,B_z,P) = (1.0, 0, 0, 0, 1.0, 0, 1.0), while the initial right-hand state is (ρ,v_x,v_y,v_z,B_y,B_z,P) = (0.125, 0, 0, 0, -1.0, 0, 0.1). Throughout the domain, B_x = 0.75, while the adiabatic index γ = 2. The solution computed on a domain with 400 zones at t = 0.08 compared to a better converged solution computed with 10^4 zones is shown in figure <ref>. §.§ Orszag-Tang Vortex A very common test in 2d for an MHD code is the compressible Orszag-Tang vortex. This problem was first studied by <cit.> and is now used as a standard comparison of MHD codes <cit.>. The setup for this problem uses a periodic box with L_X = [-0.5,0.5] and L_Y = [-0.5,0.5] and 192x192 zones. Uniform density and pressure are initialized throughout the grid with ρ = 25/36π, P = 5/12π and γ = 5/3, giving a sound speed of c_s = 1. The velocity was initialized as v_x = -v_0SIN(2π y) and v_y = v_0SIN(2π x), where v_0 = 1. The magnetic field along zone faces was derived from the vector potential defined at zone corners A_z = B_0[COS(4π x)/2 + COS(2π y)]/2π, where B_0 = 1 / (4π)^1/2, with 𝐛 = ∇×𝐀. Figure <ref> shows the resulting density, gas pressure, specific kinetic energy, and magnetic pressure at t = 0.5, as well as slices of the gas pressure at y = -0.0723 and y = -0.1875. §.§ MHD Rotor Another common MHD test problem in 2d is that of a rotating disk in a magnetized medium <cit.>. We follow the setup used by <cit.> and defined in <cit.> as “Rotor Problem 1” on a periodic domain with 400x400 zones. Distributions of density, pressure, Mach number, and magnetic pressure for the solution at t = 0.15 is shown in figure <ref>, along with slices of the y-component of the magnetic field at y = 0 and the x-component of the magnetic field at x = 0. §.§ Advection of a Field Loop A powerful test of an MHD code's ability to keep ∇·𝐁 = 0 is the advection of a weak magnetic field loop. We use a setup similar to that of GS05 for a 2d calculation. A periodic box with L_X = [-1.,1.] and L_Y [-0.5,0.5] over 256x128 zones was initialized with ρ = 1, P_gas = 1, v_x = 2, and v_y = 0.5. The magnetic field was derived from a vector potential defined at zone corners as A_z = MAX(A[R_0 - r],0) where A = 10^-3 and R_0 = 0.3. This field produces a line current through the center of the loop and a return current along R_0, but these features are unresolved on the grid. Figure <ref> shows the 2d result after two periods. We perform a 3d version of this test, also shown in Figure <ref>, following the setup used in GS08. §.§ MHD Blast Wave We performed a 3d version of the 2d magnetized strong blast wave test as defined in <cit.>. The test is performed on a periodic domain with (L_X, L_Y, L_Z) = (1, 3/2, 1) using 200x300x200 zones. The fluid is initialized at rest with ρ = 1 and a uniform magnetic field (B_x, B_y, B_z) = (10/√(2), 10/√(2), 0). The fluid has a pressure P = 1, except for in the central region within r_0 = 0.125 where P = 100. Figure <ref> shows the density, specific kinetic energy, and magnetic energy of the solution in a slice through z = 0 at t = 0.02. §.§ Circularly Polarized Alfvén Wave As a final MHD test, we show the propagation of a circular polarized Alfvén wave as described by <cit.>. This test was used by <cit.> to compare the performance of various approaches to maintaining ∇·𝐁 = 0. This test can be done in one or more dimensions, and it can be used for convergence testing as it is an exact nonlinear solution to the equations of MHD. The grid is initialized with ρ = 1, P_gas = 0.1, v_y = 0.1SIN(2π x), B_y = 0.1SIN(2π x), v_z = B_z = 0.1COS(2π x), B_x = 1, and v_x = 0. For the 2d tests we rotate these properties on the grid by an angle of θ = TAN^-1(2), while in 3d tests we perform the same rotation as 3d tests in <ref>. The grid was a periodic box with L_X = [-√(5)/2,√(5)/2] and L_Y = 0.5*L_X with 2NxN zones in 2d and (L_X, L_Y, L_Z) = (3, 3/2, 3/2) with 2NxNxN in 3d. The left panel of figure <ref> shows the convergence of the L_1 error vector norm after one wave period for 2d and 3d tests with increasing resolution, where the horizontal axis represents the number of zones across the shorter dimensions. The right panel shows, using every zone in 2d tests, profiles of the in plane transverse component of the magnetic field in the rotated frame (B_2) after five wave periods, with the horizontal axis representing the x coordinate in the rotated reference frame. The lack of scatter in these plots demonstrates that the rotated wave fronts remain coherent. § CONCLUSIONS In this paper we present the design and performance of a new hybrid MPI/OpenMP astrophysical MHD code called WOMBAT. We are developing WOMBAT for broad application in astrophysics, but especially in support of investigations of cosmological turbulence and the evolution of magnetic fields in galaxy clusters, where conductive fluid behaviors must be captured with good fidelity on a very wide range of scales. This requirement demands that WOMBAT have exceptional performance and scaling on the latest generation of HPC systems. We also argue in <ref> that the ability to scale to high thread counts is crucial to maintaining high performance for the target simulations. This is particularly important for mesh refinement and N-body extensions of WOMBAT currently in development, where load imbalance is unavoidable. This work will be presented in a follow-up to this paper. The optimization strategies incorporated into WOMBAT are based on the Patch, a the basic unit of work and domain decomposition within an MPI rank. Patches are self-contained problems with their own boundary zones and meta-data necessary to evolve them in time. These properties make Patches ideal for presenting independent work to threads within a rank. We presented the SPMD OpenMP design of WOMBAT, where only a single OpenMP parallel region exists for the duration of code execution. Threads update Patches and perform all boundary communication collaboratively with the Update and MPI-RMA Engines discussed in <ref>. We present a unique enhancement of the Cray MPICH library through a co-design effort with Cray, Inc. and the University of Minnesota. The “thread-hot” MPI-RMA feature (see <ref>) results in significant speedup of WOMBAT because of its lock-free design. We show the performance characteristics of WOMBAT on several architectures including the latest generation of Intel Xeon Phi “Knights Landing” processors. WOMBAT scaling on these architectures up to 260K threads on Blue Waters, demonstrates its capabilities and adaptability. § ACKNOWLEDGEMENTS PJM thanks Luiz DeRose (Cray) and John Levesque (Cray) for their support of this project. JD acknowledges support from the People Programme (Marie Sklodowska Curie Actions) of the European Union’s Eighth Framework Programme H2020 under REA grant agreement no. [658912]. PE is supported by the ITC and Harvard FAS Research Computing. TWJ and BJO acknowledge support from NSF grant AST1211595. CN was supported by an NSF Graduate Fellowship under Grant 000039202. We thank Cray, Inc. for use of their internal systems. Blue Waters computing resources came through a grant from the Great Lakes Consortium for Petascale Computing. The Blue Waters sustained-petascale computing project is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the state of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. § MHDTVD Integrating Equation <ref> over a volume element and over a time interval gives 𝐪_i^n+1 = 𝐪_i^n - Δ t/Δ x(𝐅_i+1/2^n+1/2 - 𝐅_i-1/2^n+1/2). In the MHDTVD method, an approximation to 𝐅_i+1/2^n+1/2, referred to as the modified flux 𝐮̅_i+1/2^n+1/2, is computed from 𝐮̅_i+1/2^n+1/2 = 1/2(𝐅(𝐪_i^n) + 𝐅(𝐪_i+1^n) ) - Δ x/2Δ t𝐟_i+1/2^n, 𝐟_i+1/2^n = ∑_k=1^7β_k,i+1/2𝐑_k,i+1/2^n, β_k,i+1/2 = Q_k(Δ t^n/Δ xa_k,i+1/2^n + γ_k,i+1/2)α_k,i+1/2 - (g_k,i + g_k,i+1), α_k,i+1/2 = 𝐋_k,i+1/2^n· (𝐪_i+1^n - 𝐪_i^n), γ_k,i+1/2 = g_k,i+1 - g_k,i/α_k,i+1/2 for α_k,i+1/2 0, 0 for α_k,i+1/2 = 0 , g_k,i = SIGN(g̃_k,i+1/2) SWEBY _limiter(g̃_k,i+1/2,g̃_k,i-1/2), g̃_k,i+1/2 = 1/2[Q_k(Δ t^n/Δ xa_k,i+1/2^n) - (Δ t^n/Δ xa_k,i+1/2^n)^2]α_k,i+1/2, Q_k(χ) = χ^2/4ϵ_k + ϵ_k for |χ| < 2ϵ_k, |χ| for |χ| ≥ 2ϵ_k. The right-handed eigenvector, 𝐑_k,i+1/2^n, and characteristics, α_k,i+1/2, are from <cit.>. The primitive variables at zone interfaces, used to construct 𝐑_k,i+1/2^n and α_k,i+1/2, come from the averaging scheme also described in <cit.>. The purpose of ϵ_k is to add a controlled amount of dissipation into each wave to ensure that Q_k(χ), referred to as the coefficient of numerical viscosity, is continuous and positive <cit.>. This eliminates spurious oscillations that can occur when there is an entropy violation across a discontinuity. The value of ϵ_k must satisfy 0 ≤ϵ_k < 0.5, and the optimal value depends on the number of dimensions and complexity of flows in the calculation. Under certain circumstances, Roe-type methods like MHDTVD will produce unphysical densities or pressures <cit.>. A typical solution to this problem is to define floor values for density and pressure that are applied when exceeded. WOMBAT uses this approach, but additionally offers a set of user-defined floor values, called the protection floor, that will automatically switch to another Riemann solver that does not have this issue. Similar to the approach of GS08, we substitute the MHDTVD fluxes with the more diffusive HLL fluxes <cit.> under the rare conditions when the protection floor is exceeded. The modified flux 𝐮̅_i+1/2^n+1/2 is computed for the HLL scheme as 𝐮̅_i+1/2^n+1/2 = b^+𝐅(𝐪_i^n) + b^-𝐅(𝐪_i+1^n)/b^+ - b^- + b^+b^-/b^+ - b^-(𝐪_i+1^n - 𝐪_i^n), b^+ = MAX{MAX(a_max,v_x,i+1^n + c_f,i+1^n),0}, b^- = MIN{MIN(a_min,v_x,i-1^n - c_f,i11^n),0}, where a_max and a_min are the maximum and minimum eigenvalues. Note that the HLL fluxes do not rely on an eigensolution to the MHD equations, which makes them more diffusive than the MHDTVD fluxes. Consequently, we apply them as infrequently as possible; so only to avoid unphysical behaviors. apj

http://arxiv.org/abs/1701.07596v3

20170126072912

Aperiodically driven integrable systems and their emergent steady states

cond-mat.str-el

=23truecm =-.4truecm

http://arxiv.org/abs/1701.08128v1

20170127174504

Evaluating a sublinear-time algorithm for the Minimum Spanning Tree Weight problem

cs.DS

Extension and restriction principles for the HRT conjecture Kasso A. Okoudjou December 30, 2023 =========================================================== We present an implementation and an experimental evaluation of an algorithm that, given a connected graph G (represented by adjacency lists), estimates in sublinear time, with a relative error e, the Minimum Spanning Tree Weight of G (see <cit.> for a theoretical exposure of the algorithm). Since the theoretical performances have already been shown and demonstrated in the above-mentioned paper of Chazelle et al. our goal is, exclusively, to experimental evaluate the algorithm and at last to present the results. Some technical insights are given on the implementation of the algorithm and on the dataset used in the test phase, hence to show how the experiment has been carried out even for reproducibility purposes; the results are then evaluated empirically and widely discussed, comparing these with the performances of the Prim algorithm and the Kruskal algorithm, launching several runs on a heterogeneous set of graphs and different theoretical models for them. We assume hereafter that the reader has knowledge about the cited paper as we will just recap the theoretical results. minimum spanning tree, sublinear time algorithms, randomized algorithm, approximation algorithm, minimum spanning tree weight, experimental evaluation 68W20, 68W25, 68R10 myheadings plain GABRIELE SANTI AND LEONARDO DE LAURENTIISEVALUATING A SUBLINEAR-TIME ALGORITHM FOR THE MST WEIGHT PROBLEM § INTRODUCTION We will discuss here some preliminary observations and assumptions. First of all, we observe that we need a set of graphs that satisfies the following points: * they should be finite and should not be multigraphs; * they should be undirected; * they should have weighted edges; * the weights on the edges should be integers; since the graph is finite, it is enough to show that there exist W such that it is the maximum weight on the edges of the graph;[more generally, it is enough to have a numerable set of values for the weights which is always true when the graph is finite] * they might contain self-loops; * they have to be connected (the graph has only one connected component); * they should be represented with adjacency lists; * they should be represented in the same manner (we need an unvarying file format). Unfortunately, the graphs and their representations, which can easily be found on the Internet, don't accomplish all of this requirements at the same time, although many standards for the file format are available. Given this observation, our choice was to use randomly generated graphs, hence to implement our own graphs generator. This gives us the opportunity to generate a wide set of connected graphs, with tunable parameters, carefully chosen looking forward to the tests; these parameters include the number of nodes, the number of edges and the edges weight, nonetheless the distribution law for the edges. The edges between the nodes are step-by-step randomly constructed, respecting the connection requirement. The different types of graphs that we use in our experimental evaluation are presented afterwards. After studying the paper we made some assumptions. One of the problem we encountered is that the theoretical algorithm assumes to have as input only graph G and to have direct access to the family of graphs G_i[we recall that G_i is an induced subgraph of G such that the maximum weight on his edges is i (e.g. G_w is exactly G)]; with “direct” we intend that no computation is required to extract G_i, which is not true. In fact, we can show easily that a lower bound for the extraction of all the family is, at least, O(m) (i.e. is linear on the number of edges). A naïve approach would cost O(m) for the extraction of G_i for a given i, hence O(w m) for the whole family; a better approach could order the edges in O(m log m) and build the family in a single pass on the edges, achieving O(m + m log m). Having as input only G, it would seem that the algorithm is responsible for the extraction of the family, but this is not desirable due to this lower bound that would sabotage the overall performance. Finally we decided to consider this cost as a part of the construction of the data structure of the graph, to be done prior to the call of the algorithm. § DESIGN STUDIES AND CHOICES §.§ Random Graphs Generator As mentioned before, our choice is to implement our own graphs generator. The aim is to generate connected random graphs with a specific set of parameters, like the number of nodes, the number of edges, the maximum edges weight and the average degree. Moreover, we want to test how our algorithm behaves in different environments, so we want to control the distribution law of the edges. Keep in mind that the graph has to satisfy all the points of the previous section, among which the connectivity requirement. Given a desired number of nodes n and e ≥ n-1 edges, the key-concepts under the implementation of our connected graphs generator are the following: * we begin by generating a random permutation of the vertices v_0, v_1, … v_n * we generate a random spanning tree by iteratively adding edges in this vector in a uniform random manner; suppose we have added τ vertices to the tree T, we randomly select a vertex s in the set { T_0, …, T_τ} and add a new edge <T_s, v_τ+1>. At the end we will have an acyclic graph with n-1 edges. * following a certain probability law, we add the remaining e - (n-1) edges note that every time we add an edge, a weight is associated to it, with a value uniformly choosen in [ 1, w ]. We even decided to use a custom file format to save the data, which is a file, standing for space separated values: every line of the file corresponds to two edges, reporting the values <v_s, v_t, w> that means source and target nodes, edge weight. Being the graph undirected, it is implicit that the edge <v_t, v_s, w> also exists. §.§ Graph Data Structures We implemented two versions of the algorithm: the first time using the well-known Boost's BGL[, <cit.>] for the graphs data structures and efficient implementations of Kruskal's and Prim's algorithms; the latter instead embodies our own implementation of both the structures and all the side algorithms. We decided to do so in order to obtain more control over the code for testing purposes and because, at the moment (version ), the BGL subgraphs framework presents some still unfixed bug. Unfortunately, our Kruskal algorithm, although using union-find structures with path compression[it is known that the Kruskal algorithm time complexity is O(m log m), but can be improved to O(m log n) using union find structures with some euristic conveniently applied], is not as fine-tuned as that proposed by the Boost's libraries; we didn't take further time on this because on the other side, our version of Prim's algorithm shows the same exact performances of the Boost version and it counts as a theoretical lower bound, being an optimal algorithm and way better than the Kruskal's solution. Our data structures have been called FastGraph for their optimization over the operation required for the test. In any way our data structures can nor do “help” the CRT[Short for B. Chazelle, R. Rubinfeld, and L. Trevisan.<cit.>] algorithm in improving his time complexity. §.§ Tuning the algorithm In the implementation of the algorithm, the graph stored in the text file is read into a structure, which is of course a representation by adjacency lists. We want to compare the performances of CRT algorithm with a standard MST algorithm that in addition computes the total weight. We want to emphasize now that the CRT algorithm is based on probabilistic assumptions; some parameters, that depend asymptotically on ε should be selected carefully in order to provide a good approximation very fast. These includes: * r, the number of vertices uniformly choosen in the “approx-number-connected-components”. This number is critical for the performances, as it directly determines the running time of the entire algorithm. * C, the large costant that we use to pick the number of vertices determinants for the application of the original paper's Lemma 4. Both these values largely affect the overall performances because they undirectly decide how many BFS will be carried out by the CRT algorithm; the BFSes represent the fundamental cost driver of the entire run. Initially in our opinion, the choice of these parameters must depend on the number of vertices in a way that they are dynamic, keeping their dependency on ε. We tried to untie this bond to n but having static values for this parameters showed poor performances and a relative error exploding too fast as we grow the input instance. As the paper says, the only requirement for r is that it is O(1/ε^2) and C ∈ O(1/ε). The demonstration reported on the original paper bound these values in a way that helped us on choosing the right function to compute them; in fact, we have that[see Theorem 7 and Theorem 8 of <cit.>] r √(w/n) < ε < 1/2, C/√(n) < ε < 1/2 hence we choose r = **√(n/w)ε - 1 /ε^2∈ O ( 1/ε^2) C = **√(n)ε - 1/ε∈ O ( 1/ε) §.§ Random Sequences Another problem we encountered was the random selection of k distinct vertices out of n with k<n; after some research, we found that this kind of problem cannot be solved in sublinear time on the number of vertices, which can't be done for the same reasons exposed in the introduction about the subgraph family. The problem here is that we can't extract with no reinsertion k values without using an external data structure; this structure has a linear cost for the maintainance that depends on n. A special case is that in which n is a prime that fulfills certain properties; the key idea is to make use of the properties of the quadratic residues of n[this is widely used in cryptography for “format preserving encryption”; see <cit.> and <cit.>]. That permits us to extract a non-repeating random value in constant time in the range 0 … n-1; sadly, this solution is not affordable here because we need a dynamic range for each call, so that we cannot fulfill such constraints for n. The solution we found is to use Fisher-Yates sequences[originally decribed in <cit.> and then redesigned for computer use in <cit.>] which permits us to prepare in advance the sequences and then get different, distinct values at each call in constant time and with dynamic bounds. The cost of the preparation of those sequences is linear and is not considered in the total cost. § IMPLEMENTATION CHOICES We choose to implement the algorithm in C++ because we wanted a language that offers the advantages of an OOP language, not least a faster development, that had a standard library and sufficiently “near” the machine to have a tighter control over the performances and the memory occupation (e.g. performances high variance due to overhead of a VM or an interpreter). We considered that absolute performances are not important here, whereas relative performances of the CRT algorithm and other MST algorithms are the focus, so finally the choice of the language is something not of the greatest importance. The IDE tool used for the implementation is JetBrains Clion. We also used GitHub, as our code versioning control system. Also, as already mentioned, we did use the Boost library to extend the STL that C++ already offers; we did implement FastGraph in place of BGL also to address memory issues in relation to the method Boost uses to store subgraphs of a graph; we in fact used an advanced method to store the family of subgraphs G_i, i = 0,1,…,w that simply store the difference of vertices between them, Δ_G_i := V(G_i) - V(G_i-1). It is always possibile to reconstruct every G_i because this family only has induced subgraphs, and this cost is not taken into account in the computation of the total time. The main function of the implementation is in the “AlgoWEB.cpp” file. Launching the program from this file allows us to run either the CRT algorithm or the Prim algorithm,or the Kruskal algorithm, and to view either the running time or the computed weight of the Minimum Spanning Tree. It takes some argument in input, namely the file containing the graph, a suitable value for ε and the path where to save the results of the single run. Besides, it is possible to run the random graph generator by the utility we developed apart. It takes many arguments in input, that you can find just by calling §.§ Random Graphs Models In order to give exhaustive results, we designed to produce 4 classes of random graphs: * Erdős-Rényi model, that builds random graphs using a uniform distribution of the edges over the vertices; his average degree d ≈2m/n; * Gaussian model, that builds random graphs with a “cluster zone” where the edges are more frequent, hence the gaussian shape of the degree distribution; we have still d ≈2m/n; * Barabási-Albert model, that builds random scale-free graphs. The average degree is d ≈m/n; * Watts-Strogatz model, that builds random graphs with small-world properties and d ≈2m/n. We want to emphasize here the fact that for the Barabási-Albert model the average degree results to be different respect to the other models; this is due to the algorithm the scientific literature offers to build such graphs. For the other models is always possible to add an arbitrary number of edges keeping the theoretical properties valid; having, on the contrary, for the Barabási-Albert model a certain probability p_k at each step to successfully add an edge, it is not possible to build a graph with an arbitrary number of edges; the user can solely give the number of vertices. But then again, the theory states that if the algorithm starts with a complete graph of m_0 vertices (hence m_0 - 1 edges), it will produce a Barabási-Albert graph whose average degree is scaled by this quantity. Our initial complete graph has m_0 = 2 vertices, so we will have d ≈2m/m_0 n = m/n. A little insight on the proof is the following: the distribution of the degree in the Barabási-Albert model is a power law with a cubic exponent. Fortunately, in that case this distribution has a well defined mean. Applying a little of arithmetic we can easily see the truthfulness of what stated previously. This difference in the models has to be bore in mind when reading the test results, because when comparing the performances over a Barabási-Albert graph with n vertices and m edges and any other one of a different model with same number of vertices and edges, we will have different average degrees. Since the CRT algorithm is designed to only depend on the latter and not on m nor n, a multiplicative factor of 2 is to be taken into account. Our implementation provide a memory-aware[it calculates the average using the progressive mean technique, avoiding arithmetic overflow] subroutine that calculates the exact value of d at each run; the values obtained so far agree with the statements above. § TESTS The following section reports the results we had launching the algorithm over a variegate dataset of graphs, as described in the next paragraph. §.§ Dataset For each random graph model listed in the previous section we decided to produce a dataset; each dataset has a “family” of graphs that differs one from each other following a pattern for the parameters. Precisely, we composed the dataset grouping sets of graphs based on the value of the parameters, following the rules: * build one set of graphs for each model (); * every set contains in turn other sets, one for each selected value of n, i.e. the number of nodes (5000, 30000, 55000, 80000, 105000, 130000, 155000, 180000, 205000, 230000); * then again, for each selected value of d, i.e. the average degree (20, 100, 200); this way we also determine the desired value for m, since we have, for our models and for the fact that m ∝ n, a proportion between the two. It's easy to see that d 2m/n, so we have the selected values for m (10, 50, 100 times n)[operatively we fixed the desired number of edges m, so d is a random variable with mean 2m/n]; * finally, a set for each selected value of w, i.e. the weight (20, 40, 60, 80). In conclusion we have × × × for a total of 4 · 10 · 3 · 4 = 480 graphs. This dataset, made of plain text files as already described, is quite heavy: 42.1 GiB of data. §.§ Runs Using the dataset just described, we used a pattern for the runs, in order to have a complete view of the behavior of the algorithm in the domain of the various parameters; every single result consists of three plots, and we obtained, inspired by the structure of the dataset: * a set of results for each model; * this set containing in turn a set for each selected value of ε (0.2, 0.3, 0.4, 0.49999); * then a set for each selected value of d/2 ≃m/n (10, 50, 100); * finally a set for each selected value of w. The first two plots report the absolute and relative error of the CRT result compared to the correct one, calculated with Prim's algorithm; the third report the trend of the used time. As we mentioned, we indeed used Kruskal too but it's not reported here for it was not faster respect to Prim's times and unuseful for the computation of the MST weight, since already done with Prim's method. We had this way a total of 3 · 4 · 3 · 4 = 144 plots, or better 48 different cases of study. A comparison between those cases we consider meaningful finally concludes our work, as reported below in section <ref>. As mentioned before, the parameters should be selected carefully in order to provide a good approximation very fast. In this sense ε plays a crucial role here: since the CRT algorithm is a probabilistic algorithm and ε is indeed the driver parameter either for performances and for the accuracy of the estimate MST weight, it does not make sense to choose values too small for this parameter, as the performances could dramatically degrades (as, indeed, is expected). So, although it could be of interest to study the right limit of 1/2, we empirically noted that values of ε below 0.2 shows undesired behaviour of the CRT algorithm, either for the computed MST weights and for the running times. This is not unusual dealing with theoretical algorithms that shows asymptotical properties; the class this algorithm belongs to is known as property testing algorithms, whose requirement is to have a query complexity much smaller than the instance size of the problem. Given that and the fact that the algorithm is not required to compute an exact value, but to estimate a value probabilistically near to the correct one[iff the property we are looking for is a probabilistically checkable proof, see <cit.>], we are not surprised that if the instance of the problem is small, the algorithm shows worse performances respect to a “deterministic” method. Because of this, results for ε < 0.2 were not comprehensive nor interesting and are not reported. Given the intrinsically probabilistic nature of the CRT algorithm, we had to launch several runs on the same graph to have a good estimate of the CRT running time. For this purpose we decided to launch 10 runs for each graph of the dataset, and to take the average running time as the estimate of the running time; the amount of runs has been decided as a compromise between having a low variance between execution time's mode and mean values, and keeping a restrained amount of tests to do over the whole dataset. For the output value of the approximated MST weight we instead took one over the ones computed, randomly, to preserve the information on the tolerance of the estimate. § RESULTS Following there are considerations about the meaningful line charts regarding the results. We know that the CRT time complexity is O(dwε^-2logdwε), where d is the average degree of a graph; on the other hand, we know that the accuracy depends on ε also, so we expect an inversely proportional relation with the running time. Therefore what we expect is that: * by increasing one or more of the parameters d, w, there should be a worsening of the average running time; * keeping all the other parameters unchanged, if we consider an increase in the value of ε there must be an improvement of the running time (as well as a worsening of the result, though); * viceversa we expect the opposite behaviour if we decrease d and/or w or decrease ε with the other parameters unchanged. Let us call the above crucial parameters. What we still need to know is what happens to the error; we should expect a direct proportion with the running times, so the above considerations could be also valid for the error. On the contrary, we'll see that this is not exactly respected. First thing to show is that the CRT algorithm is sublinear in the number of edges, hence is better than any other exact algorithm. This can be easily seen in figures from <ref> to <ref>. For the rest of the plots listed from now on it is possibile to see, above each graph, the values of the parameters for the presented run. It is interesting to note that the correct value, computed with Prim's algorithm, it's linear in the number of edges (figure <ref>); we want to stress the fact that this is not the general case, and that this trend is due to the uniform distribution of the weights over the edges. We will dissect this point more in section <ref>. Given that the CRT Algorithm respects the sub-linearity constraint, let's now see the variations that occur when selectively changing other parameters. For the sake of completeness, in figures <ref> informations about Kruskal's runs are reported, yet we won't report them in the charts that follow. §.§ Variations of crucial parameters §.§.§ Average degree d Let us now see the behaviour for the variation of d; we will initially concentrate on the running time and solely for the model. The selected values of ε and w are respectivey 0.3 and 40. As we can see in figures <ref> to <ref>, there is a worsening in the performance for small instances: an increase of the average degree d is somewhat correlated to a loss of performances to the point that our property testing algorithm needs more time that the deterministic one; still, that seems to be true under a certain dimension of the instance of the graph, so that we don't lose the truthfulness of the theoretical time complexity because for a fixed d^* it will always be possibile to find empirically a certain number of edges m^* ∝ d^* beyond which the running time function is always below C · dwε^-2logdwε for a certain C.[that comes directly from the definition of asymptotic complexity] We want here to highlight the crucial fact that the algorithm behaves better on big instances, where with “big” we refer to the parameters the performances depend on, namely d, w. Almost all the trends reported in this paper, in fact, show this initial “bad” curve and, from a certain value onward, a sublinear behaviour. We will discuss further about this in section gbb. We would speculate that the error could be related to this. Indeed in figure <ref> we can see that to an increase of d corresponds a dramatic annihilation of the error; to explain this point we use a simplified example. Let us consider a minimal complete graph[a connected graph with the least number of edges, i.e. n-1]; the algorithm launches some BFSes on a strict subset of the graph, in more steps that could be interrupted according to certain rules, based in part on a stochastic process. It is easily provable that in this kind of graphs we have a worst case of n-1 hops between to vertices u and v; if we add a random edge to this graph, the length of this path decrease at least of one hop. By induction we can hence prove that the diameter of the graph decreases as | E(G) | grows. In other words having a stronger connection within the graph (i.e. a greater probability to visit a vertex v from u in k hops of a random walk) increases the probability to have a complete view of the graph, that is more information about the MST weight. Moreover, we saw in our study of all the results showed in this paper, that the performances of the CRT are completely untied from the number of vertices n and from the number of edges m of the input graph; this suggests us also that the error is in turn driven solely by the parameters responsible of the algorithm's complexity, as the results that follow are in fact going to prove. In figure <ref> we summarize the results for the Gaussian and Small-World models, noticing that they equate the one we showed about the uniform model. This suggests that the algorithm complexity does not depend on the dimension and clustering coefficient of the graphs, being those the main differences from one model to another. In figure <ref> we summarize instead the trend of the relative error; we see here a slightly different evolution. We cannot conclude, as we did for the time complexity, that the error doesn't suffer from the different graph model. The error in fact depends on the clustering coefficient, because it is going to grow dependently on the number of not accomplished BFSes: each one of them cause in fact a loss of information. The algorithm, as well explained in <cit.>, during the BFSes phase avoid to explore nodes that shows a high degree and even stops when encounters hubs. In other words, having equals values of d in two different runs of the algorithm, we see that its time complexity trend remain the same; assuming that δ is the sample mean of the vertices degree, this tells us that the time complexity is bound to the average of δ and, since we have a growth of the error on graphs that contains hubs, we also conclude that the relative error is bound to the variance of δ. §.§ Maximum weight w Here we will manipulate the value of w, similarly to what we have already done with the average degree. Figures <ref> and <ref> are hereafter proposed; this time the other fixed values are ε = 0.4, d = 50. Still this graphs has been build using a uniform model. This time we see the error growing as w increase. So we see here a direct proportion of the execution time with the maximum weight, unlike the inverse proportion it had with d. This is due to the fact that every iteration of the subroutine that the reader can find in the original paper and remembered in pseudocode <ref>, adds a further approximation to the final result, because approximates the addend ĉ that contributes to the total approximation, that is, the final error. We see also that the dimension of the initial curve described here so far, grows proportionally to the worsening of the excution time's trend. This evidence also is observable in all the trends studied. §.§ Error tolerance ε We will test our algorithm for the values of ε = 0.2, 0.3, 0.4, 0.49999 over uniform generated graphs. As already explained, no values below 0.2 are investigated. We see in figures <ref> and <ref> the trends. As expected, we do note that the time trends tend to decrease as ε increases, since we tolerate a higher error for the computed value, so the algorithm is less aggressive on computation and takes less time. For the error trend instead we note an increase, still as expected; figures from <ref> to <ref> has to be read with attention because every function graph has a different scale. The reader must not confuse the apparent lowering of the error since it is not the case. Looking at the other graphs about the absolute error, from <ref> to <ref>, we see an expansion of the tolerance cone (light blue and ochre yellow lines) as increasing ε means admitting an higher deviation from the correct MST weight value. Here too the different scaling must not confuse about the increasing trend of the error. We see that the result is coherent with the theoretical model as the error increases with ε, but his variation is, after all, contained. §.§ Variations of graph model As a last comparison, instead of varying one by one the crucial parameters and see the time and error trends, we fix all of them and try to change the model the graph belongs to. Figures model_variation show those results. We see here that both uniform and small-world models keep the same trend for the error, but the small-world one behaves slightly better on small instances. On the contrary, the gaussian show the same time trend respect to the uniform case, but his error has a higher growth curve. The scale-free model seems to be the worse case both regarding time and error trend, but we might remember, as observed earlier at ddd, that a real term of comparison requires to considerate a scale factor of 2 as done in figure model_variation_part. We see in fact that, compared to the uniform case, the scale-free model has even a better behaviour, looking carefully at the function graph scale: but after all, both of them show sublinear complexity for instances of more than 10^7 edges, so have both the same transients as the same steady state trend. Another thing we note is that a bad trend in execution times are always bond to an explosion of the error, as we can see in the charts so far. This means that using more time to compute the value doesn't mean it will be nearer to the correct one. § A SPECIFIC CASE OF STUDY At this point, all of our graphs show a bad initial curve every time we burden the input instance; in a specific case of study this anomalous curve was so persistent that all the function graph was more than linear; we decided to deepen this special case of study, performing a longer run to see the tail of this particular trend. The results are reported on figure long_run. We can see that the original case of study hinted a sublinear trend beyond 20 millions edges instances, but we considered to investigate further, and figure <ref> confirms the sublinearity. We have a good behaviour on the error trend (figure <ref>). § CONCLUSIONS AND FINAL THOUGHTS As expected, a probabilistic algorithm like the CRT allows us to compute an approximation of the Minimum Spanning Tree weight in sublinear time on the number of edges, under certain conditions. Tunable parameters, that depends on ε, allows us to perform either a better or a worse approximation, implying respectively a very slow and a very fast computation. The choice of a small value of ε can lead to terrible running times, and for these values it does not make sense to compare the CRT algorithm with any other deterministic algorithm. For other ε values, instead, we prove the good performances of the CRT. The reader can easily view the better performances of CRT algorithm versus Prim algorithm or Kruskal algorithm watching the line charts in the previous section of this paper. More in general, we see that execution time and error depend on the number of BFSes successfully completed during the computation The more BFSes are completed, the more information the algorithm has to predict a correct value for the MST weight, but on the other hand, completing the BFSes takes time. If instead we have a lot of interrupted BFSes, we waste a large amount of time without gathering information, hence resulting in both high execution time and error. We considered so far different theoretical graph models, and we conclude that a high clustering coefficient tends to increase the probability to have interrupted BFSes. This because one of the reasons the algorithm has to interrupt the search is having encountered a hub, i.e. a vertex with a high degree. We saw in fact that when changing the model there is a slight perturbation of the trend, although it remains sublinear. The key concept is the “distance” from the hubs of the graph of the root vertex from which our BFS starts. Generally we also concluded that increasing the average degree let our algorithm gather more information, because there is a growing of the probability to visit the generic node u of our graph. On the other side, increasing the maximum weight correspond to an increase in the number of iterations our algorithm does, that leads to summing more intermediate approximated results that imply a higher final approximation. §.§ Parallel implementation We observe that the CRT algorithm lends itself very well to a parallel implementation. Indeed the majority of the algorithm's code is organized into independent sections, and in most cases they don't need to comunicate to each other. We also observe that three levels of paralellism can be achieved within the code. In the first level we parallelize each of the w independent calls to , as depicted in pseudocode <ref>, below; every of this calls internally performs r independent BFSes from r different roots, that could in turn run in parallel, achieving a second level of parallelism. Moreover, considering that in the academic world there already exist different parallel implementations of the BFS algorithm, we can use one of them to perform an additional third level of paralellism. To make it even more simpler, the number of different flows is known a priori, so a static pre-instantiation and a smart scheduling of the threads can be performed. At the first and second levels a master-slave model can be used in a fork-join structure, while in the third level a shared variable is needed between the different BFSes. As a final remark, parallelizing the BFSes could have too much overhead given that the algorithm is optimized to run them very fast and to stop the ones that seem to cost too much. § FUTURE PROSPECTS During an e-mail exchange with one of the original authors of <cit.>, Dr. Ronitt Rubinfeld, another topic of discussion and study has emerged, about the distribution of the weight on the edges. In our code, the generation of random graphs only assume a uniform distribution of the weights on the edges, i.e. a given edge can have an integer weight k ∈ [1, w] with probability 1/w. That implies a linear growth of the dimension of E(G_i), ∀ i ∈ [1, w], namely the set of G_i's edges; this is well depicted in figure <ref>, where, as i grows, the size of E(G_i) increases at each step of a quantity “near” | E(G) | /w, and it is more true as | E(G) | is big for the law of large numbers. On the other side, having a generic law of distribution for the edges weight implies having a different behavior as depicted on <ref>. This difference could be of interest because it means that the input of different subsequent iterations of will have a non regularly increasing size, or even the same size for some calls; it can be easily shown indeed that the function in <ref> is nondecreasing. Since the cost of those calls finally determines the overall cost, we might argue that this could lead to a minor difference, but at the same time think that only with another set of tests we could conclude something relevant about this observation. 10 crt B. Chazelle, R. Rubinfeld, and L. Trevisan, Approximating the minimum spanning tree weight in sublinear time. SIAM J Computing, 34, 2005. sta R. Rubinfeld, A. Shapira, Sublinear Time Algorithms. SIAM Journal on Discrete Mathematics, 2011, Vol. 25, No. 4 : pp. 1562-1588. bst Beman Dawes, David Abrahams, Rene Rivera, Boost C++ Libraries. http://www.boost.org/. fys Fisher, Ronald A.; Yates, Frank (1948) [1938], Statistical tables for biological, agricultural and medical research (3rd ed.). London: Oliver & Boyd. pp. 26–27. bbs Lenore Blum, Manuel Blum, Mike Shub; Comparison of Two Pseudo-Random Number Generators. Advances in Cryptology: Proceedings of CRYPTO '82 pp. 61-78, Plenum 1982. cafd John Black, Phillip Rogaway; Ciphers with Arbitrary Finite Domains. Topics in Cryptology - CT-RSA 2002, The Cryptographer's Track at the RSA Conference, 2002, San Jose, CA, USA, February 18-22, 2002, Proceedings, pp. 114-130 fysc Durstenfeld, R. (July 1964), “Algorithm 235: Random permutation”. Communications of the ACM.

http://arxiv.org/abs/1701.08132v2

20170127175930

Asymptotic mapping class groups of closed surfaces punctured along Cantor sets

math.GT

images/ shapes,arrows,shadows decorations.markings *namedtheorem named[1] arrows [#1]#2 footnote[#2] theoremTheorem[section] *NoNumberTheoremTheorem proposition[theorem]Proposition corollary[theorem]Corollary lemma[theorem]Lemma claim[theorem]Claim sublemma[theorem]Sublemma conjecture[theorem]Conjecture question[theorem]Question *conventionConvention *notationNotation definition construction[theorem]Construction remark[theorem]Remark case[theorem]Case definition[theorem]Definition problem[theorem]Problem example[theorem]Example quest[theorem]Question Aut Out Mod Homeo PHomeo PMod Stab HT GL int NonSep Outer Ends srank rank diam Asymptotic mapping class groups]Asymptotic mapping class groups of closed surfaces punctured along Cantor sets Universidad Autónoma de Madrid & ICMAT C. U. de Cantoblanco. 28049, Madrid, Spain aramayona@gmail.com Institut Fourier, UMR 5582, Laboratoire de Mathématiques Université Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France louis.funar@univ-grenoble-alpes.fr The first author was partially funded by grants RYC-2013-13008 and MTM2015-67781. He also acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 "RNMS: Geometric Structures and Representation Varieties" (the GEAR Network). This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 777822. We introduce subgroups _g< _g of the mapping class group (Σ_g) of a closed surface of genus g ≥ 0 with a Cantor set removed, which are extensions of Thompson's group V by a direct limit of mapping class groups of compact surfaces of genus g. We first show that both _g and _g are finitely presented, and that _g is dense in (Σ_g). We then exploit the relation with Thompson's groups to study properties _g and _g in analogy with known facts about finite-type mapping class groups. For instance, their homology coincides with the stable homology of the mapping class group of genus g, every automorphism is geometric, and every homomorphism from a higher-rank lattice has finite image. In addition, the same connection with Thompson's groups will also prove that _g and _g are not linear and do not have Kazhdan's Property (T), which represents a departure from the current knowledge about finite-type mapping class groups. [ Louis Funar December 30, 2023 ===================== § INTRODUCTION There has been a recent surge of activity around mapping class groups of infinite-type surfaces, namely those whose fundamental group is not finitely generated. These groups share many properties with their finite-type counterparts (e.g. <cit.>), but also show rather different behaviour (e.g. <cit.>). Here we will focus our attention on the mapping class group (Σ_g) of the surface Σ_g, namely the closed orientable surface of genus g≥ 0 with a Cantor set C removed. This group is related to the homeomorphism group (C) through the short exact sequence (see Section <ref> for details): 1 →(Σ_g) →(Σ_g) →(C) → 1, where (Σ_g) is the pure mapping class group, namely the subgroup of (Σ_g) whose elements fix C pointwise. In this article we study two countable subgroups _g < _g < (Σ_g), whose elements asymptotically preserve a rigid structure on Σ_g. We remark that _0 and _0 were previously introduced in <cit.> under the names and ^1/2, respectively. We now give a brief description of these groups, referring the reader to Section <ref> for a complete definition. We first need to introduce the notion of a rigid structure; for this purpose, it will be convenient to start with the genus-zero case. The reader should keep Figure <ref> in mind. A rigid structure on Σ_0 is a triple (P,𝒜, Σ_0^+), where: * P is a pants decomposition of Σ_0, * 𝒜 is a set of pairwise-disjoint properly embedded arcs such that (the closure of) every connected component of S- P, referred to as a pair of pants, is intersected by exactly three arcs in 𝒜, and * Σ_0^+ is a choice of one of the two connected components of Σ_0 - ⋃_a∈𝒜 a. Let Σ_0^1 be the result of puncturing Σ_0 once. A rigid structure on Σ_0^1 is a rigid structure on the surface (homeomorphic to Σ_0) obtained by filling in the isolated puncture of Σ_0^1. A rigid structure on Σ_g (g ≥ 1) consists of a simple closed curve α which cuts off a once-puncture surface of genus g, together with a rigid structure for the component of Σ_g - α homeomorphic to Σ_0^1. The group _g. Fix, once and for all, a rigid structure on Σ_g. A homeomorphism f:Σ_g →Σ_g is asymptotically rigid if it preserves the rigid structure outside some compact subsurface Z(f) of Σ_g; see Section <ref> for a complete definition. We define _g as the subgroup of (Σ_g) whose elements have at least one asymptotically-rigid representative. Obviously, every element of (Σ_g) with compact support belongs to _g, although the converse is not true. Indeed, denote by _c(Σ_g) the subgroup of (Σ_g) generated by compactly-supported elements, which is a direct limit of mapping class groups of compact genus-g subsurfaces of Σ_g. In Proposition <ref> we will generalize a result of <cit.> to prove that the sequence (<ref>), when restricted to _g, reads: 1 →_c(Σ_g) →_g → V → 1, where the rightmost non-trivial group is Thompson's group V (see e.g. <cit.>). This sequence reveals a fascinating connection between mapping class groups and Thompson's groups, and will be a key ingredient in the proofs of most of our results. We remark that the notation _c(Σ) is somewhat redundant, for if an element of (Σ) has compact support then it automatically belongs to (Σ). However, we will use the notation _c(Σ) to emphasize the connection with the sequence (<ref>) The group _g. The group _g is defined in a similar fashion: its elements are those mapping classes which have a representative which preserve (P,𝒜) outside some compact subsurface. Observe that _g < _g, although the inclusion is strict: for instance, a half-twist lies in _g ∖_g. For this reason, the group _g is sometimes referred to as the group of half-twists <cit.>. Using the same reasoning as above, equation (<ref>) restricts to a short exact sequence 1 →_c(Σ_g) →_g → V_2[ℤ_2] → 1, where V_2[ℤ_2] is the Higman-Thompson group V_2[ℤ_2] (see <cit.> and Section <ref>). A surprising result of Bleak-Donoven-Jonušas <cit.> asserts that V and V_2[ℤ_2] are conjugate as subgroups of (C) through an explicit homeomorphism of C (a cellular automaton). A large part of the motivation for considering _g comes from the study of smooth mapping class groups. Put a differentiable structure on the closed orientable surface S_g of genus g, and realize C as the the middle-third Cantor set on a smoothly-embedded interval on S_g. Let ^s(S_g, C) denote the smooth mapping class group of the pair (S_g,C), namely the group of isotopy classes of smooth diffeomorphisms of S_g preserving globally the Cantor set C. The following is a recent result of Neretin and the second author <cit.>: [<cit.>, Cor. 2] For every g ≥ 0, we have _g ≅^s(S_g,C). In particular, ^s(S_g,C) is countable; observe that, in stark contrast, the “topological” mapping class group (Σ_g) is uncountable. §.§ Results As we will see, the topological restrictions on the elements of _g and _g impose strong finiteness conditions on the groups. More concretely, we will prove: For every g≥ 0, _g and _g are finitely presented. We remark that the case g=0 of the above theorem was proved by Kapoudjian and the second author in <cit.>; in fact, it will serve as the base case for the inductive argument behind the proof of Theorem <ref>. In spite of the above result, we will prove that _g serves as a good approximation for the mapping class group: For every g≥ 0, _g is dense in (Σ_g). This theorem should be compared with a recent result of Patel-Vlamis <cit.> which asserts that _c(Σ_g) is dense in (Σ_g) Next, we turn our attention to the study of properties of the groups _g and _g, through the comparison with known/expected/desired properties of mapping class groups of finite-type surfaces. §.§.§ Homological stability Let S_g,n denote the surface of genus g with n boundary components. A celebrated result of Harer <cit.> asserts that, for a fixed genus g≥ 3, the k-th homology group of the mapping class group (S_g,n) is independent of n, provided that k is “sufficiently small” with respect to g (by a result of Boldsen <cit.>, k ≤ 2g/ 3 suffices). For this reason, this homology group is called the k-th stable homology group of the mapping class group of genus g. Using a translation of the proof of <cit.> to our setting, we will show: For every k ≤ 2g/3, H_k(_g, ) and H_k(_g, ) are isomorphic to the k-th stable homology group of the mapping class group of genus g. Powell <cit.> proved that (S_g) is perfect, i.e. has trivial abelianization. As a consequence of the proof of Theorem <ref>, we have: _g and _g are perfect for every g≥ 3. §.§.§ Isomorphic classification With a few well-understood exceptions, mapping class groups of finite-type surfaces are isomorphic if and only if the underlying surfaces are homeomorphic. To see this, one may compare the virtual cohomological dimension of the mapping class group <cit.> with the maximal rank of a free-abelian subgroup <cit.>. In the case of the groups _g and _g both these quantities are infinite. However, we will prove: If 0≤ g<h and 2≤ h, then there are no surjective homomorphisms _h →_g. As will become transparent, the same argument will yield that there are no surjective maps _h →_g (resp. _h →_g, and _h →_g). As a consequence of Theorem <ref>, _g ≅_h (resp. _g ≅_h) if and only if g = h. In light of this, an obvious question is: Are _g and _g isomorphic? We stress that, although the results from <cit.> might suggest a positive answer to the question above, the answer remains unknown for all values of g. In a more general situation, however, if we replace the binary Cantor set C by the set of ends of a regular tree of valence higher than 3 then the corresponding Thompson groups groups V and V[_2] might be non-isomorphic; compare with <cit.>. §.§.§ Rigidity A celebrated theorem of Ivanov <cit.> states that the mapping class group of a (sufficiently complicated) finite-type surface is rigid: every automorphism is induced by a surface homeomorphism. This has recently been extended to the infinite-type setting by Patel-Vlamis <cit.> and Bavard-Dowdall-Rafi <cit.>. Along similar lines, our next result asserts that _g and _g are also rigid. Given a group G and a subgroup H, write (G) for the automorphism group of G, and denote by N_G(H) the normalizer of H in G. We have: For every g≥ 0, (_g) ≅ N_(Σ_g)(_g) and (_g) ≅ N_(Σ_g)(_g). An immediate consequence of Ivanov's theorem in the finite-type case is that the outer automorphism group of (S_g,n) is finite; in fact, it is trivial for all but finitely many surfaces. However, in Corollary <ref> we will see that this is no longer true for the groups _g and _g. Denote by (G) the outer automorphism group of a group G, that is, the group of conjugacy classes of automorphisms of G. We will prove: For every g≥ 0, (_g) and (_g) are infinite. Let P be the pants decomposition underlying the rigid structure on Σ_0^1, and consider the element t_∞∈(Σ_0^1) obtained as the product of all (say left) half-twists about the curves of P. We further embed Σ_0^1 in Σ_g in such way that the pants decomposition P underlying the rigid structure of Σ_0^1 is sent to the pants decomposition underlying the rigid structure of Σ_g. We will show later (see Lemma <ref>) that the image of t_∞ under the homomorphism induced by the embedding Σ_0^1 →Σ_g produces an infinite-order element of N_(Σ_g)(_g) lying in the kernel of the homomorphism (_g)→(V). §.§.§ Homomorphisms from lattices Building up on work of Ivanov <cit.> and Kaimanovich-Masur <cit.>, Farb-Masur <cit.> proved that any homomorphism from a higher-rank lattice to a finite-type mapping class group has finite image (see also <cit.> for different proofs of this result). Using this result, we will prove: Let Γ be a lattice in a semisimple Lie group G of real rank at least two, where G has no compact factors isogenous to SU(1,n) or SO(1,n). For every g≥ 0, any homomorphism from Γ to _g or _g has finite image. §.§.§ Kazhdan's Property (T) A compactly generated group has Kazhdan's Property (T) if every unitary representation that has almost invariant vectors also has an invariant vector. Since we will not need any further details about Property (T), we simply refer the reader to the book <cit.> for details. It is expected <cit.> that mapping class groups of finite-type surfaces do not have Property (T). Using similar arguments to the ones in the proof of Theorem <ref>, we will observe: _g and _g do not have Kazhdan's Property (T) for any g≥ 0. §.§.§ Non-linearity A well-known open question asks whether finite-type mapping class groups are linear. The only known result in this direction is a theorem of Bigelow-Budney <cit.>, who proved that the mapping class group of the closed surface of genus two is linear, by exploiting its relation with braid groups. In sharp contrast, in Proposition <ref> we will see that _g contains an isomorphic copy of Thompson's group F, for all g≥ 0. Combining this with Theorem <ref>, we have: _g is not linear for any g≥ 0. In particular, _g is not linear either. Plan of the paper. In Section <ref> we give some basic definitions and set the notation used in the rest of the paper. In Section <ref> we will define the groups _g and _g. In Section <ref> we will explore their relation to Thompson's group V, which will be a key ingredient in the proof of many of our main results. We then proceed to prove the results mentioned in the introduction. In this direction, Theorem <ref> will be proved in Section <ref>, and Theorem <ref> in Section <ref>. Section <ref>. will deal with the proof of Theorem <ref>. In Section <ref> we will prove Theorem <ref>, while Section <ref> is concerned with the proof of Theorem <ref>. In Section <ref> we will establish Theorems <ref> and <ref>. Acknowledgements. We thank Y. Antolín, J. Bavard, L. Bowen, J. Hernández, C. Kapoudjian, T. Koberda, C. Martínez-Pérez, H. Parlier, P. Patel, J. Souto, and N. Vlamis for conversations. We are indebted to Juan Souto for suggesting the proof of Lemma <ref> and to the referee for carefully reading our paper and his numerous comments and corrections. Parts of this paper were written while the first author was visiting Yale University, to which he is grateful for its hospitality. § PRELIMINARIES In this section we recall some of the basic definitions about surfaces and their mapping class groups. §.§ Curves and surfaces Let S be a connected orientable surface, of finite or infinite topological type. If S has punctures, we will regard them either as marked points on S or as topological ends of S, and we will feel free to switch between the two viewpoints without any further mention. We will denote by S_g,n^p the compact surface of genus g≥ 0 with n≥ 0 boundary components and p≥ 0 punctures. Similarly, let Σ_g,n^p the closed surface of genus g≥ 0 with a Cantor set removed, with n≥ 0 boundary components and p≥ 0 isolated punctures. If either n or p is equal to zero we will simply omit it from the notation. By a curve on S we mean the isotopy class of a simple closed curve on S which does not bound a disk, a once-punctured disk, or an annulus whose other boundary curve is a boundary component of S. We say that two curves are disjoint if they have disjoint representatives on S; otherwise we say that they intersect. Given curves a, b⊂ S, we define their intersection number i(a,b) as the minimal number of points of intersection between representatives. Note that i(a,b) is always finite, even on a surface of infinite topological type, as curves are compact. A multicurve on S is a set of pairwise disjoint curves on S. We say that a multicurve M is locally finite if any compact subsurface of S intersects only finitely many elements of M. A pants decomposition of S is a locally-finite multicurve that is maximal with respect to inclusion; as such, its complement on S is a disjoint union of 3-holed spheres, or pairs of pants. Finally, an arc on S is a non-trivial isotopy class of properly embedded arcs on S. §.§ Mapping class group Let S be a connected orientable surface, possibly of infinite type. The mapping class group (S) is the group of isotopy classes of self-homeomorphisms of S, where homeomorphisms and isotopies are required to fix the boundary of S pointwise. We record the following immediate observation for further use: For g≥ 0 and n≥ 1, (S_g,n)<(Σ_g). In what follows, we will need to make use of the following further subgroups of (S). The pure mapping class group (S) is the subgroup of (S) whose elements fix every topological end of S. The compactly supported mapping class group _c(Σ_g) is the subgroup of (Σ_g) whose elements are the identity outside a compact subsurface of Σ_g. The following is an easy observation: Let g≥ 0. Consider any family {S_i} of compact subsurfaces of Σ_g whose union equals Σ_g, partially ordered with respect to inclusion. Then _c(Σ_g) ≅ ((S_i)). § ASYMPTOTIC MAPPING CLASS GROUPS In this section we define the groups _g and _g. We start by introducing the notion of rigid structure, which appeared originally in <cit.>. §.§ Rigid structures As in the introduction, it will be convenient to start with the genus-zero case. The reader should keep Figure <ref> in mind. A rigid structure on Σ_0 is a triple (P,A,Σ_0^+), where: * P is a pants decomposition of Σ_0, called the pants decomposition underlying the rigid structure, * A is a set of pairwise-disjoint arcs on Σ_0 such that for every pair of pants Y of P (that is, the closure of a connected component of Σ_0 - P), there are exactly three elements of A intersecting Y essentially, each connecting a different pair of boundary curves of Y, * Σ_0^+ is one of the two connected components of Σ_0 - ⋃_a ∈ A a, called the visible side of Σ_0. Observe that, up to the action of (Σ_0), there is only one rigid structure on Σ_0. It will be useful to extend the definition of rigid structure to the surface Σ_0^p obtained from Σ_0 by removing a finite collection of p≥ 1 points. In this case, we define a rigid structure on Σ_0^p as a rigid structure on the surface (homeomorphic to Σ_0) obtained from Σ_0^p by filling every isolated puncture, subject to the condition that every isolated puncture of Σ_0^p is contained in the same connected component of Σ_0 - P, where P is the pants decomposition underlying the rigid structure of Σ_0. Finally, consider the case of 1≤ g<∞. A rigid structure on Σ_g consists of a curve c ⊂Σ_g that cuts off a disk containing every puncture of Σ_g, together with a rigid structure for the planar component of Σ_g - c, namely the one homeomorphic to Σ_0^1. §.§ The groups _g and _g We now define the groups _g and _g. As mentioned above, we stress that the case g=0 was previously introduced in <cit.>. Fix, once and for all, a rigid structure on Σ_g, and write P for the pants decomposition underlying it. We say that a compact subsurface Z ⊂Σ_g is P-suited if ∂ Z ⊂ P, namely each boundary curve of Z is an element of the pants decomposition P. A homeomorphism f:Σ_g →Σ_g is asymptotically rigid if there exists a P–suited genus-g subsurface Z ⊂Σ_g with f(Z) also P–suited, and such that the restriction homeomorphism f: Σ_g - Z →Σ_g - f(Z) sends: * P ∩ (Σ_g- Z) to P ∩ (Σ_g- f(Z)), * A ∩ (Σ_g- Z) to A ∩ (Σ_g- f(Z)), * The visible side of Σ_g-Z to the visible side of Σ_g-f(Z). If we drop the last requirement we say that f is asymptotically quasi-rigid. Notation. Given an asymptotically rigid (resp. quasi-rigid) homeomorphism f as above, we will refer to the subsurface Z in the definition above as a defining subsurface for f. Observe that if f:Σ_g →Σ_g is asymptotically (quasi-) rigid and Z is a defining surface for f, then any P-suited surface containing Z is also a defining surface for f. This observation will be heavily used in the rest of the paper, without further mention. We are finally in a position to define the groups we are interested in: Let g,p≥ 0. We define _g (resp. _0^p) as the subgroup of (Σ_g) (resp. (Σ_0^p)) consisting of those elements that have at least one asymptotically rigid representative. In turn, the group _g is the subgroup of (Σ_g) consisting of those elements that have at least one asymptotically quasi-rigid representative. Observe that _c(Σ)⊂_g, by definition. However, the inclusion is proper, as a general element of _g may permute the components of the complement of every defining subsurface. The group of mapping classes of asymptotically quasi-rigid homeomorphisms of the disk punctured along a Cantor set coincides with the braided Thompson group considered by Brin <cit.> and Dehornoy <cit.>. The subgroup BV of mapping classes of asymptotically rigid homeomorphisms of the disk punctured along a Cantor set can be realized as a subgroup of _0 (see <cit.>, section 7). § THE RELATION WITH THOMPSON'S GROUPS As mentioned in the introduction, the groups _g and _g are strongly related to Thompson's groups. This is a manifestation of a more general phenomenon, which we now explain. Observe that any homeomorphism of Σ_g induces a homeomorphism of the space of ends (see, for instance, <cit.>) of Σ_g, which by definition is the Cantor set C. Thus we have a continuous homomorphism ϵ: (Σ_g) →(C), which is surjective when both homeomorphism groups are endowed with their respective compact-open topologies. Now, every Cantor set on the plane is tame, meaning that, up to homeomorphism, it is contained in some line; in particular, it is homeomorphic to the standard triadic Cantor set. A theorem of Scěpin (see e.g. <cit.>) states that any homeomorphism of the standard triadic Cantor set C⊂ [0,1]⊂^2 extends to a homeomorphism of ^2; moreover, this homeomorphism can be assumed to be the identity outside a large enough ball. In particular, we have: The homomorphism ϵ: (Σ_g) →(C) is onto. Moreover, (C) is a simple group <cit.>, and thus the connected component of the identity in (C) is trivial. Therefore, ϵ descends to a continuous surjective homomorphism (using a slight abuse of notation) ϵ: (Σ_g) →(C), where (Σ) has been endowed with the quotient topology coming from the compact-open topology on (Σ_g). Observe that the kernel of this homeomorphism is exactly the pure mapping class group (Σ_g), and thus we have a short exact sequence 1 →(Σ_g) →(Σ_g) →(C) → 1 We are now going to give the version of the exact sequence (<ref>) when (Σ_g) is replaced by _g or _g. As we will see, the subgroup which appears on the right will be (isomorphic to) Thompson's group V. We start by giving a definition of this group. §.§ Thompson's group V Recall that Thompson's group V is the group of right-continuous bijections of the unit circle that map the set of dyadic rationals to itself, are differentiable except at finitely many points, and on every interval of differentiability they are affine maps whose derivatives are powers of 2. The group V is well-known to be finitely presented, with respect to an explicit presentation. We refer the reader to the standard reference <cit.> for a thorough discussion on the different Thompson's groups. Extending results of <cit.>, we will prove: For every g≥ 0, there is a short exact sequence 1 →_c(Σ_g) →_g → V → 1. As mentioned in the introduction, Proposition <ref> is shown in <cit.> in the case when g=0, where it is also shown that it splits over Thompson's group T; compare with Proposition <ref>. In order to prove Proposition <ref>, it will be useful to work with a different incarnation of the group V (see <cit.> also). Namely, V is the group of self-transformations of the rooted 3-valent tree 𝒯 whose elements are encoded by equivalence classes of triples (T,T', σ), where T and T' are finite rooted subtrees of 𝒯 with the same number of leaves, and σ is a bijection between the set of leaves of T and T'. Such a triple extends to a transformation of 𝒯, and the equivalence relation responds to the fact that different triples may extend to the same transformation of 𝒯. Every element of V induces a homeomorphism of the space of ends of 𝒯, which is homeomorphic to the Cantor set C. Thus V < (C). Armed with this alternate description, we adapt the arguments of <cit.> in order to prove Proposition <ref> As mentioned above, the case g=0 is covered in <cit.>, so assume that g ≥ 1. Fix a rigid structure on Σ_g, and let c be the separating curve used to define it. The dual graph of the pants decomposition P underlying the rigid structure is naturally isomorphic to 𝒯, where the root corresponds to the unique pair of pants of Σ_g-P having c as boundary curve. We now define a homomorphism _g → V as follows. Let f∈_g and consider a defining subsurface Z for f. We associate to f the triple (T,T',σ) ∈ V, where T (resp. T') is the subtree of 𝒯 contained in Z (resp. f(Z)), and σ is the bijection between the sets of leaves of T and T', respectively, induced by the permutation between the sets of boundary components of Z and f(Z) given by f. At this point, one easily checks that this correspondence gives rise to a well-defined surjective homomorphism _g → V. We claim that the kernel of this homomorphism is exactly _c(Σ_g). Indeed, suppose f∈_g maps to the identity in V, and so in particular fixes every end of Σ_g. Consider a defining subsurface Z for f. Since f does not permute ends we may assume, up to replacing Z by a suitably larger defining subsurface, that every component of Σ_g -Z is mapped to itself. Therefore, after further enlarging Z if necessary, we deduce that f is the identity outside Z, and hence has compact support, as desired. Next, observe that if we replace (Σ_g) by the asymptotic mapping class group _g in equation (<ref>), we obtain: 1 →(Σ_g) ∩_g →_g →__g(C) → 1, where __g(C) denotes the image of _g in (C) under the homomorphism ϵ of (<ref>). By the same argument that we used to determine the kernel of the homomorphism _g → V, we have: For every genus g, we have (Σ_g) ∩_g = _c(Σ_g). At this point, the combination of equations (<ref>) and (<ref>) yields: __g(C) is isomorphic to Thompson's group V. In particular, we have deduced that the restriction of the short exact sequence (<ref>) to _g is precisely the sequence (<ref>). §.§ A related Higman-Thompson group We now give a brief description of the subgroup V_2[ℤ_2] of (C) that appears when restricting the exact sequence (<ref>) to _g. The elements of V_2[ℤ_2] are transformations of 𝒯 encoded by equivalence classes of tuples (T,T',σ,ε), where T, T' are subtrees, σ:∂ T→∂ T' is a bijection and ε∈ (/2)^∂ T'. The group V_2[ℤ_2] is an example of the Higman-Thompson groups V_n[G], where n∈ℕ and G a subgroup of the symmetric group on n elements <cit.>. More concretely, it is associated to the subgroup G=_2 of permutations of n elements generated by the involution exchanging j and n-j, for all j. (We remark that V = V_2[ Id] in this context.) Observe that we have an obvious inclusion V<V_2[ℤ_2]. However, by a surprising result of Bleak-Donoven-Jonušas <cit.> we have that, in fact, V ≅ V_2[ℤ_2], via an explicit element of (C) (a cellular automaton). Finally, we stress that V_2[ℤ_2] also appeared in <cit.> as the group of those homeomorphisms of a Cantor set embedded in 𝕊^2 which extend to smooth diffeomorphisms of the sphere. Using the same reasoning as in the proof of Proposition <ref>, we have: For every g≥ 0, the restriction to the sequence (<ref>) to _g yields a short exact sequence 1 →_c(Σ_g) →_g → V_2[ℤ_2] → 1. § FINITE PRESENTABILITY In this section we prove Theorem <ref>. As mentioned in the introduction, the case g=0 was settled in <cit.>, and will be a key ingredient in our proof: The group _0 is finitely presented. It will be useful for us to give a brief description of the arguments used in <cit.> for proving Theorem <ref>. The first ingredient, which will also play a central rôle here, is the following well-known result of Brown <cit.>: Let G be a group acting on a connected and simply-connected two-dimensional CW-complex X by permuting its cells. Suppose that: * The stabilizer of each vertex of X is finitely presented; * The stabilizer of every edge of X is finitely generated; * X/G is compact. Then G is finitely presented. In <cit.>, Funar-Kapoudjian applied Theorem <ref> to the action of _0 on a modification of the so-called pants complex <cit.> of Σ_0. More concretely, they first consider a graph 𝒫_0 whose vertices are (isotopy classes of) pants decompositions of Σ_0 which differ from the pants decomposition underlying the rigid structure in a finite number of curves. An edge of 𝒫_0 is given by two pants decompositions that are related by an elementary move, meaning that they differ in exactly two curves, which intersect exactly once (resp. twice) if their union fills a one-holed torus (resp. a four-holed sphere); see <cit.> for details. Using the same proof of the main result of Hatcher-Thurston <cit.>, one deduces that the graph 𝒫_0 is connected, and that it becomes a simply-connected 2-complex after gluing a 2-cell to every triangle, square, and pentagon of 𝒫_0. However, it turns out that the action of _0 on _0 is not cocompact, as there are infinitely many _0-orbits of squares <cit.>; in particular, Theorem <ref> cannot be applied to this situation. In order to overcome this, Funar-Kapoudjian construct a modification of 𝒫_0, called the reduced pants complex, by considering only two combinatorial types of squares, and show that 𝒫_0 is still simply connected. They then prove that _0 acts on 𝒫_0 satisfying all the hypotheses of Theorem <ref>, as desired. In fact, a minor variation of the arguments in <cit.> serves to prove the following strengthening of Theorem <ref>, which will be crucial for us: For every b∈∪{0}, the groups _0^b and _0^b are finitely presented. The definition of the reduced pants complex from <cit.> makes sense also for Σ_0^b, with arbitrary b. By the same arguments as in <cit.>, this complex is connected and simply-connected and, using the same reasoning as in <cit.>, the groups _0^b and _0^b each act cocompactly on it. The stabilizer of every cell is an extension of a finite permutation group by a finitely generated free-abelian group, and in particular finitely presented. At this point, the result follows from Theorem <ref>. We now turn to the proof of Theorem <ref>, whose statement we now recall: Theorem <ref> The groups _g and _g are finitely presented for every g≥ 0. In light of Theorem <ref>, it suffices to show the result for g positive. In order to do so, we will also use Brown's Theorem <ref>, this time using the action of _g (resp. _g) on the so-called cut-system complex 𝒦_g of Σ_g. We remark that cut-system complexes of finite-type surfaces were in fact used by Hatcher-Thurston <cit.> and Wajnryb <cit.> in order to compute finite presentations of mapping class groups. We now introduce the complex 𝒦_g. For concreteness, we choose to write the definitions for the surface Σ_g, although they make sense for an arbitrary connected orientable surface, of finite or infinite type. A cut system of Σ_g is a set of g non-separating curves whose union does not separate Σ_g. Let 𝒦_g be the simplicial graph whose vertices are cut systems on Σ_g, and where two cut systems are adjacent in 𝒦_g if they differ in exactly two curves, which intersect exactly once. Moreover, similar to the case of the pants complex, 𝒦_g will become a simply-connected 2-complex after gluing a 2-cell to certain circuits in 𝒦_g. Before explaining this, we need to borrow some definitions from <cit.>. A triangle in 𝒦_g consists of three pairwise-adjacent vertices of 𝒦_g; geometrically, a triangle corresponds to three curves that pairwise intersect exactly once. A square in 𝒦_g is a closed path with four vertices v_1, …,v_4 such that v_i and v_j are adjacent in 𝒦_g if and only if |i-j| =1 4; geometrically, a square corresponds to two elementary moves that occur in two disjoint one-holed tori. Finally, a pentagon in 𝒦_g consists of five vertices v_1, …,v_5 such that v_i and v_j are adjacent in 𝒦_g if and only if |i-j| =1 5; geometrically, a pentagon is determined by five curves c_1, …, c_5 on Σ_g such that both c_i and c_i+1 belong to the cut system v_i and i(c_i,c_i+2) =1, counting indices modulo 5. Armed with these definitions, we construct a 2-complex by gluing a 2-cell to every triangle, square, and pentagon of 𝒦_g. By a slight abuse of notation, we will denote the resulting complex by 𝒦_g also. The following result is essentially due to Hatcher-Thurston <cit.> and Wajnryb <cit.>: For every g≥ 1, the complex 𝒦_g is connected and simply-connected. First, Hatcher-Thurston <cit.> and Wajnryb <cit.> proved that the cut-system complex of a finite-type surface is connected and simply-connected. To see that this is also the case for 𝒦_g observe that, for every finite set of vertices A ⊂𝒦_g, the union of the curves defining the elements of A together fill a finite-type subsurface of Σ_g. We are finally in a position to prove Theorem <ref>: We prove the result for _g; the same argument, replacing every instance of by , will give the result for _g. As mentioned earlier, we are going to apply Theorem <ref> to the action of _g on 𝒦_g. First of all, Theorem <ref> tells us that 𝒦_g is connected and simply-connected. Now, the classification theorem for infinite-type surfaces <cit.> and the fact that vertices are defined by a finite set of curves, together imply that _g acts transitively on the set of vertices of 𝒦_g. Since edges and 2-cells of 𝒦_g are defined in terms of intersection numbers, and by a finite set of curves, we deduce that _g acts cocompactly on 𝒦_g. Thus, it remains to justify why the stabiliser of a vertex (resp. edge) is finitely presented (resp. finitely generated). Consider first the case of the stabiliser of a vertex u of 𝒦_g. Fix an orientation on each curve of u. Up to passing to a subgroup of finite index, and abusing notation, we may assume that every element of (u) fixes every curve of u together with its orientation. Now, cutting Σ_g open along the elements of u we obtain a surface homeomorphic to Σ_0^2g, and a short exact sequence 1→^g →(u) →_0^2g→ 1, where ^g is the group generated by the Dehn twists along the elements of u. Now, an extension of a finitely presented group by a finitely presented group is also finitely presented and thus Proposition <ref> yields that (a finite-index subgroup of) (u) is finitely presented, as desired. Observe that the surface obtained from Σ_g by cutting the surface along the curves defining an edge of 𝒦_g is homeomorphic to Σ_0^2g-1. In light of this, as above we deduce that (a finite-index subgroup of) the stabiliser of an edge e∈𝒦_g fits in a short exact sequence 1→^g-1→(e) →^2g-1_0 → 1. Therefore, the stabiliser of an edge is finitely presented, and in particular finitely generated. This finishes the proof of Theorem <ref>. § THE GROUP OF HALF-TWISTS IS DENSE The goal of this final section is to prove Theorem <ref>, whose statement we now recall: Theorem <ref> _g is dense in (Σ_g). In light of the result of Neretin and the second author <cit.> mentioned above, when g=0 this result may be interpreted as stating that homeomorphisms of the sphere minus a Cantor set may be approximated by diffeomorphisms. As it turns out, Theorem <ref> will be a consequence of a slightly stronger result, namely Theorem <ref> below. Before we state it, we need some preliminaries. Fix, once and for all, a rigid structure on Σ_g. Recall from section <ref> that the rigid structure is given by a separating curve on Σ_g, which will be denoted c(∅) for reasons that will become apparent below, plus a rigid structure on the planar component Σ_g^* of Σ_g - c(∅). Let P be the pants decomposition underlying the rigid structure on Σ_g^*. Similar to the situation in subsection <ref>, there is an infinite rooted tree 𝒯' associated to P, whose vertices are the curves of P, with the root being c(∅). As such, every vertex of 𝒯' is naturally labelled by a word w in the free semigroup F(L,R) generated by the two letters L (left) and R (right). We denote by c(w) the curve of P labelled by the word w∈ F(L,R), and write h(w) for the half-twist about c(w). In turn, this labelling induces a labelling of the set of pairs of pants of Σ_g^* by words in F(L,R). Indeed, we set P(w) to be the unique P-suited pair of pants of Σ_g^* that has c(w) as boundary component and is contained in the planar component of Σ_g - c(w). Finally, set S(-∞) to be a fixed compact genus-g subsurface of Σ_g which contains c(∅). In addition, for each w ∈ F(L,R), let S(w) be a P-suited (therefore compact) subsurface whose interior is contained in Σ_g^* - c(w), and which has c(w) as a boundary component. Observe that the choice is far from unique; however, for any such choice, the set {S(w) | w∈ F(L,R) ∪{-∞}} is proper, in the sense that every compact subsurface of Σ_g intersects only finitely many elements of this set. Because of this, we will refer to any set {S(w) | w∈ F(L,R) ∪{-∞}} as above as a proper exhaustion by subsurfaces. After all this discussion, we have the following definition: A proper sequence of mapping classes is a sequence {f_n}_n∈⊂(Σ_g), where f_n = ∏_c w∈ F(L,R) ∪{-∞} |w| ≤ n f(S(w)) h(w)^ϵ_w, where: * f(S(w)) is an element of (S(w)), * |w| denotes the length of the word w, * ϵ_w ∈{-1,0,1} for all w, * The product is ordered lexicographically, and is defined from left to right. Observe that if {f_n}_n∈⊂(Σ_g) is a proper sequence, then {f_n}_n∈⊂_g as well. In addition, since the defining subsurfaces of the f_n form a proper exhaustion of subsurfaces of Σ_g, we have: If {f_n}_n∈⊂_g is a proper sequence, then it has a limit in (Σ_g). Recall that (Σ_g) his equipped with the quotient topology coming from the compact-open topology on (Σ_g). We will prove: Every element of (Σ_g) is a limit of a proper sequence in _g. Observe that Theorem <ref> obviously implies Theorem <ref>. Let f ∈(Σ_g) be an arbitrary mapping class. We will use an inductive argument, which we call a straightening of f, to produce a proper sequence of mapping classes whose limit is f. Consider c(∅). By the classification of infinite-type surfaces <cit.>, there exist a P-suited genus-g subsurface S(-∞), and an element f_-∞ such that f_-∞ f sends c(∅) to itself. We say that f_-∞ f straightens c(∅). Next, consider the curves c(L) and c(R). Again, there exists a P-suited planar subsurface S(∅) and an element f_∅∈(S(∅)) such that f_∅ f_-∞ f sends the set {c(L),c(R)} to itself. Therefore, up to precomposing with the half-twist h(∅) about c(∅) if necessary, we may in fact assume that c(L) (resp. c(R)) is sent to itself. We say that f_∅ f_-∞ f straightens the pair of pants P(∅). We continue this process inductively to find, for all n, a proper exhaustion by (compact) P-suited subsurfaces S(w) and elements f(S(w)) ∈(S(w)) such that, setting f_n:= ∏_c w∈ F(L,R) ∪{-∞} |w| ≤ n f(S(w)) h(w)^ϵ_w, the mapping class (f_n f)_n∈ straightens every pair of pants P(w) for |w|≤ n. By Lemma <ref>, the sequence (f_n)_n∈ converges to an element f^*∈(Σ_g), in such way that f^* f fixes c(∅) and every curve of P. Moreover, by composing with an appropriate power of a Dehn twist at each step, we may assume that f^* f induces the trivial twist on each curve of P. Therefore, f^* f is an element of (S(-∞)). In other words, we have shown that, up to precomposing f_∅ with the inverse of this element, the sequence f_n converges to f in (Σ_g), as desired. § STABLE HOMOLOGY In this section, we adapt the methods of <cit.> to prove Theorem <ref>. First, observe that for n ≥ 1, gluing a pair of pants to a boundary component of S_g,n gives rise to an injective homomorphism (S_g,n) →(S_g,n+1). In particular, one has a homomorphism H_k((S_g,n),) → H_k((S_g,n+1),) between the corresponding homology groups, which has a one-sided inverse coming from the capping homomorphism <cit.> and is therefore injective. In <cit.>, Harer proved that the k-th homology group of (S_g,n) does not depend on n, provided k is sufficiently small with respect to g. The current best bound for what “sufficiently small" means is due to Boldsen <cit.> who, building up on unpublished work of Harer <cit.>, proved the following: : Let g,n,k ∈, with g≥ 0 and n, k≥ 1. Then, the homomorphism (<ref>) is an isomorphism for every k≤ 2g/3. At this point, Lemma <ref> and Theorem <ref> together imply that H_k(_c(Σ_g), ℤ) ≅ H_k((S_g,n),ℤ) for all n≥ 1, provided k ≤ 2g/3. For this reason, we will refer to the homology group H_k(_c(Σ_g), ℤ) as the stable k-th homology group of the mapping class group of genus g. We are finally in a position to give a proof of Theorem <ref>, whose statement we now recall: Theorem <ref> Let g≥ 1. For every k ≤ 2g/3, H_k(_g, ) (resp. H_k(_g, )) is isomorphic to the k-th stable homology group of the mapping class group of genus g. We adapt the proof of <cit.> to our setting. We treat the case of _g only, the other one being a direct translation. It is well-known that (S_g,n), where n≥ 1, is torsion free and has a finite dimensional classifying space, which can be taken to be manifold (see e.g. <cit.>). Thus their integral homology is of finite type and by Harer's Stability Theorem H_k(_c(Σ_g), ) is finitely generated for every k. Now, we apply the Lyndon-Hochschild-Serre spectral sequence method to the short exact sequence (<ref>), which reads 1 →_c(Σ_g) →_g → V → 1. In other words, there is a spectral sequence approximating the homology of _g whose second page is E^2_pq= H_p(V,H_q(_c(Σ_g),)). We now claim that V acts trivially on H_q(_c(Σ_g), ) for all q. Accepting this claim for the moment, it follows that the only non-zero terms of the spectral sequence above for the rational homology are those with p=0, because V is acyclic (see <cit.>). In particular, the spectral sequence for the homology of _g collapses at the second page, which implies that H_q(_g, )≅ H_q(_c(Σ_g),) for all q, as desired. Thus it remains to show: Claim. V acts trivially on H_q(_c(Σ_g), ). Since H_k(_c(Σ_g), ) is a finitely generated abelian group, we get a finite-dimensional representation V →(N, ℚ). As V is simple, it follows that this representation is either trivial or injective. On the other hand, every finitely-generated subgroup of (N, ℚ) is residually finite, while V is not. In particular, the given representation is trivial. This finishes the proof of Theorem <ref>. By the sequence (<ref>), plus the fact that _c(Σ_g) (g≥ 3) and V are both perfect, we obtain that _g and _g are perfect. § RIGIDITY The goal of this section is to prove Theorem <ref>. As mentioned above (G) will denote the automorphism group of the group G. §.§ Automorphisms of _c(Σ_g) The first ingredient in the proof of Theorem <ref> is the following: Suppose g≥ 0. Then (_c(Σ_g))= (Σ_g). We note that, prior to this work, this result had been obtained by Patel-Vlamis <cit.> for g≥ 4. Before explaining a proof of Proposition <ref>, we note the following easy observation: _c(Σ_g) is a normal subgroup of (Σ_g) for every g≥ 0. Let g∈_c(Σ_g) and let Z be a support for g. Consider an arbitrary h ∈(Σ_g), noting that any P-suited subsurface W containing h(Z) is a support for hgh^-1. Moreover, hgh^-1 induces the trivial permutation on the set of boundary components of W, and therefore hgh^-1∈_c(Σ). In order to prove Proposition <ref>, we follow Ivanov's strategy for proving that automorphisms of (sufficiently complicated) finite-type mapping class groups are conjugations. First, we observe: Every automorphism of _c(Σ_g) sends Dehn twists to Dehn twists. We stress that, if 4≤ g <∞, then Lemma <ref> quickly follows from <cit.> plus the fact that _c(Σ_g) is a direct limit of P-suited subsurfaces of Σ_g (Lemma <ref>). We now present a simpler argument, valid for arbitrary g, which was suggested to us by Juan Souto: Let ϕ∈(_c(Σ_g)). We take an arbitrary P-suited subsurface Z⊂Σ_g of genus g; in addition, we will assume that Z has at least seven boundary components (note that the latter assumption is only relevant if g<3). Since (Z) is finitely presented and _c(Σ_g) is a direct limit of pure mapping class groups of P-suited subsurfaces of genus g, by Lemma <ref>, we deduce that there exists a P-suited subsurface W⊂Σ_g such that ϕ((Z)) < (W). Let W^* be the (compact) subsurface of W supporting ϕ((Z)); in particular, this implies that ϕ((Z)) < (W^*), and that (W^*) and ϕ((Z)) have the same centralizer in _c(Σ_g). Now, observe that, given a compact subsurface Y⊂Σ_g, the rank of the center of the centralizer of (Y) in _c(Σ_g) is equal to the number of boundary components of Y. In particular, since (Z) and ϕ((Z)) are isomorphic, the discussion of the previous paragraph implies that W^* and Z have the same number of boundary components. Again, <cit.> or <cit.> imply that they have equal genus, as Z has genus g and ϕ((Z)) < (W^*). In other words, Z and W^* are homeomorphic. We claim: Claim. The isomorphism ϕ: (Z) →(W^*) is induced by a homeomorphism Z→ W^*. If Z has genus ≥ 2, then the claim is <cit.>. We now sketch an argument valid for arbitrary genus. The center C_Z of (Z) is the free abelian group generated by the Dehn twists along the boundary components of Z. Observe that ϕ induces an isomorphism (Z) /C_Z →(W^*)/C_W^*, and that (Z) /C_Z is isomorphic to the pure mapping class group of the surface which results from gluing a punctured disk to every boundary component of Z. By the work of numerous authors (see, for instance, <cit.> for a comprehensive statement), the isomorphism (<ref>) is induced by a homeomorphism f between the punctured surfaces. To see that f extends to a homeomorphism Z→ W^* inducing ϕ, we use the well-known lantern relation (see, for instance, <cit.>). Indeed, since we are assuming that Z has at least seven boundary components, given a boundary component a ⊂∂ Z we may find a collection b_1,…, b_6 of six curves, all of them essential in Z, such that the Dehn twist along a is expressed as a product of (suitable powers of) Dehn twists along the b_i's, via the lantern relation. Thus the claim follows. The claim above implies that ϕ: (Z) →(W^*) is induced by a homeomorphism Z → W^*, and in particular sends Dehn twists to Dehn twists. The lemma now follows since Z was arbitrary and every curve on Σ_g is contained in some P-suited subsurface of genus g. Continuing with the arguments towards a proof of Proposition <ref>, we next claim that every automorphism of _c(Σ_g) induces an automorphism of the curve complex 𝒞(Σ_g); we refer the reader to the articles <cit.> for various features of these complexes in the infinite-type setting. Indeed, let ϕ : _c(Σ_g) →_c(Σ_g) be an automorphism. By Lemma <ref>, given a curve c ∈Σ_g there exists a curve d such that ϕ(T_c) = T_d, where T_a denotes the (left) Dehn twist about the curve a. Since T_a = T_b if and only if a = b, we get that the curve d above is in fact unique, and hence ϕ induces a well-defined map ϕ_* : (Σ_g) →(Σ_g) by the rule ϕ_*(c) = d. Observe that the map ϕ_* is simplicial, since two Dehn twists commute if and only if the defining curves are disjoint. Moreover, it is bijective, with inverse the simplicial self-map of (Σ_g) induced by ϕ^-1. Now we need the following analogue of Ivanov's celebrated result <cit.> on automorphisms of the curve complex: The automorphism group of (Σ_g) is isomorphic to (Σ_g). We are finally in a position to prove Proposition <ref>: We wish to show that the natural homomorphism Λ: (Σ_g) →(_c(Σ_g)) is an isomorphism. We first prove that Λ is surjective. To this end, let ϕ∈(_c(Σ_g)), and consider the automorphism ϕ_* : (Σ_g) →(Σ_g) induced by ϕ. By Theorem <ref>, there exists a unique f ∈(Σ_g) with f(c) = ϕ_*(c) for all c ∈(Σ_g). Note that ψ:=f^-1ϕ is an element of (_c(Σ_g)), and that ψ_*(c) =c for all c ∈(Σ_g). Using again a direct translation of a classical argument in the finite-type setting, we will prove that the automorphism ψ is trivial, so that ϕ is in fact the conjugation by f. To this end, let h∈(Σ_g), and choose an arbitrary curve c on Σ_g. We have: ψ(hT_c h^-1) = ψ(T_h(c)) = T_ψ_*(h(c))= T_h(c). On the other hand: ψ(hT_c h^-1) =ψ(h) ψ(T_c) ψ(h)^-1 = ψ(h) T_ψ_*(c)ψ(h)^-1= T_ψ(h)(c) Combining both equations, we obtain that h(c) = ψ(h)(c) for every curve c on Σ_g. By Theorem <ref> we deduce that h = ψ(h) for every h∈_c(Σ_g). In other words, ψ is the identity on _c(Σ_g), as desired. §.§ Proof of Theorem <ref> We now embark in the proof of Theorem <ref>. The key ingredient is the following unpublished result of Kapoudjian: Let f ∈_g. Then f∈_c(Σ_g) if and only if for any finite set {f_i}⊂_g there exists h∈_g such that [h,f_iff_i^-1]=1 and f belongs to the normal closure of h in _g. Let f∈_g. Suppose first that f∈_c(Σ_g), and let {f_i}⊂_g be an arbitrary finite set. Choose a minimal support Z for f, so that f(Z)=Z and f is identity outside Z. Observe that f_i(Z) is a support for f_iff_i^-1. Then we may take h_0∈ such that h_0(Z)∩ f_i(Z)=∅ for all i, and set h=h_0fh_0^-1, which commutes with f_iff_i^-1. For the opposite direction let Z be a defining surface for f; recall this means that f sends Z to f(Z), and is rigid in the complement of Z ∪ f(Z). Up to enlarging Z if necessary, we may assume that Z has at least five boundary components. We now choose some finite set {f_i}⊂_g as in the statement, and which furthermore satisfies that the f_i all send Z to itself, and that the group generated by the f_i induces every possible permutation of the set of boundary components of Z. Seeking a contradiction, suppose that f∉_c(Σ_g). Then there exists a connected component of Σ_g-Z whose image by f is disjoint from itself. This implies that every defining surface of any h ∈_g as in the statement must be contained in Z. Furthermore, the permutation of the set of boundary components of Z induced by h must commute with every element of the conjugacy class of the (nontrivial) permutation induced by f. Now, we claim that if a permutation σ in the symmetric group on k≥ 5 elements commutes with every conjugate of a fixed nontrivial permutation σ_0, then σ is the trivial permutation. Indeed, σ commutes then with every element of the normal subgroup generated by σ_0; as k≥ 5 this normal subgroup contains the alternating group. Further, σ cannot be even as the alternating group is center-free. It follows that σ commutes with the even permutation τσ and hence σ commutes with τ, for any transposition τ. Then σ is in the center of the symmetric group which is trivial. In particular, the permutation induced by h is the trivial one, and h ∈_c(Σ_g). Hence f belongs to the normalizer of _c(Σ_g) in _g, contradicting our assumption. As an immediate consequence we get: _c(Σ_g) is a characteristic subgroup of _g. The statement of Lemma <ref> and the proof are still valid when we replace _g by _g. This implies that _c(Σ_g) is characteristic in _g also. In light of Corollary <ref>, every automorphism of _g (resp. _g) induces an automorphism of _c(Σ_g), which we have already determined in Proposition <ref>. Now, in order to calculate the automorphism group of _g and _g from this, we will make use of the following small technical result – this is surely well-known, but we include a proof for completeness. Let H be a normal subgroup of a group G, and suppose it has trivial centralizer in G. Suppose ψ: G→ G is an injective homomorphism of G such that ψ_| H= id_H. Then ψ= id_G. Since ψ is the identity on H and H is normal in G we have that, for every f∈ G and every h ∈ H, fhf^-1= ψ(fhf^-1) = ψ(f) ψ(h) ψ(f^-1) = ψ(f) h ψ(f^-1). Since h is arbitrary, it follows that ψ(f^-1)f belongs to the centralizer of H in G, and therefore ψ(f) = f, by hypothesis. Since f is also arbitrary, the result follows. The motivation for Lemma <ref> is the following claim: The centralizer of _c(Σ_g) in (Σ_g) is trivial. Suppose f∈(Σ_g) commutes with every element of _c(Σ_g), and in particular with every Dehn twist. In particular, we have that T_a= fT_a f^-1 = T_f(a), for every curve a on Σ_g. But, as we mentioned in the proof of Proposition <ref>, the Alexander method of <cit.> implies that every element of (Σ_g) that fixes every curve on Σ_g is the identity. We need the following definition before proving Theorem <ref>. Suppose Z is an orientable surface with non-empty boundary ∂ Z, where we assume that every connected component of ∂ Z has been parametrized by means of a map φ:∂ Z→ S^1, which is a homeomorphism on each component. A homeomorphism F:Z→ Z respects the boundary parametrization if φ∘ f|_∂ Z=φ. The parametrized mapping class group ^*(Z) is the group of isotopy classes of self-homeomorphisms of Z respecting the boundary parametrization, modulo isotopies which are the identity on the boundary. Observe that (Z) < ^*(Z) is a finite index normal subgroup and that, as a by-product of the definition, an element of ^*(Z) may induce a non-trivial permutation on the set of boundary components of Z. We prove the result for _g, as the case of _g is dealt with along similar lines. We first show that the natural homomorphism N_(Σ_g)(_g) →(_g) is surjective. To this end, let ϕ: _g →_g be an automorphism. By Corollary <ref>, ϕ induces an automorphism ϕ_c: _c(Σ_g) →_c(Σ_g), which is the restriction of ϕ to _c(Σ_g). Proposition <ref> implies that there exists Φ∈(Σ_g) such that ϕ_c is conjugation by Φ, denoted Ad_Φ. From the proof of Lemma <ref> for any (compact) P-suited subsurface X⊂Σ_g of genus g, there exists a compact subsurface Y=Φ(X)⊂Σ_g such that the restriction ϕ_c|_(X) sends (X) isomorphically onto (Y). Observe that for any P-suited subsurface X⊂Σ_g the group ^*(X) admits a canonical embedding into _g, by extending rigidly classes of homeomorphisms of X to classes of homeomorphisms of Σ_g. We claim: For any P-suited subsurface X⊂Σ_g of genus g, the restriction ϕ|_^*(X) sends ^*(X) isomorphically onto some copy of ^*(Φ(X)). Moreover, ϕ|_^*(X) is conjugation by Φ. As ϕ|_(X) is conjugation by Φ, we have ϕ(T_c)=T_Φ(c), when c∈∂ X is a boundary circle. The stabilizer of the set {T_c | c∈∂ X} in _g is the stabilizer of the simplex of 𝒞(Σ_g) determined by elements of ∂ X, with respect to the _g-action. We call it the stabilizer of the multicurve. Let Y be a compact subsurface of Σ_g and denote by S_∂ Y the permutation group on the set of boundary components of Y. Looking at the induced permutation on the set of boundary components, we obtain canonical homomorphisms: ^*(Y)→ S_∂ Y, ^*(Σ_g ∖(Y))→ S_∂ Y, where (Y) denotes the interior of Y. We denote by ^*(Y)× _S_∂ Y^*(Σ_g ∖(Y)) the fibred product of the two homomorphisms above, making the following diagram commutative: [ ^*(Y)× _S_∂ Y^*(Σ_g ∖(Y)) → ^*(Σ_g ∖(Y)); ↓ ↓; ^*(Y) → S_∂ Y; ] Assume now that Y has genus g. Then the stabilizer of the multicurve ∂ Y in (Σ_g) is ^*(Y)× _S_∂ Y^*(Σ_g ∖(Y)). Further, the stabilizer in _g of the multicurve ∂ X should be sent by ϕ into the stabilizer of the multicurve ∂ Y, where Y=Φ(X). Since ^*(X) stabilizes ∂ X we derive that ϕ(^*(X))⊂^*(Y)× _S_∂ Y^*(Σ_g ∖(Y)). If π_1 and π_2 denote, respectively, the projections onto the first and second factors of the fibred product then we derive a map π_1∘ϕ: ^*(X)→^*(Y), such that π_1∘ϕ|_(X) is the conjugation Ad_Φ by Φ. We claim that π_1∘ϕ=Ad_Φ. By composing with Ad_Φ^-1 this reduces to show that if a homomorphism φ:^*(X)→^*(X) restricts to identity on (X) then φ is the identity. If h∈^*(X) sends a boundary component a into b, then T_b=φ(T_h(a))=φ(hT_ah^-1)=φ(h)T_aφ(h)^-1=T_φ(h)a so that φ(h)a=b. Thus the homomorphism S_∂ X→ S_∂ X induced by φ is identity. Then the Five Lemma implies that φ is injective and Lemma <ref> gives us the desired result, as the centralizer of (X) in ^*(X) is trivial, if the complexity of the surface X is large enough. Further, the image π_2∘ϕ((X)) is trivial, since ϕ((X))=(Y). Therefore π_2∘ϕ factors through the quotient ^*(X)/(X), namely S_∂ X, and the map S_∂ X→^*(Σ_g ∖(Y))→ S_∂ Y is the isomorphism induced by the conjugation Ad_Φ. This implies that there exists a section S_∂ Y→^*(Σ∖ Y) which determines an embedding of ^*(Y) into (Σ_g) so that ϕ(^*(X)) is the image of ^*(Y). Continuing with the proof, observe that an immediate consequence of the proof of Theorem <ref> is that _g has a system of generators 𝒮 consisting uniquely of elements belonging to ^*(X_i) for finitely many P-suited subsurfaces X_i⊂Σ_g. Lemma <ref> shows that the natural map ϕ_^*(X):^*(X)→(Σ_g) coincides with Ad_Φ|_^*(X). In particular, ϕ(s)=Ad_Φ(s), for any s∈ S. Since both ϕ and Ad_Φ are homomorphisms, we derive that ϕ(s)=Ad_Φ(s) for any s∈_g. It follows that Ad_Φ(_g)⊆_g, namely Φ∈ N_(Σ_g)(_g), as claimed. To see that the homomorphism N_(Σ_g)→(_g) is injective, suppose that Φ∈(Σ_g) induces the identity automorphism of _g. Then Φ also induces the identity automorphism of _c(Σ_g). At this point, Corollary <ref> tells us that _c(Σ_g) is normal in _g, since it is characteristic. In addition, _c(Σ_g) has trivial centralizer in (Σ_g), and therefore in _g, by Lemma <ref>. Hence, Lemma <ref> implies that Φ is the identity. §.§ A note on normalizers We now explore some properties of the normalizer of _g (resp. _g) in (Σ_g); as we will see, these represent a certain departure from the case of finite-type mapping class groups. First, we have: Let G be a subgroup of (Σ_g). Assume that G∩(Σ_g) is a normal subgroup of (Σ_g). Then there is a natural surjective homomorphism N_(Σ_g)(G) → N_(C)(G/G∩(Σ_g)). As G∩(Σ_g) is a normal subgroup of G we derive from the exact sequence (<ref>) that G/G∩(Σ_g) is a subgroup of (C). Then there is a well-defined homomorphism N_(Σ_g)(G) → N_(C)(G/G∩(Σ_g)) that sends every element H of the normalizer to its restriction to the space C of ends of Σ_g. Let now h∈ N_(C)(G/G∩(Σ_g)). By the exact sequence (<ref>) h admits a lift H∈(Σ_g). Conjugation by H sends the subgroup G∩(Σ_g) into itself, because it was assumed to be a normal subgroup of (Σ_g). It follows that H∈ N_(Σ_g)(G), so that the homomorphism in the statement is surjective. By a deep theorem of Rubin (see <cit.>), if G/G∩(Σ_g) is sufficiently large, in particular if G⊇_g, then there is an isomorphism (G/G∩(Σ_g))≃ N_(C)(G/G∩(Σ_g)). A recent result <cit.> shows that the outer automorphism group (V) of V is infinite. Combining this with the lemma above, we obtain: The homomorphisms (_g)→(V) and (_g)→(V) are surjective. In particular, (_g) and (_g) are infinite. From the exact sequences (<ref>) and (<ref>) the and Corollary <ref> _g∩(Σ_g)=_g∩(Σ_g)=_c(Σ_g). Then Lemma <ref> shows that the assumptions of Lemma <ref> are satisfied. In particular we have surjective homomorphisms N_(Σ_g)(_g)→ N_(C)(V) and N_(Σ_g)(_g)→ N_(C)(V). By Rubin's Theorem N_(C)(V)≃(V). On the other hand N_(Σ_g)(_g)⊂(_g) and N_(Σ_g)(_g)⊂(_g). We remark that a theorem of Ivanov <cit.> mentioned above asserts that ((S_g,n)) is always a finite group (in fact, trivial in all but finitely many cases). Thus Corollary <ref> represents a limitation in the dictionary between asymptotic and finite-type mapping class groups. Finally, we prove that the first homomorphism of Corollary <ref> has infinite kernel: For every g≥ 0, the homomorphism (_g)→(V) has infinite kernel. As usual, we prove the result for _g only. Consider first the surface Σ_0^1, which recall that is homeomorphic to a sphere minus the union of a Cantor set and an isolated puncture. Using a totally analogous argument to that of Lemma <ref>, we deduce that there is a surjective homomorphism (_0^1)→(V) Recall the construction of the element t_∞ from Example <ref>. Let P be the pants decomposition underlying the rigid structure on Σ_0^1, and t_∞∈(Σ_0^1) obtained as the product of all (say left) half-twists about the curves of P. We claim that t_∞∈ N_(Σ_0^1)(_0^1). To see this, observe that if Z is a defining surface of an element φ∈_g, then each connected component of Σ_0^1-Z is preserved by t_∞φ t_∞^-1; moreover, this element acts as the identity on every such component. In particular, t_∞φ t_∞^-1∈_0^1 and thus t_∞∈ N_(Σ_0^1)(_0^1). Furthermore, the short exact sequence (<ref>) implies that t_∞^d∉_0^1 unless d=0, since t_∞^d fixes every end of Σ_0^1 but does not have compact support unless d=0. Hence it provides an example of an infinite order element in (_0^1). On the other hand, its image under the homomorphism (<ref>) is obviously trivial, and thus this homomorphism has infinite kernel. After all this discussion, we treat the case of Σ_g. To do so, we simply embed Σ_0^1 in Σ_g in such way that the pants decomposition P underlying the rigid structure of Σ_0^1 is sent to the pants decomposition underlying the rigid structure of Σ_g. Using the same arguments as above, we deduce that the image of t_∞ under the homomorphism induced by the embedding Σ_0^1 →Σ_g produces an infinite-order element in the kernel of the homomorphism (_g)→(V). § SURJECTIONS BETWEEN ASYMPTOTIC MAPPING CLASS GROUPS In this section we will prove Theorem <ref>. Let 0≤ g<h and h≥ 2. Seeking a contradiction, suppose there were a surjective homomorphism ϕ: _h →_g. For any f∈_c(Σ_h) there exists, by Lemma <ref>, some P-suited subsurface S_h,n⊂Σ_h such that f is in the image of (S_h,n) within _c(Σ_h). Let a∈ϕ(_c(Σ_h)), so that a=ϕ(f), for some f∈(S_h,n). For any finite set {a_i}⊂_g we can write a_i=ϕ(f_i), with f_i∈_h. According to Lemma <ref>, there exists f'∈_h such that f belongs to the normal subgroup of _h generated by f' and such that [f',f_i ff_i^-1]=1. This implies that a belongs to the normal subgroup of _g generated by b=ϕ(f') and [b,a_i a a_i^-1]=1. Lemma <ref> again implies that a∈_c(Σ_g). Restricting ϕ we obtain a homomorphism (S_h,n) →_c(Σ_g). Since (S_h,n) is finitely presented and _c(Σ_g) is a direct limit of finitely presented groups, by Lemma <ref>, we deduce that there is m∈ and a nontrivial map (S_h,n) →(S_g,m). But every such map has either a quotient of /10 as image, when h=2, or trivial image, when h≥ 3, by <cit.> or <cit.>. Let now h≥ 3. By the above ϕ(f)=1, for any f∈(Σ_h), and therefore ϕ factors through V. Let ϕ':V→_g be the induced surjective homomorphism. If we compose the projection p:_g→ V with ϕ' we obtain then a surjective homomorphism V→ V with nontrivial kernel. Since V is simple such a homomorphism should be trivial. This contradicts the surjectivity. Assume that h=2. The nested groups ϕ((S_2,n)) are quotients of /10 for every n, in particular the sequence is eventually constant and ϕ(_c(Σ_2)) is a quotient of /10. Therefore ϕ factors through the group Q=_2/[_c(Σ_2), _c(Σ_2)], yielding a surjective homomorphism ϕ':Q→_g. Observe that the restriction of the sequence (<ref>) to Q reads 1→/10=H_1(_c(Σ_2))→ Q→ V→ 1. Now, the normal subgroup K=ϕ'^-1(_c(Σ_g)) Q is infinite because it is the inverse image of an infinite subgroup by a surjective homomorphism. Therefore K cannot be contained in the kernel of the projection homomorphism p: Q→ V, which is finite. Since p(K) is a normal subgroup of V and V is simple, p(K)= V. As p is finite, we derive that Q/K must be finite. On the other hand, K has infinite index in Q, because it is the preimage of the normal subgroup _c(Σ_g) of _g by the surjective homomorphism ϕ'. Hence we get the desired contradiction. The same reasoning as above along with Remark <ref> implies that there are no surjective homomorphisms _h →_g (resp. _h →_g and _h →_g) if g<h. § HOMOMORPHISMS FROM LATTICES AND PROPERTY (T) We start this section by proving Theorem <ref>, which is a direct consequence of the existence of the short exact sequence (<ref>). Before we prove the result, recall that a discrete group G is said to have Kazhdan's Property (T) if every action of G by continuous affine isometries on a real Hilbert space has a fixed point. On the other hand, G has the Haagerup property if it admits a proper action by continuous affine isometries on a Hilbert space. It follows that if G has both properties, then G is finite. We refer the reader to the book <cit.> for a thorough exposition of these properties. We may now prove Theorem <ref>, whose statement we now recall: Theorem <ref> Let g≥ 0. Then _g and _g do not have Property (T). Recall from equation (<ref>) that there is a surjective homomorphism _g → V. It is known (see, for instance, <cit.>) that Property (T) is preserved under quotients. On the other hand, a result of Farley <cit.> asserts that V has the Haagerup property. Since V is infinite, it follows that V does not have Property (T), and thus neither does _g. The same argument yields the result for _g. We now proceed to give a proof of Theorem <ref>. The proof is again based on properties of the group V, plus a result of Farb-Masur <cit.> which asserts that every homomorphic image of a lattice in a finite-type mapping class group is finite. More concretely, they proved: Let Γ be an irreducible lattice in a semisimple Lie group of real rank at least two. For every g, n≥ 0, every homomorphism Γ→(S_g,n) has finite image. Before giving a proof of Theorem <ref>, we remind the reader of its statement: Theorem <ref> Let Γ be a lattice in a semisimple Lie group G of real rank at least two, where G has no compact factors isogenous to SU(1,n) or SO(1,n). Then, any homomorphism from Γ to _g (resp. _g) has finite image. Let G be a Lie group as in the statement. As such, G has Property (T) (see <cit.>, for instance), and therefore so does Γ (see <cit.>). For such a lattice Γ, let ϕ: Γ→_g be a homomorphism. Recall again the short exact sequence (<ref>): 1→_c(Σ_g)→_gp→ V→ 1. As in the proof of Theorem <ref>, the fact that V has the Haagerup property implies that (p ∘ϕ)(Γ) is finite. But ϕ(Γ) ∩(p) is also finite. To see this, observe that since Γ is finitely presented (see <cit.>), Lemma <ref> implies that there exists a finite-type subsurface S of Σ_g such that ϕ(Γ) ∩(p) is contained in (S). At this point, Theorem <ref> tells us that ϕ(Γ) ∩(p) is also finite, as desired. We obtain the result for _g in an analogous way. § NON-LINEARITY In this section we will prove Theorem <ref>. As mentioned in the introduction, we will do so by showing that _g contains a copy of Thompson's group F. Recall that F is the group of piecewise-linear self-homeomorphisms of [0,1] that preserve rational dyadic numbers, are differentiable except at finitely many dyadic rationals, and at each interval of differentiability the slopes are powers of 2. A strongly related group which we will also need is Thompson's group T, whose definition is the same as that of F but with the unit circle 𝕊^1 instead of [0,1]. One has the well-known inclusions F⊂ T ⊂ V; for a proof, as well as a detailed discussion on Thompson's groups, see <cit.>. We now prove: Let g≥ 0. The short exact sequence (<ref>), that is 1 →_c(Σ_g) →_g → V → 1, splits over Thompson's group F. The result is known for g=0 (see <cit.>), where in addition it is shown that the sequence (<ref>) splits over Thompson group's T, identified with the subgroup of _0 preserving the whole visible side of Σ_0. Consider now a closed disk Σ_0,1 with a Cantor set removed from its interior. The surface Σ_0,1 comes equipped with an obvious rigid structure that comes from that of Σ_0 under the natural subsurface embedding Σ_0,1↪Σ_0. The group of asymptotically rigid mapping classes of Σ_0,1 is the pure braided Thompson group BV, a subgroup of the braided Thompson group considered by Brin <cit.> and Dehornoy (<cit.>). When g>0 the map (Σ_0,1)→(Σ_g) is injective. As a consequence BV injects into _g. On the other hand it is known that BV splits over F (see <cit.>). Finally, we prove Theorem <ref>: By Proposition <ref>, B_g contains an isomorphic copy of Thompson's group F. Now, F has simple commutator subgroup <cit.>, and in particular is not residually finite. Since finitely-generated subgroups of linear groups are residually finite and F is finitely generated, we deduce that _g (and thus _g) is not linear. 99 A J. E. Andersen, Mapping Class Groups do not have Kazhdan's Property (T), arXiv:0706.2184. Anderson R. D. Anderson, The algebraic simplicity of certain groups of homeomorphisms. Amer. J. Math 80 (1958), 955–963. APV J. Aramayona, P. Patel, N. G. Vlamis, The first integral cohomology group of big mapping class groups. arXiv:1711.03132 AS J. Aramayona, J. Souto, Homomorphisms between mapping class groups, Geom. Topol. 16 (2012), 2285–2341. BDR J. Bavard, S. Dowdall, K. Rafi, Isomorphisms between big mapping class groups- I.M.R.N, in press. BeMa J. Behrstock, D. Margalit, Curve complexes and finite index subgroups of mapping class groups. Geom. Dedicata 118 (2006), 71–85. BHV B. Bekka, P. de la Harpe, A. Valette, Kazhdan's Property (T). New Mathematical Monographs, 11. Cambridge University Press, Cambridge, 2008. BF M. Bestvina, K. Fujiwara, Bounded cohomology of subgroups of mapping class groups. Geom. Topol. 6 (2002), 69–89. Bigelow S. Bigelow, R. Budney, The mapping class group of a genus two surface is linear, Algebraic and Geometric Topology, 34 (2001). BLM J. Birman, A. Lubotzky, J. McCarthy, Abelian and solvable subgroups of the mapping class groups. Duke Math. J. 50 (1983), 1107-1120. BDJ C. Bleak, C. Donoven, J. Jonušas, Some isomorphism results for Thompson like groups V_n(G), Israel J. Math. 222 (2017), 1–19. BCMNO Collin Bleak, Peter Cameron, Yonah Maissel, Andrés Navas, Feyishayo Olukoya, The further chameleon groups of Richard Thompson and Graham Higman: Automorphisms via dynamics for the Higman groups G_n,r, arXiv:1605.09302. Boldsen S. Boldsen, Improved homological stability for the mapping class group with integral or twisted coefficients. Math. Z. 270 (2012), 297–329. Bo C.-F. Bödigheimer, Configuration models for moduli spaces of Riemann surfaces with boundary, Abh. Math. Sem. Univ. Hamburg, 76 (2006) 191–233. BH M. R. Bridson, A. Haefliger. Metric Spaces of Non-Positive Curvature, Springer-Verlag. Brin M. G. Brin, The Algebra of Strand Splitting. I. A Braided Version of Thompson’s Group V, J. Group Theory 10 (2007), 757–788. Brown K. S. Brown, Cohomology of groups. Graduate Texts in Mathematics, 87. Springer-Verlag, New York-Berlin, 1982. Brown2 K. S. Brown, The geometry of finitely presented infinite simple groups. Algorithms and classification in combinatorial group theory. Math. Sci. Res. Inst. Publ., 23, Springer, New York, 1992. CFP J. W. Cannon, W. J. Floyd, W. R. Parry, Introductory notes on Richard Thompson's groups. Enseign. Math. (2) 42 (1996), 215–256. Castel F. Castel, Geometric representations of the braid groups, Astérisque 378, Paris, Soc. Math. France, 175 p. (2016). Deh P. Dehornoy, The group of parenthesized braids, Adv. Math. 205 (2006), 354–409. F-M B. Farb, H.Masur, Superrigidity and mapping class groups, Topology 37(1998), 1169–1176. FM B. Farb, D. Margalit, A primer on mapping class groups, Princeton University Press 49 (2012). FarD. Farley, Proper isometric actions of Thompson's groups on Hilbert space. Int. Math. Res. Not. 2003, no. 45. FHD. Farley, B. Hughes, Finiteness properties of some groups of local similarities, Proc. Edinburgh Math. Soc. 58 (2015), 379–402. FK2 L. Funar, C. Kapoudjian, An infinite genus mapping class group and stable cohomology. Comm. Math. Phys. 287 (2009), 787–804. FK L. Funar, C. Kapoudjian, On a universal mapping class group of genus zero. Geom. Funct. Analysis 14 (2004), 965–1012. FNe L. Funar, Y. Neretin, Diffeomorphisms groups of tame Cantor sets and Thompson-like groups, Compositio Math. 154 (2018), 1066–110. FN L. Funar, M. Nguyen, On the automorphisms group of the asymptotic pants complex of an infinite surface of genus zero, Math. Nachrichten 289(2016), 1189–1207. GS E. Ghys and V. Sergiescu, Sur un groupe remarquable de difféomorphismes du cercle, Comment. Math. Helv. 62 (1987), 185–239. Godin V. Godin, The unstable homology of the integral homology of the mapping class groups of a surface with boundary, Math. Annalen, 337 (2007), 15–60. H T. Haettel, Hyperbolic rigidity of higher rank lattices. (With an appendix by V. Guirardel and C. Horbez). Preprint, arXiv:1607.02004 Harer-unp J. L. Harer, Improved homology stability for the homology of the mapping class groups of surfaces. Preprint, DUK-M-93–09 (1993). Harer J. L. Harer, Stability of the homology of the mapping class groups of orientable surfaces. Ann. of Math. (2) 121 (1985), 215–249. Harer-vcd J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math. 84 (1986) 157–176. HT A. E. Hatcher, W. P. Thurston, A presentation for the mapping class group of a closed orientable surface, Topology, 19 (1980) 221–237. HMV J. Hernández, I. Morales, F. Valdez, Isomorphisms between curve graphs of infinite-type surfaces are geometric, arXiv:1706.03697. Ivanov N. V. Ivanov, Automorphism of complexes of curves and of Teichmüller spaces. Internat. Math. Res. Notices 14 (1997), 651–666. Ivanov-mcg N. V. Ivanov, Automorphisms of Teichmüller modular groups. In Topology and geometry – Rohlin Seminar, Lecture Notes in Math., 1346, Springer, Berlin, 1988, 199–270. Ivanov-lattice N. V. Ivanov, Teichmüller modular groups and arithmetic groups, Zapiski Nauchnnykh Seminarov Leningrad. Otdeleni Mathematicheski Institut Steklov (LOMI) Vol. 167, 1988, 95–110. KM V. Kaimanovich, H. A. Masur, The Poisson boundary of the mapping class group. Invent. Math. 125 (1996), 221–264. KS C. Kapoudjian and V. Sergiescu, An extension of the Burau representation to a mapping class group associated to Thompson's group T. In Geometry and dynamics, Contemp. Math. 389 Amer. Math. Soc., Providence, R.I., 2005, 141–164. PV P. Patel, N. Vlamis, Algebraic and topological properties of big mapping class group, arXiv:1703.02665. Powell J. Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978), 347–350. Raymond F. Raymond. The end point compactification of manifolds. Pacific J. Math. 10 (1960), 947–963. Richards I. Richards, On the classification of noncompact surfaces. Trans. Amer. Math. Soc. 106 (1963), 259–269. Rubin M. Rubin, Locally moving groups and reconstruction problems, Ordered groups and infinite permutation groups, Math. Appl., vol. 354, Kluwer Acad. Publ., Dordrecht, 1996, pp. 121–157. SW M. Szymik and N. Wahl, The homology of Higman-Thompson groups, Invent. Math. 216 (2019), 445–518. Verj A. Verjovsky, Commentaries on the paper “Solenoidal manifolds” by Dennis Sullivan, Proc. of Algebraic Methods in Geometry, Guanajuato, 2011, Journal of Singularities 9 (2014), 245–251. Waj B. Wajnryb, An elementary approach to the mapping class group of a surface. Geom. Topol. 3 (1999), 405–466. Witte D. Witte-Morris, An introduction to arithmetic groups. Deductive Press, 2015.

http://arxiv.org/abs/1702.01645v3

20170127065839

Paradoxical Stabilization of Forced Oscillations by Strong Nonlinear Friction

physics.plasm-ph

Kansai Photon Science Institute, National Institutes for Quantum and Radiological Science and Technology (QST), 8-1-7 Umemidai, Kizugawa, Kyoto 619-0215, Japan Kansai Photon Science Institute, National Institutes for Quantum and Radiological Science and Technology (QST), 8-1-7 Umemidai, Kizugawa, Kyoto 619-0215, Japan A. M. Prokhorov Institute of General Physics, the Russian Academy of Sciences, Vavilov street 38, 119991 Moscow, Russia In a dissipative dynamic system driven by an oscillating force, a strong nonlinear highly oscillatory friction force can create a quasi-steady tug, which is always directed opposite to the ponderomotive force induced due to a spatial inhomogeneity of oscillations. When the friction-induced tug exceeds the ponderomotive force, the friction stabilizes the system oscillations near the maxima of the oscillation spatial amplitude. Paradoxical Stabilization of Forced Oscillations by Strong Nonlinear Friction S. V. Bulanov January 27, 2017 ============================================================================== § INTRODUCTION In classical mechanics, Kapitza pendulum or Stephenson-Kapitza pendulum <cit.> is a statically unstable inverted pendulum whose statically unstable equilibrium position is stabilized by small fast vertical oscillations of the pivot point. This induced stability initially described by A. Stephenson <cit.> has been first explained by P. L. Kapitza <cit.>. To find theoretical reason of the induced stability Kapitza separated the pendulum motion into fast and slow oscillations and, by averaging out fast ones, found the effective potential which has minimum at the pendulum upper position, in contrast to a simple pendulum. This approach created a new concept of dynamic stabilization in mechanics <cit.>. Chelomei's pendulum provides another well known example of dynamically stabilized mechanical system <cit.>. The induced stability can be vindicated beyond the framework of the method of fast and slow motions separation <cit.>. In general, a spatial inhomogeneity of an oscillating driving force creates a quasi-steady ponderomotive force <cit.>, directed against the spatial gradient of the driving force. In the Kapitza pendulum, this ponderomotive force acts against gravitation and makes the upper position stable. A dissipation dampens oscillations around the upper equilibrium position thus further stabilizing it. When no other forces present besides the oscillating driving force, as in the case of charged particle dynamics in a standing electromagnetic wave, the ponderomotive force always repels particles from the maxima of the wave spatial amplitude <cit.>. Here we present a general model of a dissipative dynamic system driven by an oscillating force, where a strong nonlinear highly oscillatory friction creates a quasi-steady tug, which, quite counter-intuitively, is always directed opposite to the ponderomotive force and exceeds it for a sufficiently strong driving force. This leads to a seemingly paradoxical stabilization of the system oscillations near the maxima of the spatial amplitude of the driving force. It differs from the Kapitza pendulum effect in that here the stabilization factor is a nonlinear growth of the friction with the driver force, which creates a tug against the ponderomotive potential. § MODEL We consider a simple one-dimensional model of a forced oscillation with a strong nonlinear friction, given by the equation ẍ+𝒦(ℱ) ẋ =ℱ , 𝒦=νℱ^2n . Here the dot denotes differentiation with respect to time; n is a natural number. The friction coefficient 𝒦 is a non-negative function of the oscillating driving force ℱ, ℱ(x,t)=f_1(x)cosω t+f_2(x)sinω t . The model is motivated by the dynamics of a charged particle in a strong electromagnetic field, where accelerating particles lose energy and undergo a recoil due to their emission of electromagnetic radiation <cit.>. This causes a friction which nonlinearly grows with the electromagnetic field strength. Following the classical approach of Ref. <cit.>, we assume that a solution to Eq. (<ref>) can be represented as x(t)=X(t)+ξ(t) with a slowly varying function X(t) and a fast oscillating small addition ξ(t), |ξ|≪ |X|, which has a zero time average, ⟨ξ⟩=(ω/2π)∫_0^2π/ωξ(t) dt =0. Correspondingly, ⟨ x ⟩=X, ⟨ξ̇⟩=⟨ξ̈⟩ =0, < X>≈ X(t). Substituting (<ref>) into Eq. (<ref>) and expanding the functions ℱ and 𝒦 in powers of ξ as ℱ≈ F+ξ∂_X F , F=ℱ(X,t) ; 𝒦≈ K+ξ∂_X K , K=𝒦(ℱ(X,t)), we obtain Ẍ+ ξ̈+K Ẋ+K ξ̇+ ξẊ∂_X K + ξξ̇∂_X K =F+ξ∂_X F , where ∂_X is the partial derivative with respect to X (the first argument of ℱ). The time derivatives ξ̈ and ξ̇ are not small, being proportional to ω^2 and ω, respectively. They are assumed to be much greater than Ẍ and Ẋ. The friction coefficient K defined in Eq. (<ref>) is not necessarily small; it has a time-independent component κ = ⟨ K ⟩ >0 . In Eq. (<ref>), slowly varying and fast oscillating terms should cancel out separately. Neglecting the time derivatives of X, in the zeroth order approximation with respect to ξ we find for the fast oscillating term ξ̈+K ξ̇=F . Here X as an argument of functions F and K is assumed to be constant. The forced oscillation solution of Eq. (<ref>) can be cast in the form ξ=e^-κ t-Δ(t)∫_0^t e^κτ+Δ(τ) F(τ) dτ , Δ(t) = ∫_0^t [ K(τ) - κ ] dτ . The first term in the expansion with respect to Δ of the dependence given by Eq. (<ref>) ξ=(κ f_1-ω f_2) sinω t -(κ f_1+ω f_2) cosω t/ω(κ^2+ω^2) approximates the first harmonic of the solution. Averaging Eq. (<ref>) over time and taking into account that < F(X,t)>≈ 0 for nearly constant X(t), we obtain Ẍ+(κ+⟨ξ∂_X K⟩) Ẋ= ⟨ξ∂_X F ⟩ - ⟨ξξ̇∂_X K⟩ . Using here the expression for ξ, given by Eq. (<ref>), and the definitions for F and K formulated above, we obtain the equation for the slowly varying function X(t) and the average friction coefficient: Ẍ +κẊ= -∂ _X (f_1^2+f_2^2)/4(κ^2+ω^2) + n^2κ^2 ∂ _X (f_1^2+f_2^2)/(n+1)(κ^2+ω^2)^2 + κ (f_2 ∂_X f_1 - f_1 ∂_X f_2)/2ω(κ^2+ω^2)[ 2n/n+1( κ^2-ω^2/κ^2+ω^2) -1 ] , κ= 2^-2n2nnν (f_1^2+f_2^2)^n , where 2nn is the binomial coefficient. The first term on the r.h.s. of Eq. (<ref>) corresponds to the classical ponderomotive force <cit.>, modified due to the friction, the last terms represent the tug induced by the friction. In the case of f_2=0, the third term in the r.h.s. of Eq. (<ref>) vanishes. Then the tug becomes always directed opposite to the ponderomotive force. It can even exceed the latter in magnitude, when ν > 2^2n (n!)^2 (n+1)^1/2/(2n)! (4n^2-n-1)^1/2ω/f_1^2n . In this case, trajectories of the system described by the model (<ref>) drift to the local maximum of the driving force spatial amplitude. If the system were non-dissipative, the oscillations near that maximum would be destabilized by the ponderomotive force, so that the corresponding trajectories would drift against the spatial gradient of the driving force and would eventually escape to regions of a lower spatial amplitude of the driving force <cit.>. Sufficiently strong friction makes the trajectories to drift along the spatial gradient of the driving force, provided that these trajectories have already got to a region with sufficiently high driving force. This causes a seemingly paradoxical stabilization of the oscillations near the local maximum of the driving force spatial amplitude. § NUMERICAL SIMULATIONS The effect of the stabilization of oscillations due to strong friction is further demonstrated by the numerical integration of the model equation (<ref>) with the friction coefficient taken in the form 𝒦=νℱ(x,t)^4 in two cases. In the first case, shown in Fig. <ref> (a,b,c), the driving force amplitude is bell-shaped, ℱ(x,t)=f_0exp(-(x/l_0)^2) cosω t, with the width, oscillation frequency, amplitude, and friction factor equal to l_0=10, ω=1, f_0=3, and ν=0.2, respectively. In the second case, Fig. <ref> (d,e,f), the driving force is spatially periodic, ℱ(x,t)=f_0cos^2(2π x/l_0) cosω t, l_0=10, ω=1, f_0=8, and ν=0.25. In the case of bell-shaped driving force shown in Fig. <ref> (a,b,c), the trajectories, starting at t=0 from locations where the driving force spatial amplitude is relatively weak, exhibit several oscillations and then escape being pushed away by the ponderomotive force. In the region of a high spatial amplitude of the driving force, the friction-induced tug overcomes the ponderomotive force, therefore the trajectories started from this region drift towards the maximum of the driving force spatial amplitude. As they get closer to that maximum, their drift becomes slower and their oscillation amplitude decreases. In the case of spatially periodic driving force (see Fig. <ref> (d,e,f)), the trajectories, initially oscillating near the minima of the driving force spatial amplitude and having enough large oscillation amplitude, reach the region of higher driving force, where the trajectories are caught by the friction-induced tug. Eventually all such trajectories are reduced to oscillations near the maxima of the driving force spatial amplitude. In both cases the drift is slowing down near the maxima of the driving force spatial amplitude in agreement with Eq. (<ref>), because the gradient ∂_X f vanishes at the maximum of f. § LIMIT CYCLE On a trajectory asymptotically turning into periodic oscillations seen in Fig. <ref> (b,c,e,f), the driving force is almost constant, which simplifies theoretical consideration allowing more detail description of the driven oscillations. This case is described by the approximation of ℱ(x,t)=f_0 cosω t. We change variables to (τ,y) and introduce the friction parameter σ: t = τ/ω, x(t) = (f_0/ω^2)y(τ) ; σ=ν f_0^2n/ω . Then Eq. (<ref>) becomes y”(τ) + σcos^2n(τ) y'(τ) = cosτ . The general solution can be represented as y'(τ)= e^-σ S_n(τ) [ y'(0) - Y_n(0) ] + Y_n(τ) , Y_n (τ) = e^-σ S_n(τ)/e^πασ+1∫_-π/2^π/2 e^σ S_n(ζ+τ+π/2)sin(ζ + τ)dζ , S_n(τ) = ατ+ ∑_m=1^n 2nn+msin(2mτ)/2^2nm , α=2^-2n2nn . As one can see, any solution at τ→∞ tends to the limit cycle described by the function Y_n. The amplitude of the limit cycle in terms of the derivative y', A_n=max|Y_n(τ+π)-Y_n(τ)|, is A_n= 1/cosh(πασ/2)∫_-π/2^π/2 e^σ S_n(τ)cos(τ)dτ . For large σ, it decreases as a negative power of σ, e.g., A_1 ≈ 1.88σ^-2/3. Correspondingly, the oscillation amplitude decreases when the trajectory drifts towards the driving force maximum, as seen in Fig. <ref>. In the case of n=1, the function describing the limit cycle, Eq. (<ref>), can be represented as a Fourier series in terms of odd harmonics of the driving force Y_1(τ)= ∑_m=1^∞[ C_me^i(2m-1)τ+C_m^*e^-i(2m-1)τ] , where the asterix denotes complex conjugation. The sequence C_1, C_m, m≥2, representing the frequency spectrum of Y_1, is expressed in terms of modified Bessel functions of the first kind, I_k: C_1= ∑_k=-∞^∞(-1)^k+1/4k+2-iσ[ i I_k^2(σ/4) + I_k(σ/4)I_k+1(σ/4) ] , C_m = i^m I_m-1/2-iσ/4(σ/4)/I_3/2-iσ/4(σ/4)[ (2+4i/σ)C_1 +C_1^* - 2/σ] . The amplitude of the first harmonic, C_1, as a function of σ can be approximated by C_1(σ)≈σ/2/4+σ^2-δ - i[ σ/3-2/4+σ^2-δ + 16+3σ^2/(4+σ^2-δ)^2] with δ≈ 0.098. The spectral density of the function Y_1(τ), Eq. (<ref>), is |2C_m|^2; it is shown in Fig. <ref>. As one can see, the frequency spectrum contains high order harmonics according to Eqs. (<ref>-<ref>). § CONCLUSION In conclusion, in contrast to known dynamical destabilization under the action of dissipation known as dissipation-induced instabilities (see review article <cit.> and references therein) we show that a strong nonlinear friction can cause a seemingly paradoxical stabilization of forced oscillations near the maxima of the driving force spatial amplitude. In particular, such a friction occurs in the dynamics of charged particles in ultra-strong electromagnetic fields due to radiation reaction <cit.>. The threshold for the described stabilization of forced oscillations corresponds to the criterion of the importance of the radiation reaction effects <cit.>. In a standing electromagnetic waves, which can be formed in multiple high power laser configurations <cit.>, the stabilization due to a strong radiation reaction is manifested in an anomalous electron bunching near the electric field maxima <cit.>. As shown in Refs. <cit.>, electrons can be captured for many laser periods due to radiation friction impeding the ponderomotive force. When radiation reaction dominates, the electron motion in a standing wave evolves to limit cycles and strange attractors <cit.>. In the case of a circularly polarized standing wave, analytical expressions exist for the limit cycles near the electric field maxima <cit.>. A collision of multiple ultra-intense electromagnetic waves creates structurally determinate patterns in the electron phase space <cit.> due to a counterplay of the ponderomotive force and the friction-induced tug. Although in the present work we have been motivated by the intention to build up the theory of the radiative electron dynamics in the field of extremely high intensity lasers, we believe that the formulated above concept of dissipative stabilization of nonlinear dynamic systems will be useful for applications well beyond the framework of the laser-matter interaction physics <cit.>, remembering a saying of William Thomson (Lord Kelvin) “I never satisfy myself until I can make a mechanical model of a thing” <cit.>. 99 Kapitza1 P. L. Kapitza, Soviet Phys. JETP 21, 588 (1951). Kapitza2 P. L. Kapitza, Usp. Fiz. Nauk 44, 7 (1951). Stephenson A. Stephenson, Philos. Mag., Ser. 6, 15, 233 (1908). Blekhman I. I. Blekhman, Vibrational Mechanics (World Scientific Publishing, 2000). Chelomei V. N. Chelomei, Soviet Physics Doklady 28, 387 (1983). Butikov E. I. Butikov, J. Phys. A: Math. Theor. 44, 295202 (2011). LLM L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, New York, 1976), chap. 30. Licht-Lieb A. J. Lichtenberg and M. A. Lieberman, Regular and chaotic dynamics (Springer-Verlag, New York, 1992), p. 144. Destabilization R. Krechetnikov and J. E. Marsden, Rev. Mod. Phys. 79, 519 (2007). Rad-Dom A. Di Piazza, C. Müller, K. Z. Hatsagortsyan, C. H. Keitel, Rev. Mod. Phys. 84, 1177 (2012). Survey S. V. Bulanov, T. Zh. Esirkepov, S. S. Bulanov, J. K. Koga, Z. Gong, et al., arXiv:1701.03349. Multi-las S. S. Bulanov, V. D. Mur, N. B. Narozhny, J. Nees, V. S. Popov, Phys. Rev. Lett. 104, 220404, (2010). Anom-bunch A. Gonoskov, A. Bashinov, I. Gonoskov, C. Harvey, A. Ilderton, et al., Phys. Rev. Lett. 113, 014801 (2014). Fedotov A. M. Fedotov, N. V. Elkina, E. G. Gelfer, N. B. Narozhny, H. Ruhl, Phys. Rev. A 90, 053847 (2014). Attr T. Zh. Esirkepov, S. S. Bulanov, J. K. Koga, M. Kando, K. Kondo, et al., Phys. Lett. A 379, 2044 (2015). Jirka M. Jirka, O. Klimo, S. V. Bulanov, T. Zh. Esirkepov, E. Gelfer, et al., Phys. Rev. E 93, 023207 (2016). Kirk J. G. Kirk, Plasma Phys. Control. Fusion 58, 085005 (2016). Vranic M. Vranic, T. Grismayer, R. A. Fonseca, L. O. Silva, Plasma Phys. Control. Fusion 59, 014040 (2017). Gong Z. Gong, R. H. Hu, Y. R. Shou, B. Qiao, C. E. Chen, et al., Phys. Rev. E 95, 013210 (2017). MTB G. A. Mourou, T. Tajima, S. V. Bulanov, Rev. Mod. Phys. 78, 309 (2006). Marklund M. Marklund and P. Shukla, Rev. Mod. Phys. 78, 591 (2006). Kelvin W. Thomson (Lord Kelvin), Notes of Lectures on Molecular Dynamics and the Wave Theory of Light. Delivered at The Johns Hopkins University, Baltimore, by Sir William Thomson, Professor in the University of Glasgow. Stenographically Reported by A.S.Hathaway, Lately Fellow in Mathematics of The Johns Hopkins University. (Baltimore: Johns Hopkins, 1884) 270-271. equationsubsection § APPENDIX Here we present mathematical derivations for some formulae shown above. Notations for variables are the same as in the main text. The equation numbering is preserved for those equations which appear in the main text; auxiliary formulae are numbered within sections. §.§ Equations (14) and (15) In this section we derive Eqs. (<ref>) and (<ref>) from Eq. (<ref>). We use the following formulae explicitly written or assumed in the main text: F = f_1 cos(ω t) + f_2 sin(ω t), K = ν[ f_1 cos(ω t) + f_2 sin(ω t) ]^2n, 12ξ=(κ f_1-ω f_2) sinω t -(κ f_1+ω f_2) cosω t/ω(κ^2+ω^2) , 13Ẍ+(κ+⟨ξ∂_X K⟩) Ẋ= ⟨ξ∂_X F ⟩ - ⟨ξξ̇∂_X K⟩ . The time-averaged friction coefficient, defined in Eq. (8) in the main text, is κ=⟨ K⟩= ω/2π∫_0^2π/ων[ f_1 cos(ω t) + f_2 sin(ω t) ]^2n dt= 1/2π∫_0^2πν[ f_1-i f_2/2e^i τ + f_1+i f_2/2e^-i τ]^2n dτ= =ν/2π∑_m=0^2n2nm(f_1+i f_2/2)^m(f_1-i f_2/2)^2n-m∫_0^2π e^2(n-m)iτ dτ= ν/2π2nn(f_1+i f_2/2)^n(f_1-i f_2/2)^n2π . Here 2nm=(2n)!/[m!(2n-m)!] is the binomial coefficient. The last integral (marked with red) is zero except the case m=n, for which it equals 2π (therefore the sum contains the only nonzero term, for the index m=n). In this way we obtain Eq. (<ref>) from the main text: 15κ= 2^-2n2nnν (f_1^2+f_2^2)^n . In Eq. (<ref>), the term ⟨ξ∂_X K⟩ is zero because it contains only odd harmonics, ⟨ξ∂_X K⟩ = ω/2π∫_0^2π/ω ξ(t) × 2n ν[ f_1 cos(ω t) + f_2 sin(ω_t) ]^2n-1[ ∂_X f_1 cos(ω t) + ∂_X f_2 sin(ω t) ] dt =0. The next averaged term is obtained by a simple integration ⟨ξ∂_X F ⟩ = ω/2π∫_0^2π/ω(κ f_1-ω f_2) sinω t-(κ f_1+ω f_2) cosω t/ω(κ^2+ω^2)[ f_1 cos(ω t) + f_2 sin(ω_t) ] dt= = -f_1 ∂_X f_1 + f_2 ∂_X f_2/2(κ^2+ω^2) -κ(f_2 ∂_X f_1 - f_1 ∂_X f_2)/2ω(κ^2+ω^2) . The last averaged terms is -⟨ξξ̇∂_X K⟩ = -ω/2π∫_0^2π/ωξ(t)ξ̇(t)× 2n ν[ f_1 cos(ω t) + f_2 sin(ω_t) ]^2n-1[ ∂_X f_1 cos(ω t) + ∂_X f_2 sin(ω t) ] dt= = - 2nν/8ω(κ^2+ω^2)^2∑_m=0^2n-12n-1m(f_1+i f_2/2)^m(f_1-i f_2/2)^2n-1-m× ×{ (κ - iω)^2(f_1-i f_2)^2 [ (∂_X f_2 -i∂_X f_1) δ_m,n -i(∂_X f_1 -i∂_X f_2) δ_m,n+1] + . . + (κ + iω)^2(f_1+i f_2)^2 ×[ (∂_X f_2 + i∂_X f_1) δ_m,n-1 +i(∂_X f_1 + i∂_X f_2) δ_m,n-2] } = = n κ/(n+1)ω(κ^2+ω^2)^2[ 2nκω(f_1 ∂_X f_1 + f_2 ∂_X f_2) +(κ^2-ω^2)(f_2 ∂_X f_1 - f_1 ∂_X f_2) ]. Here δ_m,n is the Kronecker delta; it equals one for m=n and zero otherwise. In the last line ν is expressed in terms of κ using Eq. (<ref>). Combining the results into Eq. (<ref>) we obtain Eq. (<ref>) from the main text: 14Ẍ+κẊ= -∂ _X (f_1^2+f_2^2)/4(κ^2+ω^2) + n^2κ^2 ∂ _X (f_1^2+f_2^2)/(n+1)(κ^2+ω^2)^2 + κ (f_2 ∂_X f_1 - f_1 ∂_X f_2)/2ω(κ^2+ω^2)[ 2n/n+1( κ^2-ω^2/κ^2+ω^2) -1 ] . §.§ Equations (20-22) In this section we derive Eqs. (<ref>), (<ref>) and (<ref>), representing the solution of Eq. (<ref>): 19 y”(τ) + σcos^2n(τ) y'(τ) = cosτ . We seek the solution for the first derivative y' in the form y'(τ) = M(τ)exp(-σ∫_0^τcos^2n(η)dη) . Substituting this ansatz into Eq. (<ref>), we obtain M'(τ) = exp(σ∫_0^τcos^2n(η)dη)cos(τ) . Integrating this equation for M and substituting the result into Eq. (<ref>), we obtain a general solution of Eq. (<ref>): y'(τ) = exp(-σ∫_0^τcos^2n(η)dη) [y'(0) + ∫_0^τexp(σ∫_0^ζcos^2n(η)dη) cos(ζ) dζ] . The term in the exponent can be expanded as S_n(τ) = ∫_0^τcos^2n(η)dη = ∫_0^τ(e^iη+e^-iη/2)^2ndη =∫_0^τ∑_m=0^2n1/2^2n2nm e^i (2n-m)ηe^-i mηdη = =1/2^2n∑_m=0^2n2nm∫_0^τ e^2i (n-m)ηdη = 1/2^2n2nnτ + 1/2^2n∑_m=0 m≠n^2n2nme^2i (n-m)τ - 1/2i(n-m) = =1/2^2n2nnτ + 1/2^2n∑_m=0 m→ n-m'^n-12nme^2i (n-m)τ - 1/2i(n-m) + 1/2^2n∑_m=n+1 m→ n+m'^2n2nme^2i (n-m)τ - 1/2i(n-m) = =1/2^2n2nnτ + 1/2^2n∑_m'=1^n2nn-m'e^2i m' τ - 1/2i m' + 1/2^2n∑_m'=1^n2nn+m'e^- 2i m'τ - 1/-2i m' = =1/2^2n2nnτ + 1/2^2n∑_m=1^n2nn+me^2i m τ - e^-2i m τ/2i m =1/2^2n2nnτ + 1/2^2n∑_m=1^n2nn+msin(2m τ)/m . Here we rearranged the sums changing to new indices (marked with red) and used the identity 2nn-m=2nn+m for n≥ m≥ 0. In this way we obtain Eq. (<ref>) in the main text: S_n(τ) = ατ+ ∑_m=1^n 2nn+msin(2mτ)/2^2nm , α=2^-2n2nn . 22 Below we use the following properties of the function S_n: S_n(-τ)=S_n(τ), S_n(τ+π)=S_n(τ)+πα, S_n(π/2)=πα/2. The limit cycle is a periodic solution Y_n of Eq. (<ref>), therefore for any τ Y_n(τ) = Y_n(τ+2π). Using (<ref>) we rewrite this expression as Y_n(τ) = e^-σ S_n(τ+2π)[y'(0) + ∫_0^τ+2π e^σ S_n(ζ)cos(ζ) dζ] = = e^-2πασe^-σ S_n(τ)[ y'(0) + ∫_0^τ e^σ S_n(ζ)cos(ζ) dζ + ∫_τ^τ+2π e^σ S_n(ζ)cos(ζ) dζ] = = e^-2πασ[ Y_n(τ) + e^-σ S_n(τ)∫_τ^τ+2π e^σ S_n(ζ)cos(ζ) dζ] , which gives the formula for Y_n(τ) Y_n(τ) = e^-σ S_n(τ)/e^2πασ-1∫_τ^τ+2π e^σ S_n(ζ)cos(ζ) dζ = e^-σ S_n(τ)/e^2πασ-1∫_0^2π e^σ S_n(ζ+τ)cos(ζ+τ) dζ= =e^-σ S_n(τ)/e^2πασ-1[ ∫_0^π e^σ S_n(ζ+τ)cos(ζ+τ) dζ +∫_π^2π e^σ S_n(ζ+τ)cos(ζ+τ) dζ]= = e^-σ S_n(τ)/e^2πασ-1[ ∫_0^π e^σ S_n(ζ+τ)cos(ζ+τ) dζ -∫_0^π e^σ S_n(ζ+τ)+πασcos(ζ+τ) dζ]= = e^-σ S_n(τ)/e^2πασ-1 (1-e^πασ) ∫_0^π e^σ S_n(ζ+τ)cos(ζ+τ) dζ = e^-σ S_n(τ)/e^πασ+1∫_-π/2^π/2 e^σ S_n(ζ+τ+π/2)sin(ζ+τ) dζ . In this way we obtain Eq. (<ref>) in the main text: Y_n (τ) = e^-σ S_n(τ)/e^πασ+1∫_-π/2^π/2 e^σ S_n(η+τ+π/2)sin(η + τ)dη . 21 Accepting Y_n as a particular solution of Eq. (<ref>) and rewriting Eq. (<ref>), we obtain another form of the general solution of Eq. (<ref>), namely Eq. (<ref>): y'(τ)= e^-σ S_n(τ) [ y'(0) - Y_n(0) ] + Y_n(τ) . 20 §.§ Equation (23) In this section we derive Eq. (<ref>). The maximum difference |Y_n(τ+π)-Y_n(τ)| is reached for τ=-π/2, therefore A_n =Y_n(π/2)-Y_n(-π/2)= = e^-σ S_n(π/2)/e^πασ+1∫_-π/2^π/2 e^σ S_n(η+π)sin(η + π/2)dη - e^-σ S_n(-π/2)/e^πασ+1∫_-π/2^π/2 e^σ S_n(η)sin(η - π/2)dη = = e^-πασ/2/e^πασ+1∫_-π/2^π/2 e^σ S_n(η)+πασcos(η)dη + e^πασ/2/e^πασ+1∫_-π/2^π/2 e^σ S_n(η)cos(η)dη = =2e^πασ/2/e^πασ+1∫_-π/2^π/2 e^σ S_n(η)cos(η)dη =1/cosh(πασ/2)∫_-π/2^π/2 e^σ S_n(η)cos(η)dη . Thus we obtain Eq. (<ref>) in the main text: 23A_n= 1/cosh(πασ/2)∫_-π/2^π/2 e^σ S_n(τ)cos(τ)dτ . §.§ Equations (25) and (26) In this section we derive Eqs. (<ref>) and (<ref>) for the case of n=1. The function Y_1 is a periodic solution of the equation Y'_1(τ) + σcos^2(τ) Y_1(τ) = cosτ . It is represented as a Fourier series, which obviously should contain only odd hamonics of the driver: Y_1(τ)= ∑_m=1^∞[ C_me^i(2m-1)τ+C_m^*e^-i(2m-1)τ] , 24 where symbol “*” denotes complex conjugation. Substituting Eq. (<ref>) into Eq. (<ref>) we obtain ∑_m=1^∞[ i(2m-1) C_me^i(2m-1)τ - i(2m-1) C_m^*e^-i(2m-1)τ] + + σ(e^2iτ/4+1/2+e^-2iτ/4) ∑_m=1^∞[ C_me^i(2m-1)τ + C_m^*e^-i(2m-1)τ] =e^iτ/2+e^-iτ/2 . Rearranging the sums and collecting the Fourier coefficients of the terms corresponding to the same harmonics, we easily find the following recursive relations: C_2=-[ (2+4i/σ)C_1 +C_1^* - 2/σ] , i^-m C_m = σ/4/2(m-1/2-iσ/4)( i^-(m-1) C_m-1 - i^-(m+1) C_m+1). The last expression is the same as a recurrence identity for the modified Bessel function of the first kind <cit.>, I_μ(z) = z/2μ(I_μ-1(z)-I_μ+1(z)) for z=σ/4 and μ=m-1/2-iσ/4, and I_μ(z) ∝ i^-m C_m. The general solution of Eq. (<ref>) is a linear combination of the modified Bessel functions of the first and second kinds, where one should cancel out an unbounded term: C_m = U i^m I_m-1/2-iσ/4(σ/4). The coefficient of proportionality, U, is determined using the relation Eq. (<ref>): C_2 = -U I_3/2-iσ/4(σ/4) =-[ (2+4i/σ)C_1 +C_1^* - 2/σ] . As a result we obtain Eq. (<ref>) of the main text: C_m = i^m I_m-1/2-iσ/4(σ/4)/I_3/2-iσ/4(σ/4)[ (2+4i/σ)C_1 +C_1^* - 2/σ] . 26 The formula Eq. (<ref>) for the coefficient C_1 can be obtained using series representation for the function Y_1. For n=1, we have α=1/2, S_1(τ)=τ/2+sin(2τ)/4, and Eq. (<ref>) becomes Y_1 (τ) = e^πσ/4/e^πσ/2+1exp(-σ/4sin(2τ)) ∫_-π/2^π/2exp(σ/2η) exp(-σ/4sin(2η+2τ)) sin(η + τ)dη . Using the generating function for the modified Bessel function of the first kind <cit.>, exp[ z/2(p+1/p) ] = ∑_k=-∞^∞ I_k(z) p^k , we change exponent terms involving sin function into series in the following way exp(-σ/4sin(2ζ)) = ∑_k=-∞^∞ I_k(σ/4) i^k e^2ikζ . Then the right-hand side of Eq. (<ref>) is transformed into a double series Y_1 (τ) = e^πσ/4/e^πσ/2+1 ∑_m=-∞^∞∑_k=-∞^∞ i^m+k I_m(σ/4) I_k(σ/4) e^2imτ ∫_-π/2^π/2 e^ση/2 + 2ik(η+τ) sin(η + τ)dη . Integrating term-by-term and rearranging the sums by collecting terms involving the same harmonics, it is not difficult to obtain the following formulae Y_1(τ) = ∑_m=-∞^∞( C_m e^i (2m-1) τ + C_1-m e^-i (2m-1) τ), C_m = i^m ∑_k=-∞^∞(-1)^k+1/4k+2-iσ[ I_k(σ/4) - i I_k+1(σ/4) ] I_k+1-m(σ/4). Using Eq. (<ref>) and the symmetry property I_-k(z)=I_k(z) <cit.>, it is easy to show that C_1-m=C^*_m. For m=1 we obtain Eq. (<ref>) from the main text: C_1= ∑_k=-∞^∞(-1)^k+1/4k+2-iσ[ i I_k^2(σ/4) + I_k(σ/4)I_k+1(σ/4) ]. 25 The representation Eq. (<ref>) is equivalent to Eq. (<ref>) provided that C_1 is defined by Eq. (<ref>). Fig. <ref> shows a comparison of the numerical solution of Eq. (<ref>) for σ=20 with the analytical solution defined by Eqs. (<ref>), (<ref>), and (<ref>), where the sum in Eq. (<ref>) is cut at m=6 (which corresponds to the 2m-1=11^ th harmonic) and the sum in Eq. (<ref>) is cut at k=± 10 (i.e. only terms with the index k from -10 to 10 are taken into account). 99 math M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables (Dover Publications, 1974) chapter 9, section 6.

http://arxiv.org/abs/1701.07776v2

20170126170630

Dependent Mixtures of Geometric Weights Priors

stat.ME

Dependent Mixtures of Geometric Weights Priors Spyridon J. Hatjispyros [Corresponding author. Tel.:+30 22730 82326 E-mail address: schatz@aegean.gr ]^, *, Christos Merkatas^*, Theodoros Nicoleris^**, Stephen G. Walker^*** ^* Department of Mathematics, University of the Aegean, Karlovassi, Samos, GR-832 00, Greece. ^** Department of Economics, National and Kapodistrian University of Athens, Athens, GR-105 59, Greece. ^***Department of Mathematics, University of Texas at Austin, Austin, Texas 7812, USA. A new approach to the joint estimation of partially exchangeable observations is presented. This is achieved by constructing a model with pairwise dependence between random density functions, each of which is modeled as a mixture of geometric stick breaking processes. The claim is that mixture modeling with Pairwise Dependent Geometric Stick Breaking Process (PDGSBP) priors is sufficient for prediction and estimation purposes; that is, making the weights more exotic does not actually enlarge the support of the prior. Moreover, the corresponding Gibbs sampler for estimation is faster and easier to implement than the Dirichlet Process counterpart. Keywords: Bayesian nonparametric inference; Mixture of Dirichlet process; Geometric stick breaking weights; Geometric Stick Breaking Mixtures; Dependent Process. 1. Introduction. In Bayesian nonparametric methods, the use of priors such as the Dirichlet process (Ferguson, 1973), is justified from the assumption that the observations are exchangeable, which means the distribution of (X_1,…,X_n) coincides with the distribution of (X_π(1),…,X_π(n)), for all π∈ S(n), where S(n) is the set of permutations of {1,…,n}. However, in real life applications, data are often partially exchangeable. For example, they may consist of observations sampled from m populations, or may be sampled from an experiment conducted in m different geographical areas. This means that the joint law is invariant under permutations within the m subgroups of observations (X_j,i_j)_1≤ i_j≤ n_j, 1≤ j≤ m, so for all π_j∈ S(n_j) ((X_1,i_1)_1≤ i_1≤ n_1,…,(X_m, i_m)_1≤ i_m≤ n_m)∼ ((X_1,π_1(i_1))_1≤ i_1≤ n_1,…,(X_m, π_m(i_m))_1≤ i_m≤ n_m). When the exchangeability assumption fails one needs to use non–exchangeable priors. There has been substantial research interest following the seminal work of MacEachern (1999) in the construction of suitable dependent stochastic processes. Such then act as priors in Bayesian nonparametric models. These processes are distributions over a collection of measures indexed by values in some covariate space, such that the marginal distribution is described by a known nonparametric prior. The key idea is to induce dependence between a collection of random probability measures (_j)_1≤ j≤ m, where each _j comes from a Dirichlet process (DP) with concentration parameter c>0 and base measure P_0. Such random probability measures typically are used in mixture models to generate random density functions f(x) = ∫_ΘK(x|θ)(dθ); see Lo (1984). There is a variety of ways that a DP can be extended to dependent DP. Most of them use the stick-breaking representation (Sethuraman, 1994), that is ( · ) = ∑_k=1^∞w_kδ_θ_k( · ), where (θ_k)_k≥ 1 are independent and identically distributed from P_0 and (w_k)_k≥ 1 is a stick breaking process; so if (v_k)_k≥ 1 are independent and identically distributed from Be(1, c), a beta distribution with mean (1+c)^-1, then w_1=v_1 and for k>1, w_k=v_k∏_l<k(1-v_l). Dependence is introduced through the weights and/or the atoms. A classical example of the use of dependent DP's is the Bayesian nonparametric regression problem where a random probability measure _z is constructed for each covariate z, _z( · ) = ∑_k=1^∞ w_k(z)δ_θ_k(z)( · ), where (w_k(z),θ_k(z)) is a collection of processes indexed in z–space. Extensions to dependent DP models can be found in De Iorio et al. (2004), Griffin and Steel (2006), and Dunson and Park (2008). Recently there has been growing interest for the use of simpler random probability measures which while simpler are yet sufficient for Bayesian nonparametric density estimation. The geometric stick breaking (GSB) random probability measure (Fuentes–García, et al. 2010) has been used for density estimation and has been shown to provide an efficient alternative to DP mixture models. Some recent papers extend this nonparametric prior to a dependent nonparametric prior. In the direction of covariate dependent processes, GSB processes have been seen to provide an adequate model to the traditional dependent DP model. For example, for Bayesian regression, Fuentes–Garcia et al. (2009) propose a covariate dependent process based on random probability measures drawn from a GSB process. Mena et al. (2011) used GSB random probability measures in order to construct a purely atomic continuous time measure–valued process, useful for the analysis of time series data. In this case, the covariate z≥ 0 denotes the time that each observation is (discretely) recorded and conditionally on each observation is drawn from a time–dependent nonparametric mixture model based on GSB processes. However, to the best of our knowledge, random probability measures drawn from a GSB process, for modeling related density functions when samples from each density function are available, has not been developed in the literature. In this paper we will construct pairwise dependent random probability measures based on GSB processes. That is, we are going to model a finite collection of m random distribution functions (_j)_1≤ j≤ m, where each _j is a GSB random probability measure, such that there is a unique common component for each pair (_j,_j') with j≠ j'. We are going to use these measures in the context of GSB mixture models, generating a collection of m GSB pairwise dependent random densities (f_j(x))_1≤ j≤ m. Hence we obtain a set of random densities (f_1,…,f_m), where marginally each f_j is a random density function f_j(x) = ∫_ΘK(x|θ) _j(dθ), thus generalizing the GSB priors to a multivariate setting for partially exchangeable observations. In the problem considered here, these random density functions (f_j)_1≤ j≤ m are thought to be related or similar, e.g. perturbations of each other, and so we aim to share information between groups to improve estimation of each density, especially for those densities f_j for which the corresponding sample size n_j is small. In this direction, the main references include the work of Müller et al. (2014), Bulla et al. (2009), Kolossiatis et al. (2013) and Griffin et al. (2013); more rigorous results can be found in Lijoi et al. (2014A, 2014B). All these models have been proposed for the modeling of an arbitrary but finite number of random distribution functions, via a common part and an index specific idiosyncratic part so that for 0<p_j<1 we have _j = p_j_0 + (1-p_j)_j^*, where _0 is the common component to all other distributions and {_j^*:j=1,…,m} are the idiosyncratic parts to each _j, and _0,_j^* DP(c,P_0). In Lijoi et al. (2014B) normalized random probability measures based on the σ–stable process are used for modeling dependent mixtures. Although similar (all models coincide only for the m = 2 case), these models are different from our model which is based on pairwise dependence of a sequence of random measures (Hatjispyros et al. 2011, 2016A). We are going to provide evidence through numerical experiments that dependent GSB mixture models are an efficient alternative to pairwise dependent DP (PDDP) priors. First, we will randomize the existing PDDP model of Hatjispyros et al. (2011, 2016A), by imposing gamma priors on the concentration masses (leading to the more efficient rPDDP model). Then, for the objective comparison of execution times, we will conduct a-priori synchronized density estimation comparison studies between the randomized PDDP and the pairwise dependent GSB process (PDGSBP) models using synthetic and real data examples. This paper is organized as follows. In Section 2 we will demonstrate the construction of pairwise dependent random densities, using a dependent model suggested by Hatjispyros et al. (2011). We also demonstrate how specific choices of latent random variables can recover the model of Hatjispyros et al. and the dependent GSB model introduced in this paper. These latent variables will form the basis of a Gibbs sampler for posterior inference, given in Section 3. In Section 4 we resort to simulation. We provide comparison studies between the randomized version of the PDDP model and our newly introduced dependent GSB model, involving five cases of synthetic data and a real data set. Finally, Section 5 concludes with a summary and future work. 2. Preliminaries. We consider an infinite real valued process {X_ji:1≤ j≤ m, i≥ 1} defined over a probability space (Ω, F, P), that is partially exchangeable as in (<ref>). Let P denote the set of probability measures over ; then de Finetti proved that there exists a probability distribution Π over P^m, which satisfies P{X_ji∈ A_ji:1≤ j≤ m,1≤ i≤ n_j} = ∫_ P^m P{X_ji∈ A_ji:1≤ j≤ m,1≤ i≤ n_j | _1,…,_m} Π(d_1,…,d_m) = ∫_ P^m∏_j=1^m P{X_ji∈ A_ji:1≤ i≤ n_j | _j} Π(d_1,…,d_m) = ∫_ P^m∏_j=1^m ∏_i=1^n_j_j(A_ji) Π(d_1,…,d_m) . The de Finetti measure Π represents a prior distribution over partially exchangeable observations. We start off by describing the PDDP model, with no auxiliary variables, using only the de Finetti measure Π, marginal measures _j, then, we proceed to the definition of a randomized version of it, and to the specific details for the case of the GSB random measures. A. In Hatjispyros et al. (2011), the following hierarchical model was introduced. For m subgroups of observations {(x_ji)_1≤ i≤ n_j:1≤ j≤ m}, x_ji|θ_ji K( · |θ_ji) θ_ji|_j _j( · ) _j = ∑_l=1^mp_jl_jl, ∑_l=1^mp_jl=1, _jl=_lj _jl DP(c, P_0), 1≤ j≤ l≤ m, for some kernel density K( · | · ), concentration parameter c>0 and parametric central measure P_0 for which (_jl(dθ))=P_0(dθ). So, we have assumed that the random densities f_j(x) are dependent mixtures of the dependent random measures _j via f_j(x|_j)=∫_Θ K(x|θ)_j(dθ), or equivalently, dependent mixtures of the m independent mixtures g_jl(x| _jl)=∫_Θ K(x| ) _jl(d), l=1,…,m. To introduce the rPDDP model, we randomize the PDDP model by sampling the _jl measures from the independent Dirichlet processes DP(c_jl,P_0) and then impose gamma priors on the concentration masses, i.e. _jl DP(c_jl, P_0), c_jl G(a_jl,b_jl), 1≤ j≤ l≤ m. B. To develop a pairwise dependent geometric stick breaking version, we let the random density functions f_j(x) generated via f_j(x):=f_j(x| _j) = ∑_l=1^m p_jl g_j l(x| _jl), _j = ∑_l=1^mp_jl_jl, 1≤ j≤ m. The g_jl(x):=g_jl(x| _jl)=∫_Θ K(x| ) _jl(d) random densities are now independent mixtures of GSB processes, satisfying g_jl=g_lj, under the slightly altered definition _jl=∑_k=1^∞ q_jlkδ_θ_jlk with q_jlk=λ_jl(1-λ_jl)^k-1, λ_jl∼ h( · |ξ_jl), θ_jlk G_0, where h is a parametric density supported over the interval (0,1) depending on some parameter ξ_jl∈Ξ, and G_0 is the associated parametric central measure. The independent GSB processes {_jl: 1≤ j, l≤ m} form a matrix of random distributions with _jl=_lj. In matrix notation ℚ = (p⊗) 1, where p=(p_jl) is the matrix of random selection weights, and p⊗ is the Hadamard product of the two matrices defined as (p⊗)_jl=p_jl_jl. By letting 1 to denote the m× 1 matrix of ones it is that the jth element of vector ℚ is given by equation (<ref>). C. Following a univariate construction of geometric slice sets (Fuentes–García et al. 2010), we define the stochastic variables 𝐍=(N_ji) for 1≤ i≤ n_j and 1≤ j≤ m, where N_ji is an almost surely finite random variable of mass f_N possibly depending on parameters, associated with the sequential slice set S_ji={1,…,N_ji}. Following Hatjispyros et al. (2011, 2016a) we introduce: * The GSB mixture selection variables δ=(_̣ji); for an observation x_ji that comes from f_j(x), _̣ji selects one of the mixtures {g_jl(x):l=1,…,m}. Then the observation x_ji came from mixture g_j δ_ji(x). * The GSB clustering variables d=(d_ji); for an observation x_ji that comes from f_j(x), given δ_ji, d_ji allocates the component of the GSB mixture g_j δ_ji(x) that x_ji came from. Then the observation x_ji came from component K(x|_j_̣jid_ji). In what follows, unless otherwise specified, the random densities f_j(x) are mixtures of independent GSB mixtures. Proposition 1. Suppose that the clustering variables (d_ji) conditionally on the slice variables (N_ji) are having the discrete uniform distribution over the sets ( S_ji) that is d_ji|N_ji∼ DU( S_ji), and P{N_ji=r|_̣ji=l}=f_N(r|λ_jl), then f_j(x_ji,N_ji=r)=r^-1∑_l=1^mp_jlf_N(r|λ_jl)∑_k=1^r K(x_ji|θ_jlk), and f_j(x_ji,N_ji=r,d_ji=k|δ_ji=l) = 1 rf_N(r|λ_jl) I(k≤ r) K(x_ji|θ_jlk). The proof is given in Appendix A. The following proposition gives a multivariate analogue of equation (2) in Fuentes–García, et al. (2010): Proposition 2. Given the random set S_ji, the random functions in (<ref>) become finite mixtures of a.s. finite equally weighted mixtures of the K( · | · ) probability kernels, that is f_j(x_ji|N_ji=r)=∑_l=1^m W(r|λ_jl)∑_k=1^rr^-1K(x_ji|θ_jlk), where the probability weights { W(r|λ_jl):1≤ l≤ m} are given by W(r|λ_jl)=p_jl f_N(r|λ_jl)∑_l'=1^m p_jl' f_N(r|λ_jl'). The proof is given in Appendix A. Note that, the one–dimensional model introduced in Fuentes–García et al. (2010), under our notation attains the representation f_j(x_ji|N_ji=r,_̣ji=l)=∑_k=1^rr^-1K(x_ji|θ_jlk). 2.1 The model. Marginalizing (<ref>) with respect to the variable (N_ji, d_ji), we obtain f_j(x_ji|_̣ji=l)=∑_k=1^∞(∑_r=k^∞ r^-1f_N(r|λ_jl))K(x_ji|θ_jlk). The quantity inside the parentheses on the right-hand side of the previous equation is f_j(d_ji|_̣ji=l). Following Fuentes–García, et al. (2010), we substitute f_N(r|ł_jl) with the negative binomial distribution NB(r|2,ł_jl), i.e. f_N(r|ł_jl) = r ł_jl^2(1-ł_jl)^r-1 I(r≥ 1), so equation (<ref>) becomes f_j(x_ji|_̣ji=l)=∑_k=1^∞ q_jlk K(x_ji|θ_jlk) with q_jlk=ł_jl(1-ł_jl)^k-1, and the f_j random densities take the form of a finite mixture of GSB mixtures f_j(x_ji)=∑_l=1^m p_jl∑_k=1^∞ q_jlk K(x_ji|θ_jlk). We denote the set of observations along the m groups as x=(x_ji) and with x_j the set of observations in the jth group. The three sets of latent variables in the jth group will be denoted as N_j for the slice variables, d_j for the clustering variables, and finally _j for the set of GSB mixture allocation variables. From now on, we are going to leave the auxiliary variables unspecified; especially for _̣ji we use the notation _̣ji=(_̣ji^1,…,_̣ji^m)∈{e⃗_1,…,e⃗_m} with P{_̣ji=e⃗_l}=p_jl, where e⃗_l denotes the usual basis vector having its only nonzero component equal to 1 at position l. Hence, for a sample of size n_1 from f_1, a sample of size n_2 from f_2, etc., a sample of size n_m from f_m we can write the full likelihood as a multiple product: f( x, N, d | ) = ∏_j=1^m f( x_j, N_j, d_j | _j) = ∏_j=1^m∏_i=1^n_j I(d_ji≤ N_ji)∏_l=1^m {ł_jl^2(1-ł_jl)^N_ji-1K(x_ji| θ_j l d_ji)}^δ_ji^l. In a hierarchical fashion, using the auxiliary variables, we have for j=1,…,m and i = 1,…, n_j, x_ji, N_ji | d_ji, _̣ji, (_jr_̣ji)_1≤ r≤ m,λ_j_̣ji ∏_r=1^m{ł_jr^2(1-ł_jr)^N_ji-1K(x_ji|θ_jr d_ji)}^δ_ji^r I(N_ji≥ d_ji) d_ji | N_ji DU( S_ji), P{δ_ji=e⃗_l} = p_jl q_jik=λ_ji(1-λ_ji)^k-1, _jik G_0, k∈. 2.2 The PDGSBP covariance and correlation. In this sub–section we find the covariance and the correlation between f_j(x) and f_i(x). First we provide the following lemma. Lemma 1. Let g_(x)=∫_Θ K(x|)(d) be a random density, with =ł∑_j=1^∞ (1-ł)^j-1_̣_j and _j G_0, then [g_(x)^2] = (1 2-ł){ł∫_Θ K(x|θ)^2G_0(dθ) + 2(1-ł)(∫_Θ K(x|θ)G_0(dθ))^2}. The proof is given in Appendix A. Proposition 3. It is that Cov(f_j(x),f_i(x)) = p_ji p_ij Var(∫_Θ K(x|)_ji(d)), with Var(∫_Θ K(x|)_ji(d))=ł_ji 2-ł_ji Var(K(x|)). The proof is given in Appendix A. Suppose now that (f_j^ D(x))_1≤ j≤ m and (f_j^ G(x))_1≤ j≤ m are two collections of m DP and m GSB pairwise dependent random densities respectively, i.e. f_j^ D(x)=∑_l=1^m p_jlg_jl^ D(x) with g_jl^ D(x)=g_jl(x|_jl), and f_j^ G(x)=∑_l=1^m p_jlg_jl^ G(x) with g_jl^ G(x)=g_jl(x|_jl). Then we have the following proposition: Proposition 4. For given parameters (ł_ji), (c_ji), and matrix of selection probabilities (p_ji) it is that * The PDGSBP and rPDDP correlations are given by Corr(f_j^ G(x),f_i^ G(x)) = λ_jip_jip_ij 2-λ_ji(∑_l=1^m∑_r=1^m p_jl^2 p_ir^2λ_jlλ_ir (2-λ_jl)(2-λ_ir))^-1/2, and Corr(f_j^ D(x),f_i^ D(x)) = p_jip_ij 1+c_ji(∑_l=1^m∑_r=1^m p_jl^2 p_ir^2 (1+c_jl)(1+c_ir))^-1/2. * When λ_ji=λ and c_ji=c for all 1≤ j≤ i≤ m, the expressions for the rPDDP and PDGSBP correlations simplify to Corr(f_j^ G(x),f_i^ G(x))= Corr(f_j^ D(x),f_i^ D(x)) = p_jip_ij( ∑_l=1^m ∑_r=1^m p_jl^2 p_ir^2 )^-1/2. The proof is given in Appendix A. It is clear that, irrespective of the model, the random densities f_j(x) and f_i(x) are positively correlated whenever p_ji=p_ij=1. Similarly, the random densities f_j(x) and f_i(x) are independent (have no common part) whenever p_ji=p_ij=0. Another, less obvious feature, upon synchronization, is the ability of controlling the correlation among the models. For example, suppose that for m=2, the random densities f_1(x) and f_2(x) are dependent, and that ł_ji=(1+c_ji)^-1; then consider the expression D_12:=ł_12^2 p_12^2 p_21^2 { Corr(f_1^ G(x),f_2^ G(x))^-2 - Corr(f_1^ D(x),f_2^ D(x))^-2}. Since correlations are positive, D_12≥ 0 whenever Corr(f_1^ G(x),f_2^ G(x))≤ Corr(f_1^ D(x),f_2^ D(x)), and that D_12<0 whenever Corr(f_1^ G(x),f_2^ G(x))> Corr(f_1^ D(x),f_2^ D(x)). Then, it is not difficult to see that D_12=(p_12^2ł_12+r_1p_11^2ł_11) (p_21^2ł_12+r_2 p_22^2ł_22) -(p_12^2ł_12+p_11^2ł_11) (p_21^2ł_12+p_22^2ł_22) with r_k=(2-ł_12)/(2-ł_kk), k=1,2. We have the following cases: * ł_12>max{ł_11,ł_22} ⇔ r_1<1, r_2<1 ⇔ Corr(f_1^ G(x),f_2^ G(x))> Corr(f_1^ D(x),f_2^ D(x)). * ł_12<min{ł_11,ł_22} ⇔ r_1>1, r_2>1 ⇔ Corr(f_1^ G(x),f_2^ G(x))< Corr(f_1^ D(x),f_2^ D(x)). * ł_12=ł_11=ł_22 ⇔ r_1=r_2=1 ⇔ Corr(f_1^ G(x),f_2^ G(x))= Corr(f_1^ D(x),f_2^ D(x)). 3. The PDGSBP Gibbs sampler. In this section we will describe the PDGSBP Gibbs sampler for estimating the model. The details for the sampling algorithm of the PDDP model can be found in Hatjispyros et al. (2011, 2016A). At each iteration we will sample the variables, θ_jlk, 1≤ j ≤ l ≤ m, 1≤ k ≤ N^*, d_ji,N_ji,δ_ji, 1≤ j ≤ m, 1≤ i ≤ n_j, p_jl, 1≤ j ≤ m, 1≤ l ≤ m, with N^*=max_j,iN_ji being almost surely finite. 1. For the locations of the random measures for k=1,…,d^* where d^*=max_j,id_ji, it is that f(θ_jlk|⋯) ∝ f(θ_jlk) ∏_i=1^n_jK(x_ji|θ_jlk)^ I(_̣ji=e⃗_l, d_ji=k)∏_i=1^n_lK(x_li|θ_jlk)^ I(_̣li=e⃗_j, d_li=k) l>j , ∏_i=1^n_jK(x_ji|θ_jjk)^ I(_̣ji=e⃗_j, d_ji=k) l=j . If N^*>d^* we sample additional locations _jl,d^*+1,…,_jl,N^* independently from the prior. 2. Here we sample the allocation variables d_ji and the mixture component indicator variables δ_ji as a block. For j=1,…,m and i=1,…,n_j, we have P(d_ji=k,_̣ji=e⃗_l |N_ji=r,⋯) ∝ p_jl K(x_ji|θ_jlk) I(l≤ m) I(k≤ r). 3. The slice variables N_ji have full conditional distributions given by P(N_ji = r | _̣ji=e⃗_l,d_ji=l,⋯)∝(1-λ_jl)^r I(r≥ l), which are truncated geometric distributions over the set {l, l+1,…}. 4. The full conditional for j=1,…,m for the selection probabilities p_j=(p_j1,…,p_jm), under a Dirichlet prior f( p_j | a_j)∝∏_l=1^m p_jl^a_jl-1, with hyperparameter a_j=(a_j1,…,a_jm), is Dirichlet f( p_j |⋯) ∝ ∏_l=1^m p_jl^a_jl+∑_i=1^n_l I(δ_ji = e⃗_l) - 1. 5. Here we update the geometric probabilities (ł_jl) of the GSB measures. For 1≤ j≤ l≤ m, it is that f(ł_jl|⋯) ∝ f(ł_jl) ∏_i=1^n_j{ł_jl^2(1-ł_jl)^N_ji-1}^ I(_̣ji=e⃗_l)∏_i=1^n_l{ł_jl^2(1-ł_jl)^N_li-1}^ I(_̣li=e⃗_j) l>j ∏_i=1^n_j{ł_jj^2(1-ł_jj)^N_ji-1}^ I(_̣ji=e⃗_j) l=j . To complete the model, we assign priors to the geometric probabilities. For a fair comparison of the execution time between the two models, we apply ł_jl=(1+c_jl)^-1 transformed priors. So, by placing gamma priors c_jl∼ G(a_jl,b_jl) over the concentration masses c_jl of the PDDP model, we have f(ł_jl)= TG(ł_jl | a_jl,b_jl)∝ł_jl^-(a_jl+1)e^-b_jl/ł_jl(1-ł_jl)^a_jl-1 I(0<ł_jl<1). In the Appendix, we give the full conditionals for ł_jl's, their corresponding embedded Gibbs sampling schemes, and the sampling algorithm for the concentration masses. 3.1 The complexity of the rPDDP and PDGSBP samplers. The main difference between the two samplers in terms of execution time, comes from the blocked sampling of the clustering and the mixture indicator variables d_ji and _̣ji. The rPDDP model: The state space of the variable (d_ji,δ_ji) conditionally on the slice variable u_ji is (d_ji,δ_ji)(Ω)= ∪_l=1^m(A_w_jl(u_ji) ×{e⃗_l}), where A_w_jl(u_ji)={r∈ℕ:u_ji<w_jlr} is the a.s. finite slice set corresponding to the observation x_ji (Walker, 2007). At each iteration of the Gibbs sampler, we have m(m+1)/2 vectors of stick-breaking weights w⃗_jl, each of length N_jl^*; where N_jl^*∼ 1 + Poisson(-c_jllog u_jl^*) with c_jl being the concentration parameter of the Dirichlet process ℙ_jl and u_jl^* being the minimum of the slice variables in densities f_j and f_l. Algorithm 1 gives the blocked sampling procedure of the clustering and mixture indicator variables. An illustration of the effect of the slice variable u_ji is given in Figure 1(a). Since the weights forming the stick-breaking representation are not in an ordered form, the construction of the slice sets in step 5 of Algorithm <ref> requires a complete search in the array where the weights are stored. This operation is done in O(N_jl^*) time. For the sampling of the d_ji and _̣ji variables in step 6, the choice of their value is an element from the union ∪_l=1^m(A_w_jl(u_ji)×{e⃗_l}). This means that the rPDDP algorithm for each j, must create m slice sets which require N_jl^* comparisons each. The worst case scenario is that the sampled (d_ji,_̣ji) is the last element of ∪_l=1^m(A_w_jl(u_ji)×{e⃗_l}). Thus, the DP based procedure of sampling (d_ji,_̣ji) is of order O( m^2 n_j N_jl^* ∑_l=1^m|A_w_jl(u_ji)| ) = O(N_jl^*∑_l=1^m|A_w_jl(u_ji)|). The PDGSBP model: The state space of the variable (d_ji,δ_ji) conditionally on the slice variable N_ji is (d_ji, _̣ji)(Ω)=∪_l=1^m( S_ji×{e⃗_l}). In the GSB case, the slice variable has a different rôlee. It indicates at which random point the search for the appropriate d_ji will stop. In Figure 1(b) we illustrate this argument. In Algorithm <ref> the worst case scenario is that the sampled (d_ji,_̣ji) will be the last element of ∪_l=1^m( S_ji×{e⃗_l}). Thus, the GSB based procedure of sampling (d_ji,_̣ji) is of order 𝒪(m^2n_jN_jl) = 𝒪(N_jl). 4. Illustrations. In this section we illustrate the efficiency of the PDGSBP model. For the choice of a normal kernel (unless otherwise specified) K(x|θ) = 𝒩(x|θ) where θ=(μ,τ^-1) and τ=σ^-2 is the precision. The prior over the means and precisions of the PDGSBP (G_0) and the rPDDP model (P_0) is the independent normal-gamma measure, given by P_0(dμ,dτ)=G_0(dμ,dτ)= N(μ | μ_0,τ_0^-1) G(τ | ϵ_1,ϵ_2) dμ dτ. Attempting a noninformative prior specification (unless otherwise specified), we took μ_0=0 and τ_0=ϵ_1=ϵ_2=10^-3. For the concentration masses of the rPDDP model, a-priori, we set c_jl∼ G(a_jl,b_jl). For an objective evaluation of the execution time, of the two algorithms under different scenarios, we choose a synchronized prior specification, namely, for the geometric probabilities, we set ł_jl∼ TG(a_jl,b_jl) – the transformed gamma density given in equation (<ref>). In the appendix B, we show that such prior specifications are valid for a_jl>1. In all our numerical examples, we took a_jl=b_jl=1.1. For our numerical experiments (unless otherwise specified), the hyperparameters (_jl) of the Dirichlet priors over the matrix of the selection probabilities p=(p_jl) has been set to _jl=1. In all cases, we measure the similarity between probability distributions with the Hellinger distance. So for example, H_ G(f,f̂) and H_ D(f,f̂), will denote the Hellinger distance between the true density f and the predictive density f̂ of the PDGSBP and rPDDP algorithms, respectively. The Gibbs samplers run for 11× 10^4 iterations leaving the first 10^4 samples as a burn-in period. 4.1 Time execution efficiency of the PDGSBP model. Nested normal mixtures with a unimodal common and idiosyncratic part: Here, we choose to include all pairwise and idiosyncratic dependences in the form of unimodal equally weighted normal mixture components. The mixture components are well separated with unit variance. We define each data model M_m={f_j^(m):1≤ j≤ m} of dimension m∈{2,3,4}, based on a 4× 10 matrix M=(M_jk), with entries in the set {0,1}, having at most two ones in each column and exactly four ones in each row. When there is exactly one entry of one, the column defines an idiosyncratic part. The appearance of exactly two ones in a column defines a common component. We let the matrix M given by M= [ 1 1 1 1 0 0 0 0 0 0; 0 0 1 0 1 0 0 1 1 0; 0 1 0 0 0 1 0 1 0 1; 1 0 0 0 0 0 1 0 1 1; ], and for m∈{2,3,4}, we define M_m: f_j^(m)(x)∝∑_k=5-m^2(m+1)M_jk N(x|10(k-6),1), 1≤ j≤ m, We are taking independently samples of sizes n_j^(2)=60 from the f_j^(2)'s, n_j^(3)=120 from the f_j^(3)'s, and, n_j^(4)=200 from the f_j^(4)'s. In all cases, the PDGSBP and the rPDDP density estimations are of the same quality. In Figures 2(a)–(d) we give the histograms of the data sets for the specific case m=4, which are overladed with the kernel density estimations (KDE's) based on the predictive samples of the f_j^(4)'s coming from the PDGSBP (solid line) and the rPDDP (dashed line) models. The differences between the two models are nearly indistinguishable. The Hellinger distances between the true and the estimated densities for the case m=4 are given in table 1. In Table 2 we summarize the mean execution times (MET's) per 10^3 iterations in seconds. The PDGSBP sampler is about three times faster than the rPDDP sampler. The corresponding MET ratios for m=2,3 and 4 are 2.96, 3.04 and 3.37 respectively. We can see that the PDGSBP Gibbs sampler gives slightly faster execution times with increasing m. This will become more clear in our next simulated data example, where the average sample size per mode is being kept constant. Sparse m–scalable data set models: In this example, we attempt to create m-scalable normal mixture data sets of the lowest possible sample size. To this respect, we sample independently m groups of data sets from the densities f_j^(m)(x) ∝ N(x|(j-1)ξ,1) I(1≤ j<m)+ ∑_k=1^m-1 N(x|(k-1) ξ,1) I(j=m), with sample sizes n_j^(m)=n{ I(1≤ j<m)+(m-1) I(j=m)}. We have chosen ξ=10 and an average sample size per mode of n=20, for m∈{2,…,10}. In Figure 3 we depict the average execution times as functions of the dimension m. We can see how fast the two MET-curves diverge with increasing m. In Figure 4(a)–(j), for the specific case m=10, we give the histograms of the data sets, overladed with the KDE's based on the predictive samples of the f_j^(10)'s coming from the PDGSBP (solid line) and the rPDDP (dashed line) models. We can see that the PDGSBP and the rPDDP density estimations are of the same quality. The Hellinger distances between the true and the estimated densities for the specific case m=10 are given in Table 3. The large values of the Hellinger distances H_G(f_10^(10),f̂_10^(10))≈ H_D(f_10^(10),f̂_10^(10))≈ 0.22, are caused by the enlargement of the variances of the underrepresented modes due to the small sample size. 4.2 Normal and gamma mixture models that are not well separated. The normal mixture example: We will first consider a normal model for m=2, first appeared in Lijoi et. al (2014B). The data models for f_1 and f_2 are 7-mixtures. Their common part is a 4-mixture that is weighted differently between the two mixtures. More specifically, we sample two data sets of sample size n_1=n_2=200, independently from (f_1,f_2)=(1 2 g_11 + 1 2 g_12, 4 7 g_21 + 3 7 g_22), with g_11 = 2/7 N(-8,0.25^2) + 3/7 N(1,0.5^2) + 2/7 N(10,1) g_12 = 1/7 N(-10,0.5^2) + 3/7 N(-3,0.75^2) + 1/7 N(3,0.25^2) + 2/7 N(7, 0.25^2) g_21 = 2/8 N(-10,0.5^2) + 3/8 N(-3,0.75^2) + 2/8 N(3,0.25^2) + 1/8 N(7, 0.25^2) g_22 = 1/3 N(-6,0.5^2) + 1/3 N(-1,0.25^2) + 1/3 N(5,0.5^2). For this case, a-priori we took (μ_0,τ_0,ϵ_1,ϵ_2)=(0,10^-3,1,10^-2). In Figure 5(a)–(b) we give the histograms of the data sets, with the predictive densities of the PDGSBP and rPDDP models superimposed in black solid and black dashed curves, respectively. We can see that the PDGSBP and the rPDDP density estimations are of the same quality. In Table 4, we give the Hellinger distance between the true and the estimated densities The gamma mixture example: In this example we took m=2. The data models for f_1 and f_2 are gamma 4-mixtures. The common part is a gamma 2-mixture, weighted identically among the two mixtures. More specifically, we sample two data sets of sample size n_1=n_2=160, independently from (f_1,f_2)=(2 5 g_11 + 3 5 g_12, 7 10 g_12 + 3 10 g_22), with g_11 = 2/ 3 G( 2,1.1) + 1/ 3 G( 80,2) g_12 = 8/14 G( 10,0.9) + 6/14 G(200,8.1) g_22 = 2/ 3 G(105,3) + 1/ 3 G(500,10), Because we want to estimate the density of non negative observations, we find it more appropriate to take the kernel to be a log-normal distribution (Hatjispyros et al. 2016B). That is K(x|θ) = ℒ𝒩(x|θ) with θ=(μ,σ^2), is the log-normal density with mean exp(μ+σ^2/2). For this case, a-priori we set (μ_0,τ_0,ϵ_1,ϵ_2)=(S̅,0.5,2,0.01), S̅=1 n_1+n_2(∑_j=1^n_1log x_1j+∑_j=1^n_2log x_2j). In Figure 6(a)-(b), we display the KDE's based on the predictive samples of the two models. We can see that the PDGSBP and the rPDDP density estimations are of the same quality. In Table 5, we give the Hellinger distances. Because the common part is equally weighted among f_1 and f_2, it makes sense to display the estimations of the selection probability matrices under the two models _ G(p | (x_ji)) = [ 0.42 0.58; 0.64 0.36 ], _ D(p | (x_ji)) = [ 0.42 0.58; 0.69 0.31 ], p_ true=[ 0.4 0.6; 0.7 0.3 ]. 4.3 Borrowing of strength of the PDGSBP model. In this example we consider three populations {D_j^(s):j=1,2,3}, under three different scenarios s∈{1,2,3}. The sample sizes are always the same, namely, n_1=200, n_2=50 and n_3=200 – the second population is sampled only once. The three data sets D_1^(s), D_2^(s) and D_3^(s), are sampled independently from the normal mixtures (f_1^(s),f_2^(s),f_3^(s))= ((1-q^(s))f+q^(s)g_1, f, (1-q^(s))f+q^(s)g_2), where f =3/10 N(-10,1) + 2/10 N(-6,1) +2/10 N(6,1) + 3/10 N(10,1) g_1 = 1/2 N(-4,1) + 1/2 N(4,1) g_2 = 1/2 N(-12,1) + 1/2 N(12,1). More specifically, the three scenarios are: * For s=1, we set, q^(1)=0. This is the case where the three populations are coming from the same 4–mixture f. We depict the density estimations under the first scenario in Figures 7(a)–(c). This is the case where the small data set, benefits the most in terms of borrowing of strength. * For s=2, we set, q^(2)=1/2. The 2-mixtures g_1 and g_2 are the the idiosyncratic parts of the 6-mixtures f_1^(2) and f_3^(2), respectively. The density estimations under the second scenario are given in Figures 7(d)–(f). In this case, the strength of borrowing between the small data set and the two large data sets weakens. * For s=3 we set q^(3)=1. In this case the three populations have no common parts. The density estimations are given in Figures 7(g)–(i). This is the worst case scenario, where there is no borrowing of strength between the small and the two large data sets. The Hellinger distances between the true and the estimated densities, for the three scenarios, are given in table 6. In the second column of the Table we can see how the Hellinger distance of the estimation f̂_2^(s) and the true density f_2^(s) increases as the borrowing of strength weakens, it is that H_ G(f_2^(1),f̂_2^(1))< H_ G(f_2^(2),f̂_2^(2)) < H_ G(f_2^(3),f̂_2^(3)). 4.4 Real data example. The data set is to be found at and involves data from 310 individuals. We take the observation as SGOT (serum glutamic-oxaloacetic transaminase) level, just prior to liver transplant or death or the last observation recorded, under three conditions on the individual * The individual is dead without transplantation. * The individual had a transplant. * The individual is alive without transplantation. We normalize the means of all three data sets to zero. Since it is reasonable to assume the densities for the observations are similar for the three categories (especially for the last two), we adopt the models proposed in this paper with m = 3. The number of transplanted individuals is small (sample size of 28) so it is reasonable to borrow strength for this density from the other two. In this example, we set the hyperparameters of the Dirichlet priors for the selection probabilities to _jl= 10, j=l=1 j=l=3 1, * In Figure 8(a)–(c) we provide histograms of the real data sets and superimpose the KDE's based on the predictive samples of the PDDP and PDGSBP samplers. The two models give nearly identical density estimations. * The estimated a-posteriori selection probabilities are given below _ G(p | (x_ji)) = [ 0.61 0.23 0.16; 0.34 0.10 0.56; 0.08 0.12 0.80 ], _ D(p | (x_ji)) = [ 0.67 0.16 0.17; 0.29 0.15 0.56; 0.10 0.12 0.78 ]. By comparing the second rows of the selection matrices, we conclude that the strength of borrowing is slightly larger in the case of PDGSBP model . 5. Discussion. In this paper we have generalized the GSB process to a multidimensional dependent stochastic process which can be used as a Bayesian nonparametric prior for density estimation in the case of partially exchangeable data sets. The resulting Gibbs sampler is as accurate as its DP based counterpart, yet faster and far less complicated. The main reason for this is that the GSB sampled value of the allocation variable d_ji will be an element of the sequential slice set S_ji={1,…,N_ji}. Thus, there is no need to search the arrays of the weights; we know the state space of the clustering variables in advance. On the other hand, the sampling of d_ji in the DP based algorithm will always have one more step; the creation of the slice sets. For an objective comparison of the execution times of the two models, we have run the two samplers in an a-priori synchronized mode. This, involves the placing of G(a_jl,b_jl) priors over the DP c_jl concentration masses, leading to a more efficient version of the PDDP model introduced in Hatjispyros et al. (2011, 2016A). We have show that when the PDGSBP and PDDP models are synchronized, i.e. their parameters satisfy ł_ji=(1+c_ji)^-1, the correlation between the models can be controlled by imposing further restrictions among the ł_ji parameters. Finally, an interesting research path would be the generalization of the pairwise dependent _j measures to include all possible interactions, in the sense that _j( · )=p_j _j( · )+∑_l=2^m∑_η ∈ C_j,l,mp_j,η _η_(j)( · ) with p_j+∑_l=2^m∑_η ∈ C_j,l,mp_j,η=1, where the _j and the _η_(j)'s are independent GSB processes, C_j,l,m={(k_1,…,k_l-1):1≤ k_1<⋯<k_l-1≤ m, k_r≠ j, 1≤ r≤ m-1} and η_(j) is the ordered vector of the elements of the vector η and {j}. Now the f_j densities will be a mixture of 2^m-1 GSB mixtures, and the total number of the independent GSB processes needed to model (f_1,…,f_m) will be 2^m-1. Appendix A Proof of Proposition 1. Starting from the N_ji-augmented random densities we have f_j(x_ji,N_ji=r) = ∑_l=1^mf_j(x_ji,N_ji=r,δ_ji=l) = ∑_l=1^mp_jl f_j(x_ji,N_ji=r|δ_ji = l) = ∑_l=1^mp_jl∑_k=1^∞f_j(x_ji,N_ji=r,d_ji=k|δ_ji=l) = ∑_l=1^mp_jlf_j(N_ji=r|δ_ji=l)∑_k=1^∞f_j(d_ji=k|N_ji=r)f_j(x_ji|d_ji=k,δ_ji=l). Because f_j(N_ji=r|_̣ji=l)=f_N(r|λ_jl) and f_j(x_ji|d_ji=k,δ_ji=l)=K(x_ji|_jlk), the last equation gives f_j(x_ji,N_ji=r) = ∑_l=1^m p_jl f_N(r|λ_jl)∑_k=1^∞1 r I(k≤ r)K(x_ji|_jlk)                     = 1/r∑_l=1^mp_jlf_N(r|λ_jl)∑_k=1^r K(x_ji|θ_jlk). Augmenting further with the variables d_ji and _̣ji yields f_j(x_ji,N_ji=r,d_ji=k,_̣ji=l)=1 r p_jl f_N(r|λ_jl) I(k≤ r) K(x_ji|_jlk). Because P(_̣ji=l)=p_jl, the last equation leads to equation (<ref>) and the proposition follows. □ Proof of Proposition 2. Marginalizing the joint of x_ji and N_ji with respect to x_ji we obtain f_j(N_ji=r)=∑_l=1^m p_jlf_N(r|λ_jl). Then dividing equation (<ref>) with the probability that N_ji equals r we obtain equation (<ref>). □ Proof of Lemma 1. Because g_(x)=ł∑_j=1^∞(1-ł)^j-1K(x|_j), we have {g_(x)^2}=ł^2 {(∑_j=1^∞(1-ł)^j-1K(x|θ_j))^2} =ł^2{∑_j=1^∞(1-ł)^2j-2 [K(x|_j)^2]+2∑_k=2^∞∑_j=1^k-1(1-ł)^j+k-2 [K(x|_j)K(x|_k)]} =ł^2{∑_j=1^∞(1-ł)^2j-2[K(x|)^2]+2 ∑_k=2^∞∑_j=1^k-1(1-ł)^j+k-2[K(x|)]^2} =ł^2{1ł(2-ł)[K(x|)^2]+2 1-łł^2(2-ł)[K(x|)]^2}, which gives the desired result.□ Proof of Proposition 3. The random densities f_i(x)=∑_l=1^m p_il g_il(x) and f_j(x)=∑_l=1^m p_jl g_jl(x) depend to each other through the random measure _ji, therefore [f_i(x)f_j(x)]= [ (f_i(x)f_j(x)|_ji) ]={ [f_i(x)|_ji] [f_j(x)|_ji] }, and [f_j(x)|_ji]=∑_l≠ ip_jl [g_jl(x)]+p_jig_ji(x) =(1-p_ji) [K(x|)]+p_jig_ji(x) [f_i(x)|_ji]=∑_l≠ jp_il [g_il(x)]+p_ijg_ji(x) =(1-p_ij) [K(x|)]+p_ijg_ji(x) . Substituting back to equation (<ref>) one obtains [f_i(x)f_j(x)]=(1-p_ijp_ji) [K(x|)]^2+p_ijp_ji [g_ji(x)^2]. Using lemma 1, the last equation becomes [f_i(x)f_j(x)]=ł_jip_jip_ij 2-ł_ji{[K(x|)^2]-[K(x|)]^2}+[K(x|)]^2, or that Cov(f_j(x),f_i(x)) = ł_jip_ji p_ij 2-ł_ji Var(K(x|)). The desired result, comes from the fact that Var(∫_Θ K(x|)_ji(d)) = {ł_ji 2-ł_ji[K(x|)^2]+2(1-ł_ji) 2-ł_ji[K(x|)]^2}-[K(x|)]^2 = ł_ji 2-ł_ji([K(x|)^2]-[K(x|)]^2). □ Proof of Proposition 4. (1.) From equation (<ref>) and proposition 3, we have that Var(f_j^ G(x)) = Var(∑_l=1^m p_jlg_jl^ G(x)) =∑_l=1^m p_ji^2λ_ji 2-λ_ji Var(K(x|)). Normalizing the covariance in equation (<ref>) with the associated standard deviations, yields Corr(f_j^ G(x),f_i^ G(x)) = λ_jip_jip_ij 2-λ_ji(∑_l=1^m∑_r=1^m p_jl^2 p_ir^2λ_jlλ_ir (2-λ_jl)(2-λ_ir))^-1/2. Similarly, from proposition 1 in Hatjispyros et al. (2011), it is that Var(f_j^ D(x)) =∑_l=1^m p_ji^2 1+c_ji Var(K(x|)), and Corr(f_j^ D(x),f_i^ D(x)) = p_jip_ij 1+c_ji(∑_l=1^m∑_r=1^m p_jl^2 p_ir^2λ_jlλ_ir (1+c_jl)(1+c_ir))^-1/2. (2.) When λ_ji=λ and c_ji=c for all 1≤ j≤ i≤ m, from equations (<ref>) and (<ref>), it is clear that Corr(f_j^ G(x),f_i^ G(x))= Corr(f_j^ D(x),f_i^ D(x)) = p_jip_ij( ∑_l=1^m ∑_r=1^m p_jl^2 p_ir^2 )^-1/2. Appendix B 1. Sampling of the concentrations masses for the rPDDP model. In this case, the random densities (f_j) are represented as finite mixtures of the DP mixtures g_jl(x|_jl), where _jl∼ DP(c_jl,P_0). We randomize the concentrations by letting c_jl∼ G(a_jl,b_jl). Following West (1992) we have the following two specific cases: A. For j=l, the posterior c_jj's will be affected only by the size of the data set x_j and the number of unique clusters for which _̣ji= e_j. Letting ρ_jj=#{d_jj:_̣ji= e_j,1≤ i≤ n_j}, we have ∼̱ Be(c_jj+1, n_j) c_jj | ,̱ρ_jj ∼ π_ G(a_jj+ρ_jj, b_jj-log)̱ + (1-π_)̱ G(a_jj+ρ_jj-1, b_jj-log)̱ with the weights π_$̱ satisfyingπ_/1-π_=a_jj+ρ_jj-1/n_j(b_jj-log)̱. B. Forj≠l, the posteriorc_jl's will be affected by the size of the data setsx_jandx_land the cumulative number of unique clustersd_jifor which_̣ji=e_land the unique clustersd_lifor which_̣li=e_j. Lettingρ_jl=#{d_ji:_̣ji= e_l,1≤ i≤ n_j}+ #{d_li:_̣li= e_j,1≤ i≤ n_l},it is that ∼̱ Be(c_jl+1, n_j+n_l) c_jl | ,̱ρ_jl ∼ π_𝒢(a_jl+ρ_jl, b_jl-log)̱ + (1-π_)̱ 𝒢(a_jl+ρ_jl-1, b_jl-log)̱, with the weightsπ_$̱ satisfying π_/1-π_=a_jl+ρ_jl-1/(n_j+n_l)(b_jl-log)̱. Bear in mind that ρ_jl=0 is always a possibility, so that we impose a_jl>1. 2. Sampling of the geometric probabilities for the PDGSBP model. In this section we provide the full conditionals for the geometric probabilities λ_jl under beta conjugate and transformed gamma nonconjugate priors. We let S_jl =∑_i=1^n_j I(_̣ji= e_l) and S_jl'=∑_i=1^n_j I(_̣ji= e_l)(N_ji-1). A. For the choice of prior λ_jl∼ Be(a_jl,b_jl), for l=j it is that f(ł_jj|⋯)= Be(ł_jl| a_jj + 2 S_jj, b_jj + S_jj'), also, for l≠ j we have f(ł_jl|⋯) = Be(ł_jl| a_jl + 2(S_jl+S_lj), b_jl + S_jl' + S_lj'). B. For the choice of prior ł_jl∼ TG(a_jl,b_jl), for l=j it is that f(ł_jj|…) ∝ł_jj^2S_jj - a_jj-1(1-ł_jj)^S_jj'+ a_jj-1e^-b_jj/ł_jj I(0<ł_jj<1). To sample from this density, we include the positive auxiliary random variables ν_1 and ν_2 such that f(ł_jj,ν_1,ν_2|⋯) ∝ł_jj^2S_jj - a_jj-1 I(ν_1<(1-ł_jj)^S_jj'+ a_jj-1) I(ν_2<e^-b_jj/ł_jj) I(0<ł_jj<1). The full conditionals for ν_1,ν_2 are uniforms f(ν_1|⋯) = U(ν_1|0, (1-ł_jj)^S_jj'+ a_jj-1) and f(ν_2|⋯) = U(ν_2|0, e^-b_jj/ł_jj), and the new full conditional for λ_jj becomes f(ł_jj|ν_1,ν_2,…) ∝ł_jj^2S_jj - a_jj-1 I(-b_jjlogν_2<ł_jj<1-ν_1^1/L_jj) L_jj≥ 0 I(max{-b_jjlogν_2,1-ν_1^1/L_jj}<ł_jj<1) L_jj<0, where we have set L_jj=S_jj'+ a_jj-1. We can sample from this density using the inverse cumulative distribution function technique. Also, for l≠ j we apply the same embedded Gibbs sampling technique to the full conditional density f(ł_jl|⋯)∝ł_jl^2(S_jl+S_lj)-a_jl-1(1-ł_jl)^S_jl'+S_lj'+ a_jl-1e^-b_jl/ł_jl I(0<ł_jl<1). References. * Bulla, P., Muliere, P. and Walker, S.G. (2009). A Bayesian nonparametric estimator of a multivariate survival function. Journal of Statistical Planning and Inference 139, 3639–3648. * De Iorio, M., Müller, P., Rosner, G.L. and MacEachern, S.N. (2004). An ANOVA model for dependent random measures. Journal of the American Statistical Association 99, 205–215. * Dunson, D.B. and Park, J.H. (2008). Kernel stick–breaking processes. Biometrika 95, 307–323. * Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics 1, 209–230. * Fuentes–Garcia, R., Mena, R.H., Walker, S.G. (2009). A nonparametric dependent process for Bayesian regression Statistics and Probability Letters 79, 1112–1119. * Fuentes–Garcia, R., Mena, R.H., Walker, S.G. (2010). A new Bayesian nonparametric mixture model. Comm.Statist.Simul.Comput 39, 669–682. * Griffin, J.E. and Steel, M.F.J. (2006). Order–based dependent Dirichlet processes. Journal of the American Statistical Association 101, 179–194. * Griffin, J.E., Kolossiatis, M. and Steel, M.F.J. (2013). Comparing distributions by using dependent normalized ranom–measure mixtures. Journal of the Royal Statistical Society, Series B 75, 499–529. * Hatjispyros, S.J., Nicoleris, T. and Walker, S.G. (2011). Dependent mixtures of Dirichlet processes. Computational Statistics and Data Analysis 55, 2011–2025. * Hatjispyros, S.J., Nicoleris, T. and Walker, S.G. (2016a). Dependent random density functions with common atoms and pairwise dependence. Computational Statistics and Data Analysis 101, 236–249. * Hatjispyros, S.J., Nicoleris, T. and Walker, S.G. (2016b). Bayesian nonparametric density estimation under length bias. Communications in Statistics DOI: 10.1080/03610918.2016.1263735 * Lijoi, A., Nipoti, B. and Prüenster, I. (2014a). Bayesian inference with dependent normalized completely random measures. Bernoulli, 20, 1260–1291. * Lijoi, A., Nipoti, B. and Prüenster, I. (2014b). Dependent mixture models: clustering and borrowing information. Computational Statistics and Data Analysis 71, 17–433. * Kolossiatis, M., Griffin, J.E. and Steel, M.F.J. (2013). On Bayesian nonparametric modelling of two correlated distributions. Statistics and Computing 23, 1–15. * Lo, A.Y. (1984). On a class of Bayesian nonparametric estimates I. Density estimates. Annals of Statistics 12, 351–357. * MacEachern, S.N. (1999). Dependent nonparametric processes. In “Proceedings of the Section on Bayesian Statistical Science” pp. 50-55. American Statistical Association. * Müller, P., Quintana, F., and Rosner, G., (2004). A method for combining inference across related nonparametric Bayesian models. Journal of the Royal Statistical Society, Series B 66, 735–749. * Mena, R.H., Ruggiero, M. and Walker, S.G. (2011). Geometric stick–breaking processes for continuous–time Bayesian nonparametric modeling. Journal of Statistical Planning and Inference 141 (9), 3217–3230. * Sethuraman, J. (1994). A constructive definition of Dirichlet priors. Statistica Sinica 4 639–650. * Walker, S.G. (2007). Sampling the Dirichlet mixture model with slices Communications in Statistics 36 45–54. * West, M. (1992). Hyperparameter estimation in Dirichlet process mixture models. Technical report 92-A03, Duke University, ISDS.

http://arxiv.org/abs/1701.07574v1

20170126042428

Theory of Scanning Tunneling Spectroscopy: from Kondo Impurities to Heavy Fermion Materials

cond-mat.str-el

Department of Physics, University of Illinois at Chicago, Chicago, IL 60607, USA Kondo systems ranging from the single Kondo impurity to heavy fermion materials present us with a plethora of unconventional properties whose theoretical understanding is still one of the major open problems in condensed matter physics. Over the last few years, groundbreaking scanning tunneling spectroscopy (STS) experiments have provided unprecedented new insight into the electronic structure of Kondo systems. Interpreting the results of these experiments – the differential conductance and the quasi-particle interference spectrum – however, has been complicated by the fact that electrons tunneling from the STS tip into the system can tunnel either into the heavy magnetic moment or the light conduction band states. In this article, we briefly review the theoretical progress made in understanding how quantum interference between these two tunneling paths affects the experimental STS results. We show how this theoretical insight has allowed us to interpret the results of STS experiments on a series of heavy fermion materials providing detailed knowledge of their complex electronic structure. It is this knowledge that is a conditio sine qua non for developing a deeper understanding of the fascinating properties exhibited by heavy fermion materials, ranging from unconventional superconductivity to non-Fermi-liquid behavior in the vicinity of quantum critical points. Theory of Scanning Tunneling Spectroscopy: from Kondo Impurities to Heavy Fermion Materials Dirk K. Morr December 30, 2023 =========================================================================================== § INTRODUCTION The study of the Kondo effect <cit.> from the single magnetic impurity to heavy fermion materials <cit.>, has remained one of the most fascinating topics in condensed matter physics since its discovery more than 80 years ago <cit.>. One of the key unresolved challenges in this field is to identify the microscopic mechanism giving rise to the complex phase diagram of heavy fermion materials, and their many unconventional properties <cit.>. The most salient features of their phase diagrams are an antiferromagnetically long-range ordered phase with an associated magnetic quantum critical point (QCP) <cit.>, and a Kondo screened, heavy-Fermi-liquid region, as shown in Fig. <ref>(a) for the prototypical heavy fermion material YbRh_2Si_2 <cit.>. The great interest in heavy fermion materials arises from two intriguing phenomena associated with this QCP. Some heavy fermion materials, such as YbRh_2Si_2, possess properties <cit.> in the quantum critical region <cit.> above the QCP which violate the predictions of Landau's Fermi liquid theory – one of the cornerstones of modern condensed matter physics – and hence are labelled non-Fermi liquid (NFL) properties. Other heavy fermion materials, such as the “115" compounds <cit.>, exhibit unconventional superconducting phases close to the QCP <cit.> [see Fig. <ref>(b)]. To-date, no consensus has emerged on the microscopic origin of either of these two phenomena. While it is generally believed that the unconventional superconducting phase arises from f-electron magnetism <cit.>, the lack of detailed insight into the momentum structure of the heavy bands and of the superconducting gap has made the unambiguous identification of the pairing mechanism all but impossible. The same lack of insight into the form of the electronic and magnetic excitations in the quantum critical region has also hindered a deeper understanding of the observed NFL properties which have been attributed to the presence of massless spin fluctuations <cit.>, the competition between the antiferromagnetically ordered and Kondo screened phases <cit.>, critical fluctuations of the hybridization <cit.>, and disorder effects <cit.>. Identifying the key aspects responsible for the complex properties of heavy-fermion materials, is therefore one of the major open problems in condensed matter physics. A major breakthrough in resolving this important problem has recently been achieved by scanning tunneling spectroscopy (STS) experiments <cit.> investigating the complex electronic structure of a series of important heavy fermion materials in their normal and superconducting states. These experiments measure the spatially and energy resolved differential conductance, dI/dV, which in materials with a single electronic band is proportional to the local density of states (LDOS). An important technique employed in these experiments is quasi-particle interference (QPI) spectroscopy <cit.>. Its main idea is that the spatial oscillations induced by defects in the differential conductance, dI/dV, are dominantly 2k_F r oscillations, arising from the backscattering of electrons across the Fermi surface, and hence leading to a change of 2k_F in the electrons momentum. By fourier-transforming these real space oscillations into momentum space, one can therefore in principle map out the electronic structure of a material, as has been successfully demonstrated not only in simple metals <cit.> but also in the cuprate <cit.> and iron-based superconductors <cit.>. By measuring both the differential conductance as well as the QPI spectrum, STS experiments have investigated the electronic structure of a series of intriguing heavy fermion materials: (i) URu_2Si_2 <cit.>, which undergoes a puzzling second order phase transition at T_0 =17.5K into a state with a still unknown, hidden order parameter <cit.>, (ii) YRh_2Si_2 <cit.>, whose phase diagram exhibits a magnetic quantum critical point, and (iii) CeCoIn_5 <cit.>, a material considered to be the “hydrogen" atom for our understanding of unconventional superconductivity in heavy fermion materials. While these experiments might hold the key to understanding the complex properties of these materials, a difficulty in interpreting the experimental results arises from identifying the relation between the measured differential conductance or the QPI spectrum, and the electronic structure of heavy fermion materials <cit.>. In particular, quantum interference between electrons tunneling from the STS tip into the conduction band and into the states containing the magnetic moment (see Fig. <ref>), has rendered the interpretation of dI/dV, even for the case of single magnetic defects on metallic surfaces, quite difficult. For this reason, the dI/dV data taken near isolated magnetic defects <cit.> were often interpreted using a phenomenological expression first derived by Fano <cit.>. However, motivated by the experimental breakthroughs in performing STS experiments on heavy fermion materials, a series of theoretical studies <cit.> have recently emerged that have provided a microscopic understanding of how the interplay between the strength of the Kondo coupling, the interaction between the magnetic moments, the electronic structure of the screening conduction band, and quantum interference determines the dI/dV lineshape. These studies have also extracted the detailed momentum structure of the complex, hybridized electronic bands <cit.>, and identified the symmetry and momentum dependence of the superconducting gaps <cit.>. This in turn has enabled the development of a quantitative understanding of the microscopic mechanism underlying the emergence of unconventional superconductivity in heavy fermion materials <cit.>. The rest of the paper is organized as follows. In Sec. <ref> we review the theoretical formalism that establishes the relation between the differential conductance and the QPI spectrum measured in STS experiments, and the electronic structure of a single magnetic defect (Sec. <ref>), and of heavy fermion materials (Secs. <ref> and <ref>). In Sec. <ref> we discuss the experimental dI/dV lineshapes around isolated magnetic defects, and demonstrate how they are determined by quantum interference effects. In Sec. <ref> we review STS experiments on URu_2Si_2 (Sec. <ref>) and CeCoIn_5 (Sec. <ref>), and the novel insight they provided into the electronic structure of heavy fermion materials. In Sec. <ref>, we discuss how defects in heavy fermion materials affect their electronic structure, and give rise to hybridization waves. Finally, in Sec. <ref> we present our conclusions and provide an outlook on current and future work. § FORMALISM Quantum interference in Kondo systems and heavy fermion materials is directly tied to their multi-orbital or multi-band character. In the following, we briefly outline how the differential conductance can be computed in the presence of multiple tunneling paths, and how the quantum interference between these paths determines the dI/dV lineshape, and the corresponding quasi-particle interference spectrum. §.§ Tunneling into a single Kondo impurity To demonstrate the importance of quantum interference in determining the lineshape of the differential conductance, dI/dV, we begin by considering a system with a single magnetic impurity located on a metallic surface (see Fig. <ref>), described by the Kondo Hamiltonian <cit.> H = ∑_ k,σε_ k c^†_ k,σ c_ k,σ + J S_ R· s^c_ R , where ε_ k is the conduction band dispersion, and c^†_ k,σ (c_ k,σ) creates (annihilates) a conduction electron with spin σ and momentum k. S_ R and s^c_ R are the spin operators of the magnetic impurity and the conduction electrons at site R, respectively, and J>0 is the Kondo coupling. To describe the Kondo screening of the magnetic impurity, we use a fermionic SU(N) representation of the spin operators <cit.> via S_ r = ∑_α, β f^†_ r,αΓ_α, β f_ r,β ; s^c_ r = ∑_α, β c^†_ r,αΓ_α, β c_ r,β , where Γ = (Γ^1, ...,Γ^M) are the M=N^2-1 independent generators of SU(N) in the fundamental representation, N=2S+1 is the spin degeneracy of the magnetic moment, and f^†_ r,α, f_ r,α are the Abrikosov pseudofermion operators that represent the magnetic moment. The pseudofermion operators are subject to the constraint n_f=∑_α=1..N f^†_ r,α f_ r,α=1. Within a path integral approach, this constraint is enforced by means of a Lagrange multiplier ε_f, while the exchange interaction in Eq.(<ref>) is decoupled using a Hubbard Stratonovich transformation and introducing the hybridization field s. Here, a non-zero hybridization implies screening of the magnetic moment. By minimizing the effective action on the (static) saddle point level, we obtain two self-consistent equations (considering the case S=1/2 and hence N=2) given by s = -J/π∫_ - ∞^∞ dω n_F(ω) Im G^r_fc( R, R, ω) ; n_f = 1 = -1/π∫_ - ∞^∞ dω n_F(ω) Im G^r_ff( R, R, ω) , where n_F(ω) is the Fermi distribution function, and G^r is the full retarded Greens function arising from the hybridization process with G_ff^r( R, R, ω) = [ω + i δ -ε_f - s^2 g^r_0( R, R, ω)] ^-1 ; G_cc^r( r, r, ω) = g^r_0( r, r, ω) + g^r_0( r, R, ω) s G^r_ff( R, R, ω) s g^r_0( R, r, ω) ; G^r_cf( r, R, ω) = g^r_0( r, R, ω) s G^r_ff( R, R, ω) . Here, g^r_0 is the retarded Greens function of the unhybridized conduction electron band. In Matsubara τ-space, these Green's functions are defined via G_αβ( r^', r, τ)=-⟨ T_τα^†_ r^'(τ) β_ r(0) ⟩ (α,β=c,f). We note that Eqs.(<ref>) and (<ref>) are employed to determine the hybridization s and the renormalized energy of the f-levels, ε_f. Their solutions, together with the Green's functions of Eqs.(<ref>) - (<ref>) fully describe the many-body effects arising from the hybridization of the conduction band with the f-electron level, and the concomitant screening of the magnetic moment. To compute the differential conductance measured in STS experiments, we note that an electron tunneling from the STS tip into the system can tunnel either into a conduction electron state at r or the f-electron state at R, as schematically shown in Fig. <ref>, allowing for the emergence of quantum interference between these tunneling paths. If the STS tip is positioned above a site r of the surface, these tunneling processes are described by the Hamiltonian <cit.> H_T = t_f( r-R) ∑_σ f^†_ R,σ d_σ + ∑_ r',σ t_c( r-r') c^†_ r',σ d_σ + H.c. , where d_σ destroys an electron with spin σ in the STS tip. Here, t_c( r-r') and t_f( r-R) are the distance dependent tunneling amplitudes between the tip and a site r' on the metallic surface or the site R of the magnetic f-level, respectively. For simplicity, it is assumed that the tunneling is “on-site", i.e., t_c( r-r')=t_c δ_ r,r' and t_f( r-R)=t_f δ_ r,R since tunneling to nearest neighbor sites is strongly suppressed due to the rapid spatial decay of orbital wave-functions involved in the tunneling processes. Moreover, due to the strong Coulomb repulsion in the magnetic f-electron level, one expects that the tunneling amplitude t_f is significantly smaller than t_c even when the STS tip is positioned directly above the magnetic atom. We will see that this expectation is borne out by the theoretical analysis of the experimentally measured dI/dV lineshapes. Assuming that the STS tip is positioned above the magnetic atom at site R, the total current flowing from the STS tip into the system's conduction band and f-level is given by <cit.> I(V) = -e/ħ Re ∫_0^eVd ω/2 π[ t_c Ĝ^K_12(ω) + t_f Ĝ_13^K(ω) ] , where Ĝ^K(ω) is the Keldysh Green's function matrix that accounts for the tunneling between the tip and the system, and is given by Ĝ^K(ω) = [1̂ - Ĝ^r(ω) t̂]^-1F̂(ω) [1̂ - t̂Ĝ^a(ω) ]^-1 where F̂(ω) = 2i (1-2 n̂_F(ω) ) Im[ Ĝ^r(ω) ] ; Ĝ^r(ω) = [ G^r_t(ω) 0 0; 0 G^r_cc( R, R, ω) G^r_cf( R, R, ω); 0 G^r_fc( R, R, ω) G^r_ff( R, R, ω) ] , and the elements of Ĝ^r are given in Eqs.(<ref>) - (<ref>). Here, t̂ is the symmetric tunneling matrix that contains the non-zero tunneling elements between the tip and the system given by t̂_12 = t_c, t̂_13 = t_f. n̂_F is a diagonal matrix containing the Fermi-distribution functions of the tip, the f- and c-electron states, and G^r_t is the retarded Greens function of the tip. To gain insight into the physical quantities that govern the flow of current from the tip into the system, and ultimately determine the differential conductance, dI/dV, we consider the experimentally relevant weak-tunneling limit, t_c,t_f → 0. In this case, we expand the right hand side of Eq.(<ref>) to leading order in the tunneling elements, thus obtaining for the differential conductance dI(V)/dV = 2 π e^2/ħ N_t [t_c^2 N_c( R, V) + t_f^2 N_f( R, V) + t_c t_f N_cf( R, V) + t_f t_c N_fc( R, V) ] , where N_t, N_c and N_f are the density of states of the tip, the conduction and f-electron states, respectively, with N_c=- Im G^r_cc( R, R, V)/π and N_f=- Im G^r_ff( R, R, V)/π. Moreover, N_cf=- Im G^r_cf( R, R, V)/π and N_fc=- Im G^r_fc( R, R, V)/π represent the correlations between the f-state and the conduction electron state at R arising from the hybridization. The last two terms yield identical contributions to the differential conductance. All four terms in Eq.(<ref>) can be visualized as closed paths on which electrons tunnel from the tip into the system and back, as shown in Fig.<ref>. It is the interference between these four tunneling processes that ultimately determines the dI/dV lineshape, as discussed below. §.§ Tunneling into a Heavy Fermion Material Our starting point for the description of heavy fermion materials is the U →∞ limit of the Anderson model <cit.> which allows for charge fluctuations in the electronic levels containing the magnetic moments. The corresponding Hamiltonian is given by H = ∑_ k,σε_ k c^†_ k,σ c_ k,σ + ∑_ r,σ E_0 f^†_ r,σ f_ r,σ - V_0 ∑_ r,σ( f^†_ r,σ b_ r c_ r,σ + H.c. ) + ∑_ r,r' I_ r,r' S_ r· S_ r' , where f^†_ r,σ creates an electron with spin σ at site r in the heavy f-band, and V_0 is the (bare) hybridization between the c- and f-bands. To account for valence fluctuations between unoccupied and singly occupied f-electron sites, one introduces the slave-boson operators b^†_ r,b_ r and the constraint ∑_σ f^†_ r,σ f_ r,σ + b^†_ r b_ r = 1 which ensures an f-electron occupancy n_f<1. Moreover, I_ r,r' is the antiferromagnetic interaction between magnetic moments in the f-band. The origin of the magnetic interaction can lie either in direct exchange or arise from the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction <cit.> mediated by the conduction electrons. Insight into the complex electronic bandstructure of heavy fermion materials provided by STS experiment (see Secs.<ref>) have opened new possibilities to identify the origin of the magnetic interaction. Similar to the single Kondo impurity case, one uses the path integral approach and employs the pseudo-fermion representation for S_ r <cit.> and decouples the magnetic interaction term using a Hubbard-Stratonovich field, t_f( r,r', τ). The constraint is enforced by means of a Lagrange multiplier (ϵ_f-E_0). In the static saddle point approximation (and in the radial gauge <cit.>) one replaces b^†_ r, b_ r by their expectation value ⟨ b^†_ r⟩ = r_0( r) e^i ϕ( r) and subsumes the phase factor e^i ϕ into a redefinition of the fermionic-operators f^†,f. A condensation of the bosonic operators (i.e., r_0 ≠ 0) represents the screening of the magnetic moments. Moreover, the field t_f( r,r', τ) is replaced by its static expectation value t_f( r, r') which describes the antiferromagnetic correlations <cit.> between magnetic moments. Minimizing the effective action, one then obtains the following set of self-consistent equations s( r) = -J_0/π∫_-∞^∞ dωn_F(ω) Im G_fc( r, r,ω) ; t( r,r') = -I_ r,r'/π∫_-∞^∞ dω n_F(ω) Im G_ff( r, r^',ω) ; n_f( r) = - ∫_-∞^∞dω/π n_F(ω) Im G_ff( r, r,ω) , where n_f( r) = 1 - r_0^2( r), J_0=V^2/(ε_f-E_0)>0, and s( r)=V_0r_0( r) is the effective hybridization with s( r)=s for translationally invariant systems. Note that these self-consistent equations possess the same functional form as those for the Kondo model, Eqs.(<ref>) and (<ref>), and that within the mean-field approach described here, the self-consistent equations for the Kondo lattice model are obtained from those of the Anderson lattice model, Eqs.(<ref>) - (<ref>), in the limit r_0 → 0. Moreover, if we assume that the magnetic interaction occurs only between nearest and next-nearest-neighbor sites r, r', then for a translationally invariant system, we have t_f( r, r')=t_f1 and t_f2 for nearest and next-nearest-neighbor sites, respectively. This yields a dispersion of the heavy f-band given by ε^f_ k = -2 t_f1(cos k_x+cos k_y)-4 t_f2cos k_x cos k_y + ε_f . Moreover, in this mean-field approximation, the full Green's functions in momentum space, which describe the hybridization between the c- and f-electron bands, are given by G_ff( k,α, ω) = [(G_ff^0( k, α, ω))^-1 - s^2 G_cc^0( k, α, ω) ] ^-1 ; G_cc( k, α, ω) = [(G_cc^0( k, α, ω))^-1 - s^2 G_ff^0( k, α, ω) ] ^-1 ; G_cf( k, α, ω) = - G_cc^0( k, α, ω) s G_ff( k, α, ω) , where G_ff^0 = (ω + i Γ_f -ε^f_ k)^-1, G_cc^0 = (ω + i Γ_c -ε^c_ k)^-1, and Γ^-1_c and Γ^-1_f are the lifetimes of the c- and f-electron states, respectively. For Γ_c=Γ_f=0^+, the poles of the above Green's functions yield two energy bands with dispersion E_ k^±=ε^c_ k + ε^f_ k/2±√(( ε^c_ k - ε^f_ k/2)^2 + s^2) . For the heavy fermion material URu_2Si_2, it was argued <cit.> that the valence fluctuations occur between singly and doubly occupied f-electron sites which leads to n_f>1. In order to describe this case, it is necessary to perform a particle-hole transformation of the slave-boson Anderson Hamiltonian, in which case the constraint takes the form ∑_σ f^†_ r,σ f_ r,σ - b^†_ r b_ r = 1 and consequently n_f( r) = 1 + r_0^2( r). However, the form of the self-consistent equations, Eqs.(<ref>) - (<ref>), remains unchanged. Finally, we want to briefly mention that the last decade has also seen the development of a series of numerical approaches, such as the dynamical mean-field theory (DMFT) <cit.> or the dynamical cluster approach <cit.>, that have been successful in describing various aspects of heavy fermion materials. While these approaches are limited in their ability to describe momentum-resolved properties of these materials, they account for incoherent processes associated with the formation of the heavy Fermi liquid state. In the Anderson model, the tunneling process into a heavy fermion material is described by the Hamiltonian <cit.> H_T = ∑_ r,σ[ t_c c^†_ r,σ d_σ + t_f^(0) f^†_ r,σ b_ r d_σ + H.c. ] , where we again assume “on-site" tunneling only. Within the saddle-point approximation, the effective tunneling into the f-electron states is given by t_f=t_f^(0) r_0, and the differential conductance is obtained from Eq.(<ref>). We expect that the experimental dI/dV lineshapes should sensitively depend on whether the surface termination layer is a layer of f-moments [see Fig. <ref>(a)], in which case t_f/t_c should be larger and the dI/dV lineshape is dominated by the local electronic structure of the f-electrons, or a conduction band layer [see Fig. <ref>(b)] in which case one expects t_f/t_c to be small and dI/dV to be determined by the local electronic structure of the c-electrons. It was recently suggested that the situation might be even more complicated if the magnetic atoms do not only possess magnetic f-levels, but also conduction electron states <cit.> that directly interact with the magnetic moment via the Kondo coupling. We note in passing that t_f^(0) and t_c can in general be computed using first principle methods: this would require not only exact knowledge of the orbitals in the STM tip and the heavy fermion material that are involved in the tunneling process, but also to account for the strong Coulomb repulsion in the magnetic f-levels. Maltseva et al.<cit.> and Woelfle et al.<cit.> considered a tunneling Hamiltonian similar to that in Eq.(<ref>) to investigate the form of the differential conductance in Kondo lattice systems [see Figs. <ref>(a) and (b)] Specifically, Maltseva et al. proposed that in addition to a direct tunneling process of an electron from the tip into the conduction band, a co-tunneling process exists in which a spin-flip exchange of a tip electron with the magnetic moment occurs while tunneling into the conduction band. They demonstrated that while the differential conductance in general exhibits a hard hybridization gap, disorder will lead to a finite quasi-particle lifetime, that renders this gap soft [see Figs. <ref>(a)]. Complementary to this study, Woelfle et al. <cit.> argued that it is inelastic electron-electron scattering arising from the strong Coulomb repulsion in the f-levels that induces a finite quasi-particle lifetime, and a subsequent softening of the hybridization gap. A similar effect was also found in DMFT studies <cit.> which have shown that incoherent processes become more important with increasing temperature, leading to a smearing out of the hybridization gap and the QPI spectrum. A similar effect also arises from the interaction of conduction or f-electrons with phonons <cit.>, which can lead to a complete destruction of the heavy Fermi liquid state. We note in this regard that the existence of a hard [Fig. <ref>(c)] or soft [Fig. <ref>(d)] gap in dI/dV <cit.> – omitting for a moment finite quasi-particle lifetime effects which are expected to be small at temperatures well below the coherence temperature – depends on the existence or lack of an indirect hybridization gap in the heavy Fermi liquid bandstructure. §.§ Quasi-Particle Interference in Heavy Fermion Materials Quasi-particle interference spectroscopy has been greatly successful in providing insight into the electronic structure of simple metals <cit.> as well as unconventional superconductors <cit.>. Its basic idea is that defects or impurities elastically backscatter a particle with momentum k into a state with momentum - k (for electrons near the Fermi surface, this process is known as 2k_F-scattering). Since the momentum depends on the energy of the particle – for a free electron gas, one has | k(E)| = √(2m (E+μ)), where m is the mass of the particle, and μ the chemical potential – this backscattering process gives rise to spatial oscillations in the energy-resolved local density of states with wave-length λ = 2 π /(2 | k(E)|). Hence by Fourier transforming the spatially resolved differential conductance dI( r,V)/dV – which in a system with a single electronic band is proportional to the local density of states – into momentum space, one gains insight into the variation of | k| with E, and hence the electronic dispersion of the system. The question naturally arises of whether QPI spectroscopy can also provide insight into the more complex electronic structure of heavy fermion materials which possess at least two different electronic bands. To examine this question, one considers the elastic scattering by static impurities described by the Hamiltonian H_scatt = ∑_ r, σ U_c c^†_ r, σ c_ r, σ + U^(0)_f f^†_ r, σ b_ r f_ r, σ b^† _ r + U^(0)_cf( f^†_ r, σ b_ r c_ r, σ + H.c. ) , where the sum runs over all impurity locations. The first two terms describe the intra-band scattering within the c- and f-electron bands, while the last term represents inter-band scattering, as schematically shown in Fig. <ref>(a). Within the saddle-point approximation, the effective scattering potentials are given by U_f=U_f^(0) r_0^2 and U_cf=U_cf^(0) r_0. Using the Born approximation, we consider only the changes in the differential conductance, δ( dI( r,E=eV)/dV ) to lowest (first) order in the scattering potentials. Fourier transform of δ( dI( r,E=eV)/dV ) into momentum space, then yields the quasi-particle interference spectrum g( q,ω) ≡δ( dI( q,ω)/dV) = π e^2/ħ N_t ∑_σ=↑,↓∑_i,j=1^2 [t̂N̂_σ ( q,ω) t̂]_ij , where N̂_σ ( q,ω) =-1/π Im∫d^2 k/(2 π)^2Ĝ_σ ( k, ω) ÛĜ_σ ( k+q, ω) , with Û = [ U_c U_cf; U_fc U_f ] . Note that it is the quantum interference between the scattering processes associated with each of the scattering potentials, U_c, U_f, U_cf, and U_fc (as schematically shown in Figs.<ref>(b)-(d) for the scattering of f-electrons off a defect) that determines the form and spectral weight distribution in the resulting QPI spectrum. § QUANTUM INTERFERENCE AND DIFFERENTIAL CONDUCTANCE FOR A SINGLE KONDO IMPURITY The Kondo screening of an isolated magnetic impurity is a local process that involves conduction electrons up to a distance of the size of the Kondo screening cloud from the defect <cit.>. As scanning tunneling spectroscopy is a local probe, it is ideally suited to provide detailed insight into the complex electronic structure around the magnetic impurity, which reflects the hybridization between the conduction electron states and the state containing the magnetic moment. Madhavan et al. <cit.> therefore investigated the form of the differential conductance in the vicinity of a Co atom located on a metallic Au(111) surface (we note that though the magnetic moment of Co is located in a d-orbital, we will keep the notation of Eq.(<ref>) and refer to the pseudo-fermion states representing the magnetic moment as f-electron states). Fig. <ref>(a) shows the experimental dI/dV data <cit.>, taken when the STS tip is positioned above a magnetic Co atom. As T<T_K, the dI/dV data exhibit a characteristic hump-dip-peak structure which is a direct signature of the hybridization between the conduction band and the magnetic f-electron state of the Co impurity – and hence of the screening of the local moment – and is commonly referred to as the Kondo resonance. Overlain on the experimental results is a theoretical fit obtained by Figgins et al. <cit.> from Eq.(<ref>). This fit assumes that the screening conduction band is given by the Au(111) surface states <cit.>, and uses N=4 as required for the description of the S=3/2-spin of Co. With this input, the theoretically computed differential conductance is entirely determined by the strength of the Kondo coupling, J, and the ratio of the tunneling amplitudes t_f/t_c. While the former controls the width of the Kondo resonance, the latter governs its asymmetry. Note that even though the STS tip positioned above the Co atom, the extracted value of t_f/t_c=0.066 is small, likely due to the strong Coulomb repulsion in the f-level suppressing the tunneling process of an electron from the STS tip into this state. Fig. <ref>(b) shows the LDOS of the conduction electrons, corresponding to dI/dV in the limit t_f/t_c = 0. Its asymmetry is inconsistent with, and indeed opposite to the experimentally observed one shown in Fig. <ref>(a). Similarly, the LDOS of the f-electron state [see Fig. <ref>(c)], corresponding to dI/dV in the limit t_f/t_c →∞, exhibits a single peak, and is therefore also qualitatively different from the dI/dV lineshape observed experimentally. Figgins et al. <cit.> therefore concluded that the inclusion of both tunneling paths, and in particular that of the interference term N_cf(ω) shown in Fig. <ref>(d), is crucial in explaining the experimentally measured dI/dV curves. Moreover, as the STS tip is moved away from the Co atom, direct tunneling into the magnetic f-electron state becomes suppressed and hence t_f → 0 <cit.>. The theoretical dI/dV lineshape at a distance of r=3 a_0 from the Co atom [see Fig. <ref>(e)] obtained with t_f=0, shows the same asymmetry as the one at the site of the Co atom, and qualitatively agrees with the experimental dI/dV curve at r=5 Å <cit.> shown in Fig. <ref>(f). The microscopic origin of the asymmetry in the dI/dV lineshape does not only lie in the existence of two tunneling paths, but also in the particle-hole asymmetry of the screening conduction band. To demonstrate this, Figgins et al. <cit.> considered the case of a single magnetic impurity with a spin-1/2 moment, corresponding to N=2, and a conduction band whose Fermi wavelength λ_F = 10 a_0 is representative of the Au(111) and Cu(111) surfaces states <cit.>. For t_f=0 [solid line in Fig. <ref>(a)], the dI/dV lineshape exhibits a Kondo resonance whose asymmetry is a direct consequence of the particle-hole asymmetry of the conduction band. Indeed, reversing the latter via μ→ -μ, also leads to a reversal of the asymmetry in dI/dV [see dashed line in Fig. <ref>(a)], thus demonstrating the effect of the conduction band's particle-hole asymmetry on the dI/dV lineshape. With increasing t_f/t_c, the dI/dV lineshape undergoes a characteristic evolution, in which its asymmetry is first reversed [Figs. <ref>(b) - <ref>(c)], and subsequently, its characteristic peak-dip-hump structure is replaced by a single (asymmetric) peak [Fig. <ref>(d)]. The latter is a clear indication that as t_f/t_c becomes sufficiently large, the main contribution to dI/dV arises from the magnetic f-level. § DIFFERENTIAL CONDUCTANCE AND QUASI-PARTICLE INTERFERENCE IN HEAVY FERMION MATERIALS §.§ The Hidden Order Phase of URu_2Si_2 One of the most puzzling heavy fermion materials is URu_2Si_2 which possesses a coherence temperature of T_coh≈ 55K <cit.> and undergoes a second order phase transition at T_0 =17.5K <cit.> into a state whose microscopic nature is still unknown, and which is therefore called the hidden order phase. While the debate on the nature of this state is still ongoing <cit.>, new insight into this question has been provided by a series of scanning tunneling spectroscopy experiments <cit.> [see Fig. <ref>]. These experiments have shown that dI/dV exhibits the opening of a soft gap below T_0 [see Figs. <ref>(a) and (b)] <cit.>, and that the QPI dispersion – corresponding to that q at which |g( q,E)| exhibits a maximum for fixed E – significantly evolves through T_0 [see Figs. <ref>(c) and (d)]. In particular, the QPI dispersion exhibits a form at T ≪ T_0, which was suggested to be, at least qualitatively, consistent with that in the heavy Fermi liquid phase <cit.> of a screened Kondo (or Anderson) lattice. Yuan et al. <cit.> proposed a theoretical model to analyse these experimental findings and argued that they <cit.> reflect the emergence of a coherent Anderson lattice, and hence a heavy Fermi liquid state, below the hidden order transition. Their first evidence for this conclusion comes from the theoretical fits [see Figs. <ref>(a) and (b)] of the experimental QPI dispersions (black lines) measured by Schmidt et al. <cit.> on a U-terminated surface of a 1% Th-doped URu_2Si_2 sample <cit.>. In this sample, it is the Th atoms that scatter the conduction electrons, and induce the spatial oscillations in dI/dV that are necessary to obtain a QPI spectrum. The theoretically computed contour plots of |g( q,ω)| – obtained with U_f/U_c ≈ 0.6 and U_cf=0 from Eq.(<ref>) – in Figs. <ref>(a) and (b) reflect the existence of two hybridized bands, characteristic of the heavy Fermi liquid state. The good agreement between the maxima in the theoretical QPI contour plots and the experimental QPI dispersions allowed Yuan et al. <cit.> to extract the momentum structure of the unhybridized bands, ε^c,f_ k, of the hybridization, s, and of the hybridized bands, E_ k^±. This, in turn, enabled them to compute the change in dI/dV below T_0, i.e., δ(dI/dV)=dI/dV(T<T_0)-dI/dV(T=T_0) <cit.>, which is shown in Fig. <ref>(c) together with the experimental result <cit.>. Yuan et al. argued that the good quantitative agreement between the theoretical and experimental dI/dV lineshapes and QPI dispersions, and the consistency between these two sets of data, strongly suggests that the STS data reflect the existence of a heavy Fermi liquid state in the form of a coherent Anderson lattice of screened magnetic moments below T_0, confirming the proposal made by Schmidt et al.<cit.>. Yuan et al. further argued that the form of the QPI spectrum is determined by scattering of electrons both within and between the E_ k^±-bands with intraband scattering [see Fig. <ref>(d)] giving rise to the q_1 and q_2 branches in |q( q,ω)| shown in Figs. <ref>(a) and (b). The overlap of the energy dispersions, E_ k^±, of the two hybridized bands in the energy interval -1 ≲ω≲ 1.5 meV, allows for interband scattering with wave-vector q_3, and a corresponding q_3 branch in |q( q,ω)| [see Figs. <ref>(a) and (b)]. The q_3 branch was observed experimentally along q_y=0, thus confirming the theoretical prediction, but not along q_y=q_x. This “missing" branch is likely due to the smaller separation between the branches along this direction rendering the experimental resolution of the q_1 and q_3 branches difficult. The agreement between the theoretical and experimental QPI dispersions also provides further insight into the form of dI/dV in that it identifies the peak in dI/dV at ω = -2 meV [see arrow in Fig. <ref>(c)] as arising from the van Hove singularity of the f-electron band. The experimental QPI spectra <cit.> also provide insight into the microscopic mechanism underlying the electronic scattering by Th atoms, as the spectral weight associated with the QPI spectrum |q( q,ω)| sensitively depends on the relative strength of the scattering potentials, and hence the quantum interference between the scattering channels. To demonstrate this, Yuan et al. <cit.> contrasted the QPI spectra obtained when only one of the three scattering potentials, U_c, U_f and U_cf is non-zero [see Figs. <ref>(a) - (c)]. For intraband scattering with U_c ≠ 0 [Fig. <ref>(a)] and U_f ≠ 0 [Fig. <ref>(b)] the dominant contribution to the QPI spectrum arises from scattering between those states where the coherence factors of the c-electrons and f-electrons, respectively, are large. However, as the spectral weight in |q( q,ω)| in both cases is inconsistent with the experimentally observed QPI weight and dispersion, the latter can only be explained (as shown in Fig. <ref>) by considering intraband scattering in both the c- and f-electron bands with relative scattering strength U_f/U_c ≈ 0.6. Moreover, for interband scattering between the c and f-bands, U_cf≠ 0 [Fig. <ref>(c)], the QPI spectrum significantly deviates from the experimental QPI dispersion. In particular, interband scattering leads to only two branches in the QPI spectrum, in contrast to the three branches observed experimentally. In addition, the largest spectral weight in the QPI spectrum occurs where the experimental QPI intensity is close to a minimum [see red arrow in Fig. <ref>(c)]. These inconsistencies thus strongly suggest that the interband scattering by Th-atoms is negligible. Further evidence for the existence of a heavy Fermi liquid state below T_0 comes from the differential conductance measured by Aynajian et al. <cit.> on a U-terminated surface of pure URu_2Si_2 at T=2K and 4K, respectively [see Figs. <ref>(a) and (b)]. Starting from their analysis of the QPI spectra by Schmidt et al. <cit.>, Yuan et al.<cit.> argued that the theoretical fits of the experimental dI/dV lineshapes reproduce all of the experiment's salient features: the asymmetry and magnitude of the gap in dI/dV as well as the peak at ω≈ -0.8 meV [see arrows in Figs. <ref>(a) and (b)] which arises from the van Hove singularity of the f-electron band. Similar features were also observed by Schmidt et al. <cit.>. As these features are characteristic signatures of the hybridized band structure in the heavy Fermi liquid state, the experimental dI/dV lineshapes provide further evidence for its existence. The temperature evolution of dI/dV observed by Aynajian et al. <cit.> [see Figs. <ref>(a) and (b)] also allows one to gain insight into the microscopic mechanism that drives the emergence of the heavy Fermi liquid state below T_0. To this end, Yuan et al. showed that the observed changes in dI/dV between T=2K [Fig. <ref>(a)] and T=4K [Fig. <ref>(b)] can be solely attributed to an increasing decoherence (as described by the decoherence rate Γ_f) of the f-electron states. Increasing Γ_f even further yields the evolution of dI/dV shown in Fig. <ref>(c) which possesses the same characteristic signatures as those observed by Aynajian et al. <cit.> with increasing temperature [see Fig. <ref>(a)]: the gap in dI/dV is filled in, its magnitude remains approximately constant until one approaches the hidden order transition, and the center of the gap shifts to larger energies (a similar temperature dependence was also found by Schmidt et al. <cit.>). It is instructive to consider the evolution of r_0 and t_f1 [see Fig. <ref>(d)] with increasing Γ_f, as obtained from the self-consistent solution of Eqs.(<ref>) - (<ref>). While t_f1 varies only weakly with increasing Γ_f, r_0, and hence the effective hybridization s, is strongly suppressed and eventually vanishes at Γ_f=Γ^c_f. These results, taken together, strongly suggest that the experimentally observed formation of a heavy Fermi liquid state below T_0 is driven by a significant reduction in Γ_f at T_0 from Γ_f > Γ^c_f above T_0 to Γ_f < Γ^c_f below T_0. Chatterjee et al. <cit.> recently arrived at a similar conclusion based on the results of photoemission experiments. These conclusions demonstrate the importance of inelastic scattering process (giving rise to Γ_f) in the destruction of the coherent Anderson lattice <cit.>. An alternative explanation was proposed by Dubi and Balatsky <cit.>. They argued that a hybridization wave emerges below T_0 arising from the particle-hole pairing of an f-hole with momentum Q=0.3π/a_0 and a c-electron with momentum -Q. With increasing strength of the order parameter, V, the resulting dI/dV [Fig.<ref>(e)] develops a gap and for sufficiently large V, also exhibits a peak inside the gap, consistent with the experimental findings <cit.>. While Yuan et al. <cit.> concluded that the experimental STS data do not exhibit a direct signature of a hidden order parameter below T_0, but rather reflect the existence of a coherent heavy Fermi liquid, they argued that the deduced strong reduction in Γ_f at T_0 might be a direct signature of this order parameter. In particular, if Γ_f arises due to a coupling of the f-electrons to a fluctuating mode associated with the hidden order parameter, then the condensation of the order parameter at T_0 would significantly reduce the electron-mode coupling, thus reducing the decoherence of the f-electrons, as reflected in a suppression of Γ_f. A recent alternative explanation, ascribing the hidden order phase to the emergence of a hastatic order <cit.>, has proposed that the hidden order parameter might be “hidden" in the detailed momentum dependence of the hybridization between the f- and c-electrons. A test of this scenario will require a detailed comparison between the dI/dV lineshapes and QPI spectra following from this proposal with the experimental STS results. Finally, a comparison of the bandstructure, E_ k^±, [Fig. <ref>(d)] extracted from the STS experiments on the 1% Th-doped <cit.> and pristine URu_2Si_2 <cit.> samples, has shown that Th-doping decreases the hybridization, and hence the hybridization gap, and increases Γ_f,c, and hence the decoherence of the quasi-particles. Moreover, knowledge of the bandstructure allows one to explore the origin of the magnetic interaction, I_ r,r'. While Yuan et al. showed that the extracted I_ r,r' between nearest-neighbor sites is antiferromagnetic, the magnetic RKKY-interaction computed from the extracted bandstructure is ferromagnetic. This result suggests that the microscopic origin of the nearest-neighbor I_ r,r' in URu_2Si_2 lies in direct exchange, and not in an RKKY interaction. §.§ Differential Conductance and QPI spectroscopy in CeCoIn_5 One of the most interesting heavy fermion materials is CeCoIn_5, a member of the so-called “115"-family, whose phase diagram [see Fig. <ref>(b)] shares many of the fascinating features that are also found in the phase diagram of the cuprate <cit.> and iron-based superconductors <cit.>, such as unconventional superconductivity in proximity to antiferromagnetism. CeCoIn_5 <cit.> exhibits the largest T_c=2.3K in this family of heavy fermion materials, and has long been considered the “hydrogen atom" of heavy fermion superconductivity <cit.>. While much experimental <cit.> and theoretical effort <cit.> has focused on illuminating its unconventional properties <cit.>, and the microscopic mechanism underlying the emergence of superconductivity, no consensus has been reached to-date. A major obstacle in providing a quantitative or even qualitative explanation for its properties in the superconducting state has been the lack of insight into the material's complex electronic bandstructure <cit.>. Recent STS experiments <cit.> have therefore focused on identifying the complex electronic bandstructure of CeCoIn_5 by employing quasi-particle interference spectroscopy. In particular, STS experiments by Aynajian et al.<cit.> on CeCo(In_0.9985Hg_0.0015)_5 demonstrated that below the coherence temperature, T_coh≈ 45K <cit.>, but above T_c, the differential conductance measured on the Ce-In surface layer exhibits a typical Kondo resonance [see Fig. <ref>(a)], confirming that the material is in a heavy Fermi liquid state. A comparison of these results with those obtained on a Co termination layer show striking differences, confirming the conclusion of Sec.<ref>, that the nature of the surface termination layer should possess a strong effect on the dI/dV lineshape. In particular, while dI/dV on the Ce-In surface layer, exhibits a hump-dip-peak structure, implying that electrons from the tip tunnel predominantly into the conduction band [Fig. <ref>(a)] with a correspondingly small t_f/t_c [see theoretical fit in Fig. <ref>(c)], dI/dV on a Co termination layer shows a strong peak, which is evidence for dominant tunneling into the heavy f-electron band [Fig. <ref>(b)], and hence a large value of t_f/t_c [see theoretical fit in Fig. <ref>(d)]. An alternative explanation was recently put forth by Peters and Kawakami <cit.>. They proposed that it is only the conduction band orbital at the Ce site that directly couples to the Ce atom's magnetic moment, but not the conduction band orbitals at the In and Co sites. Using a DMFT approach, they qualitatively reproduced the dI/dV lineshapes on the Ce-In surface [Fig.<ref>(e)] and Co surface layers [Fig.<ref>(f)] of CeCoIn_5. Further evidence for the formation of a heavy Fermi liquid state is provided by the significant changes in the QPI spectrum that occur with decreasing temperature, as shown in Figs. <ref>(g) and (h). In particular, while at T=70K>T_coh, the QPI spectrum [Figs. <ref>(g)] reflects the existence of a light conduction band that crosses the Fermi energy, the onset of hybridization between the light and heavy bands at T_coh leads to a bending of this light band, as evidence by the QPI spectrum at T=20K<T_coh, shown in Fig. <ref>(h). These temperature dependent changes reflect the formation of a hybridized bandstructure characteristic for the onset of a coherent heavy Fermi liquid below T_coh. Finally, Aynajian et al.<cit.> argued that the nearly linear temperature dependence of the width of the peak observed in dI/dV on the Co surface [see Fig. <ref>(b)] is a signature of the material's proximity to a quantum critical point. More detailed insight into the complex momentum structure of the hybridized bands at a temperature T=250mK well below T_coh was provided in high-resolution QPI studies by Allan et al. <cit.>. Using the theoretical formalism of Sec. <ref>, they obtained good agreement between the experimentally observed and theoretically computed QPI dispersions [see Figs.<ref>(a) and (b)] that did not only reveal a momentum dependent hybridization [cf. Figs.<ref>(c) and (d)], but also a backbending of the heavy band [Fig.<ref>(d)], resulting in three Fermi surface sheets [see Fig.<ref>(e)]. We briefly mention that the detailed insight into the momentum structure of the heavy bands near the Fermi surface allowed Allan et al. <cit.> to extend the QPI analysis into the superconducting state. Their study revealed the momentum structure of an unconventional superconducting order parameter which is consistent with a d_x^2-y^2-wave symmetry on all three Fermi surfaces, as shown in Fig. <ref>(f). The largest superconducting gap Δ_max≈ 0.6 meV resides on the α_1 Fermi surface, followed by smaller gaps on the α_2- and β-Fermi surfaces. These results are consistent with the conclusions of Zhou et al. <cit.> based on their QPI and dI/dV data taken in the superconducting state of CeCoIn_5. Moreover, Dyke et al. <cit.> were able to use this detailed insight into the complex electronic structure of CeCoIn_5 to extract a crucial missing component in the quest for the superconducting pairing mechanism, the pairing interaction between the magnetic f-moments. This interaction, together with the detailed form of the bandstructure, allowed Dyke et al. <cit.> to solve the superconducting pairing problem, and compute a series of physical properties in the superconducting state of CeCoIn_5. The good agreement of their results with the experimental findings provides strong support for a superconducting pairing mechanism in CeCoIn_5 that is mediated by the antiferromagnetic interactions between f-electron moments. § HYBRIDIZATION WAVES AND IMPURITY STATES: EFFECT OF DEFECTS ON THE LOCAL ELECTRONIC STRUCTURE OF HEAVY FERMION MATERIALS Understanding how the formation of antiferromagnetism competes locally with the creation of a Kondo singlet, and how the magnetic and electronic degrees of freedom are coupled, both in real and momentum space, is crucial for identifying the microscopic mechanism underlying the complex phase diagram of heavy fermion materials. The great success in employing defects and impurities in the high-temperature superconductors to gain insight into their complex electronic structure <cit.>, raises the question of whether a similar approach can also be used in heavy-fermion materials to disentangle and spatially resolve their electronic and magnetic structure. To answer this question, Figgins et al. <cit.> investigated the spatial entanglement of electronic and magnetic degrees of freedom by exploring the effects of defects in the form of missing magnetic moments – Kondo holes – and non-magnetic scatterers on the local electronic structure of heavy fermion materials. They showed that a Kondo hole induces significant spatial oscillations in the hybridization [see Fig. <ref>(a)], the charge density [see Fig. <ref>(b)] and t_f1 [see Fig. <ref>(c)]. The oscillations in the hybridization and the charge density exhibit very similar spatial patterns that are nearly isotropic in space and decay exponentially with distance from the Kondo hole. This exponential decay arises from the fact that a Kondo hole induces a localized state outside the conduction band <cit.>. The origin of these spatial fluctuations is revealed by their wavelength λ_F^c/2, where λ^c_F is the Fermi wave-length of the unhybridized conduction band [see Fig. <ref>(d)], implying that they arise from 2k_F^c scattering across the Fermi surface of the unhybridized conduction band. This result is quite unexpected, since the actual Fermi surface of the hybridized system in the heavy Fermi liquid state is large, as shown in Fig. <ref>(d). However, a strong feedback effect between the conduction electron charge density (whose spatial oscillations for sufficiently small hybridization are still determined by the unhybridized conduction electron Fermi surface) and the hybridization ensures that the hybridization oscillations exhibit a wavelength of λ_F^c/2. In contrast, the spatial oscillations of Δ t_f1 shown in Fig. <ref>(c) extend predominantly along the lattice diagonal with a wavelength of λ^h_F/2= √(2) a_0, where λ^h_F is the Fermi wavelength of the hybridized Fermi surface along the diagonal [see Fig. <ref>(d)]. The spatial oscillations in t_f1 therefore arise from 2k_F^h scattering across the Fermi surface of the hybridized bands, and their spatial form is driven by the Fermi surface's strong anisotropy [Fig. <ref>(d)] which possesses a large degree of nesting along the diagonal direction. Weaker reflections of these anisotropic oscillations can also be found in Δ s, clearly demonstrating the coupling between the system's electronic and magnetic degrees of freedom. Hamidian et al. <cit.> recently investigated the effects of defects on the local electronic structure in Th-doped URu_2Si_2. To this end, they extracted the spatial variations of the direct hybridization gap, Δ_h=Δ^+_h-Δ^-_h, from the width of the Kondo resonance observed in dI/dV [see Fig. <ref>(b)]. As the hybridization gap is twice the hybridization [see Eq.(<ref>)], Hamidian et al. <cit.> were able to create a hybridization gap map that provided direct insight into the spatial variations of the hybridization induced by defects. By Fourier transforming the hybridization gap map into momentum space [see Fig.<ref>(e)], they identified the characteristic wave-vector of the hybridization oscillations as twice the Fermi wave-vector of the unhybridized conduction band. This result confirms the theoretical predictions by Figgins et al. <cit.> not only of defect-induced hybridization waves in real space, but also of their characteristic wave-length [see Fig. <ref>(a)] governed by the unhybridized conduction band. Moreover, Figgins et al. <cit.> showed that when a magnetic moment is replaced by a non-magnetic atom that induces scattering in the conduction electron band, an impurity bound state can emerge inside the hybridization gap for sufficiently large attractive scattering potential U_c<0, in contrast to the effect of a Kondo hole [see dashed black line in Fig. <ref>(a)]. Its spectroscopic signature is a sharp peak in N_c( r,ω) inside the hybridization gap [see Fig. <ref>(a)], that first emerges at the high energy side of the hybridization gap and then moves to lower energies with increasing |U_c|. The induced bound state is spatially isotropic and decays exponentially with distance from the impurity with a decay length ξ_D smaller than a lattice constant. This small value of ξ_D implies that the bound state is predominantly formed by f-electron states, as an impurity state formed by the light conduction electron states would possess a decay length a hundred times larger than the one observed. This result directly reflects the strong correlations between the light and heavy bands as the defect scatters only conduction electrons, but creates a bound state that predominantly consists of f-electron states. The predicted impurity states inside the hybridization was subsequently observed by Hamidian et al.<cit.> in Th-doped URu_2Si_2. By comparing the differential conductance far away from a Th atom [Fig. <ref>(b)] with that at a Th atom site [Fig. <ref>(c)], they concluded that the Th atom gives rise to the emergence of an impurity state inside the hybridization gap. Hamidian et al.<cit.> further observed that defects and disorder exert a strong effect on the electronic structure of the heavy Fermi liquid state, as a small concentration of 1% of Th atoms in URu_2Si_2 essentially disorders the entire electronic structure of the material. Using the theoretical formalism outlined in Secs. <ref>, Parisen Toldin et al. <cit.> found that disorder effects are enhanced in the heavy Fermi liquid state due to a strong feedback effect between the conduction electron charge density, and the hybridization, such that already an impurity concentration of 1% essentially disorders the hybridization in the entire system, as shown in Fig. <ref>(a). Parisen Toldin et al. showed that while this impurity concentration possess a pronounced effect on the electronic structure as observed in the differential conductance, its effects on the material's thermodynamic properties, such as the specific heat, are rather weak (and on the order of a few percent) in agreement with the experimentally observed changes in the specific heat of heavy fermion materials with defect concentration of 1% <cit.>. This result explains the apparent contradiction between spectroscopic and thermodynamic measurements. Moreover, Parisen Toldin et al. <cit.> compared the QPI intensity obtained from different theoretical approaches. In particular, they computed the QPI intensity, g( q,E), for a heavy Fermi liquid state with 1% of defects, whose effects on the electronic structure were self-consistently calculated [Fig. <ref>(b)]. A comparison with the QPI spectrum for a single defect, obtain within the (non-self-consistent) Born approximation showed that both approaches yield the same information regarding the allowed scattering vectors, albeit with a redistribution of their spectral weight. § CONCLUSIONS AND OUTLOOK The simultaneous development of scanning tunneling spectroscopy on heavy fermion materials, together with the theoretical framework to describe and analyse it, has provided unprecedented insight into the complex electron structure of heavy fermion materials and Kondo systems. Starting from single magnetic atoms on metallic surfaces, it was possible to demonstrate that quantum interference is a crucial element in understanding how the Kondo resonance observed in the differential conductance, is related to the changes in the local electronic structure arising from the Kondo screening process and the resulting hybridization between the conduction band and the localized magnetic moments. These advances might even provide insight <cit.> into the proposed topological nature <cit.> of the Kondo insulator SmB_6. Moreover, quasi-particle interference spectroscopy has revealed the momentum structure of the complex hybridized bands with unprecedented energy resolution, providing us with the opportunity to test theoretical models in great detail against experimental data. The extension of QPI spectroscopy to the superconducting state of CeCoIn_5 <cit.> has not only provided unique insight into the detailed momentum structure and symmetry of the superconducting order parameter, but also allowed a quantitative test of a 30-year old hypothesis for the mechanism underlying unconventional superconductivity in heavy fermion materials <cit.>. Conducting STS experiments in the superconducting state of other heavy fermion compounds, such as recent experiments by Enayat et al.<cit.> on CeCu_2Si_2, will enable us to investigate the universality of the pairing mechanism across different families of heavy fermion materials. Equally important is the success in measuring the differential conductance in YbRh_2Si_2 by Ernst et al.<cit.> which has opened the path to investigating how the electronic structure of heavy fermion materials evolves across their quantum critical points. This, in turn, might provide the missing Rosetta Stone for understanding the emergence of non-Fermi liquid behavior in the quantum critical region. All of these advances have clearly paved the way for even more exciting discoveries in the future. § ACKNOWLEDGEMENTS We would like to thank M. Allan, P. Aynajian, J.C. Davis, M. Hamidian, J. Hoffman, V. Madhavan, F. Massee, H. Manoharan, J. Van Dyke, A. Yazdani, and B. Zhou for stimulating discussions and comments. DKM gratefully acknowledges support from the U. S. Department of Energy, Office of Science, Basic Energy Sciences, under Award No. DE-FG02-05ER46225. 99 Kondo64 J. Kondo, Prog. Theor. Phys. 32, 37 (1964). Don77 S. Doniach, Physica B 91, 231 (1977). deH34 W.J. de Haas, J. de Boer, and G.J. van den Berg, Physica 1, 1115 (1934). Ste01 G.R. Stewart, Rev. Mod. Phys. 73, 797 (2001). Mon07 P. Monthoux, D. Pines, G. G. Lonzarich, Nature 450, 1177 (2007). Loh07 H. von Lohneysen, A. Rosch, M. Vojta, and P. Woelfle, Rev. Mod. Phys. 79, 1015 (2007). Si10 Q. Si and F. Steglich, Science 329, 1161 (2010). Nor11 M. R. Norman, Science 332, 196 (2011). Ste12 F. Steglich, O. Stockert, S. Wirth, C. Geibel, H. Q. Yuan, S. Kirchner and Q. Si, Preprint available at arXiv:1208.3684v1 (2012). Sca12 D. J. Scalapino, Rev. Mod. Phys. 84, 1383 (2012). ASch00 A. Schroder, G. Aeppli, R. Coldea, M. Adams, O. Stockert, H.v. Lohneysen, E. Bucher, R. Ramazashvili, and P. Coleman , Nature (London) 407, 351 (2000). Cus03 J. Custers, P. Gegenwart, H. Wilhelm, K. Neumaier, Y. Tokiwa, O Trovarelli, C Geibel, F. Steglich, C. Pepin, and P. Coleman, Nature 424, 524 (2003). Pas04 S. Paschen, T. Lühmann, S. Wirth, P. Gegenwart, O. Trovarelli, C. Geibel, F. Steglich, P. Coleman and Q. Si, Nature 432, 881 (2004). Geg05 P. Gegenwart, J. Custers, Y. Tokiwa, C. Geibel, and F. Steglich, Phys. Rev. Lett. 94, 076402 (2005). Col07 P. Coleman, preprint, arXiv:cond-mat/0612006. Loh94 H. von Lohneysen, T. Peitrus, G. Portisch, H.G. Schlager, A. Schoder, M. Sieck, and T. Trappmann, Phys. Rev. Lett. 72 3262 (1994). Map95 M.B. Maple et al., J. Low Temp. Phys. 99, 223 (1995). Tro00 O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer, F.M. Grosche, P. Gegenwart, M. Lang, G. Sparn, and F. Steglich, Phys. Rev. Lett. 85, 626 (2000). Col05b P. Coleman and A.J. Schofield, Nature 433 226 (2005). Geg08 P. Gegenwart, Q. Si, and F. Steglich, Nature Physics, 4, 186 (2008). Pet01 C. Petrovic, et al., J. Phys. Condens. Matter 13, L337 (2001). Mos01 R. Movshovich, M. Jaime, J.D. Thompson, C. Petrovic, Z. Fisk, P.G. Pagliuso, and J.L. Sarrao, Phys. Rev. Lett. 86, 5152 (2001). Iza01 K. Izawa, H. Yamaguchi, Y. Matsuda, H. Shishido, R. Settai, and Y. Onuki, Phys. Rev. Lett. 87, 057022 (2001). Sto08 C. Stock, C. Broholm, J. Hudis, H.J. Kang, and C. Petrovic, Phys. Rev. Lett. 100, 087001 (2008). Sar07 J. L. Sarrao and J. D. Thompson, J. Phs. Soc. Jpn. 76, 051013(2007). Ste79 F. Steglich, J. Aarts, C.D. Bredl, W. Liefke, D. Meschede, W. Franz, H. Schafer, Phys. Rev. Lett. 43, 1892 (1979). Mat98 N.D. Mathur, F.M. Grosche, S.R. Julian, I.R. Walker, D.M.Freye, R.K.W. Haselwimmer, and G.G.Lonzarich, Nature 394 39 (1998). Sid02 V.A. Sidorov, M. Nicklas, P.G. Pagliuso, J.L. Sarrao, Y. Bang, A.V. Balatsky, and J.D. Thompson , Phys. Rev. Lett. 89, 157004 (2002). Sto11 O. Stockert, J. Arndt, E. Faulhaber, C. Geibel, H.S. Jeevan, S. Kirchner, M. Loewenhaupt, K. Schmalzl, W. Schmidt, Q. Si, and F. Steglich, Nature Physics 7, 119 (2011). Mor95 T. Moriya and T Takimoto, J. Phys. Soc. Jpn. 64, 960 (1995). Ros97 A. Rosch, A. Schroder, O. Stockert, and H. von Lohneysen, Phys. Rev. Lett. 79, 159 (1997). Ros99 A. Rosch, Phys. Rev. Lett. 82, 4280 (1999). Abr12 E. Abrahams and P. Woelfle, PNAS 109, 3238 (2012). Si01 Q.M. Si, S. Rabello, K. Ingersent, and J.L. Smith, Nature 413, 804 (2001). Ing02 K. Ingersent and Q.M. Si, Phys. Rev. Lett. 89, 076403 (2002). Sun03 P. Sun and G. Kotliar, Phys. Rev. Lett. 91, 037209 (2003). Gre06 C. Grenzebach, F.B. Anders, G. Czycholl, and T. Pruschke, Phys. Rev. B 74, 195119 (2006). Mar10 L.C. Martin, M. Bercx, and F.F. Assaad, Phys. Rev. B 82, 245105 (2010). Tan11 D. Tanaskovic, K. Haule, G. Kotliar, and D. Dobrosavljevic, Phys. Rev. B 84, 115105 (2011). Col01 P. Coleman, C. Pepin, Q. Si, and R. Ramazashvili, J. Phys. Cond. Mat. 13, R723 (2001) and references therein. Col99 P. Coleman, Physica B 261, 353 (1999). Sen03 T. Senthil, S. Sachdev, and M. Vojta, Phys. Rev. Lett. 90, 216403 (2003). Sen04 T. Senthil, M. Vojta, and S Sachdev, Phys. Rev. B 69, 035111 (2004). Paul07 I. Paul, C. Pepin, and M.R. Norman, Phys. Rev. Lett. 98, 026402 (2007). Neto98 A. H. Castro Neto, G. Castilla, and B. A. Jones, Phys. Rev. Lett. 81, 3531 (1998). Mil02 A. J. Millis, D. K. Morr, and J. Schmalian, Phys. Rev. B 66, 174433 (2002). Sch11 A. Schroeder, S. Ubaid-Kassis,and T. Vojta, J. Phys. Cond. Mat. 23, 094205 (2011). Mir05 E. Miranda and V Dobrosavljevic, Prog. Rep. Phys. 68, 2337 (2005). Sch09 A.R. Schmidt, M.H. Hamidian, P. Wahl, F. Meier, A.V. Balatsky, J.D. Garrett, T. Williams, G.M. Luke and J.C. Davis, Nature 465, 570 (2010). Ayn10 P. Aynajian, E. H. da Silva Neto, C. V. Parker, Y. Huang, A. Pasupathy, J. Mydosh, and A. Yazdani, PNAS 107, 10383 (2010). Ern11 S. Ernst, S. Kirchner, C. Krellner, C. Geibel, G. Zwicknagl, F. Steglich, and S. Wirth, Nature 474, 362 (2011). Ham11 M.H. Hamidian, A.R. Schmidt, I.A. Firmo, M.P. Allan, P. Bradley, J.D. Garrett, T.J. Williams, G.M. Luke, Y. Dubi, A.V. Balatsky, and J.C. Davis, PNAS 108, 18233 (2011). Wahl11 P. Wahl, L. Diekhoner, M.A. Schneider, F. Treubel, C.T. Lin, and K. Kern, Phys. Rev. B 84, 245141 (2011). Ayn12 P. Aynajian, E. H. da Silva Neto, A. Gyenis, R. E. Baumbach, J. D. Thompson, Z. Fisk, E. D. Bauer, and A. Yazdani, Nature 486, 201 (2012). All13 M.P. Allan, F. Massee, D.K. Morr, J. Van Dyke, A.W. Rost, A.P. Mackenzie,C. Petrovic, and J.C. Davis, Nat. Phys. 9, 468 (2013). Zhou13 B.B. Zhou, S. Misra, E.H.D. Neto, P. Aynajian, R.E. Baumbach, J.D. Thompson, E.D. Bauer, and A. Yazdani, Nat. Phys. 9, 474 (2013). Ena15 M. Enayat, Z. Sun, A. Maldonado, H. Suderow, S. Seiro, C. Geibel, S. Wirth, F. Steglich, and P. Wahl, preprint, Phys. Rev B 93, 045123 (2016). Crom93 M.F. Crommie, C.P. Lutz, and D.M. Eigler, Nature 363, 524 (1993). Has93 Y. Hasegawa, and Ph. Avouris, Phys. Rev. Lett. 71, 1071 (1993). Hoff02 J.E. Hoffman, K. McElroy, D.H. Lee, K.M. Lang, H. Eisaki, S. Uchida, and J.C. Davis, Science 297, 1148 (2002). Han10 T. Hanaguri, S. Niitaka, K. Kuroki, and H. Takagi, Science 328, 474 (2010). Pal85 T.T.M. Palstra, A.A. Menovsky, and J.A. Mydosh, Phys. Rev. Lett. 55, 2727 (1985). Map86 M.B. Maple, et al. Phys. Rev. Lett. 56, 185 (1986). Scho87 J. Schoenes et al., Phys. Rev. B 35, 5375 (1987). Bonn88 D.A. Bonn et al., Phys. Rev. Lett. 61, 1305 (1988). Dor01 S.V. Dordevic, et al., Phys. Rev. Lett. 86, 684 (2001). Rod97 J.G. Rodrigo et al. Phys. Rev. B. 55, 14318 (1997). Esc94 R. Escudero, F. Morales, and P. Lejay, Phys. Rev. B 49, 15271 (1994). Den01 J.D. Denlinger et al., J. Elec. Spectrosc. 117–118, 347 (2001). Cox87 D.L. Cox, Phys Rev Lett 59, 1240 (1987). Cha02 P. Chandra et al., Nature 417, 831 (2002). Var06 C.M. Varma and L. Zhu, Phys. Rev. Lett. 96, 036405 (2006). Bal09 A.V. Balatsky et al., Phys. Rev. B 79, 214413 (2009). Elg09 S. Elgazzar, et al., Nature Mater. 8, 337 (2009). Pep11 C.Pepin et al., Phys.Rev.Lett.106, 106601 (2011). Hau09 K. Haule, and G. Kotliar, Nature Phys. 5, 796 (2009). Opp10 P. M. Oppeneer et al., Phys. Rev. B 82, 205103 (2010). Dubi11 Y. Dubi and A.V. Balatsky, Phys. Rev. Lett. 106, 086401 (2011). Cha13 P. Chandra, P. Coleman, and R. Flint, Nature 493, 621 (2013). Myd14 J.A. Mydosh, and P.M. Oppeneer, Phil. Mag. 94, 3642 (2014). Oka11 R. Okazaki |it et al., Science 331, 439 (2011). Kung15 H.-H. Kung et al., Science 347, 1339 (2015). Fano61 U. Fano, Phys. Rev. 124, 1866 (1961). Mad98 V. Madhavan et al., Science 280, 567 (1998). Li98 J. Li, et al., Phys. Rev. Lett. 80, 2893 (1998). Ujs00 O. Ujsaghy et al., Phys. Rev. Lett. 85, 2557 (2000). Kno02 N. Knorr et al., Phys. Rev. Lett. 88, 096804 (2002). Ali05 A. A. Aligia and A.M.Lobos, J. Phys.: Condens. Matter 17, S1095 (2005). Mad01 V. Madhavan, et al., Phys. Rev. B 64, 165412 (2001). Mal09 M. Maltseva, M. Dzero, and P. Coleman, Phys. Rev. Lett. 103, 206402 (2009). Fig10 J. Figgins and D.K. Morr, Phys. Rev. Lett. 104, 187202 (2010). Wol10 P. Wölfle, Y. Dubi, and A.V. Balatsky, Phys. Rev. Lett. 105, 246401 (2010). Fig11 J. Figgins and D. K. Morr, Phys. Rev. Lett. 107, 066401 (2011). Yuan12 T. Yuan, J. Figgins and D.K. Morr, Phys. Rev. B 86, 035129 (2012). Par13 F. Parisen Toldin, J. Figgins, S. Kirchner, and D.K. Morr, Phys. Rev. B 88, 081101(R) (2013). Dyke14 J. Van Dyke, F. Massee, M. P. Allan, J. C. Davis, C. Petrovic, and D.K. Morr, PNAS 111, 11663 (2014). Pet14 R. Peters and N. Kawakami, Phys. Rev. B 89, 041106(R) (2014). Coq69 B. Coqblin and J. R. Schrieffer, Phys. Rev. 185, 847 (1969). Read83 N. Read and D. M. Newns, J. Phys. C 16, 3273 (1983). Col83 P. Coleman, Phys. Rev. B 28, 5255 (1983). Mil87 A.J. Millis and P.A. Lee, Phys. Rev. B 35, 3394 (1987). Bic87 N. E. Bickers, Rev. Mod. Phys. 59, 845 (1987). Car71 C. Caroli et al, J. Phys. C: Solid St. Phys. 4, 916 (1971). AL D.M. Newns and N. Read, Adv. Phys. 36, 799 (1987). Kot88 G. Kotliar and J. Liu, Phys. Rev. B 38, 5142 (1988) Hew97 A. D. Hewson, The Kondo Problem to Heavy Fermions (Cambridge University Press, Cambridge, 1997). Col84 P. Coleman, Phys. Rev. B 29, 3035 (1984). Rud54 M.A. Ruderman and C. Kittel, Phys. Reve. 96, 99 (1954). Kas56 T. Kasuya, Prog. Theor. Phys. 16, 45 (1956). Yos57 K. Yosida, Phys. Rev. 106, 893 (1957). Hau10 K. Haule, C.-H. Yee, and K. Kim, Phys. Rev. B 81, 195107 (2010). Ben11 A. Benlagra, T. Pruschke, and M. Vojta, Phys. Rev. B 84, 195141 (2011). Mar08 L. C. Martin and F. F. Assaad, Phys. Rev. Lett. 101, 066404 (2008). Wu15 Wei Wu and A.-M.-S. Tremblay, Phys. Rev. X 5, 011019 (2015). Rac10 M. Raczkowski, P. Zhang, F. F. Assaad, T. Pruschke, and M. Jarrell, Phys. Rev. B 81, 054444 (2010). Bor07 L. Borda, Phys. Rev. B 75, 041307(R) (2007). Aff01 I. Affleck and P. Simon, Phys. Rev. Lett. 86, 2854 (2001). Schout09 K. Schouteden, P. Lievens, and C. Van Haesendonck, Phys. Rev. B 79, 195409 (2009). Chatt13 S. Chatterjee, J. Trinckauf, T. Hanke, D.E. Shai, J.W.Harter, T.J. Williams, G.M. Luke, K.M. Shen, and J. Geck, Phys. Rev. Lett. 110, 186401 (2013). Kei15B. Keimer et al., Nature 518, 179 (2015). Bas11 D.N. Basov and A.V. Chubukov, Nat. Phys 7, 272 (2011). Miy86 K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986). Beal86 M. T. Beal-Monod, C. Bourbonnais, and V. J. Emery, Phys. Rev. B 34, 7716 (1986). Sca86 D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 (1986). Sca87 D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B, 35 6694 (1987). Lav87 M. Lavagna, A. Millis, and P. A. Lee, Phys. Rev. Lett. 58, 266 (1987). Col89 P. Coleman and N. Andrei, J. Phys.: Condens. Matter 1, 4057 (1989) . Koh01 Y. Kohori et al., Phys. Rev. B 64, 134526 (2001). Cur12 N. J. Curro et al., Phys. Rev. B 64, 180514 (2012). Park08 W. K. Park et al., Phys. Rev. Lett. 100, 177001 (2008). Hu12 T. Hu et al., Phys. Rev. Lett. 108, 056401 (2012). Koi13 A. Koitzsch et al. Phys. Rev. B 88, 035124 (2013). Tru13 C. J. S. Truncik et al., Nature Comm. 4, 2477 (2013). Shu14 L. Shu, et al., Phys. Rev. Lett. 113, 166401 (2014). Kim15 H. Kim, M. A. Tanatar, and R. Flint, Phys. Rev. Lett. 114, 027003 (2015). Fli10 R. Flint and P. Coleman, Phys. Rev. Lett. 105, 246404 (2010). Das13 T. Das, et al., Phys. rev. B 87, 174514 (2013). Dav13 J. C. Davis and D.-H. Lee Proc. Nat. Acad. Sci. USA 110, 17623 (2013). Liu13 B. Liu, Phys. Rev. B 88, 245127 (2013). Yang14 Y.-F. Yang and D. Pines, Proc. Nat. Acad. Sci. 111, 18178 (2014). Ert15 O. Erten, R. Flint, and P. Coleman, Phys. Rev. Lett. 114, 027002 (2015). Kim01 J. S. Kim et al., Phys. Rev. B 64, 134524 (2001). Pag03 J. Paglione et al., Phys. Rev. Lett. 91, 246405 (2003). Bia03 A. Bianchi it et al., Phys. Rev. Lett. 91, 257001 (2003). Pag06 J. Paglione et al., Phys. Rev. Lett. 97, 106606 (2006). How11 L. Howald et al., J. Phys. Soc. Jpn. 80, 024710 (2011). Park05 W. K. Park, L. H. Greene, J. L. Sarrao, and J. D. Thompson, Phys. Rev. B 72, 052509 (2005). Yaz99 A. Yazdani, et al., Phys. Rev. Lett. 83, 176 (1999). Hud99 E.W. Hudson, et al., Science 285, 88 (1999). Bal06 A.V. Balatsky, I. Vekhter, and J.X. Zhu, Rev. Mod. Phys. 78, 373 (2006). Pikul A. P. Pikul, U. Stockert, A. Steppke, T. Cichorek, S. Hartmann, N. Caroca-Canales, N. Oeschler, M. Brando, C. Geibel and F. Steglich, Phys. Rev. Lett. 108, 066405 (2012). Yee13 M.M. Yee, et al., preprint, arXiv:1308.1085. Ros14 S. Rossler, et al., PNAS 111, 4798 (2014). Dze10 M. Dzero, K. Sun, V. Galitski, and P. Coleman Phys. Rev. Lett. 104, 106408 (2010).

http://arxiv.org/abs/1701.08216v1

20170127225803

Numerical analysis of electronic conductivity in graphene with resonant adsorbates: comparison of monolayer and Bernal bilayer

cond-mat.mes-hall

Numerical analysis of conductivity in bilayer graphene A. Missaoui et al. Laboratoire de Spectroscopie Atomique Moléculaire et Applications, Département de Physique, Faculté des Sciences de Tunis, Université Tunis El Manar, Campus Universitaire 1060 Tunis, Tunisia Laboratoire de Physique de la matière condensée, Département de Physique, Faculté des Sciences de Tunis, Université Tunis El Manar, Campus Universitaire 1060 Tunis, Tunisia Université Grenoble Alpes, Inst NEEL, 38042 Grenoble, France CNRS, Inst NEEL, 38042 Grenoble, France Laboratoire de Physique théorique et Modélisation, CNRS and Université de Cergy-Pontoise, 95302 Cergy-Pontoise, France We describe the electronic conductivity, as a function of the Fermi energy, in the Bernal bilayer graphene (BLG) in presence of a random distribution of vacancies that simulate resonant adsorbates. We compare it to monolayer (MLG) with the same defect concentrations. These transport properties are related to the values of fundamental length scales such as the elastic mean free path L_e, the localization length ξ and the inelastic mean free path L_i. Usually the later, which reflect the effect of inelastic scattering by phonons, strongly depends on temperature T. In BLG an additional characteristic distance l_1 exists which is the typical traveling distance between two interlayer hopping events. We find that when the concentration of defects is smaller than 1%–2%, one has l_1 ≤ L_e ≪ξ and the BLG has transport properties that differ from those of the MLG independently of L_i(T). Whereas for larger concentration of defects L_e < l_1 ≪ξ, and depending on L_i(T), the transport in the BLG can be equivalent (or not) to that of two decoupled MLG. We compare two tight-binding model Hamiltonians with and without hopping beyond the nearest neighbors. 72.15.Lh 72.15.Rn 73.20.Hb 72.80.Vp 73.23.-b Numerical analysis of electronic conductivity in graphene with resonant adsorbates: comparison of monolayer and Bernal bilayer Ahmed Missaoui1, Jouda Jemaa Khabthani2, Nejm-Eddine Jaidane1, Didier Mayou3,4 Guy Trambly de Laissardière5 January 27, 2017 ================================================================================================================================== § INTRODUCTION Graphene consists of a monolayer (MLG) carbon atoms, with sp^2 hybridization, forming a 2D honeycomb lattice, with two equivalent atoms –atom A and atom B– in a unit cell. Linear dispersion relation of the p_z electron states close to the Fermi energy induces many fascinating transport properties which give rise to potential device applications <cit.>. Few-layer graphene also present unusual properties. In particular the Bernal bilayer graphene (BLG) with AB stacking, as in graphite, breaks the atom A / atom B symmetry and leads to quadratic dispersion relation <cit.>. Electronic transport is sensitive to static defects which are for example screened charged impurities, or local defects like vacancies or adsorbates, (hydrogen, adatoms or admolecules, chemically bound to one carbon atom of the surface of graphene layer). Theoretical studies of the effects introduced by the adsorbates on the conductivity has been done for MLG (Refs. <cit.> and Refs. therein), and for BLG <cit.>. Most of them consider a standard Hamiltonian that takes only into account the hopping between the nearest neighbors orbitals. Yet some studies show the importance of hopping beyond nearest neighbors on electronic structure and transport properties <cit.>. In this paper, we present electronic properties of MLG and BLG obtained by two tight-binding (TB) models: the standard model with nearest neighbor only (TB1) and a TB model including the effect of the hopping beyond nearest neighbors (TB2). This second model, which is more realistic, predicts some differences in the transport at energies close to the resonant energy of scatters. We consider local defects, such as adsorbates or vacancies, that are resonant scatters. Local defects tend to scatter electrons in an isotropic way for each valley and lead also to strong intervalley scattering. The T matrix of a local defect usually depends strongly on the energy. In the case of simple vacancies or adsorbates (atoms or molecules) that create one covalent bound with a carbon atoms of MLG (BLG), the T matrix diverges at the energy E_MG (with TB1 model E_MG=E_D=0). For this reason, theses scatters are called resonant scatters. The adsorbate is simulated by a simple vacancy in the plane of p_z orbital as usually done <cit.>. Indeed the covalent bonding between the adsorbate and the carbon atom of graphene to which it is linked, eliminates the p_z orbital from the relevant energy window. The scatterers are distributed randomly in both planes and with the same concentration in both planes. We consider here that the up and down spins are degenerate i.e. we deal with a paramagnetic state. Indeed the existence and the effect of a magnetic state for various adsorbates or vacancies is still debated <cit.>. Let us emphasize that in the case of a magnetic state the up and down spins give two different contributions to the conductivity but the individual contribution of each spin can be analyzed from the results discussed here. We first determined the density of states (DOS) in disordered MLG and BLG in presence of static scatterers (vacancies) with various concentrations c from 0.5% to 10%. Elastic mean free path L_e, which depends on the distribution of scatters and on energy E, is also computed. From diffusive properties of wave packet in the structure, the electrical conductivity σ is computed versus E and the inelastic mean free path L_i. L_i due to electron-phonons interaction or magnetic fied. Roughly speaking, large L_i correspond to low temperature limit and small L_i to room temperature. The numerical method used takes into account all quantum effects. We show that difference between MLG and BLG is explained by considering the average distance l_1 over which a charge carrier travels in a layer between two interlayer hoppings <cit.>. As explained in this paper, l_1 ≃ 2nm and it is almost independent on the concentration of defects. Indeed BLG has some similar properties to MLG for small L_i and large c, i.e. when L_e ≃ L_i < l_1. But for small c values, L_e ≃ L_i ≥ l_1, the effects of interlayer hopping affect the electronic properties of BLG with respect to MLG case. In this case, the conductivity of the BLG varies with c like in usual metals where DOS is finite whereas MLG behaves like a semi-metals with Dirac electrons. Moreover, different regimes of transport in BLG are found depending on the values of the energy, like in MLG <cit.>. Finally the localization length ξ is computed and we study the localization regime (which can be observed experimentally in a very low temperature regime, i.e. large L_i). For the concentrations studied here, l_1 < ξ, which means that in this localization regime the coupling between the two layers plays always a significant role. Thus behavior of the BLG and MLG are different and localization length is larger in the BLG. In Sect. <ref>, the two tight binding (TB) models used are described and the corresponding density of states are discussed. Transport properties are presented in Sect. <ref>. We first describe rapidly the computational method and the relevant lengths to analyze conductivity in BLG (Sect. <ref>). Then elastic scattering length L_e (Sect. <ref>) and microscopic conductivity σ_M (Sect. <ref>) are presented. Finally quantum interference corrections to the conductivity and localization length ξ are analyzed (Sect. <ref>). Sect. 4 is devoted to concluding remarks. § ELECTRONIC STRUCTURE §.§ Tight binding Hamiltonian models BLG can be considered as two coupled MLGs with the top layer shifted by a carbon bond from the bottom layer. Consequently, BLG consists of four atoms in its unit cell, two carbons A_1, B_1 from the unit cell in the bottom layer and A_2, B_2 in the the top layer where A_2 sits at the top of A_1 (Fig. <ref>). We used a tight binding scheme (TB) <cit.>. Only p_z orbitals are taken into account since we studied the electronic properties around the Fermi energy level. Interlayer interactions are not restricted to the ppσ terms but ppπ terms have also to be introduced. The Hamiltonian has the form: Ĥ =∑_iϵ_i|i⟩⟨ i| + ∑_(i,j)t_ij|i⟩⟨ j| , where i is the orbital located at r_i and the sum runs over all neighboring i, j sites. The energy on the site is taken equal to zero for first nearest neighbor model. t_ij is the hopping element matrix between site i and site j, computed from the Slater-Koster parameters, t_ij  =  n_c^2 V_ppσ(r_ij)  +  (1 - n_c^2) V_ppπ(r_ij), where V_ppσ and V_ppπ depend from the distance r_ij and n_c is the cosines direction along the Oz axis. It is either equal to zero or to a constant because the two graphene layers have been kept flat in our model. The Slater Koster parameters are exponentially decaying function of the distance: V_ppπ(r_ij) = -γ_0 exp( q_π(1-r_ij/a) ) , V_ppσ(r_ij) = γ_1 exp( q_σ(1-r_ij/a_1) ) . It allows, according to the value of q_π, to take into account both first neighbors or second neighbors. a is the nearest neighbor distance within a layer, a=1.418 Å, and a_1 is the interlayer distance, a_1=3.349 Å. First neighbors interaction in a plane is characterized by the commonly used value γ_0=2.7 eV and the second neighbors interaction γ_0^' is set to 0.1γ_0. The ratio q_π/a in Eq. <ref> is fixed by the value of the γ_0^'. The interlayer coupling between two p_z orbitals in π configuration is γ_1, γ_1=0.48 eV, and it is fixed to obtain a good fit with ab initio calculation around Dirac energy in AA stacking and AB bernal stacking <cit.>. It is worth to note that we choose the same decay coefficient for V_ppπ and V_ppσ: q_σ/a_1 = q_π/a = log(γ_0/γ_0' )/a' - a =  2.218Å^-1, with a'=2.456 Å the distance between second neighbors in a plane. All p_z orbitals have the same on-site energy ϵ_i in two planes. In order to obtain a Dirac energy E_D equal to zero in MLG, one fixes ϵ_i equal to -0.78 eV for TB model with hopping beyond first neighbors. This is necessary because hopping beyond first neighbors breaks the electron/hole symmetry and then shifts E_ D value <cit.>. We have used this model Hamiltonian in our previous work <cit.> to study the electronic structure rotated graphene bilayer. In order to analyze the effect of hopping beyond the first-neighbor distances, we consider also the simplest TB model (TB1) with first-neighbor hopping only and the same parameters than the complete TB2 model described above. As explained in the introduction we consider that resonant adsorbates –simple atoms or molecules such as H, OH, CH_3– create a covalent bond with some carbon atoms of the BLG. To simulate this covalent bond, we assume that the p_z orbital of the carbon, that is just below the adsorbate, is removed. In our calculations the mono-vacancies are distributed at random between the both planes with a finite concentration c. §.§ Density of states Figure <ref>(b) shows the total density of states (total DOS) n(E) in BLG for different concentrations c = 0.5% to 10% of defects randomly distributed. For comparison, the DOSs of MLG with same TB Hamiltonian <cit.> are also shown (Fig. <ref>(a)). Without defects, c=0, BLG DOS and MLG DOS differ for energies E such as -γ_1 < E < γ_1 <cit.>. For small c concentrations, c< ∼ 1%, this distinction is still observed. But remarkably, MLG and BLG have very similar DOS, for concentration of defects c larger than ∼ 1%. With TB1 model (first-neighbor hopping only), states occur at energy E_ MG=0. This is reminiscent of the midgap state produced by asymmetry between the number of atoms A and B in monolayer graphene <cit.>. In agreement with previous findings for MLG <cit.> and BLG <cit.>, for large values of energies E the DOS is weakly affected by the presence of disorder. Finally near the Dirac energy there is an intermediate regime where the pseudo-gap is filled (“plateau”). With the TB2 model including all neighbors, the midgap state is no longer at E=0, but it becomes a broad peak at negative energy E_ MG. E_ MG value varies from E_ MG≃-0.2 eV to E_ MG≃-0.3 eV when defect concentration increases from c=0.5% to 10%, like in monolayer graphene (Fig. <ref>(a)) <cit.>. In the following we distinguish three cases according to energy values: (i) Sufficiently large energies, BLG DOS and MLG DOS are similar and they are not strongly modified by the presence of resonant defects. (ii) Energies in the “plateau” due to vacancies but not in the midgap states E ≠ E_ MG. (iii) Energies in the midgap states, i.e. E=E_ MG=0 for TB1 (with first-neighbor hoppings only) and E≃ E_ MG for TB2 (with hopping beyond first neighbors). These three cases correspond to different transport regimes. § TRANSPORT PROPERTIES §.§ Computational method and relevant lengths In the framework of the Kubo-Greenwood formula for electronic transport properties, the quantum diffusion coefficient D (diffusivity) and conductivity σ are computed by using the polynomial expansion method, developed by Mayou, Khanna, Roche and Triozon <cit.>. This numerical approach allows very efficient calculations by recursion (Lanczos algorithm) in real-space which take into account all quantum effects. It has been used to study quantum transport in disordered graphene, chemically doped graphene and bilayer (see Refs. <cit.> and Refs. therein). Our calculations are performed on sample containing up to a few 10^7 carbon atoms, which corresponds to typical sizes of about one micrometer square and allows to study systems with elastic mean-free length of the order of few hundred nanometers. Elastic scattering events are taken into account in the Hamiltonian, but effects of inelastic scattering by phonons at temperature T are not included in the Hamiltonian. To consider the inelastic processes, we introduce an inelastic scattering time τ_i(T) beyond which the propagation becomes diffusive due to the destruction of coherence by these inelastic processes. The effect of a magnetic field on the electron propagation is not included directly in the TB model, but a magnetic field B can have also a similar incoherent dephasing effect. This dephasing effect occurs on a length L_i(B) such that the flux of the magnetic field enclosed in the disk of radius L_i(B) is equal to the flux quantum h/e, i.e. L_i(B)≃√(h/eB). We treat these two dephasing effects in a phenomenological way through a Relaxation Time Approximation (RTA) as described here after. In the RTA, the conductivity along the x-axis is given by, <cit.> σ(E_F,τ_i) = e^2n(E_F)D(E_F,τ_i) , D(E_F,τ_i) = L_i^2(E_F,τ_i)/2τ_i , where E_F is the Fermi energy, n(E_F) is the density of states (DOS) and L_i is the inelastic mean-free path conductivity along the x-axis. L_i(E_F,τ_i) is the typical distance of propagation during the time interval τ_i for electrons at energy E. We compute the distance L_i, the diffusivity D and the conductivity σ at all inelastic scattering times τ_i and all energies E for model Hamiltonian that includes inelastic scatters distributed randomly in the super-cell. At short times τ_i –i.e. τ_i lower than elastic scattering time τ_e– the propagation is ballistic and the conductivity σ increases when τ_i increases (Fig. <ref>), L_i(E,τ_i) ≃ V_0(E) τ_i   when τ_i ≪τ_e, where V_0(E) is a velocity at the energy E and short time t. In crystals, V_0 ≥ V_B where V_B is the Boltzmann velocity (intra-band velocity) <cit.>. In BLG and MLG, V_0 and V_B have the same order of magnitude: V_0( BLG) = V_0( MLG) = √(2)V_B( MLG) <cit.>. According to the renormalization theory <cit.> in 2D systems with static defects, diffusivity D always goes to zero at very large τ_i. At each energy, the microscopic diffusivity D_M (microscopic conductivity σ_M) is defined as the maximum value of D(τ_i) (σ(τ_i)). We compute also the elastic mean-free path L_e along the x-axis, from the relation <cit.>, L_e(E) = 1/V_0(E) Max_τ_i{L_i^2(E,τ_i)/τ_i} = 2 D_M(E)/V_0(E). L_e is the average distance between two elastic scattering events. At each energy, the elastic scattering times τ_e is deduced from L_e by L_e(E) = V_0(E) τ_e(E). In our calculations τ_i and L_i are considered as adjustable parameters. Roughly speaking, when L_i ≪ L_e (τ_i ≪τ_e) the inelastic disorder dominates; it should correspond to very high temperatures. When L_i ≃ L_e (τ_i ≃τ_e), the conductivity is equal to microscopic conductivity, which should correspond to high temperature cases, typically room temperature. And when L_i ≫ L_e (τ_i ≫τ_e), quantum localization will dominates transport properties; this is the low temperature limit. In the following we therefore discuss the two important cases: L_i ≃ L_e (τ_i ≃τ_e) and L_i ≪ L_e (τ_i ≪τ_e). Figure <ref> shows the variation of the conductivity σ and the inelastic mean free path L_i versus τ_i for energies corresponding to the three previous cases (see Sect. <ref>). The first case (i) (E=1.5 eV in Fig. <ref>) corresponding to a Boltzmann transport: for large values of τ_i, the conductivity σ is almost constant as expected in a diffusive regime. This regime corresponds to energies for which the DOS is weakly affected by scatters. The third case (iii) (E=E_ MG in Fig. <ref>) is determined by the transport of the midgap states which are localized states. The latter case (ii) is an intermediary regime between the previous two: for τ_i closed to the elastic scattering time τ_e, there is a diffusive behavior where the σ(τ_i) reaches a maximum, σ_M; for larger values of τ_i, τ_i ≫τ_e, σ(τ_i) decreases progressively as expected in localization regime due to Anderson localization in 2D <cit.>. In BLG another relevant time is the average traveling time t_1 between two interlayer hoppings of the charge carriers, which is associated to an average traveling distance l_1 in a layer between two interlayer hoppings <cit.>. In perfect BLG typical values of t_1 and l_1 can be easily estimated : t_1 = ħ / Γ_1 ≃ 2 × 10^-15 s, where Γ_1 ≃ 0.4eV is the interlayer hopping parameters of the Hamiltonian, and l_1 ≃ V_m t_1 ≃ 2 nm <cit.> where V_m is the velocity in MLG, V_m ≃ V_0 ≃ 10^6 ms^-1. When there is elastic disorder such that τ_e < t_1 the value of t_1 can be modified. A simple argument may be given as follows: A Bloch state of the MLG is still coupled to Bloch states of the other layer by the same intensity, typically Γ_1, but these states are no longer eigenstates and have a typical lifetime τ_e. Because of that they have a spectral width W≃ħ /τ_e. From the Fermi Golden rule the typical time needed to jump from one layer to the other will be such that ħ/t'_1 ≃Γ_1^2/W. Therefore the new value of the interlayer hopping time t'_1 will be larger and will be such that t'_1≃ t_1 (ħ/Γ_1 τ_e). Since the propagation is diffusive on the timescale t>τ_e with diffusion coefficient D≃ V_0^2 τ_e, the new length l'_1 in presence of defects is obtained from the relation, l'_1^2/t_1' ≃ V_0^2 τ_e, and thus l_1' ≃ l_1 ≃ 2 nm depends weakly on disorder (i.e. l_1' almost independent on τ_e). As shown in the following, electronic properties of disordered BLG depend on the values of the length L_e, ξ and l_1 which are characteristic of the BLG and of the amount of elastic scatters: * For low concentration of defects, c < c_l = 1%-2%, and for E E_MG, we have, ł_1 ≤ L_e ≪ξ, and thus electronic properties of BLG are influenced by interlayer hoppings for every L_i ≥ L_e values. * For larger concentration of defects, c > c_l, one obtains, Ł_e < l_1 ≪ξ. When the effect of quantum interferences on conductivity is small, i.e. when L_e ≃ L_i < l_1, BLG behaves as two decoupled MLG. For Ł_e < l_1 ≤ L_i ≪ξ, the coupling between the two planes can affect the propagation of charge carriers before inelastic scattering makes the propagation diffusive (i.e. on the length scale L_i). In this regime, quantum corrections to transport are not the same in the BLG and in two decoupled MLG. This influences strongly the localization regime as we discuss in Sect. <ref>. §.§ Elastic mean-free path The elastic mean-free path L_e (Eq. (<ref>)) along the x-axis as a function of the E is shown figure <ref> for different values of vacancy concentrations c in MLG and BLG with both TB models. It depends on the energy even in the intermediate regime and it takes a finite and non-zero value for E=E_D but stays comparable to the distance d between adsorbates (vacancies) defined by, d ≃1/√(n_a), where n_a is the adsorbates density. Numerical results (Fig. <ref>) show that L_e values in BLG and MLG are close to each other. Moreover, L_e(E) < l_1 ≃ 2 nm (i.e. τ_e < t_1) for c > c_l ≃ 1%–2%; whereas for smaller c, L_e(E) ≥ l_1 (i.e. τ_e ≥ t_1) for c < c_l. §.§ Microscopic conductivity As shown Fig. <ref>, the static scattering events perturb strongly the wave packet propagation and a maximum value of the conductivity σ(τ_i), called microscopic conductivity σ_M, σ_M(E) = e^2n(E)D_M(E), is reached. σ_M calculated from both TB models are shown in Figs. <ref> and <ref> for different concentrations c of vacancies in MLG and BLG. According to the renormalization theory <cit.>, this value is obtained when the inelastic mean free path L_i and the elastic mean free path L_e are comparable, L_e ≃ L_i, which corresponds to τ_i ≃τ_e. As L_i and τ_i decrease when the temperature T increases, the microscopic conductivity is a good estimation of the high temperature conductivity (or room temperature conductivity). For energies corresponding the to Boltzmann regime, i.e. regime (i) described in Sec. <ref>, σ_M≃σ_B, where σ_B is calculated with the Bloch-Boltzmann approach <cit.>. In this regime the conductivity decreases with the concentration of defects. In the intermediate energy values regime (ii), the semi-classical approach fails and the behavior depends on c. From Fig. <ref>(a), for small c values, typically c< c_l ≃ 1%–2% (i.e. L_i ≃ L_e ≥ l_1), σ_M seems to reach a constant minimum value (“plateau”), but this minimum σ_M value decreases as c increases. This concentration dependence is specific to BLG and is not observed in MLG. For larger c (Fig. <ref>(b)), c>c_l (i.e. L_i ≃ L_e < l_1), σ_M reaches a minimum value independent on c value: σ_M≃ 1.2 G_0 where G_0 = 2 e^2/h. This values for BLG is <cit.> two times the universal value of the conductivity, ∼ 4e^2/(π h), expected in presence of resonant scatters in MLG <cit.>. Results with TB2 model (including hopping beyond nearest neighbors) (fig. <ref>) show that a plateau of the microscopic conductivity near the Dirac energy exists in MLG and BLG, but is not symmetric due to the symmetry breaking electron-hole. Nevertheless in this case, for c>c_l, σ_M values are still close to the universal conductivity plateau; whereas for c<c_l, the minimum value of σ_M decreases as c increases like with TB1 model. For energies in the midgap states (iii) with TB1 model (first neighbor hopping only), an anomalous behavior of the conductivity is obtained and there is a peak of σ_M at E=E_ MG=0. With TB2 (including hopping beyond nearest neighbors), this anomalous behavior is still slightly present at E ≃ E_ MG, but the change in the conductivity is rather small. Thus, as in monolayer graphene <cit.>, conduction by the midgap states is very specific to TB1 model. §.§ Quantum localization regime In this section we consider the case of large inelastic mean-free path, L_i ≫ L_e (i.e. τ_i ≫τ_e), which should corresponds to the low temperature limit. The conductivity in the intermediate energy case (ii) (see Sect. <ref>) presents localization effects that are a consequence of quantum interferences. In that case, for all vacancy concentration values c, L_i ≫ l_1 (i.e. τ_i ≫ t_1). Therefore BLG should have different localization behavior than MLG. As shown in Fig. <ref>, σ(L_i) follows the linear variation with the logarithm of the inelastic mean free path L_i, like in the case of monolayer graphene <cit.>, σ(E,L_i) = σ_0(E) - α G_0 log(L_i/L_e(E)), where G_0 = 2 e^2/h, L_e(E) is defined by Eq. (<ref>), and σ_0 values are in the range of σ_M values. For low concentration of defects c=0.5% and 1% one can estimate α≃ 0.26, and for larger c, c = 3% and 5%, α≃ 0.32. These values are close to the result found in monolayer graphene with same computational method <cit.> and close to the prediction of the perturbation theory of 2D Anderson localization for which α = 1/π <cit.>. As in the monolayer case, this linear variation of σ with log L_i is found for both models, with nearest neighbor hopping only and with hopping beyond first nearest neighbors. We finally define the localization ξ length from the expression (<ref>) by extrapolation of σ(L_i) curves (Fig. <ref>) when : σ(L_i = ξ)=0, and then ξ(E) = L_e(E) exp( σ_0(E)/α G_0). The ξ values for energies in the plateau of σ_M (i.e. case (ii) described in Sect. <ref>) are shown figure <ref>. For large concentrations of defects (c>c_l), ξ in MLG and BLG is almost independent on the energy. Moreover ξ(BLG) is always larger than ξ(MLG). This difference results from the fact that σ_0, BLG≃ 2 σ_0, MLG (Sect. <ref>) and then, for c>c_l, ( ξ(E)/L_e(E) )_BLG≃ ( ξ(E)/L_e(E) )_MLG^2 , where L_e are similar in MLG and BLG (Sec. <ref>). It is thus a multilayer effect on quantum interferences that modifies the 2D behavior with respect to MLG cases. For low defects concentration (c<c_l), interlayer coupling modifies also quantum interferences. Therefore, for every resonant scatterer concentration, quantum interference corrections to the conductivity in BLG and in MLG are not similar. § CONCLUSION To conclude we have studied numerically the quantum diffusion of charge carriers in monolayer graphene and Bernal bilayer graphene in the presence of local defects. These defects are simulated by simple vacancies randomly distributed in the structure. Among the fundamental length scales in the MLG and BLG there are the elastic mean free path L_e, the localization length ξ and the inelastic mean-free path L_i which in real systems depends on temperature or on magnetic field. For the bilayer, there is an additional length scale which is the typical distance l_1≃ 2 nm over which an electron travels in a plane before hopping to the other plane. We have compared the Bilayer and monolayer transport properties for identical concentrations of vacancies. We show that these properties can be either similar or different depending on the comparison between l_1 and the other three length scales L_e, L_i, ξ. This relation explains essentially the numerical results detailed in this paper. Our results show that for strong concentration of defects, c > c_l ≃ 1%–2%, the bilayer graphene could be equivalent to two independent disordered monolayers of graphene, because the elastic mean free path L_e is smaller than the average distance l_1. Therefore for c > c_l, the universal aspects of the conductivity are present in bilayer, as in monolayer graphene, with (TB1) or without (TB2) the hopping beyond nearest neighbors. In the high temperature limit, i.e when inelastic scattering length L_i is small, L_i ≃ L_e, the conductivity in bilayer is almost equal to two times the universal minimum plateau of microscopic conductivity in monolayer graphene (except for the Dirac energy with TB1 model that takes only into account nearest neighbor hopping). For smaller c, c < c_l, the BLG should be considered like a usual metal: with static defects the minimum of microscopic conductivity of BLG increases when the c values decreases. For the parameters studied here, the localization length ξ is larger than the traveling distance l_1 between two interlayer hopping; and therefore the BLG and MLG have different localization lengths at the same concentration, even if they have similar elastic-mean-free paths. The localization length is the largest in the BLG. In the limit, i.e L_i ≫ L_e, (which is relevant at low temperature) and for all c values, the conductivity follows a linear variation with the logarithm of L_i in MLG and BLG and for both TB1 models (nearest neighbor hopping only) and TB2 (with hopping beyond nearest neighbors), excepted for energy in the midgap states for TB1. This is in good agreement with two-dimensional Anderson localization and consistent with the expected universal behavior of conductivity of a two dimensional disordered system <cit.>. § ACKNOWLEDGMENT The authors wish to thank C. Berger, W. A. de Heer, L. Magaud, P. Mallet and J.-Y. Veuillen for fruitful discussions. The numerical calculations have been performed at Institut Néel, Grenoble, and at the Centre de Calculs (CDC), Université de Cergy-Pontoise. This work was supported by the Tunisian French Cooperation Project (Grant No. CMCU 15G1306) and the project ANR-15-CE24-0017. § AUTHOR CONTRIBUTION STATEMENT Numerical calculations have been performed by A. Missaoui and G. Trambly de Laissardière. All authors took part in drafting of the manuscript as well as in analysis and interpretation of the results. Wallace47 P. R. Wallace, Phys. Rev. 71, 622 (1947) Berger06 C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, W. A. de Heer, Science 312, 1191 (2006) Katsnelson06 M. I. Katsnelson, K. S. Novoselov, A. K. Geim, Nature Phys 2, 9 (2006) Castro09_RevModPhys A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, Rev. Mod. Phys 81, 109 (2009) Latil06 S. Latil, L. Henrard, Phys. Rev. Lett 97, 036803 (2006) Castro07 E. V. Castro, K. S. Novoselov, S. V. Morozov, N. M. R. Peres, J. M. B. Lopes dos Santos, J. Nilsson, F. Guinea, A. K. Geim, A. H. Castro Neto, Phys. Rev. Lett 99, 216802 (2007) MacdoABC F. Zhang, B. Sahu, H. Min, A. H. MacDonald, Phys. Rev. B 82, 035409 (2010) Varchon_prb08 F. Varchon, P. Mallet, J.-Y. Veuillen, L. Magaud, Phys. Rev. B 77, 235412 (2008) Ohta06 T. Ohta, A. Bostwick, T. Seyller, K. Horn, E. Rotenberg, Science 313, 951 (2006) Brihuega08 I. Brihuega, P. Mallet, C. Bena, S. Bose, C. Michaelis, L. Vitali, F. Varchon, L. Magaud, K. Kern, J. Y. Veuillen, Phys. Rev. Lett. 101, 206802 (2008) DasSarma11 S. Das Sarma, S. Adam, E. H. Hwang, E. Rossi Rev. Mod. Phys 83, 407 (2011) McCann13 E. McCann and M. Koshino, Rep. Prog. Phys. 76 056503 (2013) Ulstrup14 S. Ulstrup, J. C. Johannsen, F. Cilento, J. A. Miwa, A. Crepaldi, M. Zacchigna, C. Cacho, R. Chapman, E. Springate, S. Mammadov, F. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, P. D. C. King, P. Hofmann1, Phys. Rev. Lett. 112, 257401 (2014) V.M.Pereira V. M. Pereira, J. M. B. L. dos Santos, A. H. C. Neto, Phys. Rev. B 77, (2008) 115109 Robinson08 J. P. Robinson, H. Schomerus, L. Oroszlány, and V. I. Fal’ko, Phys. Rev. Lett. 101, 196803 (2008) Yuan10b S. Yuan, H. De Raedt, M. I. Katsnelson, Phys. Rev. B 82, 115448 (2010) Yuan12 S. Yuan, T. O. Wehling, A. I. Lichtenstein, M. I. Katsnelson, Phys. Rev. Lett. 109, 156601 (2012) Lherbier12 A. Lherbier, S. M.-M. Dubois, X. Declerck, Y.-M. Niquet, S. Roche, J.-C. Charlier, Phys. Rev. B 86, 075402 (2012) Roche12 S. Roche, N. Leconte, F. Ortmann, A. Lherbier, D. Soriano, and J.-C. Charlier, Solid States Comm. 152, 1404 (2012) Roche13 A. Cresti, F. Ortmann, T. Louvet, D. Van Tuan, S. Roche, Phys. Rev. Lett. 110, 196601 (2013) Kretinin13 A. Kretinin, G. L. Yu, R. Jalil, Y. Cao, F. Withers, A. Mishchenko, M. I. Katsnelson, K. S. Novoselov, A. K. Geim, F. Guinea, Phys. Rev. B 88, 165427 (2013) Trambly11 G. Trambly de Laissardière, D. Mayou, Modern Physics Lett. B 25, 1019 (2011) Trambly13 G. Trambly de Laissardière, D. Mayou, Phys. Rev. Lett. 111, 146601 (2013) Trambly14 G. Trambly de Laissardière, D. Mayou, Adv. Nat. Sci.: Nanosci. Nanotechnol. 5, 015007 (2014) Zhao15 P.-L. Zhao, S. Yuan, M. I. Katsnelson, H. De Raedt, Phys. Rev. B 92, 045437 (2015) Koshino06 M. Koshino, T. Ando, Phys. Rev. B 73, 245403 (2006) Adam08 S. Adam, S. Das Sarma, Phys. Rev. B 77, 115436 (2008) Koshino09 M. Koshino, New J. Phys. 11, 095010 (2009) Yuan10 S. Yuan, H. De Raedt, M. I. Katsnelson, Phys. Rev. B 82, 235409 (2010) Gonzalez10 J. W. González, H. Santos, M. Pacheco, L. Chico, L. Brey, Phys. Rev. B 81, 195406 (2010) Ferreira11 A. Ferreira, J. Viana-Gomes, J. Nilsson, E. R. Mucciolo, N. M. R. Peres, A. H. Castro Neto, Phys. Rev. B 83, 165402 (2011) VanTuan16 D. Van Tuan, S. Roche, Phys. Rev. B 93, 041403 (2016) Nair12 R. R. Nair, M. Sepioni, I.-L. Tsai, O. Lehtinen, J. Keinonen, A. V. Krasheninnikov, T. Thomson, A. K. Geim, and I. V. Grigorieva, Nature Phys. 8, 199 (2012). Scopel16 W. L. Scopel, W. S. Paz, J. C. C. Freitas, Sol. Stat. Comm. 240, 5 (2016) Snyman07 I. Snyman, C. W. J. Beenakker, Phys. Rev. B 75, 045322 (2007) Trambly12 G. Trambly de Laissardière, D. Mayou, L. Magaud, Phys. Rev. B 86, 125413 (2012) Trambly10 G. Trambly de Laissardière, D. Mayou, L. Magaud, Nano Lett. 10, 804 (2010) Trambly16 G. Trambly de Laissardière, O. F. Namarvar, D. Mayou, L. Magaud, Phys. Rev. B 93, 235135 (2016) Trambly14CRAS G. Trambly de Laissardière, D. Mayou, C. R. Physique 15, 70 (2014) Lee P. A. Lee, T. V. Ramakrishnan, Rev. Mod. Phys. 57, 287 (1985) P.W.Brouwer P. W. Brouwer, E. Racine, A. Furusaki, Y. Hatsugai, Y. Morita, and C. Mudry, Phys. Rev. B 66, 014204 (2002) Mayou88 D. Mayou, Europhysics Lett. 6, 549 (1988) Mayou95 D. Mayou and S.N. Khanna, J. Phys. I France 5, 1199 (1995) Roche97 S. Roche and D. Mayou, Phys. Rev. Lett. 79, 2518 (1997) Roche99 S. Roche and D. Mayou Phys. Rev. B 60, 322 (1999) Triozon02 F. Triozon, J. Vidal, R. Mosseri, D. Mayou, Phys. Rev. B 65, 220202 (2002)

http://arxiv.org/abs/1701.07771v3

20170126165305

Consistent Cosmic Bubble Embeddings

hep-th

9.7in 𝒦 𝒢 ℓ_Pℛ𝒬𝒩ℳ𝒲Łℒℋ̋Łℒ𝒦ØΩℬ𝒜 ℓ_P det hγδ̣ϵϕf_∞ρ̊łλκ̨μνσłλ∇̅g̅ #1 #1 #1 #1 #1 #1 #1#1αβ̱χ̧Χδ̣ΔϵεγΓηκ̨łλŁΛμνϕΦφθρ̊σ <cit.><ref>(<ref>) § §.§ §.§.§ /∇∂≡ () [ ]↔ Tr𝒪z̅ Area inside outside total sym

http://arxiv.org/abs/1701.07607v1

20170126080420

Understanding how Software Can Support the Needs of Family Caregivers for Patients with Severe Conditions

cs.CY

Tight Inefficiency Bounds for Perception-Parameterized Affine Congestion Games Pieter Kleer1 Guido Schäfer1,2 December 30, 2023 ============================================================================== empty empty In this paper, we report an extensive analysis that we performed in two scenarios where the care relation between doctor and patients are mediated by the relatives of the patients: Pediatric Palliative Care (PPC) and Nursing Homes (NH). When the patients are children or very old adults in the end of life, the provision of care often involve a family caregiver as the main point of contact for the health service. PPC and NH are characterized by emotional complexity, since incurable diseases expose the family caregivers to heavy careload and human distress. In this paper, we discuss our findings with a novel perspective, focusing on: information, coordination and social challenges that arise by dealing with such contexts; the existing technology as it is appropriated today to cope with them; and what we, as software researchers, can do to develop the right solutions.[ 2016 IEEE. To be published in The 39th International Conference on Software Engineering (ICSE 2017). Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works must be obtained from the IEEE. For more details, see the IEEE Copyright Policy. ] § BACKGROUND AND OBJECTIVES Pediatric Palliative Care (the end-of-life way of care for children with incurable diseases <cit.>) and care for elderly people in Nursing Homes are two areas of healthcare characterized by complex social and emotional challenges, in addition to medical ones <cit.>. Although the patients and diseases are very different, the two scenarios present important similarities: * Patients are typically affected by a chronic condition. This is always the case in Pediatric Palliative Care (PPC), but also Nursing Homes (NHs), due to continuous budget cuts, have been focusing more on care for persons affected by severe conditions (this is the case for Italy, where we performed our studies). Cases of people leaving a nursing home because their condition improved are a minority. For this reason, in both contexts, the treatments mainly focus on maintaining quality of life. * These care scenarios are characterized by a mediated relation between care professionals and patients where not the patient, but the family caregiver (typically the parent in PPC and the child in NH) is the person that interacts with the care structures and takes decisions. This means that the healthcare institutions take charge of both the patients and their families. * Patients are restricted to live permanently in the same building until the end-of-life (this is obvious for NH but often the case also for PPC due to the illness). An important difference is that in PPC the family also administers the care while in NH the patient is in charge of the NH staff and the family caregiver is mainly for support. In both scenarios, adults find themselves thrown into uncharted territory, managing a situation that they have never experienced before. To make things more emotionally challenging, the transition is often sudden (also in nursing homes, where many admissions come from hospitals), and may provoke tensions within the family, marks the start of a progressive health deterioration[On a personal note by the authors, this area is very emotionally draining for researchers as well <cit.>]. The relevant literature in this broad area comes from different disciplines. Healthcare studies show that patients with severe conditions are looked after by two typologies of caregivers: formal (health professional) and informal (family) caregivers <cit.>. They are co-producers of care, and their collaboration and mutual trust are essential in the care of the patients <cit.>. However, several studies highlight gaps in communication between formal and informal caregivers, revealing that often the family members have confusion and unanswered questions about the life expectancy of their relatives <cit.>. Healtcare models such as continuity of care focus on integration between caregivers to provide a coherent, transparent and predictable care service <cit.>. They support the contribution of all caregivers engaged in the care, by enhancing coordination, and focusing on the needs of the patients and their family <cit.>. This model stresses the need to work on technologies to facilitate the dynamics among all caregivers for information continuity (the need of proper and coherent information), management continuity (the need for clear protocols) and relational continuity (the need of safe relations and human support) <cit.>. Most of the existing technology studies (<cit.>) focus on solutions that foster coordination and information exchange issues. However, there is an emerging demand for technologies that help informal caregivers in both care and emotional concerns. Indeed, families caregivers are especially affected by above average burnout, depression, and stress <cit.> The recognized lack in suitable technological solutions for supporting informal caregivers is a call for actions for software researchers <cit.>. In this paper we describe the results of analyses performed over the past two years to understand which role can software applications play in helping people cope with the challenges that these contexts present. We aim in particular at understanding i) which technologies are used today by the caregivers, why, and how effective they are, and ii) how can - existing or novel - software applications better address their needs. As we will see there is space both for novel use of existing applications as well as new applications, whose requirements were not obvious to us in the beginning (and we try to focus more on these aspects). We start by describing our analysis method and then report on our findings and recommendations. § METHOD To understand the needs we carried out several studies in two different contexts in northern Italy. We based our studies mainly on qualitative methodologies, although in the NH case we also developed a data warehouse to analyze populations and processes to the extent allowed by taking information from healthcare IT system, which are very detailed in NHs. In PPC, where patients are at home as long as possible, we studied the dynamics between formal and informal caregivers of a PPC network <cit.>. We interviewed 15 families, and performed observations in the houses of three families. Data have been collected from July 2015 to March 2016 (by only one researcher, due to the sensitivity of the context). A second set of studies focused on six NHs to understand the issues and needs related to family caregivers. NHs have a larger population and we had access to a large number of subjects. The visits were conducted in the fall of 2015 and in the spring/summer of 2016, and all attended by at least three researchers, to collect different perspectives and reduce the chances of biases <cit.>. Specifically, we adopted the following research methods: [i)] * we carried in situ observations in all the contexts, to grasp the organizational and social dynamics that occur among and between family caregivers and care professionals, as well as the communication practices that take places among all the subjects involved, by also creating moments of informal discussion on the emerging issues with our informants <cit.>; * we interviewed the caregivers - formal and informal - to focus on their emotional discomfort <cit.>, and on the - technological - solutions they adopt to cope with their tasks and communication needs; * we involved several formal caregivers in some focus groups to have a deeper understanding of their perspective. C[1]>=#1X § FINDINGS The analysis of the gathered data show that there are four main areas of problems where technology can be of help (See Table <ref>). 1. Communication with the care professionals: this emerged as a major issue in both PPC and NH. In PPC, formal and informal interactions (e.g., cute photos of and information on treatment) travel on the same channel, which is typically Whatsapp. Whatsapp enhances collaboration between formal and informal caregivers, allowing real-time exchange of clinical documents (such as discharge letters and tests results) and quick remote medical consultations. Usually, the mother sends a photo or a video that shows the exacerbation of a medical condition to the members of the PPC unit by asking what to do. While this has many positive aspects (chat software is free, easy, fits into the natural daily behavior and everybody uses it), it also creates a problem in terms of lack of traceability and monitoring, unclear management of privacy, as well as communication overload (chats happen frequently and at any time) which results in the risk of losing important messages. In NH the interaction is by phone or F2F. The same problem of overload exists here, but in NH they complement much bigger problems which are i) lack of trust in the abilities and willingness of NH staff to provide care, and ii) belief that the loved one may be mistreated, due to news of criminal behaviors in NH that is sometimes reported in the national news. Furthermore, the family also feels a lack of clear and timely information. The interesting, and for us unexpected aspect in NH is that the staff, due to the interaction overload and frustrating feeling of lack of trust, are extremely supportive of any system that provides transparency into the life in a NH. Notice that, while the interaction problems with a given family tend to reduce over time, most NH (as we understood from the warehouse data) have a turnover ranging from 20 to 40% per year. This means that there are always new families to cope with. Furthermore, we learned that the staff interacts differently with the families based on their classification of "personas": with some family member they are more open and direct, with others there are more careful in the information they reveal, because of the perceived risk of over-reactions. Finally, an important finding was that NHs already have an information system which they populate in great detail, every day for every resident. So most of the information needed to provide information and transparency is there, though not in the form that can be understood by relatives (and it may not always be wise to reveal them automatically). 2. Interaction within families: PPC and NH both create very strong tensions within the family, mostly related to different emotional reactions to the problem or to disagreement in how to handle it. For example, in NH the children of the resident sometimes disagree on the choice of taking the parent to a NH, on who should go visit and on who foots the bills. We also observed frustration by family caregivers who visit more often towards those who come less often. The technology used to involve the family more in this case is again chat software, used to both inform the whole family on the situation but in part also as a tool to make relatives feel a bit "guilty" because they are not visiting as much. 3. Social support for the Family. The transition to care for a relative in chronic condition is always very painful. In addition, this transition often brings with it a social isolation because of the need (or desire) to spend time with the loved ones, but also because it can become difficult to spend time with people that do not understand what you are going through. Social support is known as a useful method for coping with traumatic situations. In PPC, family caregivers rely on Facebook groups to connect with other parents who experience the same situation from allover the world, allowing for peer-to-peer conversations to find social support, and to receive useful suggestions. However, the specificity of each illness (which in many cases is some form of rare disease) makes it difficult to find people who are living an experience similar to yours. In NH the problems are more "standardized" but the family caregivers are often relatively old themselves and do not use technology beyond, sometimes, email and chat. 4. Education and Managing Expectations. A huge source of problems and misunderstanding between family and professionals is the lack of knowledge and wrong assumptions on i) how the patient's health will evolve and ii) what the healthcare system can do about it. Very often family believes the action of the professionals should be care or rehabilitation, but this is often impossible due to the medical conditions of the person or, in the NH case, to lack of staff for performing, for example, what would be a complex physical rehab program. The problem of erroneous expectations is manifested by the fact that often the patient is not aware that their situation is permanent, even in the NH case. In this case the technology used today is essentially web browsing and searching for information, but this is sometimes the cause of the problem which is indeed fostered by the use of diverse and inconsistent sources on the Internet. For example, in NH, because there are so many "types" of NH in different countries with very different population, one may find information on the internet that does not apply to the NH at hand, but mistakenly believe it does. The same is true for many aspects of care (such as prescriptions of medicine). § OPPORTUNITIES FOR TECHNOLOGY DEVELOPMENT AND ADOPTION In this section we summarize opportunities for novel technology (or usages of existing technology that fit the problems at hand) for each problem category. We focus on what we found more interesting and surprising and omit discussions on security, privacy, data integration, usability, and other concerns the reader may expect. In family-staff interactions, by looking at the NH scenario it becomes apparent that a portal that allows relatives to view the status and activities of the relatives is both feasible and useful. It is feasible because NH staff already fills detailed information on the residents in an IT system, for internal reasons. This means that much of the information is already there. It is feasible also because the staff does want more transparency. And it is useful as relatives requested such information (and indeed they do so today, by phone). Three key requirements emerged from the analysis: The staff segments the relatives into “personas” that react to news in different ways and with whom today they use different communication strategies, and so the software must support this. Information also needs to be classified according to the level of approval required before sending it to the relatives: some information can be sent to all relatives automatically (e.g., the menu of the day, the wake up time, etc), some information requires explicit prior approval that it is ok to send, and other information needs to be edited/rewritten to avoid unnecessary concerns (The latter case also depends on the personas, and it may be different for new or “experienced” relatives). Because the relative might ask for clarifications, it is important that each staff member can have easy access to exactly what the relative has seen in the portal. An additional observation that emerged is that NHs today do not really collect information about subjective wellbeing (of residents and relatives) while it would the important to do so given that quality of life is a key aspect of care. In PPC, the opportunity lies more in taking the instant messaging paradigm and (semi-automatically) extracting messages related to coordination and administration of care. Ad hoc applications and a portal like in NH may also be proposed but it is unclear that they would be adopted, because the PPC care network is wide and ad hoc applications become effective if everybody uses them. For interactions within the family, an opportunity that emerged is the obvious extension of the portal above, where the entire family can be given access to. But what appeared even more strongly is the need to involve the family members beyond the family caregiver using the instruments they already use. For example, grandchildren of residents can be involved by pushing “involving” images or information to chat (as we experimented with telegram bots for telegram users) or Instagram, as well as add events and visit schedule to a calendar. In those PPC networks where a dedicated app is not be adopted for the reason stated above, a way to easily map whatsapp exchanges into calendars would already be beneficial. Opportunities for social support and education are instead more in terms of reusing existing technology but with better aggregation of content and people. For example, PPC would benefit for a single place that contains a set of forums, one for each rare disease, so that parents know where to go. Similarly for NH adults would benefits from illness-specific forums as well as forums related to NHs in their region, both for support but also to compare care practices and manage their expectations. All this can be integrated into a same portal and app, though the challenges here are in terms of content organization. Table <ref> summarizes the common points for each scenario. In summary, there are several directions in which we as software researchers and engineers can contribute to make a difference in this difficult and stressful context, essentially by enabling easy access to personalized information that provide transparency into care processes and information relevant to the physical and care conditions of our loved one. Verifiability: Our studies are based on a total of 35 interviews, 40 days of observation, 4 focus groups, a warehouse with data on over 4000 subjects, and document analysis of processes and health records. The work has received three ethical approvals (as available at: http://bit.ly/2eRFte8). Our data collection can be verified through a formal consultation (since data are sensitive, the consultation must be allowed by a formal permission of our ethical committees). § ACKNOWLEDGMENT This work is supported by the Trentino project "Collegamenti" that is being funded by the Province of Trento (l.p. n.6-December 13rd 1999). This project has also received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 690962. -12cm IEEEtran

http://arxiv.org/abs/1701.07974v5

20170127084919

Reinforced stochastic gradient descent for deep neural network learning

cs.LG

physhuang@gmail.com RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan taro.toyoizumi@brain.riken.jp RIKEN Brain Science Institute, Wako-shi, Saitama 351-0198, Japan Stochastic gradient descent (SGD) is a standard optimization method to minimize a training error with respect to network parameters in modern neural network learning. However, it typically suffers from proliferation of saddle points in the high-dimensional parameter space. Therefore, it is highly desirable to design an efficient algorithm to escape from these saddle points and reach a parameter region of better generalization capabilities. Here, we propose a simple extension of SGD, namely reinforced SGD, which simply adds previous first-order gradients in a stochastic manner with a probability that increases with learning time. As verified in a simple synthetic dataset, this method significantly accelerates learning compared with the original SGD. Surprisingly, it dramatically reduces over-fitting effects, even compared with state-of-the-art adaptive learning algorithm—Adam. For a benchmark handwritten digits dataset, the learning performance is comparable to Adam, yet with an extra advantage of requiring one-fold less computer memory. The reinforced SGD is also compared with SGD with fixed or adaptive momentum parameter and Nesterov's momentum, which shows that the proposed framework is able to reach a similar generalization accuracy with less computational costs. Overall, our method introduces stochastic memory into gradients, which plays an important role in understanding how gradient-based training algorithms can work and its relationship with generalization abilities of deep networks. Reinforced stochastic gradient descent for deep neural network learning Taro Toyoizumi December 30, 2023 ======================================================================= § INTRODUCTION Multilayer neural networks have achieved state-of-the-art performances in image recognition <cit.>, speech recognition, and even natural language processing <cit.>. This impressive success is based on a simple powerful stochastic gradient descent (SGD) algorithm <cit.>, and its variants. This algorithm estimates gradients of an error function based on mini-batches of an entire dataset. Gradient noise caused by mini-batches helps exploration of parameter space to some extent. However, the parameter space is highly non-convex for a typical deep network training, and finding a good path for SGD to improve generalization ability of deep neural networks is thus challenging <cit.>. As found in standard spin glass models of neural networks <cit.>, a non-convex error surface is accompanied by exponentially many local minima, which hides the (isolated) global minima and thus makes any local search algorithms easily get trapped. In addition, the error surface structure of deep networks might behave similarly to random Gaussian error surface <cit.>, which demonstrates that critical points (defined as zero-gradient points) of high error have a large number of negative eigenvalues of the corresponding Hessian matrix. Consistent with this theoretical study, empirical studies on deep network training <cit.> showed that SGD is slowed down by a proliferation of saddle points with many negative curvatures and even plateaus (eigenvalues close to zero in many directions). The prevalence of saddle points poses an obstacle to attain better generalization properties for a deep network, especially for SGD based on first-order optimization, while second-order optimization relying on Hessian-vector products is more computationally expensive <cit.>. The second order method that relies on positive-definite curvature approximations, can not follow directions with negative curvature, and is easily trapped by saddle points <cit.>. In this paper, we show a heuristic strategy to overcome the plateaus problem for SGD learning. We call this strategy reinforced SGD (R-SGD), which provides a new effective strategy to use the gradient information, i.e., to update one network parameter, an instantaneous gradient is reinforced by (accumulated) previous gradients with an increasing reinforcement probability that grows with learning time steps. In other words, the reinforcement may be turned off, and then only the instantaneous gradient is used for learning. This kind of stochastic reinforcement enhances the exploration of parameter space. The excellent performance of R-SGD is verified first on training a toy fully-connected deep network model to learn a simple non-linear mapping generated by a two-layer feedforward network, and then on a benchmark handwritten digits dataset <cit.>, in comparison to both vanilla backpropagation (BackProp) <cit.> and state-of-the-art Adam algorithm <cit.>. In the benchmark dataset, we also clarify the performance difference between R-SGD and SGD with fixed or adaptive momentum parameter <cit.> and Nesterov's momentum <cit.>. § FULLY-CONNECTED DEEP NETWORKS We consider a toy deep network model with L layers of fully-connected feedforward architecture. Each layer has n^k neurons (so-called width of layer k). We define the input as n^1-dimensional vector 𝐯, and the weight matrix 𝐖^k specifies the symmetric connections between layer k and layer k-1. The symmetry means that the same connections are used to backpropagate the error during training. A bias parameter can also be incorporated into the weight matrix by assuming an additional constant input. The output at the final layer is expressed as: 𝐲=f_L(𝐖^Lf_L-1(𝐖^L-1⋯ f_2(𝐖^2𝐯))), where f_k(·) is an element-wise sigmoid function for neurons at layer k, defined as f(x)=1/1+e^-x, unless otherwise specified (e.g., ReLU activation funtion). The network is trained to learn the target mapping generated randomly as {𝐯^m,𝐲_*^m}_m=1^M, where the input is generated from a standard normal distribution with zero mean and unit variance, and the target label y_* is generated according to the non-linear mapping y_*=f(𝐖_g𝐯), in which each entry of the data-generating matrix 𝐖_g follows independently a standard normal distribution as well. The deep network is trained to learn this non-linear mapping (continuous target labels) from a set of examples. We generate a total of 2M examples, in which the first M examples are used for training and the last M examples are used for testing to evaluate the generalization ability of the learned model. In simulations, we use deep network architecture of L=4 layers to learn the target non-linear mapping, in which the network is thus specified by n^1-n^2-n^3-n^4, with n^1 indicating the dimension of the input data and n^L the dimension of the output. We use this simple toy setting to test our idea first, and then the idea is further verified in the handwritten digits dataset. § BACKPROPAGATION AND ITS VARIANTS We first introduce the vanilla BackProp <cit.> for training the deep network defined in Sec. <ref>. We use quadratic loss (error) function defined as E=1/2ϵ^ Tϵ, where ^ T denotes a vector (matrix) transpose operation, and ϵ defines the difference between the target and actual outputs as ϵ=𝐲_*-𝐲. To backpropagate the error, we also define two associated quantities: one is the state of neurons at k-th layer defined by 𝐬^k (e.g., 𝐬^L=𝐲, 𝐬^1=𝐯), and the other is the weighted-sum input to neurons at k-th layer defined by 𝐡^k≡𝐖^k𝐬^k-1. Accordingly, we define two related gradient vectors: δ^k ≡∂ E/∂𝐬^k, κ^k ≡∂ E/∂𝐡^k, which will be used to derive the propagation equation based on the chain rule. It is straightforward to derive κ^L=-ϵ∘𝐲', where ∘ indicates the element-wise multiplication, and 𝐲' is the derivative of the non-linear transfer function with respect to its argument. By applying the chain rule, we obtain the weight update equation for the top layer as Δ𝐖^L=-ηκ^L(𝐬^L-1)^ T, where η is the learning rate, and the remaining part is the gradient information, which indicates how a small perturbation to the weight affects the change of the error computed at the top (output) layer. To update the weight parameters at lower layers, we first derive the propagating equations for gradient vectors as follows: δ^k =(𝐖^k+1)^ Tκ^k+1, κ^k =δ^k∘(𝐟_k)', where k≤ L-1. Using the above backpropagation equation, the weight at lower layers is updated as: Δ𝐖^k=-ηκ^k(𝐬^k-1)^ T, where k≤ L-1. The neural state used to update the weight parameters comes from a forward pass from the input vector to the output vector at the top layer. A forward pass combined with a backward propagation of the error forms the vanilla BackProp widely used in training deep networks given the labeled data <cit.>. To improve the training efficiency, one usually divides the entire large dataset into a set of mini-batches, each of which is used to get the average gradients across the examples within that mini-batch. One epoch corresponds to a sweep of the full dataset. The learning time is thus measured in units of epochs. For one epoch, the weight is actually updated for M/B times (B is the size of a mini-batch). This process is usually termed SGD. Here, we briefly introduce two kinds of SGD with momentum techniques. The first one is the SGD with momentum (SGDM). The learning equation is revised as ν_t =ρ_tν_t-1+𝐠_t, Δ𝐖_t =-ην_t, where 𝐠_t≡∇_𝐖E(𝐖_t-1) denotes the gradient estimated from the average over examples within the current mini-batch, and ρ_t is the momentum parameter, which can be either prefixed (ρ_t=ρ in the classical SGDM) or varied over learning steps (t). The second one is Nesterov's accelerated gradient (NAG) <cit.>, which first implements a partial update to 𝐖_t, and then uses the updated 𝐖_t to evaluate gradients, i.e., ν_t =ρ_tν_t-1-η𝐠'_t, Δ𝐖_t =ν_t, where 𝐠'_t≡∇_𝐖E(𝐖_t-1+ρ_tν_t-1). Note that the partial update takes an extra computational cost of T_maxM/B|𝐖|, where T_max denotes the maximal number of epochs, and |𝐖| denotes the total amount of network parameters. § REINFORCED STOCHASTIC GRADIENT DESCENT In the above vanilla BackProp, only current gradients are used to update the weight matrix. Therefore in a non-convex optimization, the backpropation gets easily stalled by the plateaus or saddle points on the error surface, and it is hard to escape from these regions. During training, gradients may be very noisy with large fluctuations. If update directions along some weight components are stochastically allowed to accumulate the history of gradient information, while other directions still follow the current gradients, the learning performance may be boosted. This stochastic rule of turning on accumulation may help SGD to handle the uncertainty of updating the weights. We will test this idea in the following deep neural network learning. To enable SGD to use previous gradient information, we define a stochastic process for updating modified gradient 𝐠̃_t used at each learning step as follows: (𝐠̃_t)_i← (𝐠_t)_i, with prob. 1-Γ(t), (𝐠_t)_i+(𝐠̃_t-1)_i, with prob. Γ(t). where the stochastic reinforcement is independently applied to each weight component, and 𝐠̃_t-1 contains information about the history of the evolving gradients, and the current gradients are reinforced by the previous accumulated gradients with a reinforcement probability defined by Γ(t). The stochastic rule in Eq. (<ref>) is a switch-like (all-or-none) event; its smooth averaged version given 𝐠̃_t-1 is 𝔼[𝐠̃_t|𝐠̃_t-1]=𝐠_t+Γ(t)𝐠̃_t-1, where Γ(t) is equivalent to ρ_t in SGDM with time-dependent momentum parameter. Using adaptive momentum parameter is important in boosting the learning performance of SGDM. However, the switch-like property is able to reach a better or equivalent test accuracy with fewer training steps. Comparisons will be made on the handwritten digits dataset. Eq. (<ref>) is a very simple way to re-use the previous gradient information, and forms the key component of R-SGD. We first choose Γ(t)=1-γ^t, where γ=γ_0e^-λ t_ ep. γ_0 and λ are prefixed constants, and t_ ep refers to the learning time in units of epochs. Γ(t) can be rewritten as 1-e^-t/τ_ R, where τ_ R≡-1/lnγ_0-λ t_ ep setting the time scale of the dynamics of the reinforcement probability. γ_0 is usually fixed to a value very close to one, and λ takes a small value. Therefore the current gradient has an increasing probability to be reinforced by the previous gradients, and retains its instantaneous value otherwise. This reinforcement probability is not the unique choice, e.g., 1-a_0/(t+1)^b_0 (a_0 and b_0 are constants) is also a candidate (discussed in Sec. <ref>). We show a typical trace of the reinforcement probability and γ in Fig. <ref> (a), and will test effects of hyper-parameters (γ_0,λ) on training dynamics in Sec. <ref>. Note that by setting (γ_0,λ)=(1,0), one recovers the vanilla BackProp. In all simulations, we use an exponentially-decaying learning rate η_t_ ep=η_t_ ep-1β^t_ ep, where η_0=0.8 and β=0.999, with a minimal learning rate of 0.02, unless otherwise specified. R-SGD is summarized in algorithm <ref>. The gradient used in R-SGD may be the accumulated one (over an unfixed or stochastic number of consecutive steps), which contains short or long-term memory of previous gradient information (Fig. <ref> (b)). Hence, the step-size is a sum of previous gradients over a memory length ℒ, which follows a probability P_t(ℒ) decaying with ℒ (Fig. <ref> (b)). This stochastic process is summarized by Δ𝐖_t^ R-SGD =-η_t∑_l=t-ℒ^t𝐠_l, ℒ ∼ P_t(ℒ)=(1-Γ(t-ℒ))exp(∑_l=t-ℒ+1^tlnΓ(l)), where the prefactor indicates the probability that the memory is cleaned before accumulation. This probability is normalized since ∑_ℒ=0^tP_t(ℒ)=1-∏_l=0^tΓ(l)=1, where Γ(0)=0. In the SGDM, the momentum term is deterministically added to the learning step size with coefficient ρ_t (Eq. (<ref>)). Unfolding this process, we obtain the step-size as Δ𝐖_t^ SGDM=-η_t∑_l=1^t(∏_l'=l+1:l≠ t^tρ_l'+δ_l,t)𝐠_l, where δ_l,t is a Kronecker delta function, and each gradient is weighted by a value smaller than one. To show the efficiency of R-SGD, we also compare its performance with that of a state-of-the-art stochastic optimization algorithm, namely adaptive moment estimation (Adam) <cit.>. Adam performs a running average of gradients and their second raw moments, which are used to adaptively change the learning step-size. We use heuristic parameters of Adam given in <cit.>, except that η_0=0.01 with the lowest value set to 0.001. Large η_0 as we use in R-SGD does not work in our simulations for Adam. § RESULTS AND DISCUSSION §.§ Learning performance in simple synthetic dataset We first test our idea in the simple synthetic dataset described in Sec. <ref>. We use a 4-layer deep network architecture as 100-400-200-10. Training examples are divided into mini-batches of size B=100. In simulations, we use the parameters (γ_0,λ)=(0.9995,0.0001), unless otherwise specified. Although the chosen parameters are not optimal to achieve the best performance, we still observe the outstanding performance of R-SGD. In Fig. <ref> (a), we compare the vanilla BackProp with R-SGD. We clearly see that the test performance is finally improved at 100-th epoch by a significant amount (about 77.5%). Meanwhile, the training error is also significantly lower than that of BackProp. A salient feature of R-SGD is that, at the intermediate stage, the reinforcement strategy guides SGD to escape from possible plateau regions of high error surrounding saddle points, and finally reach a region of very nice generalization properties. This process is indicated by the temporary peak in both training and test errors for R-SGD. Remarkably, even before or after this peak, there are a few less significant fluctuations in both training and test errors. These fluctuations play an important role in the exploration of the parameter space. Compared to state-of-the-art Adam, R-SGD still improves the final test performance by a significant amount (about 49.1%, see Fig. <ref> (b)). Note that Adam is able to decrease both training and test errors very quickly, but the decrease becomes slow after about 40 epochs. In contrast, R-SGD keeps decreasing both errors by a more significant amount than Adam, despite the presence of slightly significant fluctuations. Another key feature of Fig. <ref> (b) is that, a region in the parameter space with low training error does not generally have low test error. The training error reached by R-SGD is clearly higher than that of Adam, but the network architecture learned by R-SGD has nicer generalization property. This observation is consistent with a recent study of maximizing local entropy in deep networks <cit.>. We then study the effects of reinforcement parameters (γ_0,λ) on the learning performance, as shown in Fig. <ref>. If the exponential decay rate λ is large, R-SGD over-fits the data rapidly at around 17 epochs. This is because, γ decays rapidly from γ_0, and thus a stochastic fluctuation at earlier stages of learning is strongly suppressed, which limits severely the exploration ability of R-SGD in the high-dimensional parameter space. In this case, R-SGD is prone to get stuck by bad regions with poor generalization performances. However, maintaining the identical small decay rate, we find that a larger value of γ_0 leads to a smaller test error (inset of Fig. <ref>). For relatively large values of γ_0, the learning performance is not radically different. We also study the effects of training data size (M) on the learning performances. Clearly, we see from Fig. <ref>, the test error decreases with the training data size as expected. R-SGD outperforms the vanilla BackProp, and even Adam. For the simple toy model, SGDM with fixed momentum parameter could outperform Adam with a careful optimization of momentum parameter (e.g., ρ_t=0.9 ∀ t), but R-SGD still outperforms the classical SGDM (fixed momentum parameter) by about 14.3% when M=1000. By adaptively changing the momentum parameter whose value is the same as the reinforcement probability of R-SGD (i.e., a smooth averaged version of R-SGD), SGDM can reach a similar performance to that of R-SGD. Because the synthetic data in the toy model is relatively simple and the reinforcement probability used in the synthetic data is rapidly saturated to one (Fig. <ref> (a)), it is difficult to show the performance difference between R-SGD and SGDM with the adaptive momentum parameter. Therefore, for MNIST classification task in the next section, we use the reinforcement probability of Γ(t)=1-1/√(t+1), which does not rapidly approach one, and compare R-SGD with different variants of momentum-based SGD. §.§ Learning performance in MNIST dataset Finally, we evaluate the test performance of R-SGD on MNIST dataset. The MNIST handwritten digits dataset contains 60000 training images and an extra 10000 images for testing. Each image is one of ten handwritten digits (0 to 9) with 28×28 pixels. Therefore the input dimension is n^1=784. For simplicity, we choose the network structure as 784-100-200-10. Although the reinforcement probability specified by (γ_0,λ) we used in the synthetic dataset works on the MNIST dataset (see an example in Fig. <ref> (c)), we found that the reinforcement probability Γ(t)=1-1/√(t+1) works better (shortening the training time to reach a lower generalization error) for MNIST dataset, and furthermore, this choice does not saturate the reinforcement probability to one within the explored range of learning time (Fig. <ref>), which offers a nice candidate to demonstrate the performance difference between R-SGD and SGDM. Fig. <ref> shows that R-SGD improves significantly over BackProp, reaching a similar test performance to that of Adam with moderate training data size (M=10K). R-SGD achieves a test error of 0.0535±0.0021, compared with BackProp reaching 0.1047±0.0020, and Adam reaching 0.0535±0.0016 (as in Table <ref>). The test error is averaged over five independent runs (different sets of training and test examples). Note that, as training size increases (e.g., M=15K), the test performance of R-SGD becomes slightly better than that of Adam (Table <ref>). Adaptive methods such as Adam often show faster initial progress, but their performances get quickly trapped by a test error plateau <cit.>. In contrast, as shown in the inset of Fig. <ref>, R-SGD as a non-adaptive method is able to reach a lower test error by taking a few more epochs (note that Adam needs more computer memory to store the uncentered variance of the gradients). As shown in Fig. <ref> (a), SGDM with adaptive momentum parameter reaches a higher test error than R-SGD, which confirms that the switch-like event plays an important role in guiding R-SGD to a good region. In R-SGD, the reinforcement along some weight components is turned off with a finite probability, then along these directions, only the current gradients are used (the same as those used in BackProp). But the reinforcement may be turned on along these directions once again during training. In contrast, for SGDM, the momentum term with adaptive momentum parameter is always applied to update all weight components during training. This mechanism difference leads to different test performances observed in Fig. <ref> (a). The training dataset may change the error surface in practice. It seems that the performance of R-SGD is robust to this change (Table <ref>). NAG with adaptive momentum parameter ρ_t=Γ(t) learns quickly but gets trapped by a slightly higher test error. In addition, a single epoch in NAG takes an extra computational cost of M/B|𝐖| due to the partial update. For fixed-momentum-parameter SGDM (ρ_t does not change over time, unlike the reinforcement probability in R-SGD), the learning performance gets worse (Fig. <ref> (b)). When applied to a deep network with ReLU activation and cross-entropy as the objective function, R-SGD still shows competitive performance (Fig. <ref> (c)). We also use bilinear interpolation method to qualitatively analyze the test error surface of the three algorithms (BackProp, R-SGD and Adam). Using the bilinear interpolation method <cit.>, one can visualize the error surface in 3D subspace spanning four high-dimensional weight configurations. These four weight configurations defined as {𝐖_i}_i=1^4 are chosen either from one learning trajectory or from solutions obtained starting from four different random initializations. Based on these four configurations, the error function is varied as a function of a new constructed weight matrix specified by 𝐖=β(α𝐖_1+(1-α)𝐖_2))+(1-β)(α𝐖_3+(1-α)𝐖_4), where α∈[0,1] and β∈[0,1] are two control parameters. Results are shown in Fig. <ref> for one trajectory interpolation. BackProp decreases the test error slowly, and finally reaches a plateau and get stuck there. Adam decreases the error very quickly, and reaches a more apparent and lower plateau than that of BackProp. Remarkably, R-SGD first decreases the error quickly, which is followed by a plateau. The plateau is then passed and finally R-SGD reaches a region with less apparent flatness. Interpolation results of four different solutions are shown in Fig. <ref>. Clearly, BackProp can get stuck by different solutions of different qualities, depending on initializations. Both Adam and R-SGD can reach solutions of nearly the same quality, despite different initializations. However, the subspace looks sharper for R-SGD than for Adam. This rough analysis by using bilinear interpolation method implies that the high dimensional topology of weight space seen by the three algorithms might be intrinsically different. However, to characterize necessary properties of a good region with nice generalization capabilities requires a theoretical understanding of the entire high dimensional weight space in terms of analyzing some non-local quantity. Establishing a theoretical relationship between learning performances of various SGD algorithms and the intrinsic structure of the error surface is still an extremely challenging task in future studies. § CONCLUDING REMARKS In this paper, we propose a new type of effective strategy to guide SGD to overcome the plateau problem typical in deep neural network learning. This strategy takes into account previous (accumulated) gradient information in a probabilistic way when updating current weight matrix. It introduces a stochastic reinforcement to current gradients, and thus enhances the exploration ability of the original SGD. This strategy is essentially different from an independently random noise added to the gradient during the learning process <cit.>. In fact, we add time-dependent Gaussian noise to gradients during training in our simulations using default hyper-parameters <cit.>, whose performance could not be comparable to that of R-SGD within 100-epochs training (or it requires longer convergence time). A similar reinforcement strategy has been used in a single layer neural network with discrete weights <cit.>, where local fields in a belief propagation equation are reinforced. In our work, we study deep neural networks with continuous weights, and thus the gradients are reinforced. The reinforced belief propagation is conjectured to be related to local entropy maximization <cit.> in discrete neural networks. Whether R-SGD reshapes the original non-convex error surface in some way, such that searching for a parameter region of good-quality is facilitated, remains an interesting open question. We leave this for future work. In the current setting, the learning performance of R-SGD is comparable to (in MNIST) or even better than that of Adam (in the synthetic dataset), which requires one-fold more computer memory to store the uncentered variance of the gradients. The learning step-size of Adam is adaptively changed, which means that the step-size is automatically tuned to be closer to zero when there is greater uncertainty about the direction of the true gradient <cit.>, while R-SGD uses the stochastic reinforcement of the gradient information to deal with this kind of uncertainty, and shows comparable and even better performance. A recent study argued that adaptive learning methods such as Adam generalize worse than SGD or SGDM on CIFAR-10 image classification tasks <cit.>. It is thus very interesting to evaluate the performance of R-SGD on more complicated deep network model and complex datasets, which are left for future systematic studies. R-SGD may be able to avoid vanishing or exploding gradient problem typical in training a very deep network <cit.>, probably thanks to accumulation of gradients used stochastically. In addition, it may take effect in recurrent neural network training <cit.>. In fact, previous gradient information at each step can be weighted before accumulation according to its importance in guiding SGD. This is a very interesting direction for future studies on fundamental properties of R-SGD. We are grateful to the anonymous referee for many constructive comments. H.H. thanks Dr. Alireza Goudarzi for a lunch discussion which later triggered the idea of this work. This work was supported by the program for Brain Mapping by Integrated Neurotechnologies for Disease Studies (Brain/MINDS) from Japan Agency for Medical Research and development, AMED, and by RIKEN Brain Science Institute. 10 Hinton-2012imag Alex Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton. Imagenet classification with deep convolutional neural networks. In P. Bartlett, F.c.n. Pereira, C.j.c. Burges, L. Bottou, and K.q. Weinberger, editors, Advances in Neural Information Processing Systems 25, pages 1106–1114, 2012. DL-2016 Ian Goodfellow, Yoshua Bengio, and Aaron Courville. Deep Learning. MIT Press, Cambridge, MA, 2016. robbins-1951 Herbert Robbins and Sutton Monro. A stochastic approximation method. Ann. Math. Statist., 22:400–407, 1951. Bengio-2010 Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In Yee W. Teh and D. M. Titterington, editors, Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics (AISTATS-10), volume 9, pages 249–256, 2010. Huang-2014 Haiping Huang and Yoshiyuki Kabashima. Origin of the computational hardness for learning with binary synapses. Phys. Rev. E, 90:052813, 2014. Baldassi-2016 Carlo Baldassi, Christian Borgs, Jennifer T. Chayes, Alessandro Ingrosso, Carlo Lucibello, Luca Saglietti, and Riccardo Zecchina. Unreasonable effectiveness of learning neural networks: From accessible states and robust ensembles to basic algorithmic schemes. Proceedings of the National Academy of Sciences, 113(48):E7655–E7662, 2016. Bray-2007 Alan J. Bray and David S. Dean. Statistics of critical points of gaussian fields on large-dimensional spaces. Phys. Rev. Lett., 98:150201, 2007. Fyo-2007 Yan V. Fyodorov and Ian Williams. Replica symmetry breaking condition exposed by random matrix calculation of landscape complexity. Journal of Statistical Physics, 129(5):1081–1116, 2007. Lecun-2014 A. Choromanska, M. Henaff, M. Mathieu, G. Ben Arous, and Y. LeCun. The loss surfaces of multilayer networks. ArXiv e-prints 1412.0233, 2014. Dau-2014 Y. Dauphin, R. Pascanu, C. Gulcehre, K. Cho, S. Ganguli, and Y. Bengio. Identifying and attacking the saddle point problem in high-dimensional non-convex optimization. ArXiv e-prints 1406.2572, 2014. Ge-2015 R. Ge, F. Huang, C. Jin, and Y. Yuan. Escaping From Saddle Points — Online Stochastic Gradient for Tensor Decomposition. ArXiv e-prints 1503.02101, 2015. Ba-2017 Jimmy Ba, Roger Grosse, and James Martens. Distributed Second-Order Optimization using Kronecker-Factored Approximations. ICLR-2017, 2017. Lecun-1998 Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86:2278–2324, 1998. Back-1988 Rumelhart David E., Hinton Geoffrey E., and Williams Ronald J. Learning representations by back-propagating errors. Nature, 323:533–536, 1986. Adam D. Kingma and J. Ba. Adam: A method for stochastic optimization. ArXiv e-prints 1412.6980, 2014. Suts-2013 Ilya Sutskever, James Martens, George E. Dahl, and Geoffrey E. Hinton. On the importance of initialization and momentum in deep learning. In Proceedings of the 30th International Conference on International Conference on Machine Learning - Volume 28, ICML'13, pages 1139–1147, 2013. nag-2013 Y. Nesterov. Introductory lectures on convex optimization: A basic course, volume 87. Springer US, 2004. Stra-2009 Hugo Larochelle, Yoshua Bengio, Jérome Louradour, and Pascal Lamblin. Exploring strategies for training deep neural networks. J. Mach. Learn. Res., 10:1–40, 2009. EGD-2016 P. Chaudhari, A. Choromanska, S. Soatto, Y. LeCun, C. Baldassi, C. Borgs, J. Chayes, L. Sagun, and R. Zecchina. Entropy-SGD: Biasing Gradient Descent Into Wide Valleys. ArXiv e-prints 1611.01838, 2016. wilson-2017 A. C. Wilson, R. Roelofs, M. Stern, N. Srebro, and B. Recht. The Marginal Value of Adaptive Gradient Methods in Machine Learning. ArXiv e-prints 1705.08292, 2017. Im-2016 D. Jiwoong Im, M. Tao, and K. Branson. An Empirical Analysis of Deep Network Loss Surfaces. ArXiv e-prints 1612.04010, 2016. Neel-2015 A. Neelakantan, L. Vilnis, Q. V. Le, I. Sutskever, L. Kaiser, K. Kurach, and J. Martens. Adding gradient noise improves learning for very deep networks. ArXiv e-prints 1511.06807, 2015. Chau-2015 P. Chaudhari and S. Soatto. The effect of gradient noise on the energy landscape of deep networks. ArXiv e-prints 1511.06485, 2015. Zecchina-2006 A. Braunstein and R. Zecchina. Learning by message passing in networks of discrete synapses. Phys. Rev. Lett, 96:030201, 2006. Baldassi-2015 Carlo Baldassi, Alessandro Ingrosso, Carlo Lucibello, Luca Saglietti, and Riccardo Zecchina. Subdominant dense clusters allow for simple learning and high computational performance in neural networks with discrete synapses. Phys. Rev. Lett., 115:128101, 2015. Ben-2012 R. Pascanu, T. Mikolov, and Y. Bengio. On the difficulty of training Recurrent Neural Networks. ArXiv e-prints 1211.5063, 2012.

http://arxiv.org/abs/1701.08134v2

20170127181138

Reopening the Higgs portal for Singlet Scalar Dark Matter

hep-ph

1.2 i.e. e.g. et al. GeV TeV

http://arxiv.org/abs/1701.08217v3

20170127230644

Sun-Earth Lagrange reference for fundamental physics and navigation

gr-qc

Lagrange Reference Frame A. Tartaglia DISAT, Politecnico di Torino and INdAM, Corso Duca degli Abruzzi 24, 10129 Torino, Italy Tel.: +390110907328 Fax: +390110907399 angelo.tartaglia@polito.it E.C. Lorenzini Department of Industrial Engineering, University of Padova, Via Venezia 1, 35131 Padua, Italy enrico.lorenzini@unipd.it D. Lucchesi Istituto di Astrofisica e Planetologia Spaziali - Istituto Nazionale di Astrofisica (IAPS/INAF), Via Fosso del Cavaliere 100, 00133 Tor Vergata, Roma, Italy and Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Tor Vergata david.lucchesi@iaps.inaf.it G. Pucacco Department of Physics, University of Rome Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome, Italy Giuseppe.Pucacco@roma2.infn.it M.L. Ruggiero DISAT, Politecnico di Torino and INFN, Corso Duca degli Abruzzi 24, 10129 Torino, Italy matteo.ruggiero@polito.it P. Valko Department of Physics, Slovak University of Technology, Ilkovičova 3, Bratislava 812 19, Slovakia pavol.valko@stuba.sk How to use the Sun-Earth Lagrange points for fundamental physics and navigation A. Tartaglia E.C. Lorenzini D. Lucchesi G. Pucacco M.L. Ruggiero P. Valko Received: date / Accepted: date ============================================================================================== We illustrate the proposal, nicknamed LAGRANGE, to use spacecraft, located at the Sun-Earth Lagrange points, as a physical reference frame. Performing time of flight measurements of electromagnetic signals traveling on closed paths between the points, we show that it would be possible: a) to refine gravitational time delay knowledge due both to the Sun and the Earth; b) to detect the gravito-magnetic frame dragging of the Sun, so deducing information about the interior of the star; c) to check the possible existence of a galactic gravitomagnetic field, which would imply a revision of the properties of a dark matter halo; d) to set up a relativistic positioning and navigation system at the scale of the inner solar system. The paper presents estimated values for the relevant quantities and discusses the feasibility of the project analyzing the behavior of the space devices close to the Lagrange points. § INTRODUCTION We propose here to use the Lagrangian (L) points of the Sun-Earth system as a physical framework for a number of measurements related to General Relativity (GR) and possible deviations thereof. The same set of L points could furthermore be the basis for a relativistic navigation and positioning system at least at the scale of the inner Solar System. As it is well known the Lagrangian points of a gravitationally bound two-body system are a feature of Newtonian gravity. Unlike General Relativity (GR) Newton's gravity admits analytic solutions for the two-body problem; furthermore, looking for the positions, on the joint orbital plane, where the attraction of both bodies on a negligible mass test particle counterbalances exactly the centrifugal force, one finds five points where such a condition is fulfilled, with an orbital angular velocity coinciding with that of the two main bodies around their common center of mass. The traditional labelling of the five points is L_1, L_2, L_3, L_4, L_5 and the geometry of the system is as sketched in Fig. <ref>. Three points (L_1, L_2, L_3) are saddle points of the effective potential; in other words, the equilibrium there is unstable, however in the case of the Sun/Earth pair the instability is very mild. The remaining two points (L_4 and L_5) are real local minima so the equilibrium there is stable, though corresponding to a shallow potential well. When coming to a relativistic approach, even though we may guess that the situation is marginally or even negligibly different from the Newtonian case, the existence of Lagrangian points is a priori not guaranteed so that the problem needs a careful discussion. Hopefully the final conclusion is indeed that libration points (Lagrange points) still exist also in a relativistic 2 plus 1 body configuration, at least in a range of masses including the Sun/Earth pair, even though no closed solution is available for the position of such points <cit.>. The advantage of the set of the Lagrangian points is that they form a configuration rigidly rotating together with the Earth. This property has already been exploited many times for space missions, such as WMAP <cit.>, the Herschel space observatory <cit.>, Planck <cit.> (all concluded) and now Gaia <cit.>, in L_2; the Deep Space Climate Observatory <cit.>, the Solar and Heliospheric Observatory (SOHO)<cit.> and LISA Pathfinder <cit.>, in L_1. The list is not exhaustive and many more missions are planned directed again to L_1 or L_2. It is also worth mentioning that proposals have been issued to exploit, for fundamental physics, the Lagrangian points of the pair Earth-Moon <cit.>. The stability of the positions with respect to one another and to the Earth makes the Lagrangian points appropriate to work as basis for a physical reference frame at the scale of the inner solar system. Furthermore, considering the size of the polygon having the L's as vertices, we may remark that the time of flight of electromagnetic signals going from one point to another is in the order of some 10 minutes or more; such long time may act as a multiplier for the tiny asymmetries originated by angular momentum effects predicted by GR. The present paper will discuss the possibilities listed above, highlighting the advantages for fundamental physics experiments and for the positioning and guidance of spacecraft out of the terrestrial environment. In particular, we shall nickname LAGRANGE the proposal of exploiting time of flight measurements along a closed path having L points as vertices, in order to take advantage of the asymmetric propagation produced by the angular momentum of the Sun in the case of two counter-rotating electromagnetic signals. The use of one and the same loop will avoid delays due to different geometric paths for the two beams; the cancellation of the purely geometric component of the time of flight will let the above mentioned asymmetry emerge. In Section <ref> we discuss the GR time delay due both to the Sun and to the Earth; then in Subsection <ref>, we specialize the analysis to the case where emitter, central body and receiver are aligned (indeed a special case for LAGRANGE). In the calculation, the contribution of the quadrupole of the central body will be included. In Section <ref> the analysis will concern the more general configuration with a closed contour encompassing a wide area and the purpose will be the measurement of the solar gravitomagnetic effect with an accuracy better than 1%. Section <ref> evaluates the possibilities to retrieve information about a galactic gravito-magnetic field, if it is there. Section <ref> presents a Relativistic Positioning System (RPS) based on a set of emitters of electromagnetic signals located at the mentioned L-points. Since the feasibility of our proposals crucially depends on the possibility of keeping the position of each spacecraft with respect to the corresponding L and its time dependence under control, we shall discuss the issue in Section <ref>. A short conclusion closes the paper. § GRAVITATIONAL TIME DELAY A gravitational field produces a time delay and a deflection (deviation) on the propagation of electromagnetic waves. Presently, we are interested only to describe the effects of the delays in time propagation, because the effects of the deflection on the time propagation are negligible with respect to the leading contributions. The main effect depends on the mass of the central body and is fully explained in terms of the metric developed by Schwarzschild in 1916 <cit.>. Therefore, these delays are related to the gravitoelectric field of GR <cit.>. The first successful measurements having the Sun as a source were obtained by Shapiro and collaborators by means of radar-echoes from Earth to the planets Mercury and Venus <cit.>. Successively, Anderson and collaborators <cit.> repeated the measurement of the delay in the round trip time from the two spacecraft Mariner 6 and 7 orbiting the Sun. Finally, Shapiro <cit.> and Reasenberg <cit.> obtained the most accurate results with this technique by means of a transponder placed on the surface of Mars by the NASA mission Viking. The agreement between the measured delay and its general relativity prediction was around 0.1%. This kind of measurements are quite important because they allow to constrain the PPN parameter γ, which measures the space curvature per unit of mass. Currently, the best measurement of γ has been obtained by the radar tracking of the CASSINI spacecraft during a superior conjunction with the Sun along its cruise to Saturn <cit.>. Bertotti and collaborators obtained γ -1≃ 2× 10^-5. The advantage of the latter measurement relies on the Doppler tracking (not exploited before) and the multi-frequency link (both X-band and Ka-band) that allow for the plasma compensation of the solar corona. This delay, which is now known as the Shapiro time delay, represents the first GR correction to the time propagation of an electromagnetic signal between an emitter and a receiver with respect to the time of propagation that is needed in the flat spacetime of Minkowski. LAGRANGE, with its multi-spacecraft configuration, would allow the measurement of the time delay in the propagation of the electromagnetic signals in several different geometrical configurations. In the same time, it would extend the measurement of the delay not only to the effect previously mentioned, the so-called Shapiro time-delay, but also to the delay produced by the gravitomagnetic field <cit.> of the Sun and/or to that of the Earth. For instance, referring to previous Fig. <ref>, we can consider the propagation of light and the corresponding delay between the two equilateral Lagrangian points L_4 and L_5. In this case, the impact parameter b, the point of closest approach to the Sun, is equal to 0.5 AU, i.e. comparable with the other distances, avoiding the problem connected with the plasma of the solar corona, as well as the additional delay produced by the quadrupole moment of the Sun. Another very interesting geometrical configuration is the one represented by the two collinear points L_1 and L_2. The propagation of signals between these two points would allow, for the first time, a direct measurement (in the field of the Earth) of the overall delay on their propagation, as produced by the combined action of the mass and angular momentum of the Earth plus the additional delay due to its oblateness. Some of the corresponding measurements with LAGRANGE would allow to improve the current limits in gravitational physics by exploiting the present know-how and accuracy in time of flight measurements and with the present state of art in atomic clocks precision and accuracy. Conversely, other effects, in order to emerge from the noise, need an improvement in the current technology of time measurements. The LAGRANGE measurements would be based on the application of null geodesics around a spinning body in the weak field and slow motion limit (WFSM) of GR. In terms of metric, the Kerr metric will be the reference <cit.>, or, to say better, its weak field limit <cit.>, with a non-diagonal component g_0ϕ proportional to the intrinsic angular momentum (spin) J of the central body. §.§ Time delays for a configuration where emitter and receiver are aligned with the delaying object We consider a quasi-Cartesian coordinate system at the post-Newtonian level with origin in the central (deflecting and delaying) body. We consider the propagation in the z=0 plane (coincident with the plane of the ecliptic) and assume that the angular momentum J⃗ of the body is along the z-axis and that this axis is also the symmetry axis of the body (i.e. we assume cylindrical, or axial, symmetry). In particular, we assumed a standard isotropic PN approximation <cit.> where the receiver (or observer) has to be considered positioned along the positive y-axis. Under the above approximations, the line element can be written as: ds^2 = c^2dτ^2 ≃ g_00c^2dt^2 + g_xxdx^2 + g_yydy^2 + g_zzdz^2 + 2g_0xdxdt + 2g_0ydydt, where[Here g_00 represents the time-time component of the metric, while the other terms provide spatial and mixed contributions.] g_00≃ -(1+2U/c^2) g_ij≃(1-2U/c^2)δ_ij g_0x≃ 2GJ/c^2r^3(-y) g_0y≃ 2GJ/c^2r^3(x). In the above expressions, c represents the speed of light, τ the (invariant) proper time, G the Newtonian gravitational constant, J the angular momentum of the central body, r the distance in the reference plane, δ_ij the Kronecker symbol and, finally, U represents the gravitational potential[We considered only the main contribution, that arises from the first even zonal harmonic, with respect to the deviation from the spherical symmetry in the mass distribution of the Earth.] U ≃ -GM/r(1-J_2(R/r)^23(z/r)^2 -1/2), where M, R and J_2 are, respectively, the mass, radius and quadrupole moment of the body. This configuration is particularly interesting when the delays in the propagation of the electromagnetic signal are analysed in order to take care also of the effect in the time propagation produced by the quadrupole moment of the central object, besides the contributions from the gravitoelectric and gravitomagnetic fields of GR. In the case of the propagation of electromagnetic waves we need to impose the condition of null geodesics with the further condition that we restrict to the propagation in the reference plane z=0 with x=b constant and b≪ y. Therefore, Eq. (<ref>) reduces to: 0 ≃ g_00c^2dt^2 + g_yydy^2 + 2g_0ydydt. We can now solve for the coordinate time element dt from Eq. (<ref>) and integrate the final expression from the emitter position at y=-y_1 up to the receiver (or observer) position at y=+y_2 (y_1 and y_2 are positive quantities and we further assume that y_2≃ y_1). For the propagation time Δ t_prop we finally obtain: Δ t_prop≃y_2+y_1/c + 2GM/c^3ln(4y_1y_2/b^2) ±4GJ/c^4b + 2GM/c^3(R/b)^2J_2 + …, where smaller contributions to the time delay have been neglected. The first term in Eq. (<ref>) accounts for the time propagation in the flat spacetime of Special Relativity. The second term represents the contribution from the gravitoelectric field of GR in the weak field approximation: it is the Shapiro time delay. The third contribution arises from the gravitomagnetic field in the same approximation. The ± sign accounts for the chirality of this contribution: it is positive for a propagation of the signals in the same sense of rotation of the central mass, it is negative in the case of the opposite sense for the propagation. Finally, the last term represents the contribution that arises from the oblateness of the central body. The solution provided in Eq. (<ref>) implies that emitter, central body and receiver have the same x and z coordinates (with x=b, z=0) and differ only for the y coordinate (which is negative for the emitter, null for the central body and positive for the receiver). The result obtained in Eq. (<ref>) can be considered as a particular case of two results obtained in previous works <cit.>. In fact, our result coincides with that obtained in <cit.> when that work is restricted to the lensing effect which can be obtained for light propagating in their symmetry plane (coincident with our reference plane) with the transformations γ=0 and β=π/2 in their expressions and with the further condition α=0 or α=π in their final expressions for the delays due to the angular momentum and the quadrupole coefficient (see in particular their section 2).[Here γ (not to be confused with the PPN parameter commonly designated by the same symbol) and β represent two of the Euler angles that define the orientation of their symmetry plane with respect to the lens plane, while α represents the angular position of a generic light ray over the lens plane. ] Conversely our first three terms in Eq. (<ref>) coincide with equations (55), (56) and (57) of <cit.> with the transformation y_1 → -y_1 for their y_1 and the approximation y_2 ≫ b and y_1 ≫ b for their coordinates. By applying the measurements based on the propagation time determined with Eq. (<ref>) to the configuration L_1–Earth–L_2, it will be possible (at least in principle) to obtain a measurement of the Earth's quadrupole coefficient in a way independent from the usual space geodesy techniques based on the inter-satellite tracking — by means of the two twin GRACE satellites <cit.> — and from the precise orbit determination (POD) of laser-ranged satellites in orbit at a relatively high altitude, as in the case of the two LAGEOS <cit.>. Considering that the distance between the two Lagrangian points L_1 and L_2 is y_1+y_2≃ 3×10^9 m, and assuming an impact parameter b of the order of the Earth's radius R_⊕≃ 6.4× 10^6 m, for the propagation time of Eq. (<ref>) we obtain: Δ t_prop≃( 10 s) + (3.6×10^-10s) +(± 3× 10^-17s) +(3.2× 10^-14s)+ …, where the contribution of each term has been highlighted. If we consider a round trip travel for the propagation time, the smaller contribution of the gravitomagnetic field cancels out when we consider the propagation on the same side of the Earth, and the quadrupole effect can be extracted after modelling the Shapiro delay and the larger effect of the propagation time in the flat spacetime of Minkowski. The knowledge of the oblateness of the Earth is particularly important because of its long-term variations in relation to the Earth's internal structure and its mass distribution. In fact, phenomena like the melting of glaciers and ice sheets as well as mass changes in the oceans and in the atmosphere are responsible for variations in the rate of the global mass redistribution with a consequent time dependency in the quadrupole coefficient characterized by annual and interannual variations <cit.>. This kind of measurement can be initiated by Earth, the delaying body, with all the advantages of an Earth based Laboratory equipped with the best time-measuring apparatus to perform the experiment. In particular, optical clocks and lattice clocks based on Sr-atoms have reached outstanding fractional frequency instabilities down at a level of about 2× 10^-16/√(T) or less, with T the integration time <cit.>. For instance, with an integration time of about 10^4 s it is possible to reach a precision in the measurement of the quadrupole coefficient of about δ J_2/J_2≃ 3× 10^-8, comparable with the current best determinations from Earth (with calibrated errors) using the LAGEOS' data, and even better with longer integration times. Considering that the time of flight between L_1 and L_2 is of the order of 10 s, longer integration times imply of course a number of bounces back and forth between the end points of the trajectory. § SOLAR LENSE-THIRRING DRAG The Lense-Thirring effect (LT) or inertial frame dragging by a moving massive body is a feeble effect of GR, first considered by Thirring <cit.> and Lense and Thirring <cit.> in 1918, while studying the influence of rotating masses (in particular a rotating hollow massive spherical shell) on a test particle. LT may also be considered as a manifestation of gravito-magnetism i.e. of that typical component of the GR gravitational interaction resembling the magnetic field of moving charges. So far, LT has been verified experimentally in a limited number of cases. A careful analysis of the orbits of the LAGEOS and LAGEOS 2 satellites, monitored by laser ranging, evidenced the LT drag of the nodes of the orbits with a 10% accuracy <cit.><cit.>.The Gravity Probe B experiment measured the precession induced by the gravitomagnetic field of the Earth on four orbiting gyroscopes, with a 19% accuracy <cit.>. The ongoing LARES experiment (combined with the previous data from the two LAGEOS) has attained a preliminary 5% accuracy <cit.>. With a different technology, the GINGER experiment is under study and preliminary test of the technology. It is based on the use of an array of ring lasers to be located underground at the National Gran Sasso Laboratories in Italy <cit.>. Ring lasers are extremely sensitive rotation measuring devices. Their operating principle is a GR evolution of the old Sagnac effect <cit.>; what is measured in practice are frequency and amplitude of a beat between two stationary counter-propagating light beams in the ring. Rotations, either of kinematical origin or due to the chirality of the gravitational field (gravitomagnetic component), produce a right/left asymmetry of the propagation along the ring. The aim of GINGER is to verify the terrestrial LT within 1% or better. The use of the Sun-Earth Lagrangian frame would allow a measurement of the solar gravitomagnetic field (solar LT), exploiting the Sagnac approach but resorting to time of flight measurements rather than to interference phenomena or beat tones. For our purpose we may start from the external line element of a steadily rotating body in a reference frame where the main mass is at rest and the axes do not rotate with respect to the distant stars (to the quasars). As in the previous section, weak field conditions are assumed, but now, for convenience, we use polar coordinates in space. It is: ds^2=(1-2m/r) c^2 dt_0^2-dr^2/(1-2m/r)-r^2 dθ^2-r^2 (sinθ)^2 dϕ_0^2 +4j/r sin^2θc dt_0 dϕ_0 The quadrupole moment of the main body has been neglected. If M is the mass of the source, it is m=GM/c^2 with the dimension of a length. Similarly, if J is the modulus of the angular momentum of the source, it is j=GJ/c^3 with the dimension of a squared length. The index 0 labels the coordinates specific of the non-rotating, asymptotically flat reference frame. In the case of the Sun m_⊙ = 1475 m and j_⊙ = 4.7144× 10^6 m^2. It is convenient to use coordinates apt for a terrestrial (or co-orbiting with the Earth) observer. In practice we need to combine a rotation of the axes at a rate Ω corresponding to the orbital motion of the Earth, together with a boost at the tangential speed of the Earth on its orbit V <cit.>. What holds for the Earth, holds for the Lagrangian points too. Since we are considering free fall the orbital rotation rate is Keplerian, so that: Ω=c√(m/a^3); V=Ω a=c√(m/a) Here a is the radius of the orbit of the Earth (∼ 1.5× 10^11 m), m_⊙ /a ∼ 10^-8, and j_⊙/a^2 ∼ 10^-16. We may now restrict our attention to the orbital plane, so that it is θ=π/2. In the new reference frame and with the new coordinates (see the Appendix for the details) the line element is ds^2 ≃ [1+m/a(1+m/a)(1-2a/r)]c dt^2 - (1+2m/r+4m^2/r^2)dr^2 - [1-am/r^2-m^2/r^2(1-2a/r)]r^2dϕ^2 + 2[2j/ra-√(m/a)-(1-2a/r)(m/a)^3/2]a c dt dϕ The approximation has been kept to the lowest order in j and with reference to the numerical values holding for the Sun. For short we write: g_0ϕ=c [2j/ra-√(m/a)-(1-2a/r)(m/a)^3/2]a The frame is non-inertial and comoving with the laboratory; the origin remains in the center of the Sun.[It should actually be in the barycenter of the Sun-Earth pair, but the difference should be discussed among the perturbations of the spherically symmetric system.] In order to find out the time of flight of an electromagnetic signal along a given path, we may extract the time element from Eq. (<ref>) remembering that it is ds=0. In terms of a generic stationary axially symmetric space-time and referring to general coordinates, we find: c dt=-g_0idx^i±√((g_0idx^i)^2-g_00g_ijdx^i dx^j)/g_00 In order to insure an evolution towards increasing real times, the + sign must be chosen. Then we see that the term containing the square root in the right hand side of Eq. (<ref>) does not change sign when reversing the sense of motion along a given path, whereas the first term in the numerator does. Since we are interested in the asymmetries in the propagation we consider the difference between the right- and left-handed time of flight along the same elementary section of the path; in this way the square root cancels and the other term doubles. Finally we integrate along the whole closed space trajectory and express the result in terms of the proper time of the observer. The total time of flight asymmetry turns out to be <cit.>: c δτ=-2√(g_00)∮g_0i/g_00dx^i §.§ Application to a Lagrangian polygon Casting into Eq. (<ref>) the information extracted from Eq. (<ref>) and preserving the solar weak field approximation, we get: cδτ ≃ -2{1+m/2a(1-2a/r)[1+m/2a(3/2+a/r)]}× ∮(2j/r^2)rdϕ ≃ -4∮j/rdϕ Suppose now that the closed path is a polygon, whose edges are light rays. Of course the corresponding null trajectories will be affected by the gravitational lensing due to the mass of the Sun. However we know that the angular deviation due to the lensing effect is proportional to m_⊙, so that its influence in the calculation of (<ref>) is negligible. In practice we may assume the space trajectories of electromagnetic signals to be straight; the typical equation is simple: b/r=cos(ϕ-Φ) The closest distance from the straight line to the center of the system is b and the azimuth of the closest point is Φ. Suppose for example that a signal, propagating in the ecliptic plane, goes from position A, with coordinates r_A and ϕ_A to the arrival point B with coordinates r_B and ϕ_B. We easily work out the contribution of this stretch to the integral (<ref>): c δτ_AB≃ 4j/b(sin(ϕ_B-Φ)-sin(ϕ_A-Φ)) Let us apply the above result to a triangular loop having L_4, L_2 and L_5 at the corners. The coordinates in the plane of the ecliptic, measuring the angles from the Sun-Earth line, are: L_4 : r_4=a, ϕ_4=π/3 L_2 : r_2=a+a_2,ϕ_2=0 L_5 : r_5=a, ϕ_5=-π/3 It is a_2∼ 1.5× 10^9 m, so that a_2/a∼ 10^-2. The minimum distance between the L_4-L_2 (or L_5-L_2) line and the center of the system is b_24=b_25=(a+a_2)cos(π/6+√(3)/2a_2/a) Numerically: b_24=b_25∼ 1.3× 10^11 m. The angular coordinate of the minimum distance point, on one side or the other, is Φ_24=-Φ_25≃π/6+√(3)/2a_2/a. Coming to the L_4-L_5 line, it is b_45=acosπ/6≃ 7.5× 10^10 and Φ_45=0. Summing up, and considering the full triangle, we have: δτ_245 = 2δτ_52+δτ_45 ≃ 8j√(3)sin√(3)/2a_2/a/c (a+a_2)cos(π/6+√(3)/2a_2/a) -8j/√(3)c a Eq. (<ref>) may be approximated to first order in a_2/a i.e. at the % level: δτ≃ 8√(3)ja_2/c a^2-8j/√(3)c a Finally, casting numbers in, we obtain (in seconds): δτ_245≃ 4.30× 10^-13 The total expected time of flight asymmetry is well within the range of measurability, at least in terrestrial laboratory conditions. The challenge is to measure it in space. §.§ Retrievable information on the interior of the Sun Besides making use of Sun's angular momentum as a source of a LT field for a basic science experiment, i.e. for another precise test of GR, there are other, some even truly practical, reasons for such observations. It is widely believed that the observed differential rotation of the Sun triggers a near-surface layer of rotational shear, known as tachocline, where large-scale dipole magnetic fields are generated by dynamo action, ultimately leading to the 11-year solar cycle of sunspots <cit.>. Crucial to the possible role of the tachocline in the dynamo are its location and depth. While Sun's photosphere can be directly observed and also neutrinos provide some direct information about processes in the core of our star, the tachocline is not directly observable. Until now, all available information about this boundary layer between the radiative interior and the differentially rotating outer convective zone, have been collected via helioseismology observations, mainly using the Solar and Heliospheric Orbiter (SOHO) and the Solar Dynamics Observatory (SDO) probes <cit.>. The estimated location of the shear layer at Sun's equator is (0.693±0.002) R_⊙, i.e. beneath the convection zone base, and with a width of 0.04 R_⊙. Using Sun's density profile based on the Standard Solar Model <cit.>, the tachocline itself should contribute at the level ∼ 0.5 % to the total angular momentum of the Sun, i.e. to the source of the LT field. Although such precision of the solar LT field determination lies at the very edge of the expected LAGRANGE project sensitivity, a periodic low frequency temporal variation of the LT field strength would open another window for Sun interior studies. § RELEVANCE OF THE MEASUREMENT OF A POSSIBLE GALACTIC GRAVITOMAGNETIC FIELD When addressing the effects of rotating massive bodies (Sun) on a local space-time geometry in our planetary system, it is rational to consider also possible analogous effects originating from larger structures dynamics, i.e. from our Galaxy or even more. The main reason is that fields associated with metric tensor components (g_0i or g_0ϕ as used in Eq.s (<ref>) and (<ref>)) might mimic the effects typically associated to the presence of dark matter (DM), i.e. additional centripetal or centrifugal acceleration (a_c ∝ v B_LT) and gravitational lensing (effective refraction index n ∝ 1-A_LT). The gravito-magnetic field potential (A_LT) and field strength (B_LT), rather than metric tensor components, are used for clearer analogy only. The best studied and quantified DM problem is related to the dynamic stability of dwarf and spiral galaxies. In the case of the Milky Way (MW) the distribution of the accounted for luminous mass in stars (∼ 5× 10^10 M_⊙), nonluminous interstellar gas and dust (∼ 5× 10^9 M_⊙), central black hole (∼ 4×10^6 M_⊙)[In the center of our galaxy, there is an extremely dense compact object (Sagittarius A*) most probably consisting of a black hole.] and central bulge (∼4.5×10^9 M_⊙) is not compatible with the observed nearly flat rotation curves [v(r)=const] of stars and gas in the disk <cit.>. The same property of rotation curves for spiral galaxies has been confirmed for star-free, edge regions, via radio emission observations of neutral hydrogen <cit.>. For the MW, the mutual gravitational attraction of stars, central black hole and interstellar dust provide a significant part (nearly all) of the required centripetal force at small distances (up to 5 kpc) from the center, while flat rotation curves at larger distances (above 10 kpc) undeniably point towards some other source of centripetal force <cit.>. The typical approach to address this problem is to postulate the presence of a large galactic halo, extending beyond 30 kpc, consisting of massive nonluminous particles with isothermal spherical distribution. In such models, the stabilizing effect of DM is being contemplated through a static gravitational field (i.e. the g_00 component of the metric tensor) due to the mass of the invisible halo. Depending on the peculiar DM model, the total mass of the MW may be entirely dominated by the dark halo and could reach values ranging from ∼5.2×10^11 M_⊙ <cit.> up to ∼1.5×10^12 M_⊙ <cit.>. Using a naive but straightforward example, a LT field of ∼ 8.9×10^ - 16 s^ - 1 strength, would suffice to provide all necessary centripetal acceleration to account for the motion of the Sun around the MW center with a tangential velocity 220 km/s at 8 kpc radius. According to a realistic MW mass distribution model <cit.>, an additional force component is needed, to account for 30 km/s of the total v_LSR = 220 km/s orbital velocity.[LSR stands for Local Standard of Rest.] A local value of the LT field B_LT∼ 2.2 × 10^ - 16 s^ - 1 would account for this additional centripetal force. Similar values could be deduced for the M31 galaxy <cit.> in which the rotational velocity term associated with the DM scenario provides a linearly growing contribution to the rotation curve with a slope of 1.2 × 10^ - 16 s^ - 1. This result could also be interpreted as the influence of a homogenous LT field, perpendicular to the plane of the disk of the galaxy, with an identical intensity of B_LT∼ 1.2 × 10^ - 16 s^ - 1. The hypothesized LT field strengths are weaker than the current experimental possibilities <cit.> but well within the LAGRANGE project scope. On the contrary, the galactic LT field strengths, calculated from known baryonic mass-velocity data, are typically much weaker: only ∼ 2.6 × 10^ - 22 s^ - 1 for the MW at the Sun distance from the center. Taking into account that none of the recent experiments have been able to detect the physical nature of DM (see overview in <cit.>) and a clear observational evidence of a strong correlation between galactic baryonic content and plateau velocity of the rotation curves (v_p), expressed through the Tully-Fisher relation (M_B ∝ v_p^4) <cit.> for more than 100 rotationally supported galaxies of different masses and morphologies <cit.>, constitutes a justified reason to consider alternatives to standard DM scenarios and assess the possible presence of local LT fields with strength in the 10^ - 16 ÷ 10^ - 20 s^ - 1 range. Overwhelming share of non-baryonic energy-mass density, in coincidence with observed clustering of visible matter over a larger volume in the observable universe, suggests that other potential sources of LT fields, with strengths within the interval of interest, are viable. In that respect, even a residual primordial LT field, originating from the initial singularity and the subsequent fast evolution processes (inflation era) cannot be excluded. Presumed primordial LT fields would imprint on the CMB spectrum in a similar way as DM and would influence large scale structures evolution into characteristic filaments with congregated clusters of galaxies and large voids in between <cit.>. Similar filaments and voids structures and other relevant analogies are commonly observed in solid state systems <cit.>. Filaments of vortex lines of quantum vortices in superfluid helium or magnetic flux bundles in superconducting materials are the best examples. Although such simple analogies cannot guarantee they would have something in common with large scale structures in the universe, the existence of forces inside and among such filaments, originating from the interaction with the bulk of the medium (space-time voids), looks a lot like DM and dark energy (DE) effects. Therefore local LT field search (measurement) might open another window into the DM and DE problem. § RELATIVISTIC POSITIONING The solution adopted for global positioning on Earth or in its vicinity is mainly based on the GPS method and on the GPS, GLONASS, Galileo and other present or future satellites dedicated constellations. Without entering into a discussion of the strengths and weaknesses of that approach, it is easily agreed that it cannot be extended beyond the near terrestrial environment or, at least, that the application to space navigation is an opportunity to reconsider the whole method, especially regarding the way to account for the effects of special and general relativity. An intrinsically relativistic positioning system (RPS) has been proposed and is described in <cit.>. It is based on the local timing of at least four remote independent sources of electromagnetic pulses; the essence of the method is graphically presented in Fig. <ref>. Successive pulses (but they could also be periodic equal phase hypersurfaces) cover space-time by a regular four-dimensional lattice. The world-line of an observer intersects the walls of successive cells of the lattice; the proper time interval measured by the observer between consecutive crossings provides the basic information. Counting the pulses (after identifying the various sources) and applying a simple linear algorithm it is possible to calculate the coordinates (including time) of the receiver in the fiducial reference frame <cit.>. The dimensionless coordinates along the light cone of a source are expressed as the sum of an integer part n_a (the subscript a labels the sources) and of a fractional part X_a. The integer is obtained just counting the successive arrivals of the pulses; the fractional part is given by a simple linear algorithm applied to sequences of arrival times in the proper reference of the observer <cit.>. Projecting the n_a+X_a light cone coordinates onto the axes of the fiducial reference frame finally produces the practical coordinates we are interested in. Of course the sources may be orbiting satellites, but in that case you have to know with the best possible accuracy, the position of each satellite (i.e. its real orbit) while time passes. The situation would be far simpler if the position of the emitter were fixed in the fiducial reference frame. This possibility is implemented in nature if the signals come, for instance, from pulsars: their positions in the sky are indeed fixed or slowly moving at a well known rate; furthermore pulsars are also very good clocks, even better, in the long term, than our atomic clocks. An exercise application of the RPS, using pulsars, is presented in <cit.>. The inconvenience of pulsars is that their pulses are extremely weak so that large antennas are needed and special techniques must be implemented in order to identify and extract the signal from an overwhelming noise. Such troubles can be removed placing artificial "pulsars" in points that keep rigidly their positions in an appropriate reference system. That is indeed the case of the Lagrangian points. L_1, L_2, L_4 and L_5, equipped with emitters of regular pulses would form a very interesting basis for a physical reference frame co-orbiting with the Earth. L_3 has not been considered because it is located in the opposite side of the Sun with respect to the Earth, so being invisible from our planet. An important feature of the system is that the distances between the reference points range between 1.5 million km approximately (from L_1 or L_2 to the Earth) to 150 million km (from the Earth to L_4 or L_5). Such large values dramatically reduce the effect of the geometric dilution which renders GPS (and the other terrestrial positioning systems) useless when extended away from our planet. Of course all Lagrangian points lie in a plane and that is usually the case also for most space missions, but the size of the base and using four L points reduces the problem of geometric dilution within distances of a few AU, excepting limited "wakes" along the lines containing a couple of emitters. The spacecraft carrying the emitter devices could in general not coincide with the corresponding Lagrange point, but would rather orbit around the point on stable (L_4 and L_5) or on halo or weakly unstable Lissajous orbits (L_1 and L_2). The final accuracy of the positioning would depend mainly on the accuracy with which the instantaneous position on the orbit is known; we are discussing this issue in the next section. The other limiting factor for the final result is the quality of the clock used by the receiver: in principle a clock fit for a 10^ - 10s accuracy attains also a centimeter accuracy in determining a travelled distance. § ORBITAL DYNAMICS AROUND THE L POINTS The transmitting/transponding spacecraft of LAGRANGE will be placed in orbits around the collinear and triangular Lagrangian points. The motion around the Lagrangian points of a small body is described by the classical solution of the Restricted Three Body Problem (RTBP) <cit.>. The assumptions that underline the RTBP are that the orbiting body has a negligible mass with respect to the two primaries, in our case Earth and Sun, and the primaries follow circular orbits. We further restrict our preliminary assessments to the planar case of the RTBP. It is well known that motion around a collinear point is always (weakly) unstable while the stability of motion about the triangular points depends on the mass ratio of the two primaries. In the case of the Sun-Earth system (and any other combination of mass ratios in the solar system) the motion is stable. LAGRANGE will require spacecraft placed in orbit around L_1, L_2, L_4 and L_5. The planar motion in the proximity of the Sun-Earth L_4 and L_5 (where a linear approximation holds) follow orbits that have two frequency components: a faster motion with a 1-year period and a much slower one with a 156-year period. The planar motion around a collinear point is characterized by a couple of complex-conjugate eigenvalues associated with a "stable" manifold and a couple of positive real roots associated with an unstable manifold that produces a divergent motion of the spacecraft. Initial conditions can be chosen in such a way as to excite only the complex conjugate eigenvalues in order to minimize the instability that however will eventually be excited by non-gravitational perturbations or earlier on by imperfect initial conditions. For the Sun-Earth system, the period of the "weakly-unstable" planar orbit is half a year. Including also the out-of-plane component one gets a second but very close period. The different frequencies give rise to orbits that describe quasi-periodic Lissajous figures or, for sufficiently large amplitude, inclined "halo" orbits. Fig. <ref> shows the velocity components of a 10-km wide orbit in the stable manifold of L_1. Due to the small amplitude, an orbit with the same amplitude around L_2 has very similar velocity values. Also for the motion around a triangular point, initial conditions can be chosen so as to excite one eigen-frequency, e.g., the fast one (see Fig. <ref> showing the velocity components of a 10-km wide orbit at L_4). In the case of the Sun-Earth system and within the linear approximation, the resulting in-plane and out-of-plane frequencies are practically equal (i.e., with a period of one year) thereby producing a quasi-periodic orbit. The orbital motion around the Lagrange points will cause a change of the length of the radio-wave path that will overlap with the change associated with the chirality typical of the Lense-Thirring effect. The question is how to discern one from the other. The flight time of the radio-waves to cover the L_2-L_4-L_5-L_2 circuit is about 2000 s. During that time the position change of a realistically-sized orbit around a Lagrangian point is greater than the change of the radio-wave circuit path associated with the Lense-Thirring effect. If the motion around the Lagrangian points were to be purely periodic and with a period that is a fraction of the total duration of the signal data taking (i.e., in order to cover a number of orbital cycles), then this Keplerian-type motion could be resolved from the measured data by frequency analysis. The question that remains to be addressed is how to remove the quasi-periodic, gravity-related components associated with the motion of the spacecraft around the Lagrangian points from the signal, in other words, how to distinguish the (gravity-related) relativistic signal from the non-relativistic secular drift. Both motions on the stable manifold of the collinear points and around the triangular points can be reconstructed with good accuracy by properly taking into account non-linear effects <cit.>. Analytic series expansions are obtained in the case of the spatial circular RTBP so to include also out-of-plane motion. Semi-analytical solutions can be further implemented when including more general features like the eccentricity of the primaries <cit.>. In both approaches, the quality of the prediction of the time evolution of small amplitude orbits is determined by the order N of the perturbation expansions. The relative error is given by the Nth power of a perturbative parameter proportional to the amplitude. One avenue worth exploring for removing the "Keplerian drift" may also hinge on the different behaviors of the relativistic and non-relativistic secular or quasi-secular terms: the Lense-Thirring term grows steadily with time while the Keplerian drift manifests itself as a growth of the orbital amplitude about the Lagrangian points, e.g., either the secular drift of the orbit around L_1 and L_2 (the orbit slowly spiraling out) or in the case of a triangular point a residual component of the low-frequency term for the orbit around L_4 and L_5. In addition to the above points, one should also consider the tracking accuracy in order to separate the relativistic secular signature from the classical effects on the orbit of a spacecraft over the measurement time. The higher possible accuracy is obtained by integrating the Doppler measurements over short arcs. The figure of merit of Doppler measurements in the time domain is well described by means of the Allan deviation σ_y <cit.>. For instance, considering an Allan deviation of about 3× 10^-15 (with a reasonable integration time of about 1000 s), already reached in the case of the tracking of the CASSINI spacecraft <cit.>, the accuracy in the range-rate measurements is about c×σ_y ∼ 9×10^-4 mm/s, that corresponds to an error in the position of a spacecraft of about 1.8 mm on the ∼ 2000 s time span of a single measurement. This error is about 18 times larger than the precision required on the knowledge of the length of the radiowave circuit (i.e., on the relative position of the satellites) to be compatible with a differential time measurement of 4.3×10^-13 s. However, by considering n of such short arcs, while both the Lense-Thirring effect to be measured and the knowledge of the relative position between the satellites grow by the same factor n, the overall orbit determination accuracy remains at the 1.8 mm level, making the measurement possible over a time span of about 7.5 days. By improving the Allan deviation by a factor of three, i.e. σ_y ∼ 1×10^-15, which is possible by current technology, the measurement of the Lense-Thirring effect can be obtained on an overall time span less than 1 day. § CONCLUSION We have illustrated the proposal of using the system of the Lagrange points of the Sun-Earth system for various experiments and applications. The possibility to measure relativistic time delays both from the Sun and from the Earth has been discussed and the worked out numerical values show that the measurements would be within the range of possibilities offered by current technologies; the experiment would also lend the opportunity to determine the size of the contribution of the quadrupole moment, J_2, both of the Sun and of the Earth. Another proposal we have put forth is the measurement of the inertial frame dragging (Lense-Thirring effect) caused by the angular momentum of the Sun. The technique to be exploited is molded on the Sagnac effect, determining the time of flight asymmetry along a closed path whose edges are the L points, travelled in opposite directions by electromagnetic signals. We have seen that using, for instance, L_2, L_4 and L_5, the time of flight difference would be in the order of a few 10^-13 s, again within the feasibility range of existing technologies. The direct detection of the LT effect of the Sun, besides adding a measurement of a gravito-magnetic phenomenon per se to the experiments made in circumterrestrial, or planned in terrestrial, environments, would give the possibility to extract interesting information on the interior of the Sun. We have also discussed the relevance of a possible detection of a galactic gravito-magnetic field; its presence could be evidenced by the envisioned L-points configuration at the scale of an AU. A special interest of a possible galactic LT effect is connected with the dark matter halo of the Milky Way, its consistency and, possibly, angular momentum. Passing to a practical application of the L-points set, we have presented and commented a relativistic positioning system at the scale of the full orbit of the Earth. Once more the configuration of the system, its stability in time and its being tied to the orbital motion of the Earth, lend the opportunity of building a positioning and navigation system that could profitably be used by all future space missions, at least in the inner solar system. Of course all the above is possible provided one can know the actual position of each spacecraft with respect to its L-point and keep track of it in time. We have also discussed this fundamental issue and we have seen that a measurement strategy based on continuous data acquisition during few days runs would permit to extract the information on the time of flight asymmetry with the required accuracy. Once LAGRANGE would have been deployed, there would then indeed be many more opportunities it could offer for fundamental physics, depending on the equipment one would be able to put on board the spacecraft. It is just the case to mention the possibility to detect gravitational waves (GW). The size of the experimental setup would indeed be adequate. An option would be to exploit signals exchanged between the L-points adopting a zero-area Sagnac interferometer <cit.>. Furthermore, considering again the size and adopting this time a wide area configuration of the light paths (as the triangle L_2-L_4-L_5), we should also remember that GW's do carry angular momentum also. The response of the system would strongly depend on the relative orientation, but in principle a GW impinging orthogonally on the ecliptic plane, should superpose a transient asymmetry of the times of flight on the continuous signal due to the solar (and galactic) LT-drag. Summing up, the idea of using a set of the Lagrangian points of the Sun-Earth system (from two, to four at a time) and measuring the flight times of electromagnetic signals exchanged between spacecraft located in the L-points, turns out to be in the range of existing technologies and is appealing. It would be very fruitful for fundamental physics experiments related to tests of GR and possible deviations from it, giving also information concerning the Sun, the Earth and the Milky Way. To pass from proposal to reality an undoubtedly huge effort is required to set up the missions needed to carry and locate the spacecraft at the L-points (which could be done progressively, performing different experiments gradually while the stations are launched); to properly equip them; then to control the system and perform the measurements. We think it could be rewarding to try. 99 Perdomo2016 O. M. Perdomo, Existence and stability of Lagrangian points in the relativistic restricted three body problem, arXive:1601.00924v1 (2016) Bennett2013 C. L. Bennett et al., Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results, The Astrophysical Journal Supplement Series, 208, 20B (2013) ESA2013 ESA, Herschel, Herschel Brochure, ESA (2013) Adam2016 R. Adam and et al., Planck 2015 results - I. Overview of products and scientific results, Astronomy and Astrophysics, 594, 1-38 (2016) ESA2017 ESA, Gaia overview, published on line on the site of ESA (2017) NASA2017 NASA, Deep Space Climate Observatory, published on line on the site of NASA (2017) Baldwin2015 E. Baldwin, B. Fleck and D. Müller, Two decades of observing the Sun, European Space Agency Bulletin, 163, 1-23 (2015) ESA2015 ESA, LISA PATHFINDER - First Steps to observing Gravitational Waves from Space, ESA Brochure, BR-323, 1-16 (2015) Battista2015 E. Battista et al., Earth-moon Lagrangian points as a test bed for general relativity and effective field theories of gravity, Physical Review D, 92, 064045 (2015) Schwarzschild1916 K. Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsberichte der Koeniglich Preussischen Akademie der Wissenschaften, Berlin, Preussischen Akademie der Wissenschaften, 189-196 (1916) Thorne1988 K. S. Thorne, Gravitomagnetism, jets in quasars, and the Stanford Gyroscope Experiment, Ed.s J.D. Fairbank, B.S. Deaver Jr, C.W.F. Everitt and P.F.M ichelson, Near Zero: New Frontiers of Physics, W. H. Freeman and Company, New York, 573-586 (1988) Shapiro1971 I. I. Shapiro et al., Fourth test of general relativity - new radar result, Physical Review Letters, 26, 1132-1135 (1971) Anderson1975 J. D. Anderson et al., Experimental test of general relativity using time-delay data from Mariner 6 and Mariner 7, The Astrophysical Journal, 200, 221-233 (1975) Shapiro1977 I. I. Shapiro et al., The Viking Relativity Experiment, Journal of Geophysical Research, 82, 4329-4334 (1977) Reasenberg1979 R. D. Reasenberg et al., Viking relativity experiment - Verification of signal retardation by solar gravity, The Astrophysical Journal Letters, 234, 219-221 (1979) Bertotti2003 B. Bertotti, L. Iess and P. Tortora, A test of general relativity using radio links with the Cassini spacecraft, Nature, 425, 374-376 (2003) Kerr1963 R. P. Kerr, Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics, Physical Review Letters, 11, 237-238 (1963) Ohanian2013 H. C. Ohanian and R. Ruffini, Gravitation and Spacetime, Cambridge University Press (2013) Will1993 C. M. Will, Theory and Experiment in Gravitational Physics, Cambridge University Press (1993) Ciufolini2002 I. Ciufolini and F. Ricci, Time delay due to spin and gravitational lensing, Classical and Quantum Gravity, 19, 3863-3874 (2002) Ruggiero2002 M. L. Ruggiero and A. Tartaglia, Gravitomagnetic effects, Il Nuovo Cimento B, 117, 743-768 (2002) Reigber2005 Ch. Reigber et al., An Earth gravity field model complete to degree and order 150 from GRACE: EIGEN-GRACE02S, Journal of Geodynamics, 39, 1-10 (2005) Cheng2013 M. Cheng, B. D. Tapley and J. C. Ries, Deceleration in the Earth’s oblateness, Journal of Geophysical Research, 118, 740-747 (2013) Al-Masoudi2015 Ali Al-Masoudi et al., Noise and instability of an optical lattice clock, Physical Review A, 92, 063814 (2015) Nicholson2015 T. L. Nicholson et al., Systematic evaluation of an atomic clock at 2 × 10^−18 total uncertainty, Nature Communications, 6, 6896 (2015) Thirring1918 H. Thirring, Über die Wirkung rotierender ferner Massen in der Einsteinschen Gravitationstheorie, Physikalische Zeitschrift, 19, 33 (1918) Lense1918 J. Lense and H. Thirring, Über den Einfluß der Eigenrotation der Zentralkörper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie, Physikalische Zeitschrift, 19, 156 (1918) Ciufolini2004 I. Ciufolini and E.C. Pavlis, A confirmation of the general relativistic prediction of the Lense–Thirring effect, Nature, 431, 958-960 (2004) Ciufolini2011 I. Ciufolini et al., Testing gravitational physics with satellite laser ranging, European Physical Journal Plus, 126, 1-19 (2011) Everitt2011 C. W. F. Everitt et al., Gravity Probe B: Final Results of a Space Experiment to Test General Relativity, Physical Review Letters, 106, 221101 (2011) Ciufolini2016 I. Ciufolini et al., A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model, The European Physical Journal C, 76, 120 (2016) Bosi2011 F. Bosi et al., Measuring gravitomagnetic effects by a multi-ring-laser gyroscope, Physical Review D, 84, 122002-1–122002-23 (2011) Divirgilio2014 A. DiVirgilio et al., A ring lasers array for fundamental physics, Comptes Rendus Physique, 15, 866-874 (2014) Tartaglia2017 A. Tartaglia et al., Testing general relativity by means of ring lasers, European Physical Journal Plus, 132, 73 (2017) Sagnac1913 M. G. Sagnac, A ring lasers array for fundamental physics, Comptes Rendus de l'Académie des Sciences, 157, 708-710 (1913) Divirgilio2010 A. DiVirgilio et al., A LASER Gyroscope System to detect the Gravito-Magnetic Effect on Earth, International Journal of Modern Physics D, 19, 2331-2343 (2010) Ruggiero2015 M.L. Ruggiero and A. Tartaglia, A note on the Sagnac effect for matter beams, European Physical Journal Plus, 130, 90 (2015) Hughes2012 D. W. Hughes, R. Rosner and N. O. Weiss, The Solar Tachocline, Cambridge University Press (2012) SOHO2004 ESA, Helio- and Asteroseismology: Towards a Golden Future in Proceedings of the SOHO 14 / GONG 2004 Workshop, Pub. ESA (2004) Christensen2002 J. Christensen-Dalsgaard, Helioseismology, Reviews of Modern Physics, 74, 1073-1129 (2002) Bahcall1989 J. N. Bahcall, Neutrino Astrophysics, Cambridge University Press (1989) Turck2016 S. Turck-Chièze, The Standard Solar Model and beyond, Journal of Physics: Conference Series, 665, 012078 (2016) Binney2008 J. Binney and S. Tremaine, Galactic Dynamics, Princeton University Press (2008) Sofue2001 Y. Sofue and V. Rubin, Rotation Curves of Spiral Galaxies, Annual Review of Astronomy and Astrophysics, 39, 137 (2001) Iocco2015 F. Iocco, M. Pato and G. Bertone, Evidence for dark matter in the inner Milky Way, Nature Physics, 11, 245-248 (2015) Eadie2016 G. M. Eadie and W. E. Harris, Bayesian Mass Estimates of the Milky Way: The Dark and Light Sides of Parameter Assumptions, The Astrophysical Journal, 829, 108 (2016) McMillan2011 P. J. McMillan, Mass models of the Milky Way, Monthly Notices of the Royal Astronomical Society, 414, 2446 (2011) Gerhard2002 O. Gerhard, Mass distribution in our Galaxy, Space Science Reviews, 100, 129-138 (2002) Carignan2006 C. Carignan et al., The Extended H I Rotation Curve and Mass Distribution of M31, The Astrophysical Journal Letters, 641, L109-L112 (2006) Strigari2013 L. E. Strigari, Galactic searches for dark matter, Physics Reports, 531, 1 (2013) Tully1977 R. B. Tully and J. R. Fisher, A new method of determining distances to galaxies, Astronomy and Astrophysics, 54, 661-673 (1977) McGaugh2016 S. S. McGaugh, F. Lelli and J.M. Schombert, Radial Acceleration Relation in Rotationally Supported Galaxies, Physical Review Letters, 117, 201101 (2016) Tully2014 R. B. Tully et al., The Laniakea supercluster of galaxies, Nature, 513, 71 (2014) Volovik2003 G. E. Volovik, The Universe in a Helium Droplet, Clarendon Press (2003) Tartaglia2013 A. Tartaglia, Relativistic space-time positioning: principles and strategies, Acta Futura, 7, 111-124 (2013) Tartaglia2011 A. Tartaglia, M.L. Ruggiero and E. Capolongo, A null frame for spacetime positioning by means of pulsating sources, Advances in Space Research, 47, 645-653 (2011) Ruggiero2011 M. L. Ruggiero, E. Capolongo and A. Tartaglia, Pulsars as Celestial Beacons to Detect the Motion of the Earth, International Journal of Modern Physics D, 20, 1025-1038 (2011) Szebehely1967 V. Szebehely, Theory of Orbits - The Restricted Problem of Three Bodies, Academic Press (1967) Richardson1980 D. L. Richardson, Analytic construction of periodic orbits about the collinear points, Celestial Mechanics and Dynamical Astronomy, 22, 241-253 (1980) Celletti2015 A. Celletti, G. Pucacco and D. Stella, Lissajous and Halo orbits in the restricted three-body problem, Journal of Nonlinear Science, 25, 343-370 (2015) Paez2015 R. Páez and U. Locatelli, Trojan dynamics well approximated by a new Hamiltonian normal form, Monthly Notices of the Royal Academy of Sciences, 453, 2177-2188 (2015) Paez2016 R. Páez, U. Locatelli and C. Efthymiopoulos, New Hamiltonian expansions adapted to the Trojan problem, Celestial Mechanics and Dynamical Astronomy, 126, 519-541 (2015) Allan1988 D. W. Allan et al., Standard Terminology for Fundamental Frequency and Time Metrology, in Proceedings of the 42nd Annual Frequency Control Symposium, Baltimore, MD, June 1-4, 419-425 (1988) Abbate2002 S. F. Abbate and et al., The Cassini Gravitational wave experiment, in SPIE Annual Meeting, Hawaii (2002) Sun1996 K.-S. Sun et al., Sagnac Interferometer for Gravitational-Wave Detection, Physical Review Letters, 76, 3053-3056 (1996)

http://arxiv.org/abs/1701.07934v2

20170127034109

G-Warm inflation

gr-qc

http://arxiv.org/abs/1701.07454v1

20170125191623

Unifying microscopic and continuum treatments of van der Waals and Casimir interactions

cond-mat.mes-hall

Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4–6, 14195, Berlin, Germany Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4–6, 14195, Berlin, Germany Physics and Materials Science Research Unit, University of Luxembourg, L-1511 Luxembourg Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA We present an approach for computing long-range van der Waals (vdW) interactions between complex molecular systems and arbitrarily shaped macroscopic bodies, melding atomistic treatments of electronic fluctuations based on density functional theory in the former, with continuum descriptions of strongly shape-dependent electromagnetic fields in the latter, thus capturing many-body and multiple scattering effects to all orders. Such a theory is especially important when considering vdW interactions at mesoscopic scales, i.e. between molecules and structured surfaces with features on the scale of molecular sizes, in which case the finite sizes, complex shapes, and resulting nonlocal electronic excitations of molecules are strongly influenced by electromagnetic retardation and wave effects that depend crucially on the shapes of surrounding macroscopic bodies. We show that these effects together can modify vdW interactions by orders of magnitude compared to previous treatments based on Casimir–Polder or non-retarded approximations, which are valid only at macroscopically large or atomic-scale separations, respectively. Unifying microscopic and continuum treatments of van der Waals and Casimir interactions Alejandro W. Rodriguez December 30, 2023 ======================================================================================= Van der Waals (vdW) interactions play an essential role in non-covalent phenomena throughout biology, chemistry, and condensed-matter physics <cit.>. It has long been known that vdW interactions among a system of polarizable atoms are not pairwise-additive but instead strongly depend on geometric and material properties <cit.>. However, only recently developed theoretical methods have made it possible to account for short-range quantum interactions in addition to long-range many-body screening in molecular ensembles <cit.>, demonstrating that nonlocal many-body effects cannot be captured by simple, pairwise-additive descriptions; these calculations typically neglect electromagnetic retardation effects in molecular systems. Simultaneously, recent theoretical and experimental work has characterized dipolar Casimir–Polder interactions between macroscopic metallic or dielectric objects and atoms, molecules, or Bose–Einstein condensates, further extending to nonzero temperatures, dynamical situations, and fluctuations in excited states (as in so-called Rydberg atoms) <cit.>. Yet, while theoretical treatments have thus far accounted for the full electrodynamic response of macroscopic bodies (including retardation), they often treat molecules as point dipoles of some effective bulk permittivities or as collections of noninteracting atomic dipoles, ignoring finite size and other many-body electromagnetic effects. In this paper, motivated by the aforementioned theoretical developments <cit.>, we describe an approach that seamlessly connects atomistic descriptions of large molecules to continuum descriptions of arbitrary macroscopic bodies, characterizing their mutual vdW interactions. In particular, while molecules that are very close to macroscopic objects require atomistic descriptions of the latter, and very large molecules that are far from macroscopic objects require consideration of the contributions of vibrational (in addition to electronic) resonances to the vdW interaction energy, we focus on a mesoscopic regime involving molecular sizes and separations on the order of 1–100 nm, where macroscopic objects can be treated continuously for the purposes of computing electromagnetic field responses (and molecular vibrational resonances can be neglected), yet electromagnetic retardation in conjunction with the finite sizes, nontrivial shapes, and nonlocal electronic correlations of large molecules need to be self-consistently considered to accurately characterize vdW interactions. We specifically investigate interactions among various large molecules and gold surfaces, and show that the effect of nonlocal polarization correlations, encapsulated in the ratio of retarded, many-body (RMB) to pairwise vdW energies (or forces), causes relative deviations from pairwise treatments ranging from 20% to over 3 orders of magnitude; further differences of over an order of magnitude are observed when retardation or finite size effects are neglected. The basis of our work is an equation for the long-range dispersive vdW energy of a system of polarizable bodies, consisting of N microscopic bodies (molecules), labeled by k and described by electric susceptibilities 𝕍_k, and a collection of continuum bodies (an environment) described by a collective, macroscopic susceptibility 𝕍_env, displayed schematically in schematic. The energy of such a collection of bodies can be obtained from the scattering framework <cit.> and written as an integral over imaginary frequency ω = ξ, ℰ = ħ/2π∫_0^∞dξ ln[(𝕋_∞𝕋^-1)], in terms of T-operators that depend on the bodies' susceptibilities as well as on the homogeneous electric Green's function 𝔾_0(ξ, x⃗, x⃗') = (∇⊗∇ - ξ^2/c^2𝕀)e^-ξ |x⃗ - x⃗'|/c/4π |x⃗ - x⃗'| (including retardation) mediating electromagnetic interactions; they encode the scattering properties of the various bodies, and are given by, 𝕋 = (𝕀 - (𝕍 + 𝕍_env) 𝔾_0)^-1 (𝕍 + 𝕍_env), where 𝕍 = ∑_k 𝕍_k; 𝕋_∞ = 𝕋_env∏_k𝕋_k, written in terms of 𝕋_k(env) = (𝕀 - 𝕍_k(env)𝔾_0)^-1𝕍_k(env), encodes the scattering response of the bodies in isolation from one another <cit.>. The energy in (<ref>) treats microscopic and macroscopic bodies on an equal footing, yet the key to its accurate evaluation lies in appropriately representing the degrees of freedom (DOFs) of each entity. Typically, macroscopic environments are well described by continuum susceptibilities 𝕍_env, whose response can be expanded in a basis of incoming and outgoing propagating planewaves, as is typical of the scattering framework <cit.>, or via localized functions, e.g. tetrahedral mesh elements, in brute-force formulations <cit.>. Microscopic bodies, on the other hand, generally require quantum descriptions, but recent work has shown that one can accurately represent their response 𝕍_k = ∑_p α_p |f_p⟩⟨f_p| through bases {|f_p⟩} of either exponentially localized (for insulators) or polynomially delocalized (for metals) functions <cit.>, that accurately capture multipolar interactions among electronic wavefunctions <cit.>. For molecules with finite electronic gaps, the bare response is well described by sums over dipolar ground-state oscillator densities <cit.>, f_p (iξ, x⃗) = (√(2π)σ_p(iξ))^-3exp(- (x⃗ - x⃗_p)^2 /2σ_p^2(iξ)), centered at the locations x⃗_p of each atom p, normalized such that ∫d^3x⃗ f_p = 1, and featuring a Gaussian width that, rather than being phenomenological <cit.>, depends on the atomic polarizability via σ_p (iξ) = (α_p (iξ)/√(72π^3))^1/3 <cit.>. The isotropic atomic polarizabilities α_p are computed via density functional theory, as in recent works <cit.>, which include short-range electrostatic, hybridization, and quantum exchange effects. Since microscopic and macroscopic bodies are assumed to be disjoint, it is more efficient to partition the T-operators into blocks belonging to either molecules or macroscopic objects, allowing a trace over the macroscopic DOFs. The definitions of 𝕋_k(env) imply 𝕋_k(env)^-1 = 𝕍_k(env)^-1 - 𝔾_0, which means that the relevant T-operators can be written as: 𝕋^-1 = [ 𝕋_mol^-1 -𝔾_0; -𝔾_0 𝕋_env^-1 ], 𝕋_∞ = [ 𝕋_mol,∞ 0; 0 𝕋_env ] thus partitioning the molecular and macroscopic (environmental) DOFs. These depend on the molecular T-operators 𝕋_mol^-1 = [ 𝕋_1^-1 -𝔾_0 … -𝔾_0; -𝔾_0 𝕋_2^-1 … -𝔾_0; ⋮ ⋮ ⋱ ⋮; -𝔾_0 -𝔾_0 … 𝕋_N^-1, ] with 𝕋_mol,∞ = ∏_k 𝕋_k, which are in turn partitioned into blocks for each of the N molecular bodies. Given this, the product in the determinant can be evaluated as: (𝕋_∞𝕋^-1) = (𝕋_mol,∞𝕋_mol^-1) (𝕀 - 𝔾_0𝕋_env𝔾_0𝕋_mol) = (𝕋_mol,∞𝕋_mol^-1) (𝕀 - 𝔾_env𝕍) × (𝕀 - 𝔾_0𝕍)^-1 where we used the property 𝔾_0𝕋_k(env) = (𝕀 - 𝔾_0𝕍_k(env))^-1 - 𝕀, and consolidated the scattering properties of the macroscopic bodies into the operator 𝔾_env = 𝔾_0 (𝕀 - 𝕍_env𝔾_0)^-1, which solves [∇×∇× + ξ^2/c^2(𝕀 + 𝕍_env)] 𝔾_env = -ξ^2/c^2𝕀 for an imaginary frequency ω = ξ, thereby encoding the macroscopic DOFs purely in the electric field response; this can be solved via any number of state-of-the-art analytical or numerical classical electrodynamic techniques <cit.>, including but not limited to scattering <cit.> and finite-difference <cit.> methods. Moreover, as the molecules are all disjoint, then (𝕋_mol,∞𝕋_mol^-1) = (𝕀 - 𝔾_0𝕍) ∏_k (𝕀 - 𝔾_0𝕍_k)^-1. Putting all of these identities together yields the following expression for the energy: ℰ = ħ/2π∫_0^∞dξ ln[(𝕄𝕄_∞^-1)] where 𝕄 = 𝕀 - 𝔾_env𝕍 and 𝕄_∞ = ∏_k (𝕀 - 𝔾_0𝕍_k). The above log-determinant formula for the energy includes retardation by construction and accounts for many-body screening and multiple scattering to all orders, thereby ensuring full consideration of finite size, complex shape effects, and collective polarization excitations (see supplement for an alternate equivalent derivation including all of these effects). Moreover, existing sophisticated techniques for modeling molecular and electromagnetic-field responses come together in the operator products 𝔾𝕍_k; when represented in the p-dimensional molecular basis {|f_p⟩}, their block matrix elements are of the form: f_p | 𝔾𝕍_k f_q = α_q∫d^3x⃗ d^3x⃗' f_p (x⃗) 𝔾 (x⃗, x⃗') f_q (x⃗') (see supplement for more details). The equivalence of (<ref>) and (<ref>) captures the seamless unification of ideas and methods previously confined to either atomistic vdW or continuum Casimir physics <cit.>: (<ref>) is similar to prior log-determinant expressions used to describe molecular interactions in vacuum <cit.>, except that 𝔾_0 and 𝔾_env are replaced by nonretarded (quasistatic) vacuum fields 𝔾_0(ξ = 0). We demonstrate the importance of all of these effects by comparing the vdW energies (or forces) obtained from (<ref>) to those from pairwise or other approximate treatments in a number of configurations, consisting of one or two molecules above either a gold half-space or a conical gold tip. While the Green's function of the half-plate can be computed analytically <cit.>, the latter is computed using brute-force numerical techniques <cit.>, with the dielectric function of gold taken from <cit.>. We specifically study a C_500-fullerene of radius 1 nm, a 250 atom 30 nm-long linear carbyne wire, and a 1944 atom-large 2.6 nm × 2.9 nm × 5.5 nm protein associated with human Huntington's disease <cit.>. We further compare the RMB energy from (<ref>) to typical approximations used in the literature: the non-retarded vdW energy ℰ_0, obtained by evaluating (<ref>) with 𝔾_0 and 𝔾_env replaced by their respective quasistatic (ξ = 0) responses, and the Casimir–Polder (CP) energy, ℰ_CP = -ħ/2π∫_0^∞ dξ [α·𝔾_env·(𝕀 + 1/2α·𝔾_env) ] which ignores finite size effects by instead contracting the dressed susceptibility of the molecular ensemble into effective dipolar polarizabilities, α = ⊕_k ∑_p,q⟨ f_p | (𝕀 - 𝕍_k𝔾_0)^-1𝕍_k f_q⟩, thus neglecting higher-order many-body interactions among the different molecules and surfaces. Finally, we define a pairwise interaction energy, ℰ_PWS = -ħ/2π∫_0^∞dξ [∑_k𝕍_k𝔾_env(𝕀 + 1/2∑_l ≠ k𝕍_l𝔾_env) ] which, as in (<ref>), is obtained as a lowest-order expansion of (<ref>) in the scattering; this captures both finite size and retardation but ignores all high-order many-body interactions, with the sums over k, l running over either individual or pairs of molecules. When comparing non-retarded and CP energies to their corresponding pairwise approximations, it suffices to take the quasistatic limit in (<ref>) and to let (𝕀 - 𝕍_k 𝔾_0)^-1→𝕀 for the effective polarizability α in (<ref>), respectively. onemol shows the RMB to pairwise energy ratio ℰ/ℰ_PWS of various configurations (insets), with the fullerene interaction (blue line) found to vary only slightly, attaining a maximum of 1.16 at z ≈ 10 nm; such a small discrepancy stems from the small size and isotropic shape of the fullerene, which limits possible nonlocal correlations in its polarization response. Even weaker relative correlations are observed in the case of the protein (green line), which despite its greater size, number of atoms, and chemical complexity, has a reduced response compared to semi-metallic carbon allotropes <cit.>. To separate the various many-body effects, the inset of onemol compares the RMB power law ∂ln(ℰ)/∂ln(z) of the fullerene interaction to its counterparts when neglecting either finite size or retardation. As expected, both approximations become accurate in their corresponding regimes of validity, with the power law asymptoting to -4 and -1.9 at large and small z, respectively, but fail in the intermediate, mesoscopic regime z ≈ 10 nm. Even larger discrepancies arise in the case of the wire, whose large size and highly anisotropic shape support long-wavelength collective fluctuations. We find that the absolute values of both ℰ_0 (dashed red) and ℰ_CP (dotted red) for the parallel wire overestimate ℰ by factors of 3–7 [onemol(a)] due to the slower decay of the Green's function in the former and lack of screening over the length (or modes) of the wire in the latter. The corresponding energy ratios, however, behave differently in that the effect of screening is strongest in the quasistatic limit, which ends up greatly dampening the many-body excitations relative to pairwise approximations and hence leads to smaller non-retarded energy ratios; in contrast, by construction CP ignores many-body interactions with the surface and thus screening has a much weaker impact relative to the pairwise approximation, leading to larger CP energy ratios. At intermediate z ≈ 10 nm of the order of the wire length, ℰ/ℰ_PWS≈ 30, with the approximate energy ratios deviating by 20%. Similar results are observed in the case of a wire in the perpendicular orientation (black lines), with the pairwise energy leading to slightly larger discrepancies at short separations due to the screening and decreasing impact of atoms farther away from the plate. We now investigate the mutual vdW interactions among two fullerenes or parallel wires oriented either parallel or perpendicular to the gold plate [twomol], focusing primarily on horizontal separations d on the order of molecular sizes, where many-body and finite size effects are strongest. Especially in the case of two wires, the pairwise approximation is shown to fail by many orders of magnitude, with the largest energy ratios occurring at asymptotically large z, i.e. for two molecules in vacuum, while at small z a decreasing ratio reflects the dominant interactions (and screening) of the individual molecules with the plate. The transition and competition between the two limiting behaviors occurs at mesoscopic z ∼ d, and is more clearly visible from the plots in twomol(lower inset), which show ℰ/ℰ_PWS versus d at several values of z. In particular, in the case of parallel wires at mesoscopic z = 10 nm, the competition leads to a nonmonotonic energy ratio, with the maximum of 70 occurring at intermediate d ≈ 3 nm. Comparisons against non-retarded and CP approximations illustrate behaviors similar to the previous case of a single wire, with each under- and over-estimating the ratios by approximately 20% and 30%, respectively. Also shown in twomol(upper inset) is the ratio of the horizontal force F_y = -∂ℰ/∂ y on the wires to its pairwise counterpart, plotted against z for parallel wires at d = 10 nm. Note that by construction, F_y,PWS is independent of z and thus, the system experiences an absolute decrease in the force due to the screening induced by the plate. Comparing F_y,0 and F_y,CP, one finds the surprising result that in contrast to the energy ratio of a single molecule, the screening by the plate makes retardation more rather than less relevant to the force at small z, leading to an ≈ 10% decrease in the force magnitude. Finally, we consider the energy of a molecule above a gold conical tip [cone] by comparing it to that of a gold plate at the same vertical separation z, with 𝔾_env in the former computed through the use of a free, surface-integral Maxwell solver, SCUFF-EM <cit.>. The finite cone has a base diameter of 54 nm and a height of 50 nm from the base to the bottom of a hemispherical tip of diameter 20 nm. The ratio decreases with increasing z, with the energy scaling as z^-6 at asymptotically large separations (not shown) as the finite sizes of the cone and molecule become irrelevant and their interactions dipolar. (Note that a decreasing ratio is expected also for a semi-infinite cone due to its smaller effective area and hence stronger decay compared to a plate.) The ratios at small z for the fullerene and perpendicular wire approach 1 since in this limit, their small horizontal sizes allow the hemispherical tip, which effectively acts like a plate at such short separations, to dominate the interaction. By contrast, the ratio in the case of a parallel wire is non-monotonic, decreasing with at short separations since in this configuration, the wire excitations in the limit z → 0 still sample the finite curvature of the tip and conical slope, leading to a different asymptotic power law. In conclusion, we have demonstrated a unifying approach to computing vdW interactions among molecules and macroscopic bodies that accounts for many-body and multiple-scattering effects to all orders. By comparing against commonplace pairwise, CP, and non-retarded approximations, we quantified the impact of nonlocality, finite size, and retardation on the vdW energy between molecules and either a planar or conical macroscopic body. We have consistently found larger deviations in approximate interactions for long, semi-metallic molecules such as carbyne wires, whereas compact, insulating molecules such as many proteins are reasonably well-described as effectively dilute dielectric particles, allowing these low-order approximations to be more valid. In the future, one might consider more complex macroscopic bodies, such as periodic gratings <cit.> that may elicit larger differences between RMB and approximate interactions even for compact biomolecules, as well as extend these results to incorporate the effects of infrared molecular resonances <cit.>. This material is based upon work supported by the National Science Foundation under Grant No. DMR-1454836 and by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1148900. § APPROXIMATIONS Our calculations above make two related approximations related to Gaussian damping. First, we approximate (<ref>) as ⟨ f_p|𝔾𝕍_k f_q⟩≈α_q∫ d^3x⃗' 𝔾 (ξ, x⃗_p, x⃗') f_p+q (ξ, x⃗') where f_p + q is the same as f_q, but with √(2)σ_p replaced with √(σ_p^2 + σ_q^2), in line with <cit.>; this effectively approximates the Galerkin discretization by a collocation method, with the basis Gaussian functions f_p + q acquiring modified widths. Secondly, for computational convenience, we consider only scattered fields (Green's functions) from dipolar rather than Gaussian sources, which is justified so long as the atoms are several widths (angstroms) away from the macroscopic surfaces. § VDW ENERGY VIA FLUCTUATION–DISSIPATION THEOREM We provide a heuristic derivation of the retarded, many-body (RMB) vdW energy of a general collection of molecular or macroscopic bodies, requiring only that they be disjoint and have no correlations in the polarization response between bodies. Each body k is described by an electric susceptibility 𝕍_k, relating its polarization to the total electric field via |P⃗_k⟩ = 𝕍_k|E⃗⟩; these susceptibilities account for short-range quantum and electrostatic correlations, allowing us to focus solely on long-range electrodynamic correlations when considering the vdW energy. [When neglecting retardation, the charge density and electric potential are more frequently used, so the density response is written in terms of the susceptibility as χ(ω, x⃗, x⃗') = ∑_i,j∂_i∂_j (𝕍_k)_ij (ω, x⃗, x⃗').] Our derivation follows analysis <cit.> based on the fluctuation–dissipation theorem; we note previous demonstrations <cit.> of its equivalence to the summation of ground-state energies of the coupled molecular system <cit.>. Following RosaPRA2011, the assembly of the constituents of all bodies from infinite separation into the final configuration defining 𝕍 = ∑_k𝕍_k can be considered the result of an adiabatic change in the particle–field coupling strength λ∈ [0, 1], in which case the energy of the system can be written as, ℰ = -∫_0^∞ dω ∫_0^1 dλ/λ P⃗|E⃗_ZP, per the Feynman–Hellmann theorem <cit.>. Here, |E⃗⟩ denotes zero-point fluctuating electric fields, |P⃗⟩ = 𝕍|E⃗⟩ is the induced polarization, and _ZP denotes the quantum statistical average over zero-point fluctuations. The connection to scattering problems comes from the well-known fluctuation–dissipation theorem <cit.>, |E⃗⟩⟨E⃗|_ZP = ħ/π𝔾, which expresses field fluctuations in terms of the Green's function 𝔾 of the system. The latter solves Maxwell's equations and can be written in terms of the susceptibility as 𝔾 = (𝕀 - 𝔾_0𝕍)^-1𝔾_0 <cit.>. Exploiting the analyticity of 𝕍 and 𝔾_0 in the complex-ω plane <cit.> and performing a Wick rotation of the energy integral from real to imaginary frequency ω = ξ, leads to a simplified expression for the energy [The entirety of this derivation is identical to that of past work employing the so-called adiabatic connection fluctuation–dissipation (ACFD) framework, but using the vector polarization, tensor electric susceptibility, and tensorial vacuum Green's function instead of the scalar charge density, density response, or Coulomb potential, in order to account for retardation. It is therefore not a coincidence that the log-determinant frequency integrand is so similar in form to past expressions for the vdW energy of a single body.], ℰ = ħ/2π∫_0^∞ dξ ln((𝕀 - 𝔾_0𝕍)) where we rescaled the response functions 𝔾_0 and 𝕍 by the coupling constant λ and integrated over λ. The net interaction energy among the bodies is found by subtracting self-energies of the form in (<ref>), replacing 𝕍 by 𝕍_k separately for each k. This allows recasting the net vdW interaction energy as (<ref>) in terms of scattering operators: 𝕄 = 𝕀 - 𝔾_0𝕍 𝕄_∞ = ∏_k (𝕀 - 𝔾_0𝕍_k). If the system considered consists of N molecular bodies labeled k, and an arbitrary number of macroscopic bodies collectively described by 𝕍_env, then one can write 𝕄 = 𝕀 - 𝔾_0𝕍_env - 𝔾_0𝕍 𝕄_∞ = (𝕀 - 𝔾_0𝕍_env) ∏_k (𝕀 - 𝔾_0𝕍_k), where 𝕍 = ∑_k𝕍_k only runs over the molecular bodies. Multiplying 𝕄𝕄_∞^-1 produces terms of the form (𝕀 - 𝔾_0𝕍_env)^-1𝔾_0≡𝔾_env, which is just the electric field response due to 𝕍_env alone and can be computed via analytical or numerical formulations of continuum electrodynamics. Redefining 𝕄' = 𝕀 - 𝔾_env𝕍 𝕄'_∞ = ∏_k (𝕀 - 𝔾_0𝕍_k), and dropping primes, these new operators can then be substituted into (<ref>) to obtain the net vdW interaction energy among N molecules and a general macroscopic environment.

http://arxiv.org/abs/1701.07522v3

20170125235951

Joint Uplink-Downlink Cell Associations for Interference Networks with Local Connectivity

cs.IT

Joint Uplink-Downlink Cell Associations for Interference Networks with Local Connectivity Manik Singhal and Aly El Gamal ECE Department, Purdue University Email: {msingha,elgamala}@purdue.edu Received: date / Accepted: date ========================================================================================================== We study information theoretic models of interference networks that consist of K Base Station (BS) - Mobile Terminal (MT) pairs. Each BS is connected to the MT carrying the same index as well as L following MTs, where the connectivity parameter L ≥ 1. We fix the value of L and study large networks as K goes to infinity. We assume that each MT can be associated with N_c BSs, and these associations are determined by a cloud-based controller that has a global view of the network. An MT has to be associated with a BS, in order for the BS to transmit its message in the downlink, or decode its message in the uplink. In previous work, the cell associations that maximize the average uplink-downlink per user degrees of freedom (puDoF) were identified for the case when L=1. Further, when only the downlink is considered, the problem was settled for all values of L when we are restricted to use only zero-forcing interference cancellation schemes. In this work, we first propose puDoF inner bounds for arbitrary values of L when only the uplink is considered, and characterize the uplink puDoF value when only zero-forcing schemes are allowed. We then introduce new achievable average uplink-downlink puDoF values. We show that the new scheme is optimal for the range when N_c ≤L/2 when we restrict our attention to zero forcing schemes. Additionally we conjecture that the having unity puDoF during uplink is optimal when N_c ≥ L. § INTRODUCTION The fifth generation of cellular networks is expected to bring new paradigms to wireless communications, that exploit recent technological advancements like cloud computing and cooperative communication (also known as Coordinated Multi-Point or CoMP). In particular, the rising interest in Cloud Radio Access Networks (C-RAN) (see e.g., <cit.>-<cit.>) holds a promise for such new paradigms. These paradigms require new information theoretic frameworks to identify fundamental limits and suggest insights that are backed by rigorous analysis. The focus of this work is to identify associations between cell edge mobile terminals and base stations, that maximize the average rate across both uplink and downlink sessions, while allowing for associating one mobile terminal with more than one base station and using cooperative transmission and reception schemes between base stations in the downlink and uplink sessions, respectively. With a cloud-based controller, optimal decisions for these associations can take into account the whole network topology, with the goal of maximizing a sum rate function. Cloud-based CoMP communication is a promising new technology that could significantly enhance the rates of cell edge users (see <cit.> for an overview of CoMP). In <cit.>, an information theoretic model was studied where cooperation was allowed between transmitters, as well as between receivers (CoMP transmission and reception). CoMP transmission and reception in cellular networks are applicable in the downlink and uplink, respectively. The model in <cit.> assumed that each message can be available at M_t transmitters and can be decoded through M_r received signals. It was shown that full Degrees of Freedom (DoF) can be achieved if M_t+M_r ≥ K+1, where K is the number of transmitter-receiver paris (users) in the network. Recently in <cit.>, alternative frameworks for cooperation in both downlink and uplink were introduced. The new frameworks are based on the concept of message passing between base stations. In the downlink, quantized versions of the analog transmit signals are being shared between base station transmitters. The supporting key idea is that information about multiple messages can be shared from one transmitter to another with the cost of sharing only one whole message (of the order of log P, where P is the transmit power), if we only share information needed to cancel the interference caused by the messages at unintended receivers, through dirty paper coding (see <cit.>). In the uplink, decoded messages are shared from one base station receiver to another, where they are used to cancel interference. It was shown in <cit.> that there is a duality in this framework between schemes that are used in the downlink and those that are used for the uplink, with the clear advantage that the same backhaul infrastructure can be used to support both scenarios. In this work, we first characterize the puDoF of message passing decoding in the uplink of locally connected interference networks when N_c < L/2. We then consider the problem of jointly optimizing the assignment of messages over the backhaul to maximize the average puDoF across both downlink and uplink sessions. We assume that each base station can be associated with N_c mobile terminals, and that an association is needed whenever a mobile terminal's message is used by a base station in either the downlink or the uplink. This usage of a message could be either for delivering the message in downlink, decoding the message in uplink, or for interference cancellation. This problem was first considered in <cit.>, where the average puDoF was characterized for the case when L=1. Here, we consider general values of L, and first show how our new result for the uplink settles the average puDoF problem when N_c ≤L/2. We then tackle this problem when N_c > L, by fixing the uplink scheme to the optimal uplink-only scheme, that associates each mobile terminal with the L+1 base stations connected to it, and characterize the optimal downlink scheme under this constraint. The intuition behind this step is that full DoF is achieved in the uplink when N_c > L through associating each mobile terminal with all L+1 base stations connected to it: any change in that cell association is expected to decrease the uplink puDoF with a factor greater than the gain achieved for the downlink puDoF. When considering this work, it is important to note that the assumptions in a theoretical framework need not reflect directly a practical setting, but are rather used to define a tractable problem whose solution can lead to constructive insights. For example, it was shown in <cit.> that imposing a downlink backhaul constraint where each message can be available at a specified maximum number of transmitters (maximum transmit set size constraint), can lead to solutions that are also useful to solve the more difficult and more relevant to practice problem, where an average transmit set size constraint is used instead of the maximum. Also, in <cit.>, it was shown that solutions obtained for the locally connected network models, that are considered in this work, can be used to obtain solutions for the more practical cellular network models, by viewing the cellular network as a set of interfering locally connected subnetworks and designing a fractional reuse scheme that avoids interference across subnetworks. §.§ Prior Work In <cit.>, the considered problem was studied for Wyner's linear interference networks (channel model was introduced in <cit.>). The optimal message assignment and puDoF value were characterized. Linear networks form the special case of our problem when L=1. Here, all our results are for general values of the connectivity parameter L. Also, in <cit.>, the downlink part of our problem was considered, and the optimal message assignment (cell association) and puDoF value were characterized for general values of the connectivity parameter L, when we restrict our attention to zero-forcing (or interference avoidance) scheme. §.§ Document Organization In Section <ref>, we present the problem setup. In Section <ref>, we discuss previous work on zero-forcing CoMP transmission schemes for the downlink. We then present bounds for the puDoF of the uplink in Section <ref>, and prove the converse in Sections  <ref> and  <ref>. In Section <ref>, we present new achievable puDoF values when the average of the uplink and downlink is considred. We finally present concluding remarks in Section <ref>. § SYSTEM MODEL AND NOTATION For each of the downlink and uplink sessions, we use the standard model for the K-user interference channel with single-antenna transmitters and receivers, Y_i(t) = ∑_j=1^K H_i,j(t) X_j(t) + Z_i(t), where t is the time index, X_j(t) is the transmitted signal of transmitter j, Y_i(t) is the received signal at receiver i, Z_i(t) is the zero mean unit variance Gaussian noise at receiver i, and H_i,j(t) is the channel coefficient from transmitter j to receiver i over time slot t. We remove the time index in the rest of the paper for brevity unless it is needed. The signals Y_i and X_i correspond to the receive and transmit signals at the i^th base station and mobile terminal in the uplink, respectively, and the i^th mobile terminal and base station in the downlink, respectively. For consistency of notation, we will always refer to H_i,j as the channel coefficient between mobile terminal i and base station j. §.§ Channel Model We consider the following locally connected interference network. The mobile terminal with index i is connected to base stations {i,i-1,⋯,i-L}, except the first L mobile terminals, which are connected only to all the base stations with a similar or lower index. More precisely, H_i,j = 0 iff i ∉{j,j+1,⋯,j+L},∀ i,j ∈ [K], and all non-zero channel coefficients are drawn from a continuous joint distribution. Finally, we assume that global channel state information is available at all mobile terminals and base stations. §.§ Cell Association For each i ∈ [K], let C_i ⊆ [K] be the set of base stations, with which mobile terminal i is associated, i.e., those base stations that carry the terminal's message in the downlink and will have its decoded message for the uplink. The transmitters in C_i cooperatively transmit the message (word) W_i to mobile terminal i in the downlink. In the uplink, one of the base station receivers in C_i will decode W_i and pass it to the remaining receivers in the set. We consider a cell association constraint that bounds the cardinality of the set C_i by a number N_c; this constraint is one way to capture a limited backhaul capacity constraint where not all messages can be exchanged over the backhaul. | C_i| ≤ N_c, ∀ i∈ [K]. We would like to stress on the fact that we only allow full messages to be shared over the backhaul. More specifically, splitting messages into parts and sharing them as in <cit.>, or sharing of quantized signals as in <cit.> is not allowed. §.§ Degrees of Freedom Let P be the average transmit power constraint at each transmitter, and let W_i denote the alphabet for message W_i. Then the rates R_i(P) = log| W_i|/n are achievable if the decoding error probabilities of all messages can be simultaneously made arbitrarily small for a large enough coding block length n, and this holds for almost all channel realizations. The degrees of freedom d_i, i∈[K], are defined as d_i=lim_P →∞R_i(P)/log P. The DoF region D is the closure of the set of all achievable DoF tuples. The total number of degrees of freedom (η) is the maximum value of the sum of the achievable degrees of freedom, η=max_ D∑_i ∈ [K] d_i. For a K-user locally connected with connectivity parameter L, we define η(K,L,N_c) as the best achievable η on average taken over both downlink and uplink sessions over all choices of transmit sets satisfying the backhaul load constraint in (<ref>). In order to simplify our analysis, we define the asymptotic per user DoF (puDoF) τ(L,N_c) to measure how η(K,L,N_c) scales with K while all other parameters are fixed, τ(L,N_c) = lim_K→∞η(K,L,N_c)/K. We further define τ_D (L,N_c) and τ_U (L,N_c) as the puDoF when we optimize only for the downlink and uplink session, respectively. §.§ Interference Avoidance Schemes We consider in this work the class of interference avoidance schemes, where every receiver is either active or inactive. An active receiver can observe its desired signal with no interference. In the downlink, we are considering cooperative zero-forcing where a message's interference is cancelled over the air through cooperating transmitters. In the uplink, we are considering message passing decoding where a decoded message is passed through a cooperating receiver to other receivers wishing to remove the message's interference. We add the superscript zf to the puDoF symbol when we impose the constraint that the coding scheme that can be used has to be a zero-forcing scheme. For example, τ_U^zf(L,N_c) denotes the puDoF value when considering only the uplink and impose the restriction to zero-forcing schemes. § PRIOR WORK: DOWNLINK-ONLY SCHEME In <cit.>, the considered setting was studied for only downlink transmission. When restricting our choice of coding scheme to zero-forcing schemes, the puDoF value was characterized as, τ_D^zf(L,N_c)=2N_c/2N_c+L, and the achieving cell association was found to be the following. The network is split into subnetworks; each with consecutive 2N_c+L transmitter-receiver pairs. The last L transmitters in each subnetwork are inactive to avoid inter-subnetwork interference. The zero-forcing scheme aims to deliver 2N_c messages free of interference in each subnetwork, so that the acheived puDoF value is as in (<ref>). In order to do that with a cooperation constraint that limits each message to be available at N_c transmitters, we create two Multiple Input Single Output (MISO) Broadcast Channels (BC) within each subnetwork; each with N_c transmitter-receiver pairs, and ensure that interference across these channels is eliminated. We now discuss the cell association in the first subnetwork, noting that the remaining subnetworks follow an analogous pattern. The first MISO BC consists of the first N_c transmitter-receiver pairs. For each i∈{1,2,⋯,N_c}, message W_i is associated with base stations with indices in the following set, C_i={i,i+1,⋯,N_c}. The second MISO BC consists of the N_c transmitters with indices in the set {N_c+1,N_c+2,⋯,2N_c} and the N_c receivers with indices in the set {N_c+L+1,N_c+L+2,⋯,2N_c+L}. Note that the middle L receivers in each subnetwork are deactivated to eliminate interference between the two MISO BCs. For each i∈{N_c+L+1,N_c+L+2,⋯,2N_c+L}, message W_i is associated with transmitters that have indices in the set C_i={i-L,i-L-1,⋯,N_c+1}. It was shown in <cit.> that the puDoF value of (<ref>) achieved by this scheme is that best achievable value in the downlink using the imposed cooperation constraint and zero-forcing schemes. § UPLINK-ONLY SCHEME We discuss in this section backhaul designs that optimize only the uplink rate, and consider only zero-forcing coding schemes. More precisely, we show that the following theorem holds. The asymptotic puDoF for the uplink when considering message passing schemes is characterised by the following equation: τ_U^zf(L,N_c) = 1 L + 1 ≤ N_c, N_c + 1/L + 2 L/2≤ N_c ≤ L, 2N_c/2N_c + L 1 ≤ N_c ≤L/2 - 1. The cell association that is used to achieve the above is as follows. When N_c ≥ L+1, each mobile terminal is associated with the L + 1 base stations connected to it. The last base station, with index K, in the network decodes the last message and then passes it on to the L other base stations connected to the K^th mobile terminal, eliminating all interference caused by that mobile terminal. Each preceding base station then decodes its message and passes it on to the other base stations, eliminating the interference caused by the message. Thus, one degree of freedom is achieved for each user. In the second range L/2≤ N_c ≤ L, the cell association that is used to achieve a puDoF value of N_c + 1/L + 2 is as follows. The network is split into subnetworks, each with consecutive L+2 transmitter-receiver pairs. In each subnetwork, the last N_c + 1 words are decoded. For each i ∈{L+2, L+1, ⋯, L + 2 - N_c + 1}, message W_i is associated with base stations {i,i-1,⋯,L + 2 - N_c + 1}⊆ C_i. Thus the last N_c words are decoded. The base stations with indices in the set {2,3,⋯,L + 2 - N_c} are inactive as there is interference from the last transmitter in the subnetwork which cannot be eliminated. The first base station decodes W_L+2-N_c. To eliminate the interference caused by the transmitters in the set S = {L + 2 - N_c + 1, L + 2 - N_c + 2, ⋯, L + 1} at the first base station of the subnetwork, we add the first base station to each C_i, ∀ i ∈ S. Now for messages with indices in the set S, we have used β_i = 2 + i - (L + 2 - N_c + 1) associations up to this point; the factor of two comes from the base station resolving W_i and the first base station of the subnetwork. But each transmitter with indices in the set S\{L + 1} also interferes with the subnetwork directly preceding this subnetwork. ∀ i ∈ S\{L + 1}, the message W_i interferes with the bottom L + 1 - i base stations of the preceding subnetwork, which is precisely the number of associations left for the respective message i.e. N_c - β_i = L + 1 - i, thus inter-subnetwork interference can be eliminated at those base stations. In the third range 1 ≤ N_c ≤L/2 - 1, the cell association that is used to achieve the lower bound of 2N_c/2N_c + L is similar to the one described in Section <ref> for the downlink. The network is split into disjoint subnetworks; each with consecutive 2N_c + L transmitter-receiver pairs. For the uplink, we consider two sets of indices for transmitters A_T = {1,2,⋯, N_c} and B_T = {N_c + L + 1,N_c + L + 2⋯, 2N_c + L}, and corresponding sets of receivers A_R = {1,2,⋯, N_c} and B_R = {N_c + 1,N_c + L + 2⋯, 2N_c}. For each i ∈ A_T, the message W_i is associated with the receivers receiving it in A_R. Receiver i decodes W_i and the other associations in C_i exist for eliminating interference. Similarly For each j ∈ B_T, the message W_j is associated with the receivers receiving it in B_R, but now receiver j - L decodes W_j and the other associations in C_j are for eliminating interference. We observe that if we were not restricted to the zero-forcing coding scheme then for the third range, we could achieve 1/2 puDoF using asymptotic interference alignment <cit.>, which is higher than the value achieved by zero-forcing. The next sections complete the proof of Theorem <ref> § CONVERSE PROOF WHEN L/2≤ N_C ≤ L In this section, we provide a converse proof for the second range of (<ref>). More precisely, we show that the following holds. τ_U^zf (L,N_c) = N_c+1/L+2, L/2≤ N_c ≤ L. We start by proving the case when N_c=L, The optimal zero-forcing puDoF for the uplink can be characterised as: τ_U^zf(L,L) = L + 1/L + 2. We begin by dividing the network into subnetworks of L + 2 consecutive transmitters-receiver pairs. We observe that in any subnetwork, if we have N_c + 1=L+1 consecutive active receivers (base stations), then the transmitter connected to all these receivers must be inactive, because a message's interference cannot be canceled at N_c or more receivers. Let Γ_BS be the set of subnetworks where all N_c + 2 receivers are active, and Φ_BS be the set of subnetworks with at most N_c active receivers. Similarly, let Γ_MT and Φ_MT be the subnetworks with N_c+2 active transmitters and at most N_c active transmitters, with respect to order. To be able to achieve a higher puDoF than (<ref>), it must be true that both conditions hold: | Γ_BS|>| Φ_BS| and | Γ_MT|>| Φ_MT|. Now note that for any subnetwork that belongs to Γ_BS, at most N_c transmitters will be active, because the interference caused by any message cannot be canceled at N_c or more receivers. Hence Γ_BS⊆Φ_MT. Further, the same logic applies to conclude that for any subnetwork with N_c+1 active receivers, the number of active transmitters is at most N_c+1, and hence Γ_MT⊆Φ_BS. It follows that if |Γ_BS|>|Φ_BS|, then |Γ_MT| < |Φ_MT|, and hence the statement in (<ref>) is proved. To aid in the next step we define MT-BS pairs (m_i, b_j) as decoding pairs if W_i is decoded at base station j. To prove that τ_U^zf(L,N_c)=N_c + 1/L + 2 when L/2≤ N_c < L, we use the following lemmas : For any zero-forcing scheme, one of the following is true for any two decoding pairs (m_i_1, b_j_1) and (m_i_2, b_j_2): j_2 ∉{i_1, i_1 -1, ⋯, i_1 - L} or j_1 ∉{i_2, i_2 -1, ⋯, i_2 - L}. If the claim were not true, i.e. j_2 ∈{i_1, i_1 -1, ⋯, i_1 - L} and j_1 ∈{i_2, i_2 -1, ⋯, i_2 - L} then W_i_1 and W_i_2 would interfere with one another and could not be decoded using the zero-forcing scheme. This is a consequence of the work done in  <cit.>. For any set L ⊆ [K] of L + 1 consecutive indices, a maximum of N_c mobile terminals with indices in L can be decoded at base stations with indices in L for any zero-forcing scheme. We prove this claim by contradiction. If N_c + 1 or more mobile terminals with indices in L are decoded at base stations with indices in L, then at least one of the mobile terminals would be associated with more than N_c base stations. This violates the constraint in  (<ref>). From Lemma <ref>, we have the following corollary: For any two decoding pairs (m_i_1, b_j_1) and (m_i_2, b_j_2) in a zero-forcing scheme, if i_1 > i_2 then j_1 > j_2 and vice versa. Immediately from Lemma <ref> we have that subnetwork only decoding, i.e. transmissions from a subnetwork are decoded in the same subnetwork, can only decode at most N_c + 1 words in each subnetwork of L+2 consecutive BS-MT pairs. Our proof will be based on the concept that to break the inner bound described in (<ref>), at least one subnetwork of L+2 consecutive MT-BS pairs must have more than N_c + 1 active mobile terminals. And all such subnetworks must borrow base stations from the subnetwork above it to decode words corresponding to its own mobile terminals. This happens because for a consecutive set of L+2 mobile terminals, the only base stations that can help decode their transmissions are the corresponding base stations or the other L base stations with preceding indices that are connected to the set. To aid in the writing we define : α_k = (L+2) × (k-1), here α_k denotes the first index of each subnetwork L_k. In this sense L_k is topologically below L_k-1, i.e. mobile terminals from L_k are connected to some base stations in L_k-1. Additionally MT_i denotes mobile terminal i, and BS_j denotes base station j We use the above lemmas and definitions to define a best case scenario for inter-subnetwork interference. A best case scenario is where the interference from one subnetwork's (e.g., L_k) mobile terminals to another subnetwork's (e.g., L_k-1) base stations is focused on the bottom most base stations. This is defined as the best case scenario because from Lemma <ref>, we know that for L_k-1's own mobile terminals to be decoded in L_k-1 , we need base stations that are indexed outside the range of the interference from the mobile terminals of L_k. We also define that if there exists decoding pairs (m_i, b_j) such that i ≥α_k and j < α_k, i.e. the mobile terminal is in L_k and the base station is in L_k-1, then L_k borrows a base station from L_k-1. Similarly, if there exists certain consecutive base stations in L_k-1 indexed by (α_k - μ, α_k - μ + 1, ⋯α_k - 1 ) such that no words can be decoded here in the zero-forcing scheme due to the cooperation constraint being met in L_k, we say that L_k blocks μ base stations in L_k-1. We introduce two new variables x and δ. Here x defines the number of extra mobile terminals (beyond N_c+1) active in a subnetwork of L+2 consecutive mobile terminals and base stations, and δ defines the number of base stations that L_k borrows from L_k-1 to help decode words from L_k. When L/2≤ N_c ≤ L it follows from the network topology and the defined cooperation constraint that we have 1 ≤ x,δ≤ N_c. We want to show that τ_U^zf(L,N_c) ≤N_c + 1/L + 2 when L > N_c ≥L/2. It follows from the pigeonhole principle that to break this bound, there must be at least one subnetwork (say L_k) where we have N_c + 1 + x mobile terminals active. Now by Lemma <ref>, we have that L_k must borrow at least x base stations from L_k-1, thus x ≤δ≤ N_c. We now consider possible cases for the value of δ. When δ = 1, thus x = 1, so L_k has N_c + 2 active mobile terminals. As L_k is borrowing one base station, say base station j, N_c + 1 words must have been decoded in L_k. By Lemma <ref>, there exists at least one decoding pair (m_i,b_n) where i,n ≥α_k, such that b_n is not connected to the highest indexed active mobile terminal in L_k. Due to the size of the subnetwork, this forces n = α_k. Hence, mobile terminal i's transmission is decoded at the first base station of L_k. By Lemma <ref>, this implies that j ∉{i, i-1, ... i-L}. It follows that the best case scenario occurs when i = α_k + (L + 2 - (N_c + 1)), making j ≤α_k - N_c = α_k-1 + L+2-N_c. Let the number of available base stations left in L_k-1 be θ. As N_c ≥L/2, it follows that θ≤ L + 2 - N_c ≤ N_c + 2. Additionally, due to the borrowed base station, the number of associations allowed for MT_α_k-1 + L+1 (the last mobile terminal in L_k-1) has effectively reduced by one. From Lemma <ref>, we have that a maximum of N_c mobile terminals can be decoded in L_k-1. It follows that either the average number of active mobile terminals over the two subnetworks is still N_c + 1 per subnetwork, or L_k-1 will have to borrow at least one base station from L_k-2. We do not consider the former case, as we just have to restart our argument from L_k-2 because all subnetworks with higher indexes will have an average of N_c +1 active mobile terminals per subnetwork. Hence, we only consider the latter case where L_k-1 borrows at least one base station from L_k-2. As base station α_k-1 is being used in L_k-1, the lowest possible indexed base station that L_k-1 borrows from L_k-2 is base station α_k-2 + (L+2 - N_c). Therefore the argument for L_k-2 borrowing base stations from L_k-3 is exactly the same as the argument shown for L_k-1 borrowing from L_k-2. It follows that this borrowing will continue till either we stop borrowing at some subnetwork L_i, where i < k, or L_1 needs to borrow at least one more base station, which is not possible. If L_i does not borrow from L_i-1, we have that L_i and L_k have at most N_c and N_c + 1 + 1 active mobile terminals, respectively, and all other subnetworks between them have at most N_c + 1 active mobile terminals, resulting in an average of N_c +1 active mobile terminals per subnetwork over these k-i subnetworks. Thus we can discard them as they do not break the inner bound and start the same argument over from L_i-1. If we continue borrowing till L_1, we have that L_1 and L_k have at most N_c and N_c + 1 + 1 active mobile terminals respectively and all other subnetworks have at most N_c + 1 active mobile terminals, resulting that the average number of active mobile terminals over the whole network is N_c + 1 per subnetwork which implies that τ_U^zf(L,N_c) ≤N_c + 1/L + 2. This presents the simplest case for our iterative argument. When δ > 1, we have a similar argument as described in the previous paragraph. By Lemma <ref>, we have that the borrowed base stations in L_k-1 will have to send the associations downwards, i.e. the lowest indexed borrowed base station in L_k-1 will have to be exclusively connected to the lowest indexed active mobile terminal of L_k. As the index of the lowest active mobile terminal in L_k is at most α_k + (L + 2 - (N_c + 1 + x)) - 1, we have that the index of the lowest borrowed base station in L_k-1 is α_k-1 + (L+3 - N_c - x). Therefore the number of available base stations in L_k-1 can be expressed as L + 3 - N_c - x. These available base stations must at least decode N_c + 1 + (1 - x) mobile terminals' transmissions to have an average greater than N_c + 1 active mobile terminals per subnetwork over L_k and L_k-1 without L_k-1 borrowing base stations from L_k-2. This cannot happen when L + 3 - N_c - x < N_c + 1 + 1 - x, which is only possible when N_c > L+1/2. Hence, the condition N_c > L+1/2 implies that L_k-1 has to borrow at least one base station from L_k-2, which presents an iterative argument as the one shown when δ=1. Now we consider the case when L/2≤ N_c ≤L+1/2. By Lemma <ref>, we also have that the maximum number of mobile terminals from L_k-1 decoded in L_k-1's available base stations is N_c. As we only need N_c + 2 - x active mobile terminals decoded to break the inner bound defined, L_k-1 will not have to borrow from L_k-2 when x ≥ 2. At least N_c + 2 - x mobile terminals' transmissions must be decoded in L_k-1, but MT_α_k-1 + L+1 has its associations reduced by δ≥ x. Using MT_α_k-1 + L+1, a maximum of N_c - δ + 1 ≤ N_c + 1 - x transmitted words can be decoded within L_k-1, which will lead us to have L_k-1 borrowing at least one base station from L_k-2. This presents another iterative argument, akin to the one shown above. In order to achieve a case where L_k-1 does not have to borrow base stations from L_k-2, our best case scenario guides us to find the first mobile terminal in L_k-1, which is connected to at most x - 2 base stations that are being borrowed by L_k, but still connected to at least N_c + 2 - x available base stations in L_k-1. Assume that the index of that mobile terminal is α_k-1 + ν. Clearly, ν≤ (L + 2 - N_c - x) + (x - 2) = L - N_c. So in L_k-1 we have N_c + 2 - x active mobile terminals without borrowing from L_k-2, but mobile terminal α_k-1 + ν has already used up all its associations and it is connected to some base stations in L_k-2, specifically at least N_c base stations. Hence, L_k-2 has a maximum L + 2 - N_ c ≤ N_c + 2 base stations available to decode more transmissions, and we need at least N_c + 1 words to be decoded here, which can be done, but this would imply that at least two mobile terminals are associated with N_c base stations. These two mobile terminals are indexed higher than κ, where κ = α_k-2 + L + 1 - (N_c + 1). Hence, L_k-2 blocks at least N_c of the bottom L base stations in L_k-3, and one can see that each further subnetwork blocks at least one base station from the preceding subnetwork for the average number of active mobile terminals per subnetwork to remain above N_c+1. If say L_i does not block any base stations in L_i-1, then L_i can have at most N_c active mobile terminals decoded in L_i. It follows that either L_i borrows from L_i-1 or only has N_c active mobile terminals. If L_i borrows from L_i-1 we have a similar iterative argument as shown above. Otherwise, L_i has only N_c active mobile terminals, making the average number of active mobile terminals through the considered k-i subnetworks N_c + 1 per subnetwork. Hence, each subnetwork continues blocking base stations in the preceding subnetwork and the extra active mobile terminals in the whole network does not scale and is fixed by the constant x, which shows that the average number of active mobile terminals asymptotically approaches N_c + 1 for every subnetwork of size L + 2. We have shown that if any subnetwork has more than N_c + 1 active mobile terminals when L ≥ N_c ≥L/2, either the number of extra active mobile terminals do not scale with size of the network, or the average over the whole network remains bounded by N_c + 1 active mobile terminals per subnetwork. This forces that the average number of decoded words per subnetwork is at most N_c + 1, implying that the asymptotic puDoF during the uplink using zero forcing, τ_U^zf(L,N_c) ≤N_c + 1/L + 2. We have shown in Section <ref> that τ_U^zf(L,N_c) ≥N_c + 1/L + 2, implying that τ_U^zf(L,N_c) = N_c + 1/L + 2 whenever L/2≤ N_c ≤ L. The proof of (<ref>) is thus complete. § CONVERSE PROOF FOR UPLINK WHEN N_C < L/2 In this section, we provide a converse proof for the third range of (<ref>). More precisely, we show that the following holds. τ_U^zf (L,N_c) = 2N_c/2N_c + L, N_c < L/2. Similar to Section <ref> Our proof will be based on the concept that to break the inner bound described in (<ref>), at least one subnetwork of 2N_c + L consecutive MT-BS pairs must have more than 2N_c active mobile terminals. And all such subnetworks must either borrow or block base stations from the subnetwork above it to decode words corresponding to its own mobile terminals. This happens because for a consecutive set of 2N_c + L mobile terminals, the only base stations that can help decode their transmissions are the corresponding base stations or the other L base stations with preceding indices that are connected to the set. Using the Lemmas and definitions from Section <ref> we start our proof. We present cases on x, and δ. Firstly we notice as each subnetwork has 2N_c + L mobile-terminals, base-stations pairs so there is a possibility of decoding more then 2N_c transmissions using subnetwork-only-decoding (SO-decoding) in L_k. We first show that only a maximum of 2N_c + 1 transmissions can be decoded using SO-decoding.By Lemma <ref>, to decode 2N_c + 2 transmissions using SO-decoding you need a subnetwork of 2*L + 1 mobile-terminal and base stations pairs which is larger than 2N_c + L. Thus L_k can decode at most 2N_c + 1 using SO-decoding Thus, our first case is x = 1 and δ = 0. In L_k we have that the highest indexed base stations that can decode the 2N_c + 1 transmissions are the L + 1 + N_c + 1 = L + N_c + 2 highest indexed base stations. But to transmit and decode 2N_c +1 transmissions in L_k there will be at least 3 mobile terminals that achieve the maximum number of associations. By topology, only the lowest indexed such terminal is the one that blocks base stations in L_k-1. Specifically it blocks L - 2N_c + 3 base stations, thus only 4N_c - 3 base stations are left to decode at least 2N_c + 1 - 1 base stations. From these 4N_c - 3, L_k-1 needs at least L + 2 - (L - 2N_c + 3) base stations to decode N_c + 1 transmissions. But at least 2 of the mobile terminals transmitting to these base stations would meet its max constraint, and if you utilize exactly L + 2 - (L - 2N_c + 3) base stations then by Corollary <ref> only 1 of the highest L - 2N_c + 3 can be transmitting. Similar to above we only consider the lower indexed mobile terminal which reaches its maximum association constraint. This mobile terminal is indexed at most α_k-1 + 4N_c - 5, which results that including the base stations which are decoding transmissions and those which cant due association constraints at least L + 2 of the 4N_c - 3 base stations are used up. We are left with 4N_c - 3 - (L + 2) = 4N_c - L - 5 ≤ 2N_c - 6 base stations as N_c ≤L - 1/2. Now these 2N_c - 6 base stations must decode at least N_C - 1 transmissions, thus at least N_c - 2 of these transmissions must come from lowest indexed 2N_c - 7 mobile terminals, and the other transmission is transmitted from at most the α_k-1 + (2N_c + L - (L + 2)) - 1 = α_k-1 + (2N_c - 3). Which forces that in L_k-2 the highest indexed L + 1 - (2N_c - 2) = L - 2N_c + 3 can decode at most 1 transmission and the highest indexed L + 1 - (2N_c - 7) = L - 2N_c + 8 can decode at most 2 transmissions. So in L_k-2 the remaining 2N_c + L - (L - 2N_c + 8) = 4N_c - 8 base stations must decode at least 2N_c - 2. Using a similar argument as above L + 2 of these available base stations are used to decode N_c + 1 transmissions. So in L_k-2, 4N_c - 8 - (L+2) = 4N_c - L - 10 ≤ 2N_c - 11 of the lowest indexed base stations must decode at least N_c - 3 transmissions. So compared to L_k-1 where the lowest indexed 2N_c - 6 base stations had to decode N_c - 1 transmissions, in L_k-2 the lowest indexed 2N_c - 11 have to decode at least N_c - 3 transmissions, so even though the number of needed transmissions decreased by 2, the number of available base stations decreased by 5. This propagation of interference would continue through all preceding subnetworks and the number of available base stations would keep decreasing faster than the number of transmissions to be decoded. So either we reach L_1 or we stop this propagation of interference at some L_i, where i < k. If the latter happens, then in L_i one could decode at most N_c + 1, which would bring the average number of decoded words between the k - i subnetworks to less than 2N_c/2N_c + L, so we just restart our argument from L_i - 1. If the former happens then the number of extra decoded transmissions did not scale with the network size, and thus the asymptotic puDOF is still 2N_c/2N_c + L. If x = 1, and δ = 1, one observes that the interference from L_k to L_k-1 is actually worse than as described above. This is due to the fact that the extra active mobile terminal's transmission will be either decoded at one of the highest indexed L - 2N_c + 3 base stations of L_k-1 or a lower indexed base station. This either causes the same interference as described above or by Lemma 1 the effective max constraint of some of the higher indexed mobile terminals is reduced, which is worse than before. Thus a similar argument follows. If x = 1, and δ > 1, then by Lemma 1, we have that the borrowed base stations in L_k-1 would be lower indexed than the highest indexed L - 2N_c + 3 base stations, thus the interference is worse than the first argument, which leads to the same conclusion that the asymptotic puDoF is still 2N_c/2N_c + L. Now if x > 1, by Lemma 1, and the first argument we have that either the highest indexed L - 2N_c + 3 base stations are blocked in L_k-1 and some lower indexed base stations are borrowed, or all borrowed base stations have a lower index than the highest indexed L - 2N_c + 3 base stations, which from above arguments leads us to the same conclusion that the asymptotic puDoF is still 2N_c/2N_c + L. So we have shown that We have shown that if any subnetwork has more than 2N_c active mobile terminals when N_c < L/2, either the number of extra active mobile terminals do not scale with size of the network, or the average over the whole network remains bounded by 2N_c active mobile terminals per subnetwork. This forces that the average number of decoded words per subnetwork is at most 2N_c, implying that the asymptotic puDoF during the uplink using zero forcing, τ_U^zf(L,N_c) ≤2N_c/2N_c + L. We have shown that τ_U^zf(L,N_c) ≥2N_c/2N_c + L, implying that τ_U^zf(L,N_c) = 2N_c/2N_c + L whenever N_c < L/2. § AVERAGE UPLINK-DOWNLINK DEGREES OF FREEDOM In <cit.>, the puDoF value τ(L=1,N_c) was characterized. Here, we present zero-forcing schemes, with the goal of optimizing the average rate across both uplink and downlink for arbitrary values of L ≥ 2. We propose the following theorem The average uplink-downlink puDoF that can be achieved utilizing the interference avoidance schemes described in Section <ref> is characterized by τ^zf(L,N_c) ≥1/2(1 + γ_D(N_c, L)) L + 1≤ N_c, 2N_c/2N_c + L 1 ≤ N_c ≤L, where γ_D(N_c, L) is the downlink component of the puDoF when N_c ≥ L+1, and is given by γ_D(L,N_c) = 2(⌈L + 1/2⌉ + N_c - (L + 1))/L + 2(⌈L + 1/2⌉ + N_c - (L + 1)). The coding scheme that achieves the inner bound for the second range of (<ref>) is essentially the union of the scheme described in Section <ref> and the scheme that achieves the third range of (<ref>). The network is split into disjoint subnetworks; each with consecutive 2N_c + L transmitter-receiver pairs. We consider two sets of base stations A_BS = {1,2,⋯, N_c} and B_BS = {N_c + 1,N_c + 2⋯, 2N_c}, and two sets of mobile terminals A_MT = {1,2,⋯, N_c} and B_MT = {N_c + L + 1,N_c + L + 2⋯, 2N_c + L}. Now for each i ∈ A_MT, C_i= A_BS. Similarly for each j ∈ B_MT, C_j= B_BS. Thus, for the downlink and uplink, we can get the optimal puDoF described in Sections <ref> and <ref> when N_c < L/2. For the case where N_c ≥ L + 1, the coding scheme that achieves the inner bound in (<ref>) is as follows. First, we associate each mobile terminal with the L+1 base stations connected to it. This achieves the puDoF value of unity during the uplink in the same way as the scheme that achieves it in Section <ref>. Hence, we know so far that C_i⊇{i, i-1, i-2, ⋯, i - L}∩[K], ∀ i∈[K]. When sending messages from base stations to mobile terminals, the cooperative zero-forcing scheme works in a "downward" fashion as shown in  <cit.>. Due to the network topology, the uplink message passing scheme that achieves the unity puDoF works in an "upward" manner as shown in Section <ref>. So to maximize the downlink puDoF, we need to find a coding scheme that optimizes these opposing trends. We define C_i^D as the set of extra associations that the downlink scheme requires for MT i. Thus, ∀ i ∈ [ K] we have that C_i = C_i^D ∪{i, i-1, ⋯, i - L}. For the downlink, we divide the network into disjoint subnetworks; each consists of L + 2(⌈L + 1/2⌉ + N_c - (L + 1)) consecutive transmitter-receiver pairs. We define ϵ = ⌈L+1/2⌉, and κ = ϵ + N_c - (L+1). The cell association has a repeated pattern every 2κ + L BS-MT pairs, and hence, it suffices to describe it for the first 2κ+L BS-MT pairs. We consider two cases based on the parity of the connectivity parameter L. If L is odd, we partition the indices of mobile terminals in the subnetwork into three sets: S_1 = {ϵ, ϵ + 1, ⋯, ϵ + κ -1}, S_2 = {2ϵ + κ, 2ϵ + κ + 1 ⋯, 2ϵ + 2κ -1}, S_3 = {1, 2, ⋯, L + 2κ}∖ ( S_1 ∪ S_2). The mobile terminals indexed in S_3 are kept inactive. The cell associations for downlink are given by the following description. C_i^D= {1, 2, ⋯, κ - 1}, ∀ i ∈ S_1, {ϵ + κ, ϵ + κ + 1, ⋯, ϵ + 2κ - 1}, ∀ i ∈ S_2. If L is even, we partition the indices of mobile terminals in the subnetwork into three sets: S'_1 = {ϵ, ϵ + 1, ⋯, ϵ + κ -1}, S'_2 = {2ϵ + κ - 1, 2ϵ + κ + 1 ⋯, 2ϵ + 2κ -2}, S'_3 = {1, 2, ⋯, L + 2κ}∖ ( S_1 ∪ S_2). The mobile terminals indexed in S_3' are kept inactive. The cell associations are given by the following description. C_i^D= {1, 2, ⋯, κ - 1}, ∀ i ∈ S'_1, {ϵ + κ, ϵ + κ + 1, ⋯, ϵ + 2κ - 1}, ∀ i ∈ S'_2. So If L is odd we have a subnetwork of L + 2κ transmitter-receiver pairs and we decode (ϵ + κ - 1 - ϵ + 1) + (2ϵ + 2κ -2 - (2ϵ + κ) + 1)= 2κ words during the downlink, and hence our average puDoF during the downlink is 2κ/L + 2κ = 2(L+1/2 + (N_c - (L+1)))/L + 2(L+1/2 + (N_c - (L+1))). A similar argument follows for the case when L is even. We have that (<ref>) is valid thus Theorem <ref> holds. Figures <ref> and <ref> serve as examples for the average uplink-downlink inner bounds defined in this section . In the case of L = 1, the optimal puDoF is characterized in <cit.>. The findings there coincide with our findings, as L = 1 we find that for N_c ≥ L + 1, it directly implies that ϵ = 1 , andκ = (N_c - L) = N_c - 1 which results in γ_D(N_c, L) = 2(N_c - 1)/2(N_c -1) + 1. § AVERAGE UPLINK-DOWNLINK DOF WITH FULL UPLINK DOF We show that the downlink puDoF as described in (<ref>) is optimal when we have unity DoF for uplink, i.e., each mobile terminal is associated with all the base stations connected to it. In other words, we are restricted in this section to cell associations that satisfy the following definition. We say that an association scheme is called a Full coverage association if each mobile terminal is associated with all the base stations connected to it. More precisely, ∀i∈ K, {i, i-1, ⋯ i-L}∈ C_i. We then have the following theorem: Optimal downlink puDoF when we have a full coverage association and N_c > L is characterized as γ_D(L,N_c) = 2(⌈L + 1/2⌉ + (N_c - L + 1))/L + 2(⌈L + 1/2⌉ + (N_c - L + 1)) . To help in proving Theorem <ref>, we define the following: ϵ = ⌈L+1/2⌉ and κ = ϵ + N_c - (L+1). In order to prove Theorem <ref>, we first break up the network into subnetworks of 2κ + L consecutive base station (transmitter) and mobile terminal (receiver) pairs. Label each subnetwork L_k such that the lowest index of a base station and mobile terminal in L_k is k×(2κ + L). Our proof will be based on the concept that to break the puDoF described in Theorem <ref>, at least one subnetwork of 2κ + L consecutive MT-BS pairs must have more than 2κ + 1 active mobile terminals. First, we show that without borrowing base stations from preceding subnetworks, surpassing the puDoF value in Theorem <ref> is impossible. To do so, we first define a special class of downlink schemes and then formulate a lemma. We say that a downlink scheme relies on Subnetwork-only downlink decoding if transmissions from a subnetwork can only be decoded in the same subnetwork. Utilizing subnetwork-only downlink when we already have a full coverage association scheme, a subnetwork can decode a maximum of 2κ words. Furthermore, for a subnetwork to decode 2κ words it must be that the last receiver in the subnetwork is active. We prove Lemma <ref> by contradiction. Say 2κ + 1 words are decoded, then at least 2κ + 1 transmitters are active in the subnetwork. So there exists at least one active transmitter, say BS i, indexed between the indices of two sets of κ active transmitters. Let ν_1 and ν_2 be the sets of indices of the active transmitters above and below i, with respect to order. Also, let x = maxν_1, and y = minν_2. We first observe that the smallest possible value for x is κ and similarly the largest possible value of y is κ + ϵ, so we have i ∈ν_3 = {κ +1, κ + 2, ⋯, κ + ϵ -1}. For there to be 2κ + 1 words decoded words in the subnetwork, we also need 2κ + 1 active receivers, thus we can make corresponding disjoint index sets ν'_1, ν'_2, and ν'_3. Where receivers with indices in ν'_1 receive the first κ words, receivers with indices in ν'_2 receive the last κ words, and ν'_3 is the set of indices of receivers that can decode the extra word, say W_j. Now, x' = maxν'_1 and y' = minν'_2, so we observe that the smallest possible value for x' is κ + ϵ -1 and the largest possible value of y' is κ + L. Hence, we have j ∈ν'_3 = {κ + ϵ, ⋯κ + L -1}∩{i, i+1, ⋯ i+L}. We then observe that at least ϵ + 1 active receivers indexed in the set ν' = ν'_1 ∪ν'_2 observe the extra transmitted signal. To aid in writing, we define μ'_j as the index set of receivers listening to transmitters which are transmitting W_j. Hence, for every receiver indexed μ'_j there must be a unique active transmitter indexed in ν = ν_1 ∪ν_3 such that the transmitter is associated with W_j to deliver the message at MT j and cancel interference at all other receivers indexed in ν'; call this set of indices μ_j. With just the full association scheme there are at most ϵ -1 such active transmitters, thus we must activate more transmitters indexed in ν_3, or add more active transmitters indexed in ν_1 ∪ν_2 to C_j. Either of those actions increases the size of μ'_j thus we must then increase the size of μ_j to cancel interference. So to cancel the interference of the extra transmitter we will always introduce more interference in the channel. So we cannot get more than 2κ words decoded through Subnetwork-only decoding. Additionally if the last receiver was not active we would have that the largest possible value of y' is κ + L - 1 which would imply that it would be observing transmissions from BS x, and to erase that interference an extra transmitter indexed in {κ + 1, ⋯ y'} must be activated during the downlink and associated with everything that BS x is associated with during the downlink. But this would mean at least one of the messages associated with BS x during the downlink would be associated with more than N_c base stations overall (uplink and downlink), which is not possible, thus the last active receiver must be active. This method of reasoning validates Lemma <ref> If Theorem <ref> were not true then we would have that at least one subnetwork in the entire network must decode at least 2κ + x words. We present cases on x. If x=1, let the first subnetwork that decodes 2κ + 1 words be L_k. Lemma <ref> would then imply that at least one base station is active in L_k-1 such that this base station is either canceling the interference induced from the extra base stations active in L_k or is the extra base station that is carrying the extra word for L_k, but this implies that in L_k-1 at least the last mobile terminal cannot be active, which using Lemma <ref> implies that at most 2κ -1 words can be decoded in L_k-1. Thus the average over the two subnetworks is still 2κ words decoded per subnetwork. When x>1, one observes that the maximum number of base stations that can help from L_k-1 when trying to decode words in L_k is L - 1. These extra L - 1 base stations are being used to cancel interference induced from extra base stations in L_k. That would imply that the preceding subnetwork would only help with decoding a maximum of L + N_c - (L+1) words, so the smallest index of an active receiver in L_k that is being helped by transmitters in L_k-1 is at most L + N_c - (L+1), which implies that the next active receiver must have an index that is at least 2L + N_c - L+1, which leaves at most κ + ϵ - L receivers to decode at least 2κ + 1 - (L + N_c - (L+1)) = κ + ϵ - L + 1 words. Thus, in order to decode 2κ + x words where x > 1 in subnetwork L_k, we require that κ + ϵ - L receivers decode at least κ + ϵ - L + 1 words. Clearly this is impossible. Thus, only an average of 2κ words per subnetwork can be decoded, which results in 2κ/2κ + L words decoded per receiver, which is exactly what Theorem <ref> states. § CONCLUDING REMARKS In this work, we presented an effort to understand optimal cell association decisions in locally connected interference networks, focusing on optimizing for the average uplink-downlink puDoF problem. We consider a backhaul constraint that allows for associating each mobile terminal with N_c base stations (cells), and an interference network where each base station is connected to a corresponding mobile terminal as well as L mobile terminals with succeeding indices. We characterized the optimal association and puDoF for the uplink problem when zero-forcing schemes are considered. We also found that the characterization of the optimal association for the average uplink-downlink puDoF problem when N_c < L/2 follows from our uplink characterization and previous work for the downlink problem. We also presented the optimal zero-forcing downlink scheme if we fix the uplink scheme to the uplink-only-optimal scheme when N_c ≥ L+1. We conjecture that it is in fact optimal to have full DoF in the uplink when N_c ≥ L+1, and hence it would follow that the presented cell association and average puDoF are optimal in this case. IEEEtran 10 CRAN S. Veetil, K. Kuchi and R. K. Ganti. (2015, Dec.). Performance of cloud radio access networks. [Online]. Available: http://arxiv.org/pdf/1512.05904v1.pdf CRAN-2 A. Checko, H. L. Christiansen, Y. Yan, L. Scolari, G. Kardaras, M. S. Berger and L. Dittmann, “Cloud RAN for mobile networks - a technology overview," IEEE Communication Surveys Tutorials, vol. 17, no. 1, pp. 405-426, First Quart. 2015. CRAN-3 China Mobile, “Next generation fronthaul interface," White Paper, Oct. 2015. CRAN-4 The 5G Infrastructure Public Private Partnership. (2015, Jan.). 5G-Xhaul Project. [Online]. Available: https://5g-ppp.eu/5g-xhaul/ CRAN-Simeone O. Simeone, A. Maeder, M. Peng, O. Sahin and W. Yu. (2015, Dec.). Cloud radio access network: Virtualizing wireless access for dense heterogeneous systems. [Online]. Available: http://arxiv.org/abs/1512.07743. CRAN-Simeone-2 S. -H. Park, O. Simeone and S. Shamai. (2016, Jan.). Joint optimization of cloud and edge processing for fog radio access networks. [Online]. Available: http://arxiv.org/abs/1601.02460. CoMP-book P. Marsch and G. P. Fettweis, Coordinated Multi-Point in Mobile Communications: From Theory to Practice, 1st ed. New York, NY: Cambridge, 2011. Annapureddy-ElGamal-Veeravalli-IT12 V. S. Annapureddy, A. El Gamal, V. V. Veeravalli, ““Degrees of Freedom of Interference Channels with CoMP Transmission and Reception," IEEE Trans. Inf. Theory, vol. 58, no. 9, pp. 5740-5760, Sep. 2012. Ntranos-arXiv14 V. Ntranos, M. Maddah-Ali, G. Caire. (2014, Jul.). On uplink-downlink duality for cellular IA. [Online]. Available: https://arxiv.org/abs/1407.3538. DPC M. H. M. Costa, “Writing on dirty paper (corresp.)," IEEE Trans. Inf. Theory, vol. 29, pp. 439-441, May 1983. ElGamal-Annapureddy-Veeravalli-IT14 A. El Gamal, V. S. Annapureddy, and V. V. Veeravalli, “Interference channels with coordinated multi-point transmission: Degrees of freedom, message assignment, and fractional reuse", IEEE Trans. Inf. Theory, vol. 60, no. 6, pp. 3483-3498, Mar. 2014. Bande-ElGamal-Veeravalli-arXiv16 M. Bande, A. El Gamal, V. V. Veeravalli. (2016, Oct.). Degrees of Freedom in Wireless Interference Networks with Cooperative Transmission and Backhaul Load Constraints. [Online]. Available: https://arxiv.org/abs/1610.09453. ElGamal-ISIT16 A. El Gamal, “Cell associations that maximize the average uplink-downlink degrees of freedom," in Proc. IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, Jul. 2016. Wyner A. Wyner, “Shannon-theoretic approach to a Gaussian cellular multiple-access channel,” IEEE Trans. Inf. Theory, vol. 40, no. 5, pp. 1713-1727, Nov. 1994. Wigger M. Wigger, R. Timo and S. Shamai (2016, Mar.). Conferencing in Wyner's Asymmetric Interference Network: Effect of Number of Rounds. [Online]. Available: http://arxiv.org/abs/1603.05540 Cadambe-IA V. Cadambe and S. A. Jafar, “Interference alignment and degrees of freedom of the K-user interference channel,” IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3425-3441, Aug. 2008. Ntranos-arxiv-CIA V. Ntranos, M. Maddah-Ali, G. Caire. (2014, Feb.). Cellular Interference Alignment [Online]. Available: https://arxiv.org/abs/1402.3119.

http://arxiv.org/abs/1701.08179v1

20170127193725

Balancing and Walking Using Full Dynamics LQR Control With Contact Constraints

cs.RO

Analysis and Measurement of the Transfer Matrix of a 9-cell, 1.3-GHz Superconducting Cavity A. Halavanau^1,2, N. Eddy^2, D. Edstrom Jr.^2, E. Harms^2, A. Lunin^2, P. Piot^1,2, A. Romanov^2, J. Ruan^2, N. Solyak^2, V. Shiltsev^2 ^1 Department of Physics and Northern Illinois Center for Accelerator & Detector Development, Northern Illinois University, DeKalb, IL 60115, USA ^2 Fermi National Accelerator Laboratory, Batavia, IL 60510, USA December 30, 2023 ==================================================================================================================================================================================================================================================================================================================================================================== empty empty Torque control algorithms which consider robot dynamics and contact constraints are important for creating dynamic behaviors for humanoids. As computational power increases, algorithms tend to also increase in complexity. However, it is not clear how much complexity is really required to create controllers which exhibit good performance. In this paper, we study the capabilities of a simple approach based on contact consistent LQR controllers designed around key poses to control various tasks on a humanoid robot. We present extensive experimental results on a hydraulic, torque controlled humanoid performing balancing and stepping tasks. This feedback control approach captures the necessary synergies between the DoFs of the robot to guarantee good control performance. We show that for the considered tasks, it is only necessary to re-linearize the dynamics of the robot at different contact configurations and that increasing the number of LQR controllers along desired trajectories does not improve performance. Our result suggest that very simple controllers can yield good performance competitive with current state of the art, but more complex, optimization-based whole-body controllers. A video of the experiments can be found at <https://youtu.be/5T08CNKV1hw> § INTRODUCTION Biped robots that are expected to locomote in human environments require whole-body controllers that can offer precise tracking and well-defined disturbance rejection behavior. Although walking is a complex task involving both hybrid dynamics and underactuation, the level of controller complexity required to execute such a task is unclear. In recent years, optimal control strategies have seen success both in simulation and on real systems for torque controlled humanoids. Previous work, <cit.>, have utilized Quadratic Programs (QPs) to compute inverse dynamics control optimized over a variety of constraints (e.g. dynamic consistency, joint tracking, friction cones, etc.). Trajectories are often planned in operational space and then converted to joint torques using the QPs. The problem can further be organized into hierarchies to solve whole-body control problems according to a set priority in goals such that tasks of higher priority will always be achieved first <cit.>. Unfortunately, along with the growing flexibility of these methods comes added computational overhead, complexity in tuning, and a lack of theoretical disturbance rejection metrics (such as the gain and phase margin of classical controls). In linear control theory, the infinite horizon linear quadratic regulator (LQR) controller is the optimal solution for tracking a steady state pose with quadratic cost on state error and control effort. LQR controllers have also been extensively used to locally stabilize non-linear systems. In our previous work <cit.>, we proposed a contact-consistent LQR control design for humanoid robots. The advantage of the controller is that it explicitly takes into account the coupling between the different joints to create optimal feedback controllers for whole-body coordination. Additionally, this control policy is computationally lightweight and demonstrates robust push recovery behavior competitive with more sophisticated balance controllers which use QP solvers for inverse dynamics <cit.>, rejecting impulses up to 11.7 Ns with peak forces of 650 N while in double support without stepping. Despite this good performance, previous work was limited to a single contact scenario using a single linearization of the dynamics. More recently, <cit.> proposed to use a similar approach to stabilize constrained systems in order to track dynamic behaviors. Feedback gains, although computed, are not directly executed. Instead, walking trajectories are stabilized by computing the cost-to-go of the time varying LQR problem. This cost is then used as the objective in an inverse dynamics controller, which is implemented using a QP solved at each control cycle. This optimization-based approach allows for the use of additional constraints such as torque limits and contact friction cones but has a relatively high computational complexity compared to a simple LQR design. Further, only results in simulation were presented. In this work, we extend our previous approach on contact-consistent LQR control and demonstrate that it can be used for more complex scenarios including switching contacts. In contrast to approaches which solve a QP at each control cycle, we explore the idea of using only a small number of LQR controllers computed from a set of predefined robot poses. Our hypothesis is that only a small number of LQR controllers corresponding to a contact-consistent linearization of the dynamics at key poses are necessary to stabilize complex tasks. We focus predominantly on real robot experiments: balancing under disturbances caused by upper body motion, single leg balancing, and switching between multiple linearizations and contact conditions for walking. Additionally, we experimentally study how the number of re-linearizations of the dynamics along a trajectory affects performance. As hypothesized, our experimental results suggest that very few LQR controllers around key poses (typically at different contact configuration) are sufficient to stabilize a wide variety of tasks, and increasing the number of controllers does not improve tracking. These results suggest that a simple set of LQR controllers with low computational complexity can be used to control a wide range of humanoid motions. § PROBLEM FORMULATION AND CONTROL DESIGN This work extends the approach developed in <cit.>, where we derived a full state LQR using the full dynamics of the robot. The main difference with our previous work is the method used to enforce contact consistency. Previously we used a kinetic energy argument to project the dynamics, which was only valid for linearization at zero velocity states. The projection developed below is now valid around arbitrary states which was necessary for linearization along planned trajectories. The equations of motion for a floating-base robot with n total degrees of freedom (DoFs) including the floating base and m contact constraints can be written as M(θ) θ̈ + C(θ,θ̇)θ̇+ g(θ) = S^Tτ +J_c^T(θ) f_c, where the variables are defined as Contact with the environment is represented through contact constraints on the endeffector as [ J_c 0; J̇_̇ċ J_c; ][ θ̇; θ̈ ] = [ 0; 0 ]. Eq. (<ref>) enforces that an endeffector in contact must have zero velocity and zero acceleration. We numerically linearize the above dynamics to determine a linear time invariant (LTI) system of the form: ẋ = Ax +Bu. Section II of <cit.> contains a detailed discussion of the linearization process as well as how the dynamics are reformulated such that the contact forces are chosen to be consistent with the contact constraint. As mentioned in <cit.>, the resulting linear system in uncontrollable. Because of this, and small errors due to numerical precision during the linearization process, the resulting matrix is commonly ill-condition. This make it difficult for off-the-shelf programs (e.g. minreal in Matlab) to numerically eliminate the uncontrollable states of the system, as required to solve the LQR problem. To resolve this we ensure that the constraints are embedded in the linearized dynamics by projecting the linearized system into the nullspace of the kinematic constraint of Eq. (<ref>) using the projection matrix N, N = null( [ J_c 0; J̇_̇ċ J_c; ]). Here, N ∈ℝ^n × (n-m) is an orthonormal basis for the nullspace of the constraint equation which maps the linearized dynamics to the minimal system as follows. This method of projecting the dynamics into the constrained subspace is more general than previously shown in <cit.> because it is mathematically valid for linearizing around non-static poses, whereas the previous approach was not. This process enables one to linearize around predefined poses or the current state of the robot at each point along a trajectory as for the time varying LQR problem. The optimal feedback gain matrix K_m are computed by minimizing the following cost function in the reduced state space J = ∫_0^∞ (x_m^TQ_mx_m+u^TR_mu)dt We then map the gains from the minimal system, K_m, back to the full system as K = K_m N^T. The resulting controller for the humanoid robot is thus τ = τ_0 - Kx. Where τ_0 is the vector of joint torques that compensate the dynamics of the robot around the linearized state <cit.>. For example, for a linearization at zero velocity poses, it corresponds to a gravity compensation term. § EXPERIMENTS In this section, we present experiments which evaluate the performance of the LQR control framework in a number of different scenarios. Section <ref> demonstrates the use of the controller to stabilize the robot during upper body disturbances. Section <ref> presents results for robust disturbance rejection in single support. Section <ref> investigates the number of LQR linearizations necessary to track a side-to-side motion. Finally, sections <ref> and <ref> present a framework and results for static walking control in simulation and on the real robot, respectively. All robot experiments were conducted on the lower body of a hydraulic Sarcos humanoid using joint torque controllers as described in <cit.> with slight modifications as discussed in Section <ref>. At the joint position level, we combined moving average filtering onboard the motor controller cards (which run at 5Khz) with offboard second-order Butterworth filtering at 1Khz. The moving average filter used a window size of 16 measurements; the Butterworth filter had a cutoff frequency of 40 Hz. This combination was determined empirically to remove noise without incurring a large delay, improving feedback control stability overall. All experiments used a foot size of 12.5 cm by 25 cm. §.§ Linearizing Around Different Postures and Contacts By visualizing the resulting gain matrices from linearizing around different poses and contact conditions as shown in Fig. <ref>, we gain insight on the balancing strategy by inspecting the coupling between joint states and output torques. This type of analysis is not readily available for other controllers which output joint torques rather that a local feedback policy. High costs on the diagonals indicate decoupled joint tracking while off diagonal terms indicate coupling between different states (note that a perfectly diagonal matrix would correspond to a standard independent joint PD controller). Figure <ref> shows different poses and contact conditions used in this study and provides values of the Frobenius norm of the difference between gain matrices with respect to those for the centered posture. We can see that the gains change only slightly when shifting to the side compared to when the contact constraint changes, indicating that switching constraints has a much larger effect on the LQR solution. This insight is further explored experimentally in Section <ref>. In general, the diagonal terms produced by the LQR controller are lower than the independent joint PD controller typically used on the robot. The Frobenius norm of the LQR gain matrices used on the robot and independent joint PD matrix are approximately 650 and 1,500 respectively. §.§ Disturbances Caused By Upper Body Motion In <cit.>, the LQR controller was evaluated through external pushes on the robot with impulses of up to 11.7 Ns along the sagittal plane. While disturbances are often external in nature, they can also originate from the motion of the robot itself. It is a common scenario for a humanoid to engage in tasks that decouple upper and lower body goals. The upper body may be moving around and interacting with the world while the lower body is purely focused on balancing. To simulate this, a mass of 10kg was added to the torso joint and moved through sinusoidal motions of different frequencies. The LQR controller was able to balance for motions of the upper body moving up to 0.8 Hz in the sagittal plane and 0.5 Hz in the frontal plane with an amplitude of 0.1 rad. Plots of the sagittal disturbance tests for slow and fast sinusoids are shown in Fig. <ref>. §.§ Single Leg Balancing In the case of locomotion over very rough or complex terrain, stepping to maintain stability may not always be possible. An example of this is the problem of crossing stepping stones; there are a finite number of safe regions to step and the robot must be able to reject disturbances or correct for error while in single support so it can plan and execute safe stepping motions. In the following experiment, we test the ability of the LQR controller to balance in single support while being externally perturbed in different locations. Referring to the gain matrices in Fig. <ref>, one can see that the balancing strategy for single support is drastically different than that for double support. In the gain matrix, it is clear that the support leg has a large amount of coupling (shown by off diagonal terms) to help the robot balance, while the swing leg essentially becomes decoupled from the support leg (primarily diagonal terms) with some coupling relating the hip position to base error (i.e. the swing leg can be moved at the hip to help correct for errors measured in the base). This balancing strategy can be seen on the robot in <https://youtu.be/5T08CNKV1hw>. §.§ Switching Between Multiple Linearizations Intuitively, using more linearizations should better capture the dynamics of the robot. However, the gain matrices produced by varying the pose changed very little compared to those produced by changing the contact constraints. To quantify the effect of using multiple linearizations, we tracked a side-to-side motion and transitioned between sets of gains, again using the control diagram shown in Fig. <ref>. The root mean squared error (RMSE) for CoM tracking was calculated for two different motion speeds, as summarized in Table <ref>. From the experiments, we saw no clear advantage to using a larger number of linearizations. We believe that this indicates that for the scenarios presented, that there are other bottlenecks on the real robot (such as sensor noise and torque tracking) that dominate the system's performance. We thus conclude that using multiple linearizations for a given contact state gives no advantage over using a single linearization on the real robot. §.§ Walking: Multiple Linearizations and Contact Switching §.§.§ Simulation While any walking planner could be used, we chose to generate walking trajectories offline. In doing so, we rely solely on the ability of the proposed LQR control architecture to robustly track the original planned trajectory. We used the preview control approach proposed by <cit.> based on a ZMP stability criterion. From the resulting CoM and predefined foot step trajectories, we generate desired joint space trajectories using inverse kinematics and feed-forward torques from gravity compensation. During double support, we exploit torque redundancy to optimize contact forces to shift the robot's weight between successive stance legs <cit.>. We found that this weight distribution optimization is crucial in order to achieve the desired motion and ZMP. From the preceding results, we know that the LQR formulation behaves well for tasks without contact switching. Switching contact conditions based solely on the planned trajectory often cause instability because the real contact with the environment never follows the exact timing of the planned trajectory. For the walking task, it thus proved essential to estimate the contact state using both the measured normal force and the planned trajectory. To ensure that contacts are created robustly during the swing leg touch down, trajectories were designed such that the foot contacted the ground with nonzero velocity. Furthermore, to deal with joint torque discontinuities at contact transitions, caused if one were to simply switch from one LQR controller to another, we use a heuristic approach to blend the gains produced by the LQR controllers. When the contact condition changed, we quickly interpolate to an independent joint PD controller from the previous LQR gain matrix and then interpolate to the new LQR gain matrix. The gain matrices were selected by considering the estimated contact condition of the endeffectors and the minimum norm of the current posture to the pre-selected linearized postures. The control scheme used for walking is shown Fig. <ref>. In simulation, the robot could precisely track the prescribed ZMP trajectory and planned footsteps for a wide range of timings and step parameters using only three linearizations (one for double support and one for each single support pose) as shown in Fig. <ref>. The close tracking of the open loop trajectory over a large number of steps demonstrates the effectiveness of using only a small number of linearizations. §.§.§ Real Robot Expanding on the simulation results, we tested the proposed control architecture on the real robot to walk-in-place. Fig. <ref> shows snapshots from the walking experiment as well as the CoP and CoM relative to the feet for a series of nine steps. The maximum walking speed we were able to achieve had a 1.5s SSP and a 1.8s DSP. As the robot walked at faster rates, the placement of of the foot steps became less precise and the performance degraded because of inconsistencies with the pre-planed trajectory. If the feet land in the wrong location, the desired pose would become kinematically infeasible and the LQR controller would be physically unable to drive the error to zero. We believe that replanning the trajectory online would alleviate this issue and allow for both faster and more stable walking. § DISCUSSION The experiments of the preceding section demonstrate both the strengths and shortcomings of the whole body LQR control approach for a number of different tasks. We showed in the experiments that a relatively small number of linearizations were necessary to control stepping tasks and that increasing the number of linearization along the tracked trajectories did not necessarily improve performance on the real robot. Our experiments suggest that linearizations are only necessary at different contact configurations and that a complete gain schedule along planned trajectories, where gains change every control cycle, might not provide additional robustness in real robot experiments. It is also important to notice that we could not achieve stepping and balancing tasks by merely using an independent joint PD controller and that taking into account joint coupling improves significantly performance. Properly damping the system was one of the biggest limitations to obtain robust behaviors on the robot. LQR controllers tend to create aggressive feedback on joint velocity which can create issues on the robot due to excessive noise in the velocity signals. While this problem does not exist in simulations, it becomes quickly visible on the real robot. While a combination of onboard and offboard filtering helped, we expect that using a model-based filter with additional sensors such as IMUs can allow to increase the control bandwidth <cit.>. Using control design techniques that explicitly take into account measurement uncertainty might also help addressing this issue <cit.>, published in WAFR 2016. Additionally, we found that a significant issue in achieving stable LQR control was the performance of the low level torque feedback control loop. In our previous work, we tuned aggressive low-level torque controllers using velocity compensation to eliminate natural actuator damping and ensure good force tracking. While this worked sufficiently in other inverse dynamics approaches <cit.>, we found that a too aggressive velocity compensation gain in the low-level controller tended to destabilize the higher-level LQR controller. In our experiments, velocity compensation was removed and torque feedback gains were lowered. While torque tracking bandwidth was slightly reduced, this allowed to use higher feedback gains for the LQR controller and significantly improved performance. The trade-off between force tracking performance and higher level control performance is extensively discussed in Focchi et. al <cit.>. Contact transitions were also difficult to control and accurate contact estimation was an important element of the control architecture. Indeed, gains change drastically when contact conditions changes and poor estimation of this can lead to unstable behaviors (e.g. when switching to a double support control when the robot is still in single support). In addition, our heuristic used to smooth transitioning between gains at each contact sequence during the fast contact transition also significantly increased performance. While controlling contact transitions was not an issue in simulation, it remains an important issue for implementation on real robots. § CONCLUSION In this paper we formulated a time invariant LQR controller and show on a real robot that this computationally lightweight control policy can be used to combine a small number of linearizations to create complex motion. By using only a single linearization around a key pose projected into contact constraints, we were able to stabilize upper body motion for slow (0.2 Hz) and fast (0.8 Hz) motions and balance on one foot and while rejecting pushes of up to 8.2 Ns with peak forces of 433N without stepping. We then explored using multiple linearization to tracked side-to-side motions and found that on the real robot there was no measurable advantage for using a higher number of linearizations. Finally, using linearizations around each contact situation (double support and both single support poses) was sufficient enough to track a ZMP walking trajectory when coupled with a contact estimator that helped transition between contact switching. These results highlights both the ability to control complex motion with relatively simple control policies and also the need to evaluate modern algorithms on real hardware. On the robot, we observed that the aggressive stabilizing commands LQR produces could be problematic with high sensor noise and limited control bandwidth. In the future, we plan to incorporate joint state estimation <cit.>, focus on robustifying contact transitions and test the control framework in combination with online model predictive control planners. IEEEtran

http://arxiv.org/abs/1701.08208v1

20170127221033

Energy-Efficient Memories using Magneto-Electric Switching of Ferromagnets

cs.ET

Energy-Efficient Memories using Magneto-Electric Switching of Ferromagnets Akhilesh Jaiswal, Indranil Chakraborty and Kaushik Roy, Fellow, IEEE A. Jaiswal, I. Chakraborty and K. Roy are with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN , 47907 USA e-mail: jaiswal@purdue.edu; ichakra@purdue.edu;  kaushik@purdue.edu. The work was supported in part by, C-SPIN, a MARCO and DARPA sponsored StarNet center, by the Semiconductor Research Corporation, the National Science Foundation, Intel Corporation and by the DoD Vannevar Bush Fellowship. January 27, 2017 ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Voltage driven magneto-electric (ME) switching of ferro-magnets has shown potential for future low-energy spintronic memories. In this paper, we first analyze two different ME devices viz. ME-MTJ and ME-XNOR device with respect to writability, readability and switching speed. Our analysis is based on a coupled magnetization dynamics and electron transport model. Subsequently, we show that the decoupled read/write path of ME-MTJs can be utilized to construct an energy-efficient dual port memory. Further, we also propose a novel content addressable memory (CAM) exploiting the compact XNOR operation enabled by ME-XNOR device. Magneto-electric effect, CAM, dual port, memory, XNOR, LLG. § INTRODUCTION Magneto-resistive memories based on current driven Spin Transfer Torque (STT) <cit.>, have attracted immense research interest due to their non-volatility, almost unlimited endurance and area-efficiency <cit.>. However, STT based memories suffer from inherent low switching speed and high write-energy consumption <cit.>. Recently, voltage induced Magneto-Electric (ME) effect, has shown potential for fast and energy-efficient switching of ferromagnets <cit.>. Many device proposals for memory <cit.>, <cit.> and logic applications <cit.> of the ME effect can be found in the literature. In this paper, we explore two different ME devices - i) ME magnetic tunnel junctions (ME-MTJs) <cit.> and ii) ME-XNOR device <cit.>. We analyze the ME devices with respect to writability, readability and switching speed using a coupled magnetization dynamics and transport model. Further, we propose two novel energy-efficient memories - i) a dual port memory and ii) a content addressable memory (CAM), using the aforementioned ME devices. § ME EFFECT Various single phase <cit.> and composite multi-ferroic materials <cit.> have been experimentally demonstrated to exhibit the ME effect. ME effect is due to exchange bias coupling in single phase materials <cit.> and is usually due to strain coupling <cit.> in case of composite materials. For example, in single phase BiFeO_3 due to the coupling between the ferro-electric polarization, the (anti) ferromagnetism of BiFeO_3, and the ferromagnetism of an underlying nano-magnet, the magnetization of the nano-magnet can be switched by application of an electric field <cit.>. Similarly, strain coupled magnetization reversal in PMN-PT has been proposed in <cit.> . Note, since multi-ferroics in general and ME effect in particular, is currently an area of intense research investigation, we do not follow a particular material set or experiment. Rather, in this work, we treat the ME effect by a generic parameter referred to as the magneto-electric co-efficient (α_ME) <cit.> (explained later in the manuscript). Such an abstraction of the ME effect is justified, since the aim of the present paper is not to explore the various physical phenomenons driving the ME effect. Instead we intend to examine the implications of ME based devices with focus on memory applications. § ME DEVICES UNDER CONSIDERATION We consider two ME based devices - ME-MTJ <cit.> and ME-XNOR <cit.>, with focus on memory applications. ME-MTJ consists of an MTJ in contact with an ME oxide underlayer as shown in Fig. <ref>(a). The MTJ itself is composed of a pinned layer (PL), a free layer (FL) and an oxide spacer (usually MgO <cit.>). Depending on the orientations of the free and the pinned layer the ME-MTJ can be in either low resistance parallel (P) state or high resistance anti-parallel (AP) state. The normalized difference in the resistances of the AP and P state is expressed by the tunnel magneto-resistance (TMR) ratio of the MTJ. In order to switch the ME-MTJ from P (AP) to AP (P) state a positive (negative) voltage exceeding a certain threshold needs to be applied on terminal 1 in Fig. <ref>(a). The metal contact to the ME oxide, the ME oxide itself and the free layer of the MTJ can be considered as a capacitor. On the other hand, the value stored in the ME-MTJ can be read by sensing the resistance between terminals 1 and 2. In Fig. <ref>(b) we show the ME-XNOR device. The ME-XNOR device consists of two free layers separated by MgO and in contact with respective ME oxides. If the voltage polarity on the terminals 1 and 2 are the same, the MTJ stack would be in P state (measured between terminals 3 and 4), while a different voltage polarity on the two terminals would lead to an AP state. Thus, the proposed device emulates an XNOR functionality. ME-XNOR device in previous works have been used for logic applications <cit.>. In this work, we would later show that ME-XNOR device can be used to construct an energy efficient CAM. In the next section, we describe the simulation model. § DEVICE MODELING Under mono-domain approximation, magnetization dynamics can be modeled using the LLG equation, proposed by Landau, Lifshitz and Gilbert, as shown below <cit.>, <cit.> ∂m̂/∂ t=-|γ| m̂× H_EFF+αm̂×∂m̂/∂ t where H_EFF is the effective magnetic field. H_EFF is the sum of the demagnetization field <cit.>, <cit.>, the interface anisotropy field <cit.> and any other external field. m̂ is the unit magnetization vector, γ is the gyromagnetic ratio and α is the Gilbert damping constant. The thermal noise is modeled using the Brown's model <cit.> and is accounted for by expressing a contributing field to H_EFF as H⃗_thermal = ζ⃗√(2α kT/|γ|M_SVdt), where ζ⃗ is a vector with components that are zero mean Gaussian random variables with standard deviation of 1. V is volume of the free layer, T is the temperature and k is the Boltzmann's constant and dt is time step. The ME effect can be included in H_EFF by writing the ME field as <cit.> H_ME = 1/μ_0α_MEE = 1/μ_0α_MEV_ME/t_ME, where the magneto-electric constant is α_ME <cit.>, E is the electric field and V_ME is the voltage across the ME capacitor. Equation (1) can be solved numerically through the Heun's method <cit.>. In addition, we used the Non Equilibrium Green's Function (NEGF) formalism <cit.> for estimation of the resistance of the MTJ stack. § DEVICE CHARACTERISTICS §.§ Writability Writing into ME devices is accomplished by application of appropriate voltages across the ME capacitor. An important parameter that dictates the write voltage and hence the write energy is the magneto-electric co-efficient (α_ME). α_ME is the ratio of magnetic field generated per unit applied electric field <cit.>. Experimentally, various ME materials have shown α_ME in the range 0.1/c to 1/c (c is speed of light) <cit.>. In Fig. <ref> (a), we show a typical magnetization switching curve and in Fig. <ref> (b) we plot the switching probability as a function of voltage across the ME capacitor for different values of α_ME. It can be seen, ME materials with high α_ME are desirable for achieving low write energy. §.§ Readability In a memory configuration, a CMOS transistor is used in series with the storage device. Therefore, the bit-cell TMR i.e. the TMR of the device with the series resistance of the CMOS transistor is a more relevant metric for the sensing margin as opposed to the device TMR. In Fig. <ref>(a), we have shown the bit-cell TMR as a function of MgO thickness assuming a 45nm PTM <cit.> transistor in series with varying W/L (width/length) ratios. It can be seen a higher value of MgO thickness is required to increase the bit-cell TMR and reduce the parasitic effect of the transistor series resistance <cit.>. For the ME devices, due to the decoupled read/write paths, the thickness of the MgO oxide can be increased without degrading the write efficiency (which is dictated by the ME oxide). Thus, the decoupled read/write paths for ME devices allows for better sensing due to increased bit-cell TMR. §.§ Switching Speed Though, a detailed switching dynamics for ME devices is still under research investigation <cit.>, yet it is expected that ME switching would be much faster as compared to STT switching <cit.>. This is because ME switching dynamics behaves as if the magnetization direction is being switched by an external field which does not require an incubation delay <cit.> to initiate the switching process. In Fig. <ref>(b) we have shown a typical 3D trajectory of the ME switching mechanism, based on the model presented in section IV. It can be seen if the applied electric field is strong enough, the magnetization vector starts switching without any initial incubation delay. In our simulations for an α_ME of 1/c, complete reversal was obtained within 500ps. § ME MEMORY DESIGN §.§ ME Dual Port Memory The proposed dual port memory using ME-MTJs is shown in Fig. <ref>. Each bit-cell consists of one ME-MTJ and two transistors. The transistor connected to WWLs are the write transistors and those connected to RWLs are the read transistors. Data can be written into the ME-MTJs by activating the write transistors of a particular row and applying appropriate write voltages (positive or negative) on WBLs. Similarly, for reading out the data, the read transistors of a given row are activated and a read voltage is applied on RBLs. The current flowing through the bit-cell is then compared with a reference to sense the current state of the ME-MTJ. A dual port memory is characterized by simultaneous read and write operations i.e. while one row of the memory array is being read simultaneously another row of the memory array can be written into, thereby, improving the memory throughput <cit.>. The dual port nature of the proposed ME-MTJ memory can be explained as follows. Let us consider row-1 in Fig. <ref> is being written into. The write transistors corresponding to row-1 would be activated and by application of proper voltages on WBLs, a P or an AP state can be written into the ME-MTJs. Simultaneously, the read transistors corresponding to row-2 are activated and by sensing the current flowing through the RBLs, the state of the ME-MTJs connected to row-2 can be sensed. Our simulations indicate, write energy consumption per bit of 0.072 fJ for α_ME = 1/c and read energy consumption of 1.3fJ for read voltage of 200mV and read time of 0.5ns. For the present proposal ME switching enables two orders of magnitude improvement in write energy and 8x improvement in switching speed as compared to STT based MTJs <cit.>, in addition to improved TMR and throughput. §.§ ME CAM The ME-XNOR based CAM cell is shown in Fig. 5 (a). The function of M1 is to selectively provide the ME-oxide capacitor with a ground connection when Data Input Line (D_in) is activated. In the read circuit, a reference MTJ (Ref_MTJ) forms a voltage divider with the resistance of the MTJ (R_MTJ). The match signal is obtained at the drain of p-MOS M2 (denoted by node match), where a low voltage indicates a match is obtained and vice-versa. The node match is pre-charged to V_DD. The strengths of the n-MOS and the p-MOS transistors, connected to the match line, are adjusted such that even one activated p-MOS in a row is enough to maintain the output node in its pre-charged state. The operation of the circuit can be divided into three modes: i) Write Mode, ii) Data Input Mode and iii) Read Mode. To write data in the lower (upper) ferromagnet, a write pulse corresponding to bit `1' (positive voltage) and `0' (negative voltage), respectively, is applied on the BL (D_in) with the WL (DWL) activated. If the digital bit written in the lower ferromagnet is same as the data to be matched (stored in the upper ferromagnet), the MTJ switches to low resistance state. Finally in the read mode, a read pulse of 1 V (V_READ) is applied for the read process. The output of the inverter goes `high' only if the MTJ is in low resistance state indicating that the bit written in the top magnet in mode (ii) matches the bit stored in the bottom magnet. Matching of all bits in a row turns all the p-MOS OFF and match goes low, indicating that a match is found. The write and read energy per bit was found to be 0.072 fJ and 15 fJ, respectively, indicating two orders of magnitude improvement in write energy and comparable read energy as compared to previous works as in <cit.>. § CONCLUSION The prospects of achieving voltage driven switching of magnetization has renewed the interest for future low-power non-volatile spintronic memories. In this paper, we first analyze the writability, readability and switching speed of devices based on ME effect. Further, we propose two energy efficient memories using the ME devices. The proposed dual port memory allows for energy-efficient write operations in addition to faster speed, improved TMR and throughput. The proposed CAM requires lesser number of transistors due to the compact XNOR operation enabled by the ME XNOR device, resulting in an area-efficient as well as energy-efficient CAM. IEEEtran 10 url@samestyle toshiba H. Noguchi, K. Ikegami, K. Kushida, K. Abe, S. Itai, S. Takaya, N. Shimomura, J. Ito, A. Kawasumi, H. Hara et al., “7.5 a 3.3 ns-access-time 71.2μw/mhz 1mb embedded stt-mram using physically eliminated read-disturb scheme and normally-off memory architecture,” in 2015 IEEE International Solid-State Circuits Conference-(ISSCC) Digest of Technical Papers.1em plus 0.5em minus 0.4emIEEE, 2015, pp. 1–3. fong_bit X. Fong, S. H. Choday, and K. Roy, “Bit-cell level optimization for non-volatile memories using magnetic tunnel junctions and spin-transfer torque switching,” IEEE Transactions on Nanotechnology, vol. 11, no. 1, pp. 172–181, 2012. jaiswal_sca A. Jaiswal, X. Fong, and K. Roy, “Comprehensive scaling analysis of current induced switching in magnetic memories based on in-plane and perpendicular anisotropies,” Journal on emerging and selected topics in circuits and systems. Revival_ME M. Fiebig, “Revival of the magnetoelectric effect,” Journal of Physics D: Applied Physics, vol. 38, no. 8, p. R123, 2005. me_mem_1 N. Sharma, A. Marshall, J. Bird, and P. Dowben, “Magneto-electric magnetic tunnel junction as process adder for non-volatile memory applications,” in 2015 IEEE Dallas Circuits and Systems Conference (DCAS), Oct 2015, pp. 1–4. me_mem_2 J.-M. Hu, Z. Li, L.-Q. Chen, and C.-W. Nan, “High-density magnetoresistive random access memory operating at ultralow voltage at room temperature,” Nature communications, vol. 2, p. 553, 2011. me_logic_1 N. Sharma, A. Marshall, J. Bird, and P. Dowben, “Magneto-electric magnetic tunnel junction logic devices,” in Energy Efficient Electronic Systems (E3S), 2015 Fourth Berkeley Symposium on.1em plus 0.5em minus 0.4emIEEE, 2015, pp. 1–3. Intel_ME S. Manipatruni, D. E. Nikonov, and I. A. Young, “Spin-orbit logic with magnetoelectric nodes: A scalable charge mediated nonvolatile spintronic logic,” arXiv preprint arXiv:1512.05428, 2015. mesl A. Jaiswal and K. Roy, “Mesl: Proposal for a non-volatile cascadable magneto-electric spin logic,” arXiv preprint arXiv:1611.07738, 2016. BFO_story R. Ramesh, “Electric field control of ferromagnetism using multi-ferroics: the bismuth ferrite story,” Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 372, no. 2009, p. 20120437, 2014. strain S. Wu, J. Miao, X. Xu, W. Yan, R. Reeve, X. Zhang, and Y. Jiang, “Strain-mediated electric-field control of exchange bias in a co90fe10/bifeo3/srruo3/pmn-pt heterostructure,” Scientific reports, vol. 5, 2015. ramesh_exchange S. Wu, S. A. Cybart, D. Yi, J. M. Parker, R. Ramesh, and R. Dynes, “Full electric control of exchange bias,” Physical review letters, vol. 110, no. 6, p. 067202, 2013. determinis J. Heron, J. Bosse, Q. He, Y. Gao, M. Trassin, L. Ye, J. Clarkson, C. Wang, J. Liu, S. Salahuddin et al., “Deterministic switching of ferromagnetism at room temperature using an electric field,” Nature, vol. 516, no. 7531, pp. 370–373, 2014. ikeda S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. Gan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno, “A perpendicular-anisotropy cofeb–mgo magnetic tunnel junction,” Nature materials, vol. 9, no. 9, pp. 721–724, 2010. why_mgo W. Butler, X.-G. Zhang, T. Schulthess, and J. MacLaren, “Spin-dependent tunneling conductance of fe| mgo| fe sandwiches,” Physical Review B, vol. 63, no. 5, p. 054416, 2001. llg T. L. Gilbert, “A phenomenological theory of damping in ferromagnetic materials,” IEEE Transactions on Magnetics, vol. 40, no. 6, pp. 3443–3449, 2004. llg_thesis M. d'Aquino, “Nonlinear magnetization dynamics in thin-films and nanoparticles,” Ph.D. dissertation, Università degli Studi di Napoli Federico II, 2005. demag Z. Wang, G. Yu, X. Liu, B. Zhang, X. Chen, and W. Lu, “Magnetization characteristic of ferromagnetic thin strip by measuring anisotropic magnetoresistance and ferromagnetic resonance,” Solid State Communications, vol. 182, pp. 10–13, 2014. demag_f A. Aharoni, “Demagnetizing factors for rectangular ferromagnetic prisms,” Journal of applied physics, vol. 83, no. 6, pp. 3432–3434, 1998. Brown W. F. Brown Jr, “Thermal fluctuations of a single-domain particle,” Journal of Applied Physics, vol. 34, no. 4, pp. 1319–1320, 1963. nikonov_bench D. E. Nikonov and I. A. Young, “Benchmarking spintronic logic devices based on magnetoelectric oxides,” Journal of Materials Research, vol. 29, no. 18, pp. 2109–2115, 2014. Heun W. Scholz, T. Schrefl, and J. Fidler, “Micromagnetic simulation of thermally activated switching in fine particles,” Journal of Magnetism and Magnetic Materials, vol. 233, no. 3, pp. 296–304, 2001. knack X. Fong, S. K. Gupta, N. N. Mojumder, S. H. Choday, C. Augustine, and K. Roy, “Knack: A hybrid spin-charge mixed-mode simulator for evaluating different genres of spin-transfer torque mram bit-cells,” in 2011 International Conference on Simulation of Semiconductor Processes and Devices.1em plus 0.5em minus 0.4emIEEE, 2011, pp. 51–54. PTM [Online] http://ptm.asu.edu/. amiri P. K. Amiri, Z. Zeng, P. Upadhyaya, G. Rowlands, H. Zhao, I. Krivorotov, J.-P. Wang, H. Jiang, J. Katine, J. Langer et al., “Low write-energy magnetic tunnel junctions for high-speed spin-transfer-torque mram,” IEEE Electron Device Letters, vol. 32, no. 1, pp. 57–59, 2011. incubation N. N. Mojumder and K. Roy, “Proposal for switching current reduction using reference layer with tilted magnetic anisotropy in magnetic tunnel junctions for spin-transfer torque (stt) mram,” IEEE Transactions on Electron Devices, vol. 59, no. 11, pp. 3054–3060, 2012. yusung Y. Seo, K. W. Kwon, X. Fong, and K. Roy, “High performance and energy-efficient on-chip cache using dual port (1r/1w) spin-orbit torque mram,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 6, no. 3, pp. 293–304, Sept 2016. toshibaa H. Noguchi, K. Ikegami, N. Shimomura, T. Tetsufumi, J. Ito, and S. Fujita, “Highly reliable and low-power nonvolatile cache memory with advanced perpendicular stt-mram for high-performance cpu,” in 2014 Symposium on VLSI Circuits Digest of Technical Papers, June 2014, pp. 1–2. vcma_cam K. L. Wang, H. Lee, and P. K. Amiri, “Magnetoelectric random access memory-based circuit design by using voltage-controlled magnetic anisotropy in magnetic tunnel junctions,” IEEE Transactions on Nanotechnology, vol. 14, no. 6, pp. 992–997, 2015.

http://arxiv.org/abs/1701.07725v1

20170126144621

The olivine-dominated composition of the Eureka family of Mars Trojan asteroids

astro-ph.EP

firstpage–lastpage Regularized characteristic boundary conditions for the Lattice-Boltzmann methods at high Reynolds number flows [ 18 november 2016 ============================================================================================================== We have used the XSHOOTER echelle spectrograph on the European South Observatory (ESO) Very Large Telescope (VLT) to obtain UVB-VIS-NIR (ultraviolet-blue (UVB), visible (VIS) and near-infrared (NIR)) reflectance spectra of two members of the Eureka family of L5 Mars Trojans in order to test a genetic relationship to Eureka. In addition to obtaining spectra, we also carried out VRI photometry of one of the VLT targets using the 2-m telescope at the Bulgarian National Astronomical Observatory – Rozhen and the two-channel focal reducer. We found that these asteroids belong to the olivine-dominated A, or S_ a, taxonomic class. As Eureka itself is also an olivine-dominated asteroid, it is likely that all family asteroids share a common origin and composition. We discuss the significance of these results in terms of the origin of the martian Trojan population. planets and satellites: individual: Mars – minor planets, asteroids: individual: Trojan asteroids – techniques: imaging spectroscopy – techniques: photometric § INTRODUCTION The so-called Mars Trojans are asteroids located in the L4 or L5 Lagrangian points of Mars. They are thought to have been there since the very early phases of the Solar System's history. There were nine confirmed Martian Trojans in total as of 2015, of which eight were at L5 and 1 at L4. Occupants of the Mars Trojan clouds might represent a small set of survivors of an early generation of planetesimals from which the inner Solar System was built. The long-term stability of the Mars Trojan clouds has been extensively investigated by <cit.>. According to these authors, in the L4 and L5 regions of Mars, there are combinations of orbital eccentricity and inclination that are stable over time-scales comparable to the Solar System's age. Of the eight L5 Trojans, seven (including Eureka) form the Eureka family, whose existence has been pointed out by <cit.> and <cit.>. Due to its compactness, this is probably a genetic family rather than a random grouping of orbits. It formed sometime in the last Gyr <cit.>. The members of the Eureka asteroid family are also stable Trojans. For this reason, the discovery that asteroid (5261) Eureka belongs to the rare A taxonomic class <cit.> is very interesting. The spectral reflectance properties of A-class asteroids have been interpreted as diagnostic of an overall composition rich in the mineral olivine [(^++,^++)_2_4], a magnesium iron silicate that is a primary component of the Earth's mantle and – supposedly – of the other terrestrial planets. The low abundance of olivine-rich objects among asteroids is an old conundrum. In particular, the existence of metal meteorites suggests that during the early phases of the Solar System's history, several planetesimals reached sizes sufficient for complete thermal differentiation due to the heat produced by the decay of short-lived isotopes, primarily . Metal meteorites are interpreted in this scenario as fragments of the metal-rich cores of such differentiated bodies, exposed after their complete collisional disruption. In this context, one should expect that olivine-rich meteorites and asteroids should be common because olivine is a primary component of the mantles of differentiated bodies. However, this prediction is not borne out of the observations. Asteroids belonging to the olivine-rich A class, or even to the fairly similar S_ a sub-class of the S taxonomic complex <cit.>, are quite rare. At the end of the 1990s, several authors proposed that the original olivine-rich asteroids were 'battered to bits' by collisions and had thus disappeared long ago <cit.>. Whatever the cause of such disappearance, collisional evolution or, perhaps more likely, massive removal of the original planetesimals accreted in the main belt, as predicted by the Grand Tack and Nice models of early evolution of the Solar System <cit.>, only a small number of survivors belonging to the A class can be found among the current asteroid population. As a consequence, the discovery of an A-class asteroid confined in a special dynamical environment within the terrestrial planet region suggests that the other members of the Eureka family also deserve a careful investigation. Based on these considerations, we began an observing campaign to obtain information on the spectral reflectance properties of these objects. This is a challenging task because, in spite of their relative proximity to the Earth, Mars Trojans are faint. Their observation therefore requires the use of large-aperture instruments. In our investigation, we used XSHOOTER, the first European South Observatory (ESO) second-generation instrument developed for the Very Large Telescope <cit.>, which is currently mounted on VLT Unit 2 Kueyen. XSHOOTER is a spectrograph capable of obtaining spectra from 300 to 2480 nm in a single shot at high spectral resolution (from 3000 up to 10 000). In the following sections we present the spectra we obtained for two members of the Eureka family, and compare them with the available reflectance spectra for Eureka itself. We also obtained some limited spectrophotometric information by carrying out multiband VRI photometry. We used the two-channel focal reducer – Rozhen (FoReRo2) of the 2m Ritchey-Chrétien-Coudé telescope at the Bulgarian National Astronomical Observatory (BNAO) to obtain (V-R) and (V-I) colours of a member of the Eureka family, also one of our two targets observed with XSHOOTER. The colour indices were compared with the results obtained by <cit.>, and with the average values of the colour indices for different asteroid taxonomic classes. This paper is organised as follows. In Section <ref>, we present our VLT and Rozhen observations. The adopted procedures for data reduction are described in Section <ref>, and the results in Section <ref>, separately for spectroscopy and multiband photometry. Our main conclusions and a general discussion of our results is given in Section <ref>. § OBSERVATIONS We used XSHOOTER during two nights, FoReRo2 during two nights as well and ACAM for 1 night. The objects that were observed are as follows: XSHOOTER: (385250) 2001 DH47, (311999) 2007 NS2 and spectral solar analogue star HD 67010; FoReRo2: (385250) 2001 DH47, (289) Nenetta [an asteroid with olivine-dominated surface <cit.>] and Stetson standard field L101; ACAM: (3819) Robinson [an asteroid with olivine-dominated surface <cit.>] and Stetson standard field L104. Details of our observations are presented in Tables <ref>–<ref>. §.§ Spectroscopy §.§.§ XSHOOTER This instrument is an echelle spectrograph with an almost fixed spectral setup. The observer can choose between SLIT (and slit width) and IFU (integral field unit) modes. Here we used the SLIT mode. A detailed description of the instrument is available at <cit.> and ESO's webpage[http://www.eso.org/sci/facilities/paranal/instruments/xshooter.html]. This spectrograph has the ability to simultaneously obtain data over the entire 300–2480 nm spectral range by splitting the incoming light from the telescope into three beams, each sent to a different arm: ultraviolet–blue (UVB), visible (VIS), and near-infrared (NIR). Using two dichroic filters, the light is first sent to the UVB arm, then to the VIS arm, and finally the remaining light arrives at the NIR arm. The disadvantage of this choice of optical light path is the high thermal background in the K-region of the NIR spectrum. The observations are presented in Table <ref>. Observations were done in nodding mode to facilitate subsequent sky signal estimation and subtraction. §.§ VRI Photometry §.§.§ FoReRo2 The instrument is a two-channel focal reducer that adapts the imaging elements of the detector to the characteristic size of the object or the seeing disk. FoReRo2 was built mainly for observations of cometary plasma but has been proved suitable for many other tasks <cit.>. Behind the RC (Cassegrain) focus, the light beam is recollimated by a lens collimator. A dichroic mirror reflects the blue part of the spectrum and transmits the red part. For each channel, camera lenses form reduced images of the Cassegrain focal plane, which are recorded by two CCD systems. Filters are placed into the parallel beam after colour separation. For this investigation we used VRI standard filters; therefore, only the red channel of the instrument was in operation. The observations are presented in Table <ref>. §.§.§ ACAM The ACAM instrument of the 4.2-m William Herschel Telescope (WHT) at the Observatorio del Roque de los Muchachos (La Palma, Spain) was used to obtain SDSS photometry of asteroid (3819) Robinson. The observations are presented in Table <ref>. § DATA REDUCTION §.§ XSHOOTER To reduce the XSHOOTER spectra we first processed the data through the ESO esoreflex pipeline version 2.6.8. All spectra were reduced under the assumption of point-like sources and had their instrument signature removed, i.e. de-biased, flat-fielded, wavelength-calibrated, order-merged, extracted, sky-subtracted and, finally, flux-calibrated. To achieve a signal-to-noise ratio (S/N) sufficient for scientific analysis the reduced 1D spectra for both the asteroid and solar analogue standard stars were then rebinned in 20 nm steps. Subsequently, the spectrum of the asteroid was divided by the solar analogue spectrum, the result normalised to unity at 550 nm and again smoothed using a floating average with different wavelength steps in the VIS and NIR regions. Finally, we carried out running sigma clipping of the data with a 20 nm window in wavelength and a 3 σ criterion. During this stage we excluded sections of the spectra affected by high telluric line contamination in the following wavelength intervals: 0.90–1.00, 1.35–1.50 and 1.80–1.90 μm, as suggested by <cit.> and <cit.>. §.§ FoReRo2 All imaging data had their instrument signature removed as well by de-biasing and flat-fielding. The images in I were de-fringed using the median I image combined from all I images for the night. Standard daophot aperture photometry with aperture size 2×full width at half-maximum (FWHM) was performed. Aperture correction, which was measured through a large aperture using the growth-curve method <cit.>, was applied in order to place the instrumental magnitudes of the observed point source objects, which were measured through a small aperture (2×FWHM), on the same system as those of the standard stars. Stetson standard field L101, observed on three different airmasses, was used for obtaining extinction coefficients in each filter. Linear regression fits for magnitude–magnitude and colour–colour calibration were performed as well. Then, the instrumental magnitudes of the asteroid targets were absolutely calibrated using coefficients from those fits, and, finally, their true (V-R) and (V-I) colours were computed. §.§ ACAM All imaging data had their instrument signature removed as well by de-biasing and flat-fielding. The same calibration procedure as the one for FoReRo2 was performed using Stetson standard field L104. The photometry performed in SDSS-g'r'i' filters was converted to VRI using relations presented by <cit.>. § RESULTS §.§ Spectroscopy The resulting XSHOOTER spectra are represented by black lines in Figs <ref>–<ref>. On Figs <ref> and <ref>, we show, apart from the reflectance spectra of our two targets, a spectrum of (5261) Eureka (blue) obtained on 2015May 19 <cit.> and average spectra for the A (red), S (green) and S_ a (purple) classes in the <cit.> taxonomy produced from data available in the PDS data base <cit.> as point-by-point means of 6, 144 and 2 asteroids respectively. We find a 1 μm absorption feature for both Eureka family members where the reflectance minimum is located slightly longward of 1 μm. This is also the case for the A- and S_ a-type spectra and a diagnostic feature of an olivine-rich surface. The regions with high telluric lines contamination, excluded from our analysis, (see Section <ref>), do not affect distinguishing between the S, A and S_ a classes, as the 1 μm feature in S-class spectra is below 0.9 μm. We also note the higher and flatter reflectance of the S-class spectrum in the region of 1.0-1.5 μm, which distinguishes it from the other spectra. Overall, the two asteroid spectra are in better agreement with an S_ a than an A taxonomy, both in the visible and the infrared (IR). Eureka, also an S_ a-type asteroid in the classification scheme[The observed coincidence between the S_ a spectrum and that of Eureka is not accidental; this asteroid is one of only two objects whose spectra define this taxonomic class], appears to be somewhat more reflective than 311999 and 385250 from ∼0.6 up to at least 1.3 μm. Eureka's 1 μm feature is unique among A/S_ a-types. It is the only object with that particular band shape. The spectra of the two new objects, which we classify as S_ a, though they have a lower S/N ratio, appear consistent with Eureka's. At 2μm the reflectivities of the three objects are indistinguishable from each other; although possibly a consequence of the lower S/N ratio, we believe it to be a real feature of the spectrum. Seeking to strengthen the case for the presence of olivine, we compared our spectra with a number of laboratory olivine spectra from the relab data base. Exposure of asteroid surfaces to the space environment changes their optical properties. Therefore we applied a simple space weathering correction to the laboratory spectra prior to the comparison by dividing the olivine relab spectra with a first order polynomial. Through visual inspection, we found that spectra MS-CMP-042-A and MS-CMP-014 (green lines in Figs <ref> and <ref> altered by our space-weathering correction, red lines) are reasonable matches to the asteroid spectra. The first laboratory spectrum corresponds to unprocessed pure olivine; the second, also of pure olivine, is altered by laser irradiation to simulate micrometeoroid impacts. Although it is difficult to obtain a good match over the entire wavelength interval covered by our spectra, by restricting ourselves to the visible part. and the simple space weathering model, we find that our best match (red lines) fits the visible part of our spectra quite well, particularly the minor absorption features around 0.63 and 0.8 μm. Progressing further along this line of investigation probably requires a more refined model of space-weathering effects, which is beyond the scope of this paper. §.§ VRI Photometry Visible colour photometry can be used as a consistency check and also to eliminate candidate spectroscopic classes. Its advantage is that it is generally applicable to fainter objects than spectroscopy. On the other hand, unambiguous taxonomic classification and mineralogical interpretation typically requires detailed knowledge of the reflectance as a function of wavelength as well as extending observations into the IR. Our purpose here is to identify, through direct measurements, the domain that Eureka, its family members and asteroids of a similar mineralogical composition occupy in colour space with a view to possible future observations of the fainter Eureka family asteroids. The (V-R) and (V-I) colour indices for one of the objects, (385250) 2001 DH47, were derived from three series of 10 exposures, acquired in I, R, and V filters with FoReRo2 at BNAO. As some images were of inferior quality, only eight exposures in each filter were used, from which average values were computed. Because of the faintness of the asteroid target, we used relatively long exposures of 300 s. This resulted in a duty cycle of 318 s and extended the observing run to 3 h. On 2016 January 8 we carried out photometric observations of (385250) 2001 DH47. A rotation period of P≈ 4.0 ± 0.8 h and a peak-to-peak amplitude of 0.6 mag were derived <cit.>. The asteroid light variation could potentially introduce systematic effects in the colour indices. We used those parameters to understand the effect of the asteroid lightcurves in determining the colour indices obtained from the Rozhen data. Due to the relatively high uncertainty in the rotation period P and a long time interval between the January 8 and February 6 observations, we could not determine accurate rotation phases for our exposures. In addition, the January 8 observations were performed at the solar phase angle of 30.^∘4, while on February 6 the phase angle was 8.^∘2. Assuming the classic amplitude–phase relationship with m=0.02 mag deg^-1 <cit.>, we estimated the lightcurve amplitude of (385250) 2001 DH47 on February 6 to be 0.4 mag. A simulated lightcurve of (385250) 2001 DH47, with a simple sinusoidal shape, rotation period P=4 h, and peak-to-peak amplitude A=0.4 mag, is presented in Fig. <ref>, with the relative times of the I, R, and V exposures superimposed on the lightcurve. The rotation phase of the first point is arbitrary, but the intervals between consecutive points reflect the actual timings of the exposures. In this model, the average brightness in the I, R, and V filters would be 0.084, -0.080, and 0.102, respectively, and the colour indices would be , . In reality, the rotation phase of the first point of the observing sequence is unknown, so we repeated the computations with phase shifts from 0.02 to 1 in steps of 0.025. Also, for each phase shift, we computed colour indices while changing the rotation period from 3.2 h to 4.8 h, in steps of 0.1 h. In this way, we obtain 39× 17= 663 pairs of colour indices (V-I), and (V-R), which allowed us to estimate the systematic uncertainty of the colour indices computed from the Rozhen observations due to asteroid brightness variation. To estimate the statistical uncertainties resulting from the random scatter of the measurements we used formal sigma values given by the daophot package for each single V, R, I instrumental magnitude. These were then propagated to final estimates of the (V-I) and (V-R) colours, resulting in sigma values of 0.05 and 0.04 mag, respectively. The former is slightly greater than the latter due to the greater noise of the I-filter measurements. Fig. <ref> presents the (V-R) and (V-I) colour indices of (385250) 2001 DH47, compared with the average colours of all taxonomic classes in the <cit.> classification scheme as given in <cit.> as well as the S_ a and S_ r classes in the <cit.> classification. The systematic uncertainty due to the lightcurve is shown as an ellipse. The random measurement uncertainties of the colour indices are represented by the error bars. S_ a and S_ r classes in the <cit.> taxonomy – distinct from the classes of the same name in the <cit.> taxonomy – are intermediate between S and A, and S and R, respectively. The visible (0.44–0.92 μm) spectra that define these classes have a very steep ultraviolet slope shortward of 0.7 μm. The 1 μm absorption feature while deep and clearly visible in S_ r, is shallow and not well defined in S_ a, which shows that it might be shifted above 1 μm and can be associated with a presence of olivine. To compute colours representing the S_ a and S_ r classes we searched for asteroids belonging to those classes in the SMASS II data base. SDSS colours for five S_ a and three S_ r objects included in the SDSS data base of moving objects <cit.> were converted into the VRI system using the same procedure as for the ACAM measurements. Our colours for the Main Belt asteroids (289) Nenetta and (3819) Robinson are included as well. Both of these asteroids have been mineralogically classified as olivine-dominated [defined as S(I)-type in the <cit.> classification] based on their 0.5–2.5 μm spectra <cit.>. Based on visible (0.44–0.92 μm) spectra alone, <cit.> classified (289) Nenetta as an A-type asteroid and (3819) Robinson as an S_ r-type asteroid. Interestingly, Eureka was also classified as S_ r in their work. Our (V-R) colour of (3819) Robinson is lower than that of (289) Nenetta, consistent with the somewhat shallower slope shortward of the reflectance peak for that spectral class as compared to the A type. We therefore conclude that the colour of (385250) 2001 DH47, after taking into account rotation-related photometric effects, is consistent with the spectroscopic determination of its taxonomy. Taking into account the asteroids' rotational brightness changes will be important in our future attempts to constrain the taxonomic class through colour photometry. § DISCUSSION AND CONCLUSIONS Our spectroscopic and spectro-photometric data confirm that three members of the Eureka family in the L5 Mars Trojan cloud – including the previously-observed Eureka – exhibit properties that are best interpreted in terms of a high surface abundance of olivine. Objects sharing the same property, taxonomically classified as members of the A and S_ a classes in the <cit.> taxonomy, are quite unusual among the asteroid population. As mentioned in the Introduction, there are reasons to believe that olivine-rich bodies were common among planetesimals accreted in the inner regions of the Solar System during the early phases of planetary formation. The current underabundance of such objects may well indicate that most of those that were originally present have been lost – so called "missing mantle problem" <cit.>. If this interpretation is correct, then it is interesting that we find a small group of these bodies in one of the Mars Trojan clouds. Due to the particular dynamical environment that ensures orbital stability over long time-scales, the Martian Trojan clouds are one of the few places where one would expect to find samples of the first generation of planetesimals accreted in this region. In other words, these asteroids might well be samples of the original building blocks that came together to form Mars and of the other terrestrial planets. The common fate of these bodies elsewhere seems to have been a nearly complete removal, possibly the result of intense collisional evolution <cit.>. It is also possible that the dynamical excitation of planetesimals in the current main belt during an early phase of migration of the giant planets <cit.> is implicated in the disappearance of these objects. The fact that these olivine-rich asteroids belong to a group (the Eureka family) of objects that may share a common origin is also interesting. Recent numerical modelling of the family's evolution under planetary gravitational perturbations and the Yarkovsky effect led <cit.> to conclude that the group is likely a genetic family formed roughly in the last Gyr of the Solar System's history. Whether or not the family sprung off a common parent – a proto-Eureka – has an obvious bearing on the relative abundance of olivine-rich material near Mars in the early Solar System. It is also important to view this in the context of the apparent compositional diversity of the Martian Trojan population overall. <cit.> obtained visible spectra of Eureka, (101429) 1998 VF31 at L5 and (121514) 1999 UJ7 at L4, later complemented by NIR spectral coverage for the first two asteroids <cit.>. They found that the latter two asteroids do not share Eureka's taxonomy and concluded that all these objects were once parts of larger bodies that formed separately in different locations of the Solar System. Future investigations of objects in Mars' Trojan clouds should include high-S/N spectra in the visual and NIR spectral regions to help us better understand the compositional relationships. Extending the Mars Trojan inventory down to smaller sizes and determining their rotational characteristics would also help evaluate the stability of these objects in the size regime where non-gravitational perturbations such as the Yarkovsky effect become important. § ACKNOWLEDGEMENTS This work was supported via a grant (ST/M000834/1) from the UK Science and Technology Facilities Council. Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme 296.C-5030 (PI: A. Christou). We gratefully acknowledge observing grant support from the Institute of Astronomy and Rozhen National Astronomical Observatory, Bulgarian Academy of Sciences. We thank Colin Snodgrass for kindly agreeing to observe asteroid 3819 with ACAM@WHT during program ITP6, and Maxime Devogele for his suggestion to compare our asteroid spectra with laboratory olivine spectra. Astronomical research at the Armagh Observatory and Planetarium is grant-aided by the Northern Ireland Department for Communities (DfC). This research utilises spectra acquired from the NASA textscrelab facility at Brown University. Eureka spectral data utilised in this publication were obtained and made available by the The MIT-UH-IRTF Joint Campaign for NEO Reconnaissance. The IRTF is operated by the University of Hawaii under Cooperative Agreement no. NCC 5-538 with the National Aeronautics and Space Administration, Office of Space Science, Planetary Astronomy Program. The MIT component of this work is supported by NASA grant 09-NEOO009-0001, and by the National Science Foundation under Grants Nos. 0506716 and 0907766. mnras

http://arxiv.org/abs/1701.07983v3

20170127093709

Weak order in averaging principle for stochastic differential equations with jumps

math.PR

Bengong Zhang College of Mathematics and Computer Science, Wuhan Textile University benyan1219@126.com Hongbo Fu College of Mathematics and Computer Science, Wuhan Textile University hbfu@wtu.edu.cn Li Wan College of Mathematics and Computer Science, Wuhan Textile University wanlinju@aliyun.com Jicheng Liu School of Mathematics and Statistics, Huazhong University of Science and Technology jcliu@hust.edu.cn Weak order in averaging principle for stochastic differential equations with jumps Bengong ZhangHongbo Fu Li Wan Jicheng Liu Received: date / Accepted: date ======================================================================================= The present article deals with the averaging principle for a two-time-scale system of jump-diffusion stochastic differential equation. Under suitable conditions, the weak error is expanded in powers of timescale parameter. It is proved that the rate of weak convergence to the averaged dynamics is of order 1. This reveals the rate of weak convergence is essentially twice that of strong convergence. 60H10 70K70 § INTRODUCTION We consider a two-time-scale system of jump-diffusion stochastic differential equation in form of dX^ϵ_t=a(X_t^ϵ, Y_t^ϵ)dt+b(X_t^ϵ)d B_t+c(X_t-^ϵ)d P_t, X_0^ϵ=x, dY^ϵ_t=1/ϵf(X_t^ϵ, Y_t^ϵ)dt+1/√(ϵ)g(X_t^ϵ, Y_t^ϵ)d W_t+h(X_t-^ϵ,Y_t-^ϵ)d N^ϵ_t, Y_0^ϵ=y, where X_t^ϵ∈ℝ^n, Y_t^ϵ∈ℝ^m, the drift functions a(x, y)∈ℝ^n, f(x,y)∈ℝ^m, the diffusion functions b(x )∈ℝ^n× d_1, c(x)∈ℝ^n, g(x,y)∈ℝ^m× d_2 and h(x,y)∈ℝ^m. B_t and W_t are the vectors of d_1, d_2-dimensional independent Brownian motions on a complete stochastic base (Ω, ℱ,ℱ_t, ℙ), respectively. P_t is a scalar simple Poisson process with intensity λ_1, and N_t^ϵ is a scalar simple Poisson process with intensity λ_2/ϵ. The positive parameter ϵ is small and describes the ratio of time scales between X^ϵ_t and Y^ϵ_t. Systems (<ref>)-(<ref>) with two time scales occur frequently in applications including chemical kinetics, signal processing, complex fluids and financial engineering. With the separation of time scale, we can view the state variable of the system as being divided into two parts: the “slow" variable X^ϵ_t and the “fast" variable Y^ϵ_t. It is often the case that we are interested only in the dynamics of slow component. Then a simplified equation, which is independent of fast variable and possesses the essential features of the system, is highly desirable. Such a simplified equation is often constructed by averaging procedure as in <cit.> for deterministic ordinary differential equations, as well as the further development <cit.> for stochastic differential equations with continuous Gaussian processes. As far as averaging for stochastic dynamical systems in infinite dimensional space is concerned, it is worthwhile to quote the important works of <cit.> and also the works of <cit.>. For related works on averaging for multivalued stochastic differential equations we refer the reader to <cit.>. In order to derive the averaged dynamics of the system (<ref>)-(<ref>), we introduce the fast motion equation with a frozen slow component x∈ℝ^n in form of dY_t^x= f(x, Y_t^x)dt+g(x, Y_t^x)d W_t+h(x,Y^x_t-)d N_t, Y_0^x=y, whose solution is denoted by Y_t^ϵ(y). Under suitable conditions on f,g and h, Y_t^ϵ(y) induces a unique invariant measure μ^x(dy) on ℝ^m, which is ergodic and ensures the averaged equation: dX̅_t=a̅(X̅_t)dt+b(X̅_t)dB_t+c(X̅_t-)dP_t, X̅_0 =x, where the averaging nonlinearity is defined by setting a̅(x ) = ∫_ℝ^ma(x,y)μ^x(dy) = lim_t→ +∞𝔼 a(x, Y_t^x(y)). In <cit.>, it was shown that under the above conditions the slow motion X^ϵ_t converges strongly to the solution X̅_t of the above averaged equation with jumps. The order of convergence 1/2 in strong sense was provided in <cit.>. To our best knowledge, there is no existing literature to address the weak order in averaging principle for jump diffusion stochastic differential systems. In fact, it is fair to say that the weak convergence in stochastic averaging theory of systems driven by jump noise is not fully developed yet, although some strong approximation results on the rate of strong convergence were obtained <cit.>. Therefore, we aim to study this problem in this paper. Here we are interested in the rate of weak convergence of the averaging dynamics to the true solution of slow motion X^ϵ_t. In other word, we will determine the order, with respect to timescale parameter ϵ, of weak deviation between original solution to slow equation and the solution of the corresponding averaged equation. The main technique we adapted is to find an expansion with respect to ϵ of the solutions of the Kolmogorov equations associated with the jump diffusion system. The solvability of the Poisson equation associated with the generator of frozen equation provides an expression for the coefficients of the expansion. As a result, the boundedness for the coefficients of expansion can be proved by smoothing effect of the corresponding transition semigroup in the space of bounded and uniformly continuous functions, where some regular conditions is needed on drift and diffusion term. Our result shows that the weak convergence rate to be 1 even when there are jump components in the system. It is the main contribution of this work. We would like to stress that asymptotic method was first applied by Bréhier <cit.> to an averaging result for stochastic reaction-diffusion equations in the case of Gaussian noise of additive type was included only in the fast motion. However, the extension of this argument is not straightforward. The method used in the proof of weak order in <cit.> is strictly related to the differentiability in time of averaged process. Therefore, once the noise is introduced in the slow equation, difficulties will arise and the procedure becomes more complicated. Our result in this paper bridges such a gap, in which the slow and the fast motions are both perturbed by noise with jumps. The rest of the paper is structured as follows. Section 2 is devoted to notations, assumptions and summarize preliminary results. The ergodicity of fast process and the averaged dynamics of system with jumps is introduced in Section 3. Then the main result of this article, which is derived via the asymptotic expansions and uniform error estimates, is presented in Section 4. Finally, we give the appendix in section 5. It should be pointed out that the letter C below with or without subscripts will denote generic positive constants independent of ϵ in the whole paper. § ASSUMPTIONS AND PRELIMINARY RESULTS For any integer d, the scalar product and norm on d-dimensional Euclidean space ℝ^d are denoted by (·,·)_ℝ^d and ·_ℝ^d, respectively. For any integer k, we denote by C_b^k(ℝ^d,ℝ) the space of all k-times differentiable functions on ℝ^d, which have bounded uniformly continuous derivatives up to the k-th order. In what follows, we shall assume that the drift and diffusion coefficients arising in the system fulfill the following conditions. (A1) The mappings a(x,y), b(x), c(x), f(x,y), g(x,y) and h(x,y) are of class C^2 and have bounded first and second derivatives. Moreover, we assume that a(x,y), b(x) and c(x) are bounded. (A2) There exists a constant α>0 such that for any x∈ℝ^n, y∈ℝ^m it holds y^Tg(x,y)g^T(x,y)y≥αy_ℝ^m. (A3) There exists a constant β>0 such that for any y_1, y_2∈ℝ^m and x∈ℝ^n it holds (y_1-y_2, f(x,y_1)-f(x,y_2)+λ_2(h(x,y_1)-h(x,y_2)))_ℝ^m +g(x,y_1)-g(x,y_2)^2_ℝ^m+λ_2|h(x,y_1)-h(x,y_2)|^2 ≤ -βy_1-y_2^2_ℝ^m. Notice that from (A1) it immediately follows that the following directional derivatives exist and are controlled: D_xa(x,y)· k_1_ℝ^n≤ Lk_1_ℝ^n, D_ya(x,y)· l_1_ℝ^n≤ Ll_1_ℝ^m, D_xx^2a(x,y)·(k_1,k_2)_ℝ^n≤ Lk_1_ℝ^nk_2_ℝ^n, D_yy^2a(x,y)·(l_1,l_2)_ℝ^n≤ Ll_1_ℝ^ml_2_ℝ^m, where L is a constant independent of x,y,k_1, k_2, l_1 and l_2. For differentiability of mappings b, c,f, g and h we possess the analogous results. For examples, we have D^2_xxb(x)· (k_1,k_2)_ℝ^n≤ L k_1_ℝ^nk_2_ℝ^n, k_1,k_2∈ℝ^n, D^2_yyf(x,y)· (l_1,l_2)_ℝ^m≤ L l_1_ℝ^ml_2_ℝ^m, l_1,l_2∈ℝ^m. As far as the assumption (A2) is concerned, it is a sort of non-degeneracy condition and it is assumed in order to have the regularizing effect of the Markov transition semigroup associated with the fast dynamics. Assumption (A3) is the dissipative condition which determines how the fast equation converges to its equilibrium state. As assumption (A1) holds, for any ϵ>0 and any initial conditions x∈ℝ^n and y∈ℝ^m, system (<ref>)-(<ref>) admits a unique solution, which, in order to emphasize the dependence on the initial data, is denoted by (X_t^ϵ(x,y), Y_t^ϵ(x,y)). Moreover the following lemma holds (for a proof see e.g. <cit.>). Under the assumptions (A1), (A2) and (A3), for any x∈ℝ^n, y∈ℝ^m and ϵ>0 we have 𝔼X_t^ϵ(x,y)^2_ℝ^n≤ C_T(1+x^2_ℝ^n+y^2_ℝ^m), t∈ [0, T] and 𝔼Y_t^ϵ(x,y)^2_ℝ^n≤ C_T(1+x^2_ℝ^m+y^2_ℝ^m), t∈ [0, T]. § FROZEN EQUATION AND AVERAGED EQUATION Fixing ϵ=1, we consider the fast equation with frozen slow component x∈ℝ^n, dY_t^x(y)= f(x, Y_t^x(y))dt+g(x, Y_t^x(y))d W_t+h(x,Y^x_t-(y))d N_t, Y_0^x=y. Under assumptions (A1)-(A3), such a problem has a unique solution, which satisfies <cit.>: 𝔼Y_t^x(y)^2_ℝ^m≤ C(1+x^2_ℝ^n+e^-β ty^2_ℝ^m), t≥ 0. Let Y_t^x( y') be the solution of problem (<ref>) with initial value Y_0^x=y', the Itô formula implies that for any t≥0, 𝔼Y_t^x(y)-Y_t^x (y')_ℝ^m^2≤y-y'_ℝ^m^2e^-β t. Moreover, as discussion in <cit.> and <cit.>, equation (<ref>) admits a unique ergodic invariant measure μ^x satisfying ∫_ℝ^my_ℝ^m^2μ^x(dy)≤ C(1+x^2_ℝ^n). Then, by averaging the coefficient a with respect to the invariant measure μ^x, we can define an ℝ^n-valued mapping a̅(x):=∫_ℝ^ma(x,y)μ^x(dy), x∈ℝ^n. Due to assumption (A1), it is easily to check that a̅(x) is 2-times differentiable with bounded derivatives, and hence it is Lipschitz-continuous such that a̅(x_1)-a̅(x_2)_ℝ^n≤ Cx_1-x_2_ℝ^n, x_1, x_2∈ℝ^n. According to invariant property of μ^x, (<ref>) and assumption (A1), we have 𝔼a(x, Y_t^x( y))-a̅(x)^2_ℝ^n = ∫_ℝ^m𝔼(a(x, Y_t^x (y))-a(x, Y_t^x(z)))μ^x(dz) ^2_ℝ^n ≤ ∫_ℝ^m𝔼Y_t^x (y)-Y_t^x(z)_ℝ^m^2μ^x(dz) ≤ e^-β t∫_ℝ^my-z^2_ℝ^mμ^x(dz) ≤ Ce^-β t(1+x^2_ℝ^n+y^2_ℝ^m). Now we can introduce the effective dynamical system dX̅_t(x)=a̅(X̅_t(x))dt+b(X̅_t(x))dB_t+c(X̅_t-(x))d P_t, X̅_0 =x. As the coefficients a̅, b and c are Lipschitz-continuous, this equation admits a unique solution such that 𝔼X̅_t(x)^2_ℝ^n≤ C_T(1+x^2_ℝ^n), t∈ [0, T]. With the above assumptions and notations we have the following result, which is a direct consequence of Lemma <ref>, Lemma <ref> and Lemma <ref>. Assume that x∈ℝ^n and y∈ℝ^m, Then, under assumptions (A1), (A2) and (A3), for any T>0 and ϕ∈ C_b^3(ℝ^n,ℝ), there exists a constant C_T,ϕ,x,y such that |𝔼ϕ(X^ϵ_T(x,y))-𝔼ϕ(X̅_T(x))|≤ C_T,ϕ,x,yϵ. As a consequence, it can be claimed that the weak order in averaging principle for jump-diffusion stochastic systems is 1. § ASYMPTOTIC EXPANSION Let ϕ∈ C_b^3(ℝ^n, ℝ) and define a function u^ϵ(t, x,y):[0, T]×ℝ^n×ℝ^m→ℝ by u^ϵ(t, x,y)=𝔼ϕ(X_t^ϵ(x,y)). We are now ready to seek an expansion formula for u^ϵ(t, x,y) with respect to ϵ with the form u^ϵ(t,x,y)=u_0(t,x,y)+ϵ u_1(t,x,y)+r^ϵ(t,x,y), where u_0 and u_1 are smooth functions which will be constructed below, and r^ϵ is the remainder term. To this end, let us recall the Kolmogorov operator corresponding to the slow motion equation, with a frozen fast component y∈ℝ^m, which is a second order operator taking form ℒ_1Φ(x) = (a(x,y),D_x Φ(x))_ℝ^n+1/2Tr[D_xx^2 Φ(x) · b(x ) b^T(x )] +λ_1(Φ(x+c(x ))-Φ(x)), Φ∈ C_b^2(ℝ^n, ℝ). For any frozen slow component x∈ℝ^m, the Kolmogorov operator for equation (<ref>) is given by ℒ_2Ψ(y) = (f(x,y),D_y Ψ(y))_ℝ^m+1/2Tr[ D_yy^2Ψ(y) · g(x,y) g^T(x,y)] +λ_2(Ψ(y+h(x,y))-Ψ(y)), Ψ∈ C_b^2(ℝ^m, ℝ). We set ℒ^ϵ:=ℒ_1+1/ϵℒ_2. It is known u^ϵ(t,x,y) solves the equation ∂/∂ tu^ϵ(t, x, y)=ℒ^ϵ u^ϵ(t, x, y), u^ϵ(0, x,y)=ϕ(x), Also recall the Kolmogorov operator associated with the averaged equation (<ref>) is defined as ℒ̅Φ(x) = (a̅(x ),D_x Φ(x))_ℝ^n+1/2Tr[ D_xx^2Φ(x) · b(x ) b^T(x )] +λ_1(Φ(x+c(x))-Φ(x)), Φ∈ C_b^2(ℝ^n, ℝ). If we set u̅(t, x)=𝔼ϕ(X̅_t(x)), we have ∂/∂ tu̅(t, x)=ℒ̅u̅(t, x), u̅(0, x)=ϕ(x). §.§ The leading term Let us begin with constructing the leading term. By substituting expansion (<ref>) into (<ref>), we see that ∂ u_0/∂ t+ϵ∂ u_1/∂ t+∂ r^ϵ/∂ t = ℒ_1u_0+ϵℒ_1u_1+ℒ_1r^ϵ + 1/ϵℒ_2u_0+ℒ_2u_1+1/ϵℒ_2r^ϵ. By equating powers of ϵ, we obtain the following system of equations: ℒ_2u_0=0, ∂ u_0/∂ t=ℒ_1u_0+ℒ_2u_1. According to (<ref>), we can conclude u_0 does not depend on y, that is u_0(t,x, y)=u_0(t,x). We also impose the initial condition u_0(0,x)=ϕ(x). Note that ℒ_2 is the generator of a Markov process defined by equation (<ref>), which admits a unique invariant measure μ^x, we have ∫_ℝ^mℒ_2u_1(t,x,y)μ^x(dy)=0. Thanks to (<ref>), this yields ∂ u_0/∂ t(t,x) = ∫_ℝ^m∂ u_0/∂ t(t,x)μ^x(dy) = ∫_ℝ^mℒ_1u_0(t,x)μ^x(dy) = ∫_ℝ^m(a(x,y), D_x u_0(t,x))_ℝ^nμ^x(dy) +1/2Tr[D_xx^2u_0(t,x)· b(x ) b^T(x )] +λ_1(u_0(x+c(x ))-u_0(x)) = ℒ̅u_0(t,x), so that u_0 and u̅ are described by the same evolutionary equation. By uniqueness argument, we easily have the following lemma: Under assumptions (A1), (A2) and (A3), for any x∈ℝ^n, y∈ℝ^m and T>0, we have u_0(T,x,y)=u̅(T,x). §.§ Construction of u_1 According to Lemma <ref>, (<ref>) and (<ref>), we get ℒ̅u̅=ℒ_1u̅+ℒ_2u_1, which means that ℒ_2u_1(t,x,y) = (a̅(x)-a(x,y),D_xu̅(t,x))_ℝ^n := -ρ(t,x,y), where ρ is of class C^2 with respect to y, with uniformly bounded derivatives. Moreover, for any t≥ 0 and x∈ℝ^n, the equality (<ref>) guarantees that ∫_ℝ^mρ(t,x,y)μ^x(dy)=0. For any y∈ℝ^m and s>0 we have ∂/∂ s𝒫_sρ(t,x,y) = (f(x,y),D_y [𝒫_sρ(t,x,y)])_ℝ^m +1/2Tr[D_yy^2[𝒫_sρ(t,x,y)]· g(x,y) g^T(x,y)] +λ_2(𝒫_s[ρ(t,x,y+h(x,y))]-𝒫_s[ρ(t,x,y)]), here 𝒫_s[ρ(t,x,y)]:=𝔼ρ(t, x,Y^x_s(y)). Recalling that μ^x is the unique invariant measure corresponding to Markov process Y^x_t(y) defined by equation (<ref>), from Lemma <ref> we infer that |𝔼ρ(t, x,Y^x_s(y))-∫_ℝ^mρ(t,x,z)μ^x(dz)| =|∫_ℝ^m𝔼[ρ(t, x,Y^x_s(y))- ρ(t, x,Y^x_s(z))]μ^x(dz)| ≤∫_ℝ^m|𝔼(a(x,Y^x_s(z))- a(x,Y^x_s(y)), D_xu̅(t,x))_ℝ^n|μ^x(dz) ≤ C∫_ℝ^m𝔼Y_s^x(z)- Y_s^x(y)_ℝ^nμ^x(dz). Now it follows from (<ref>) and (<ref>) that |𝔼ρ(t, x,Y_s^x(y))-∫_ℝ^mρ(t,x,z)μ^x(dz)| ≤ C(1+x_ℝ^n+y_ℝ^m)e^-β/2s, which implies lim_s→+∞𝔼ρ(t, x,Y_s^x(y))=∫_ℝ^mρ(t,x,z)μ^x(dz)=0. With the aid of the above limit, we can deduce from (<ref>) that (f(x,y),D_y ∫_0^+∞[𝒫_sρ(t,x,y)]ds)_ℝ^m +1/2Tr[D_yy^2∫_0^+∞[𝒫_sρ(t,x,y)]· g(x,y) g^T(x,y)ds] +λ_2(∫_0^+∞𝒫_s[ρ(t,x,y+h(x,y))]ds-∫_0^+∞𝒫_s[ρ(t,x,y)]ds) =∫_0^+∞∂/∂ s𝒫_s[ρ(t,x,y)]ds =lim_s→+∞𝔼ρ(t, x,Y_s^x(y))-ρ(t,x,y) =∫_ℝ^mρ(t,x,z)μ^x(dz)-ρ(t,x,y) =-ρ(t,x,y), which implies ℒ_2(∫_0^+∞𝒫_sρ(t,x,y) ds)=-ρ(t,x,y). Therefore, u_1(t,x,y):=∫_0^+∞𝔼ρ(t,x,Y^x_s(y))ds is the solution to equation (<ref>). Under assumptions (A1), (A2) and (A3), for any x∈ℝ^n, y∈ℝ^m and T>0, we have |u_1(t,x,y)|≤ C_T(1 +x_ℝ^n+y_ℝ^m), t∈[0, T]. By (<ref>), we have u_1(t,x,y)=∫_0^+∞𝔼(a̅(x)- a(x,Y_s^x(y)), D_xu̅(t,x))_ℝ^n ds, so that |u_1(t,x,y)| ≤ ∫_0^+∞a̅(x)- 𝔼[a(x,Y_s^x(y))]_ℝ^n ·D_xu̅(t,x)_ℝ^n ds. Therefore, from Lemma <ref> and (<ref>), we get |u_1(t,x,y)| ≤ C_T(1+x_ℝ^n +y_ℝ^m )∫_0^+∞e^-β/2 sds ≤ C_T(1+x_ℝ^n +y_ℝ^m ). §.§ Determination of remainder r^ϵ We now turn to the construction for remainder term r^ϵ. It is known that (∂_t-ℒ^ϵ)u^ϵ=0, which, together with (<ref>) and (<ref>), implies (∂_t-ℒ^ϵ)r^ϵ = -(∂_t-ℒ^ϵ) u_0-ϵ(∂_t-ℒ^ϵ)u_1 = -(∂_t-1/ϵℒ_2-ℒ_1)u_0-ϵ(∂_t-1/ϵℒ_2-ℒ_1)u_1 = ϵ(ℒ_1u_1-∂_tu_1). In order to estimate the remainder term r^ϵ we need the following two lemmas. Under assumptions (A1), (A2) and (A3), for any x∈ℝ^n, y∈ℝ^m and T>0, we have |∂ u_1/∂ t(t,x,y)|≤ C_T(1+x_ℝ^n+y_ℝ^m). In view of (<ref>), we get ∂ u_1/∂ t(t,x,y)=∫_0^+∞𝔼(a̅(x)- a(x,Y_s^x(y)), ∂/∂ tD_xu̅(t,x))_ℝ^nds. By Lemma <ref> introduced in Section <ref>, we have |∂ u_1/∂ t(t,x,y)| ≤ ∫_0^+∞𝔼(a̅(x)- a(x,Y_s^x(y))_ℝ^n·∂/∂ tD_xu̅(t,x)_ℝ^n)ds ≤ C_T∫_0^+∞𝔼a̅(x)- a(x,Y_s^x(y))_ℝ^nds, so that from (<ref>) we have |∂ u_1/∂ t(t,x,y)|≤ C_T(1+x_ℝ^n+y_ℝ^m). Under assumptions (A1), (A2) and (A3), for any x∈ℝ^n, y∈ℝ^m and T>0, we have |ℒ_1u_1(t,x,y)|≤ C_T(1+x_ℝ^n+y_ℝ^m), t∈ [0, T]. Recalling that u_1(t,x,y) is the solution of equation (<ref>) and equality (<ref>) holds, we have ℒ_1u_1(t,x,y) = (a(x,y),D_x u_1(t,x,y))_ℝ^n + 1/2Tr[D^2_xxu_1(t,x,y)· b(x ) b^T(x )] + λ_1[u_1(t,x+c(x ),y)-u_1(t,x,y)], and then, in order to prove the boundedness of ℒ_1u_1, we have to estimate the three terms arising in the right hand side of above equality. Step 1: Estimate of (a(x,y),D_x u_1(t,x,y))_ℝ^n. For any k∈ℝ^n, we have D_xu_1(t,x,y)· k = ∫_0^+∞(D_x(a̅(x)- 𝔼a(x,Y_s^x(y)))· k,D_xu̅(t,x))_ℝ^n ds +∫_0^+∞(a̅(x)-𝔼 a(x,Y_s^x(y)),D^2_xxu̅(t,x)· k)_ℝ^n ds =: I_1(t,x,y,k)+I_2(t,x,y,k). By Lemma <ref> and <ref>, we infer that |I_1(t,x,y,k)| ≤ D_xu̅(t,x)_ℝ^n∫_0^+∞D_x(a̅(x)- 𝔼a(x,Y_s^x(y)))· k_ℝ^nds ≤ C_Tk_ℝ^n(1+x_ℝ^n+y_ℝ^m)∫_0^+∞e^-β/2sds ≤ C_Tk_ℝ^n(1+x_ℝ^n+y_ℝ^m). By Lemma <ref> and inequality (<ref>), we obtain |I_2(t,x,y,k)| ≤ C_Tk_ℝ^n∫_0^+∞a̅(x)-𝔼 a(x,Y_s^x(y))_ℝ^nds ≤ C_Tk_ℝ^n(1+x_ℝ^n+y_ℝ^m) ∫_0^+∞e^-β/2sds ≤ C_Tk_ℝ^n(1+x_ℝ^n+y_ℝ^m). This, together with (<ref>), implies D_xu_1(t,x,y)· k≤ C_Tk_ℝ^n(1+x_ℝ^n+y_ℝ^m), and then, as a(x,y) is bounded, it follows |(a(x,y),D_x u_1(t,x,y))_ℝ^n| ≤ C_Ta(x,y)_ℝ^n(1+x_ℝ^n+y_ℝ^m) ≤ C_T (1+x_ℝ^n+y_ℝ^m). Step 2: Estimate of Tr[D^2_xxu_1(t,x,y)· b(x ) b^T(x )]. Since u_1(t,x,y) is given by the representation formula (<ref>), for any k_1,k_2∈ℝ^n we have D^2_xxu_1(t,x,y)· (k_1, k_2) =∫_0^+∞𝔼(D^2_xx(a̅(x)- a(x,Y_s^x(y)))· (k_1, k_2),D_xu̅(t,x))_ℝ^n ds +∫_0^+∞𝔼(D_x(a̅(x)- a(x,Y_s^x(y)))· k_1,D^2_xxu̅(t,x)· k_2)_ℝ^n ds +∫_0^+∞𝔼(D_x(a̅(x)- a(x,Y_s^x(y)))· k_2,D^2_xxu̅(t,x)· k_1)_ℝ^n ds +∫_0^+∞𝔼(a̅(x)- a(x,Y_s^x(y)), D^3_xxxu̅(t,x)· (k_1,k_2))_ℝ^n ds :=∑_i=1^4J_i(t,x,y,k_1,k_2). Thanks to Lemma <ref> and Lemma <ref> we get |J_1(t,x,y,k_1,k_2)| ≤ ∫_0^+∞|𝔼(D^2_xx(a̅(x)- a(x,Y_s^x(y)))· (k_1,k_2),D_xu̅(t,x))_ℝ^n | ds ≤ C_T(1+x_ℝ^n+y_ℝ^m)k_1_ℝ^n k_2_ℝ^n ∫_0^+∞e^-β/2sds ≤ C_T(1+x_ℝ^n+y_ℝ^m)k_1_ℝ^n k_2_ℝ^n . By Lemma <ref> and (<ref>) we infer that |J_2(t,x,y,k_1,k_2)| ≤ ∫_0^+∞|𝔼(D_x(a̅(x)- a(x,Y_s^x(y)))· k_1,D^2_xxu̅(t,x)· k_2)_ℝ^n |ds ≤ C_T(1+x_ℝ^n+y_ℝ^m)k_1_ℝ^n k_2_ℝ^n ∫_0^+∞e^-β/2sds ≤ C_T(1+x_ℝ^n+y_ℝ^m)k_1_ℝ^n k_2_ℝ^n . With a similar argument we can also show that |J_3(t,x,y,k_1,k_2)| ≤ C_T(1+x_ℝ^n+y_ℝ^m)k_1_ℝ^n k_2_ℝ^n . By making use of Lemma <ref> and (<ref>), we get |J_4(t,x,y,k_1, k_2)| ≤ C_Tk_1_ℝ^n·k_2_ℝ^n·(1+x_ℝ^n+y_ℝ^m)∫_0^+∞ e^-β/2sds ≤ C_Tk_1_ℝ^n·k_2_ℝ^n(1+x_ℝ^n+y_ℝ^m). In view of the above estimates (<ref>), (<ref>), (<ref>) and (<ref>), we can conclude that there exists a constant C_T such that |D^2_xxu_1(t,x,y)· (k_1, k_2)|≤ C_T k_1_ℝ^n·k_2_ℝ^n(1+x_ℝ^n+y_ℝ^m), t∈[0,T], which means that for fixed y∈ℝ^m and t∈ [0, T], D^2_xxu_1(t,x,y) _L( ℝ^n,ℝ)≤ C_T(1+x_ℝ^n+y_ℝ^m), where ·_L(ℝ^n,ℝ) denotes the usual operator norm on Banach space consisting of bounded and linear operators from ℝ^n to ℝ. As the diffusion function g is bounded, we get Tr(D^2_xxu_1(t,x,y)gg^T) ≤ C_TD^2_xxu_1(t,x,y)_L( ℝ^n,ℝ) ≤ C_T(1+x_ℝ^n+y_ℝ^m). Step 3: Estimate of λ_1[u_1(t,x+c(x ),y)-u_1(t,x,y)]. By Lemma <ref> and boundedness condition of c(x), we directly have |λ_1[u_1(t,x+c(x ),y)-u_1(t,x,y)]| λ_1[|u_1(t,x+c(x ),y)|+|u_1(t,x,y)|] ≤ C_T(1 +x_ℝ^n+y_ℝ^m), t∈ [0, T]. Finally, it is now easy to gather all previous estimates for terms in (<ref>) and conclude |ℒ_1u_1(t,x,y)|≤ C_T(1+x_ℝ^n+y_ℝ^m), t∈ [0, T]. Under the conditions of Lemma <ref>, for any T>0, x∈ℝ^n and y ∈ℝ^m, we have |r^ϵ(T,x,y)|≤ C_Tϵ(1+x_ℝ^n+y_ℝ^m). By a variation of constant formula, we write the equation (<ref>) in its integral form r^ϵ(T,x,y) = 𝔼[r^ϵ(0,X^ϵ_T(x,y),Y^ϵ_T(x, y))] + ϵ[∫_0^T𝔼(ℒ_1u_1-∂ u_1/∂ s)(s, X^ϵ_T-s(x,y),Y^ϵ_T-s(x, y)) ds]. Since u^ϵ and u̅ satisfy the same initial condition, we have |r^ϵ(0, x,y)| = |u^ϵ(0,x,y)-u̅(0,x)-ϵ u_1(0,x,y)| = ϵ |u_1(0,x,y)|, so that, thanks to (<ref>), (<ref>) and (<ref>) we have 𝔼[r^ϵ(0,X^ϵ_T(x,y),Y^ϵ_T (x,y)]≤ Cϵ(1+x_ℝ^n+ y_ℝ^m). Using Lemma <ref> and Lemma <ref> yields 𝔼[(ℒ_1u_1-∂ u_1/∂ s)(s, X^ϵ_T-s(x,y),Y^ϵ_T-s (x,y))] ≤ C 𝔼 (1+X^ϵ_T-s(x,y)+Y^ϵ_T-s(x,y)), and, according to (<ref>) and (<ref>), this implies that 𝔼[∫_0^T(ℒ_1u_1-∂ u_1/∂ s)(s, X^ϵ_T-s(x,y), Y^ϵ_T-s(x,y)) ds] ≤ C_T (1+x_ℝ^n+ y_ℝ^m). The last inequality together with (<ref>) yields |r^ϵ(T,x,y)|≤ϵ C_T (1+x_ℝ^n+ y_ℝ^m). § APPENDIX In this appendix we collect some technical results to which we appeal in the proofs of the main results in Section 4 . For any T>0, there exists a constant C_T>0 such that for any x,k∈ℝ^n and t∈ [0, T], we have |D_xu̅(t,x)· k| ≤ C_Tk_ℝ^n. Observe that for any k∈ℝ^n, D_xu̅(t,x)· k = 𝔼[Dϕ(X̅_t(x))·η ^k,x_t] = 𝔼(ϕ'(X̅_t(x)),η^k,x_t)_ℝ^n, where η^k,x_t denotes the first mean-square derivative of X̅_t(x) with respect to x∈ℝ^n along the direction k∈ℝ^n, then we have dη^k,x_t=D_xa̅(X̅_t(x))·η^k,x_tdt+D_xb(X̅_t(x))·η^k,x_td B_t +D_xc(X̅_t-(x))·η^k,x_t-d P_t, η ^k,x_0=k. This means that η ^k, x_t is the solution of the integral equation η^k, x_t = k+∫_0^tD_xa̅(X̅_s(x))·η^k, x_s ds+∫_0^tD_xb(X̅_s(x))·η^k, x_sdB_s +∫_0^tD_xc(X̅_s-(x))·η_s-^k, xdP_s and then thanks to assumption (A1), we get 𝔼η ^k,x_t^2_ℝ^n≤ C_Tk^2_ℝ^n +C_T∫_0^t𝔼η ^k,x_s^2_ℝ^n ds. Then by Gronwall lemma it follows that 𝔼η ^k,x_t^2_ℝ^n≤ C_Tk ^2_ℝ^n, t∈ [0, T], so that |D_xu̅(t,x)· k|≤ C_T k_ℝ^n. Next, we introduce an analogous result for the second derivative of u̅(t,x). For any T>0, there exists a constant C_T>0 such that for any x,k_1,k_2∈ℝ^n and t∈ [0, T], we have |D^2_xxu̅(t,x)·(k_1,k_2)|≤ C_Tk_1_ℝ^n·k_2_ℝ^n. For any k_1, k_2 ∈ℝ^n, we have D^2_xxu̅(t,x)·(k_1,k_2) = 𝔼[ϕ”(X̅_t(x))·(η^k_1,x_t,η^k_2,x_t) + ϕ'(X̅_t(x))·ξ^k_1,k_2,x_t], where ξ^k_1,k_2,x_t is the solution of the second variation equation corresponding to the averaged equation, which may be rewritten in the following form: ξ^k_1,k_2,x_t = ∫_0^t[D_xa̅ (X̅_s(x))·ξ^k_1,k_2,x_s+D_xx^2a̅ (X̅_s(x))·(η^k_1,x_s,η^k_2,x_s)]ds + ∫_0^t[D_xx^2b (X̅_s(x))·(η^k_1,x_s,η^k_2,x_s)+D_xb(X̅_s(x))·ξ^k_1,k_2,x_s]dB_s + ∫_0^t[D_xx^2c(X̅_s-(x))·(η^k_1,x_s-,η^k_2,x_s-)+D_xc(X̅_s-(x))·ξ^k_1,k_2,x_s-]dP_s. Thus, by assumption (A1) and (<ref>) we have 𝔼ξ^k_1,k_2,x_t^2_ℝ^n ≤ C_T∫_0^t({𝔼η^k_1,x_s^2_ℝ^n}^1/2{𝔼η^k_2,x_s^2_ℝ^n}^1/2+𝔼ξ^k_1,k_2,x_s^2_ℝ^n)ds ≤ C_Tk_1_ℝ^n·k_2_ℝ^n +C_T∫_0^t𝔼ξ^k_1,k_2,x_s^2_ℝ^nds. By applying the Gronwall lemma we have 𝔼ζ^k_1,k_2,x_t^2_ℝ^n≤ C_Tk_1_ℝ^n·k_2_ℝ^n. Returning to (<ref>), we can get |D^2_xxu̅(t,x)·(k_1,k_2)|≤ C_T h_1_ℝ^n·k_2_ℝ^n. By using the analogous arguments used before, we can prove the following estimate for the third order derivative of u̅(t,x) with respect to x. For any T>0, there exists a constant C_T>0 such that for any x,k_1,k_2,k_3∈ℝ^n and t∈ [0, T], we have |D^3_xxxu̅(t,x)·(k_1,k_2,k_3)|≤ C_T k_1_ℝ^n·k_2_ℝ^n·k_3_ℝ^n. The following lemma states boundedness for the first derivative of a̅(x)-𝔼a(x, Y^x_t(y)) with respect to x. There exists a constant C>0 such that for any x∈ℝ^n, y∈ℝ^m, k∈ℝ^n and t>0 it holds D_x (a̅(x)-𝔼a(x, Y^x_t(y)))· k_ℝ^n≤ Ce^-β/2tk_ℝ^n(1+x_ℝ^n+y_ℝ^m). The proof is a modification of the proof of <cit.>. For any t_0>0, we set ã_t_0(x,y,t)=â(x,y,t)-â(x,y,t+t_0), where â(x,y,t):=𝔼a(x, Y^x_t(y)). Then we have lim_t_0→ +∞ã_t_0(x,y,t)=𝔼a(x, Y^x_t(y))-a̅(x). By Markov property, we have ã_t_0(x,y,t) = â(x,y,t)-𝔼a(x,Y_t+t_0^x(y)) = â(x,y,t)-𝔼â(x, Y_t_0^x(y),t) Due to assumption (A1), for any k∈ℝ^n we have D_xã_t_0(x,y,t)· k = D_xâ(x,y,t)· k-𝔼D_x(â(x, Y_t_0^x (y),t))· k = â_x'(x,y,t)· k-𝔼â_x'(x, Y_t_0^x(y),t)· k -𝔼â_y'(x, Y_t_0^x(y),t)·(D_xY_t_0^x(y)· k), where the symbols â_x' and â_y' denote the directional derivatives with respect to x and y, respectively. Note that the first derivative ζ_t^x,y, k=D_xY_t^x(y)· k, at the point x and along the direction k∈ℝ^n, is the solution of equation dζ_t^x, y,k = ( f_x'(x, Y_t^x(y))· k+f_y'(x, Y_t^x(y))·ζ_t^x, y,k)dt +(g_x'(x, Y_t^x(y))· k+g_y'(x, Y_t^x(y))·ζ_t^x, y,k)dW_t +(h_x'(x, Y_t-^x(y))· k+h'_y(x, Y_t-^x(y))·ζ_t-^x, y,k)dN_t with initial data ζ_0^x,y, k=0. Hence, by assumption (A1), it is straightforward to check 𝔼ζ_t^x,y, k_ℝ^m≤ Ck_ℝ^n for any t≥ 0. Note that for any y_1, y_2∈ℝ^m, we have â(x,y_1, t)-â(x,y_2,t)_ℝ^n = 𝔼a(x, Y_t^x(y_1))-𝔼a(x, Y_t^x(y_2))_ℝ^n ≤ C𝔼Y_t^x(y_1)-Y_t^x(y_2)_ℝ^m ≤ Ce^-β/2ty_1-y_2_ℝ^m, where (<ref>) was used to obtain the last inequality. This means that â_y'(x, y,t)· l_ℝ^m≤ Ce^-β/2tl_ℝ^m, l∈ℝ^m. From (<ref>) and (<ref>), we obtain 𝔼[â_y'(x, Y_t_0^x(y),t)·(D_xY_t_0^x(y)· k)]_ℝ^m =𝔼[â_y'(x, Y_t_0^x(y),t)· (ζ_t_0^x,y, k )]_ℝ^m ≤ C e^-β/2tk_ℝ^n. Then, by easy calculations, we have â_x'(x,y_1,t)· k-â_x'(x,y_2,t)· k =𝔼(a_x'(x, Y_t^x(y_1)))· k-𝔼(a_x'(x, Y_t^x(y_2)))· k +𝔼(a_y'(x, Y_t^x(y_1))·ζ_t^x,y_1, k-a_y'(x, Y_t^x(y_2))·ζ_t^x,y_2, k) = 𝔼(a_x'(x, Y_t^x(y_1)))· k-𝔼(a_x'(x, Y_t^x(y_2)))· k +𝔼([a_y'(x, Y_t^x(y_1))-a_y'(x, Y_t^x(y_2))]·ζ_t^x,y_1, k) +𝔼(a_y'(x, Y_t^x(y_2))·(ζ_t^x,y_1, k-ζ_t^x,y_2, k)) := ∑_i=1^3𝒩_i(t,x,y_1,y_2, k). Now, we estimate the three terms in the right hand side of above equality. Concerning 𝒩_1(t,x,y_1,y_2, k) we have 𝒩_1(t,x,y_1,y_2, k)_ℝ^n ≤𝔼(a_x'(x, Y_t^x(y_1)))· k-(a_x'(x, Y_t^x(y_2)))· k_ℝ^n ≤ C𝔼Y_t^x(y_1)-Y_t^x(y_2)_ℝ^m·k_ℝ^n ≤ Ce^-β/2ty_1-y_2_ℝ^m·k_ℝ^n. Next, by assumption (A1) we get 𝒩_2(t,x,y_1,y_2, k)_ℝ^n ≤𝔼[a_y'(x, Y_t^x(y_1))-a_y'(x, Y_t^x(y_2))]·ζ_t^x,y_1, k_ℝ^n ≤ C{𝔼ζ_t^x,y_1, k^2_ℝ^m}^1/2·{𝔼Y_t^x(y_1)-Y_t^x(y_2)^2_ℝ^m}^1/2 ≤ C e^-β/2tk_ℝ^n·y_1-y_2_ℝ^m. For the third term, by making use of assumption (A1) again, we can infer that 𝒩_3(t,x,y_1,y_2, k)_ℝ^n ≤𝔼a_y'(x, Y_t^x(y_2))·(ζ_t^x,y_1, k-ζ_t^x,y_2, k)_ℝ^n ≤ C𝔼ζ_t^x,y_1, k-ζ_t^x,y_2, k_ℝ^m ≤ C e^-β/2ty_1-y_2_ℝ^m·k_ℝ^n. Now, returning to (<ref>) and taking into account of (<ref>), (<ref>) and (<ref>), we get â_x'(x,y_1,t)· k-â_x'(x,y_2,t)· k ≤ C e^-β/2ty_1-y_2_ℝ^m·k_ℝ^n, which leads to â_x'(x,y,t)· h- 𝔼â_x'(x,Y_t_0^x(y),t)· k_ℝ^n ≤ C e^-β/2t(1+y_ℝ^m+Y_t_0^x(y)_ℝ^m)·k_ℝ^n ≤ e^-β/2t(1+x_ℝ^n+y_ℝ^m)·k_ℝ^n, where we used the inequality (<ref>). Returning to (<ref>), by (<ref>) and (<ref>) we conclude that D_xã_t_0(x,y,t)· k_ℝ^n≤ Ce^-β/2t(1+x_ℝ^n+y_ℝ^m)k_ℝ^n. Taking the limit as t_0→ +∞ we obtain D_x (a̅(x)-𝔼a(x, Y^x_t(y)))_ℝ^n≤ Ce^-β/2tk_ℝ^n(1+x_ℝ^n+y_ℝ^m). Proceeding with similar arguments above we obtain the following higher order differentiability. There exists a constant C>0 such that for any x, k_1, k_2∈ℝ^n, y∈ℝ^m and t>0 it holds D^2_xx(a̅(x)-𝔼a(x, Y^x_t(y)))(k_1,k_2)_ℝ^n ≤ Ce^-β/2tk_1_ℝ^nk_2_ℝ^n(1+x_ℝ^n+y_ℝ^m). Finally, we introduce the following auxiliary result. There exists a constant C>0 such that for any x, k∈ℝ^n, y∈ℝ^m and t>0 it holds ∂/∂ tD_xu̅(t,x)· k_ℝ^n≤ Ck_ℝ^n. For simplicity of presentation, we will prove it for the 1-dimensional case. The multi-dimensional situation can be treated similarly, only notations are somewhat involved. In this case we only need to show |∂/∂ t∂/∂ xu̅(t,x)|≤ C. Actually, for any ϕ∈ C_b^3(ℝ,ℝ) we have ∂/∂ xu̅(t,x)=∂/∂ x𝔼ϕ(X̅_t(x))=𝔼(ϕ'(X̅_t(x))·∂/∂ xX̅_t(x)). If we define ς^x_t:=∂/∂ xX̅_t(x), we have ς^x_t = 1+∫_0^ta̅'(X̅_s(x))·ς^x_s ds+∫_0^tb'(X̅_s(x))·ς^x_sdB_s +∫_0^tc'(X̅_s-(x))·ς_s-^xdP_s. The boundedness of a̅', b' and c' guarantees 𝔼|ς^x_t|^2≤ C_T, t∈ [0, T]. By using Itô formula we have 𝔼[ϕ'(X̅_t(x))·ς_t^x] =ϕ'(x)+𝔼∫_0^t[ϕ'(X̅_s(x))a̅'(X̅_s(x))ς_s^x +ς_s^xϕ”(X̅_s(x))a̅(X̅_s(x))]ds +𝔼∫_0^tb'(X̅_s(x)))ς_s^xϕ”(X̅_s(x))b(X̅_s(x))ds +1/2𝔼∫_0^tς_s^xϕ”'(X̅_s(x))b^2(X̅_s(x))ds +λ_1𝔼∫_0^tϕ'(X̅_s(x))c'(X̅_s-(x))ς_s^xds +λ_1𝔼∫_0^tς_s^x[ϕ'(X̅_s-(x)+c(X̅_s-(x)))-ϕ'(X̅_s-(x))]ds +λ_1𝔼∫_0^tc'(X̅_s-(x))ς_s^x[ϕ'(X̅_s-(x)+c(X̅_s-(x)))-ϕ'(X̅_s-(x))]ds. Since ϕ belongs to C_b^3(ℝ, ℝ), from the assumption (A1) it follows that for any t∈ [0, T], |∂/∂ t[∂/∂ xu̅(t,x)]| = |∂/∂ t𝔼[ϕ'(X̅_t(x))·ς_t^x]| ≤ C|𝔼ς_t^x|, then, by taking (<ref>) into account, one would easily arrive at (<ref>). § ACKNOWLEDGMENTS We would like to thank Professor Jinqiao Duan for helpful discussions and comments. Hongbo Fu is supported by Natural Science Foundation of Hubei Province (No. 2018CFB688), NSF of China (No. 11301403) and Chinese Scholarship Council (No. [2015]5104). Bengong Zhang is supported by NSF of China (No. 11401448). Li Wan is supported by NSF of China (No. 61573011). Jicheng Liu is supported by NSF of China (No. 11271013). § COMPETING INTERESTS The authors declare that they have no competing interests. § AUTHORS CONTRIBUTIONS The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript. 99 Bao J. Bao, G. Yin, C. Yuan, Two-time-scale stochastic partial differential equations driven by α-stable noises: Averaging principles, Bernoulli 23 (1) (2017) 645-669. Bogoliubov N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-linear Oscillations, Gordon & Breach Science Publishers, New York, 1961. Brehier C. E. Bréhier, Strong and weak orders in averging for SPDEs, Stochastic Process. Appl. 122 (2012) 2553-2593. Cerrai1 S. Cerrai, M. I. Freidlin, Averaging principle for a class of stochastic reaction-diffusion equations, Proba. Theory Related Fields. 144 (2009) 137-177. Cerrai2 S. Cerrai, A Khasminkii type averaging principle for stochastic reaction-diffusion equations, Ann. Appl. Probab. 19 (3) (2009) 899-948. Cerrai-Siam S. Cerrai, Averaging principle for systems of reaction-diffusion equations with polynomial nonlinearities perturbed by multiplicative type noise, SIAM J. Math. Anal. 43(6) (2011) 2482-2518. Freidlin-Wentzell1 M. I. Freidlin, A. D. Wentzell, Random Perturbation of Dynamical Systems, 2nd ed., Springer-Verlag, New York, 1998. Freidlin-Wentzell2 M. I. Freidlin, A. D. Wentzell, Long-time behavior of weakly coupled oscillators, J. Stat. Phys 123 (2006) 1311-1337. Fu-Liu H. Fu, L. Wan, Y. Wang, J. Liu, Strong convergence rate in averaging principle for stochastic FitzHug-Nagumo system with two time-scales, J. Math. Anal. Appl. 416 (2014) 609-628. Fu-Liu-2 H. Fu, L. Wan, J. Liu, Strong convergence in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales, Stochastic Process. Appl. 125 (2015) 3255-3279. Givon D. Givon, Strong convergence rate for two-time-scale jump-diffusion stochastic differential systems, SIAM Mul. Mod. Simu. 6 (2007) 577-594. Guo Z. Guo, L^p (p≥2)-strong convergence in averaging principle for multivalued stochastic differential equation with non-Lipschitz coefficients, Adv. Difference Equ. 2017: 386. Khas R. Z. Khasminskii, On the principle of averaging the Itô stochastic differential equations (Russian), kibernetika 4 (1968) 260-279. Kifer1 Y. Kifer, Some recent advance in averaging, Modern dynamical systems and applications, Cambridge University Press, Cambridge, UK, 2004, pp. 385-403. Kifer2 Y. Kifer, Diffusion approximation for slow motion in fully coupled averaging, Proba. Theor. Relat. Fields, 129 (2004) 157-181. Kifer3 Y. Kifer, Another proof of the averaging principle for fully coupled dynamical systems with hyperbolic fast motions, Discrete Contin. Dyn. Syst. 13 (2005) 1187-1201. LiuDi1 D. Liu, Strong convergence rate of principle of averaging for jump-diffusion processes, Front. Math. China 7(2) (2012) 305-320. Vere1 A. Y. Veretennikov, On the averaging principle for systems of stochastic differential equations, Mathematics of the USSR-Sbornik 69 (1991) 271-284. Vere2 A. Y. Veretennikov, On large deviations in the averaging principle for SDEs with full dependence, Ann. Probab. 27 (1999) 284-296. Volosov V. M. Volosov, Averaging in systems of ordinary differential equations, Russian mathematical surveys 17 (1962) 1-126. Xujie J. Xu, L^p-strong convergence of the averaging principle for slowCfast SPDEs with jumps, J. Math. Anal. Appl. 445 (2017) 342-373. Xujie2 J. Xu, J. Liu, An averaging principle for multivalued stochastic differential equations, Stochastic Anal. Appl. 32 (2014) 962-974. Xu Y. Xu, J. Duan, W. Xu, An averaging principle for stochastic dynamical systems with Lévy noise, Physica D 240(17) (2011) 1395-1401. Xu3 Y. Xu, B. Pei, Y. Li, Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Math. Methods Appl. Sci. 11(38) (2014) 2120-2131. Wainrib G. Wainrib, Double averaging principle for periodically forced slow-fast stochastic systems, Electron. Commun. Probab. 18 (51) (2013) 1-12. wangwei W. Wang, A. J. Roberts, Average and deviation for slow-fast stochastic partial differential equations, J. Differential Equations 253 (2012) 1265-1286

http://arxiv.org/abs/1701.07772v1

20170126165420

Loschmidt Echo in Many-Body Localized Phase

cond-mat.dis-nn

=1 [1] ifundefinedalias@#1 #1 nameusealias@#1 enenglish Englishenglish

http://arxiv.org/abs/1701.07774v2

20170126165601

Adaptively Detecting Malicious Queries in Web Attacks

cs.CR

[pages=1-last]main.pdf

http://arxiv.org/abs/1701.07644v4

20170126103934

The Diffuse Radiation Field at High Galactic Latitudes

astro-ph.GA

Department of Physics, Christ, Bengaluru 560 029, India akshaya.subbanna@gmail.com Indian Institute of Astrophysics, Bengaluru 560 034, India jmurthy@yahoo.com Department of Physics, Christ, Bengaluru 560 029, India ravichandran.s@christuniversity.in Henry A. Rowland Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA henry@jhu.edu Department of Physics, Astronomy and Geosciences, Towson University, Towson, MD 21252, USA joverduin@towson.edu We have used observations of the North and South Galactic poles to study the diffuse ultraviolet background at locations where the Galactic light is expected to be at a minimum. We find offsets of 230 – 290 in the FUV (1531 Å) and 480 – 580 in the NUV (2361 Å). Of this, approximately 120 can be ascribed to dust scattered light and another 110 (190 in the NUV) to extragalactic radiation. The remaining radiation is, as yet, unidentified and amounts to 120 – 180 in the FUV and 300 – 400 in the NUV. We find that molecular hydrogen fluorescence contributes to the FUV when the 100 surface brightness is greater than 1.08 MJy sr^-1. § INTRODUCTION The diffuse radiation at high latitudes is, by definition, a combination of the diffuse Galactic light (DGL) and the extragalactic background light (EBL). The largest component of the DGL at low latitudes is the light from Galactic plane stars scattered by interstellar dust <cit.> but this will be at a minimum at the poles where there is little dust. Thus much of the diffuse light at the poles might be expected to be from the EBL <cit.>. As a result, there were many observations of the cosmic ultraviolet background at the pole and we have listed them in Table <ref>. The typical surface brightness was 200 – 300 ph cm^-2 s^-1 sr^-1 Å^-1 (hereafter photon units) in the far ultraviolet (FUV: 1300 – 1800 Å) and 300 – 600 in the near ultraviolet (NUV: 1800 – 3200 Å). The EBL is comprised of several parts with the most significant being the integrated light of galaxies which <cit.> found to be 60 – 81 (FUV) and 121 – 181 (NUV). These values are model-dependent but differ by no more than about 20 <cit.>. There may be smaller contributions from the integrated light of QSOs (16 – 30 : <cit.>) and the IGM ( 20 : <cit.>) for a total EBL of 96 – 131 in the FUV and 157 – 231 in the NUV. Phenomenological models of the cosmic spectral energy distribution are increasingly consistent with observational data and semi-analytic models, except in the ultraviolet, where they differ by as much as 100 photon units <cit.>. A good review of the current state of uncertainty in ultraviolet EBL intensity may be found in <cit.>. <cit.> has argued strongly that there is an additional component to the DGL, unrelated to dust-scattered starlight. Much of the evidence for this component comes from observations of the Galactic poles in the FUV from <cit.>. We have used an improved reduction of the diffuse background <cit.> with a Monte Carlo model for the dust scattered light <cit.> to further explore the background in the vicinity of both Galactic poles in the far-ultraviolet (FUV: 1531 Å) and the near-ultraviolet (NUV: 2361 Å). § DATA The mission <cit.> took observations covering most of the sky in the FUV and NUV bands. An observation consisted of one or more visits with exposure times of 100 – 1000 seconds each which could be added together to reach total integration times of as high as 100,000 seconds. The original data from the mission were distributed as FITS (Flexible Image Transport System) files with a pixel size of 1.5”. <cit.> masked out the stars, rebinned to 2' pixels and subtracted the foreground emission <cit.> to produce a map of the diffuse background over the sky. We have used the visit-level data from <cit.>, available from the High Level Science Products (HLSP) data repository[https://archive.stsci.edu/prepds/uv-bkgd/] at the Space Telescope Science Institute, to study the diffuse emission at the Galactic poles. We further rebinned the original 2' bins of <cit.> by a factor of 3 (into 6' bins) to improve the signal-to-noise and the resultant maps are shown for the North Galactic pole (NGP) in Fig. <ref> and the South Galactic pole (SGP) in Fig. <ref> along with the 100 maps from <cit.>, also rebinned to 6' pixels. Although one might expect a good correlation between the FUV and the NUV and between both UV bands and the IR <cit.>, there is much less structure in the NUV image than in the 100 images or, indeed, in the FUV. Given that these are archival data, the number of visits and the exposure times per field fluctuate wildly but with most of the field observed in multiple visits. The deepest observation was the Subaru Deep Field <cit.>, which was targeted by <cit.> as part of the overall saturation coverage of that region by a number of different observatories. There were a total of 99 different visits in the FUV and 169 in the NUV with exposure times from 80 – 1700 seconds for each visit. The cumulative exposure times over the three years from Apr. 2004 to May 2007 is 83,031 seconds in the FUV and 164,369 seconds in the NUV. The primary source of uncertainty in the derived astrophysical background is the foreground emission (airglow in both bands and zodiacal light in the NUV), which is comparable to the astrophysical emission at high Galactic latitudes. We have tested the foreground subtraction by tracking the background surface brightness of a single 6' bin over all the visits in the Subaru field (Fig. <ref>). There are variations in both bands which, despite the missing FUV observations, are obviously correlated (r = 0.9). These are manifested as an increase in the overall background level of the image which we believe are due to changes in the radiation environment around the spacecraft but could not find any obvious trigger, either terrestrial or solar. The mean value of the background over all the visits in a 6' pixel is 346 ± 41 in the FUV and 563 ± 55 in the NUV and we have adopted these uncertainties in our analysis. We took the individual visits and added them into polar grids (Fig. <ref> and <ref>), weighting each visit by its exposure time. Most of the field was covered by multiple visits and we assumed that the diffuse surface brightness in a given field was comprised of a constant DGL + EBL with any difference between visits being due to the uncharacterized foreground discussed above. We subtracted this difference from each visit, effectively setting the median level of the diffuse surface brightness to the minimum over all visits. There is a bright point in the top of the NUV image of the SGP (Fig. <ref>) due to nebulosity around the fifth magnitude star HD 224990 (B3V). We have not included those points in our analysis. Bright points in the FUV images are due to artifacts around hot stars and are not used in the analysis. § RESULTS §.§ UV-IR Correlations Both the UV and the 100 surface brightness track the presence of dust and should be linearly correlated at high Galactic latitudes where the optical depth is low. We have plotted the observed correlations in Fig. <ref> and tabulated them in Table <ref>. The UV does indeed correlate with the IR but not as well as one might expect, as is apparent from a visual comparison of the images in Fig. <ref> and <ref>. The bright IR features such as Markkanen's Cloud <cit.> are readily seen in the FUV at both poles but are not prominent in the NUV. We noted an inflection point in the FUV/IR ratio in the NGP at an IR surface brightness of 1.08 MJy sr^-1 (Fig. <ref>). We performed an F-Test <cit.> to investigate whether the additional term was justified and found an F-value of 1325 which is significant at greater than a 99.9% level. <cit.>, perhaps coincidentally, found a similar inflection point at a 100 surface brightness of 0.8 MJy sr^-1 in Pioneer optical data, which they identified with the cosmic infrared background (CIB: <cit.>). In this scenario, both the CIB and the UV offset would represent that part of the background which is not correlated with interstellar dust. However, we would then expect a similar inflection point in the FUV data at the SGP or in the NUV at either pole which is not seen. Another possibility is that the change in slope is due to molecular hydrogen (H_2) fluorescence <cit.> in the Werner bands kicking in at a 100 surface brightness of 1.08 MJy sr^-1 (logN_H = 20.2). Canonically, H_2 is only formed at column densities greater than logN_H = 20.5 – 20.7 <cit.>, when self-absorption protects the molecules from dissociation by ultraviolet photons. <cit.> have found that the fraction of the total diffuse radiation in the form of fluorescent Werner band emission from molecular hydrogen is 5-10% of the total observed surface brightness at the poles, or about 30 . These observations were averaged over 10 – 15 degrees at the poles and we find that the putative Werner band emission in the data is about 60 , not too far off from their observations. <cit.> and <cit.> have found significant molecular gas at high latitudes at column densities of 20.2 < logN_H < 20.5, which <cit.> attributed to a clumpy medium with the molecular gas concentrated in high density cirrus clouds. Unfortunately, we do not have the spectroscopic information needed to further investigate the emission and cannot further constrain the source of the rise in the FUV. §.§ Zero-Points The diffuse radiation at the poles is likely to be dominated by the EBL and the observed baseline will therefore place an upper limit on the EBL. The y intercepts for the FUV are 288 in the NGP and 241 in the SGP with the corresponding values for the NUV being 531 and 579 in the NGP and SGP, respectively. Taken at face value, these are upper limits for the EBL and match well with earlier determinations of the background at the poles (Table <ref>), including with results from <cit.>, <cit.>, and <cit.>. As an independent check, we have calculated the slopes and offsets using the E(B - V) from <cit.> finding very similar offsets (Table <ref>). However, as discussed in the Introduction, the expected limits on the EBL are 96 – 131 in the FUV and 157 – 231 the NUV, or about half the observed value in the FUV and about one third in the NUV. This offset has been noted before (Table <ref>) but with no definite identification <cit.>. §.§ Correlation with E(B - V) Much of the Ultraviolet Virgo Cluster Survey (GUVICS: <cit.>) falls within our area and our extracted diffuse values are in excellent agreement in the areas of overlap, despite independent approaches to the extraction of the diffuse radiation from the observations. <cit.> subtracted what they termed as “any emission not related to the cirrus” from the EBL and from unknown Galactic sources, possibly including “a very diffuse cirrus contribution” and then derived a linear relationship between the FUV (in ) and the reddening of E(B - V) = 0.02378 + 8.77× 10^-5× (FUV-315), where 315 was their offset. They suggested that the diffuse UV background could be used to calculate the E(B - V) at a higher spatial resolution and precision than either the data <cit.> or the data <cit.>. This method does indeed show promise and we have attempted the same with our data over both poles (Fig. <ref>) using Planck reddening. We found relations of E(B - V) = 0.01124 + 1.119 × 10^-4× (FUV - 250) over the much larger area we observe in the NGP and E(B - V) = 0.01288 + 4.6841 × 10^-5× (FUV - 212) in the SGP. As <cit.> point out, the FUV emission is dependent on the geometry of the stars and the dust and care has to be taken when using the data to predict extinction over the sky. § MODELING MILKY WAY RADIATION Most of the DGL at low Galactic latitudes is unequivocally due to the scattering of the light of hot stars from interstellar dust and we have applied the model developed by <cit.> to predict the amount of Galactic dust-scattered radiation in the polar regions. This model uses a Monte Carlo process to track photons emitted from stars with location and spectral type from the Hipparcos catalog <cit.> and stellar spectra from <cit.>. The dust was taken from the 3-dimensional extinction map derived from PanSTARRS data by <cit.> with an angular resolution of about 14' at the poles. The gaps in the <cit.> map were filled using the reddening map given by <cit.> with a scale height of 125 pc <cit.>. Our modeled dust distribution is shown in Fig. <ref> for both poles and is similar to the IR maps shown in Fig. <ref> and Fig. <ref>, respectively. The distribution of the extinction with distance (along a specific line of sight) is shown in Fig. <ref> and is consistent with a scale height of 125 pc <cit.> and a cavity of about 50 pc radius around the Sun <cit.>. We assumed the scattering function of <cit.> with the albedo (a) and phase function asymmetry factor (g = < cos θ >) as free parameters. The dust at both poles has been extensively investigated through polarization measurements <cit.>. The polarization in the NGP was divided into two regions: Area I and Area II <cit.>, approximately corresponding to with the 100 surface brightness and the polarization being larger in Area II. The overall extinction in both poles is low with minimum values close to zero <cit.>, except for limited areas where clouds are seen in the IR maps with peak values of E(B - V) from 0.02 – 0.04 <cit.>. <cit.> found that the polarization was correlated with the IR maps with the caveat that the polarization maps probed the dust to a distance of about 400 pc while the IR emission measured the dust along the entire line of sight. <cit.> note that there may be some dusty structures extending to high positive latitudes within Area I, as suggested by the distribution of dark and molecular clouds, in addition to the diffuse dust. In general, we find that our dust model is in agreement with the polarization observations. We have run our scattering model for a range of optical constants with representative results shown in Fig. <ref> and <ref>. The major dust features are clearly visible in the models but the brightness is much less than that observed unless the grains scatter isotropically (Fig. <ref>). Because most of the photons at the poles come from Galactic plane stars <cit.>, the earliest papers did indeed find that g = 0. It is now generally accepted <cit.> that the optical constants are close to a = 0.4; g = 0.6 in the FUV and a = 0.4; g = 0.5 in the NUV and we have used those models to fit the observed emission at each pole. There is too much noise in both the models and the data to compare on a pixel-by-pixel level and we have rather integrated both as a function of the 100 values from <cit.> in Figures <ref> and <ref>. The fit is good in both poles and both bands with best-fit offsets of 233 and 234 in the FUV in the NGP and SGP, respectively and offsets of 485 and 538 in the NUV in the NGP and SGP, respectively. These are not far different from the zero-point offsets in Table <ref>. We had previously noted the inflection point in the FUV-IR correlation at 1.08 MJy sr^-1 in the NGP; a comparison with the models shows that it is present in both poles in the FUV. As discussed above, this may be due to fluorescence from the Werner bands of molecular hydrogen. § LIGHT FROM DARK MATTER? The continued presence of this unexplained excess in the diffuse background prompts us to briefly consider possible connections to nonstandard physics. Leading particle dark-matter candidates such as supersymmetric WIMPs or axions produce photons by annihilation or decay, but not at UV energies <cit.>. Another possibility is offered by primordial black holes (PBHs), which emit Hawking radiation with an approximately blackbody spectrum peaking at the characteristic energy E=ħ c^3/(8π G M) for PBHs of mass M. Thus a background with E=7 eV (midway between our FUV and NUV energies) might be associated with PBHs of characteristic mass M≈ 2 × 10^21 g. This value coincides with one of three narrow remaining theoretically allowed PBH mass windows <cit.>, a so far unremarked coincidence that we find intriguing enough to explore briefly here. A plausible production mechanism for PBHs with masses close to this range has been identified by <cit.>. The question is whether PBHs of this kind could contribute significantly to the unexplained excess identified above, whose bolometric intensity Q_u=4π I_λλ≈ 5× 10^-5 erg s^-1 cm^-2 with I_λ≈ 180 photon units at λ≈ 2000 Å. PBH luminosity is very low, L<∼ 2×10^-55L_⊙(M/M_⊙)^-2≈ 6× 10^7 erg s^-1 <cit.>. If these PBHs make up the cold dark matter in the halo of the Milky Way, then their local density ρ≈ 0.008 M_⊙ pc^-3 <cit.>. If they are distributed uniformly, then the nearest one is located at a distance r̅=(ρ/M)^-1/3≈ 100 AU. Its intensity Q=L/(4πr̅^2)≈ 2×10^-24 erg s^-1 cm^-2 as seen by us is far too low to account for Q_u. Alternatively, the total number of PBHs in the halo is N=M_h/M≈ 1× 10^24 where M_h≈ 1× 10^12M_⊙ <cit.>. If these are clustered near the Galactic center at R=8 kpc, then the halo intensity Q_h=N L/(4π R^2)≈2× 10^-19 erg s^-1 cm^-2. This is still 15 orders of magnitude too small. More realistically, if the PBH halo extends beyond the Sun and can be regarded as approximately uniform in the solar vicinity, then Q_h= LR≈7× 10^-17 erg s^-1 cm^-2 where luminosity density L=Lρ/M≈ 2× 10^-33 erg s^-1 cm^-3. This still falls short of Q_u by 12 orders of magnitude, a discrepancy that cannot plausibly be attributed to non-uniformity in the PBH distribution. We infer that PBHs are not likely to contribute significantly to the astrophysical background, a conclusion reinforced by others <cit.>. The failure of this explanation, of course, only deepens the mystery. § CONCLUSIONS We have used data to study the diffuse ultraviolet background at both the North and South Galactic poles with two primary results: * There is an excess emission (over the DGL and the EBL) of 120 – 180 in the FUV and 300 – 400 in the NUV. Offsets in the UV emission have always been observed at the poles (Table <ref>) but it has not been apparent how to attribute it to the different contributors. Although we do not know its origin, we can affirm that the excess emission cannot be accounted for by current models of the DGL and EBL. * We find that there is a change in the FUV-IR correlation at a 100 surface brightness of 1.08 MJy sr^-1 (Fig. <ref> and <ref>). We believe that the most likely explanation for this is molecular hydrogen fluorescence indicating that self-shielding occurs at a column density of logN_H = 20.2. We believe that the study of the Galactic poles will prove to be fruitful in differentiating between the Galactic and extragalactic (and terrestrial) components. Deep spectroscopy of the poles, including of cirrus features, would have been invaluable in separating the components but that seems unlikely in the near future with a dearth of UV missions expected. In its absence, we will continue our in-depth study of diffuse emission with . We thank Prof. Berdyugin and Teerikorpi for clarifying the polarization results in the poles. Part of this research has been supported by the Department of Science and Technology under Grant IR/S2/PU-006/2012. This research has made use of NASA's Astrophysics Data System Bibliographic Services. We have used the GnuDataLanguage (http://gnudatalanguage.sourceforge.net/index.php) for the analysis of this data. The data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NNX09AF08G and by other grants and contracts. yahapj

http://arxiv.org/abs/1701.07971v4

20170127083924

The nature of spin excitations in the one-third magnetization plateau phase of Ba$_3$CoSb$_2$O$_9$

cond-mat.str-el

Reinforced stochastic gradient descent for deep neural network learning Taro Toyoizumi December 30, 2023 ======================================================================= 23pt Magnetization plateaus in quantum magnets—where bosonic quasiparticles crystallize into emergent spin superlattices—are spectacular yet simple examples of collective quantum phenomena escaping classical description. While magnetization plateaus have been observed in a number of spin-1/2 antiferromagnets, the description of their magnetic excitations remains an open theoretical and experimental challenge. Here, we investigate the dynamical properties of the triangular-lattice spin-1/2 antiferromagnet Ba_3CoSb_2O_9 in its one-third magnetization plateau phase using a combination of nonlinear spin-wave theory and neutron scattering measurements. The agreement between our theoretical treatment and the experimental data demonstrates that magnons behave semiclassically in the plateau in spite of the purely quantum origin of the underlying magnetic structure. This allows for a quantitative determination of Ba_3CoSb_2O_9 exchange parameters. We discuss the implication of our results to the deviations from semiclassical behavior observed in zero-field spin dynamics of the same material and conclude they must have an intrinsic origin. 18pt Quantum fluctuations favor collinear spin order in frustrated magnets <cit.>, which can be qualitatively different from the classical limit (S →∞) <cit.>. In particular, quantum effects can produce magnetization plateaus <cit.>, where the magnetization is pinned at a fraction of its saturation value. Magnetization plateaus can be interpreted as crystalline states of bosonic particles, and are naturally stabilized by easy-axis exchange anisotropy, which acts as strong off-site repulsion <cit.>. However, the situation is less evident and more intriguing for isotropic Heisenberg magnets, which typically have no plateaus in the classical limit. In a seminal work, Chubukov and Golosov predicted the 1/3 magnetization plateau in the quantum triangular lattice Heisenberg antiferromagnet (TLHAFM), corresponding to an up-up-down (UUD) state <cit.>. Their predictions were confirmed by numerical studies <cit.> and extended to plateaus in other models <cit.>. Experimentally, the 1/3 plateau has been observed in the spin-1/2 isosceles triangular lattice material Cs_2CuBr_4, <cit.> as well as in the equilateral triangular lattice materials RbFe(MoO_4)_2 (S=5/2) <cit.> and Ba_3CoSb_2O_9 (effective S= 1/2) <cit.>. Notwithstanding the progress in the search of quantum plateaus, much less is known about their excitation spectra. Given that they are stabilized by quantum fluctuations, it is natural to ask if these fluctuations strongly affect the excitation spectrum. The qualitative difference between the plateau and the classical orderings may appear to invalidate spin-wave theory. For instance, the UUD state in the equilateral TLHAFM is not a classical ground state unless the magnetic field H is fine-tuned <cit.>. Consequently, a naive spin wave treatment is doomed to instability. On the other hand, spin wave theory builds on the assumption of an ordered moment |⟨𝐒_𝐫⟩| close to the full moment. Given that a sizable reduction of |⟨𝐒_𝐫⟩| is unlikely within the plateau because of the gapped nature of the spectrum, a spin wave description could be adequate. Although this may seem in conflict with the order-by-disorder mechanism <cit.> stabilizing the plateau <cit.>, this phenomenon is produced by the zero-point energy correction E_ zp = (1/2)∑_𝐪ω_𝐪 + O(S^0) (ω_𝐪 is the spin wave dispersion), which does not necessarily produce a large moment size reduction. One of our goals is to resolve this seemingly contradictory situation. Recently, Alicea developed a method to fix the unphysical spin-wave instability <cit.>. This proposal awaits experimental verification because the excitation spectrum has not been measured over the entire Brillouin zone for any fluctuation-induced plateau. We will demonstrate that the modified nonlinear spin wave (NLSW) approach indeed reproduces the magnetic excitation spectrum of Ba_3CoSb_2O_9 within the 1/3 plateau <cit.>. The resulting model parameters confirm that the anomalous zero-field dynamics reported in two independent experiments <cit.> must be intrinsic and non-classical. In this , we present a comprehensive study of magnon excitations in the 1/3 magnetization plateau phase of a quasi-two-dimensional (quasi-2D) TLHAFM with easy-plane exchange anisotropy. Our study combines NLSW theory with in-field inelastic neutron scattering (INS) measurements of Ba_3CoSb_2O_9. The Hamiltonian is = J ∑_⟨𝐫𝐫'⟩( S^x_𝐫 S^x_𝐫' + S^y_𝐫 S^y_𝐫' + Δ S^z_𝐫 S^z_𝐫') + J_c ∑_𝐫( S^x_𝐫 S^x_𝐫+𝐜/2 + S^y_𝐫 S^y_𝐫+𝐜/2 + Δ S^z_𝐫 S^z_𝐫+𝐜/2) - ∑_𝐫 S_𝐫^x, where ⟨𝐫𝐫'⟩ runs over in-plane nearest-neighbor (NN) sites of the stacked triangular lattice and 𝐜/2 corresponds to the interlayer spacing (Fig. <ref>a). J (J_c) is the antiferromagnetic intralayer (interlayer) NN exchange and 0 ≤Δ < 1. The magnetic field is in the in-plane (x) direction (we use a spin-space coordinate frame where x and y are in the ab plane and z is parallel to c). = g_⊥ H is the reduced field and g_⊥ is the in-plane g-tensor component. This model describes Ba_3CoSb_2O_9 (Fig. <ref>b), which comprises triangular layers of effective spin 1/2 moments arising from the 𝒥 = 1/2 Kramers doublet of Co^2+ in a perfect octahedral ligand field. Excited multiplets are separated by a gap of 200–300 K due to spin-orbit coupling, which is much larger than the Néel temperature T_N = 3.8 K. Below T = T_N, the material develops conventional 120^∘ ordering with wavevector 𝐐 = (1/3,1/3,1) <cit.>. Experiments confirmed a 1/3 magnetization plateau for 𝐇∥ ab (Fig. <ref>c) <cit.>, which is robust down to the lowest temperatures. We compute the dynamical spin structure factor using NLSW theory in the 1/3 plateau phase. We also provide neutron diffraction evidence of the UUD state within the 1/3 plateau of Ba_3CoSb_2O_9, along with maps of the excitation spectrum obtained from INS. The excellent agreement between theory and experiment demonstrates the semiclassical nature of magnons within the 1/3 plateau phase, despite the quantum fluctuation-induced nature of the ground state ordering. § RESULTS §.§ Quantum-mechanical stabilization of the plateau in quasi-2D TLHAFMs. While experimental observations show that deviations from the ideal 2D TLHAFM are small in Ba_3CoSb_2O_9 <cit.>, a simple variational analysis shows that any J_c > 0 is enough to destabilize the UUD state classically. Thus, a naive spin wave treatment leads to instability for J_c > 0. However, the gapped nature of the spectrum <cit.> implies that this phase must have a finite range of stability in quasi-2D materials. This situation must be quite generic among fluctuation-induced plateaus, as they normally require special conditions to be a classical ground state <cit.>. To put this into a proper semiclassical framework, we apply Alicea 's trick originally applied to a distorted triangular lattice <cit.>. Basically, we make a “detour” in the parameter space with the additional 1/S-axis quantifying the quantum effect (Fig. <ref>). Instead of expanding the Hamiltonian around S →∞ for the actual model parameters, we start from the special point, J_c = 0, = 3 J S, and a given value of 0 ≤Δ≤ 1, that includes the UUD state in its classical ground state manifold. Assuming the spin structure in Fig. <ref>a, we define S^x_𝐫 = S̃^z_𝐫,   S^y_𝐫 = S̃^y_𝐫,   S^z_𝐫 = -S̃^x_𝐫, for 𝐫∈ A_e, B_e, A_o, and C_o and S^x_𝐫 = -S̃^z_𝐫,   S^y_𝐫 = S̃^y_𝐫,   S^z_𝐫 = S̃^x_𝐫, for 𝐫∈ C_e, B_o. Introducing the Holstein-Primakoff bosons, a^(†)_μ,𝐫, with 1 ≤μ≤ 6 being the sublattice index for A_e, B_e, C_e, A_o, B_o, and C_o in this order, we have S̃^z_𝐫 = S - a^†_μ,𝐫 a^ _μ,𝐫, S̃^+_𝐫 = S̃^x_𝐫 + i S̃^y_𝐫≈√(2S)( 1 - a^†_μ,𝐫 a^ _μ,𝐫/4S) a^ _μ,𝐫, and S̃^-_𝐫 = ( S̃^+_𝐫)^† for 𝐫∈μ, truncating higher order terms irrelevant for the quartic interaction. We evaluate magnon self-energies arising from decoupling of the quartic term. As shown in Fig. <ref>b, the linear spin wave (LSW) spectrum for J_c = 0 and = 3 J S features two q-linear gapless branches at 𝐪 = 0, both of which are gapped out by the magnon-magnon interaction (Fig. <ref>c). Small deviations from J_c = 0 and = 3 J S do not affect the local stability of the UUD state because the gap must close continuously. Thus, we can investigate the excitation spectrum of quasi-2D systems for fields near = 3 J S. Figures <ref>d and <ref>e show the spectra for shifted by ± 10% from = 3 J S, where we still keep J_c = 0. For < 3 J S, a band-touching and subsequent hybridization appear between the middle and the top bands around 𝐪=(1/6,1/6) (Fig. <ref>d). For > 3 J S, a level-crossing between the middle and bottom bands appears at around 𝐪=0 (Fig. <ref>e). A small J_c > 0 splits the three branches into six (Figs. <ref>f and <ref>g). Figures <ref>a and <ref>b show the reduction of the sublattice ordered moments for S = 1/2, J_c = 0, 0.09J, and selected values of Δ. We find |δ⟨S^x_μ⟩|/S ≲ 30% throughout the local stability range of the plateau. Thus, our semiclassical approach is fully justified within the plateau phase. Figures <ref>c and <ref>d show the field dependence of the staggered magnetization, M_UUD = 1/6(⟨S^x_A_e⟩ + ⟨S^x_B_e⟩ - ⟨S^x_C_e⟩ + ⟨S^x_A_o⟩ - ⟨S^x_B_o⟩ + ⟨S^x_C_o⟩), which is almost field-independent; a slightly enhanced field-independence appears for small Δ. Similarly, while the magnetization is not conserved for Δ 1, it is nearly pinned at 1/3 for the most part of the plateau (Figs. <ref>e and <ref>f). §.§ UUD state in Ba_3CoSb_2O_9. Next we show experimental evidence for the UUD state in Ba_3CoSb_2O_9 by neutron diffraction measurements within the plateau phase for field applied along the [1,-1,0] direction. We used the same single crystals reported in Refs. <cit.>, grown by the traveling-solvent floating-zone technique and characterized by neutron diffraction, magnetic susceptibility, and heat capacity measurements. The space group is P6_3/mmc, with the lattice constants a= b = 5.8562 Åand c = 14.4561 Å. The site-disorder between Co^2+ and Sb^5+ is negligible with the standard deviation of 1%, as reported elsewhere <cit.>. The magnetic and structural properties are consistent with previous reports and confirm high quality of the crystals <cit.>. These crystals were oriented in the (h,h,l) scattering plane. The magnetic Bragg peaks at (1/3, 1/3, 0) and (1/3, 1/3, 1) were measured at T = 1.5 K (Figs. <ref>a and <ref>b). The large intensity at both (1/3, 1/3, 0) and (1/3, 1/3, 1) confirms the UUD state at μ_0 H ≥ 9.8 T <cit.> (Fig. <ref>c). The estimated ordered moment is 1.65(3) at 10 T and 1.80(9) at 10.9 T. They correspond to 85(2)% and 93(5)% of the full moment <cit.>, roughly coinciding with the predicted range (Fig. <ref>d). This diffraction pattern can be contrasted with that of the 120^∘ state, characterized by a combination of the large intenisty at (1/3, 1/3, 1) and lack of one at (1/3, 1/3, 0). Our diffraction result is fully consistent with previous nuclear magnetic resonance (NMR) <cit.> and magnetization measurements <cit.>. §.§ Excitation spectrum. We now turn to the dynamical properties in the UUD phase. Figures <ref>a–<ref>c show the INS intensity I(𝐪,ω) ≡ k_ i/k_ f (d^2σ/dΩdE_ f) along high-symmetry directions. The applied magnetic field μ_0 H = 10.5 T is relatively close to the transition field μ_0 H_c1 = 9.8 T <cit.> bordering on the low-field coplanar ordered phase <cit.>, while the temperature T = 0.5 K is low enough compared to T_N≈ 5 K <cit.> for the UUD phase at this magnetic field. The in-plane dispersion shown in Fig. <ref>a comprises a seemingly gapless branch at 𝐪=(1/3, 1/3, -1) (Fig. <ref>c), and two gapped modes centered around 1.6 meV and 2.7 meV. Due to the interlayer coupling, each mode corresponds to two non-degenerate branches. As their splitting is below the instrumental resolution, we simply refer to them as ω_1, ω_2 and ω_3, unless otherwise mentioned (Fig. <ref>). The dispersions along the c-direction are nearly flat, as shown in Figs. <ref>b and <ref>c for q=(1/2, 1/2, l) and q=(1/3, 1/3, l), respectively, reflecting the quasi-2D lattice <cit.>. Among the spin wave modes along 𝐪=(1/2, 1/2, l) and 𝐪=(1/3, 1/3, l) in Figs. <ref>b and <ref>c, ω_1 for 𝐪=(1/2, 1/2, l) displays a relatively sharp spectral line. As discussed below, most of the broadening stems from the different intensities of the split modes due to finite J_c. Comparing the experiment against the NLSW calculation, we find that the features of the in-plane spectrum in Fig. <ref>a are roughly captured by the theoretical calculation near the low-field onset of the plateau in Fig. <ref>f (= 2.7 J S ≈ 1.03). This observation is in accord with the fact that the applied field (μ_0 H = 10.5 T) is close to μ_0 H_c1 = 9.8 T <cit.>. To refine the quantitative comparison, we calculate the scattering intensity I_tot(𝐪,ω) ≡ (γ r_0/2)^2 | F(𝐪) |^2 ∑_α (1 - q̂^αq̂^α) g^2_α𝒮^αα(𝐪,ω) where F(𝐪) denotes the magnetic form factor of Co^2+ corrected with the orbital contribution, (γ r_0/2)^2 is a constant, q̂^α = q^α/|𝐪|, and 𝒮^αα(𝐪,ω) are the diagonal components of the dynamical structure factor evaluated at 10.5 T; off-diagonal components are zero due to symmetry. Defining the UUD order as shown in Fig. <ref>a, transverse spin fluctuations related to single-magnon excitations appear in 𝒮^yy and 𝒮^zz, while longitudinal spin fluctuations corresponding to the two-magnon continuum appear in the inelastic part of 𝒮^xx, denoted as 𝒮_∥. Accordingly, I_tot(𝐪,ω) can be separated into transverse I_⊥ and longitudinal I_∥ contributions. To compare with our experiments, the theoretical intensity is convoluted with momentum binning effects (only for I_⊥) and empirical instrumental energy resolution. Figures <ref>d–<ref>f show the calculated I_⊥(𝐪,ω), along the same high-symmetry paths as the experimental results in Figs. <ref>a–<ref>c, for , , , and . The agreement between theory and experiment is excellent. When deriving these estimates, J is controlled by the saturation field μ_0 H_sat = 32.8 T for 𝐇∥ĉ <cit.>. To obtain the best fit, we also analyzed the field dependence of ω_1, ω_2 and ω_3 (Fig. <ref>). Remarkably, the calculation in Figs. <ref>d–<ref>f reproduces the dispersions almost quantitatively. It predicts a gapped ω_1 mode, although the gap is below experimental resolution. The smallness of the gap is simply due to proximity to H_c1. For each ω_i, the band splitting due to J_c yields pairs of poles ω^±_i dispersing with a phase difference of π in the out-of-triangular-plane direction (Figs. <ref>e and <ref>f). For each pair, however, one pole has a vanishing intensity for q=(1/2, 1/2, l). Consequently, ω_1 along this direction is free from any extrinsic broadening caused by overlapping branches (Fig. <ref>e), yielding a relatively sharp spectral line (Fig. <ref>h). The corresponding bandwidth ≈ 0.2 meV (Fig. <ref>b) provides a correct estimate for J_c. By contrast, for q=(1/3, 1/3, l), all six ω^±_i branches have non-zero intensity, which leads to broadened spectra and less obvious dispersion along l (Figs. <ref>c and <ref>f). The field-dependence of ω_1–ω_3 at 𝐪= (1/3, 1/3, 1) is extracted from constant-q scans at T = 0.1 K for selected fields 10.5 T–13.5 T within the plateau (Fig. <ref>a). By fitting the field-dependence of the low-energy branches of ω_1,2, which become gapless at a plateau edge, we obtain the quoted model parameters. The field dependence is reproduced fairly well (Figs. <ref>b and <ref>c), although the calculation slightly underestimates ω_3. We find ω_1 and ω_3 (ω_2) increase (decreases) almost linearly in H, while the ω_1 and ω_2 branches cross around 12.6 T. The softening of ω_1 (ω_2) at the lower (higher) transition field induces the Y-like (V-like) state, respectively <cit.>. The nonlinearity of the first excitation gap near these transitions (visible only in the calculation) is due to the anisotropy; there is no U(1) symmetry along the field direction for Δ 1. § DISCUSSION Our work has mapped out the excitation spectrum in the 1/3 plateau—a manifestation of quantum order-by-disorder effect—in Ba_3CoSb_2O_9. Despite the quantum-mechanical origin of the ground state ordering, we have unambiguously demonstrated the semiclassical nature of magnons in this phase. In fact, the calculated reduction of the sublattice magnetization, δ S_μ = S - |⟨𝐒_𝐫⟩| with 𝐫∈μ, is relatively small (Fig. <ref>b): δ S_A_e = δ S_A_o = 0.083, δ S_B_e = δ S_C_o = 0.073, and δ S_C_e = δ S_B_o = 0.14 at 10.5 T for the quoted model parameters. This is consistent with the very weak intensity of the two-magnon continuum (Figs. <ref>g and <ref>h). This semiclassical behavior is protected by the excitation gap induced by anharmonicity of the spin waves (magnon-magnon interaction). We note that a perfect collinear magnetic order does not break any continuous symmetry even for Δ = 1, i.e., there is no gapless Nambu-Goldstone mode. The collinearity also means that three-magnon processes are not allowed <cit.>. The gap is robust against perturbations, such as anisotropies, lattice deformations <cit.>, or biquadratic couplings for S > 1/2 (a ferroquadrupolar coupling can stabilize the plateau even classically <cit.>). Thus, we expect the semiclassical nature of the excitation spectrum to be common to other 2D and quasi-2D realizations of fluctuation-induced plateaus, such as the 1/3 plateau in the spin-5/2 material RbFe(MoO_4)_2 <cit.>. Meanwhile, it will be interesting to examine the validity of the semiclassical approach in quasi-1D TLHAFMs, such as Cs_2CuBr_4 <cit.>, where quantum fluctuations are expected to be stronger. Finally, we discuss the implications of our results for the zero-field dynamical properties of the same material, where recent experiments revealed unexpected phenomena, such as broadening of the magnon peaks indescribable by conventional spin-wave theory, large intensity of the high-energy continuum <cit.>, and the extension thereof to anomalously high frequencies <cit.>. Specifically, Ref. <cit.> reported magnon spectral-line broadened throughout the entire Brillouin zone, significantly beyond instrumental resolution, and a high frequency (≳ 2 meV) excitation continuum with an almost comparable spectral weight as single-magnon modes. All of these experimental observations indicate strong quantum effects. Given that the spin Hamiltonian has been reliably determined from our study of the plateau phase, it is interesting to reexamine if a semiclassical treatment of this Hamiltonian can account for the zero-field anomalies. A semiclassical treatment can only explain the line broadening in terms of magnon decay <cit.>. NLSW theory at H = 0 describes the spin fluctuations around the 120^∘ ordered state by incorporating single-to-two magnon decay at the leading order O(S^0). At this order, the two-magnon continuum is evaluated by convoluting LSW frequencies. The self-energies include Hartree-Fock decoupling terms as well as the bubble Feynman diagrams comprising a pair of cubic vertices Γ_3 ∼ O(S^1/2) <cit.>, with the latter computed with the off-shell treatment. The most crucial one corresponds to the single-to-two magnon decay (see the inset of Fig. <ref>a), Σ(𝐪,ω) = 1/2N∑_𝐤|Γ_3(𝐤, 𝐪 - 𝐤; 𝐪)|^2/ω - ω^H=0_𝐤 - ω^H=0_𝐪 - 𝐤 + i0, where ω^H=0_𝐤 denotes the zero-field magnon dispersion. We show the zero-field dynamical structure factor, 𝒮_H=0^tot(𝐪,ω), at the M point for representative parameters in Figs. <ref>a–<ref>d. The NLSW result for the ideal TLHAFM (J_c = 0 and Δ = 1) exhibits sizable broadening and a strong two-magnon continuum <cit.> (see also Fig. <ref>e). However, a slight deviation from Δ = 1 renders the decay process ineffective because the kinematic condition, ω^H=0_𝐪 = ω^H=0_𝐤 + ω^H=0_𝐪 - 𝐤, can no longer be fulfilled in 2D for any decay vertex over the entire Brillouin zone if Δ≲ 0.92 <cit.>. This situation can be inferred from the result for J_c = 0 and Δ = 0.85, where the two-magnon continuum is pushed to higher frequencies, detached from the single-magnon peaks. In fact, the sharp magnon lines are free from broadening. The suppression of decay results from gapping out one of the two Nambu-Goldstone modes upon lowering the Hamiltonian symmetry from SU(2) to U(1), which greatly reduces the phase space for magnon decay. The interlayer coupling renders the single-magnon peaks even sharper and the continuum even weaker (Figs <ref>c and <ref>d). To determine whether the anomalous zero-field spin dynamics can be explained by a conventional 1/S expansion, it is crucial to estimate Δ very accurately. Previous experiments reported Δ = 0.95 (low-field electron spin resonance experiments compared with LSW theory <cit.>) and Δ = 0.89 (zero-field INS experiments compared with NLSW theory <cit.>). However, the NLSW calculation reported a large renormalization of the magnon bandwidth (≈ 40% reduction relative to the LSW theory) <cit.>, suggesting that the previous estimates of Δ may be inaccurate. Particularly, given that Δ is extracted by fitting the induced gap ∝√(1 - Δ), the LSW approximation underestimates 1 - Δ (deviation from the isotropic exchange) because it overestimates the proportionality constant <cit.>. Figures <ref>d and <ref>f show 𝒮_H=0^tot(𝐪,ω) for and . We find that 𝒮_H=0^tot(𝐪,ω) remains essentially semiclassical, with sharp magnon lines and a weak continuum, which deviates significantly from the recent results of INS experiments <cit.>. We thus conclude that the Hamiltonian that reproduces the plateau dynamics fails to do so at H = 0 within the spin wave theory, even after taking magnon-magnon interactions into account at the 1/S level. We also mention that the breakdown of the kinematic condition for single-to-two magnon decay also implies the breakdown of the condition for magnon decay into an arbitrary number of magnons <cit.>. Thus, the semiclassical picture of weakly interacting magnons is likely inadequate to simultaneously explain the low-energy dispersions and the intrinsic incoherent features (such as the high-intensity continuum and the line-broadening) observed in Ba_3CoSb_2O_9 at H = 0. One may wonder if extrinsic effects can explain these experimental observations. It is possible for exchange disorder to produce continuous excitations as in the effective spin-1/2 triangular antiferromagnet YbMgGaO_4 <cit.>. However, our single crystals are the high-quality samples used in Refs. <cit.>, essentially free from Co^2+-Sb^5+ site-disorder. Indeed, our crystals show only one sharp peak at 3.6 K in the zero-field specific heat <cit.> in contrast to the previous reports of multiple peaks <cit.>, which may indicate multi-domain structure. Another possible extrinsic effect is the magnon-phonon coupling, that has been invoked to explain the measured spectrum of the spin-3/2 TLHAFM CuCrO_2 <cit.>. However, if that effect were present at zero field, it should also be present in the UUD state. The fact that Eq. (<ref>) reproduces the measured excitation spectrum of the UUD state suggests that the magnon-phonon coupling is negligibly small (a similar line of reasoning can also be applied to the effect of disorder). Indeed, we have also measured the phonon spectrum of Ba_3CoSb_2O_9 in zero field by INS and found no strong signal of magnon-phonon coupling. Our results then suggest that the dynamics of the spin-1/2 TLHAFM is dominated by intrinsic quantum mechanical effects that escape a semiclassical spin-wave description. This situation is analogous to the (π,0) wave-vector anomaly observed in various spin-1/2 square-lattice Heisenberg antiferromagnets, <cit.> but now extending to the entire Brillouin zone in the triangular lattice. Given recent theoretical success on the square-lattice <cit.>, our results motivate new non-perturbative studies of the spin-1/2 TLHAFM. § METHODS §.§ Neutron scattering measurements. The neutron diffraction data under magnetic fields applied in the [1,-1,0] direction were obtained by using CG-4C cold triple-axis spectrometer with the neutron energy fixed at 5.0 meV at High Flux Isotope Reactor (HFIR), Oak Ridge National Laboratory (ORNL). The nuclear structure of the crystal was determined at the HB-3A four-circle neutron diffractometer at HFIR, ORNL and then was used to fit the nuclear reflections measured at the CG-4C to confirm that the data reduction is valid. Only the scale factor was refined for fitting the nuclear reflections collected at CG-4C and was also used to scale the moment size for the magnetic structure refinement. 14 magnetic Bragg peaks collected at CG-4C at 10 T were used for the magnetic structure refinement. The UUD spin configuration with the spins along the field direction was found to best fit the data. The nuclear and magnetic structure refinements were carried out using FullProf Suite <cit.>. Our inelastic neutron scattering experiments were carried out with the Multi Axis Crystal Spectrometer (MACS) <cit.> at NIST Center for Neutron Research (NCNR), NIST, and the cold neutron triple-axis spectrometer (V2-FLEXX) <cit.> at Helmholtz-Zentrum Berlin (HZB). The final energies were fixed at 3 meV and 5 meV on the MACS and 3.0 meV on V2-FLEXX. §.§ Constraint on J due to the saturation field. An exact expression for the saturation field for 𝐇∥ĉ, H_sat, can be obtained from the level crossing condition between the fully polarized state and the ground state in the single-spin-flip sector. From the corresponding expression, we obtain: J = g_∥ H_sat S^-1/ 3 + 6 Δ + 2 (1 + Δ) (J_c/J) , where g_∥ = 3.87 and μ_0 H_sat = 32.8 T <cit.>. §.§ Variational analysis on classical instability of the 1/3 plateau in quasi-2D TLHAFMs. We show that the UUD state is not the classical ground state in the presence of the antiferromagnetic interlayer exchange J_c > 0. To verify that the classical ground space for J_c = 0 acquires accidental degeneracy in the in-plane magnetic field, we rewrite Eq. (<ref>) as = J/2∑_( 𝐒_,A + 𝐒_,B + 𝐒_,C - /3J𝐱̂)^2 - (1 - Δ) J ∑_⟨𝐫𝐫'⟩ S^z_𝐫 S^z_𝐫' + J_c ∑_𝐫( S^x_𝐫 S^x_𝐫+𝐜̂/2 + S^y_𝐫 S^y_𝐫+𝐜̂/2 + Δ S^z_𝐫 S^z_𝐫+𝐜̂/2) + const., where the summation of ∑_ is taken over the corner-sharing triangles in each layer, with 𝐫 = (,μ) (μ=A,B,C) denoting the sublattice sites in each triangle. This simply provides an alternative view of each triangular lattice layer (Fig. <ref>a). 𝐱̂ is the unit vector in the x or field direction. The easy-plane anisotropy forces every spin of the classical ground state to lie in the ab plane and the second term in Eq. (<ref>) has no contribution. Hence, for J_c = 0, any three-sublattice spin configuration satisfying S^z_𝐫 = 0 and 𝐒_,A + 𝐒_,B + 𝐒_,C = /3J𝐱̂,  ∀, is a classical ground state, where we momentarily regard 𝐒_,μ as three-component classical spins of length S. Since there are only two conditions corresponding to the x and y components of Eq. (<ref>), whereas three angular variables are needed to specify the three-sublattice state in the ab plane, the classical ground state manifold for J_c = 0 retains an accidental degeneracy, similar to the well-known case of the Heisenberg model (Δ = 1) <cit.>. The UUD state is the classical ground state only for = 3 J S. The classical instability of the UUD state for J_c > 0 can be demonstrated by a variational analysis. The UUD state in the 3D lattice enforces frustration for one third of the antiferromagnetic interlayer bonds, inducing large variance of the interlayer interaction. As shown in Fig. <ref>a, only two of the three spin pairs along the 𝐜 axis per magnetic unit cell can be antiferromagnetically aligned, as favored by J_c, while the last one has to be aligned ferromagnetically. To seek for a better classical solution, we consider a deformation of the spin configuration parameterized by 0 ≤θ≤π at = 3 J S, such that the spin structure becomes noncollinear within the ab plane (Fig. <ref>b). Because the magnetization in each layer is fixed at S/3 per spin, the sum of the energies associated with the intralayer interaction and the Zeeman coupling is unchanged under this deformation. In the meantime, the energy per magnetic unit cell of the interlayer coupling is varied as E_c(θ) = 2 J_c S^2 ( cos2θ - 2 cosθ). We find that E_c(θ) is minimized at θ = π/3 for J_c>0, corresponding to a saddle point. This is a rather good approximation of the actual classical ground state for small J_c > 0, as can be demonstrated by direct minimization of the classical energy obtained from Eq. (<ref>). The crucial observation is that the classical ground state differs from the θ = 0 UUD state. §.§ NLSW calculation for the UUD state. We summarize the derivation of the spin wave spectrum in the quasi-2D TLHAFM with easy-plane anisotropy [see Eq. (<ref>)]. As discussed in the main text, we first work on the 2D limit J_c = 0 exactly at = 3JS, and a given value of 0 ≤Δ≤ 1, which are the conditions for the UUD state to be the classical ground state. Defining the UUD state as shown in Fig. <ref>a, we introduce the Holstein-Primakoff bosons, a^(†)_μ,𝐫 as in Eqs. (<ref>)–(<ref>). After performing a Fourier transformation, a^ _μ,𝐤 = (1/)^1/2∑_𝐫∈μ e^-i𝐤·𝐫 a_μ,𝐫, where = N/6 is the number of magnetic unit cells (six spins for each) and N is the total number of spins, we obtain the quadratic Hamiltonian as the sum of even layers (sublattices ) and odd layers (sublattices A_o–C_o) contributions: = + , where the constant term has been dropped. Here, = S/2∑_𝐤∈RBZ[ (𝐚^†_𝐤)^T (𝐚^ _-𝐤)^T ][ H^0_11,𝐤 H^0_12,𝐤; H^0_21,𝐤 H^0_22,𝐤 ][ 𝐚^ _𝐤; 𝐚^†_-𝐤 ], with H^0_11,𝐤 = H^0_22,𝐤, H^0_12,𝐤 = H^0_21,𝐤, where the summation over 𝐤 is taken in the reduced Brillouin zone (RBZ) corresponding to the magnetic unit cell of the UUD state. From now on, we will denote this summation as ∑_𝐤. We have introduced vector notation for the operators, 𝐚^_𝐤 = [ a^ _A_e,𝐤; a^ _B_e,𝐤; a^ _C_e,𝐤 ]≡[ a^ _1,𝐤; a^ _2,𝐤; a^ _3,𝐤 ],  𝐚^†_-𝐤 = [ a^†_A_e,-𝐤; a^†_B_e,-𝐤; a^†_C_e,-𝐤 ]≡[ a^†_1,-𝐤; a^†_2,-𝐤; a^†_3,-𝐤 ], and matrix notation for the quadratic coefficients, H^0_11,𝐤 = [ S^-1 3/2J(1 + Δ) γ^_ k 3/2J(1 - Δ) γ^_- k; 3/2J(1 + Δ) γ^_- k S^-1 3/2J(1 - Δ) γ^_ k; 3/2J(1 - Δ) γ^_ k 3/2J(1 - Δ) γ^_- k 6J - S^-1 ], [1] H^0_12,𝐤 = [ 0 -3/2J(1 - Δ) γ^_ k -3/2J(1 + Δ) γ^_- k; -3/2J(1 - Δ) γ^_- k 0 -3/2J(1 + Δ) γ^_ k; -3/2J(1 + Δ) γ^_ k -3/2J(1 + Δ) γ^_- k 0 ], with γ_𝐤 = 1/3 (e^i𝐤·𝐚 + e^i𝐤·𝐛 + e^-i𝐤·(𝐚 + 𝐛)). Similarly, we have = S/2∑_𝐤[ (𝐚̅^†_𝐤)^T (𝐚̅^ _-𝐤)^T ][ H̅^0_11,𝐤 H̅^0_12,𝐤; H̅^0_21,𝐤 H̅^0_22,𝐤 ][ 𝐚̅^ _𝐤; 𝐚̅^†_-𝐤 ], with 𝐚̅^_𝐤 = [ a^ _A_o,𝐤; a^ _B_o,𝐤; a^ _C_o,𝐤 ]≡[ a^ _4,𝐤; a^ _5,𝐤; a^ _6,𝐤 ],  𝐚̅^†_-𝐤 = [ a^†_A_o,-𝐤; a^†_B_o,-𝐤; a^†_C_o,-𝐤 ]≡[ a^†_4,-𝐤; a^†_5,-𝐤; a^†_6,-𝐤 ], and H̅^0_11,𝐤 = H̅^0_22,𝐤 = [ 0 1 0; 0 0 1; 1 0 0 ] H^0_11,𝐤[ 0 0 1; 1 0 0; 0 1 0 ], H̅^0_12,𝐤 = H̅^0_21,𝐤 = [ 0 1 0; 0 0 1; 1 0 0 ] H^0_12,𝐤[ 0 0 1; 1 0 0; 0 1 0 ]. The excitation spectrum of retains two relativistic modes at 𝐤 = 0 (Fig. <ref>b). Below, we include nonlinear terms to gap out these excitations. At this stage, the nonlinear terms correspond to the mean-field (MF) decoupling of the intra-layer quartic terms. Once we obtain such a MF Hamiltonian with the gapped spectrum, the deviation from the fine-tuned magnetic field = 3JS and interlayer coupling (as well as some other perturbation, if any) can be included. Here, the additional term contains both LSW and NLSW terms. To proceed, we first define the following mean-fields (MFs) symmetrized by using translational and rotational invariance: ρ^_μ = 1/∑_𝐫∈μa^†_μ,𝐫 a^ _μ,𝐫, δ^_μ = 1/∑_𝐫∈μ(a^ _μ,𝐫)^2, [1] ξ^_μν = 1/3∑_𝐫∈μ∑_η̂_μν^a^†_μ,𝐫 a^ _ν,𝐫+η̂_μν^, [1] ζ^_μν = 1/3∑_𝐫∈μ∑_η̂_μν^a^ _μ,𝐫 a^ _ν,𝐫+η̂_μν^, where η̂_μν^ represents the in-plane displacement vector connecting sites 𝐫∈μ to a nearest neighbor site in sublattice ν. The mean values ... are evaluated with the ground state of . The MFs for odd (even) layers are obtained from those for even (odd) layers as ρ^_A_o = ρ^_B_e,  ρ^_B_o = ρ^_C_e,  ρ^_C_o = ρ^_A_e,  [1] δ^_A_o = δ^_B_e,  δ^_B_o = δ^_C_e,  δ^_C_o = δ^_A_e,  [1] ξ^_A_oB_o = ξ^_B_eC_e,  ξ^_B_oC_o = ξ^_C_eA_e,  ξ^_C_oA_o = ξ^_A_eB_e,  [1] ζ^_A_oB_o = ζ^_B_eC_e,  ζ^_B_oC_o = ζ^_C_eA_e,  ζ^_C_oA_o = ζ^_A_eB_e. By collecting all the contributions mentioned above, we obtain the NLSW Hamiltonian, = S/2∑_𝐤[ (𝐚^†_𝐤)^T (𝐚̅^†_𝐤)^T (𝐚^ _-𝐤)^T (𝐚̅^ _-𝐤)^T ][ 𝐤^ 𝐤^ 𝐤' 𝐤'; (𝐤^)_^† 𝐤^ (-𝐤')_^T 𝐤'; (-𝐤')_^∗ (-𝐤')_^∗ (-𝐤^)_^∗ (-𝐤^)_^∗; (𝐤')_^† (-𝐤')_^∗ (-𝐤^)_^T (-𝐤^)_^∗ ][ 𝐚^ _𝐤; 𝐚̅^ _𝐤; 𝐚^†_-𝐤; 𝐚̅^†_-𝐤 ], where 𝐤^ = H^0_11,𝐤 + [ -2 J_c 0 0; 0 2 J_c 0; 0 0 2 J_c ] + S^-1[ μ^A_e_MF + 2 J_c ρ^_A_o (t^A_eB_e_MF)^∗γ^_𝐤 t^C_eA_e_MFγ^_-𝐤; t^A_eB_e_MFγ^_-𝐤 μ^B_e_MF -2 J_c ρ^_B_o (t^B_eC_e_MF)^∗γ^_𝐤; (t^C_eA_e_MF)^∗γ^_𝐤 t^B_eC_e_MFγ^_-𝐤 μ^C_e_MF - 2 J_c ρ^_C_o ], [1] 𝐤^ = H̅^0_11,𝐤 + [ -2 J_c 0 0; 0 2 J_c 0; 0 0 2 J_c ] + S^-1[ μ^A_o_MF + 2 J_c ρ^_A_e (t^A_oB_o_MF)^∗γ^_𝐤 t^C_oA_o_MFγ^_-𝐤; t^A_oB_o_MFγ^_-𝐤 μ^B_o_MF - 2 J_c ρ^_B_e (t^B_oC_o_MF)^∗γ^_𝐤; (t^C_oA_o_MF)^∗γ^_𝐤 t^B_oC_o_MFγ^_-𝐤 μ^C_o_MF - 2 J_c ρ^_C_e ], [1] 𝐤' = H^0_12,𝐤 + S^-1[ Γ^A_e_MF g^A_eB_e_MFγ^_𝐤 g^C_eA_e_MFγ^_-𝐤; g^A_eB_e_MFγ^_-𝐤 Γ^B_e_MF g^B_eC_e_MFγ^_𝐤; g^C_eA_e_MFγ^_𝐤 g^B_eC_e_MFγ^_-𝐤 Γ^C_MF ], [1] 𝐤' = H̅^0_12,𝐤 + S^-1[ Γ^A_o_MF g^A_oB_o_MFγ^_𝐤 g^C_oA_o_MFγ^_-𝐤; g^A_oB_o_MFγ^_-𝐤 Γ^B_o_MF g^B_oC_o_MFγ^_𝐤; g^C_oA_o_MFγ^_𝐤 g^B_oC_o_MFγ^_-𝐤 Γ^C_o_MF ], [1] 𝐤^ = cos k_3 [ J_c (1 + Δ) 0 0; 0 J_c (1 - Δ) 0; 0 0 J_c(1 - Δ) ] + S^-1cos k_3 [ (t^A_eA_o_MF)^∗ 0 0; 0 (t^B_eB_o_MF)^∗ 0; 0 0 (t^C_eC_o_MF)^∗ ], [1] 𝐤' = cos k_3 [ -J_c (1 - Δ) 0 0; 0 -J_c (1 + Δ) 0; 0 0 -J_c(1 + Δ) ] + S^-1cos k_3 [ g_MF^A_eA_o 0 0; 0 g_MF^B_eB_o 0; 0 0 g_MF^C_eC_o ]. Here the MF parameters are given as follows. First, those associated with the intralayer coupling are μ^A_e_MF = 3J [ ρ^_B_e - ρ^_C_e - 1 + Δ/2( ξ^_A_eB_e - ζ^_C_eA_e) - 1 - Δ/2( ξ^_C_eA_e - ζ^_A_eB_e) ], [1] μ^B_e_MF = 3J [ ρ^_A_e - ρ^_C_e - 1 + Δ/2( ξ^_A_eB_e - ζ^_B_eC_e) - 1 - Δ/2( ξ^_B_eC_e - ζ^_A_eB_e) ], [1] μ^C_e_MF = 3J [ -ρ^_A_e - ρ^_B_e + 1 + Δ/2( ζ^_B_eC_e + ζ^_C_eA_e) - 1 - Δ/2( ξ^_B_eC_e + ξ^_C_eA_e) ], [1] t^A_eB_e_MF = 3J [ ξ^_A_eB_e - 1 + Δ/4( ρ^_A_e + ρ^_B_e) + 1 - Δ/8( δ^∗_A_e + δ^_B_e) ], [1] t^B_eC_e_MF = 3J [ -ξ^_B_eC_e + 1 + Δ/8( δ^∗_B_e + δ^_C_e) - 1 - Δ/4( ρ^_B_e + ρ^_C_e) ], [1] t^C_eA_e_MF = 3J [ -ξ^_C_eA_e + 1 + Δ/8( δ^∗_C_e + δ^_A_e) - 1 - Δ/4( ρ^_C_e + ρ^_A_e) ], [1] Γ^A_e_MF = 3J/2[ 1 + Δ/2(ξ^_C_eA_e - ζ^_A_eB_e) + 1 - Δ/2(ξ^∗_A_eB_e - ζ^_C_eA_e) ], [1] Γ^B_e_MF = 3J/2[ 1 + Δ/2(ξ^∗_B_eC_e - ζ^_A_eB_e) + 1 - Δ/2(ξ^_A_eB_e - ζ^_B_eC_e) ], [1] Γ^C_e_MF = 3J/2[ 1 + Δ/2(ξ^_B_eC_e - ξ^∗_C_eA_e) - 1 - Δ/2(ζ^_B_eC_e + ζ^_C_eA_e) ], [1] g^A_eB_e_MF = 3J [ ζ^_A_eB_e - 1 + Δ/8( δ^_A_e + δ^_B_e) + 1 - Δ/4( ρ^_A_e + ρ^_B_e) ], [1] g^B_eC_e_MF = 3J [ -ζ^_B_eC_e + 1 + Δ/4( ρ^_B + ρ^_C) - 1 - Δ/8( δ^_B + δ^_C) ], [1] g^C_eA_e_MF = 3J [ -ζ^_C_eA_e + 1 + Δ/4( ρ^_C_e + ρ^_A_e) - 1 - Δ/8( δ^_C_e + δ^_A_e) ], for even layers and μ_MF^A_o = μ_MF^B_e,  μ_MF^B_o = μ_MF^C_e,  μ_MF^C_o = μ_MF^A_e,   t_MF^A_oB_o = t_MF^B_eC_e,   t_MF^B_oC_o = t_MF^C_eA_e,   t_MF^C_oA_o = t_MF^A_eB_e, Γ_MF^A_o = Γ_MF^B_e,  Γ_MF^B_o = Γ_MF^C_e,  Γ_MF^C_o = Γ_MF^A_e, [1] g_MF^A_oB_o = g_MF^B_eC_e,   g_MF^B_oC_o = g_MF^C_eA_e,   g_MF^C_oA_o = g_MF^A_eB_e, for odd layers. Similarly, the new MF parameters associated with the interlayer coupling are t^A_eA_o_MF = J_c [ - 1 + Δ/2( ρ^_A_e + ρ^_A_o) + 1 - Δ/4( δ^∗_A_e + δ^_A_o) ], t^B_eB_o_MF = J_c [ 1 + Δ/4( δ^∗_B_e + δ^_B_o) - 1 - Δ/2( ρ^_B_e + ρ^_B_o) ], t^C_eC_o_MF = J_c [ 1 + Δ/4( δ^∗_C_e + δ^_C_o) - 1 - Δ/2( ρ^_C_e + ρ^_C_o) ], g_MF^A_eA_o = J_c [ - 1 + Δ/4( δ^_A_e + δ^_A_o) + 1 - Δ/2( ρ^_A_e + ρ^_A_o) ], [1] g_MF^B_eB_o = J_c [ 1 + Δ/2( ρ^_B_e + ρ^_B_o) - 1 - Δ/4( δ^_B_e + δ^_B_o) ], g_MF^C_eC_o = J_c [ 1 + Δ/2( ρ^_C_e + ρ^_C_o) - 1 - Δ/4( δ^_C_e + δ^_C_o) ]. Figure <ref> shows the Δ-dependence of these MF parameters. Because these MF parameters are real valued, the coefficient matrix in Eq. (<ref>) has the form, H_NLSW = [ P^_𝐤 Q^_𝐤; Q^_𝐤 P^_𝐤 ], with P^_𝐤 = [ 𝐤^ 𝐤^; 𝐤^ 𝐤^ ],      Q^_𝐤 = [ 𝐤' 𝐤'; 𝐤' 𝐤' ]. This form can be diagonalized by a Bogoliubov transformation, [ 𝐚^ _𝐤; 𝐚̅^ _𝐤; 𝐚^†_-𝐤; 𝐚̅^†_-𝐤 ] = ([ ; 3c.5U_𝐤 3c.5V_𝐤; ; 3c.5V_𝐤 3c.5U_𝐤 ]) ([ ; .5α^ _𝐤; ; .5α^†_-𝐤 ]), where α^ _𝐤 (α^†_-𝐤) is the 6-component vector comprising the annihilation (creation) operators of Bogoliubov bosons. The transformation matrices satisfy U_𝐤^μκ = (U_-𝐤^μκ)^∗ and V_𝐤^μκ = (V_-𝐤^μκ)^∗. The poles, ω^_κ,𝐤, are the square-roots of the eigenvalues of S^2(P_𝐤 + Q_𝐤) (P_𝐤 - Q_𝐤). When calculating the sublattice magnetization, the reduction of the ordered moment relative to the classical value S corresponds to the local magnon density. With the phase factors for each sublattice, c^_A_e = c^_B_e = -c^_C_e = c^_A_o = -c^_B_o = c^_C_o = 1 (see Fig. <ref>a), we have ⟨S^x_𝐫⟩ = c^_μ( S - ⟨ a^†_μ,𝐫 a^ _μ,𝐫⟩) = c^_μ( S - 1/∑_𝐤∑_κ| V_𝐤^μκ|^2 ), for site 𝐫 in sublattice μ. The dynamical spin structure factor is defined by 𝒮^αα(𝐪,ω) = ∫_-∞^∞dt/2π e^iω t1/N∑_𝐫,𝐫' e^-i𝐪· (𝐫 - 𝐫')⟨ S_𝐫^α(t) S_𝐫'^α(0) ⟩ = ∑_n δ(ω - ω_n) |⟨0| S_𝐪^α|n⟩|^2, where S_𝐪^α = N^-1/2∑_𝐫 S_𝐫^α e^-i𝐪·𝐫 and |n⟩ and ω_n denote the nth excited state and its excitation energy, respectively. The longitudinal spin component is S_𝐪^x = √(N)/3S (δ_𝐪,0 + δ_𝐪,𝐐 + δ_𝐪,-𝐐) + δ S_𝐪^x, with 𝐐 = (1/3,1/3,1) and δ S_𝐪^x = - √(1/N)∑_μ,𝐤 c^_μ a^†_μ,𝐤 - 𝐪 a^ _μ,𝐤, We truncate the expansions of the transverse spin components at the lowest order: S_𝐪^y ≈ -i √(S/12)∑_μ( a^ _μ,𝐪 - a^†_μ,-𝐪), S_𝐪^z ≈√(S/12)∑_μ (-c^_μ) ( a^ _μ,𝐪 + a^†_μ,-𝐪). The transverse components of the dynamical structure factor, 𝒮_⊥(𝐪,ω) = 𝒮^yy(𝐪,ω) + 𝒮^zz(𝐪,ω), reveal the magnon dispersion, 𝒮^yy(𝐪,ω) = ∑_n δ(ω - ω_n) |⟨0| S_𝐪^y|n⟩|^2 ≈S/12∑_κδ(ω - ω^_κ,𝐪) |∑_μ(U_𝐪^μκ - V_𝐪^μκ) |^2, 𝒮^zz(𝐪,ω) = ∑_n δ(ω - ω_n) |⟨0| S_𝐪^z|n⟩|^2 ≈S/12∑_κδ(ω - ω^_κ,𝐪) |∑_μ c^_μ(U_𝐪^μκ + V_𝐪^μκ) |^2. Meanwhile, 𝒮^xx(𝐪,ω) comprises the elastic contribution and the longitudinal fluctuations, 𝒮_∥(𝐪,ω) = ∑_n δ(ω - ω_n) |⟨0|δ S_𝐪^x|n⟩|^2, which can be evaluated by using Wick's theorem. The result at T = 0 is 𝒮_∥(𝐪,ω) = Θ(ω) N^-1∑_𝐤∑_κ,λ,μ,ν c^_μ c^_νA_μν;κλ(𝐤;𝐪) δ(ω - ω^_κ,-𝐤+𝐪 - ω^_λ,𝐤), where A_μν;κλ(𝐤;𝐪) = 1/2[ ( U_𝐤-𝐪^μκ)^∗ V_𝐤^μλ + ( V_𝐤-𝐪^μκ)^∗ U_𝐤^μλ] [ U_𝐤-𝐪^νκ( V_𝐤^νλ)^∗ + V_𝐤-𝐪^νκ(U_𝐤^νλ)^∗]. §.§ Data Availability. All relevant data are available from the corresponding authors upon reasonable request. naturemag_noURL § END NOTES §.§ Acknowledgments We thank H. Tanaka, A. Chernyshev, and O. Starykh for valuable discussions. J.M. acknowledges the support of the Ministry of Science and Technology of China (2016YFA0300500). Y.K. acknowledges the financial support by JSPS Grants-in-Aid for Scientific Research under Grant No. JP16H02206. The work at Georgia Tech was supported by ORAU's Ralph E. Powe Junior Faculty Enhancement Award (M. Mourigal). H.D.Z. acknowledges support from NSF-DMR-1350002. The work performed in NHMFL was supported by NSF-DMR-1157490 and the State of Florida. We are grateful for the access to the neutron beam time at the neutron facilities at NCNR, BER-II at Helmholtz-Zentrum Berlin and HFIR operated by ORNL. The research at HFIR at ORNL was sponsored by the Scientific User Facilities Division (T.H., H.B.C., and M. Matsuda), Office of Basic Energy Sciences, U.S. DOE. This work utilized MACS supported in part by the National Science Foundation under Agreement No. DMR-1508249. §.§ Author contributions Y.K. and J.M. conceived the project. H.D.Z. prepared the samples. T.H., Y.Q., D.L.Q., Z.L., H.B.C., M. Matsuda, L.G., M. Mourigal, and J.M. performed the neutron scattering experiments. E.S.C. measured the magnetization. Y.K., L.G., C.D.B., and M. Mourigal performed the NLSW calculations. Y.K., J.M., M. Mourigal, and C.D.B. wrote the manuscript with comments from all the authors. §.§ Competing financial interests The authors declare no competing financial interests. §.§ Correspondence and materials Correspondence and requests for materials should be addressed to Y.K. (yoshitomo.kamiya@riken.jp) and J.M. (jma3@sjtu.edu.cn).

http://arxiv.org/abs/1701.07605v1

20170126075658

Lattice coding for Rician fading channels from Hadamard rotations

cs.IT

Lattice coding for Rician fading channels from Hadamard rotations Alex Karrila^1, Niko R. Väisänen^1, David Karpuk^1, Member, IEEE, and Camilla Hollanti^1, Member, IEEE ^1A. Karrila, N. R. Väisänen, D. Karpuk and C. Hollanti are with the Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, FI-00076 AALTO, Espoo, Finland. Emails: firstname.(letter.)lastname@aalto.fi. D. Karpuk was supported by Academy of Finland grant #268364. Grant acknowledgements? December 30, 2023 =================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== In this paper, we study lattice coding for Rician fading wireless channels. This is motivated in particular by preliminary studies suggesting the Rician fading model for millimeter-wavelength wireless communications. We restrict to lattice codes arising from rotations of ^n, and to a single-input single-output (SISO) channel. We observe that several lattice design criteria suggest the optimality of Hadamard rotations. For instance, we prove that Hadamard rotations maximize the diamond-packing density among all rotated ^n lattices. Finally, we provide simulations to show that Hadamard rotations outperform optimal algebraic rotations and cross-packing lattices in the Rician channel. algebraic rotations, diamond packings, Hadamard rotations, lattice code design, orthogonal lattices, reliability, Rician fading, single-input single-output (SISO) channels § INTRODUCTION Reliability is a key issue in designing wireless communications, since the channels are vulnerable to distortions. Reliability is typically improved by simultaneous error-correction coding and physical-layer design, with the tradeoff of decoding complexity and information rate. Orthogonal lattice codes are a highly conventional physical-layer design in all types of wireless channels. Such codes provide for fast vector decoding based on solving a closest-vector problem. In addition, the Gray mapping from bit vectors to lattice vectors guarantees a beneficial conversion of vector decoding errors to bit decoding errors. Thus, the fundamental question in physical-layer reliability is to find the orthogonal lattice, or equivalently, the rotation, that provides a low rate of vector decoding errors. Algebraic rotations are known to provide a solution in Rayleigh fading single-input single-output (SISO), see <cit.> for a good overview, as well as in multiple-input multiple-output (MIMO) channels <cit.>. The case is nevertheless not closed: research on future fifth generation (5G) communication networks calls for studying optimal rotations for the Rician channel. Namely, instead of the usual 700MHz–2.6GHz, an extension to millimeter waves (e.g. 28GHz) is anticipated <cit.>. Using millimeter waves provides both advantages and challenges. Using traditional spectrum allows for the transmission of data over a longer distance but at lower capacity, whereas millimeter wave offers greater bandwidth, but over shorter distances. However, the channel models for such new frequency spectra are not yet fully understood. For this reason, it is unclear at the moment what kind of modulation and encoding of the data will be most useful for energy efficiency and signal robustness. Nevertheless, tentative studies <cit.> show that the scale channel coefficients are Rician distributed. In this paper, we show that several alternative design approaches suggest the optimality of Hadamard rotations in Rician SISO channels. We show experimentally that Hadamard rotations outperform the algebraic rotations of <cit.> and the cross-packing lattices of <cit.> over the Rician channel. The Rician channel is indexed by a parameter K>0, with K=0 being the Rayleigh channel and K →∞ the Gaussian channel. While the algebraic lattices offer better performance at K = 0, they are outperformed by the Hadamard rotations already at small K. We present results only for K = 20 for the sake of compactness, but similar results were obtained for several K. Lattices from Hadamard rotations has previously been proposed for certain fading channels in <cit.>, and as an alternative to OFDM for optical channels in <cit.>, to give a few applications. Nevertheless, it seems that their surprisingly good performance in the Rician case has not been noticed before. §.§.§ Organization In Section <ref>, we provide the necessary background on lattices, Hadamard rotations, and the Rician channel. The design approaches based on error probability estimates and sphere packings are presented in Sections <ref>–<ref>, and approaches based on diamond packings and diversity estimates in Section <ref>. Simulation results are provided in Section <ref>. § PRELIMINARIES §.§ Lattices A lattice is a discrete additive subgroup of ^n. We assume familiarity with the basic concepts related to lattices and lattice codes, and refer the unaccustomed reader to <cit.> or <cit.>. We point out that we consistently work in the column vector convention. We are going to be interested in the following class of lattices. A full-rank lattice Λ⊂^n is well-rounded (WR) if its minimal vectors span ^n. The minimal vectors of a WR lattice Λ are not guaranteed to generate Λ <cit.>. WR lattices are of interest here mainly due to their relation to the sphere-packing problem. Namely, all local maxima of the sphere packing density and hence in particular the sphere-packing optimal lattices are well-rounded: in a non-WR lattice, one can shrink the orthogonal complement of the minimal vectors to obtain a lattice with same minimal norm but smaller volume. The following is a partial converse to this statement: Let Λ⊂^n be a WR full lattice, scaled to unit volume. Then, the sphere packing density of Λ is minimized if and only if Λ is a rotation of ^n. Any n linearly independent minimal vectors of Λ generate a WR sublattice Λ' whose sphere-packing density is smaller or equal to that of Λ. Thus, it suffices to prove the claim for the WR lattice Λ' generated by its minimal vectors. But the claim is then immediate from Hadamard's inequality. §.§ Hadamard matrices A (real) Hadamard matrix is a square matrix whose all entries are ± 1 and whose column vectors are orthogonal. The orthogonality condition can be equivalently cast as W^T W = n I. Thus, W/√(n) is an orthogonal matrix, called a Hadamard rotation. Hadamard matrices are conjectured to exist in all dimensions divisible by four, and known to exist in all such dimensions relevant for lattice coding purposes. The Kronecker product of two Hadamard matrices yields a third one. Based on this fact, Sylvester's construction is the simplest way to obtain Hadamard matrices in dimensions that are powers of two, defined inductively by W_2 = [ 1 1; 1 -1 ] , W_2^k+1 = W_2 ⊗ W_2^k. We denote Hadamard matrices by W and Hadamard rotations by U, often working with W to avoid normalization constants. §.§ Fading SISO channels and the Rice distribution We consider a single-input-single-output (SISO) wireless channel. We assume perfect channel state information (CSI) at the receiver and no CSI at the transmitter. Such a channel is modeled by the real channel equation 𝐲 = (𝐡) 𝐱 + 𝐯, where 𝐱∈^n and 𝐲∈^n are the transmitted and received vectors, respectively, and 𝐡∈_+^n and 𝐯∈^n are mutually independent random vectors modeling fading and noise, respectively. We assume Gaussian noise, 𝐯∼𝒩(0, σ^2 I), and an interleaved channel where the h_i are i.i.d. Our primary interest lies in Rician distributed h_i, modeling a fading with a line of sight and scattering routes. The strength of the line of sight is captured by the Rician factor K ≥ 0 indexing the different Rice distributions defined by the density ρ(h)=2h(1+K)e^ -K-h^2(1+K) I_0 (2h √( K^2+K ) ), where I_0 is the zeroth-order Bessel function of first kind. The case K=0 is the well-studied Rayleigh fading channel, while in the limit K →∞, h_i becomes deterministically one and we obtain the additive white Gaussian noise (AWGN) channel. § LATTICE DESIGN CRITERIA IN RICIAN CHANNELS In this section, we provide some motivating computations for the optimality of Hadamard rotations in physical-layer designs. The design criteria rely on the h_i having small variance, and thus being concentrated around their mean. This is in contrast with the Rayleigh fading channel, in which deep fades (some h_i≈ 0) are a major cause of decoding errors. The two most conventional ways to design physical-layer reliability in fading channels are minimizing the PEP bound (<ref>) or, in the AWGN channel, maximizing the sphere-packing density of the lattice. We study analogous design approaches in channel models where deep fades are not the primary cause of decoding errors, such as the Rician channel[This assumption is validated for the Rician channel in Section <ref>.], by studying the pairwise error probability (PEP) estimate and the sphere-packing at a near-average fade. We find an agreement of the two approaches, both suggesting the optimality of Hadamard rotations. §.§ Pairwise error probability The standard PEP bound states that the probability P of a vector decoding error is bounded by P ≤1/2∑_𝐭∈Λ∖{0}𝔼{exp( - ‖(𝐡) 𝐭‖^2 /8 σ^2 ) } , where Λ is the code lattice, σ^2 the noise variance, and the expectation is over 𝐡. In the deep-fade dominated Rayleigh fading channel, this expectation is analyzed by fixing the distribution of the fading h_i and expanding around σ^2 = 0. To study a noise dominated channel, let us fix the noise σ^2 and “expand around (h^2_i) = 0”: denoting (‖(𝐡) 𝐭‖^2 - 𝔼{‖(𝐡) 𝐭‖^2 } )/8 σ^2 = ϵ, the exponential in (<ref>) becomes exp( - ‖(𝐡) 𝐭‖^2 /8 σ^2 ) = exp( - 𝔼{‖(𝐡) 𝐭‖^2 }/8 σ^2 ) e^-ϵ = exp( - 𝔼{‖(𝐡) 𝐭‖^2 }/8 σ^2 ) (1 - ϵ + ϵ^2/2 - …) Neglecting the higher-order terms represented by the ellipses and substituting 𝔼{‖(𝐡) 𝐭‖^2 } = 𝔼{ h^2 }‖𝐭‖^2, we approximate the PEP bound as 1/2∑_𝐭∈Λ∖{0}𝔼{exp( - ‖(𝐡) 𝐭‖^2 /8 σ^2 ) } ≈ 1/2∑_𝐭∈Λ∖{0} e^ -𝔼{ h^2 }‖𝐭‖^2/8 σ^2[ 1 + (h^2) /2(8 σ^2 )^2‖𝐭‖_4^4], where ‖·‖_p denotes the usual L^p vector norm. Since (h^2) was assumed small, the minimal vectors of Λ dominate the series above, and their L^2 norm should be maximized and L^4 norm minimized in order to minimize the error probability. In other words, we should first maximize the packing density of Λ and then rotate it so that the minimal vectors are parallel to [± 1, …, ± 1]^T. If Λ is a rotation of ^n this condition is satisfied if and only if it is a Hadamard rotation. §.§ Sphere packings Regarding the fading channel as an instantaneous Gaussian channel, we should maximize the packing density of the randomly faded lattice (𝐡) Λ. First, the average norm of a given lattice vector 𝐭∈Λ is after fading 𝔼{‖(𝐡) 𝐭‖^2 } = 𝔼{ h^2 }‖𝐭‖^2. This tells us to maximize the packing radius of Λ, but does not differentiate between rotations of Λ. Next, the random norms ‖( 𝐡 ) 𝐭‖^2, especially the shortest ones, should be stabilized around their expectation [ ‖(𝐡) 𝐭‖^2 ]. Hence, we should minimize the variance ( ‖(𝐡) 𝐭‖^2 / [ ‖(𝐡) 𝐭‖^2 ] ) = (h_i^2)/[h_i^2]^2‖𝐭‖_4^4/‖𝐭‖_2^4. The conclusions are identical to those reached from the criteria derived from the PEP; specifically, if Λ is a rotation of ^n and 𝐭 a minimal vector of Λ, then the above quantity will be minimized exactly when it is a Hadamard rotation. More generally, we could expand the class of lattices we are interested in to include non-orthogonal lattices. In particular, there exist many well-rounded lattices (which necessarily have good sphere packings) all of whose minimal vectors are parallel to [ ± 1, …, ± 1]^T. For example, the body-centered cubic lattice in ^3, generated by the vectors [ 1, 1, 1]^T, [ 1, -1, -1]^T, and [ -1, -1, 1]^T, or its tensor product with any Hadamard lattice. § SPHERE-PACKING DENSITY OF HADAMARD LATTICES IN RICIAN FADING CHANNELS In this section, we provide a probabilistic estimate for the sphere-packing density of a Hadamard rotated unit lattice after fading. We work with the unnormalized Hadamard matrices W ∈^n × n, and denote the lattices of interest by Λ = W ^n and Λ_h = (𝐡) W ^n. Notice that the natural generators of the faded lattice Λ_h are of the form [± h_1 … , ± h_n]^T. If they are minimal vectors of Λ_h, then Λ_h is well-rounded and, by Lemma <ref>, Λ_h has a good sphere packing. In this section, we compute the probability of this event in low dimensions. For any realization of 𝐡, there is a minimal vector (𝐡) W ω of the faded lattice Λ_h, where the integer lattice coordinates ω∈^n either satisfy ‖ω‖^2 < n, or ω is a row of the matrix W. Let w_j be the j^th row of W, and ω∈^n some arbitrary lattice coordinates. We compare the lengths of the lattice Λ_h vectors (𝐡) W w_j and (𝐡) W ω. First, W w_j = n 𝐞_𝐣, so for some j, we have ‖ (𝐡) W w_j ‖^2 = n^2 min_1 ≤ i ≤ m h_i^2. Next, denote W ω = 𝐳 and note that ‖ (𝐡) W ω‖^2 = ∑_k = 1^n h_k^2 z_k^2 ≥ ‖𝐳‖^2 min_1 ≤ k ≤ n h_k^2 = n ‖ω‖^2 min_1 ≤ k ≤ n h_k^2, where the last step used the fact that W/√(n) is a rotation matrix. Now (𝐡) W ω must satisfy ‖ω‖^2 < n to be shorter than (𝐡) W w_j. Let us denote by C the event that the natural generators (𝐡)W𝐞_i of a faded Hadamard lattices Λ_h = (𝐡) W ^n are minimal vectors. Then, ℙ{ C } is given by integrating the joint density of h_ 1^2,...,h_n^2 over a cone. For fixed dimension, the previous lemma gives finitely many lattice coordinates ω that can yield minimal vectors. Then, (𝐡) W 𝐞_j are minimal vectors if and only if all such ω satisfy ‖ (𝐡) W ω‖^2 ≥ ∑_k = 1^n h_k^2 , a linear inequality in h_1^2,...,h_n^2. This corollary allows for numerical computations, as illustrated in Figure <ref>. In particular, when h_i are taken Rician with parameter K, the probability (1- 𝐏{ C }) of Λ_h not being WR decays exponentially in K. In four dimensions, the exponential decay is faster than in two. This leads to two conclusions. First, especially in larger dimensions, one can expect Hadamard rotations to perform well in Rician channels where K is sufficiently away from zero. Second, the decoding errors in such a setup mainly occur due to large noise, rather than deep fades. This supports the standing assumption of “noise-dominated errors” in the design approach suggesting Hadamard rotations in Section <ref>. § CONNECTIONS TO OTHER DESIGN CRITERIA Aside from the above criteria derived from the PEP and sphere packing density, there have recently been more subtle design criteria introduced for lattice codes over fading channels, wherein deep fades are not the primary cause of errors. These include the cross-packing density <cit.> and local diversity <cit.>. In this section, we present results which study how well Hadamard rotations of ^n satisfy these design criteria. §.§ Diamond packings Recently, <cit.> designed lattices for Rician channels by maximizing their cross-packing density (equivalently, maximizing cross-packing radius), where the crosses consist of axis-aligned line segments. This was motivated by approximating the shape of a contour surfaces of the terms in the Rician PEP estimate (<ref>) by a cross polyomino, i.e. an appropriately thickened cross. Analogously, approximating the shape of the contour surfaces by the convex hulls of the n-dimensional crosses, also known as L^1 norm balls or diamonds, one ends up designing lattices based on their diamond packing density. We find that the Hadamard rotations maximize the diamond packing density of the rotated ^n lattices. Let R ∈^n be a rotation matrix. Then, the minimal L^1 norm of the rotated ^n lattice R ^n satisfies min_𝐭∈ R^n 𝐭0‖𝐭‖_1 ≤√(n), with equality if and only if R is a Hadamard rotation. To prove the inequality, let 𝐞 be any elementary basis vector. Take 𝐭 = R 𝐞 a rotated basis vector. Recall now the relation of L^2 and L^1 norms on ^n: ‖𝐭‖_1 ≤√(n)‖𝐭‖_2, with equality if and only if 𝐭 is parallel to [± 1, …, ± 1 ]^T. This implies that min_𝐭∈ R^n 𝐭0‖𝐭‖_1 ≤√(n), and the equality is possibly reached only if R is Hadamard. To prove that the inequality is sharp for Hadamard rotations, notice that ‖𝐭‖_1 ≥ a if and only if 𝐭·𝐬≥ a for some sign vector 𝐬 = [± 1, …, ± 1 ]^T. Now, let 𝐭 be any nonzero vector of a Hadamard rotated lattice U^n, and let 𝐮 be a basis vector of the lattice U^n (i.e., a column of U) with a nonzero (integer) coefficient in 𝐭∈ U^n. Choosing 𝐬 = ±√(n)𝐮, the orthogonality of the Hadamard basis implies 𝐭·𝐬≥√(n), so indeed ‖𝐭‖_1 ≥√(n) for all nonzero 𝐭∈ U^n. §.§ Local diversity In <cit.> the design of reliable lattices in low signal-to-ratio (SNR) Rayleigh fading channels was considered, and the authors deduced that Hadamard rotations are optimal within a certain one-parameter Lie group of rotations. They explained the appearance Hadamard rotations, which contrary to conventional algebraic rotations are not fully diverse, by local diversity of Hadamard lattices, i.e., a tradeoff between diversity and length of the lattice vectors. We prove the following sharp local diversity estimate for Hadamard rotations. Let U ∈^n × n be a Hadamard rotation and let 𝐭 = U ω be of diversity k > 0. Then, k ‖𝐭‖^2 ≥ n. Since 𝐭 has diversity k, we have ‖𝐭‖_2 ≥‖𝐭‖_1 / √(k). Substituting the minimal L^1 norm ‖𝐭‖_1 ≥√(n), we obtain ‖𝐭‖_2^2 ≥ n / k as desired. § SIMULATIONS We now present simulation results to confirm our previous findings. The key parameters for our simulations are the dimension n of the signal constellation and its order q per dimension (so that the number of constellation points is q^n), the parameter K of the Rician distribution, and the volume-to-noise ratio (VNR) which defines the variance of the noise σ_n^2 by the formula VNR = (Λ)^2/n/(8σ_n^2). In simulations for n = 2 we used q=8, and for n = 4 we used q=4. §.§ Setup details Let M denote the generator matrix of the simulated code lattice Λ and S the signal constellation. To construct the signal constellation, we start from a finite region of ℤ^n described by S” = { (x_1, …, x_n) | x_i ∈ℤ, 0 ≤ x_i < q ∀ i }. Then, we center such a region by setting S' = S” - (q-1)/2. Finally, our signal constellation S is a image of S' under the generator matrix, S = MS'. The channel simulations were based on first generating a uniform random constellation vector 𝐱, then generating the Rician fading and Gaussian noise vectors of the channel equation (<ref>) to obtain the received vector 𝐲, and finally solving the closest vector problem 𝐱̂ = min_𝐭∈ S ||𝐲 - diag(𝐡)𝐭||^2. The decoding is correct if 𝐱̂ = 𝐱. §.§ Results §.§.§ Rotations We first compared three different rotations of ^n in dimension n = 4. These rotations are the identity rotation, the best known algebraic rotation <cit.>, and the Hadamard rotation from Sylvester's construction. We simulated the performance of these rotations in the Rician channel over a large range of K values. In Fig. <ref> we plot error rate as a function of VNR, for the Rician parameter K=20. The Hadamard rotations perform slightly better than the algebraic rotations over the whole VNR range, which supports the results of Section <ref>. Similar simulations for other values of K produced comparable results. To see how the performance of the different lattices varies with the Rician parameter K, we plot in Figure <ref> the error rates as a function of K for fixed VNR. We see that in dimension n = 4 there is a value for K, namely K≈ 4.4, after which the Hadamard rotation performs better than the other two simulated rotations. An analogous simulation in dimension n = 2 produced similar results, with the critical value of K being K≈ 7.2. §.§.§ Cross-packing lattices Recently, <cit.> designed a family of cross-packing lattices for Rician channels, indexed by a parameter t, of which t=1 and t=2 performed best. Let us compare the Hadamard rotations against these lattices (normalized to unit volume). Note that the cross-packing lattices are not orthogonal, so comparing vector error rates in the codes, as plotted, is not equivalent to comparing bit error rates. In Fig. <ref> we present simulation results which compare the performance of the Hadamard rotation to the cross-packing lattices in dimension n = 2, which show that the Hadamard rotation offers a modest improvement over the cross-packing lattices. § CONCLUSIONS Motivated by applications to 5G and millimeter wave communications, we studied lattice codebook design for the Rician fading channel. It was found that Hadamard rotations of ^n satisfy the design criteria derived from the corresponding PEP and the sphere-packing density. Two particularly attractive features of Hadamard lattices are that they often retain good sphere-packing properties after experiencing fading, and that they are maximizers of the diamond packing density among rotations of ^n. Simulations were provided which demonstrate that Hadamard lattices outperform other lattice constructions in the Rician channel, such as algebraic rotations and cross-packing lattices. 10 Viterbo F. Oggier and E. Viterbo, “Algebraic Number Theory and Code Design for Rayleigh Fading Channels,” Foundations and Trends in Communications and Information Theory, vol. 1, no. 3, pp. 333–415, Dec. 2004. Viterbo-taulukot E. Viterbo, “Table of Best Known Full-diversity Rotations”, www.ecse.monash.edu.au/staff/eviterbo/rotations/rotations.html. Oggier F. Oggier, J.-C. Belfiore, and E. Viterbo, “Cyclic Division Algebras: A Tool for Space–Time Coding”, Foundations and Trends in Communications and Information Theory, vol. 4, no. 1, pp 1–95, Nov 2007. webmagazine1 J. Best, “The race to 5G: Inside the fight for the future of mobile as we know it”. www.techrepublic.com/article/does-the-world-really-need-5g/ webmagazine2 S. Deng et al., “Small Wavelengths – Big Potential: Millimeter Wave Propagation Measurements for 5G”. www.microwavejournal.com/articles/23274-small-wavelengths-big-potential-millimeter-wave-propagation-measurements-for-5g?v=preview Rappaport M. K. Samimi, S. Sun, T. S. Rappaport, “MIMO Channel Modeling and Capacity Analysis for 5G Millimeter-Wave Wireless Systems”, 10th European Conference on Antennas and Propagation (EuCAP 2016), April 2016, Davos, Switzerland. ViterboITW A. Sakzad, A.-L. Trautmann, and E. Viterbo, “Cross-packing Lattices for the Rician Fading Channel”, 2015 IEEE Information Theory Workshop (ITW), Jerusalem, 2015, pp. 1–5. HadamardPrecoding1 X. Yuan, C. Xu, L. Ping and X. Lin, “Precoder Design for Multiuser MIMO ISI Channels Based on Iterative LMMSE Detection,” in IEEE Journal of Selected Topics in Signal Processing, vol. 3, no. 6, pp. 1118-1128, Dec. 2009. HadamardPrecoding2 M. Noshad and M. Brandt-Pearce, “Hadamard coded modulation: An alternative to OFDM for wireless optical communications,” 2014 IEEE Global Communications Conference, Austin, TX, 2014, pp. 2102-2107. Sloane J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed. New York: Springer-Verlag, 1998. Nguyen P. Q. Nguyen, The LLL Algorithm: Survey and Applications. Springer Berlin Heidelberg, 2010, ch. Hermite’s Constant and Lattice Algorithms, pp. 19–69. Karpuk-Hollanti D. A. Karpuk and C. Hollanti, “Locally Diverse Constellations From the Special Orthogonal Group,” IEEE Trans. Wireless Commun., vol. 15, no. 6, pp. 4426–4437, May 2016.

http://arxiv.org/abs/1701.07967v1

20170127082250

Heavy-tailed random walks, buffered queues and hidden large deviations

math.PR

Heavy-tailed buffered queues H. Bernhard B. Das Singapore University of Technology and Design Singapore University of Technology and Design, Pillar of Engineering Systems and Design, 8 Somapah Road, Singapore 487372 e1 e2 It is well-known that large deviations of random walks driven by independent and identically distributed heavy-tailed random variables are governed by the so-called principle of one large jump. We note that further subtleties hold for such random walks in the large deviation scale which we call hidden large deviation. We apply this idea in the context of queueing processes with heavy-tailed service times and study approximations of severe congestion times for (buffered) queues. We conclude with simulated examples to verify our results. [class=AMS] [Primary ]60F10 60G50 60G70 [; secondary ]60B10 62G32 buffered queues heavy-tails large deviations regular variation § INTRODUCTION Stochastic processes with heavy-tailed components as building blocks are of interest in many areas of application, including, but not restricted to, hydrology <cit.>, finance, insurance and risk management <cit.>, tele-traffic data <cit.>, queueing theory <cit.>, social networks and random graphs <cit.>. The notion of heavy-tails in applied probability is often studied under the paradigm of regular variation. In this paper we concentrate on understanding subtle properties of heavy-tailed random walks which enables us to understand structures of simple GI/G/1 queues with heavy-tailed service times under certain regularity conditions. In particular, we establish how queueing congestion may occur (which we define in terms of long intense periods), not only because of one large jump, but also in terms of further jumps occurring in the process. It is well-known that if {Z_i}_i≥ 1 are iid zero mean regularly varying random variables, then large deviations of their partial sums S_n = ∑_i=1^n Z_i are essentially due to one of the random variables Z_i attaining a large value, see <cit.> for further details. Early results on this notion popularly known as the principle of one large jump were obtained in <cit.>. More formally, the notion of one large jump in this case can be written as |S_n| > x = n|Z_1|>x (1+ o(1)), x>b_n for some choice of b_n ↑∞ as n→∞. Similar large deviation principles (LDPs) have been obtained in <cit.> under the more general assumption of the random variables being sub-exponential. Moving forward from random variables, the notion of regular variation of càdlàg processes has been characterized in <cit.>. It was aptly noted in <cit.> that large deviations for such processes with heavy-tailed margins are very closely related to the notion of regular variation. A precise large deviation result for partial sum processes on the space [0,1] of càdlàg functions was provided in <cit.>; in fact this result was obtained for d-dimensional processes. In particular for d=1, the authors establish that if S^n =(S_⌊ nt⌋)_t∈[0,1] is the càdlàg embedding of {S_k}_k=1^n into [0,1] with S_0=0, for suitably chosen sequences γ_n>0 and λ_n ↑∞ one can observe that γ_n S^n /λ_n ∈ · w^#→μ(· ), n→∞, for a non-null measure μ, where w^#→ denotes convergence in the space of boundedly finite measures on _0, see <cit.> and <cit.> for further details on the space and w^#-convergence. In particular the result shows that an appropriate choice of scaling is γ_n = [n Z_1>λ_n]^-1 and the limit measure μ concentrates all its mass on step functions with exactly one jump discontinuity, which essentially retrieves the one large jump principle. Now it seems likely that there are possibilities - albeit rarer than the above case - that a large deviation of S_n may occur because two or more of the random variables {Z_i}_i=1^n were large. The probabilities of these events, although negligible under the scaling γ_n = [n Z_1>λ_n]^-1, are not exactly zero. Consequently, in this paper we aim to recover the rates at which such deviations happen and examine their structure. Furthermore our goal is to use such results in the context of queueing to understand the behavior of what we call long intense periods in a large-deviation-type event. Analysis of hidden behavior of regularly varying sequences on ^d <cit.> and more recently on ^∞ and Lévy processes on [0,1] <cit.> has been conducted under the name hidden regular variation. Connections between hidden regular variation and elements of the classical large deviations framework have been established recently in <cit.>. In this paper, our first contribution is to extend the large deviation principle in (<ref>) to hidden large deviations in the spirit of hidden regular variation. We establish that the most probable way a large deviation event occurs, which is not the result of only one random variable being large, is actually when two random variables are large; resulting in a non-null limit measure as in (<ref>) concentrating on processes having two jump discontinuities. For our analysis we use the framework proposed in <cit.> and the notion of convergence used here is known as -convergence which is closely related to the w^#-convergence of boundedly finite measures and developed in <cit.>. We briefly recall the required background on regular variation and -convergence in Section <ref>. The results on hidden large deviations of random walks are dealt with in Section <ref>, where the key result is obtained in Theorem <ref>. Queues with heavy-tailed service times have been of interest to researchers for a few years <cit.>. Our interest lies in figuring out when do we see long busy or intense periods in a queue. In <cit.>, the author shows that for a GI/G/1 queue with heavy-tailed service times, the most likely way a long busy period occurs is when one big service requirement arrives at the beginning of the busy period and the queue drifts back to zero linearly thereafter. Consequently, a large deviation of a queueing process also looks exactly the same. <cit.> studies the steady state loss in buffered queues and shows that for large buffers K the steady state loss can be approximated by the expected loss due to one arrival A filling the buffer completely starting zero, that is the expected loss is approximately [A-K]; see <cit.> for similar results concerning fluid queues. Equipped with a hidden large deviation principle for random walks, in Section <ref> we study queuing processes with heavy-tailed service times and finite capacity, which is a natural model to assume in many contexts. We define a long intense period as the fraction of time a queue with buffer capacity K>0 spends continuously above a level θ K, θ∈ (0,1) for one sojourn and study the length of the longest such period for a given observation horizon. A closely related notion of long strange segments <cit.> has been consequently investigated in <cit.>, which examines the length of time the average process value spends in an unexpected regime. Considering hidden large deviations in such a setting provides more insight since we observe that the first large deviation approximation gives only a crude estimate of the distribution of the length of intense periods for large buffers. In Theorem <ref> we derive an approximation to the distribution of the length of long intense periods in queues with large buffer sizes and conduct a simulation study in Section <ref> to show the effectiveness of the result. Finally, future directions are indicated and conclusions drawn in Section <ref>. § NOTATIONS AND BACKGROUND In this section we provide a summary of frequently used notations and concepts along with a review of material necessary for the results in the following sections. We mostly adhere to the notations and definitions introduced in <cit.>. §.§ Basic notations A few notations and concepts are summarized here. Detailed discussions are in the references provided. Unless otherwise specified, capital letters like X,Z, S with various subscripts and superscripts are reserved for real-valued (and sometimes vector-valued) random variables, whereas bold-symboled capital letters like ,, (again with various subscripts and superscripts) denote vector- or function-valued random elements. Small letters in bold, like , are vectors in a suitable Euclidean space where =(z_1,…,z_n) if ∈^n. p0.17 p0.77 β Regularly varying functions with index β∈: that is, f:ℝ_+↦ℝ_+ satisfying lim_t→∞f(tx)/f(t)=x^β, for x>0. We abuse notation and write X∈-α for regularly varying random variables as in Definition <ref>. In case of positive random variables, this is equivalent to requiring the tail of the cumulative distribution function to satisfy the limit relation above. (𝕊\ℂ) (𝕊, ℂ) = (𝕊\ℂ) is the set of Borel measures on 𝕊\ℂ that are finite on sets bounded away from ℂ. μ_n→μ Convergence in (𝕊\ℂ); see Definition <ref>. U_j^↑ {∈[0,1]^j: 0≤ u_1 < … < u_j ≤ 1 }. =([0,1],) Space of all real-valued càdlàg functions on [0,1] equipped with the Skorohod J_1 metric. d_J_1 Metric on . If Λ denotes the class of strictly increasing continuous functions λ: [0,1] → [0,1] with λ(0)=0, λ(1)=1, then for f,g∈, we define d_J_1 (f,g) := inf_λ∈Λmax{sup_t∈[0,1] |f(t)-g∘λ (t)|, sup_t∈ [0,1] |λ(t)-t| } = inf_λ∈Λf-g∘λ∨λ-e where e(t)=t, ∀ t∈ [0,1]. See <cit.> for further details on the J_1-topology. Note that similar definitions could be worked out if we take _M =([0,M],) in place of . _=j Space of all real-valued step functions on [0,1] with exactly j jumps, j≥ 1. Assume _=0 is the space containing only the constant function at 0. Moreover, _=j⊂. _≤ j Space of all step functions on [0,1] with j or less jumps, _≤ j=⋃_k=0^j_=k. k (λ) {∈^n: |{i: |z_i| > λ}| = k} for λ>0, 1≤ k≤ n. It denotes the subset of ^n where exactly k co-ordinates are above λ > 0 in absolute value. Moreover ^n = ⋃_k=0^n k (λ). See Section <ref>. ν_α^j Product measure on (\{0})^j: ν_α×…×ν_α_j times with ν_α as defined in (<ref>). §.§ Convergence in We state our results as convergence in , a mode of convergence closely related to standard weak convergence of probability measures. The idea is as follows: to allow for hidden large deviations we need to exclude the set of possible large deviations from the space we consider in order to keep the limit measure non-degenerate in that region. This is similar to how large deviations avoid the law of large numbers for zero mean random variates: we need to exclude 0 from the non-negative real line to obtain a limit measure for |Z_1 + ⋯ + Z_n| ≥ nz, z> 0. Convergence in follows the same principle but we allow for the removal of an arbitrary closed set. As a consequence, we can define convergence in càdlàg spaces where we exclude certain types of step functions which form a closed set in . In particular, let 𝕊 be a complete separable metric space, ℬ(𝕊) the collection of Borel sets on 𝕊 and ℂ⊂𝕊 a closed subset of 𝕊. Then we denote (𝕊, ℂ) = (𝕊\ℂ) the set of Borel measures on 𝕊\ℂ which are finite on sets bounded away from ℂ, that is, the collection of sets A ∈ℬ(𝕊) such that inf{ d(x,y): x ∈ℂ, y ∈ A}>0 where d denotes the metric on 𝕊. Finally, we call a sequence {μ_n }_n≥ 0 convergent if the assigned values converge for a suitable class of test functions or sets. Denoting 𝕆 = 𝕊\ℂ the support set we use the notation := (𝕊, ℂ) as eponym for the mode of convergence and use the following definition of -convergence. A sequence of measures {μ_n}_n≥ 0⊂ converges to μ_0 ∈ if for all closed sets F and open sets G in ℬ() which are bounded away from ℂ we have lim sup_n→∞μ_n (F) ≤μ_0(F), lim inf_n →∞μ_n(G) ≥μ_0(G). We write μ_n →μ_0 in as n →∞, or simply μ_n →μ_0. The definition above only states -convergence in terms of open and closed subsets of 𝕆. Theorem 2.1 of <cit.> provides several alternative characterizations of convergence in . The corresponding version of a continuous mapping theorem in follows as Theorem 2.3 in the same publication. We state the continuous mapping theorem again for the sake of convenience. We denote a second space with the same properties as 𝕊 by 𝕊' and similarly add dashes to the corresponding elements of the second space. Denote h: 𝕊→𝕊' a measurable mapping such that for all sets A' ∈ℬ(𝕊') ∩ h(𝕊\ℂ) bounded away from ℂ' also h^-1(A') is bounded away from ℂ. Then μ_n →μ in (𝕊, ℂ) implies μ_n∘ h^-1→μ∘ h^-1 in (𝕊',ℂ') if the measure μ attains zero mass on the set of discontinuity points of h. Relationship between -convergence and w^# convergence <cit.> state their version of a large deviation result for random walks in terms of w^# convergence, working with boundedly finite measures instead. Specifically they consider the space _0 = (0,∞] ×𝕊_, where 𝕊_ is the unit sphere in . The metric on the radius is defined as d_(0,∞](x,y):= |1/x - 1/y|, thus making any set not bounded away from the zero function (in the usual J_1 metric) unbounded in the modified space. In turn, one may work with convergence of boundedly finite measures. Unfortunately it is not immediately clear how to extend this theory to allow for the removal of more than just the zero function, whereas convergence in on the contrary is specifically designed for this purpose. §.§ Regular variation and heavy-tailed large deviations A measurable function f: (0,∞) → (0,∞) is regularly varying at infinity with index α∈ if lim_t→∞f(tx)/f(t) = x^α for all x>0. A sequence of positive numbers {a_n}_n≥ 1 is regularly varying with index α∈ if lim_n→∞a_[cn]/a_n = c^α for all c>0. Regular variation of unbounded random variables thus usually is defined in terms of regular variation of the tail of the corresponding cumulative distribution functions at infinity; see <cit.>, <cit.>, or <cit.> for related properties and examples. We work with an equivalent definition stated in terms of -convergence <cit.>. A random variable X is regularly varying at infinity if there exists a regularly varying sequence γ_n and a non-zero measure μ∈(\{0}) such that γ_n X/n ∈·→μ(·), n→∞, in (\{0}). Since the measure μ satisfies the scaling property μ(sA) = s^-αμ(A), s>0, A ∈\{0} for some α≥ 0 we write X∈-α. For this paper we assume α>0. Moreover, for X∈-α, we also assume that the following condition is satisfied. lim_n→∞X>n/|X|>n = p, lim_n→∞X<-n/|X|>n = 1-p:=q. for some 0≤ p ≤ 1. This is called the tail balance condition. For univariate X∈-α we stick to this choice unless otherwise specified. We also denote by ν_α, the following measure on \{0} for x>0,y>0, ν_α((-∞,-y) ∪ (x,∞)) = qy^-α + px^-α. The limit measure μ in Definition <ref> is equal to ν_α if we choose the sequence γ_n to be [|X| > n]^-1. Now, a heavy-tailed large deviation principle for real-valued random variables can be defined as follows in terms of -convergence. A sequence of random variables {X_n}_n≥ 1, with X_n → 0 in probability, satisfies a heavy-tailed large deviation principle if there exists a positive sequence γ_n ↑∞ and a non-zero measure μ∈(\{0}) such that as n→∞ γ_n X_n ∈·→μ(·), in (\{0}). The similarity between the definitions of regular variation and heavy-tailed large deviation principle (LDP) is quite evident here. One salient difference is that regular variation is defined for a single random element, whereas an LDP, for a sequence of random elements. The special case of X_n = X/n shows that regular variation is a specific form of heavy tailed LDP according to our definition. The definition of an LDP implies that X_n ∈·→ 0 as n→∞ for all sets in \{0}. For our purposes we give a more general definition of LDPs for random elements on a metric space 𝕊. Additionally we do no longer restrict to removing the zero element, but an arbitrary closed set ℂ⊂𝕊. The random elements {_n}_n≥ 1⊂𝕊 satisfy a (heavy-tailed) large deviation principle on 𝕊\ℂ for a closed set ℂ⊂𝕊 if there exists a positive sequence γ_n ↑∞, and a non-zero measure μ∈(𝕊\ℂ) such that as n→∞ γ_n _n∈ · →μ(·), in (𝕊\ℂ). We write _n∈LD( γ_n,μ,𝕊\ℂ). The definition of heavy-tailed LDP as given in <cit.> is equivalent to Definition <ref> for stochastic processes with sample paths in . It has been observed, especially in the case of heavy-tailed random walks, that the limit measure μ obtained in the LDP, concentrates only on step functions with one jump; see <cit.>. Hence we may ask whether a different structure is observable if we do not allow one jump functions to be in the support of the limit measure for an LDP. Essentially, we are asking how often do we see events which are not governed by one jump in the space . The same question can be asked iteratively by removing the support set of a new found limit measure and examining the hidden structure of rarer and rarer events. Hence a sequence of large deviation principles can be defined here. The random elements {_n}_n≥ 1⊂𝕊 satisfy a sequence of (heavy-tailed) large deviation principles if there exists an increasing sequence {ℂ^(j)}_j≥ 1 of closed subsets of 𝕊 (i.e. ℂ^(k)⊃ℂ^(j) for k>j≥ 1), positive sequences γ_n^(j)↑_n ∞, j≥ 1 with γ_n^(j+1) / γ_n^(j)→_n ∞, and non-zero measures μ^(j)∈(𝕊\ℂ^(j)), j ≥ 1 such that _n ∈LD(γ_n^(j),μ^(j),𝕊\ℂ^(j)), j≥ 1. A similar definition could be stated for random elements satisfying only a finite number J≥ j ≥ 1 of large deviation principles. The limit measure μ^(j) necessarily satisfies μ^(j)( (ℂ^(j+1))^c )=0 for all j≥ 1. More precisely, the measure concentrates on ℂ^(j+1)\ℂ^(j). Thus, the k level LDP uncovers the structure of rare events which were hidden (i.e. negligible) under the scaling of the preceding j level LDPs of the sequence with j<k. We state a few useful facts about regularly varying functions that will be used repeatedly in the text. All of these are contained among others in Appendix B of <cit.>. In the following let f ∈-α_f,g ∈-α_g be two regularly varying functions. Then the sign of the regular variation exponent determines the behaviour at infinity: α_f>0: lim_x→∞ f(x) = 0 α_f<0: lim_x→∞ f(x) = ∞ Furthermore the product and composition of two regularly varying functions is also regularly varying fg ∈ℛ𝒱_-α_f -α_g g(x)→∞: f ∘ g ∈ℛ𝒱_α_fα_g Finally, we will repeatedly appeal to Karamata's theorem, especially the part concerning the tail integral, namely if α_f>1 then α_f >1: lim_x →∞xf(x)/∫_x^∞ f(t) t = α_f -1, α_f <1: lim_x →∞xf(x)/∫_x_0^x f(t) t = 1-α_f. § HIDDEN LARGE DEVIATIONS AND RANDOM WALKS Equipped with the terminology and tools in Section <ref>, we proceed to understand the structure of heavy-tailed random walks in this section. We look at heavy-tailed random walks as elements of . The key result for hidden LDPs for heavy-tailed random walks is in Theorem <ref>. §.§ Bounds on sums of random variables For random variables Z_1, …, Z_n, denote their sum by S_n= Z_1+…+Z_n. Here S_n denotes the n-th step of a random walk. One of the key tools to bound movements in the random walk caused by “small” realizations will be Bernstein's inequality, see <cit.>. Let Z_1, … , Z_n be iid bounded random variables with zero mean, variance Z_1=σ^2 and |Z_1|≤ M. Then for any t>0, P(|S_n| ≥ t) < 2exp{- t^2/ 2nσ^2 + 2/3Mt}. We use this exponential bound on the absolute value of the sum to bound the probability of a large deviation in the sum of regularly varying random variables happening due to many variables attaining a small but non-negligible value. This bound, as we see hence, turns out to be exponential rather than polynomial in the deviation level λ_n. Let {Z_i}_i=1^∞ be a sequence of iid random variables with Z_1 ∈ℛ𝒱_-α, α >0. In case the expectation exists, we assume it to be zero. Denote S_n= ∑_k=1^n Z_k and let λ_n ∈ℛ𝒱_ρ be a regularly varying sequence such that in case Z_1<∞, we have ρ > 1/2 Z_1=∞, we have αρ >1. Then for any δ>0 and ε_0>0 small enough, there exists a constant c>0 such that for large enough n, |S_n| > δλ_n, |Z_i| ≤λ_n^1-ε_0, ∀ i≤ n < 2exp(-cλ_n^ε_0 ). When Z_1 has finite variance, the condition, ρ > 1/2, guarantees that λ_n↑∞ fast enough such that we avoid the central limit regime. When Z_1 has infinite variance, then αρ>1 ensures that the probability of at least one of the variables exceeding a large threshold on the scale of λ_n still tends to zero. We transform S_n by making the Z_i's bounded (the bound still depending on n); and then apply Lemma <ref> appropriately to obtain the result. First observe that given δ>0, for small enough ε_0 and large enough n, A_n := |S_n| > δλ_n, |Z_i| ≤λ_n^1-ε_0∀ 1≤ i≤ n ≤| ∑_i=1^n Z_i |Z_i|≤λ_n^1-ε_0| > δλ_n ≤| ∑_i=1^n (Z_i |Z_i|≤λ_n^1-ε_0 - Z_i |Z_i|≤λ_n^1-ε_0)| > δ/2λ_n , where in the last inequality above we used the following bound, valid for large enough n and some constant c>0. | n/λ_nZ_1 |Z_1|≤λ_n^1-ε_0| ≤n/λ_n|Z_1| |Z_1|≤λ_n^1-ε_0 ≤n/λ_n|Z_1| |Z_1|≤λ_n ≤ c n |Z_1|>λ_n. Now using Lemma <ref> (Bernstein's inequality) to bound the sum of n zero mean random variables bounded in absolute value by M = 2 λ_n^1-ε_0, we obtain that for large enough n A_n ≤ 2 exp( - (δ/2λ_n)^2/2nZ_1 |Z_1|≤λ_n^1-ε_0 + 4/3λ_n^1-ε_0δ/2λ_n ) ≤ 2exp( -λ_n^ε_0c_1/c_2 + β(n)), where c_1,c_2 are positive constants and β(n) = 2nZ_1 |Z_1|≤λ_n^1-ε_0/λ_n^2-ε_0. Next we show that β(n)→ 0 as n→∞ which will imply that for large enough n, there is a ζ>0 such that A_n≤ 2exp( -λ_n^ε_0c_1/c_2 + ζ) =2exp( -cλ_n^ε_0), where c=c_1/c_2+ζ, and thus the lemma is proven. We show β(n) → 0 for three different cases as follows. * If α∈ (0,2) (implying infinite variance and αρ>1), using Karamata's theorem <cit.> for small enough ε_0, large enough n and constant C>0 we have β(n) ≤2n/λ_n^2-ε_0[ Z_1^2|Z_1|≤λ_n^1-ε_0] ∼2n/λ_n^2-ε_0× C λ_n^2(1-ε_0)|Z_1|>λ_n^1-ε_0 ∼ 2 C nλ_n^-ε_0|Z_1|>λ_n^1-ε_0→ 0 (n→∞). * If Z_1 < ∞, then β(n)≤ Cn/λ_n^2-ε_0 for some C>0 and hence vanishes as n →∞ for small enough ε_0>0. * If α =2 and Z_1=∞, then the variance is a slowly varying function. Again, for ε_0 small enough nλ_n^-2+ε_0→ 0 at a polynomial rate and hence β(n)→ 0. §.§ Random walks embedded in [0,1] We embed the random walk S_n=Z_1+…+Z_n in =[0,1] and discuss its large deviations. Let ^(n) = (Z_1,…,Z_n) where Z_i's are iid realizations from Z_1. For t∈ [0,1] and k ∈{1,…,n} define the functions _k^(n)(t) = Z_kk/n≤ t on . Now define ^(n)(t) := ∑_k=1^n ^(n)_k (t) = ∑_k=1^⌊ nt ⌋ Z_k =S_⌊ nt ⌋ to be the embedding of the random walk induced by Z_1, …, Z_n into the space . Moreover let 0 (λ) := {∈^n: |z_i| ≤λ, ∀ i}. The following corollary is a consequence of Lemma <ref>. Under the conditions of Lemma <ref>, for any δ>0 and ε_0>0 small enough, there exist a constant c>0 such that for large enough n, P(sup_t∈ [0,1] |^(n)| > δλ_n , ^(n)∈0(λ_n^1-ε_0)) ≤ 2exp(-cλ_n^ε_0/2) Observe that from Lemma <ref>, for any δ>0 and ε_0>0 small enough and for large enough n, sup_t∈ [0,1] |^(n)| > δλ_n , ^(n)∈0(λ_n^1-ε_0) = sup_t∈ [0,1] |S_⌊ nt ⌋| > δλ_n , Z_i≤λ_n^1-ε_0, ∀ i≤ n ≤sup_1≤ k ≤ n |S_k| > δλ_n , Z_i≤λ_n^1-ε_0, ∀ i≤ n ≤ 2 nexp(-cλ_n^ε_0) ≤ 2exp(-cλ_n^ε_0/2) for some constant c>0. Now we define functions which relate vectors in ^n to càdlàg step functions in . For integers j∈, denote U_j^↑ := {∈[0,1]^j: 0≤ u_1 < … < u_j ≤ 1 } and define functions h_j: × U_j^↑→, h_j((,))(t) := ∑_i=1^j z_iu_i≤ t. The maps h_j allow us to define the collection of functions with exactly j jumps as subsets of . Hence we define the following classes of càdlàg functions. _=0 :={x∈: x(t)=0, 0≤ t ≤ 1} = {the zero function in [0,1]}, _=j := {x∈: x(t) = h_j(,)(t) , 0≤ t≤ 1, (,)∈× U_j^↑) }, _≤ j := ⋃_i=0^j _=i = {càdlàg step functions with j or less jumps}. The map h_j: × U_j^↑↦ is continuous for j∈. The proof which is similar to Lemma 5.3 in <cit.> is skipped here. The function h_j maps points in × U_j^↑ to functions in _=j⊂, which are càdlàg functions in [0,1] with exactly j jumps. Hence for any F⊂ bounded away from _≤ (j-1), we have h_j^-1(F) = h_j^-1(F∩_=j). Hence h_j∘ h_j^-1 (F) = F∩_=j. The following result extends the large deviation result of <cit.> in the setting of <cit.> in order to obtain what we think of as hidden large deviations. The special case of j=1 in Theorem <ref> corresponds to <cit.>. The Lebesgue measure (in ^j) is denoted _j and ν_α^j is the j-fold product measure of ν_α (again in ^j) with ν_α as defined in (<ref>). Let j≥ 1. Under the conditions of Lemma <ref> and subsequent notations, for λ_n→∞ as n→∞, γ_n^(j)^(n)/λ_n ∈ · → (ν_α^j ×_j) ∘ h_j^-1 (·) in 𝕄(\{_≤ (j-1)}) as n→∞, where γ_n^(j) = [ n |Z_1|>λ_n]^-j. Note that under the conditions of Lemma <ref> we have ^(n)/λ_n → 0 (0 ∈) in probability. In other words, the theorem states that the random element _n = ^(n)/λ_n satisfies a sequence of LDPs on {_≤ j}_j≥ 1. That is, the large deviations of the random walk concentrate on step functions with an increasing number of steps at increasingly faster rates. In particular, for polynomially bounded rates γ_n large deviations of partial sum processes of iid regularly varying random variables will always concentrate on step functions, among all functions in . Naturally \_≤ j are not the only possible spaces to look at; and other types of LDPs might hold for dependent processes; we do not investigate such possibilities here. Additionally we do not investigate large deviations on \⋃_j=0^∞_=j in this paper. We show convergence in according to Definition <ref> for (<ref>), starting with the upper bound for closed sets. The idea is to dissect the space ^n, which contains the first n elements of the random walk, into a union of n disjoint sets that define which dimensions are allowed to be “big”. For any k=0,1,…, n, and λ>0, define k (λ) = {∈^n: |{i: |z_i| > λ}| = k}. Hence k ⊂^n are all points in ^n which have exactly k-co-ordinates with absolute value greater than λ. Clearly _n = ⋃_k=0^n k (λ). Upper bound Let F⊂ be a closed set, bounded away from _≤ (j-1). Then for small ϵ_0>0, ^(n)/λ_n ∈ F = ^(n)/λ_n ∈ F, ^(n)∈⋃_k=0^n k(λ_n^1-ε_0) = ∑_i=0^n^(n)/λ_n ∈ F, ^(n)∈ k(λ_n^1-ε_0) =: ∑_i=0^nB_i. We show that when multiplied by γ_n^(j), all the probabilities above are negligible except B_j. Now, since F was chosen to be bounded away from _≤ (j-1), there exists a δ_0>0, such that all elements of F have a minimum distance δ_0 to step functions with at most j-1 jumps. In particular F is bounded away from the zero element in . 1. Bounding B_0 Using Corollary <ref>, we have constants c_0>0 and ϵ_0>0 such that, B_0 ≤sup |^(n)(t)/λ_n| > δ_0/2, |Z_i| ≤λ_n^1-ε_0 ≤ 2exp(- c_0λ_n^ε_0). Hence this term is exponentially bounded and goes to 0 when multiplied by γ_n^(j). 2. Bounding B_i for 1≤ i ≤ j-1 For i∈:={1,2,…,n}, denote by (i)={={k_1,…,k_i}: 1≤ k_1<…<k_i≤ n}, all possible subsets of size i of the index set . We will show that B_i for 1≤ i ≤ j-1 are also exponentially bounded. With the same δ_0 as previously chosen, we have B_i = ^(n)/λ_n ∈ F, ^(n)∈ i(λ_n^1-ε_0) = ∑_∈(i)^(n)/λ_n ∈ F, |Z_l| > λ_n^1-ε_0, ∀ l ∈, |Z_l| ≤λ_n^1-ε_0 , ∀ł∈\ ≤∑_∈(i)(sup_t ∈ [0,1]|^(n)(t)-∑_m= 1^i_k_m^(n)(t) | > λ_n δ_0/2, . |Z_l| > λ_n^1-ε_0, ∀ l ∈, |Z_l| ≤λ_n^1-ε_0 , ∀ l ∈\) ≤∑_∈(i)(sup_t ∈ [0,1]|∑_l∈\_l^(n)(t) | > λ_n δ_0/2, |Z_l| ≤λ_n^1-ε_0 , ∀ l ∈\) = ∑_∈(i)( S_ (n-i)^*() > λ_n δ_0/2, |Z_l| ≤λ_n^1-ε_0 , ∀ l ∈\), where S_(n-i)^*() = sup_t∈[0,1]| ∑_l∈\^(n)_l(t)| sup_t∈[0,1]| ∑_l=1^n-i^(n)_l (t) | = sup_1≤ l ≤ n-i |S_l| for any ∈(i). Since the size of the set (i) is |(i)| = ni, we have B_i ≤ ni ( sup_1≤ j ≤ n-i|S_j| > λ_n δ_0/2, |Z_l| ≤λ_n^1-ε_0 , 1≤ l ≤ n-i ), ≤ 2 ni exp(-c_iλ_n^ε_0/2), for some c_i>0 according to Corollary <ref>. Since our choice of γ_n^(j) = [ n |Z_1|>λ_n]^-j, clearly γ_n^(j)∑_i=1^j-1B_i→ 0, as n→∞. 3. Bounding B_i for j+1≤ i ≤ n We bound the quantity γ_n^(j)∑_i=j+1^nB_i together. We argue that when multiplied with γ_n^(j), the probability of events with more than j large jumps is also negligible. Observe that γ_n^(j)∑_i=j+1^nB_i ≤γ_n^(j)∃ k_1,…, k_j+1∈ : |Z_k_i| >λ_n^1-ε_0, i=1,…,j+1 = γ_n^(j)nj+1|Z_1| > λ_n^1-ε_0^j+1 ≤ c n |Z_1|> λ_n^1-ε_0^j+1/|Z_1| > λ_n^j =:f_n, for some c>0. Now f_n is a regularly varying sequence with parameter r_0:= 1 -(j+1)ρα + ε_0ρ(j+1)α + jρα = (1-αρ) + (j+1)ε_0ρα, see <cit.> for details on operations on regular variation. Since by choice αρ>1, for small enough ε_0, we have r_0<0. Hence γ_n^(j)∑_i=j+1^nB_i≤ f_n→ 0 as n →∞. 4. Bounding B_j Finally, we are left with the term γ_n^(j)B_j which is the non-negligible contributing term for large n. For δ>0, let F^δ := {x ∈: d_J_1(x,F)≤δ} with δ small enough such that F^δ is still bounded away from _≤(j-1). B_j = ^(n)/λ_n ∈ F, ^(n)∈ i(λ_n^1-ε_0) = ∑_∈(j)^(n)/λ_n ∈ F, |Z_l| > λ_n^1-ε_0, ∀ l ∈, |Z_k| ≤λ_n^1-ε_0 , ∀ l ∈\ ≤∑_∈(j)P(sup_t ∈ [0,1]|^(n)(t)-∑_m=1^j_k_m^(n)(t) | ≤λ_n δ, ^(n)/λ_n ∈ F , . |Z_l| > λ_n^1-ε_0, ∀ l ∈, |Z_l| ≤λ_n^1-ε_0 , ∀ l ∈\) + ∑_∈(j)P(sup_t ∈ [0,1]|^(n)(t)-∑_m=1^j_k_m^(n)(t) | > λ_n δ, ^(n)/λ_n ∈ F , . |Z_l| > λ_n^1-ε_0, ∀ l ∈, |Z_l| ≤λ_n^1-ε_0 , ∀ l ∈\) ≤∑_∈(j)∑_m=1^j^(n)_k_m/λ_n ∈ F^δ + njsup_t∈ [0,1]| ∑_l ∈\^(n)_l|> λ_nδ, |Z_l| ≤λ_n^1-ε_0 , ∀ l ∈\ = P_j,1^(n) + P_j,2^(n). Now, using Lemma <ref>, and arguments similar to the one for bounding B_i for 1≤ i ≤ j-1, we can check that the quantity P_j,2^(n) is negligible at rate γ_n^(j) and hence γ_n^(j)P_j,2^(n)→ 0 as n→∞. In the remaining term P_j,1^(n), we use the inverse of the map h_j to measure the probability. For any closed set F^*⊂, define (J(F^*),T(F^*)) :=h_j^-1(F^*∩_=j) ⊂× U_j^↑ to be the pre-image of F^* ∩_=j under the map h_j broken into the jump part and the time part. Also note that due to the continuity of h_j (Lemma <ref>), since F^δ is closed, the pre-image of F^δ is also closed. Clearly, ∑_m=1^j^(n)_k_m/λ_n ∈ F^δ is equivalent to ∑_m=1^j^(n)_k_m/λ_n ∈ F^δ∩_=j, as F^δ is bounded away from _≤ j-1. Thus, P_j,1^(n) = ∑_∈(j)∑_m=1^j^(n)_k_m/λ_n ∈ F^δ = ∑_∈(j)∑_m=1^j^(n)_k_m/λ_n ∈ F^δ∩_=j = ∑_∈(j)∑_m=1^j Z_k_mk_m/n(t)/λ_n ∈ F^δ∩_=j = ∑_1≤ k_1, …,k_j≤ n P( (Z_k_m/λ_n)_1≤ m≤ j∈ J(F^δ)) 1( (k_1/n,…,k_j/n)∈ T(F^δ) ) = P((Z_1,…,Z_j)/λ_n∈ J(F^δ) ) ∑_1≤ k_1< …< k_j ≤ n1( (k_1/n,…,k_j/n)∈ T(F^δ) ). Note that as n→∞, |Z_1|>λ_n^-jP((Z_1,…,Z_j)/λ_n∈ J(F^δ) ) →ν_α^j(J(F^δ)). Similarly, for T(F^δ), we obtain for n→∞, n^-j∑_1≤ k_1< …< k_j ≤ n1( (k_1/n,…,k_j/n)∈ T(F^δ) ) → Leb_j(T(F^δ)). Hence using (<ref>) and (<ref>), we have as n→∞, γ_n^(j) P_j,1^(n)→ (ν_α^j×_j)(J(F^δ),T(F^δ)) = (ν_α^j ×_j) ∘ h_j^-1 (F^δ). Therefore lim sup_n→∞γ_n^(j)B_j≤ (ν_α^j ×_j) ∘ h_j^-1 (F^δ), for δ>0. Summing up all the bounds we obtained, we have lim sup_n→∞γ_n^(j)^(n)/λ_n ∈ F≤ (ν_α^j ×_j)(h_j^-1(F^δ)). The continuity of h_j ensures h_j^-1(F) = ⋂_δ>0 h_j^-1(F^δ) and hence letting δ→ 0 gives us the required upper bound lim sup_n→∞γ_n^(j)^(n)/λ_n ∈ F≤ (ν_α^j ×_j) ∘ h_j^-1 (F). Lower bound Let G be open and bounded away from _≤ (j-1). Now define, G^-δ⊂ G, G^-δ = {f∈ G: d_J_1(f,g) < δ implies g∈ G}. Choose δ small enough such that G^-δ is non-empty. It is still open and bounded away from _≤ (j-1). Searching for a lower bound, we shrink the set G to its bare minimum, ^(n)/λ_n ∈ G ≥∑_1≤ k_1< …< k_j ≤ n∑_i=1^j _k_i^(n)/λ_n ∈ G^-δ, sup |^(n)-∑_i=1^j _k_i^(n)| < λ_nδ = ∑_1≤ k_1< …< k_j ≤ n∑_i=1^j _k_i^(n)/λ_n ∈ G^-δsup |^(n)-∑_i=1^j _k_i^(n)| < λ_nδ The second factor converges to one since S_n/λ_n → 0 in probability as n→∞. For the first factor, we proceed in the same fashion as for P_j,1^(n) above. Note that instead of our functions being in [0,1], we can easily extend Theorem <ref> to càdlàg functions in _M=[0,M] for some number M>0, with minor modifications to the proof. Hence all the results obtained in this section hold if we amend the definitions of the spaces ,_=j,_≤ j accordingly. Without loss of generality we refer to these results as if they hold for _M and its appropriate subsets from now on. §.§ Random walks with a constant drift The conclusion in Theorem <ref> assumes that the random variables are centred. In case Z_1≠ 0, we can use the theorem to infer information about the deviations from the mean for a process created with iid variables Z_1^* = Z_1- Z_1. Nevertheless, if we assume α>1 and Z_1≠ 0 we may look at the random walk with drift. By setting λ_n = n we are able to preserve the drift in the limit. Theorem <ref> can be modified to incorporate a drift term; for this we require two further lemmas as given below. Recall that e denotes the identity function on the respective domain [0,M]. Let f : [0,M] → be continuous. Then the map ϕ_f: _M →_M ϕ_f: x ↦ x+f is continuous in the J_1-topology. Let ε>0. Suppose x,y ∈_M and d_J_1(x,y)<δ. Then d_J_1(ϕ_f(x), ϕ_f(y)) = inf_λ x + f - (y + f) ∘λ∨ e- λ ≤inf_λ( x - y ∘λ∨e-λ + f-f∘λ∨ e-λ) ≤ 2 δ + f - f∘λ_min∨ 2δ , where λ_min denotes a time-shift close to the infimum of the distance of x and y. We may bound the fluctuation of f by the modulus of continuity w_f(δ)= sup_|s-t| ≤δ |f(s) - f(t)| which tends to zero as δ→ 0 to obtain d_J_1(ϕ_f(x), ϕ_f(y)) ≤ 4δ + w_f(2δ). Let {μ_n}_n≥ 1 be a sequence of finite measures on 𝕊 and μ_0 ∈(𝕊\ℂ). Suppose μ_n →μ_0 in as n →∞. Additionally assume there is an addition operation such that (𝕊,+) forms a group. Let { y_n }_n≥ 1⊂𝕊 be a sequence with y_n → 0. For y∈𝕊 denote s_y: A ↦ A-y, A ∈ℬ(𝕊) the map shifting sets by an element y. Then μ_n ∘ s_y_n→μ_0 , n →∞ in . Let F ⊆𝕊 be closed and bounded away from ℂ. Let δ >0 such that F^δ is still bounded away form ℂ. For n large enough d(y_n,0)<δ (where d denotes the metric on 𝕊) and hence μ_n ∘ s_y_n (F) ≤μ_n(F^δ). A similar argument holds for open sets G bounded away form ℂ. Letting δ→ 0 proves the result. Let {Z_i}_i=1^∞ be a sequence of iid random variables with Z_1 ∈ℛ𝒱_-α, α>1. Denote m = Z_1 and define h_j^m: (\{0})^j × U_j^↑→, h_j^m((,))(t) := ∑_i=1^j z_iu_i≤ t + mt, and correspondingly _=j^m := h_j^m (^j\{0}× U_j^↑). Then, as n →∞, γ_n^(j)^(n)/n ∈·→ (ν_α^j ×_j) ∘(h_j^m)^-1 (·), in 𝕄(\_≤ (j-1)^m). The space _=j^m is defined as the space of step functions with exactly j discontinuities and a constant drift term “mt”. In particular, _=0^m = { x(t) = mt, t ∈ [0,M]}. Theorem <ref> allowed for scalings λ_n that are growing fast enough such that ^(n)/λ_n stays close to zero for large n. Note that in Corollary <ref> we restrict to α>1 and specify λ_n = n to preserve the drift term. Necessarily, we observe for sets A bounded away from _=0^m that ^(n)/n ∈ A→ 0 as n →∞. Hence we examine (a sequence of) large deviation principles on \_≤ j-1^m. The proof to the corollary is an application of Lemmas <ref> and <ref>, and the continuous mapping argument of Theorem <ref>. By Theorem <ref> we have γ_n^(j)(^(n) - ⌊ ne ⌋ m )/n ∈·→ (ν_α^j ×_j ) ∘ h_j^-1(·), n→∞, in 𝕄(\_≤ (j-1)). Continuous mapping yields γ_n^(j)(^(n) - ⌊ ne ⌋ m )/n + me ∈·→ (ν_α^j ×_j ) ∘ (h_j^m)^-1(·), n→∞, in 𝕄(\_≤ (j-1)^m) by virtue of Lemma <ref>. The result then follows by Lemma <ref>. § APPLICATION TO FINITE BUFFER QUEUES In this section we apply the results of Theorem <ref> to the modified Lindeley recursion; see (<ref>) below. This formula is usually interpreted as describing the evolution of the queue length in a queue with finite buffer. First we derive large deviation principles for what we call long intense periods, defined as the maximum time a queue-size process spends continuously above a certain threshold. These LDPs are derived in a limit where both, the threshold level and the buffer size, approach infinity while the arrival process is sped up appropriately. Second, we present a simulation study which combines two of the derived LDPs to provide a simple analytical approximation and explanation for the empirical distribution of extremal lengths of long intense periods. §.§ Queueing processes We study recursions of the form Q_n^K = min{max{Q_n-1^K + A_n - C_n,0},K}, with Q_0 ≥ 0, n ≥ 1 where {A_n}_n ≥ 1 and {C_n}_n≥ 1 are two sequences of iid non-negative random variables. This is the modified version of Lindley's recursion <cit.> to accommodate queues with finite buffers of size K. The recursion in (<ref>) can be interpreted in many different ways. For example in the context of network traffic, A_n may be interpreted as the number of packets arriving in the time interval C_n-C_n-1, whereas Q_n-1 describes the amount of work previously in the buffer of a single server processing work at a fixed rate. Any number of packets arriving at a full buffer are immediately discarded. For example, <cit.> studies (<ref>) under the assumption that ∫_0^x A_1>z z/ A_1 follows a subexponential distribution to conclude that the stationary loss rate is essentially due to one large observation when the buffer size approaches infinity. That is (Q_n^K +A_n+1 - C_n+1 - B) ∨ 0 = (A-K) ∨ 0(1 + o(1)), K→∞. Sample path LDPs for queueing processes with both infinite and finite buffers are studied in <cit.> mostly under the assumption that the moment generating function exists. We work with regularly varying random variables A_1 ∈-α, α>0 throughout which do not satisfy this assumption. To study the queueing recursion (<ref>) we follow the continuous mapping approach. First we define a suitable embedding of the sequences {A_n} and {C_n} in the space _M and then employ a continuous map to obtain a process which agrees with the queueing recursion at specified discrete time stamps. See for example <cit.>, <cit.> or <cit.> for more on this approach applied to queueing processes. This map is usually called reflection or Skorohod map. We briefly recall the required results. For a process x ∈_M with x(0) = 0 we call {v(t), l(t), u(t)} the solution to the Skorohod problem if v(t) ∈ [0,K] and v(t) = x(t) + l(t) - u(t), ∫_0^∞ v(t) l(t) = 0, ∫_0^∞ (K-v(t)) u(t) = 0, and both l,u are non-negative non-decreasing functions. Denote ψ_0^K: _M →_M the reflection map on the interval [0,K] as ψ_0^K : x ↦ v, where v denotes the resulting regulated process of the Skorohod problem. This map is (Lipschitz-) continuous on _M equipped with the J_1 metric, see e.g. Lemma 4.6 of <cit.>. To facilitate the discussion let C_n = c, n≥ 1 for some c>0 and denote (t) := ∑_i=1^∞ (A_i-c) t≥ i, t ≥ 0 the embedding of the random walk induced by A_n - c in _M. Then ^K := ψ_0^K() is an embedding of Q_n^K into _M satisfying ^K(t) = Q_t^K for t ∈_0. Consequently we call ^K a queueing process with buffer K. We could also work with other embeddings as for example (t):= ∑_n=1^∞ A_nt ≥ n - ct, t ≥ 0 and define _B^K := ψ_0^K() to allow for more nuanced interpretations of the queueing process . But since this work focuses on scaled versions of the queueing process with both time t and space ^K(t) scaled appropriately, the exact form of the interpolation is mostly irrelevant for the limit. §.§ Long intense periods We adopt the position that a queueing process with the queue size close to the buffer K corresponds to an undesirable state. In such a state the service quality (of which ^K(t) is a proxy) is perceived as suboptimal. In the following we introduce and study the longest period an observed queueing process spends above a certain threshold θ K during the observation horizon [0,M]. We call such intervals long intense periods and investigate their length. For a càdlàg function x ∈_M and a fixed level η∈_+ we define L^η: _M →_+ x ↦sup_0≤ s < t ≤ M{ t-s: x(u) >η ∀ u ∈ (s,t) }. For a queueing process with buffer K we call L^θ K(^K) the length of the intense period at level θ∈ (0,1). How useful is it to calculate large deviations for long intense periods in queues? Can we gain more insight into waiting times in queues with this information To illustrate the applicability of such results, we use a simulation study which investigates the distribution of long intense periods for large threshold levels θ∈ (0,1). The object of our simulation study is a queueing process ^K(t) with N=50000 arrival variables following a power law distribution with tail index α=1.44 and expectation m = 0.5. The queue has a finite buffer K=20000 and any additional service requirements will be lost. We assume the server works at a fixed rate c=1 with A_i describing the amount of service requirements arriving in one unit of time. We study long intense periods above the level θ=0.85, that is ^K(t)> 17000 is considered intense. The queueing process is observed on [0,M] with M=N. With this example we treat service time distributions that still have finite means but infinite variance. The particular α value corresponds to the tail parameter of file sizes in Internet traffic reported in <cit.>. Specifically we consider the arrival distribution A_1 > z = ( z/(α-1) m + 1 )^-α , z>0. Trivially A_1 >z∈-α. Figure <ref> contains a histogram of the realized lengths of the long intense periods in queueing processes with the parameters above. It is based on 22000 observations which exhibit a strictly positive long intense period. That is, Figure <ref> shows a histogram of L^θ K()|L^θ K() >0. We would like to understand the shape of the histogram that we observe here; why is there a peak in the middle and a decay afterwards? We revisit the histogram at the end of this section, accompanied by an explanation for its shape, based on (hidden) large deviations. §.§ Large deviations for long intense periods We work out the corresponding sequence of large deviation principles for long intense periods of queueing processes. Let A_i, i≥ 1 be a sequence of iid non-negative regularly varying random variables with A_1 ∈-α, α>1. Assume c>m:=A_1 and define the queueing process ^K,(n)(t):= ^nK(nt)/n, t∈ [0,M], n ≥ 1 with ^K defined as in (<ref>). Denote κ :=1-θ/c-m K. The intense periods L_n:= L^θ K (^K,(n)) of the queueing process ^K,(n) observed on [0,M] satisfy a sequence of large deviation principles on [0,M] \ [0,(j-1)κ ] with the limit measure μ^(j)_L concentrating it's mass on ( (j-1)κ, jκ]. Specifically, L_n ∈γ_n^(j), μ^(j)_L, [0,M] \ [0,(j-1)κ], 1 ≤ j ≤⌊M/κ⌋, where the limit measure is given by μ^(j)_L = (ν_α^j ×_j) ∘(h^m-c_j)^-1∘( ψ_0^K )^-1∘( L^θ K)^-1. The assumption c>A_1 ensures that the process will drift in the negative direction on average, such that the process being close to its buffer is actually a rare event. At the first level for j=1 the theorem states that the long intense periods of a queueing process with buffer K and negative drift may be approximated by summing over all one-jump functions that contain a jump of size at least θ K. Since the measure concentrates on one-jump functions, the maximum attainable long intense period is attained by a single jump that exceeds the buffer limit K, with the process drifting in negative direction at rate m-c afterwards. Thus, latest at time κ after the jump the process will leave the intense region, no matter the size of the jump. Measuring the longest connected interval of time spent above a certain threshold is not a continuous operation for càdlàg processes. For example consider for M>2 the function x∈_M x(t) := 1-t if t ∈ [0,1) 2-t if t ∈ [1,M]. Adding a small constant via ϕ_c(x)(t):= x(t) +c we obtain for c<0: L^0(ϕ_c(x)) = (1-|c|)∧ 0 but L^0(x) = L^0(ϕ_0(x))=2, while at the same time ϕ_c(x) → x as c→ 0. Consequently L is not continuous. Nevertheless L^θ K is continuous almost everywhere with respect to the limit measure μ = ν_α×_1∘(h_1^m)^-1∘( ψ_0^K ) ^-1 on _M as the only way to obtain a discontinuity is through the jump at the end of the long intense interval. But the jump position is uniformly distributed hence the measure of that set is zero. Additionally, for our purposes, there is no need to consider functions outside the support of μ. We need the following lemma to prove Theorem <ref>. Let j ∈. Denote E_j := {x ∈_M: [ [ x(t)=0] OR [x(t+s) = x(t) - s(c-m) , |s| small enough]; for all but j points t. Additionally x(t) ≥ x(t-) ∀ t∈ [0,M]. ]}, D_j := {[ x ∈ E_j: ∃ t ∈{discontinuity points of x}; such that x(t-) = θ K OR x(t)=θ K OR t ∈{0,M} ]}. Then L^θ K is continuous on E_j\ D_j. Note that E_j contains all càdlàg functions which contain exactly j positive jumps and decrease at rate c-m otherwise, regulated to take values in [0,K]. The set D_j further restricts to those functions whose jumps are bounded away from the critical level θ K. To show the claim we need to introduce additional machinery. Namely we define an intense period as a period during which the function x ∈ E_j stays continuously above the critical level θ K and enumerate all such periods. Subsequently we show that the length of each such period cannot change much in case x is not perturbed too much. Denote _sL(l,x,v), _tL(l,x,v) : _+ ×_M × [0,M] →_+ _sL(l,x,v) := inf{u ∈ (v,M]: x(u) > l }, _tL(l,x,v) := inf{u ∈ ( _sL(l,x,v),M]: x(u)<l OR u=M }. We assume - as is usually the case - that inf∅ = ∞. Next we recursively record the start and end times of what we call intense periods, starting at zero. s_1,t_1 := _sL(l,x,0), _tL(l,x,0), s_i,t_i := _sL(l,x,s_i-1), _tL(l,x, s_i-1), i≥ 2, n_x := max{i: s_i < ∞} In case the tuple s_i,t_i is finite - either both or none are - we call t_i-s_i the length of the i^th intense period of x. Note that for x ∈ E_j there are exactly n_x intense periods, with 0≤ n_x ≤ j. The length of the longest of these corresponds to what we defined above in (<ref>) as the length of the long intense period of x. The enumeration of intense periods is well-defined for any càdlàg function on [0,M], although their number need no longer be finite, nor must the linear enumeration from zero capture all instances of the function exceeding a level l. Let x ∈ E_j \ D_j. Then the set of time points at which x is above the critical level can be partitioned as { u: x(u) > θ K} = ⋃_i=1^n_x [s_i,t_i). Since all jump discontinuities of x have values bounded away from the critical level, all of the intervals [s_i,t_i) and [t_i,s_i+1) are of positive length. Moreover, denoting Δ^θ K_J(x) := min{ |x(u^-) - θ K|∧ |x(u) - θ K | ∧ u-0 ∧ M-u : u is a discontinuity point of x}, we find that for all 0<δ < Δ^θ K_J(x), x(u) ∈ (θ K -δ, θ K + δ) ⇔ u ∈(t_i - δ/c-m, t_i + δ/c-m)∩ [0,M] for some 1≤ i ≤ n_x. Next we show that for all ε>0, small enough such that ε(c-m)< Δ^θ K_J(x), there exists a ζ such that whenever d_J_1(x,y) < ζ we have [ u ∈ [s_i+ε,t_i-ε) ⇒ y(u) > θ K ,; u ∈ [t_i + ε,s_i+1-ε) ⇒ y(u) < θ K. ] This implies that any y close enough to x has similar intense periods as x, ignoring any negligible intense periods of y. Hence, L^θ K is continuous at x in (_M,d_J_1). We proceed by showing that the above claim holds for ζ = ε ((c-m)∧ 1)/3. Then there exists a λ∈Λ such that x- y∘λ < ε ((c-m)∧ 1)/2, λ - e < ε ((c-m)∧ 1)/2. Now (<ref>) combined with (<ref>) (where δ = ε(c-m)/2) implies u ∈ [s_i , t_i-ε/2) ⇒ (y ∘λ)(u)> θ K + ε (c-m)/2 - ε((c-m)∧1)/2≥θ K, u ∈ (t_i-ε/2,s_i+1) ⇒ (y ∘λ)(u)< θ K. Accounting for the time change introduced through λ, we infer from (<ref>) that the last two implications in (<ref>) hold when the two intervals get reduced by a further ε/2 on each side. In turn this proves the statement in (<ref>) and thus continuity of L^θ K on E_j\ D_j for all j≥ 1. We apply the continuous mapping argument in Theorem <ref> twice. First, using the Skorohod map of (<ref>), the large deviations result in Corollary <ref> and continuous mapping yield γ_n^(j)Q^nK(nt)/n ∈·→ (ν_α×_j) ∘(h^m-c_j)^-1∘( ψ_0^K )^-1 (·), n →∞ in ( ψ_0^K ( _M \ (_M)^m-c_≤ j-1) ). This is due to the definition in (<ref>) satisfying ψ_0^K (x/n) = ψ_0^nK(x)/n. The queueing map ψ_0^K preserves the number of jumps and hence satisfies the “bounded away” condition of Theorem <ref>. It is immediate from the definition of the Skorohod problem that ψ_0^K(_M \ (_M)^m-c_≤ j-1) ⊆ E_j. Additionally, note that μ_Q^(j) (D_j) = 0, j≥ 1 as (h^m-c_j)^-1∘( ψ_0^K )^-1(D_j) ⊂^2j is not of full dimension; the condition of having x(t) or x(t^-) = θ K amounts to imposing restrictions linking the time and value of a jump through relations of the form x(t_i) - (t_i-t_i-1) (c-m) + J(t_i) = θ K, where J(t_i) denotes the size of the j jump. Hence Lemma <ref> above combined with a second application of Theorem <ref> yields the result. §.§ Calculating explicit limit measures In the following we compute the limit measures μ_L^(1) and μ_L^(2). We assume M> 2κ throughout. For j=1 we obtain μ^(1)_L( (l,∞) ) = (M-l) ( l(c-m) + θ K)^-α if l ∈ (0,κ] 0 otherwise. In other words, the measure μ_L^(1) is the sum of a point mass at l = κ with value K^-α(M-κ) and an absolutely continuous part on (0,κ). Considering this initial large deviations estimate on its own we would approximate ( L^θ K( ^K) > κ ) ≈ 0. For any finite buffer non-limit scenario this may be too coarse. A more refined estimate based on hidden large deviations allows for more accuracy. Namely on [0,M] \ [0,κ ] we have γ_n^(2) L^θ nK(^nK(nt)) ∈·→μ_L^(2)(·), n →∞, in ( [0,M] \ [0,κ]), which concentrates on (κ,2κ]. This again can be explained by the rate γ_n^(2) only allowing for at most two jumps in the random walk. Any intense period with length L<κ is more likely to happen due to one jump hence processes containing only one jump must be excluded in the hidden large deviation principle. Long intense periods with length L>2κ are not possible since the maximum length is achieved if the buffer is filled at some initial time t_0< M-2κ starting the long intense period and an additional jump at time t_0 + κ of size at least (1-θ)K. We compute the limit measure for the events {L > l}, l ∈ (κ,2κ]: μ_L,θ,K^(2) ( (l,∞) ) = μ^(2) ( {All two jump functions with L>l}) = (M-l) ∫_θ K^∞ν_α ( j_1) ∫_l-κ^(K∧ j_1 - θ K)/(c-m) u_2 ∫_l(c-m) - ( K∧ j_1 - θ K)^∞ν_α( j_2) = (M-l) ∫_θ K^∞ν_α( j_1) K∧ j_1 - θ K /c-m > l-κK∧ j_1 - θ K /c-m - (l-κ)/( l(c-m) - (K ∧ j_1 - θ K) )^α = M-l/c-m∫_θ K + l(c-m) - (1-θ)K^K α x^-α-1x - θ K - l(c-m) + (1-θ K) /( l(c-m) + θ K - x) ^α x + M-l/c-mK^-α2(1-θ) K - l(c-m) /( l(c-m) - (1-θ) K )^α . Further limit measures can be computed but the explicit derivation becomes more cumbersome as the level increases. §.§ Simulation study - combining the first two LDPs In this section we provide some insights on the practical relevance of hidden large deviations. We find that in the setting of the simulation study described in Example <ref> we are able to numerically validate the rate and limit measure of hidden large deviations. The previous section established large deviation principles for long intense periods for any interval [(j-1)κ, jκ] with j≤⌊ M/κ⌋, each with its own rate. And indeed, for the theory of LDPs we may only treat these limit measures separately due to the different magnitudes of the rates γ_n^(j). In practice however, for any finite observation period of a queue with finite buffer size, several of the limit measures might be relevant for a single statistic. The simulation study will examine the interplay of different rates in a single probability estimate. We proceed to construct the two estimates involving the first and second level LDPs separately. One jump According to traditional large deviation estimates for heavy tailed queueing processes with “large” buffers, the long intense period will be due to a single jump reaching above the threshold level θ K and the queue drifting in direction -(c-m) thereafter. To use Theorem <ref> we need to choose a queue sequence number n. We thus obtain the following approximation. L^θ K( ^K)> l = L^θ K/n(^K/n,(n)) > l/n = L_n > l/n ≈1/γ_n^(1)μ^(1)_L((l/n,∞)) = n^αA_1>n (M-l) (l(c-m) + θ K)^-α if l ∈ (0, κ] 0 otherwise . The point mass at κ yields L^θ K(^K) ∈ (κ - ε ,κ + ε)≈(n/K)^αA_1 > n (M-κ). One or two jumps The two jump measure can be approximated in the same fashion as the approximation for one jump above using equation (<ref>) instead. To get a single estimate for the distribution of the long intense periods of the queueing process ^K we propose to combine the two estimates into a single approximation. L^θ K( ^K) > l≈ nA_1>nμ_L^(1)( (l/n,∞) ) if l ∈ (0,κ] n^2 A_1>n^2 μ^(2)_L( ( l/n,∞) ) if l ∈ ( κ, 2 κ] 0 otherwise, where the buffer size K and observation horizon M are scaled accordingly in the limit measures. In Figure <ref> we plot the same histogram as in Figure <ref>, and view it as an estimate of the density L^θ K( ^K) ∈ l | L^θ K( ^K)> 0, l>0. The limit measure μ^(1)_L puts zero mass on values beyond the vertical red line which marks the location of the point mass of μ^(1)_L. Due to n being finite we expect some values immediately to the right of the point mass as caused by only a finite number of random variables approximating the mean rate of decrease for the queue content. Nevertheless, concerning the values on the far right we believe an explanation via Hidden Large deviations (HLD) is best suited for the distribution of L^θ K( ^K). Hence we add the estimate in (<ref>) to the plot. To visualize the point mass we fix two ε_1,ε_2>0 such that the point mass at κ gets distributed over the area (κ-ε_1 , κ + ε_2). Outside of this region we approximate the measure with the corresponding densities. Additionally we provide a plot of the tail of the distribution on a log scale to better visualize the fit for the hidden large deviation estimate. The figures clearly show how our hidden large deviation estimates closely approximate the histogram observed (more clearly in the right plot in Figure <ref>.) § CONCLUSION AND FURTHER REMARKS We provide limit measures for successively rarer large deviations of random walks with regularly varying iid increments. Scaling time and space appropriately we are able to obtain limit measures for large deviations of queueing processes which preserve the drift term in the limit. In the final section we showed that hidden large deviations at the second level, though happening at the squared rate of the first large deviation, are numerically observable. Clearly, our (hidden) large deviation estimate performs quite well to approximate the histogram - even at the tail (on a log scale.) For future directions of study, one can explore large deviations on a space \⋃_j=1^∞_=j which we have not ventured into. The choice of our space of convergence was governed by jumps in heavy-tailed processes. Namely, we expect all of the random variables to attain values on the same scale. The exact structure of such deviations remains largely an open question. Similarly, we did not explore into situations where the “iid" assumption gets relaxed. A j level LDP happens at a rate which is the j power of the rate of the first LDP. This clearly is a consequence of the independence among the random variables driving the random walk. Future work on weakly dependent variables are under current investigation. § ACKNOWLEDGEMENTS We would like to thank Parthanil Roy for interesting discussions on the preliminary ideas of (hidden) large deviations. Additionally, we gratefully acknowledge support from MOE Tier 2 grant MOE-2013-T2-1-158. imsart-nameyear

http://arxiv.org/abs/1701.07648v2

20170126104743

Microscopic Aspects of Magnetic Lattice Demagnetizing Factors

cond-mat.mtrl-sci

Department of Physics, Royal Institute of Technology, SE-106 91 Stockholm, Sweden London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, 17-19 Gordon Street, London, WC1H OAH, U.K. Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada Canadian Institute for Advanced Research, 180 Dundas St. W., Toronto, Ontario, M5G 1Z8, Canada Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario, N2L 2Y5, Canada London Centre for Nanotechnology and Department of Physics and Astronomy, University College London, 17-19 Gordon Street, London, WC1H OAH, U.K. Department of Physics, Royal Institute of Technology, SE-106 91 Stockholm, Sweden The demagnetizing factor N is of both conceptual interest and practical importance. Considering localized magnetic moments on a lattice, we show that for non-ellipsoidal samples, N depends on the spin dimensionality (Ising, XY, or Heisenberg) and orientation, as well as the sample shape and susceptibility. The generality of this result is demonstrated by means of a recursive analytic calculation as well as detailed Monte Carlo simulations of realistic model spin Hamiltonians. As an important check and application, we also make an accurate experimental determination of N for a representative collective paramagnet (i.e. the Dy_2Ti_2O_7 spin ice compound) and show that the temperature dependence of the experimentally determined N agrees closely with our theoretical calculations. Our conclusion is that the well established practice of approximating the true sample shape with corresponding ellipsoids for systems with long-range interactions will in many cases overlook important effects stemming from the microscopic aspects of the system under consideration. Microscopic Aspects of Magnetic Lattice Demagnetizing Factors P. Henelius Received December 14, 2016; accepted January 23, 2017 ============================================================= § INTRODUCTION Long-range interactions are important in many areas of science, from cosmology, through the gravitational interaction, to biology, through Coulomb's law. A long-range interaction may be defined in d spatial dimensions by its two-body potential V(r) scaling with distance r as r^-α where α≤ d <cit.>. The paramount problem in such systems is how to integrate V(r) over an extended system. Following Newton and Euler, the analysis of general systems has been largely based on the exact solutions for spheres and ellipsoids <cit.>. This raises the question of whether approximating other shapes to corresponding ellipsoids <cit.> just neglects uninteresting details or whether there are crucial properties that are lost in the approximation. The demagnetizing problem in magnetic systems is a natural setting for exploring this question since it is accessible and of intrinsic importance in experiments, and constitutes a paragon for exploring the thermodynamics of long-range interacting systems <cit.>. Demagnetizing effects are also important in superconductors, while analogues occur, for example, in electric systems <cit.> (depolarizing factor), in the problem of strain fields around inclusions <cit.>, and in the treatment of avalanching systems in confined geometries <cit.>. In an applied magnetic field H_ ext = B_ ext/μ_0, the thermodynamic energy of an ellipsoid of volume V and magnetic moment m acquires a contribution E_mag=(μ_0/2) N m^2/V, where N is the demagnetizing factor. After subtracting E_mag from the total energy, differentiation with respect to the magnetization, M ≡ m/V, defines the internal field ≡+, where H_ d = - NM is the demagnetizing field. The intrinsic magnetic susceptibility = ∂ M/ ∂ is a shape-independent material property derived from the experimentally determined susceptibility ≡∂ M/ ∂ through 1/= 1/ - N . The determination of N is a fundamental problem that dates back to the work of Poisson and Maxwell <cit.>. In the 1940s, Osborn <cit.> and Stoner <cit.> tabulated N for general ellipsoids, while more recently, Aharoni <cit.> treated cuboids in the →0 limit. These highly cited papers bear witness to the importance of accurately computable and easily accessible demagnetizing factors. Given that i) it was realized already in the 1920s that N for a non-ellipsoidal sample is a function not only of the sample shape, but also of χ_int itself <cit.>, and that ii) many experiments are routinely performed not on ellipsoids but on cuboids <cit.>, it is perhaps remarkable that it was only very recently that the χ-dependence of N was calculated for cuboids away from the →0 limit <cit.>. The existence of demagnetizing factors for cuboids suggests that their thermodynamics may be formulated in terms of an internal field, with corrections that become dependent on both shape and temperature <cit.> (through χ_int). In this work, we have found that, for magnetic lattices, the demagnetizing factor of cuboids depends also on the local spin symmetry and allowed orientations of the magnetic moments. With reference to the question posed at the very beginning, our result illustrates a case where a long-range interaction integrates in a qualitatively different way for a cuboid and an ellipsoid, such that the discrete microscopic nature of the system matters in the former case but not in the latter. We are aware of only a few previous studies where effects of such discreteness have been discussed  <cit.>. Our interest in this problem was spurred by the recent experimental observation of anomalous demagnetizing effects in the spin ice material Dy_2Ti_2O_7 <cit.>. One may ask whether small differences in the estimated N really matter for exposing important physics. The answer is found in Eq. (<ref>). If ≪ 1, then is insensitive to the precise value of N. However, in many physical systems that display unusual and interesting magnetic phenomena, is large, and becomes a sensitive function of N. Examples include the spin ice materials Dy_2Ti_2O_7 and Ho_2Ti_2O_7, which support magnetic monopole excitations <cit.>, and LiHo_1-xY_xF_4 which displays ultra-slow relaxation <cit.>. Important demagnetizing effects are manifest when an accurately directed field is required: for example in experiments on the elusive Kasteleyn transition <cit.>, sub-lattice pinning <cit.> and multiple field-driven transitions <cit.>; or else for disentangling the in- and out-of-phase frequency response <cit.>. In such cases, quantitative conclusions and accurate tests of theory depend, through χ_ int, on an accurate knowledge of N. Our work illustrates how this may be achieved. The rest of the paper is organized as follows. In <ref> we discuss how to determine N experimentally. In <ref> we introduce an iterative method for obtaining N, and we consider in <ref> a Monte Carlo calculation of N. Finally, we close the paper with a discussion in <ref>. For details regarding the experimental and numerical procedures we refer the reader to Appendices <ref>-<ref>. The effects of short-range interactions are considered in some detail in Appendix <ref>. § EXPERIMENTAL DETERMINATION OF N To illustrate the importance of the demagnetizing correction, and to test the theory presented below, we first present the experimental determination of N for a particular case. The localized-moment paramagnet Dy_2Ti_2O_7 (a spin ice) is well-suited to this purpose as it has a large susceptibility, is crystallographically well-defined (in the cubic space group Fd 3̅ m) with no evidence of crystal distortion <cit.>, and can be accurately cut into high-quality single crystal samples of different shape. Since its spin Hamiltonian has been established in great detail <cit.>, it is convenient to adopt Dy_2Ti_2O_7 as a model system for studying the demagnetizing factor. A sphere of diameter 4 mm and a cube of dimensions 2× 2× 2 mm^3, with edges precisely oriented along the cubic crystallographic axes [100], [010] and [001] directions, were commercially hand-cut from different larger crystals of Dy_2Ti_2O_7 provided by D. Prabhakaran <cit.> (see Ref. <cit.>). The cube was epi-polished on all sides <cit.>. Crystal shape, orientation, and experimental conditions were carefully controlled to minimize measurement errors; see Appendix <ref>. The experimental susceptibilities of both the sphere and the cube (χ_exp^sphere, χ_exp^cube) were determined from measurements of the magnetic moment. Setting the demagnetizing factor of the sphere to N_ sphere = 1/3, that of the cube was determined through Eq. (<ref>), i.e., N_ cube=1/χ_exp^cube -1/χ_exp^sphere + N_ sphere. In order to match the susceptibility of the cube and sphere in the high-T limit, χ_exp^cube was shifted by about 1% (χ_exp^cube→χ_exp^cube/1.0074) before calculating N_ cube. Fig. <ref> shows how the experimental N_ cube departs significantly from the 1/3 value when χ≳ 1. This is the main experimental result of our study. The inset of the figure compares the uncorrected susceptibility data and the data derived from assuming N = 1/3 for both samples. The predicted theoretical continuation of the experimental data below 2 K (dashed curves) is based on a generalized version of the dipolar spin ice model <cit.>. § DETERMINATION OF N VIA AN ITERATIVE METHOD In this section we introduce an iterative method to calculate the on-site field distribution inside a linear magnetic material placed in a uniform magnetic field. In the iterative algorithm we first assume that equals and calculate the induced local magnetization for an assumed . This magnetization generates a demagnetizing field that, in turn, modifies . The resulting field-magnetization equations are iterated until convergence. With the converged field and magnetization distributions in hand, one then computes N. To proceed, we consider a sample of volume V with magnetic moments. As a first case, we focus on Ising moments _i=m_iμ_B, where is the unit vector in the local Ising direction at site i, and m_i is dimensionless. We first determine the component of the local field along the Ising moment at site i, =_i ·, which is the sum of three contributions: = + + , which we now discuss one by one. First, the dipolar field at site i produced by all the other point magnetic dipoles within the sample, ≡_i^dip·, is given by the familiar form <cit.> = μ_0 μ_B/4π∑_j i(3 (·_ij)(·_ij) -·/r_ij^3 )m_j. Second, we consider an external field in the global direction, ^ext=B^ext, with = B^extcosθ_i, where cosθ_i≡·, the angle between the direction of the Ising axis at site i and the direction of ^ext. Third, is the contribution from the self-field, . In the classic case of a single point dipole <cit.>, a term 2/3μ_0 μ_Bδ() must be added to ensure that the average magnetic field in a sphere containing the dipole gives the correct macroscopic field. Similarly, we add a self-field to ensure that the internal magnetic field in a uniformly magnetized sample has the expected value, for example =2/3μ_0 for a uniformly magnetized sphere or cube <cit.>. Note that one should, in general, treat the limit of a uniformly magnetized non-ellipsoidal sample with some care. In this work, we are primarily concerned with paramagnetic samples in the linear response regime, where a weak magnetic field induces a magnetization proportional to it, as in a typical χ measurement. For a non-ellipsoidal sample, the induced magnetization is in general non-uniform, except in the χ→0 limit. In this limit, vanishes and, as a consequence, and M are uniform. Our goal is therefore to determine the self-field so that the magnetic field has the expected value in the χ→0 limit. We demonstrate the basic idea with two examples. We first take a cubic sample with all moments aligned in the global direction. In this case , and are all aligned with the direction for which the field equation =μ_0(+) reduces to B^z=μ_0(M^z-N_0M^z)=2/3μ_0M^z, where N_0=1/3 is the χ→0 limit of N for a cube <cit.>. If we consider a simple cubic lattice, it is well known that the lattice sum vanishes <cit.>. This implies that B^z,self=2/3μ_0M^z must be incorporated to ensure the expected net B^∥ field value. As a second example, we consider the case of a lattice where all the Ising axes are tilted by the same angle θ_i=θ with respect to the z-axis, with half the spins tilted to the right and half to the left so that there is no net magnetization in the or directions. The total , , and fields are again in the -direction, but what should the B^ field parallel to the magnetic moments be? From =μ_0(+), it follows that , is generated by two terms, which we discuss separately. We begin with the term generated directly by , namely ^1=μ_0, or B^1,z=μ_0 M^z=μ_0M^cosθ, where M^ is the magnetization in the local Ising directions, M^=V^-1∑_i=1^_i·. This equation is satisfied by B^1,=μ_0M^. The second term, B^2,z=μ_0H^z=-μ_0N_0M^z=-μ_0N_0M^cosθ is generated by . The field along the magnetic moment is thus B^2,=-μ_0N_0M^cos^2θ, and the net self-field becomes = B^1,+B^2,= μ_0μ_B/V[1- N_0cos^2θ]m_i, which is valid when the dipolar lattice sum, Eq. (<ref>), vanishes and when the average is along ^ext. For the case in which the lattice sum does not vanish, it must be subtracted from the self-field in order to ensure the expected net field value. <ref> give the local field in terms of the set of local magnetizations, {m_i}. With the local fields available we next consider the reverse relation that yields the {m_i} induced by . Using =χ (linear media), we get =μ_0(+/χ)=μ_0χ+1/χ, leading to m_i=V/(/+1)/μ_0μ_B, where is the local susceptibility in the direction, M^= H^. We can now proceed to iterate the expressions for in Eq. (<ref>) and m_i in Eq. (<ref>) until convergence, and then calculate N from Eq. (<ref>), where is given by =^zz= (∂ M^z/∂ H^z_ext)_T =μ_0μ_B/V B^ext∑_i=1^m_icosθ, where we are still considering site independent tilt angles, θ_i=θ. The intrinsic susceptibility, , expresses the relation between ^ext and induced under Ewald, or tin foil, boundary conditions <cit.>, which eliminate demagnetizing fields and correspond to the N = 0 limit. As a result, both and are responses to an internal field. While measures the response in the direction of , measures the response along the local Ising axis . With · =cosθ, H_ext^zcosθ induces a magnetization M^= H_ext^zcosθ. This magnetization, in turn, has a component M^z=M^cosθ= H_ext^zcos^2θ along , and therefore =cos^2θ. To sum up, once the converged and m_i distributions have been determined, N is calculated using Eq. (<ref>), N =[μ_0μ_B/V B^ext∑_i=1^m_icosθ]^-1 - 1/cos^2θ. In Fig. <ref>, we include N calculated for the pyrochlore lattice using the iterative method (red line), and the main theoretical result is shown in Fig. <ref>, where N is displayed as a function of for cubic samples of the simple cubic (sc) and body centered cubic (bcc) lattices with the Ising direction parallel to ^ext (cosθ=1). Results (not shown) for a tetragonal lattice, relevant to LiHoF_4 <cit.>, are found to be identical to the sc case. We also display results for a bcc lattice with spins pointing in the [101] and [1̅ 0 1 ] directions (cosθ=1/√(2)), and a pyrochlore lattice (cosθ=1/sqrt3) built from the conventional cubic unit cell <cit.>. Finally, we include results for the dipolar model with spherically symmetric Heisenberg spins on an sc lattice. § DETERMINATION OF N VIA MONTE CARLO SIMULATIONS With the iterative method, we are able to reach relatively large system sizes of 𝒪(10^6) spins. To verify that this method, which is mean-field like and does not include fluctuations in the m_i's, gives the same result as a full statistical calculation for a given spin Hamiltonian, we have also calculated N using Monte Carlo (MC) simulations for several representative cases (see Fig.  <ref>). For a single data point, the MC approach requires 𝒪(10^5) core hours <cit.> to reach the necessary precision for 𝒪(10^4) moments. Since the iterative formulation contains an internal susceptibility, but no explicit temperature, T, it is necessary to tune either the MC T, or the iterative method χ_int, so that the MC susceptibility calculated using Ewald boundary conditions, χ_int^MC, matches the susceptibility from the iterative calculation. We have chosen to adjust the MC temperature, T, in order to tune to the desired value. In other words, and to emphasize, we do not compare a temperature-dependent mean-field theory calculation with a MC calculation at the same nominal temperature, a calculation which would not generally yield the same N in the thermodynamic limit. For details concerning the numerical methods, we refer the reader to Appendices <ref> - <ref>. For definitiveness, we use the magnetostatic dipolar Hamiltonian ℋ=μ_0μ^2/4π∑_i > jΛ_ijσ_iσ_j, where σ_i=± 1, μ is the magnetic moment and Λ_ij=[ (·)-3 (·_ij) (·_ij)]/r_ij^3, and χ^zz, in zero field, is determined according to χ^zz=∂ M^z/∂ H^z=μ_0μ^2/k_TV⟨(∑_i=1^σ_icosθ)^2⟩. Using Ewald boundary conditions, we obtain , while open boundary conditions yield , with N obtained from Eq. (<ref>). Results for the MC method are shown in Fig. <ref>. All MC and iterative results have been extrapolated to infinite system size and, in Fig. <ref>, we compare the system-size dependence of the iterative and MC methods. Results for open boundary conditions are extrapolated using the form a+b/L+c/L^2 where the leading 1/L term represents a surface to volume ratio effect, while we use a+b/L^3+c/L^6 for periodic boundary conditions, with the leading 1/L^3-term representing the inverse volume of the system. These functions yield the best fit to the data, but we find that the extrapolated value of N is rather insensitive to the precise fitting function, see Appendix <ref>. § DISCUSSION The key results of this study are threefold. First, we find quantitative agreement between two theoretical methods – iterative and MC (Fig. <ref>) – and experiment (Fig. <ref>), demonstrating that our methods are sound. Second, the explicit T dependence of N for a cuboid has been verified for a real material (Fig. <ref>). Finally, N is found to depend on the symmetry and direction of the moments (Fig. <ref>). The sc, bcc and LiHoF_4 lattices with collinear Ising spins yield the same N, indicating that N is not directly sensitive to the lattice. However, turning the local Ising axes away from causes a more rapid decrease of N with increasing . The pyrochlore lattice with tilt angle cosθ=1/√(3) yields a smaller N than the bcc lattice with cosθ=1/√(2) for >0. The spin ice pyrochlore lattice and the dipolar model with Heisenberg spins yield the same result as the continuum method of Chen et al. <cit.>, and we conjecture that models with isotropic χ will generally follow this behavior <cit.>. Exchange interactions, even when known in detail (e.g., for Dy_2Ti_2O_7 <cit.>), have not been included in our theoretical models. This is because demagnetizing fields arise solely from the long-range dipolar interactions. The thermodynamic limit for short-range models is well-defined <cit.>, and inclusion of short-range interactions does not alter the thermodynamic limit results for N; see Appendix <ref>. Thermal fluctuations also appear irrelevant in this limit. For ellipsoids, N is calculated from averaged macroscopic fields that do not include thermal fluctuations and, similarly, our mean-field like iterative method captures the essential demagnetizing effects also for cuboids. However, in the non-universal approach to the thermodynamic limit (Fig. <ref>), there is an expected and significant finite-size difference between the iterative and the MC methods. What are the experimental implications of our results? If an accurate measurement of is required, then the corrections to N(χ→0) identified here may be dramatic for χ≳ N. For example, in the case of Dy_2Ti_2O_7, T χ_ (T) features a peak, which is easily shifted outside the experimental temperature window by application of the ordinary χ=0 demagnetizing correction (see Fig. <ref> and Ref.  <cit.>). More generally, while the demagnetizing correction is readily controlled for needles or ellipsoids, it is not always easy to prepare real samples with these ideal shapes. This is particularly true of non-metallic and often brittle samples – e.g., spin ice <cit.> and LiHoF_4 <cit.> – which have become of significant interest in recent years. Therefore, insofar as cuboidal samples are often the most practical to prepare and control, the best approach may be to use them alongside the theoretical corrections identified in this work. Our methods are general and valid for localized-moment magnets independently of details like interaction range and spin dimensionality, and the iterative method can be generalized to non-cuboids. The iterative method could also prove useful for calculating demagnetizing effects in aggregate systems, such as biomedically relevant dispersions of magnetic nanoparticles <cit.>. In conclusion, considering the demagnetizing problem as a paradigm for the study of long-range interactions, our results confirm that N may be defined for cuboids such that their free energy includes a term F_mag = (μ_0/2) V N(T) M^2 <cit.> where M is thermodynamically conjugate to H_ int. By going beyond Maxwell's continuum theory, we show that N depends not only on sample shape and χ, but also on microscopic factors: the spin dimensionality and local spin anisotropy. Given that microscopic details affect even such a fundamental and well-studied macroscopic property as N, it is interesting to ask how they could affect the thermodynamics of more general long-range interacting systems. We thank D. Prabhakaran for providing crystals from which the samples were cut, and Tom Fennell and Jeffrey Rau for useful discussions. The simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the Center for High Performance Computing (PDC) at the Royal Institute of Technology (KTH). M.T. and P.H. are supported by the Swedish Research Council (2013-03968), M.T. is grateful for funding from Stiftelsen Olle Engkvist Byggmästare (2014/807), and L.B. is supported by The Leverhulme Trust through the Early Career Fellowship program (ECF2014-284). The work at the University of Waterloo was supported by the Canada Research Chair program (M.J.P.G., Tier 1). This research was supported in part by the Perimeter Institute for Theoretical Physics. Research at the Perimeter Institute is supported by the Government of Canada through Innovation, Science, and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation, and Science. § SUSCEPTIBILITY MEASUREMENT The magnetic susceptibility was measured using a Quantum Design SQUID magnetometer and the crystals were positioned in a cylindrical plastic tube to ensure a uniform magnetic environment. Measurements were performed in the RSO (reciprocating sample option) operating mode to achieve better sensitivity by eliminating low frequency noise. The position of the sample was carefully optimized to minimize misalignment with respect to the applied magnetic field. In particular, the sphere was measured at different positions and orientations in order to confirm the isotropic response and to fully reproduce the results of <cit.>. Similarly, the cube, with edges cut along [001], [010] and [001], was measured with the field aligned along all three orientations giving equivalent results, as would be expected. Different measurements were made on each sample and orientation: low-field susceptibility (at μ_0H_0 = 0.0025, 0.005 and 0.01 T) in field-cooled (FC) versus zero-field-cooled (ZFC) protocol. In addition, magnetic field sweeps at fixed temperature were performed in order to evaluate the susceptibility accurately and confirm the low-field linear response approximation. The FC versus ZFC susceptibility measurements involved cooling the sample to 1.8 K in zero field, applying the weak magnetic field, measuring the susceptibility while warming up to 350 K, cooling to 1.8 K again and finally re-measuring the susceptibility while warming. Before switching the magnetic field off, field scans with small steps were performed in order to estimate the absolute susceptibilities. As expected, and previously reported <cit.>, no difference between field-cooled and zero-field-cooled magnetization was observed in this temperature range. The magnetization of each sample was averaged over all six repetitions (three fields, two scans each) to minimize the influence of noise. § ITERATIVE METHOD The iterative method was implemented using a form of trivial parallelization, in which the local field at all sites is calculated in parallel for a given magnetic moment distribution. An MPI allgather call <cit.> is used in order to achieve good strong scaling <cit.> when run on many processors, a necessity in order to reach 𝒪(10^6) spins used in this study. The number of iterative steps required to reach convergence increases with increasing susceptibility but is 𝒪(10^2) regardless of the number of spins. Therefore, internode communication is not a bottleneck even though we gather and broadcast a vector equal to the length of the number of spins at every iterative step. A typical run for the largest system sizes (2× 10^6 spins) and 1024 cores <cit.> takes around 6 hours and requires roughly 400 communications when the intrinsic susceptibility, ∼ 10. § MONTE CARLO METHOD The Monte Carlo (MC) method used in this study is mostly based on the Metropolis-Hastings single-spin flip algorithm <cit.> applied to Ising spins. The exception is a loop algorithm <cit.>, which we applied to the dipolar spin ice Hamiltonian in addition to the single-spin flip algorithm. § EXTRAPOLATION TO INFINITE SYSTEM SIZE The approach to the infinite system size limit of the demagnetizing factor N in the iterative and MC methods is illustrated in Fig. 3 in the main text. Fig. 3 was generated by selecting three MC temperatures (16 K, 10 K and 3.5 K), and calculating the susceptibilities extrapolated to infinite system size for these temperatures (χ_int^MC= 1.00, 1.82 and 7.53). The iterative method calculations were performed with these susceptibility values for all system sizes, and the MC T was also kept the same for all system sizes. The functional forms used for the extrapolation also deserve further comments. For the open boundary case, the leading term is of the form 1/L, the surface to volume ratio. This is numerically confirmed in Table <ref>, where the first column gives the fitting function, the second the root-mean-square error (RMSE), and the third column the extrapolated value of N. The MC susceptibility is calculated with open boundary conditions for χ_int= 1.82 (red squares in Fig. 3 in the main text). The smallest RMSE is found in the first and last row of Table <ref>, both with a leading 1/L term. All data for open boundary conditions in this study have been transformed using the form a+b/L^1+c/L^2, marked in red in Table <ref>. In Table <ref>, the corresponding data for periodic boundary conditions are shown, and we note that the RMSE and extrapolated N are not very sensitive to the precise form of the extrapolation function, but the minimum RMSE is found for the function a+b/L^3+c/L^6, which represents an expansion in inverse volume of the surface-free system. All data for periodic boundary conditions in this study have been transformed using the form a+b/L^3+c/L^6, marked in red in Table <ref>. § SHORT-RANGE EXCHANGE INTERACTIONS As discussed in the main text, models with short-range interactions have a well defined shape-independent thermodynamic limit and adding exchange interactions to the dipolar Hamiltonian does not alter the demagnetizing factor. We illustrate this explicitly by a numerical MC simulation of the so-called dipolar spin ice model, which has been found to reproduce a number of properties of the Dy_2Ti_2O_7 and Ho_2Ti_2O_7 dipolar spin ice materials <cit.>. The Hamiltonian for this model consists of the dipolar term defined in <ref> and an exchange term of the form ℋ_=∑_i>j J_ij· σ_iσ_j, where the strength of the dipolar interaction is given by D=μ_0μ^2/4π r_nn^3k_B with r_nn being the nearest-neighbor distance and k_B the Boltzmann's constant. The matrix J_ij is the exchange interaction strength between particle i and j. Here we consider first (J_1), second (J_2) and third-nearest-neighbor (J_3) exchange interactions. In Fig. <ref> we show the demagnetizing factor for this model with parameters (D=1.3224 K, J_1=3.41 K, J_2=-0.14 K, and J_3=0.025 K, see Ref. <cit.>) and the same model with no exchange interaction (D=1.3224 K, J_1=J_2=J_3=0 K). We expect the infinite system size susceptibility to be dependent on boundary conditions, as shown in Fig. <ref>, while the difference of the inverse susceptibilities (demagnetizing factor N) is independent of boundary conditions, as shown in Fig. <ref>. Hence, we expect no entangling between the dipolar part and the exchange part for the determination of N when both are present. 55 fxundefined [1] ifx#1 fnum [1] #1firstoftwo secondoftwo fx [1] #1firstoftwo secondoftwo noop [0]secondoftwo ref[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0] rl [1]href #1 @bib@innerbibempty [Campa et al.(2009)Campa, Dauxois, and Ruffo]campa09 author author A. Campa, author T. Dauxois, and author S. Ruffo, @noop journal journal Phys. Rep. volume 480, pages 57 (year 2009)NoStop [Osborn(1945)]osborn45 author author J. A. Osborn, @noop journal journal Phys. Rev. volume 67, pages 351 (year 1945)NoStop [Stoner(1945)]stoner45 author author E. C. Stoner, @noop journal journal Philos. Mag. volume 36, pages 803 (year 1945)NoStop [Beleggia et al.(2006a)Beleggia, De Graef, and Millev]beleggia author author M. Beleggia, author M. De Graef, and author Y. T. Millev, @noop journal journal Philos. Mag. volume 86, pages 2451 (year 2006a)NoStop [Beleggia et al.(2006b)Beleggia, De Graef, and Millev]graef author author M. Beleggia, author M. De Graef, and author Y. T. Millev, @noop journal journal J. Phys. D volume 39, pages 891 (year 2006b)NoStop [Di Fratta(2016)]difratta16 author author G. Di Fratta, @noop journal journal Proc. R. Soc. A volume 472, pages 0197 (year 2016)NoStop [Aharoni(1998)]aharoni98 author author A. Aharoni, @noop journal journal J. Appl. Phys. volume 83, pages 3432 (year 1998)NoStop [Ponomareva et al.(2005)Ponomareva, Naumov, Kornev, Fu, and Bellaiche]ponomareva05 author author I. Ponomareva, author I. I. Naumov, author I. Kornev, author H. Fu, and author L. Bellaiche, @noop journal journal Phys. Rev. B volume 72, pages 140102 (year 2005)NoStop [Eshelby(1957)]eshelby57 author author J. D. Eshelby, @noop journal journal Proc. R. Soc. A volume 241, pages 376 (year 1957)NoStop [Durin and Zapperi(2000)]durin00 author author G. Durin and author S. Zapperi, @noop journal journal J. Appl. Phys. volume 87, pages 7031 (year 2000)NoStop [Csikor et al.(2007)Csikor, Motz, Weygand, Zaiser, and Zapperi]csikor07 author author F. F. Csikor, author C. Motz, author D. Weygand, author M. Zaiser, and author S. Zapperi, @noop journal journal Science volume 318, pages 251 (year 2007)NoStop [Laurson et al.(2013)Laurson, Illa, Santucci, Tallakstad, Måløy, and Alava]alava13 author author L. Laurson, author X. Illa, author S. Santucci, author K.-T. Tallakstad, author K. J. Måløy, and author M. J. Alava, @noop journal journal Nat. Comm. volume 4, pages 2927 (year 2013)NoStop [Chen et al.(1991)Chen, Brug, and Goldfarb]chen91 author author D.-X. Chen, author J. A. Brug, and author R. B. Goldfarb, @noop journal journal IEEE Trans. Magn. volume 27, pages 3601 (year 1991)NoStop [Würschmidt(1923)]wuhr23 author author J. Würschmidt, @noop journal journal Z. Phys. volume 12, pages 128 (year 1923)NoStop [Stäblein and Schlechtweg(1935)]stab35 author author F. Stäblein and author H. Schlechtweg, @noop journal journal Z. Phys. volume 95, pages 630 (year 1935)NoStop [Quilliam et al.(2008)Quilliam, Meng, Mugford, and Kycia]quil08 author author J. A. Quilliam, author S. Meng, author C. G. A. Mugford, and author J. B. Kycia, @noop journal journal Phys. Rev. Lett. volume 101, pages 187204 (year 2008)NoStop [Higashinaka and Maeno(2005)]Higa_112 author author R. Higashinaka and author Y. Maeno, @noop journal journal Phys. Rev. Lett. volume 95, pages 237208 (year 2005)NoStop [Chen et al.(2002)Chen, Pardo, and Sanchez]chen02 author author D.-X. Chen, author E. Pardo, and author A. Sanchez, @noop journal journal IEEE Trans. Magn. volume 38, pages 1742 (year 2002)NoStop [Chen et al.(2005)Chen, Pardo, and Sanchez]chen05 author author D.-X. Chen, author E. Pardo, and author A. Sanchez, @noop journal journal IEEE Trans. Magn. volume 41, pages 2077 (year 2005)NoStop [Christensen and Mørup(1983)]morup83 author author P. H. Christensen and author S. Mørup, @noop journal journal J. Magn. Magn. Mater. volume 35, pages 130 (year 1983)NoStop [Vedmedenko et al.(2003)Vedmedenko, Oepen, and Kirschner]vedmed02 author author E. Y. Vedmedenko, author H. P. Oepen, and author J. Kirschner, @noop journal journal J. Magn. Magn. Mater. volume 256, pages 237 (year 2003)NoStop [Millev et al.(2003)Millev, Vedmedenko, and Oepen]millev03 author author Y. T. Millev, author E. Vedmedenko, and author H. P. Oepen, @noop journal journal J. Phys. D volume 36, pages 2945 (year 2003)NoStop [Bovo et al.(2013)Bovo, Jaubert, Holdsworth, and Bramwell]bovo13 author author L. Bovo, author L. D. C. Jaubert, author P. C. W. Holdsworth, and author S. T. Bramwell, @noop journal journal J. Phys.: Condens. Matter volume 25, pages 386002 (year 2013)NoStop [Castelnovo et al.(2008)Castelnovo, Moessner, and Sondhi]Castel08 author author C. Castelnovo, author R. Moessner, and author S. L. Sondhi, @noop journal journal Nature volume 451, pages 42 (year 2008)NoStop [Biltmo and Henelius(2012)]bilt12 author author A. Biltmo and author P. Henelius, @noop journal journal Nat. Comm. volume 3, pages 857 (year 2012)NoStop [Fennell et al.(2007)Fennell, Bramwell, McMorrow, Manuel, and Wildes]fennell07 author author T. Fennell, author S. T. Bramwell, author D. F. McMorrow, author P. Manuel, and author A. R. Wildes, @noop journal journal Nat. Phys. volume 3, pages 566 (year 2007)NoStop [Ruff et al.(2005)Ruff, Melko, and Gingras]Ruff2005 author author J. P. C. Ruff, author R. G. Melko, and author M. J. P. Gingras, @noop journal journal Phys. Rev. Lett. volume 95, pages 097202 (year 2005)NoStop [Sato et al.(2006)Sato, Matsuhira, Tayama, Hiroi, Takagi, and Sakakibara]Sato1 author author H. Sato, author K. Matsuhira, author T. Tayama, author Z. Hiroi, author S. Takagi, and author T. Sakakibara, @noop journal journal Journal of Physics: Condensed Matter volume 18, pages L297 (year 2006)NoStop [Sato et al.(2007)Sato, Matsuhira, Sakakibara, Tayama, Hiroi, and Takagi]Sato2 author author H. Sato, author K. Matsuhira, author T. Sakakibara, author T. Tayama, author Z. Hiroi, and author S. Takagi, @noop journal journal Journal of Physics: Condensed Matter volume 19, pages 145272 (year 2007)NoStop [Schiffer et al.(1994)Schiffer, Ramirez, Huse, and Valentino]Schiffer1994 author author P. Schiffer, author A. P. Ramirez, author D. A. Huse, and author A. J. Valentino, @noop journal journal Phys. Rev. Lett. volume 73, pages 2500 (year 1994)NoStop [Quilliam et al.(2011)Quilliam, Yaraskavitch, Dabkowska, Gaulin, and Kycia]Quilliam2011 author author J. A. Quilliam, author L. R. Yaraskavitch, author H. A. Dabkowska, author B. D. Gaulin, and author J. B. Kycia, @noop journal journal Phys. Rev. B volume 83, pages 094424 (year 2011)NoStop [Ruminy et al.(2016)Ruminy, Groitl, Keller, , and Fennell]ruminy16 author author M. Ruminy, author F. Groitl, author T. Keller, , and author T. Fennell, @noop journal journal Phys. Rev. B volume 94, pages 174406 (year 2016)NoStop [Yavors'kii et al.(2008)Yavors'kii, Fennell, Gingras, and Bramwell]Yavo08 author author T. Yavors'kii, author T. Fennell, author M. J. P. Gingras, and author S. T. Bramwell, @noop journal journal Phys. Rev. Lett. volume 101, pages 037204 (year 2008)NoStop [Henelius et al.(2016)Henelius, Lin, Enjalran, Hao, Rau, Altosaar, Flicker, Yavors'kii, and Gingras]hene16 author author P. Henelius, author T. Lin, author M. Enjalran, author Z. Hao, author J. G. Rau, author J. Altosaar, author F. Flicker, author T. Yavors'kii, and author M. J. P. Gingras, @noop journal journal Phys. Rev. B volume 93, pages 024402 (year 2016)NoStop [Prabhakaran and Boothroyd(2011)]Prabhak author author D. Prabhakaran and author A. Boothroyd, @noop journal journal Crys. Growth volume 318, pages 1053 (year 2011)NoStop [epi()]epi @noop note SurfaceNet GmbH, http://www.surfacenet.deNoStop [Griffiths(1999)]griffiths author author D. J. Griffiths, @noop title Introduction to Electrodynamics, edition 3rd ed. (publisher Prentice-Hall, year 1999)NoStop [Griffiths(1982)]GriffPaper82 author author D. J. Griffiths, @noop journal journal American Journal of Physics volume 50, pages 698 (year 1982)NoStop [Jackson(1975)]jack author author J. D. Jackson, @noop title Classical Electrodynamics, edition 2nd ed. (publisher John Wiley & Sons, year 1975)NoStop [Ewald(1921)]ewal21 author author P. P. Ewald, @noop journal journal Ann. Phys. volume 369, pages 253 (year 1921)NoStop [Gingras and Henelius(2011)]gingras11 author author M. J. P. Gingras and author P. Henelius, @noop journal journal J. Phys.: Conf. Ser. volume 320, pages 012001 (year 2011)NoStop [Melko and Gingras(2004)]melko04 author author R. G. Melko and author M. J. P. Gingras, @noop journal journal J. Phys.: Condens. Matter volume 16, pages R1277 (year 2004)NoStop [pro()]process @noop note Intel Xeon E5-2698v3 processors with Cray Aries interconnect between nodesNoStop [Levy(1968)]levy68 author author P. Levy, @noop journal journal Phys. Rev. volume 170, pages 595 (year 1968)NoStop [Ruelle(1963)]ruelle63 author author D. Ruelle, @noop journal journal Helv. Phys. Acta volume 36, pages 183 (year 1963)NoStop [Campa et al.(2014)Campa, Dauxios, Fanelli, and Ruffo]campa14 author author A. Campa, author T. Dauxios, author D. Fanelli, and author S. Ruffo, @noop title Physics of long-range interacting systems, edition 1st ed. (publisher Oxford University Press, year 2014)NoStop [Gingras(2011)]gingras11a author author M. J. P. Gingras, title Introduction to frustrated magnetism, (publisher Springer (Ed. Lacroix, Mendels & Mila), year 2011) Chap. chapter Spin IceNoStop [Sánchez et al.(2017)Sánchez, Mendoza Zélis, Arciniegas, Pasquevich, and Fernández van Raap]sanchez17 author author F. H. Sánchez, author P. Mendoza Zélis, author M. L. Arciniegas, author G. A. Pasquevich, and author M. B. Fernández van Raap, @noop journal journal Phys. Rev. B volume 95, pages 134421 (year 2017)NoStop [Rhodes and Rowlands(1954)]rhodes54 author author P. Rhodes and author G. Rowlands, @noop journal journal Proc. Leeds Phil. Lit. Soc. volume 6, pages 191 (year 1954)NoStop [Bramwell et al.(2000)Bramwell, Field, Harris, and Parkin]Bramwell00 author author S. T. Bramwell, author M. N. Field, author M. J. Harris, and author I. P. Parkin, http://stacks.iop.org/0953-8984/12/i=4/a=308 journal journal Journal of Physics: Condensed Matter volume 12, pages 483 (year 2000)NoStop [mpi()]mpi @noop note Open MPI v2.0.0 documentation. https://www.open-mpi.org/doc/v2.0/NoStop [Kunkel et al.(2016)Kunkel, Balaji, and Dongarra]scaling author author J. Kunkel, author P. Balaji, and author J. Dongarra, https://books.google.se/books?id=kRJkDAAAQBAJ title High performance computing: 31st international conference, (year 2016), note ISC High Performance 2016, Frankfurt, Germany, June 19-23, 2016, ProceedingsNoStop [Hastings(1970)]Hast author author W. K. Hastings, @noop journal journal Biometrika volume 57, pages 97 (year 1970)NoStop [Melko et al.(2001)Melko, den Hertog, and Gingras]loopalgo author author R. G. Melko, author B. C. den Hertog, and author M. J. P. Gingras, 10.1103/PhysRevLett.87.067203 journal journal Phys. Rev. Lett. volume 87, pages 067203 (year 2001)NoStop [den Hertog and Gingras(2000)]Hertog2000 author author B. C. den Hertog and author M. J. P. Gingras, @noop journal journal Phys. Rev. Lett. volume 84, pages 3430 (year 2000)NoStop

http://arxiv.org/abs/1701.08136v2

20170127181800

Multiband full-bandwidth anisotropic Eliashberg theory of interfacial electron-phonon coupling and high-T$_c$ superconductivity in FeSe/SrTiO$_3$

cond-mat.supr-con

alex.aperis@physics.uu.se peter.oppeneer@physics.uu.se Department of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-75120 Uppsala, Sweden 0.4cm We examine the impact of interfacial phonons on the superconducting state of FeSe/SrTiO_3 developing a materials' specific multiband, full bandwidth, anisotropic Eliashberg theory for this system. Our selfconsistent calculations highlight the importance of the interfacial electron-phonon interaction, which is hidden behind the seemingly weak coupling constant λ_m=0.4, in mediating the high-T_ c, and explain other puzzling experimental observations like the s-wave symmetry and replica bands. We discover that the formation of replica bands has a T_ c decreasing effect that is nevertheless compensated by deep Fermi-sea Cooper pairing which has a T_ c enhancing effect. We predict a strong coupling dip-hump signature in the tunneling spectra due to the interfacial coupling. Multiband, full bandwidth anisotropic Eliashberg theory of interfacial electron-phonon coupling and high-T_ c superconductivity in FeSe/SrTiO_3 Peter M. Oppeneer December 30, 2023 =============================================================================================================================================== Superconductivity in monolayer-thick FeSe on SrTiO_3 reaches amazingly high transition temperatures of typically T_ c=50–70 K <cit.> and up to 100 K <cit.>, much higher than the 8-K value of bulk FeSe <cit.>. A coupling between SrTiO_3 phonons and FeSe electrons occurs at the FeSe/SrTiO_3 interface, which manifests itself as electron replica bands <cit.>. The value of this coupling is estimated by experiments to be around 0.4, thus it is commonly believed to moderately enhance T_ c but not be enough to explain it <cit.>. The superconducting state in iron-based superconductors is customarily associated with residual spin fluctuations due to the remnant quasi-nesting between electron and hole Fermi sheets that give rise to a sign alternating gap <cit.>. However, for FeSe/STO the situation is markedly different. Charge transfer at the interface induces electron doping in FeSe <cit.>, leading to a distinct Fermi surface consisting of only two electron sheets around the corners of the tetragonal Brillouin zone (M point) <cit.>. The observed anisotropic superconducting gap has a more conventional form with plain s-wave symmetry <cit.> and is thus nodeless in the entire Brillouin zone. Furthermore, angular resolved photoemission spectroscopy (ARPES) measurements <cit.> reveal an interface-induced electron-phonon interaction (EPI) between FeSe electrons and polar STO phonons that is strongly peaked at the q=0 phonon wavevector <cit.>. There is growing experimental evidence for the pivotal role of such interfacial phonons in engineering high-T_ c heterostructures that involve FeSe <cit.> or even FeAs <cit.> monolayers. Although ab initio calculations confirm the existence of small-q phonons as a strictly interfacial phenomenon in FeSe/STO <cit.> and indicate the importance of the coupling between substrate phonons and FeSe electrons <cit.>, the estimated low value of the electron-phonon coupling constant (λ≤ 0.4) has been widely considered insufficient to explain the impressive T_ c enhancement unless another, dominant pairing mechanism is at play <cit.>. On the other hand, Eliashberg calculations within a single band model suggest that interfacial phonons may lead to the high T_ c with a coupling of merely half of that estimated by experiments <cit.>. However, a materials' specific theory of superconductivity that can account for the interplay between multiple bands, doping and small-q phonons has not yet been developed. It remains therefore unsolved to what extend and how such phonons contribute to the peculiar superconductivity in FeSe/STO. Here, we present the first anisotropic, full bandwidth multiband Eliashberg calculations dedicated to unveil the influence of the interfacial electron-phonon coupling in FeSe/STO. Our theory extends on previous single band approaches <cit.> by establishing a microscopic description of superconductivity in this system on a materials' specific level, thus paving the way toward more realistic calculations of higher accuracy. Our selfconsistent results provide unambiguous support for the dominant contribution of these phonons to the high T_ c and to further enigmatic experimental observations, and allows us to shed light on novel aspects of the mechanism responsible for the high-T_ c. Remarkably, bands not crossing the Fermi level provide an additional Cooper pairing channel that enhances T_ c and places the value of the gap over T_ c ratio in the strong coupling regime. In stark contrast to previous proposals, our here predicted deep Fermi sea Cooper pairing does not depend upon interband scattering processes mediated by bosons at large wavevectors, like e.g. antiferromagnetic spin fluctuations <cit.> or the conventional short-ranged in real space EPI <cit.>. In the presence of a strong inhomogeneous dielectric background, the EPI develops a pronounced forward scattering peak <cit.>. Here, the interface-induced EPI is modelled by a dispersionless mode at ħΩ=81 meV <cit.> coupled to FeSe electrons via the functional form g( q)=g_0 exp(-| q|/q_c) with q_c=0.3a^-1 <cit.>, with a the FeSe lattice constant. This is the only mediator of superconductivity in our theory. In what follows, we do not take into account explicitly the effect of Coulomb repulsion on superconductivity, the implications of which are discussed further below. For the electron dispersions of monolayer FeSe we use a recently derived ten-band tight-binding bandstructure <cit.>. Since the FeSe doping level is not a priori known, the electron filling is chosen such that the bottom of the electron bands around the M point in the Brillouin zone are at -50 meV as observed in experiment <cit.>. We determine the value of the electron-phonon scattering strength g_0, by requiring that the ARPES replica bands are reproduced at their observed energies <cit.>. In this way, we circumvent the need for treating screening effects at the FeSe/STO interface explicitly <cit.> and our determined value for the EPI strength may considered as the overall strength of the resulting effective EPI. We obtain g_0=728 meV, which is significantly close to the ab initio calculated value for anatase TiO_2 <cit.>, but a bit lower. This discrepancy may be understood as due to an enhanced screening effect at the FeSe/STO interface <cit.>. We solve the three coupled Migdal-Eliashberg equations for Δ( k, ω), Z( k,ω), and χ( k,ω), describing the superconductivity order parameter, electron mass and chemical potential renormalizations, respectively, selfconsistently with full bandwidth, momentum and energy dependence, while taking care to keep the electron occupancy n fixed throughout the calculations <cit.>. The latter quantity measures the electron filling (the case of half-filling corresponds to n=1). We find n ≈0.8, indicating that the system is in the electron-doped regime. We also note that when forward-scattering processes dominate the EPI, the Migdal theorem holds even in non-adiabatic cases <cit.>. Figure <ref>(a) shows the calculated electron spectral function at T=10 K for the whole energy bandwidth and momenta along the M-Γ -M high-symmetry line of the folded Brillouin zone, which is measured in ARPES experiments. Comparison of the spectral function with the bare tight-binding band structure <cit.> used as input in our calculations (shown by white dashed lines) reveals that the interfacial small-q phonon modifies the FeSe electrons in a manifest way. Several shake-off effects in the band structure take place over the whole bandwidth, including the appearance of new bands near -160 meV. The opening of a superconducting gap around the Fermi level can be seen. Most of the predicted shake-off effects in the band structure have not yet been observed by experiment <cit.>. However, the replica bands appearing near the M point (zoom-in shown in Fig. <ref>(b) at ∼110 meV distance from the main electron bands that form the Fermi surface have been experimentally resolved <cit.>. Figure <ref>(c) highlights that not only the position of the peaks but also the peak ratio, A_2/A_1=0.17, agrees well with experiment <cit.>. Notably, within solely phononic small-q theory we obtain the superconducting T_ c=61 K, in good agreement with experiment <cit.>, with a temperature dependence of the gap edge as shown in Fig. <ref>(d). The momentum dependence of the calculated superconducting gap Δ ( k_ F) is shown in Fig. <ref>(a). It has s-wave symmetry and is moderately anisotropic (∼25%) with gap values that vary from 8-11 meV over the Fermi surface with an average value of 10 meV. These values are in agreement with experiments <cit.> although the location of the gap maxima seems to somewhat deviate from those experimentally measured (e.g. <cit.>). The resulting anisotropy of the gap is a consequence of the small-q form of the interfacial EPI (cf. <cit.>). Taking the maximum required excitation energy at the gap edge to calculate the gap over T_ c ratio, we obtain the strong coupling (non-BCS) value Δ/k_ B T_ c=2.1 (in contrast to the BCS value Δ / k_ B T_ c=1.76). The chemical potential renormalization χ( k_ F), shown in Fig. <ref>(b), has an anisotropic momentum dependence with an average Fermi surface value of ⟨χ( k_ F)⟩=5.9 meV. The fact that χ( k_ F) even changes sign at certain Fermi surface points indicates the highly non-trivial role this quantity plays in shaping the quasiparticle band structure of the monolayer. In contrast, the mass renormalization function Z( k_ F), shown in Figs. <ref>(c), (d), is rather isotropic with an average Fermi surface value ⟨ Z( k_ F)⟩=1.37 and 1.40 for temperature below and above T_ c, respectively. This quantity is related to the electron-phonon coupling constant λ_m, by ⟨ Z ( k) ⟩ _ k_ F |_ T>T_c=1+λ_m which in our case yields λ_m= 0.4. This weak coupling value matches remarkably well to experiments <cit.>. Also, in the superconducting state λ^ 10 K_m≈0.37 and satisfies A_2/A_1≈λ^ 10 K_m/2 (Fig. <ref>(c)) <cit.>. A plain calculation of T_ c with our obtained value of λ_m=0.4 in McMillan's formula gives T_ c=17 K. On the other hand, using our numerical results in the two T_ c formulas recently proposed for interfacial phonon-mediated superconductivity in FeSe/STO <cit.>, yields T_ c=272–283 K and 117.5 K, respectively (note that the T_ c equation in <cit.> is derived in the q_c→ 0 limit). These estimations are in stark contrast to the T_ c=61 K obtained here by our selfconsistent Eliashberg theory, which thus resolves the controversy between a seemingly weak λ_m and high-T_ c superconductivity in FeSe/STO. To elucidate further the mechanism of T_ c enhancement in FeSe/STO, we carried out a series of simulations where we first solve the usual momentum-dependent Eliashberg equations for electrons only at the Fermi surface of FeSe/STO and then perform full-bandwidth calculations while sequentially adding more bands until we recover the full bandwidth multiband calculation. Our findings are summarized in Fig. <ref>(a). In the first case (blue symbol in Fig. <ref>(a) where no electronic spectral rearrangement is allowed and thus no replica bands can form, we find λ_m=0.63. We note that this value equals the one given by the standard formula, λ=⟨λ_ q⟩_ k_ F^, k^'_ F=0.63 (with λ_ q the momentum-dependent electron-phonon coupling <cit.>). Therefore, the obtained T_ c=60.6 K is the maximum T_ c reachable by Fermi-surface Cooper pairing. Inclusion of the complete contribution of the two bands that form the Fermi surface (purple square in Fig. <ref>(a) leads to λ_m=0.4 and T_ c=56.8 K. Compared with the previous case, here a part of the electron-phonon coupling strength is consumed in mediating the electronic spectral weight transfer from the bands crossing the Fermi level to the replica bands. The effective interaction left available for Cooper-pair mediation is weaker and concomitantly so is the T_ c. This weak coupling picture is further witnessed by the near-BCS value of the calculated ratio Δ/k_ B T_c=1.8. Remarkably, turning on contributions from near-Fermi-level bands (red symbols in Fig. <ref>(a) gradually increases T_ c to 61.2 K (green square) but without affecting the value of λ_m. This behavior indicates that these bands contribute to superconductivity. To quantify this remarkable finding, we projected the superconducting gap function Δ( k,E) on the different electronic bands of FeSe/STO as shown in Fig. <ref>(b). Since Δ( k,E) is a measure of the Cooper-pair binding energy, it is positive for Cooper pairing with an s-wave gap and negative otherwise <cit.>. Figure <ref>(b) clearly shows that superconductivity in FeSe/STO stems not only from Fermi surface regions around M, but also from regions around the Γ-X and X-M directions of the Brillouin zone. Although in the latter regions Δ( k,E) is in the μeV range (Fig. <ref>(c)), the resulting net contribution is enough to overcompensate the T_ c decreasing effect of the replica band formation. It is also enough to raise the Δ/k_ B T_c ratio to 2.1 and thus provide a strong coupling phenomenology. We emphasize that our predicted deep Fermi sea Cooper pairing is markedly different from previous suggestions of pairing through incipient bands <cit.>. Here, the mediating interaction is not only phonon-driven but more importantly, it is local in momentum space due to its small-q shape, thus it relies explicitly on intraband processes. However, by the nature of our full bandwidth Eliashberg theory, different bands are coupled implicitly via the frequency sector by the interfacial EPI, due to the large characteristic energy scale of the latter, in some sense reminiscent of the incipient band scenario. Our findings thus generalize the usual picture where Cooper pairing relies on near Fermi surface electrons and prove that deep Fermi-sea Cooper pairing is possible in multiband systems <cit.>. The recent puzzling superconductivity observed in doped LiFeAs without a Fermi surface <cit.> is plausibly explained within our picture. Scanning tunneling spectroscopy (STS) measurements represent another key experimental feature reported for FeSe/STO <cit.>. To compare to STS data we have calculated the differential conductance spectrum dI/dV which, at low temperatures, is proportional to the superconducting density of states. The tunneling spectrum, calculated at T=10 K, is shown in Fig. <ref>. Zooming-in to the low energy regime (Fig. <ref>(b)) reveals the opening of an s-wave superconducting gap in the tunneling spectra that starts to close already around ±5 meV and exhibits main coherence peaks at ±11 meV, with secondary peaks a few meV's higher. The calculated spectrum is in excellent agreement with STS measurements <cit.>. The position of the main coherence peaks in Fig. <ref>(b) coincides with the maximum of the gap-edge on the Fermi surface whereas the closing of the gap beginning at 5 meV is consistent with the minimum gap-edge of 8 meV (see Fig. <ref>(a)) in combination with the finite broadening of ∼3 meV used in our calculations. Figure <ref>(a) shows the calculated tunneling spectra, normalized to the normal state values, at an intermediate energy range. Remarkably, we find superconductivity related structures in the spectrum up to energies almost 30 times higher than the superconducting gap itself. Although it is well established that such non-BCS behavior is the hallmark of strong-coupling superconductivity <cit.>, this is unexpected here given the seemingly weak coupling constant of the system. In strongly coupled superconductors, the structure of the spectral function of the mediating bosons can be visible in the tunneling spectrum <cit.>. Here, we predict that the interfacial phonon mode should manifest itself as two kinks around ±91 meV (Fig. <ref>(a)), whose location coincides with ±(Ω+Δ) where Δ is the average gap-edge value. Furthermore, at higher energies we predict a distinct dip-hump structure in the spectra with a dip at 150 meV and a hump at 220 meV (Figs. <ref>(a),(c)). Analyzing this additional strong coupling feature we find that it originates from the competition between the real and imaginary components of the superconducting gap function at an energy scale that is larger than the characteristic boson frequencies <cit.>. The energy location of the dip and the hump depends on the coupling strength <cit.>, and more specifically, on the Δ/k_ B T_c ratio <cit.>. For comparison, within isotropic Eliashberg theory assuming an Einstein phonon at ħΩ=81 meV, we find that our predicted dip-hump spectrum in FeSe/STO can only be fitted when the obtained gap over T_ c ratio matches the one in FeSe/STO Δ/k_ B T_c=2.1 but with a strong-coupling isotropic value λ _iso=1.6 (Fig. <ref>(c)). The very good quantitative agreement between experiment and our selfconsistent calculations for FeSe/STO provides a consistent picture where the interfacial phonons drive the superconductivity. For that picture to be complete one needs to incorporate the pair-breaking effect of the Coulomb interaction on the T_ c. Inclusion of the latter effect into the full bandwidth Eliashberg calculations on an equal footing with the EPI requires knowledge of the frequency dependent renormalization of the Coulomb interaction throughout the system's bandwidth and is out of the scope of the present work. By approximating the Coulomb repulsion through the pseudopotential term μ^* <cit.>, we estimate that for μ^*= 0.1 and 0.14, T_ c=33 K and 26 K, respectively. However, the presence of an additional low-energy attractive channel due to the intrinsic EPI in FeSe monolayer <cit.>, although not sufficient to mediate the high-T_ c on its own, can balance the T_ c decrease due to Coulomb repulsion. We find that inclusion of the intrinsic EPI of freestanding monolayer FeSe <cit.> leads to T_ c=57 K and 51 K for μ^*= 0.1 and 0.14, respectively. In conclusion, our first of its kind full-bandwidth multiband theory shows that the interfacial EPI in FeSe/STO with a seemingly weak λ_m=0.4, explains key experimental facts like the replica bands, superconducting gap and tunneling spectra while also producing the correct T_ c in the absence of any significant Coulomb pair-breaking. Our explicit calculations unveil the T_ c increasing effect of deep Fermi-sea Cooper pairing and the T_ c decreasing effect of replica-band formation, and suggest new pathways to engineer high T_ c's. A definite confirmation for the former effect will be the observation of a dip-hump feature in the tunneling spectra, which will also serve as an additional fingerprint of the decisive involvement of the interfacial EPI in mediating the high-T_ c. On a fundamental level, our findings put to the question the current standard perception of the efficiency of EPI in mediating high-T_ c superconductivity and, whether Fermi-surface restricted theory is sufficient to capture the superconductivity of other doped high-T_ c materials. We thank G. Varelogiannis for fruitful discussions. This work has been supported by the Swedish Research Council (VR), the Röntgen-Ångström Cluster, and the Swedish National Infrastructure for Computing (SNIC). apsrev4-1 51 fxundefined [1] ifx#1 fnum [1] #1firstoftwo secondoftwo fx [1] #1firstoftwo secondoftwo noop [0]secondoftwo ref[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0] rl [1]href #1 @bib@innerbibempty [Qing-Yan et al.(2012)Qing-Yan, Zhi, Wen-Hao, Zuo-Cheng, Jin-Song, Wei, Hao, Yun-Bo, Peng, Kai, Jing, Can-Li, Ke, Jin-Feng, Shuai-Hua, Ya-Yu, Li-Li, Xi, Xu-Cun, and Qi-Kun]Qing-Yan2012 author author W. Qing-Yan, author L. Zhi, author Z. Wen-Hao, author Z. Zuo-Cheng, author Z. Jin-Song, author L. Wei, author D. Hao, author O. Yun-Bo, author D. Peng, author C. Kai, author W. Jing, author S. Can-Li, author H. Ke, author J. Jin-Feng, author J. Shuai-Hua, author W. Ya-Yu, author W. Li-Li, author C. Xi, author M. Xu-Cun, and author X. Qi-Kun, http://stacks.iop.org/0256-307X/29/i=3/a=037402 journal journal Chin. Phys. Lett. volume 29, pages 037402 (year 2012)NoStop [Liu et al.(2012)Liu, Zhang, Mou, He, Ou, Wang, Li, Wang, Zhao, He, Peng, Liu, Chen, Yu, Liu, Dong, Zhang, Chen, Xu, Hu, Chen, Ma, Xue, and Zhou]Liu2012 author author D. Liu, author W. Zhang, author D. Mou, author J. He, author Y.-B. Ou, author Q.-Y. Wang, author Z. Li, author L. Wang, author L. Zhao, author S. He, author Y. Peng, author X. Liu, author C. Chen, author L. Yu, author G. Liu, author X. Dong, author J. Zhang, author C. Chen, author Z. Xu, author J. Hu, author X. Chen, author X. Ma, author Q. Xue, and author X. Zhou, http://dx.doi.org/10.1038/ncomms1946 journal journal Nat. Commun. volume 3, pages 931 (year 2012)NoStop [He et al.(2013)He, He, Zhang, Zhao, Liu, Liu, Mou, Ou, Wang, Li, Wang, Peng, Liu, Chen, Yu, Liu, Dong, Zhang, Chen, Xu, Chen, Ma, Xue, and Zhou]He2013 author author S. He, author J. He, author W. Zhang, author L. Zhao, author D. Liu, author X. Liu, author D. Mou, author Y.-B. Ou, author Q.-Y. Wang, author Z. Li, author L. Wang, author Y. Peng, author Y. Liu, author C. Chen, author L. Yu, author G. Liu, author X. Dong, author J. Zhang, author C. Chen, author Z. Xu, author X. Chen, author X. Ma, author Q. Xue, and author X. J. Zhou, http://dx.doi.org/10.1038/nmat3648 journal journal Nat. Mater. volume 12, pages 605 (year 2013)NoStop [Tan et al.(2013)Tan, Zhang, Xia, Ye, Chen, Xie, Peng, Xu, Fan, Xu, Jiang, Zhang, Lai, Xiang, Hu, Xie, and Feng]Tan2013 author author S. Tan, author Y. Zhang, author M. Xia, author Z. Ye, author F. Chen, author X. Xie, author R. Peng, author D. Xu, author Q. Fan, author H. Xu, author J. Jiang, author T. Zhang, author X. Lai, author T. Xiang, author J. Hu, author B. Xie, and author D. Feng, http://dx.doi.org/10.1038/nmat3654 journal journal Nat. Mater. volume 12, pages 634 (year 2013)NoStop [Peng et al.(2014)Peng, Xu, Tan, Cao, Xia, Shen, Huang, Wen, Song, Zhang, Xie, Gong, and Feng]Peng2014a author author R. Peng, author H. C. Xu, author S. Y. Tan, author H. Y. Cao, author M. Xia, author X. P. Shen, author Z. C. Huang, author C. Wen, author Q. Song, author T. Zhang, author B. P. Xie, author X. G. Gong, and author D. L. Feng, http://dx.doi.org/10.1038/ncomms6044 journal journal Nat. Commun. volume 5, pages 5044 (year 2014)NoStop [Lee et al.(2014)Lee, Schmitt, Moore, Johnston, Cui, Li, Yi, Liu, Hashimoto, Zhang, Lu, Devereaux, Lee, and Shen]Lee2014 author author J. J. Lee, author F. T. Schmitt, author R. G. Moore, author S. Johnston, author Y.-T. Cui, author W. Li, author M. Yi, author Z. K. Liu, author M. Hashimoto, author Y. Zhang, author D. H. Lu, author T. P. Devereaux, author D.-H. Lee, and author Z.-X. Shen, http://dx.doi.org/10.1038/nature13894 journal journal Nature volume 515, pages 245 (year 2014)NoStop [Ge et al.(2015)Ge, Liu, Liu, Gao, Qian, Xue, Liu, and Jia]Ge2015 author author J.-F. Ge, author Z.-L. Liu, author C. Liu, author C.-L. Gao, author D. Qian, author Q.-K. Xue, author Y. Liu, and author J.-F. Jia, http://dx.doi.org/10.1038/nmat4153 journal journal Nat. Mater. volume 14, pages 285 (year 2015)NoStop [Hsu et al.(2008)Hsu, Luo, Yeh, Chen, Huang, Wu, Lee, Huang, Chu, Yan, and Wu]Hsu2008 author author F.-C. Hsu, author J.-Y. Luo, author K.-W. Yeh, author T.-K. Chen, author T.-W. Huang, author P. M. Wu, author Y.-C. Lee, author Y.-L. Huang, author Y.-Y. Chu, author D.-C. Yan, and author M.-K. Wu, 10.1073/pnas.0807325105 journal journal Proc. Natl. Acad. Sci. USA volume 105, pages 14262 (year 2008)NoStop [Ding et al.(2016)Ding, Lv, Zhao, Wang, Wang, Song, Chen, Ma, and Xue]Ding2016 author author H. Ding, author Y.-F. Lv, author K. Zhao, author W.-L. Wang, author L. Wang, author C.-L. Song, author X. Chen, author X.-C. Ma, and author Q.-K. Xue, 10.1103/PhysRevLett.117.067001 journal journal Phys. Rev. Lett. volume 117, pages 067001 (year 2016)NoStop [Tian et al.(2016)Tian, Zhang, Li, Wu, Wu, Sun, Zhou, Wang, Ma, Xue, and Zhao]Tian2016 author author Y. C. Tian, author W. H. Zhang, author F. S. Li, author Y. L. Wu, author Q. Wu, author F. Sun, author G. Y. Zhou, author L. Wang, author X. Ma, author Q.-K. Xue, and author J. Zhao, 10.1103/PhysRevLett.116.107001 journal journal Phys. Rev. Lett. volume 116, pages 107001 (year 2016)NoStop [Mazin et al.(2008)Mazin, Singh, Johannes, and Du]Mazin2008 author author I. I. Mazin, author D. J. Singh, author M. D. Johannes, and author M. H. Du, 10.1103/PhysRevLett.101.057003 journal journal Phys. Rev. Lett. volume 101, pages 057003 (year 2008)NoStop [Li et al.(2014)Li, Xing, Huang, and Xing]Li2014 author author B. Li, author Z. W. Xing, author G. Q. Huang, and author D. Y. Xing, http://dx.doi.org/10.1063/1.4876750 journal journal J. Appl. Phys. volume 115, eid 193907 (year 2014)NoStop [Fan et al.(2015)Fan, Zhang, Liu, Yan, Ren, Peng, Xu, Xie, Hu, Zhang, and Feng]Fan2015 author author Q. Fan, author W. H. Zhang, author X. Liu, author Y. J. Yan, author M. Q. Ren, author R. Peng, author H. C. Xu, author B. P. Xie, author J. P. Hu, author T. Zhang, and author D. L. Feng, http://dx.doi.org/10.1038/nphys3450 journal journal Nat. Phys. volume 11, pages 946 (year 2015)NoStop [Aperis et al.(2011)Aperis, Kotetes, Varelogiannis, and Oppeneer]Aperis2011 author author A. Aperis, author P. Kotetes, author G. Varelogiannis, and author P. M. Oppeneer, 10.1103/PhysRevB.83.092505 journal journal Phys. Rev. B volume 83, pages 092505 (year 2011)NoStop [Weger and Peter(1999)]Weger1999 author author M. Weger and author M. Peter, http://dx.doi.org/10.1016/S0921-4534(99)00065-9 journal journal Physica C volume 317, pages 252 (year 1999)NoStop [Varelogiannis(1998)]Varelogiannis1998 author author G. Varelogiannis, 10.1103/PhysRevB.57.13743 journal journal Phys. Rev. B volume 57, pages 13743 (year 1998)NoStop [Kulić and Zeyher(1994)]Kulifmmodecuteclseci1994 author author M. L. Kulić and author R. Zeyher, 10.1103/PhysRevB.49.4395 journal journal Phys. Rev. B volume 49, pages 4395 (year 1994)NoStop [Abrikosov(1994)]Abrikosov1994 author author A. A. Abrikosov, http://dx.doi.org/10.1016/0921-4534(94)90131-7 journal journal Physica C volume 222, pages 191 (year 1994)NoStop [Rebec et al.(2017)Rebec, Jia, Zhang, Hashimoto, Lu, Moore, and Shen]Rebec2017 author author S. N. Rebec, author T. Jia, author C. Zhang, author M. Hashimoto, author D.-H. Lu, author R. G. Moore, and author Z.-X. Shen, 10.1103/PhysRevLett.118.067002 journal journal Phys. Rev. Lett. volume 118, pages 067002 (year 2017)NoStop [Choi et al.(2017)Choi, Johnston, Jang, Koepernik, Nakatsukasa, Ok, Lee, Choi, Lee, Akbari, Semertzidis, Bang, Kim, and Lee]Choi2017 author author S. Choi, author S. Johnston, author W.-J. Jang, author K. Koepernik, author K. Nakatsukasa, author J. M. Ok, author H.-J. Lee, author H. W. Choi, author A. T. Lee, author A. Akbari, author Y. K. Semertzidis, author Y. Bang, author J. S. Kim, and author J. Lee, 10.1103/PhysRevLett.119.107003 journal journal Phys. Rev. Lett. volume 119, pages 107003 (year 2017)NoStop [Xie et al.(2015)Xie, Cao, Zhou, Chen, Xiang, and Gong]Xie2015 author author Y. Xie, author H.-Y. Cao, author Y. Zhou, author S. Chen, author H. Xiang, and author X.-G. Gong, http://dx.doi.org/10.1038/srep10011 journal journal Sci. Rep. volume 5, pages 10011 (year 2015)NoStop [Wang et al.(2016a)Wang, Linscheid, Berlijn, and Johnston]Wang2016b author author Y. Wang, author A. Linscheid, author T. Berlijn, and author S. Johnston, 10.1103/PhysRevB.93.134513 journal journal Phys. Rev. B volume 93, pages 134513 (year 2016a)NoStop [Zhou and Millis(2016)]Zhou2016a author author Y. Zhou and author A. J. Millis, 10.1103/PhysRevB.93.224506 journal journal Phys. Rev. B volume 93, pages 224506 (year 2016)NoStop [Huang and Pickett(2017)]Huang2017 author author Y. N. Huang and author W. E. Pickett, 10.1103/PhysRevB.95.165107 journal journal Phys. Rev. B volume 95, pages 165107 (year 2017)NoStop [Rademaker et al.(2016)Rademaker, Wang, Berlijn, and Johnston]Rademaker2016 author author L. Rademaker, author Y. Wang, author T. Berlijn, and author S. Johnston, http://stacks.iop.org/1367-2630/18/i=2/a=022001 journal journal New J. Phys. volume 18, pages 022001 (year 2016)NoStop [Wang et al.(2016b)Wang, Nakatsukasa, Rademaker, Berlijn, and Johnston]Wang2016c author author Y. Wang, author K. Nakatsukasa, author L. Rademaker, author T. Berlijn, and author S. Johnston, @noop journal journal Supercond. Sci. Technol. volume 29, pages 054009 (year 2016b)NoStop [Bang(2014)]Bang2014 author author Y. Bang, http://stacks.iop.org/1367-2630/16/i=2/a=023029 journal journal New J. Phys. volume 16, pages 023029 (year 2014)NoStop [Linscheid et al.(2016)Linscheid, Maiti, Wang, Johnston, and Hirschfeld]Linscheid2016a author author A. Linscheid, author S. Maiti, author Y. Wang, author S. Johnston, and author P. J. Hirschfeld, 10.1103/PhysRevLett.117.077003 journal journal Phys. Rev. Lett. volume 117, pages 077003 (year 2016)NoStop [Chen et al.(2015)Chen, Maiti, Linscheid, and Hirschfeld]Chen2015 author author X. Chen, author S. Maiti, author A. Linscheid, and author P. J. Hirschfeld, 10.1103/PhysRevB.92.224514 journal journal Phys. Rev. B volume 92, pages 224514 (year 2015)NoStop [Weger et al.(1996)Weger, Peter, and Pitaevskii]Weger1996 author author M. Weger, author M. Peter, and author L. P. Pitaevskii, 10.1007/s002570050248 journal journal Z. Physik B volume 101, pages 573 (year 1996)NoStop [Hao and Hu(2014)]Hao2014 author author N. Hao and author J. Hu, 10.1103/PhysRevX.4.031053 journal journal Phys. Rev. X volume 4, pages 031053 (year 2014)NoStop [SM1()]SM1 @noop note See Supplemental Material at [url] for details, which include Refs. <cit.>NoStop [Gor'kov(2016)]Gorkov2016 author author L. P. Gor'kov, 10.1103/PhysRevB.93.060507 journal journal Phys. Rev. B volume 93, pages 060507 (year 2016)NoStop [Verdi and Giustino(2015)]Verdi2015 author author C. Verdi and author F. Giustino, 10.1103/PhysRevLett.115.176401 journal journal Phys. Rev. Lett. volume 115, pages 176401 (year 2015)NoStop [Abrikosov(2005)]Abrikosov2005 author author A. A. Abrikosov, http://dx.doi.org/10.1016/j.physc.2005.05.006 journal journal Physica C volume 424, pages 149 (year 2005)NoStop [Varelogiannis et al.(1996)Varelogiannis, Perali, Cappelluti, and Pietronero]Varelogiannis1996 author author G. Varelogiannis, author A. Perali, author E. Cappelluti, and author L. Pietronero, 10.1103/PhysRevB.54.R6877 journal journal Phys. Rev. B volume 54, pages R6877 (year 1996)NoStop [Kulić and Dolgov(2017)]Kulic2017 author author M. L. Kulić and author O. V. Dolgov, http://stacks.iop.org/1367-2630/19/i=1/a=013020 journal journal New J. Phys. volume 19, pages 013020 (year 2017)NoStop [Scalapino et al.(1966)Scalapino, Schrieffer, and Wilkins]Scalapino1966 author author D. J. Scalapino, author J. R. Schrieffer, and author J. W. Wilkins, 10.1103/PhysRev.148.263 journal journal Phys. Rev. volume 148, pages 263 (year 1966)NoStop [Miao et al.(2015)Miao, Qian, Shi, Richard, Kim, Hoesch, Xing, Wang, Jin, Hu, and Ding]Miao2015 author author H. Miao, author T. Qian, author X. Shi, author P. Richard, author T. K. Kim, author M. Hoesch, author L. Y. Xing, author X.-C. Wang, author C.-Q. Jin, author J.-P. Hu, and author H. Ding, http://dx.doi.org/10.1038/ncomms7056 journal journal Nat. Commun. volume 6, pages 6056 (year 2015)NoStop [Tang et al.(2016)Tang, Liu, Zhou, Li, Ding, Li, Zhang, Li, Song, Ji, He, Wang, Ma, and Xue]Tang2016 author author C. Tang, author C. Liu, author G. Zhou, author F. Li, author H. Ding, author Z. Li, author D. Zhang, author Z. Li, author C. Song, author S. Ji, author K. He, author L. Wang, author X. Ma, and author Q.-K. Xue, 10.1103/PhysRevB.93.020507 journal journal Phys. Rev. B volume 93, pages 020507 (year 2016)NoStop [Varelogiannis(1995)]Varelogiannis1995 author author G. Varelogiannis, 10.1103/PhysRevB.51.1381 journal journal Phys. Rev. B volume 51, pages 1381 (year 1995)NoStop [Subedi et al.(2008)Subedi, Zhang, Singh, and Du]Subedi2008 author author A. Subedi, author L. Zhang, author D. J. Singh, and author M. H. Du, 10.1103/PhysRevB.78.134514 journal journal Phys. Rev. B volume 78, pages 134514 (year 2008)NoStop [Koufos et al.(2014)Koufos, Papaconstantopoulos, and Mehl]Koufos2014 author author A. P. Koufos, author D. A. Papaconstantopoulos, and author M. J. Mehl, 10.1103/PhysRevB.89.035150 journal journal Phys. Rev. B volume 89, pages 035150 (year 2014)NoStop [Coh et al.(2015)Coh, Cohen, and Louie]1367-2630-17-7-073027 author author S. Coh, author M. L. Cohen, and author S. G. Louie, http://stacks.iop.org/1367-2630/17/i=7/a=073027 journal journal New J. Phys. volume 17, pages 073027 (year 2015)NoStop [Aperis et al.(2015)Aperis, Maldonado, and Oppeneer]Aperis2015 author author A. Aperis, author P. Maldonado, and author P. M. Oppeneer, 10.1103/PhysRevB.92.054516 journal journal Phys. Rev. B volume 92, pages 054516 (year 2015)NoStop [Bekaert et al.(2016)Bekaert, Vercauteren, Aperis, Komendová, Prozorov, Partoens, and Milošševi ćć]Bekaert2016 author author J. Bekaert, author S. Vercauteren, author A. Aperis, author L. Komendová, author R. Prozorov, author B. Partoens, and author M. V. Milošševi ćć, 10.1103/PhysRevB.94.144506 journal journal Phys. Rev. B volume 94, pages 144506 (year 2016)NoStop [Eschrig and Koepernik(2009)]Eschrig2009 author author H. Eschrig and author K. Koepernik, 10.1103/PhysRevB.80.104503 journal journal Phys. Rev. B volume 80, pages 104503 (year 2009)NoStop [Marsiglio et al.(1988)Marsiglio, Schossmann, and Carbotte]Marsiglio1988 author author F. Marsiglio, author M. Schossmann, and author J. P. Carbotte, 10.1103/PhysRevB.37.4965 journal journal Phys. Rev. B volume 37, pages 4965 (year 1988)NoStop [Degiorgi et al.(1994)Degiorgi, Nicol, Klein, Grüner, Wachter, Huang, Wiley, and Kaner]Degiorgi1994 author author L. Degiorgi, author E. J. Nicol, author O. Klein, author G. Grüner, author P. Wachter, author S.-M. Huang, author J. Wiley, and author R. B. Kaner, 10.1103/PhysRevB.49.7012 journal journal Phys. Rev. B volume 49, pages 7012 (year 1994)NoStop [Choi et al.(2002)Choi, Roundy, Sun, Cohen, and Louie]Choi2002 author author H. J. Choi, author D. Roundy, author H. Sun, author M. L. Cohen, and author S. G. Louie, @noop journal journal Nature volume 418, pages 758 (year 2002)NoStop [Margine and Giustino(2013)]Margine2013 author author E. R. Margine and author F. Giustino, @noop journal journal Phys. Rev. B volume 87, pages 024505 (year 2013)NoStop

http://arxiv.org/abs/1701.07619v1

20170126090225

Optimization Methods for Dirichlet Control Problems

math.OC

Optimization Methods for Dirichlet Control Problems Mariano Mateos Dpto. de Matemáticas, Universidad de Oviedo, Campus de Gijón, 33203 Gijón, Spain Email: mmateos@uniovi.es. January 2017 =============================================================================================================================== We discuss several optimization procedures to solve finite element approximations of linear-quadratic Dirichlet optimal control problems governed by an elliptic partial differential equation posed on a 2D or 3D Lipschitz domain. The control is discretized explicitely using continuous piecewise linear approximations. Unconstrained, control-constrained, state-constrained and control-and-state constrained problems are analyzed. A preconditioned conjugate method for a reduced problem in the control variable is proposed to solve the unconstrained problem, whereas semismooth Newton methods are discussed for the solution of constrained problems. State constraints are treated via a Moreau-Yosida penalization. Convergence is studied for both the continuous problems and the finite dimensional approximations. In the finite dimensional case, we are able to show convergence of the optimization procedures even in the absence of Tikhonov regularization parameter. Computational aspects are also treated and several numerical examples are included to illustrate the theoretical results. Dirichlet optimal control, discretization, constrained optimization, preconditionate conjugate gradient, semismooth Newton methods 49J20, 49M05, 49M15, 65N30 myheadings plain M. Mateos Optimization for Dirichlet Control Problems § INTRODUCTION Only in the last ten years it has been possible to develop a systematic study of Dirichlet optimal control problems governed by elliptic equations, which started with the seminal paper <cit.>. In that work a 2D control-constrained problem governed by a semilinear equation posed on a convex polygonal domain is studied. Several other works have been published about numerical error estimates; see <cit.> or <cit.> for a variational approach to control-constrained problems posed on smooth 2D or 3D domains, <cit.> for a superconvergence result on the state approximation for unconstrained 2D problems, <cit.> for control-and-state constrained problems posed on convex polygonal domains. In <cit.> the authors study the control-constrained problem in a 2D smooth convex domain taking into account the problems derived by the boundary approximation and in <cit.> an apparent paradox between <cit.> and <cit.> is explained. The regularity of the solution in possibly nonconvex polygonal plane domains is studied in <cit.>; see also the introduction of that paper for further references about related problems. In the recent publication <cit.>, error estimates for a Dirichlet control problem governed by a parabolic equation are obtained. In this work, the spatial discretization of the control is studied in both the cases of continuous piecewise linear finite elements and variational approach. Just before that, several works dealing with efficient optimization methods for control or state constrained problems had appeared; in <cit.> the semismooth Newton method is thoroughly studied. Since all these papers about optimization are previous to the papers about the numerical analysis of Dirichlet control problems, very little or no reference is made in them to their applicability to the problems we are going to deal with. Only in <cit.> two different Dirichlet control problems with more regular solutions than the ones we treat here are studied in the infinite-dimensional case. Let us also mention that in <cit.> the Dirichlet boundary condition is replaced by a Robin penalization. For this kind of penalization the methods developed in the aforementioned references are directly applicable. In <cit.>, the semismooth Newton method is studied in the context of the variational approach of the control. Although the authors only exemplify their results through distributed and Robin boundary control, a combination of their results and some of the results we present in Section <ref> can be applied to Dirichlet control. See Remark <ref> below. In this work we describe optimization methods for Dirichlet control problems in both the infinite and finite dimensional cases. Convergence proofs, examples and practical implementation details are discussed for all the algorithms through the paper. In Section <ref> we state the problem and prove that it is well posed in Lipschitz domains; see Lemma <ref>. So far, only smooth, convex, polygonal or polyhedral domains had been studied. Next, we discretize the problem. As is usual in control problems, we have three choices to discretize the control: the first option is not to discretize the control, using a variational discretization as introduced in <cit.> for distributed problems; as a second option we may use piecewise constant functions; and finally we can use continuous piecewise linear functions. The choice is not trivial because first order optimality conditions for Dirichlet control problems involve the normal derivative of the adjoint state. If we discretize directly the optimality system using e.g. continuous piecewise linear finite elements, we would obtain two different approximations of the control: the trace of the discrete state would be continuous piecewise linear and the normal derivative of the discrete adjoint state would be piecewise constant. In <cit.> and in <cit.> this difficulty is solved using a mixed formulation of the state equation which is discretized with the lowest order Raviart-Thomas element. In <cit.> a convergence proof for this kind of discretization is given. In this way, both the trace of the discrete state and the normal derivative of the discrete adjoint state are piecewise constant functions, so the identification of both with the discrete control is meaningful. The discretization of the control in the presence of control constraints is carried out in these papers using a variational discretization. This kind of approximation may be convenient when the gradient of the state is the variable of interest, since this quantity is treated as one of the variables of the problem. Another approach, in finite dimension, is to use the variational discrete normal derivative of the discrete adjoint state introduced in <cit.>, which is a continuous piecewise linear function. Doing so, both the trace of the discrete state and the normal derivative of the discrete adjoint state are continuous piecewise linear functions, so the identification of both with the discrete control is meaningful. Following this idea, in <cit.> the control is not discretized for the control-constrained case, but a variational approach is followed. In this work we investigate optimization methods for the case of discretizing the control explicitely using continuous piecewise linear functions. The end of Section <ref> is devoted to describe with some detail some computational aspects that will be important in the rest of the work. We next provide in Section <ref> an efficient method to solve the unconstrained problem. We propose in Subsection <ref> a preconditioned conjugate gradient (pcg in the rest of the work) method for a reduced problem in the spirit of <cit.>. We are able to prove convergence of the conjugate gradient even for the case where the Tikhonov regularization parameter ν vanishes. In sections <ref>, <ref> and <ref> we study the convergence of the semismooth Newton method for the constrained problems and write practical algorithms for the solution of the finite dimensional approximations. In Section <ref> we deal with the control-constrained problem. In many of the aforementioned references about the semismooth Newton method for control-constrained problems the authors study convergence of an abstract problem or an infinite dimensional problem. For distributed and Neumann control problems the results are immediately applicable to the finite dimensional approximations because the controls can be discretized using piecewise constant functions, and therefore the variational inequality arising from first order necessary optimality conditions can be written in an element-wise form. The same idea applies when we are dealing with a variational approach as proposed in <cit.> or a mixed formulation, as studied in <cit.>. When we use continuous piecewise linear elements, the variational inequality cannot be written in a point-wise or elementwise form; see (<ref>) and Remark <ref>. We include the analysis of Newton methods for both the original problem and the discretized one. In Section <ref> we study the state-constrained problem using a Moreau-Yosida penalization. Since there are no control constraints, the analysis of the semismooth Newton method for the infinite dimensional problem is applicable to the finite dimensional one, so we do not need to repeat it. We prove that the finite-element approximation of the penalized problems converge to the solution of the penalized problem. This result cannot be deduced straightforward from the ones in the literature since the penalized functional is not of class C^2. A continuation strategy as proposed in <cit.> is developed. Finally, in Section <ref> we discuss the problem with both control and state constraints. It is well known that the main difficulty with Dirichlet control problems is the low regularity of the solutions. This regularity and related error estimates are, so far, well established in 2D polygonal domains <cit.> and 2D or 3D smooth domains <cit.> but there is not, up to our best knowledge, a general study in 3D polyhedral domains. Although the main focus of this work is on optimization methods, we also study the regularity of the solution and error estimates of the approximations in one example case in a polyhedron; see Example <ref>. § STATEMENT OF THE PROBLEM AND FIRST PROPERTIES Let Ω⊂ℝ^d, d=2 or d=3, be a bounded domain with Lipschitz boundary Γ and in this domain consider a target state y_Ω∈ L^2(Ω). Consider also the continuous linear operator S:L^2(Γ)→ L^2(Ω) such that y=Su if and only if y is the solution in the transposition sense (see Definition <ref> below) of -Δ y = 0Ω, y=uΓ. Let ω⊂Ω be a domain such that ω̅⊂Ω and define two pairs of functions α, β∈ C(Γ) and a, b∈ C(ω̅) such that α(x)<β(x) for all x∈Γ and a(x)<b(x) for all x∈ω̅. For some fixed regularization parameter ν≥0, define J(u) =1/2Su-y_Ω^2_L^2(Ω)+ν/2u^2_L^2(Γ) and consider the sets U_α,β={u∈ L^2(Γ): α(x)≤ u(x)≤β(x)x∈Γ}, and K_a,b={y∈ L^2(Ω)∩ C(ω̅): a(x)≤ y(x)≤β(x)x∈ω̅}. In this work we will study optimization procedures for the following four Dirichlet control problems: (P^U)min_u∈ L^2(Γ) J(u), (P^C)min_u∈ U_α,βJ(u), (P^S)min_Su∈ K_a,bJ(u), (P^CS)min_[ u∈ U_α,β; Su∈ K_a,b ]J(u), namely the unconstrained, control-constrained, state-constrained and control-and-state constrained problems. Almost all the literature related to these problems is written using the state equation (<ref>). There would be no problem in taking into account an equation of the form Ay :=-∑_i=1^d∂_i(a_i,j∂_j y) + a_0 y = FΩ, y = u + GΓ with regular enough F, G, a_0≥ 0 and a_i,j=a_j,i satysfing an uniform ellipticity condition. Let us state precisely what we mean by solution in the transposition sense. Consider the space Φ = {ϕ: ϕ∈ H^1_0(Ω)Δϕ∈ L^2(Ω)}. This is a Banach space with the graph norm ϕ_Φ = ϕ_H^1(Ω)+ Δϕ_L^2(Ω). Further, the functions in this space satisfy ∂_n ϕ∈ L^2(Γ). This is known to be true for smooth domains; convex domains, see <cit.>; plane polygonal domains, see <cit.>; or polyhedral domains, see <cit.> and the usual trace theorem. We have not been able to find a proof of this fact for general Lipschitz domains. In <cit.> the regularity ϕ∈ H^3/2(Ω) is proved. Nevertheless, as the authors notice in page 165 of this reference, the trace theorem is not valid neither in H^3/2(Ω) nor in H^1/2(Ω), so we cannot deduce immediately that ∂_nϕ∈ L^2(Γ). The results in <cit.> imply that the usual trace result can be extended to harmonic functions in the limit cases. We show next how to take advantage of this to prove that ∂_nϕ∈ L^2(Γ) in Lipschitz domains. Regarding the analysis of semismooth Newton methods –see Lemma <ref> below– we also prove L^q(Γ) regularity for some q>2. Let Ω⊂ℝ^d, d=2 or d=3, be a bounded domain with Lipschitz boundary Γ and consider ϕ∈Φ. Then, there exists q_0>2 depending on the domain such that, for 2≤ q<q_0, we have ∂_n ϕ∈ L^q(Γ) and ∂_n ϕ_L^q(Γ)≤ C Δϕ_L^2(Ω). If, further, Ω is smooth or convex or polygonal or polyhedral, then there exists t>0 such that ∂_n ϕ∈ H^t(Γ) and ∂_n ϕ_H^t(Γ)≤ C Δϕ_L^2(Ω). Denote z=-Δϕ, extend z by zero to ℝ^d, consider the Newtonian potential centered at the origin N(x) and define w = z * N, the convolution product of z and N. Then w∈ H^2(Ω) and ∇ w∈ H^1(Ω)^d, so it is clear that Tr (∇ w)∈ H^1/2(Γ)^d↪ L^q(Γ)^d for all q<∞ if d=2 and all q≤ 4 if d=3, since the dimension of Γ is d-1. This implies that: (a) ∂_n w = ∇ w· n∈ L^q(Γ) because the boundary Γ is Lipschitz and therefore it has a unit normal vector defined almost everywhere; and there exists C>0 such that ∂_n w_L^q(Γ)≤ CΔϕ_L^2(Ω); (b) g = Tr(w)∈ W^1,q(Γ) due precisely to the definition of W^1,q(Γ), and there exists C>0 such that g_W^1,q(Γ)≤ C Δϕ_L^2(Ω). Define now v∈ H^1(Ω) the unique solution of -Δ v = 0 in Ω, v = g on Γ. Using <cit.>, we have that there exists q_0>2 such that if 2≤ q <q_0, then the nontangential maximal function of the gradient of v satisfies M(∇ v)∈ L^q(Γ) and there exists C>0 such that M(∇ v)_L^q(Γ)≤ Cg_W^1,q(Γ). As is pointed out in <cit.>, this implies that ∇ v has nontangential limit a.e. on Γ and we can define the normal derivative of v at a point s∈Γ as the nontangential limit as x→ s of ∇ v(x)· n(s). For a precise definition of nontangential limit and nontangential maximal function see, e.g., the introduction of the work <cit.>. This, together with inequalities (<ref>) and (<ref>) imply that ∂_n v ∈ L^q(Ω) and ∂_n v_L^q(Γ)≤ C Δϕ_L^2(Ω). So we have that ϕ = v-w and we can define in a natural way ∂_n ϕ = ∂_n v -∂_n w∈ L^q(Γ). The estimate (<ref>) follows from (<ref>) and (<ref>). For smooth, convex, polygonal or polyhedral domains, the second result follows from the regularity ϕ∈ H^3/2+t(Ω) and the usual trace theorem for ∇ϕ. See <cit.> for convex domains, <cit.> for plane polygonal domains and <cit.> for polyhedral domains. We will say that y∈ L^2(Ω) is the solution in the transposition sense of (<ref>) if (y,-Δϕ)_Ω = -(u,∂_n ϕ)_Γϕ∈Φ. Here and in the rest of the work (·,·)_X stands for the standard inner product in L^2(X). The adjoint operator of S is S^*:L^2(Ω)→ L^2(Γ) defined by S^*z = -∂_nϕ, where ϕ is the unique weak solution of -Δϕ = zΩ, ϕ = 0Γ. We can write now J(u) = 1/2(Su-y_Ω,Su-y_Ω)_Ω+ν/2(u,u)_Γ =1/2(S^*Su+ν u,u)_Γ - (S^*y_Ω,u)_Γ + c_Ω where c_Ω=0.5 y_Ω_L^2(Ω)^2 is a constant. From this expression, we can easily compute the derivative of J at a point u∈ L^2(Γ) in the direction v∈ L^2(Γ): J'(u)v=(S^*Su+ν u,v)_Γ -(S^*y_Ω,v)_Γ. For later use, we will define now for every u∈ L^2(Γ), y_u=Su∈ H^1/2(Γ) and φ_u∈ H^1_0( Ω) the unique solution of -Δφ_u = y_u-y_ΩΩ, φ_u=0Γ. §.§ Discretization. To discretize the problems. consider {𝒯_h}_h a regular family of triangulations of Ω̅. To simplify the notation, we will suppose that Γ is polygonal or polyhedral. Related to the mesh, we will call N the number of nodes and {x_j}_j=1^N the nodes of the mesh. We define the sets of interior indexes, boundary indexes and indexes in ω̅ as I = {j:x_j∈Ω}, B={j:x_j∈Γ} and J = {j:x_j∈ω̅}. For the discretization of the state and the adjoint state we use the space of linear finite elements Y_h⊂ H^1(Ω), Y_h={y_h∈ C(Ω̅) y_h∈ P^1(T) ∀ T∈𝒯_h}. As usual, we will abbreviate Y_h0=Y_h∩ H^1_0(Ω). For the control we use the space U_h of continuous piecewise linear functions on Γ U_h = {u_h∈ C(Γ) u_h∈ P^1(T∩Γ) ∀ T∈𝒯_h}. Notice that the elements of U_h are the traces of the elements of Y_h. We will denote I_h:C(Ω̅)→ Y_h or I_h:C(Γ)→ U_h the interpolation operator related to these spaces and Π_h:L^2(Ω)→ Y_h or Π_h:L^2(Γ)→ U_h the projection onto this spaces in the L^2 sense. For all y∈ L^2(Ω) and all u∈ L^2(Γ): (Π_h y,y_h)_Ω = (y,y_h)_Ω ∀ y_h∈ Y_h (Π_h u,u_h)_Γ = (u,u_h)_Γ ∀ u_h∈ U_h. We discretize the state equation following the work by Berggren <cit.>: define S_h:L^2(Γ)→ L^2(Ω) such that for u∈ L^2(Γ), y_h=S_hu if and only if y_h∈ Y_h is the unique solution of a(y_h, z_h)=0 ∀ z_h∈ Y_h0, (y_h,v_h)_Γ = (u,v_h)_Γ ∀ v_h∈ U_h, where a(·,·) is the bilinear form associated to the operator in the PDE. In the case of the Laplace operator a(y,z)=∫_Ω∇^T y∇ z dx. The discrete functional is thus defined as J_h(u)=1/2(S_hu-y_Ω,S_hu-y_Ω)_Ω+ν/2(u,u)_Γ. We define now U^h_α,β = {u_h∈ U_h: α(x_j)≤ u_h(x_j)≤β(x_j) ∀ j∈B}, and K^h_a,b = {y_h∈ Y_h: a(x_j)≤ y_h(x_j)≤ b(x_j) ∀ j∈J}. The discrete problems reads thus as (P_h^U)min_u_h∈ U_h J_h(u_h), (P_h^C)min_u∈ U^h_α,βJ_h(u_h), (P_h^S)min_S_hu_h∈ K^h_a,bJ_h(u_h), (P_h^CS)min_[ u_h∈ U^h_α,β; S_hu_h∈ K^h_a,b ]J_h(u_h). The adjoint operator of S_h is given by the discrete variational normal derivative. See <cit.>. S_h^*:L^2(Ω)→ L^2(Γ) and w_h=S_h^*y if w_h∈ U_h satisfies (w_h,z_h)_Γ = (y,z_h)_Ω - a(z_h,ϕ_h)z_h∈ Y_h, where ϕ_h∈ Y_h0 is the unique solution of a(z_h, ϕ_h) = (y,z_h)_Ω z_h∈ Y_h0. It is customary to write S_h^*y=-∂_n^hϕ_h. For u∈ L^2(Γ) and y∈ L^2(Ω), we have (S_hu,y)_Ω = (u,S^*_h y)_Γ. We can then write J_h(u) =1/2(S^*_hS_hu+ν u,u)_Γ -(S_h^* y_Ω,u)_Γ+c_Ω. The computation of the derivative of J_h at a point u∈ L^2(Γ) in the direction v∈ L^2(Γ) is then obtained with J'_h(u)v =(S_h^*S_hu+ν u,v)_Γ - (S^*_hy_Ω,v)_Γ. Since our final objective is to optimize in U_h, let us see in more detail how to make the computations in this space. Consider {e_j}_1^N the nodal basis in Y_h, where N is the dimension of Y_h and satisfies N=N_I+N_B, the latter being respectively the number of interior and boundary nodes. With an abuse of notation, we will also denote e_j the restriction of e_j to Γ. Define the usual finite element stress, mass and boundary mass matrices by K_i,j=a( e_i,e_j), M_i,j=(e_i,e_j)_Ω, B_i,j =(e_i,e_j)_Γ1≤ i,j≤ N. We will also use I∈ℝ^N× N for the identity matrix, O∈ℝ^N× N for the zero matrix and usually refer to submatrices as K_I,I or K_I,: defined by taking the rows or columns designed by the sets of indexes in the subscripts, the semicolon meaning “all the indexes”. For instace, the usual boundary mass matrix is B_B,B. Given u_h=∑_j∈Bu_j e_j, we denote u∈ℝ^N_B× 1 the vector (u_1,…,u_N_B)^T and for y_h=∑_j=1^N y_je_j we denote y∈ℝ^N× 1 the vector (y_1,…,y_N)^T. Using (<ref>), we have that y_h=S_hu_h iff [[ K_I,I K_I,B; O_B,I B_B,B ]] [ [ y_I; y_B ]] = [ [ 0; B_B,Bu ]]. Since B_B,B is nonsingular, we can write this as [ K_I,Iy_I = -K_I,Bu,; y_B = u. ] If we define S∈ℝ^N_B× N as S=[[ K_I,I K_I,B; O_B,I I_B,B ]]^-1I_:,B we have that y_h=S_hu_h if and only if y=Su. Given y_h∈ Y_h, let ϕ_h=∑_j∈Iϕ_j e_j be the solution of (<ref>) for y=y_h. Denoting ϕ∈ℝ^N_I× 1 the corresponding vector, it can be computed as the solution of K_I,Iϕ = M_I,:y. Then we could compute w∈ℝ^N_B× 1, the vector whose components are the coefficients of the (minus) discrete normal derivative -∂_n^hϕ_h = S_h^*y_h=w_h=∑_j∈Bw_je_j solving the system B_B,Bw = M_B,:y - K_B,Iϕ. Formally, we can also write that w = B_B,B^-1S^TMy. To finish this section we also define the matrix A∈ℝ^N_B,× N_B and the vector f∈ℝ^N_B× 1 by A_i,j= (S_h^*S_h e_i+ν e_i,e_j)_Γ f_i= (S^*_hy_Ω,e_i)_Γ. We have that J_h(u_h)=1/2u^TAu - f^Tu+c_Ω. We also note that A = S^TMS+νB_BB. To compute the vector f, we consider the projection of y_Ω onto Y_h in the L^2(Ω) sense, y_Ω,h = Π_h y_Ω and denote y_Ω∈ℝ^N× 1, the vector whose j-th component is y_Ω,h(x_j), and f=S^TMy_Ω. So for u_h,v_h∈ U_h, the latter represented by the vector v, we can compute J'(u_h)v_h=v^TB_B,Bw +νv^T B_B,Bu-v^Tf. Notice that applying the equality (<ref>), the explicit computation of w is not necessary, and we can write J'(u_h)v_h=v^T(M_B,:y- K_B,Iϕ +νB_B,Bu-f). § UNCONSTRAINED PROBLEM Problem (P^U) has a unique solution u̅∈ L^2(Γ) that satisfies J'(u̅)=0. For every 0<h<h_0, problem (P^U_h) has also a unique solution u̅_h that satisfies J'_h(u̅_h)=0. The problems being convex, these conditions are also sufficient. Moreover it is known that u̅_h→u̅ strongly in L^2(Γ) and also error estimates are available in terms of the mesh size h if ν>0 and the domain is either 2D and polygonal or smooth. See <cit.>. Let us comment two different approaches for the solution of the FEM approximation of (P^U). §.§ Solve the optimality system for the state, the adjoint state and the control. We write first order optimality conditions for (P^U_h). There exists some h_0>0 such that for every 0<h<h_0, there exist unique u̅_h∈ U_h, y̅_h∈ Y_h and φ̅_h∈ Y_h0 satisfying the optimality system: a(y̅_h,z_h) = 0 z_h∈ Y_h0, y̅_h ≡ u̅_hΓ, a( z_h, φ̅_h) = (y̅_h-y_Ω,z_h)_Ωz_h∈ Y_h0, νu̅_h ≡ ∂_n^hφ̅_hΓ. Taking into account the definition of discrete variational normal derivative and relations (<ref>)–(<ref>), we can write this optimality system as [[ K_I,I O_I,B K_I,B O_I,I; O_B,I I_B,B -I_B,B O_B,I; -M_I,I -M_I,B O_I,B K_I,I; M_B,I M_B,B νB_B,B -K_B,I ]] [[ y_I; y_B; u; φ_I ]] = [[ 0_I; 0_B; -M_I,:y_Ω; M_B,:y_Ω ]]. We may eliminate u with the boundary condition, and reorder the equations to get the linear system [[ M+νB -K_:,I; -K_I,: O_I,I ]] [[ y; φ_I ]] = [[ My_Ω; 0 ]]. Notice that system (<ref>) is solvable even for ν = 0. Solving this equation would completely solve the problem for the unconstrained case. When the discretization is very fine, the number of unknowns may make the solution of the system by direct methods too difficult. The preconditioned conjugate gradient method for this kind of linear systems is studied by Schöberl and Zulehner in <cit.> and Herzog and Sachs in <cit.>. A preconditioner can be built using matrices representing scalars products in Y_h and Y_h0. Following Algorithm 1 in the last-mentioned reference, at each iteration three linear systems must be solved: two of size N and one of size N_I. In <cit.>, the systems are solved using a multigrid method. Reduced problems in the adjoint state variable and related pcg have also been studied in <cit.>. Nevertheless, the structure of the matrix we have in (<ref>) is different to the one treated in that reference and application of their results is not straightforward. §.§ Use an iterative method to solve a reduced problem in the control variable. Let us see another way of solving (P^U_h). Using (<ref>), first order optimality conditions can also be written as Au = f. The matrix A is symmetric and positive definite for all ν≥ 0 and there exists C>0 independent of h and ν such that its condition number is bounded by κ( A ) ≤ C λ_N_B(B_B,B)/λ_1(M)λ_N(M) +νλ_N_B(B_B,B)/λ_1(M) +νλ_1(B_B,B), where 0<λ_1(B_B,B)< λ_N_B(B_B,B) and 0<λ_1(M)<λ_N(M) are respectively the smallest and greatest eigenvalues of the matrices B_B,B and M. It is clear from (<ref>) that A is symmetric. The mass matrices M and B_B,B are symmetric and positive definite and therefore λ_1(M)>0 and λ_1(B_B,B)>0. Since the boundary components of Su are exactly u, we have that Su_ℝ^N≥u_ℝ^N_B and hence u^TAu = u^T(S^TMS+νB_B,B) u = (Su)^TM(Su) +νu^T B_B,Bu)≥ (λ_1(M)+ νλ_1(B_B,B)) u_ℝ^N_B^2 so A is positive definite. From <cit.>, we know that there exists some C>0 such that S_h u_h_L^2(Ω)≤√(C)u_h_L^2(Γ). Since S_hu_h_L^2(Ω)^2 = (Su)^TM(Su)≥λ_1(M)Su_ℝ^N^2 and u_h_L^2(Γ)^2 = uB_B,Bu≤λ_N_B(B_B,B)u_ℝ^N_B^2, we obtain u^TAu = (Su)^TM(Su) +νu^T B_B,Bu) ≤ (C λ_N_B(B_B,B)/λ_1(M)λ_N(M)+ νλ_N_B(B_B,B)) u_ℝ^N_B^2. The bound for the condition number follows directly from the last inequality and (<ref>). For a graded family of mesh with grading parameter 0<μ≤ 1 (see <cit.>), we usually define h=N^-1/d, N being the number of nodes of the mesh and d the dimension of Ω. The case μ=1 corresponds to that of a quasi-uniform mesh family. If 𝒯_h is a family of graded meshes with grading parameter 0<μ≤ 1, then there exists C>0 independent of h and ν such that κ(A)≤ C h^(2d-1)(1-1/μ)1 + ν h^(1-1/μ)d/h^1/μ+ν. In particular, for a quasi-uniform family, there exists C>0 such that κ(A)≤ C1+ν/ν+h. from <cit.> and <cit.>, we have that there exists constants 0<C_1<C_2 λ_N_B(B_B,B)≤ C_2 h^d-1,λ_1(B_B,B)≥ C_1 h^d-1/μ,λ_N(M)≤ C_2 h^d,λ_1(M) ≥ C_1 h^d/μ Estimate (<ref>) follows then from (<ref>). Estimate (<ref>) follows from (<ref>) for μ=1. The explicit computation of the matrix A is out of the question, since it requires the solution of 2N_B systems of size N_I. Much better performance can be achieved using an iterative method which only requires the computation of d=Au. This is shown in Algorithm <ref>. Preconditioned conjugate gradient methods can be carried out at the cost of one evaluation of Au per iteration. The computation of f can be done in a similar way, but of course it only must be computed once; see Algorithm <ref>. To finish this section, let us say a word about a preconditioner to solve (<ref>). In practice, matrices like B_B,B or P=M_B,B+νB_B,B make acceptable preconditioners. This is specially important when using graded meshes; see Example <ref> below. Notice that at each pcg iterate, we only need to solve two linear systems, each of size N_I, to compute Au, plus another one of size N_B to compute P^-1r. All the code for all the examples has been made by the author using Matlab R2015b and has been run on a desktop PC with Windows 10 Pro 64bits with 16GB RAM DIMM 1333Mhz on an Intel Core i7 CP 870@2.93Ghz. In this example we compare the different execution times, t_D and t_P, that we obtain when we solve the optimality system using a direct solver for equation (<ref>) or using an iterative method, the preconditioned conjugate gradient method in our case, to solve the reduced problem (<ref>). We have used respectively Matlab builtin commands and . We use the example in <cit.>, where Ω the interior of the convex hull of the points (-0.5,-0.5), (0.5,-0.5), (0.5,0), (0,0.5), (-0.5,0.5); y_Ω≡ 1 and ν=1. A rough initial mesh is built with Matlab PDEToolbox (version 2.1) commands and set to 0.2 and subsequent nested meshes are obtained by regular diadic refinement using . We use our own code to assemble the matrices K, M and B. For the pcg we use as initial guess the null vector, a relative tolerance of 1E-10 and the preconditioner P = M_B,B+νB_B,B. At each iteration we have to solve the linear systems (<ref>) and (<ref>). To do this, we first obtain the Cholesky factors of K_I,I so that at each iteration we only have to solve triangular systems. The interior mesh nodes were ordered using a symmetric approximate minimum degree permutation in order to minimize the number of nonzero elements in the Cholesky factors of K_I,I using Matlab command . An analogous reordering of the boundary nodes is done to optimize the sparsity pattern of the Cholesky factors of the preconditioner. This reorderings and factorizations take more or less the same cputime than each single iteration of the pcg. This time is included in t_P. For reference and possible double-check, we also report on the optimal value of the functional. We have marked with ∞ the experiments in which we have run out of RAM. From the data in Table <ref> it is clear that the proposed iterative method for the reduced problem (<ref>) has an overall better performance for Dirichlet control problems. We have also coded Algorithm 1 in <cit.>, though the practical comparison with the other two methods is less clear. The main difficulty is the choice of appropriate scalar products. For instance, we have tried (using the notation in <cit.>) X=K+B, X=K+M or X=K+M+B for the scalar product in Y_h and Q=K_I,I for the scalar product in Y_h0, but we have observed that the number of iterations needed to achieve the prescribed accuracy grows as the number of unknowns N increases. For the first three meshes in the 2D example described in Table <ref> and ν=1 we obtain 126, 173 and 236 iterations respectively. In this example we show the effect of the use of P=M_B,B+νB_B,B as a preconditioner to solve (<ref>). In this case ν = 0.01, Ω = {x=re^iθ∈ℝ^2: r<1, 0<θ<11π/12}, and we use a mesh family graded at the origin with parameter μ = 13/33≈ 0.4; see <cit.>. In Table <ref> we compare the number of iterations used to reach the prescribed accuracy of 1E-10 without and with preconditioner. This example shows very clearly the effectiveness of this strategy of preconditioning. In this example we show that the number of pcg iterations for each fixed h is independent of the value of ν≥ 0 for all ν<ν_0. We take the same domain and meshes as in the 2D problem in Example <ref> and set y_Ω(x)=|x|^2, so that J(u̅)>0 even for ν=0. The results are summarized in Table <ref>. A 3D example in a polyhedron. Up to our best knowledge, there is not a general theory about regularity of the solutions or approximation results for 3D Dirichlet control problems posed on polyhedral domains. In <cit.>, the authors study a semi-discrete or variational approach to 3D problems on regular domains. Although the semi-discrete approach coincides with the full approach for unconstrained problems, we cannot profit their results since the regularity of the solutions in a smooth domain is (see <cit.>) much higher than the one we may obtain in polyhedral domains. For this example we will take Ω the unit cube (-0.5,0.5)^3, y_Ω≡ 1 and ν = 1. First, we obtain a regularity result for the solution of our problem. Let Ω⊂ℝ^3 be the interior of a rectangular parallelepiped and y_Ω∈ L^p(Ω) for some 3<p<+∞. Consider u̅ the solution of problem (P^U). Then, u̅∈ W^1-1/p(Γ), y̅∈ W^1,p(Ω) and φ̅∈ W^2,p(Ω). Moreover, u̅≡ 0 on the edges and corners of Γ. Since u̅∈ L^2(Γ), using transposition and interpolation it is clear that y∈ H^1/2(Ω). Classical Sobolev embedding in 3D leads to y∈ L^3(Ω). Since Ω is a parallelepiped and y_Ω is regular enough, <cit.> states that φ̅∈ W^2,3(Ω), and hence ∂_nφ̅∈Π_i W^2/3,3(Γ_i), where Γ_i, i=1:6, are the faces of Γ. This does not imply immediately that u̅ belongs to W^2/3,3(Γ) because 2/3· 3=2, which is the topological dimension of Γ, and some integral compatibility condition should be imposed on the edges; but for all q<3, it is true that u̅∈ W^1-1/q,q(Γ), and therefore y̅∈ W^1,q(Ω). Using again Sobolev embeddings, we have that y̅∈ L^s(Ω) for all s<+∞ and hence y̅-y_Ω∈ L^p(Ω). Applying once again <cit.>, we have that φ̅∈ W^2,p(Ω). Now we have that ∂_nφ̅∈Π_i W^1-1/p,p(Γ_i) and we can prove that if we define ∂_nφ̅=0 on the edges of Ω, then we obtain a continuous function. To do this, we use a similar argument to the one used in <cit.> for 2D problems. Since p>3, φ̅∈ C^1(Ω̅). Consider two faces A and B with a common edge AB and let τ^1_X, τ^2_X be two linearly independent vectors tangent to face X (X=A or X=B) such that τ^1_A=τ^1_B is also tangent to the edge AB. Since φ̅= 0 on Γ, we have that for every x∈ A, ∇φ̅(x)·τ^1_A = ∇φ̅(x)·τ^2_A = 0 and for every x∈ B, ∇φ̅(x)·τ^1_B = ∇φ̅(x)·τ^2_B = 0. So for x∈ AB, we have that ∇φ̅(x)=0 and therefore ∇φ̅(x)· n can be seen as a continuous function if we set its value to 0 on the edges, despite the jump discontinuities of the normal vector n. So u̅ is continuous and hence u̅∈ W^1-1/p,p(Γ). The regularity of the optimal state follows from the trace theorem; see e.g. <cit.>. Using this regularity, an error estimate can be proved for the example problem. Let Ω be a rectangular parallelepiped and y_Ω∈ L^p(Ω) for some 3<p<+∞. Consider u̅ be the solution of problem (P^U) and u̅_h be the solution of (P^U_h). Then there exists some h_0>0 and C>0 such that for all 0<h<h_0 u̅-u̅_h_L^2(Γ)≤ C h^1-1/p. The proof follows the same guidelines as those of <cit.> or <cit.> and thus will be omitted. An interesting remark is that for 2D problems, we can deduce uniform convergence for the controls using the L^2(Γ) estimate and an inverse inequality, since the boundary is 1D. This does not work for 3D problems with the error estimate at hand, since now Γ is 2D. To solve the problem we have used a regular mesh of identical cubes of size h, each of them divided into 6 tetrahedra according to the Kuhn triangulation of the cube (see e.g. <cit.>). Up to our best knowledge, the current version of the Matlab PDEToolbox (version 2.1) computes an approximation of B using the barycenter formula. Although this does not affect the order of convergence for the FEM, this matrix plays a central role in Dirichlet control problems (for instance, B_B,B is not singular, but the barycenter approximation may be singular), so we have computed it in an exact way (using the mid-sides formula). Mesh data, computation times and optimal values are displayed in Table <ref>. § CONTROL CONSTRAINED PROBLEMS Problem (P^C) has a unique solution u̅∈ L^2(Γ) that satisfies J'(u̅)(u-u̅)≥ 0 for all u∈ U_α,β. For every 0<h<h_0, problem (P^C_h) has also a unique solution u̅_h that satisfies J'_h(u̅_h)(u_h-u̅_h)≥ 0 for all u_h∈ U^h_α,β. The problems being convex, these conditions are also sufficient. Moreover it is known that u̅_h→u̅ strongly in L^2(Γ) and also error estimates are available in terms of the mesh size h when ν>0 and the domain is convex in 2D. See <cit.>. In <cit.> the smooth (2D and 3D) case is treated using variational approach, whose optimization process is different from the one we are presenting in the work at hand: in <cit.>, the problem is solved using a fixed point method, convergent for ν large enough; see <cit.> and Remark <ref> below for the convergence of the semismooth Newton method. §.§ Continuous problem We can formulate first order optimality conditions as: there exist unique u̅∈ L^2(Γ), y̅∈ H^1/2(Ω) and φ̅∈ H^1_0(Ω) such that -Δy̅ = 0xxxxΩ, y̅=u̅Γ, -Δφ̅ = y̅-y_ΩΩ, φ̅=0Γ, (-∂_nφ̅+ νu̅, u-u̅) ≥ 0xxxx u∈ U_α,β. We first describe a semismooth Newton method to solve this optimality system and prove a convergence result for it (see Theorem <ref>). Next, we will reformulate the optimality system in terms of the Lagrange multipliers related to the constraints. We will see that this approach is better suited to the discrete problem, but has the drawback that the Newton method related to it is not semismooth; nonetheless, we will prove a convergence result for it; cf. Theorem <ref>. To facilitate the notation, we will skip the lower bound α, and will work only with the constraint u≤β. The variational inequality (<ref>) is a projection in L^2(Γ). In this case, it is equivalent to a pointwise projection: νu̅(x)=min{νβ(x),∂_nφ̅(x)}Γ. In order to analyze the semismooth Newton method to solve the optimality system, we define G:L^2(Γ)→ L^2(Γ) by G(u) = ν u-min{νβ,-S^*(S u-y_Ω)}. Thus, solving the optimality system is equivalent to solving G(u)=0. Given u∈ L^2(Γ), we define the sets of active and free points related to u as Γ_A(u)={x∈Γ: -S^*(S u-y_Ω)>νβ}, Γ_F(u) = Γ∖Γ_A(u). Abusing notation and when this does not lead to confusion, we will often drop the u and only write Γ_A and Γ_F. (Although it is customary to use the word “inactive”, we have preferred to use “free” since we have already used the letter I for the identity matrix I and the interior nodes I.) χ_B will denote the characteristic function of a set B. The relation G:L^2(Γ)→ L^2(Γ) is slantly differentiable in the sense stated in <cit.> and ∂^CLG semismooth in the sense stated in <cit.>, where ∂^CLG is Clarke's generalized differential <cit.>. A slanting functional M(u)∈ℒ(L^2(Γ),L^2(Γ)) is given by M(u)v = ν v+χ_Γ_F(u)S^*Sv for all v∈ L^2(Γ). Finally, if ν>0, M(u) has an inverse uniformly bounded in ℒ(L^2(Γ),L^2(Γ)) for all u∈ L^2(Γ). Using (<ref>) in Lemma <ref>, we have that there exists some q>2 such that S^*S∈ℒ(L^2(Γ),L^q(Γ)). The slant differentibility then follows directly from <cit.> and the semismoothness from <cit.>; see also <cit.>. The expression for the slanting functional follows from the slant derivative of the function min(0,·) and the chain rule <cit.>. Finally, given z∈ L^2(Γ) M(u)v = z{[ ν v = z Γ_A(u),; ν v = z -S^*Sv Γ_F(u). ]. This equations can be read as -Δ y =0Ω, y =vΓ, -Δφ = yΩ, φ =0Γ, ν v = z Γ_A(u), ν v = z + ∂_nφΓ_F(u). Let us define w=χ_Γ_F(u)v. Taking into account the definition of S and S^*, we have that ν v = ν w + zχ_Γ_A(u) and -Δ y_w =0Ω, y_w =wΓ, -Δφ_w = y_wΩ, φ_w =0Γ, w = 0 Γ_A(u), ν w = z + ∂_nφ_w - S^*S z χ_Γ_A(u)Γ_F(u). On one hand, using the continuity of S^* (cf. (<ref>)) and of S, we have S^*S z χ_Γ_A(u)_L^2(Γ)≤ C z_L^2(Γ_A(u))≤ C z_L^2(Γ). On the other hand, using the definition of solution in the transposition sense, we have that (y_w,y_w)_Ω = -(w,∂_nφ_w)_Γ = - ν (w,w)_Γ + (w,z)_Γ - (w,S^*Sz χ_Γ_A(u))_Γ. So νw^2_L^2(Γ) = (w,z)_Γ - (w,S^*S z χ_Γ_A(u))_Γ- y_w^2_L^2(Ω) ≤ w_L^2(Γ)z_L^2(Γ) + w_L^2(Γ)S^*Sz χ_Γ_A(u)_L^2(Γ) ≤ Cw_L^2(Γ)z_L^2(Γ). And we get w_L^2(Γ)≤ C/νz_L^2(Γ). Taking into account the definition of w and the condition for v on the active set, we get that v_L^2(Γ)≤ C/νz_L^2(Γ), where C is independent of u, and hence M(u) has a uniformly bounded inverse for each ν>0. Notice that in the infinite dimensional case, if ν=0 then M(u)v = χ_Γ_F(u)S^*Sv. In practical cases, it is known (cf. (<ref>) or <cit.>) that there exists t>0 such that S^*Sv∈ H^t(Γ) which is compactly embedded in L^2(Γ), and hence M(u) does not have a bounded inverse. Given a current iterate u∈ L^2(Γ), we may compute next iterate u^+ using a semismooth Newton method as follows: M(u)(u^+-u) = -G(u). Writing this in detail leads to ν u^+-ν u +χ_Γ_F(u)S^*S(u^+-u) = -ν u+min{νβ,-S^*(Su-y_Ω)}, which means that, if ν>0, ν u^+-ν u = -ν u+νβ⇒ u^+ = βΓ_A(u), and u^+ = -1/νS^*(S u^+-y_Ω)Γ_F(u). This equations can be read as -Δ y^+ = 0Ω, y^+ = u^+Γ, -Δφ^+ = y^+-y_ΩΩ, φ^+ = 0Γ, u^+ = βΓ_A(u), u^+ = 1/ν∂_nφ^+Γ_F(u). With all these considerations, we can write a semismooth Newton method to solve the optimality system in the infinite dimensional case. The semismooth Newton method described in Algorithm <ref> converges q-superlinealy to u̅ provided u^0 is close enough to u̅ in the sense of L^2(Γ). Once we have proved Lemma <ref>, this result is a direct consequence of <cit.>; see also <cit.> or <cit.>. The discrete version of the variational inequality (<ref>) –see (<ref>) below– does not have a pointwise version analog to (<ref>). A more convenient approach to the continuous problem, from the point of view of the discretized problem, is obtained using Lagrange multipliers associated to the bound control constraints. The con of this approach is that we do not obtain a semismooth Newton method, in the sense that the involved functions are known not to be semismooth. It must noticed that, when the variational discretization of the control is used, <cit.>, the discrete optimal control is obtained as the pointwise projection of the discrete optimal adjoint state as in (<ref>), so convergence of the semismooth Newton method for such a discretization would follow from Theorem 4.2 and Lemma <ref> with S replaced by S_h. To simplify the notation, we will reduce the exposition to the case of having only an upper bound u≤β on Γ. Condition (<ref>) can be replaced by the following pair of equations: there exists also λ̅∈ L^2(Γ) such that equation (<ref>) can be written as νu̅(x) = ∂_nφ̅(x)-λ̅(x)Γ. u̅(x)≤β(x), λ̅(x)≥ 0, λ̅(x)(u̅(x)-β(x)) = 0x∈Γ. Condition (<ref>) can be written as λ(x)-max{0,λ(x)+c(u(x)-β(x))} =0x∈Γ, which is true for any c>0. Since u∈ L^2(Γ) appears inside the max operation, it is known that the operator described in (<ref>) is not semismooth (following <cit.>) or has not a slant derivative (as in <cit.>), so it is not clear that Newton's method applied to (<ref>), (<ref>), (<ref>) converges. Define F_c(u,λ) = ([ ν u+S^*Su-S^*y_Ω+λ; λ-max{0,λ+cu-cβ} ]) and for each pair control-multiplier (u,λ)∈ L^2(Γ)× L^2(Γ), set Γ_A(u,λ) = {x∈Γ: 1/cλ+u>β}, Γ_F(u,λ) = Γ∖Γ_A(u,λ). We have that ([ ν I+S^*S I; - cχ_Γ_A(u,λ) I χ_Γ_F(u,λ) I ])∈∂^CL F_c(u,λ). So, given a current iterate (u,λ), a Newton-like iterate to obtain (u^+,λ^+) reads as ν u^+ = -S^*(Su^+-y_Ω) - λ^+, u^+ = βΓ_A(u,λ), λ^+ =0 Γ_F(u,λ). It is remarkable that the parameter c only appears in this equations hidden in the definition of the active set. If c=ν>0, the sequence u^k generated by the Newton-like method described in Algorithm <ref> converges q-superlinearly to u̅ in L^2(Γ) provided u^0 is close enough to u̅ in the sense of L^2(Γ) and λ^0=S^*y_Ω-(S^*Su^0 + ν u^0). In this case we cannot apply directly the results in <cit.>; see also <cit.> or <cit.> since u∈ L^2(Γ) appears inside the max operation, and it is known that the operator described in (<ref>) is not semismooth (following <cit.>) or has not a slant derivative (as in <cit.>). We will follow instead the method of proof of <cit.> and we will show that the sequence u^k generated by Algorithm <ref> is exactly the same as the one generated by Algorithm <ref>, which we will call ũ^k. Writing in detail (<ref>) we obtain -Δ y^+ = 0 Ω, y^+ = u^+Γ, -Δφ^+ = y^+-y_ΩΩ, φ^+ = 0Γ, ν u^+ = ∂_nφ^+ -λ^+Γ. Using now (<ref>) and (<ref>) we obtain -Δ y^+ = 0Ω, y^+ = u^+Γ, -Δφ^+ = y^+-y_ΩΩ, φ^+ = 0Γ, u^+ = βΓ_A(u,λ), u^+ = 1/ν∂_nφ^+Γ_F(u,λ), λ^+ = ∂_nφ^+-νβΓ_A(u,λ), λ^+=0Γ_F(u,λ). Notice that (<ref>), (<ref>) and (<ref>) is exactly as (<ref>), (<ref>) and (<ref>), provided that Γ_A(u,λ)= Γ_A(u). To finish, let us prove by induction that u^k=ũ^k and Γ_A(u^k,λ^k)= Γ_A(ũ^k) for all k∈ℕ∪{0}. For k=0 it is clear from the definition of the active sets and the choice of λ^0 made in the assumption of this theorem. Suppose now u^k=ũ^k and Γ_A(u^k,λ^k)= Γ_A(ũ^k). It is clear that u^k+1=ũ^k+1. On the other hand, from (<ref>) and (<ref>) and the choice c=ν we have that c u^k+1+λ^k+1= - S^*S(u^k+1-y_Ω), and therefore Γ_A(u^k+1,λ^k+1)= Γ_A(ũ^k+1). The result, hence, follows from Theorem <ref>. We want to remark here that writting the Newton step as in (<ref>), (<ref>), (<ref>) and (<ref>) we obtain exactly the same kind of algorithm as the one described in <cit.> §.§ Finite dimensional approximation Let us focus now on the finite dimensional approximation. For every 0<h<h_0, there exist a unique solution u̅_h∈ U_h of (P_h^C) and unique, y̅_h∈ Y_h and φ̅_h∈ Y_h0 such that a(y̅_h, z_h) = 0 z_h∈ Y_h0, y̅_h ≡ u̅_hΓ, a( z_h,φ̅_h) = (y̅_h-y_Ω,z_h)_Ωz_h∈ Y_h0, (-∂_n^hφ̅_h+νu̅_h,u_h-u̅_h) ≥ 0u_h∈ U^h_α,β. This way of writting the optimality system is useful to obtain error estimates (cf. <cit.>). Nevertheless, we cannot deduce a pointwise projection formula for the optimal control from the variational inequality. Think of this naive example. Let Γ=[-1,1] and take a mesh with nodes {-1,0,1}. For every u_h,v_h∈ U_h, we have that (u_h,v_h)_Γ = u^T Bv, where B is the mass matrix B=1/6([ 2 1 0; 1 4 1; 0 1 2 ]). Consider α = -∞, β = 0, and take , e.g., w =(-2,1,1)^T. Then we have that (-w_h+u̅_h,u_h-u̅_h)≥ 0 ∀ u_h∈ U^h_α,βu̅_h = -1.5e_1 + 0 e_2 + 0 e_3u = (-1.5,0,0)^T, but min(w,0)= (-2,0,0)^T, and we get different results with the L^2(Γ) projection and the pointwise projection. To circumvent this difficulty, we rewrite conditions (<ref>)–(<ref>) in order to use standard semismooth Newton method. Taking into account (<ref>), we can write (P^C_h) as [ min1/2u^T Au -f^Tu; α≤u≤β, ] where A and f are defined in (<ref>) and (<ref>) and α,β∈ℝ^N_B× 1 are the vectors whose j-component are respectively α(x_j) and β(x_j) and the inequalities are understood componentwise. In order to simplify the notation, we will restrict ourselves again to the case without lower bound. The optimality system can thus be written in the following form: if u is the solution of problem (<ref>), then there exists a unique λ∈ℝ^N_B× 1 such that [ Au+λ -f = 0,; λ≥ 0, u≤β, λ^T(u-β) = 0. ] Notice that although we may name λ̅_h(x) = ∑_j∈Bλ_j e_j(x) for x∈Γ to obtain a Lagrange multiplier λ̅_h∈ U_h, the complementarity condition with respect to the dot product in ℝ^N_B, does not imply that (λ̅_h,u̅_h-β)_Γ =0, which shows again the convenience of using (<ref>) instead of (<ref>)–(<ref>). To continue, we rewrite again the second condition in (<ref>) to obtain [ Au+λ -f = 0,; λ-max(0,λ+c(u-β)) = 0. ] Notice that (<ref>) looks like a discrete version of the optimality system formed by (<ref>), (<ref>) and (<ref>). Acting in an analogous way as we did for the continuous problem, we define F_h,c(u,λ) = ([ Au+λ -f; λ-max(0,λ+c(u-β)) ]). and for every pair (u,λ), we define the sets of active and free indexes as A(u,λ) = {j:λ_j+c(u_j-β(x_j))> 0},F(u,λ) = {j:λ_j+c(u_j-β(x_j))≤ 0}. Abusing notation and when this does not lead to confusion, we will often just write A and F. Notice that A∪F=B for every possible pair control-multiplier. The function F_h,c is slantly differentiable. A slanting function for F_h,c is M_h,c(u,λ) = ( [ A I_B,B; -c I_B,AI_A,B I_B,FI_F,B ]). Finally, for all ν≥ 0, the inverse of M_h,c(u,λ) is uniformly bounded w.r.t (u,λ). In finite dimension, the function max{0,·} is semismooth due to Rademacher's theorem (see e.g. <cit.> or <cit.>). A forward computation shows that M_h,c is a slanting function. Let us prove the uniform boundedness of the inverse. Given z∈ℝ^N_B and η∈ℝ^N_B, we have that M_h,c(u,λ) [[ v; μ ]] = [[ z; η ]]{[ Av+μ = z,; -cv_A = η_A,; μ_F = η_F. ]. We write the last equalities as Av+ [ [ μ_A; 0_F ]] = z- [ [ 0_A; η_F ]] =:ζ, -cv_A = η_A. This is the optimality system of the equality constrained optimization problem v = min_w∈ℝ^N_B1/2w^TAw - ζ^Tw, subject to w_A = -η_A/c. Writing w = I_ℬ,ℱw_F + I_ℬ,𝒜w_A and taking into account the equality constraint, we have that v_F is the solution of the following unconstrained optimization problem v_F = min_w_F1/2w_F^T (I_F,BAI_B,F) w_F - (I_F,B( ζ + AI_B,Aη_A/c ))^T w_F and therefore v_F is the solution of the following linear system I_F,BAI_B,Fv_F = I_F,B( ζ +AI_B,Aη_A /c) Since A is symmetric and positive definite, so is I_F,BAI_B,F, and its smallest eigenvalue is bounded from below by 0<λ_1(M)+νλ_1(B_B,B); see (<ref>). Therefore, the previous system is solvable and there exists a constant C>0, that may depend on h, ν and c, but is independent of u, such that v_F≤ C (z+η). From this it is straight to deduce that v+ μ≤ C (z+η) and hence M_h,c has a uniformly bounded inverse. With these considerations, given a current iterate (u,λ) with active and free index sets A=A(u,λ) and F=F(u,λ), we can compute the next step of Newton's method (u^+,λ^+) solving [ u^+ = min_u∈ℝ^N_B× 11/2u^T Au -f^Tu; xxxxu_A= β_A.; λ^+ = f -Au^+. ] At each iteration, this is equivalent to solving the following unconstrained optimization problem in the lower-dimensional space ℝ^N_F× 1: [ u^+_F = min_u_F∈ℝ^N_F× 11/2u_F^T (I_F,BAI_B,F) u_F - (I_F,B( f -AI_B,Aβ_A ))^T u_F; u^+_A= β_A; λ^+ = f -Au^+ ] Again the preconditioned gradient method works fine to solve this problem. A good preconditioner in practice is P = M_F,F+νB_F,F. Alternatively, taking into account the definition of A and f (see also Algorithms <ref> and <ref> respectively), we may write one step of the semismooth Newton algorithm as [ [[ M+νB -K_:,I I_:,A; -K_I,: O_I,I O_I,A; I_A,: O_A,I O_A,A ]] [[ y^+; φ^+_I; λ^+_A ]] = [[ My_Ω; 0_I; I_A,Bβ ]]; u^+ = y^+_B; λ^+_F = 0_F ] As for the unconstrained problem, direct methods for small size problems or preconditioned conjugate gradient techniques described in <cit.> can be applied to solve this system at each Newton step. Notice that the only information from iteration k used to compute iteration k+1 is the set of active indexes. Therefore, when A(u_k+1,λ_k+1)=A(u_k,λ_k) we have reached an stationary point. This is usually the criterion used to stop the semismooth Newton method. Another consequence of this is that to initiate the algorithm, in principle we do not not need an initial guess u^0 and λ^0, but only an initial guess for the active set. We include nevertheless an initial guess for the control variable in Algorithm <ref> because we use u_k as the initial guess for the pcg to obtain u_k+1. The following convergence result is a direct consequence of Lemma <ref> and <cit.>; see also <cit.> or <cit.>. The sequence u_k generated by Algorithm <ref> converges q-superlinearly to u, the solution of (<ref>). We resume the 2D problem described in Example <ref> and the 3D problem described in Example <ref>, with the upper constraint β≡0.16. We test Algorithm <ref>. Following the tip of Theorem <ref>, we have taken the parameter c=ν and λ_0=f-Au_0. Nevertheless, we have not been able to observe any problem for different values of c. The seed is set to u_0=0. To solve the optimality system of the unconstrained optimization problem at each Newton iterate, we have used the preconditioned conjugate gradient method for (<ref>) with initial guess I_F,Bu_k. The use of the reduced problem is even more advisable in this case than it was in Examples <ref> and <ref>, because the size of the system (<ref>) is N+N_I+N_A, which in any case is greater or equal than the size of the system (<ref>), which is N+N_I. In Tables <ref> and <ref> and we report on the number of Newton iterations for each mesh size as well as the total number of conjugate gradient iterations. For reference, we also report on the number of active nodes and the optimal solution of the discrete problem being approximated. Choosing the initial guess for Algorithm <ref>. If possible, it is a good idea to select an initial point for the Newton method close to the solution. If we are dealing with the problem for some mesh size h, a good candidate for the initial iteration is I_hu̅_h_-1, the solution in a coarser mesh with h_-1≥ h. This idea can be iterated with a mesh family with parameters h_-M,…,h_0=h. The computation time should decrease for fine meshes provided that the interpolation can be carried out in an effective way. For instance, using nested meshes. Solving the last 2D problem in Table <ref> takes a cputime of 620 seconds. Using a nested iteration, we solve the problem with 5120 boundary nodes in 436 seconds, with just 2 Newton iterates and 7 conjugate gradient iterates at the finest level. The times include mesh generations and matrix assembly. To generate the meshes and interpolate the solution, we have used Matlab PDEToolbox command . In the 3D case (see Table <ref>), solving the problem for h=2^-6 takes 681 seconds. Using a nested mesh strategy this time is reduced to 404 seconds with 2 Newton iterates and 9 pcg iterations at the finest level. In our example we have been able to make an efficient interpolation of the solution at the previous level just using Matlab's . Absence of Tikhonov parameter In the same 2D pentagonal domain, we take again β≡ 0.16, but now α≡ -1.2, ν = 0, y_Ω≡ 1 if x_1>0.25, and y_Ω≡ -1 if x_1<0.25. We are able to solve the finite dimensional problem, but up to our best knowledge, there is no proof available about the convergence of the discrete optimal solutions to the continuous optimal solution. To solve the problem we follow a nested mesh strategy. The number of pcg iterations per Newton iterate grow as h tends to zero, as is to be expected from Corollary <ref>; see Table <ref>. The constraints have been chosen in such way that we seemingly find both a bang-bang part and a singular arc. We have sketched a plot of the discrete optimal control for h=0.2× 2^-8 in Figure <ref>. The boundary has been stretched on a 1D line and its corners have been marked with circles on the graph of u̅_h. § STATE CONSTRAINTS In the rest of the work, we will suppose that d=2, Ω is convex and Γ is polygonal. According to <cit.>, if problem (P^S) admits a feasible Slater point, then it has a unique solution u̅∈ H^1/2(Γ) and there exist y̅∈ H^1(Ω)∩ C(ω̅), φ̅∈ W^1,t_0(Ω) for all t<2 and two nonnegative measures μ̅^+, μ̅^-∈ℳ(ω̅) such that -Δy̅ = 0Ω, y̅ = u̅Γ, -Δφ̅= y̅-y_Ω+μ̅^+-μ̅^-Ω, φ̅= 0Γ, u̅(x) = 1/ν∂_nφ̅(x)Γ, ⟨μ̅^+-μ̅^-,y-y̅⟩≤ 0 ∀ y∈ K_a,b, and μ̅^+⊂{y̅ = b}, μ̅^-⊂{y̅ = a}. In this case the adjoint state equation (<ref>) must be understood in the transposition sense, <cit.>, and ⟨·,·⟩ denotes the duality product between ℳ(ω̅) and C(ω̅). As is pointed out in <cit.>, or <cit.>, the PDAS or semismooth Newton methods described in the previous section are not applicable to this problem, since the multiplier is a measure. In <cit.> a Moreau-Yosida regularization is proposed. This strategy is further investigated in <cit.>. Let us briefly describe this method. Again, to simplify the notation, we will do the exposition for the unilateral constraint y≤ b in ω̅. Given a shift function μ^*∈ L^q(ω) for some q>2 and a parameter γ>0, we will solve the unconstrained problem (Q^γ) min_u∈ L^2(Γ) J(u) +1/2γ∫_ω̅max{0,μ^*+γ(Su-b)}^2 dx. This problem has a unique solution u^γ∈ H^1/2(Γ) for every γ >0. Since the functional is of class C^1, first order optimality conditions can be derived directly from the work <cit.>: there exist unique y^γ∈ H^1(Ω) and φ^γ∈ H^1_0(Ω) such that -Δ y^γ = 0Ω, y^γ = u^γΓ, -Δφ^γ = y^γ-y_Ω+χ_ω̅max{0,μ^*+γ(y^γ-b)}Ω, φ^γ = 0Γ, ν u^γ = ∂_nφ^γΓ. The semismooth Newton method to solve (<ref>), (<ref>), (<ref>) converges locally q-superlinearly. We may write the system (<ref>), (<ref>), (<ref>) as the equation G(u)=0, where G:L^2(Γ)→ L^2(Γ) is given by G(u) = ν u+S^*S u-S^*y_Ω +S^*( χ_ω̅max{0,μ^*+γ(Su-b)}). Using the regularity results in <cit.>, together with the election of μ^* in L^q(ω) we have that μ^*+γ(Su-b)∈ L^q(Ω) for some q>2, and hence G is semismooth in the sense stated in <cit.>; see Theorems 2.13 and 2.10(c) in the aforementioned reference. Define now ω_A(u)={x∈ω̅: μ^*+γ(Su-b) > 0}. A slant differential of G(u) is given by the expression M(u) v = ν v + S^*Sv +γ S^*χ_ω_A(u) Sv. Let us see that the inverse of M(u) is uniformly bounded in ℒ(L^2(Γ),L^2(Γ)) for all u∈ L^2(Γ). For any z∈ L^2(Γ), the equation M(u)v=z can be read as -Δ y = 0Ω, y=vΓ, -Δφ = y+γχ_ω_A(u) yΩ, φ=0Γ, ν v = ∂_nφ + zΓ. Using the definition of solution in the transposition sense and the last equation of this system, we have that (y,y+γχ_ω_A(u) y)_Ω = -(v,∂_nφ) = -ν(v,v)_Γ+ν(v,z)_Γ and hence νv_L^2(Γ)^2 = ν(v,z)_Γ-∫_Ω y^2 -γ∫_ω_A(u)y^2≤νv_L^2(Γ)z_L^2(Γ). So we have that v_L^2(Γ)≤z_L^2(Γ) and therefore M(u) is uniformly bounded in ℒ(L^2(Γ),L^2(Γ)) for all u∈ L^2(Γ). When γ→+∞, u^γ→u̅; see <cit.> or <cit.>. Let us turn now to the discrete problem (P^S_h). Consider the space ℳ_h⊂ℳ(ω̅) which is spanned by the Dirac measures corresponding to the nodes {x_j}_j∈J, where J={j x_j∈ω̅}. Following <cit.>, if the continuous problem (P^S) has a regular feasible Slater point, then (P^S_h) has a unique solution u̅_h. Moreover there exist unique y̅_h∈ K_a,b^h, φ̅_h∈ Y_h,0 and two nonnegative measures μ̅_h^+, μ̅_h^-∈ℳ_h such that a( y̅_h,z_h) =0 ∀ z_h∈ Y_h0, y̅_h = u̅_hΓ, a( z_h,φ̅_h) =(y̅_h-y_Ω,z_h)+ ⟨μ̅_h^+-μ̅_h^-,z_h⟩ ∀ z_h∈ Y_h0, ⟨μ̅_h^+-μ̅_h^-,y_h-y̅_h⟩ ≤ 0 ∀ y_h∈ K_a,b^h, νu̅_h =∂_n^hφ̅_hΓ, and μ̅_h^+⊂{x_j: y̅(x_j)=b(x_j)}, μ̅_h^-⊂{x_j: y̅(x_j)=a(x_j)}. Since we are dealing with a finite dimensional problem and the max function is known to be semismooth in finite dimension, we could think about applying directly a semismooth Newton method as described in <cit.>. Nevertheless, the other assumption fails here: the slant derivative may not have an inverse. Let us show this. Consider F(u,μ) = ([ Au-f+S^Tμ; μ -max{ 0,μ+γ(Su-b) ]). It is clear that the optimality system (<ref>)–(<ref>) is equivalent to the equation F(u,μ)=0. We define the sets of active and free nodes related to a pair control-multiplier as A(u_h,μ) = {j∈J:μ_j+γ(S_h(u_h)(x_j)-b_j) >0}, F = J∖A. The slant derivative of F(u,μ) is given by M(u,μ) = ([ A S^T; -γI_J,AI_A,:S I_J,FI_F,J ]). The inverse of M(u,μ) may not exist for some (u,μ). For any (z,δ)∈ℝ^N_B×ℝ^N_J, we have that M(u,μ)([ v; η ]) = ([ z; δ ]) {[ Av + S^Tη = z,; -γI_J,AI_A,:Sv + I_J,FI_F,Jη = δ. ]. From the second equation we have that γI_A,:Sv = - I_A,Jδ. If N_A > N_B we have a linear system with more equations than variables, and hence it will not be consistent for at least one value of δ and M will not have an inverse. Instead, we use a Moreau-Yosida penalization of (P^S_h). Again we will write only the case of unilateral upper constraint. For some shift function μ^*∈ Y_h such that μ^*(x_j)=0 if j∉J, and a parameter γ>0, a direct discretization of (Q^γ) could be (Q̃^γ_h)min_u_h∈ U_hJ_h(u_h)+1/2γ∫_ω̅max{0,μ^*+γ(S_hu_h-b)}^2dx. Problem (Q̃^γ_h) has a unique solution u^γ_h∈ U_h that converges to u^γ as h→ 0. Since the functional is not of class C^2, the problem does not fit exactly into the framework of <cit.>, so we will give a sketch of the proof of this fact. Problem (Q̃^γ_h) has a unique solution u^γ_h∈ U_h. Moreover, there exist unique y_h^γ∈ Y_h and φ_h^γ∈ Y_h0 such that a( y^γ_h,z_h) =0 ∀ z_h∈ Y_h0, y^γ_h = u^γ_hΓ, a(z_h,φ^γ_h) =( y^γ_h-y_Ω+ max{0,μ^*+γ(y_h^γ-b)},z_h)_Ω ∀ z_h∈ Y_h0, ν u^γ_h =∂_n^hφ^γ_hΓ. Finally, u_h^γ→ u^γ in L^2(Γ) and there exists some λ>0 such that u^γ-u^γ_h_L^2(Γ)≤ C h^λ. Existence and uniqueness of solution follows immediately from the coerciveness of the discrete penalized functional. Since this discrete functional is also of class C^1, first order optimality conditions are standard. Consider two controls u_1,u_2∈ L^2(Γ) and its related states y_i and adjoint states φ_i w.r.t. problem (Q^γ). Using the definition of solution in the transposition sense for the states and both the state and adjoint state equations (<ref>) and (<ref>) we have (shorting z_i = μ^*+γ(y_i-b), i=1,2) (-∂_nφ_1 + ∂_nφ_2,u_1-u_2)_Γ = (y_1-y_2,-Δφ_1+Δφ_2)_Ω = (y_1-y_2, y_1+χ_ωmax{0,z_1}-y_2-χ_ωmax{0,z_2})_Ω = y_1-y_2^2_L^2(Ω)+1/γ(z_1 - z_2,max{0,z_1}-max{0,z_2})_ω ≥ 0. The rest is quite standard: νu^γ- u^γ_h_L^2(Γ)^2 ≤ (-∂_nφ^γ+ν u^γ+∂_nφ_u_h^γ- ν u_h^γ, u^γ-u_h^γ)_Γ = (-∂_nφ^γ+∂_n^hφ_h^γ+ν u^γ-ν u_h^γ, u^γ- u_h^γ)_Γ +(-∂_n^hφ_h^γ+∂_nφ_u_h^γ,u^γ- u_h^γ)_Γ. For fixed γ, u_h^γ is uniformly bounded in L^2(Γ) because u_h^γ_L^2(Γ)^2≤ 1/νy_Ω_L^2(Ω)^2 + 1/γ/νmax{0,μ^*-γ b}_L^2(Ω)^2, and therefore using <cit.> we have that -∂_n^hφ_h^γ+∂_nφ_u_h^γ≤ C h^λ for some λ>0. For the first addend we obtain directly 0 using first order optimality conditions (<ref>) and (<ref>). A practical way of computing the integral in the penalty term is to use the lumped mass matrix L∈ℝ^N× N. Since we are going to use it only for integrals in ω̅, we define it as L_i,j = 0i≠ j, L_j,j = 0j∉J, L_j,j= ∑_k=1^N M_k,j.j∈J. Therefore, we will be solving (Q^γ_h)min_u_h∈ U_hJ_h(u_h)+1/2γ∑_j∈JL_j,jmax{0,μ^*(x_j)+γ(S_hu_h(x_j)-b(x_j))}^2. First order optimality conditions for this problem read like a( y^γ_h,z_h) =0 ∀ z_h∈ Y_h0, y^γ_h = u^γ_hΓ, a( z_h,φ^γ_h) =( y^γ_h-y_Ω,z_h)_Ω +∑_j∈JL_j,jmax{0,μ^*(x_j)+γ(y_h^γ(x_j)-b(x_j))}z_h(x_j) ∀ z_h∈ Y_h0, ν u^γ_h =∂_n^hφ^γ_hΓ. This nonlinear system can be solved using a semismooth Newton method. To this end, we will use the nonlinear operator G_h:ℝ^N_B→ℝ^N_B defined by G_h(u) = Au-f+ S^TLmax{0,μ^*+γ(Su-b)}. Thus, solving (<ref>)–(<ref>) is equivalent to solving G_h(u)=0. For every u_h∈ U_h we define sets of active and free nodes as A_ω(u_h,γ,h) = {j∈J: μ^*_j+γ(S_hu_h(x_j)-b_j) > 0}, F_ω(u_h,γ,h) =I∖A_ω(u_h,γ,h). Abusing notation we will drop some or all of the arguments or will use the vector notation when this does not lead to confusion, e.g. A_ω, A_ω(u). Notice that if μ^*≡ 0, then A_ω(u_h,γ,h) is independent of γ. For the sake of notation, it is also convenient to define for every set A_ω⊆J the diagonal matrix H(A_ω)∈ℝ^N× N such that H_i,j=δ_i,j{[ 0 j∉A_ω; L_j,j j∈A_ω. ]. Abusing notation, we will often write H(u)=H(A_ω(u)) or even we will write H when this does not lead to confusion. The proof of the following result is as the corresponding one in infinite dimension. G_h(u) is slantly differentiable, a slant differential is given by M(u)v = Av+γS^TH(u) Sv and it has a uniformly bounded inverse w.r.t. u. Notice that using this H notation, we can write G_h(u) = Au-f+ S^TH(u)(μ^*+γ(Su-b)), and therefore, for a given u, and denoting H=H(u), one Newton step reads like Au^+ +γS^THSu^+ = f+ S^TH(γb-μ^*). Let us comment that for the computation of w = S^THy for some y∈ℝ^N, first we solve K_I,Iϕ_s = H_I,:y, and, next, we have w = H_B,:y-K_B,Iϕ_s = -K_B,Iϕ_s because H is diagonal and its nonzero components correspond to nodes that lie in ω̅, and are hence interior to Ω. Again a preconditioned conjugate gradient method can be used to solve this system, provided an efficient way of computing d=(A + γS^THS) v and the second member of the system. Notice that in each of the algorithms <ref> and <ref> the computation of ϕ can be done with just one system solve. Alternatively, the solution to (<ref>) can be obtained solving [ [[ M+νB+γH -K_:,I; -K_I,: O_I,I ]] [[ y^+; φ_I^+ ]] = [[ My_Ω+H(γb-μ^*)); 0 ]]; u^+ = y^+_B. ] Finally, the semismooth Newton algorithm to solve our problem reads as: As we noticed in Remark <ref>, the only information from iteration k used to compute iteration k+1 is the set of active indexes. Therefore, when A_ω(u_k+1)=A_ω(u_k) we have reached an stationary point. This is usually the criterion used to stop the semismooth Newton method. Nevertheless, as we will see below in the context of algorithms <ref> and <ref>, the main use for the solution of (Q_h^γ) will be to provide an initial guess for the next step in those procedures, so it does not seem necessary to solve exactly (Q_h^γ) in general. Following <cit.>, we may implement the following stopping criterion for Algorithm <ref>. After step <ref>, we compute an approximation of the multiplier μ_k+1∈ℝ^N as μ_k+1,j = 0j∉A_ω,k, μ_k+1,j = μ^*_j+γ(y_k+1,j-b_j)j∈A_ω,k and we may stop the algorithm if μ_k+1-max(μ^*+γ(y_k+1-b),0_L^2(Ω)< ε_λ. In practice, this quantity gives a good measure of the change in the active set between iterates as well as the unfeasibility combined with the penalization parameter γ. We state the convergence result for the Newton method. It follows directly from Theorem <ref> The semismooth Newton method to solve (Q^γ_h) described in Algorithm <ref> converges locally q-superlinearly. In all the examples below we have taken the shift μ^*=0. In general it is not a good idea to solve directly (Q_h^γ) for some γ big enough. The resulting intermediate problems are usually very ill conditioned. We resume the 2D problem taken from <cit.> and described in Example <ref>. As in <cit.>, we take ω the ball centered at (-0.1,-0.1) with radius 0.2 and b≡ 0.15. We use Algorithm <ref> taking u_0=0. The results are summarized in Table <ref>. We measure the unfeasibility of the state as the maximum constraint violation mcv(y_h) = max(y_h-b,0)_L^∞(ω̅). The first thing that can be observed is that γ cannot be too big w.r.t. the problem size. Although the algorithm should converge in finite time, a value too big for γ will make the active sets of the intermediate steps fluctuate, and in practice it may not stop. This is what happens in this case with meshes coarser than the first one exposed in the table. On the other hand, it can also be noticed that the amount of Newton iterations looks mesh independent for h small enough. This was to be expected because the infinite dimensional version of the method is convergent (cf. Theorem <ref>). Nevertheless, the computational effort can be better measured by the total number of conjugate gradient iterations made when we solve (<ref>) in step <ref> of Algorithm <ref>. Solving the solution in the finest mesh takes 8246 seconds. In <cit.> a continuation strategy is proposed. This reduces considerably the computational effort. We will use subscripts for the iteration number of the semismooth Newton method described in Algorithm <ref> and superscripts for the numbering of the iterates of the continuation strategy provided in Algorithm <ref>. For some sequence {τ^n}_n≥ 1 such that τ^n>1 for all n≥ 1, we have There are still three important details to be explained about Algorithm <ref>: first and most important, the choice of the initial guess u^0; next, the appropriate values for τ^n; and finally, a suitable stopping criterion. To obtain a good initial guess, we will solve the problem in a coarser mesh with a smaller value of γ. Since we are going to deal with meshes of different sizes, we will use the notation A_ω(u_h,γ,h) when needed. We also recall that I_h is the pointwise interpolation operator from C(Γ̅) onto U_h. Given (h,γ^0), we fix M>0 and pick a finite sequence (h_n,γ_n)_n=0^M such that h_n is decreasing, h_M = h, γ_n is increasing and γ_M≤γ^0. Although Algorithm <ref> is meaningful for non-nested meshes, the computational effort needed to make the interpolation in step <ref> in non-nested meshes can be considerable, specially for 3D problems. In <cit.> some criteria are given to choose τ^n+1>1. In practice, if τ^n+1 is very small the algorithm would not advance; if it is very big, we would lose the advantage given by the continuation strategy. In <cit.> the authors stop the continuation strategy if residuals related to the state, the adjoint state and the multiplier are smaller than a certain tolerance of order O(h^2). In our case, since the problem is linear quadratic, the residuals related to the state and the adjoint state are zero (at least up to roundoff error). The residual related to the multiplier can be computed as r_d = ∑_j∈A_ω^n+1L_j,j (y^n+1_j-b_j). As an alternative, a tolerance for the maximum constraint violation e_∞ can be used stop if mcv(y_h^γ_n)≤ e_∞. With all these considerations, we propose the following algorithm. Fix a mesh sequence such that {h_j}_j=0^M is decreasing, γ_0>0, C>0, u_h_0∈ U_h_0, n_max∈ℕ and a sequence {τ^n}_n=1^n_max+1 such that τ^n>1 for all n. Now we apply Algorithm <ref> for γ_0=1, τ^n=10 and a family of 9 nested meshes of sizes h_j=0.2*2^-j, j=0:8, the coarsest for h_0 with 52 nodes and the finest for h_8 with 2689537 nodes. A summary of the results can be found in Table <ref>. The total computation time was 1570 seconds, which is a significant improvement compared with the 8246 seconds used by Algorithm <ref> (see Example <ref>). Determination of (the support of) the Lagrange multipliers of (P^S). Comparison of the adjoint state equations for (P_h^S) and (Q_h^γ) leads to the approximation formula μ̅_h^+ ≈∑_j∈A_ω(u_h^γ)L_j,j (μ^*_j +γ(y^γ_h(x_j)-b_j))δ_x_j. Since ω̅⊂Ω, it is very common that the Lagrange multipliers of the original problem are finite sums of Dirac measures centered at points on the boundary of ω, say μ̅^+ = ∑_k=1^n μ_kδ_X_k for some n>0 and X_k∈∂ω. Increasing γ will have as a result a better approximation of the Lagrange multiplier and its support. Notice, nevertheless, this will done with great effort and the approximation of both the control, the state and the functional optimal value will not improve in a significant way. In Example 5.7 the number of active nodes for the last mesh is 748. All of them are on ∂ω but they are not isolated. We can see in Table <ref> how the number of active nodes decreases as γ increases, with little variation of the functional or the state. § CONTROL AND STATE CONSTRAINTS Again according to <cit.>, if problem (P^CS) admits a feasible Slater point, then it has a unique solution u̅∈ H^1/2(Γ) and there exist y̅∈ H^1(Ω)∩ C(ω̅), φ̅∈ W^1,t_0(Ω) for all t<2 and two nonnegative measures μ̅^+, μ̅^-∈ℳ(ω̅) such that -Δy̅ = 0Ω, y̅ = u̅Γ, -Δφ̅= y̅-y_Ω+μ̅^+-μ̅^-Ω, φ̅= 0Γ, u̅(x) = min{β(x),max{α(x),1/ν∂_nφ̅(x)}}Γ, ⟨μ̅^+-μ̅^-,y-y̅⟩≤ 0 ∀ y∈ K_a,b and μ̅^+⊂{y̅ = b}, μ̅^-⊂{y̅ = a}. As we said in the previous section, a semismooth Newton strategy for this problem is meaningless, so instead we are going to deal with a Moreau-Yosida approximation. As we did in the previous sections, we will consider only unilateral constraints u≤β on Γ and y≤ b in ω̅ to simplify the notation. For a shift function μ^*∈ L^q(ω) for some q>2 and a parameter γ>0, we consider the problem (Q^C,γ) min_u∈ U_-∞,βJ(u)+1/2γ∫_ω̅max{0,μ^*+γ(Su-b)}^2dx. This problem has a unique solution u^γ∈ H^1/2(Γ). Moreover, there exist y^γ∈ H^1(Ω), φ^γ∈ H^s(Ω), s>3/2, such that -Δ y^γ = 0Ω, y^γ = u^γΓ, -Δφ^γ = y^γ-y_Ω+max{0,μ^*+γ(y^γ-b)}Ω, φ^γ = 0Γ, (-∂_nφ^γ+ν u^γ,u-u^γ) ≥ 0u∈ U_-∞,β. Define G^γ(u) = ν u-min{νβ,-S^*Su+S^*y_Ω-S^*χ_ω̅max{0,μ^*+γ(Su-b)}}. It is clear that u^γ is the unique solution of (Q^c,γ) if and only if G^γ(u^γ)=0. For some fixed shift function μ^*∈ L^q(ω), q>2, consider the active sets Γ_A(u,γ) = {x∈Γ: -S^*Su+S^*y_Ω-S^*χ_ωmax{0,μ^*+γ(Su-b)>νβ} and ω_A(u,γ) = {x∈ω̅: μ^*+γ(Su-b)>0}. A slant differential of G^γ(u) is given by M^γ(u)v = ν v+χ_Γ_A(u,γ)S^*(1+ γχ_ω_A(u,γ))Sv. G^γ is slantly differentiable, M^γ(u) is a slant differential of G^γ and for every fixed ν>0, M^γ(u) has an inverse in ℒ(L^2(Γ),L^2(Γ)) uniformly bounded for all u∈ L^2(Γ). The semismooth Newton method M^γ(u^+-u) = -G^γ(u) converges q-superlinearly. The proof follows the same lines as those of Lemma <ref> and Theorems <ref> and <ref> As we did for the pure control-constrained case, to deal with a problem better suited to the finite dimensional case, we write the optimality condition (<ref>) with the help of a Lagrange multiplier. There exists λ^γ∈ L^2(Γ) such that, for any c>0, ν u^γ = ∂_nφ^γ-λ^γΓ, λ^γ= max{0,λ^γ+c(u^γ-β)}Γ. Define now F_c^γ(u,λ) = ( [ ν u + S^*Su-S^*y_Ω + S^*χ_ω̅max{0,μ^*+γ(Su-b)}+λ; λ - max{0,λ + cu -cβ} ]) We have that ([ ν I+S^*(1+ γχ_ω_A(u,γ))S I; cχ_Γ_A(u,λ)I χ_Γ_F(u,λ)I ])∈∂^CLF_c^γ(u,λ) and hence a Newton-like method to solve F_c^γ(u,λ)=0 is given by ν u^+ = -S^*(1+ γχ_ω_A(u,γ))Su^+-y_Ω) -λ^+, u^+ = βΓ_A(u,λ), λ^+ = 0Γ_F(u,λ). Although F_c^γ is known not to be slantly differentiable, it can be proved as in Theorem <ref> that the sequence generated by the above described Newton-like method to solve F_c^γ(u,λ)=0 is the same than the one generated by the semismooth Newton method to solve G^γ(u)=0 provided we take the same initial guess u_0, c=ν and λ_0=S^*(y_Ω-(1+ γχ_ω_A(u_0,γ))Su_0)-ν u_0. Let us turn now to the finite dimensional problem. We can write the approximation of problem (Q^C,γ_h) as a constrained optimization problem in ℝ^N_B,1: (Q^C,γ_h){[ min1/2u^TAu-f^Tu+ 1/2γmax{0,μ^*+γ(Su-b)}^TLmax{0,μ^*+γ(Su-b)}; u≤β. ]. Existence and uniqueness of solution of this problem, as well as error estimates for the difference u^γ-u^γ_h_L^2(Γ) can be proved as we did for (Q̃^γ) in Lemma <ref>. Since we have control constraints, instead of the point-wise interpolation, the Casas-Raymond interpolate <cit.> should be used in the last step. First order optimality conditions read as [ Au+S^TLmax{0,μ^*+γ(Su-b)}+λ = f,; λ-max(0,λ+c(u-β)) = 0. ] Using the definitions (<ref>) and (<ref>) for the active sets of indexes A(u,λ) and A_ω(u,γ,h) and the matrix H related to A_ω, we have that one step of Newton's method can be written as [ u^+_F = min_u_F1/2u_F^T (I_F,B(A + γS^THS)I_B,F) u_F; ; xxxxxx - ( I_F,B ( f - AI_B,Aβ_A - S^TH ( μ^* +γ (SI_B,Aβ_A -b) ) ) )^T u_F; u^+_A = β_A; λ^+ = f - Au^+ - S^TH ( μ^* +γ (Su^+ - b)) ] or, alternatively, as [ [[ M+νB+γH -K_:,I I_:,A; -K_I,: O_I,I O_I,A; I_B,: O_B,I O_B,A; ]] [[ y^+; φ_I^+; λ^+ ]] = [[ My_Ω+H(γb-μ^*); 0; I_A,Bβ ]]; u^+ = y^+_B; λ^+_F = 0. ] An adaptation of algorithms <ref> and <ref> to solve (Q^C,γ_h) is straightforward, and so is an adaptation of Algorithm <ref> to use a continuation strategy together with a nested mesh strategy. We repeat Example <ref> adding the control constraint u≤ 0.16. We obtain the results summarized in Table <ref>. The total computation time was 2894 seconds. Although this may seem a lot of time, we have to remember that we are solving an optimal control problem with pointwise constraints in both the control and the state using a mesh with almost 2.7E+06 nodes and 5.4E+06 elements to test the algorithm. § FUNDING The author was partially supported by the Spanish Ministerio de Ciencia e Innovación under Project MTM2014-57531-P. 10 Casas-Raymond2006 Casas E, Raymond JP. Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J Control Optim. 2006;45(5):1586–1611. <http://dx.doi.org/10.1137/050626600>. Deckelnick-Gunther-Hinze2009 Deckelnick K, Günther A, Hinze M. Finite element approximation of Dirichlet boundary control for elliptic PDEs on two- and three-dimensional curved domains. SIAM J Control Optim. 2009;48(4):2798–2819. <http://dx.doi.org/10.1137/080735369>. Gunther2010 Günther A. Structure exploiting Galerkin schemes for optimal control of PDEs with constraints on the involved variables [dissertation]. Universität Hamburg; 2010. <http://ediss.sub.uni-hamburg.de/volltexte/2010/4926/pdf/PhDThesis_Guenther.pdf>. MayRannacherVexler2013 May S, Rannacher R, Vexler B. Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems. SIAM J Control Optim. 2013;51(3):2585–2611. <http://dx.doi.org/10.1137/080735734>. Mateos-Neitzel2015 Mateos M, Neitzel I. Dirichlet control of elliptic state constrained problems. Comput Optim Appl. 2016;63(3):825–853. <http://dx.doi.org/10.1007/s10589-015-9784-y>. Casas-Sokolowski2010 Casas E, Sokolowski J. Approximation of boundary control problems on curved domains. SIAM J Control Optim. 2010;48(6):3746–3780. <http://dx.doi.org/10.1137/090761550>. Casas-Gunther-Mateos2011 Casas E, Günther A, Mateos M. A paradox in the approximation of Dirichlet control problems in curved domains. SIAM J Control Optim. 2011;49(5):1998–2007. <http://dx.doi.org/10.1137/100794882>. AMPR2015 Apel T, Mateos M, Pfefferer J, et al. On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains. SIAM J Control Optim. 2015;53(6):3620–3641. <http://dx.doi.org/10.1137/140994186>. Gong-Hinze-Zhou2016 Gong W, Hinze M, Zhou Z. Finite element method and a priori error estimates for Dirichlet boundary control problems governed by parabolic PDEs. J Sci Comput. 2016;66(3):941–967. <http://dx.doi.org/10.1007/s10915-015-0051-2>. Chen-Nashed-Qi2000 Chen X, Nashed Z, Qi L. Smoothing methods and semismooth methods for nondifferentiable operator equations. SIAM J Numer Anal. 2000;38(4):1200–1216 (electronic). <http://dx.doi.org/10.1137/S0036142999356719>. Bergounioux-Ito-Kunisch1999 Bergounioux M, Ito K, Kunisch K. Primal-dual strategy for constrained optimal control problems. SIAM J Control Optim. 1999;37(4):1176–1194 (electronic). <http://dx.doi.org/10.1137/S0363012997328609>. Bergounioux-Kunisch2002 Bergounioux Mt, Kunisch K. Primal-dual strategy for state-constrained optimal control problems. Comput Optim Appl. 2002;22(2):193–224. <http://dx.doi.org/10.1023/A:1015489608037>. Kunisch-Rosch2002 Kunisch K, Rösch A. Primal-dual active set strategy for a general class of constrained optimal control problems. SIAM J Optim. 2002;13(2):321–334. <http://dx.doi.org/10.1137/S1052623499358008>. Ulbrich2003 Ulbrich M. Semismooth Newton methods for operator equations in function spaces. SIAM J Optim. 2002;13(3):805–842 (2003). <http://dx.doi.org/10.1137/S1052623400371569>. Hintermuller-Ito-Kunisch2003 Hintermüller M, Ito K, Kunisch K. The primal-dual active set strategy as a semismooth Newton method. SIAM J Optim. 2003;13(3):865–888 (2003). <http://dx.doi.org/10.1137/S1052623401383558>. Hintermuller-Kunisch2006 Hintermüller M, Kunisch K. Feasible and noninterior path-following in constrained minimization with low multiplier regularity. SIAM J Control Optim. 2006;45(4):1198–1221. <http://dx.doi.org/10.1137/050637480>. Casas-Mateos-Raymond-2009 Casas E, Mateos M, Raymond JP. Penalization of Dirichlet optimal control problems. ESAIM Control Optim Calc Var. 2009;15(4):782–809. <http://dx.doi.org/10.1051/cocv:2008049>. Hinze-Vierling2012 Hinze M, Vierling M. The semi-smooth Newton method for variationally discretized control constrained elliptic optimal control problems; implementation, convergence and globalization. Optim Methods Softw. 2012;27(6):933–950. <http://dx.doi.org/10.1080/10556788.2012.676046>. Hinze2005 Hinze M. A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput Optim Appl. 2005;30(1):45–61. <http://dx.doi.org/10.1007/s10589-005-4559-5>. Hinze_Pinnau_Ulbrich_Ulbrich2009 Hinze M, Pinnau R, Ulbrich M, et al. Optimization with PDE constraints. (Mathematical Modelling: Theory and Applications; Vol. 23). New York: Springer; 2009. Gong-Yan2011 Gong W, Yan N. Mixed finite element method for Dirichlet boundary control problem governed by elliptic PDEs. SIAM J Control Optim. 2011;49(3):984–1014. <http://dx.doi.org/10.1137/100795632>. Casas-Mateos-Troeltz2005 Casas E, Mateos M, Tröltzsch F. Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput Optim Appl. 2005;31(2):193–219. <http://dx.doi.org/10.1007/s10589-005-2180-2>. Grisvard1992 Grisvard P. Singularities in boundary value problems. (Recherches en Mathématiques Appliquées [Research in Applied Mathematics]; Vol. 22). Masson, Paris; Springer-Verlag, Berlin; 1992. Jerison-Kenig1995 Jerison D, Kenig C. The inhomogeneous Dirichlet problem in Lipschitz domains. J Funct Anal. 1995;130(1):161–219. <http://dx.doi.org/10.1006/jfan.1995.1067>. Jerison-Kenig1981N Jerison DS, Kenig CE. The Neumann problem on Lipschitz domains. Bull Amer Math Soc (NS). 1981;4(2):203–207. <http://dx.doi.org/10.1090/S0273-0979-1981-14884-9>. Dahlberg-Kenig1987 Dahlberg BEJ, Kenig CE. Hardy spaces and the Neumann problem in L^p for Laplace's equation in Lipschitz domains. Ann of Math (2). 1987;125(3):437–465. <http://dx.doi.org/10.2307/1971407>. Berggren2004 Berggren M. Approximations of very weak solutions to boundary-value problems. SIAM J Numer Anal. 2004;42(2):860–877 (electronic). <http://dx.doi.org/10.1137/S0036142903382048>. AMPR2017 Apel T, Mateos M, Pfefferer J, et al. Error estimates for Dirichlet control problems in polygonal domains. 2017;In preparation. Schoberl-Zulehner2007 Schöberl J, Zulehner W. Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems. SIAM J Matrix Anal Appl. 2007;29(3):752–773 (electronic). <http://dx.doi.org/10.1137/060660977>. Herzog-Sachs2010 Herzog R, Sachs E. Preconditioned conjugate gradient method for optimal control problems with control and state constraints. SIAM J Matrix Anal Appl. 2010;31(5):2291–2317. <http://dx.doi.org/10.1137/090779127>. Schiela-Ulbrich2014 Schiela A, Ulbrich S. Operator preconditioning for a class of inequality constrained optimal control problems. SIAM J Optim. 2014;24(1):435–466. <http://dx.doi.org/10.1137/120877532>. ASW1996 Apel T, Sändig AM, Whiteman JR. Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains. Math Methods Appl Sci. 1996;19(1):63–85. Kamenski-Huang-Xu2014 Kamenski L, Huang W, Xu H. Conditioning of finite element equations with arbitrary anisotropic meshes. Math Comp. 2014;83(289):2187–2211. <http://dx.doi.org/10.1090/S0025-5718-2014-02822-6>. Dauge1989 Dauge M. Problèmes mixtes pour le laplacien dans des domaines polyédraux courbes. C R Acad Sci Paris Sér I Math. 1989;309(8):553–558. nevcas2012direct Nečas J. Direct methods in the theory of elliptic equations. Corrected 2nd ed. Springer Berlin Heidelberg; 2012. Monographs and Studies in Mathematics. Plaza2007 Plaza A. The eight-tetrahedra longest-edge partition and Kuhn triangulations. Comput Math Appl. 2007;54(3):427–433. <http://dx.doi.org/10.1016/j.camwa.2007.01.023>. Clarke1983 Clarke F. Optimization and nonsmooth analysis. Toronto: John Wiley & Sons; 1983. Ito-Kunisch2003 Ito K, Kunisch K. Semi-smooth Newton methods for state-constrained optimal control problems. Systems Control Lett. 2003;50(3):221–228. <http://dx.doi.org/10.1016/S0167-6911(03)00156-7>.

http://arxiv.org/abs/1701.07456v1

20170125192907

Control Allocation for Wide Area Coordinated Damping

cs.SY

Control Allocation for Wide Area Coordinated Damping M. Ehsan Raoufat, Student Member, IEEE, Kevin Tomsovic, Fellow, IEEE, and Seddik M. Djouadi, Member, IEEE This work was supported in part by the National Science Foundation under grant No CNS-1239366, and in part by the Engineering Research Center Program of the National Science Foundation and the Department of Energy under NSF Award Number EEC-1041877 and the CURENT Industry Partnership Program. M. Ehsan Raoufat, Kevin Tomsovic and Seddik M. Djouadi are with the Min H. Kao Department of Electrical Engineering and Computer Science, The University of Tennessee, Knoxville, TN 37996 USA (e-mail: mraoufat@utk.edu). A. Mahmoodzadeha.mahmoodzadeh@iau-boukan.ac.ir, B. MalekolkalamiB.Malakolkalami@uok.ac.ir ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ In this work, a modal-based sparse control allocation (CA) is proposed for coordinated and fault-tolerant wide-area damping controllers (WADCs). In our proposed method, the supervisory CA only communicates with necessary actuators to achieve the required damping performance and in case of actuator failures (e.g., due to loss of communication or scheduling), capabilities of the remaining actuators are fully used before the nominal performance is degraded. This method offers the advantages of modular design where WADC is initially designed to achieve satisfactory damping without the detailed knowledge of actuators. In the next step, CA is designed to manage actuator failures and limitations without the need to redesign the nominal WADC. The proposed approach is applied to a modified 286-bus Western Electricity Coordinating Council (WECC) system to verify the feasibility on a complex power system. Simulation results indicate the effectiveness of the proposed method in coordinating multiple actuators and building resiliency. Inter-area oscillations, Western Electricity Coordinating Council, wide-area damping controller, coordinated control, fault-tolerant control, sparse control allocation. § INTRODUCTION Inter-area oscillations have been identified as a major problem faced by most power systems and stability of these oscillations are of vital concern due to the potential for equipment damage and resulting restrictions on available transmission capacity between different areas <cit.>. With recent advances in wide-area measurement systems (WAMS), inter-area modes can be observed globally and wide-area damping controllers (WADCs) can be deployed to enhance the stability <cit.>. Multiple design techniques and methodologies have been reported for damping inter-area oscillations including designs for supplementary control of generator excitation <cit.>, FACTS devices <cit.> and renewable energy sources <cit.>. However, very few such systems have been deployed in practice partly due to high level of robustness and reliability requirements for any closed loop power system controls. Traditional power system topology is changing and a large number of small-scale renewable sources are being installed throughout the system. In this aspect, spatial distributions of wind farms are crucial to reduce the need for new transmission infrastructure. These wind farms could be selected as WADC actuators and contribute to damping inter-area oscillations though active/reactive power modulation <cit.>. In contrast with a large wind farm in a concentrated location, deployment of multiple small-scale wind farms will require special techniques for actuator coordination as none could be used individually to achieve adequate damping. Moreover, the availability of these weather dependent renewable resources could pose design challenges for reliability of critical controllers. Considering reactive power modulation in Type 4 wind turbines (i.e., full converter asynchronous generators), the amount of available reactive power depends upon the operating mode, converter rating and grid code requirements. This may mean that some WADC actuators become temporarily unavailable (failed) or have more limited capabilities. Moreover, communication failures such as packet loss, excessive time delay and cyber-attacks may also lead to failures in these geographically-dispersed actuators. Thus, developing robust controllers to accommodate such failures and maintain the system stability is an important challenge in deploying WADCs. In this paper, a sparse control allocation (CA) method is developed to optimally coordinate a set of actuators to damp the inter-area modes and achieve a fault-tolerant WADC. In our approach, the damping controller is designed based on a fault-free model and the supervisory CA distributes the control signals to necessary actuators based on the desired control actions, total cost, effects on different modes of the system and actuator constraints. This paper generalizes the previous methods on control allocation <cit.> by considering the temporal sparsity and the effects of virtual control on the modal system. This technique allows us to give the highest priority to the control efforts associated with the critical inter-area modes. In <cit.>, an attempt to coordinate multiple wind farms was addressed but without considering the effects of actuator failures, capabilities and limits. This paper also extends <cit.> in which unavailability of WADC actuators has not been considered. Feasibility of the proposed approach has been verified on a modified 286-bus Western Electricity Coordinating Council (WECC) system with multiple small-scale wind farms. This paper is organized as follows. In Section II, a modular control allocation technique is developed for system with redundant actuators and a multi-objective synthesis is presented as one method to design damping controller. Preliminaries on dynamic modeling of a WECC system with distributed wind farms are described in Section IV. Nonlinear time-domain simulations are presented in Section V to demonstrate the effectiveness of the proposed method in coordinating multiple actuators. Concluding remarks are given in Section VI. § MODAL-BASED SPARSE CONTROL ALLOCATION FOR WIDE-AREA DAMPING Control allocation can be used to coordinate a redundant set of actuators for a class of over-actuated systems in which the number of actuators (m) exceeds the number of states (n) <cit.>. Here, we consider model-based redundancy in the actuators as physical redundancy (e.g., replicating an actuator) is not cost effective in power systems. The assumption of redundancy rank (B)=n<m needs to be satisfied to guarantee a set of admissible control signals <cit.>, where B ∈ R^n × m is the control input matrix. However, for power systems like many other practical systems, this assumption is not necessarily valid for the full-order system. In this work, the control allocation problem is formulated based on the reduced-order model and it is assumed that this model accurately represents the dominant contribution of different actuators to the inter-area modes of interest. The Hankel norm approximation <cit.> can be used to obtain the reduced-order model and the order of model reduction can be determined by examining the Hankel singular values. Considering the reduced-order system with state variables x_r∈ R^n, using an appropriate transformation z=ψ x_r where ψ∈ R^n × n, the realization in modal form can be written as rCl ż(t) = Λz(t)+ψB_r u(t) y(t) = C_r ψ^-1 z(t) Λ = [ ι_1 0 0 …; 0 σ_1 ω_1; 0 -ω_1 σ_1; ⋮ ⋱ ] where Λ=ψ A_r ψ^-1 is a block diagonal matrix whose elements are eigenvalues of A_r (assuming no repeated eigenvalues), u∈ R^m denotes the input and y∈ R^p is the measured output. Real eigenvalue ι_i appears on diagonal and complex conjugate eigenvalues σ_i ±ω_ij appear as a 2-by-2 block on the diagonal of Λ. By introducing the virtual control input v∈ R^n, the system equations can be expressed as rCl ż(t) = Λz(t)+I_n v(t) y(t) = C_r ψ^-1 z(t) v(t) = ψB_r u(t) which decomposes the system into two parts and leads to a modular design where WADC generates the virtual control signal v and control allocator distributes the effort among the available actuators. Matrix ψ is full rank and rank (ψ B_r) =n < m, hence ψ B_r has null space of dimension m-n in which u can be perturbed without affecting the response §.§ Wide-area Damping Controller Design We designed a multi-objective damping controller based on LMI optimization technique introduced in <cit.> but our approach to the CA can accommodate other control approaches. The controller is designed based on the reduced order model (<ref>) and (<ref>) to avoid feasibility problems and realize practical low-order controllers. Further details of this approach to design WADC can be found in <cit.>. The damping controller designed by the above methodology can be written as: rCl ẋ_k(t) = A_kx_k(t)+B_k y(t) v(t) = C_kx_k(t)+D_k y(t) Although the above WADC is designed using robust control methods, failure in the communication links or in the actuators will lead to poor damping performance. §.§ Modal-based Sparse Control Allocation Based on the order of the reduced model, the system can now represent an over-actuated system and the problem of modal-based sparse control allocation with proper filtering to reduce the variations can be represented as follows rCl u_tmin ‖W_u u_t ‖^2_2 + ‖W_s (u_t-u_t-T_s) ‖^2_2 + λ‖u_t ‖_1 s. t. ψB_r u_t=v_t u_min ≤u_t ≤u_max where W_u and W_s are positive definite matrices, usually diagonal, and represent the weighting for distributions and variations in the control signal, respectively. The term ‖ u_t ‖_1= ∑_i=1^m| u_t,i| denotes the ℓ_1 norm of control vector u_t and λ≥ 0 is the regularization parameter. Virtual control input v_t is derived from the nominal WADC at time t and T_s denotes the time step. The key feature of the proposed control allocation strategy is that the temporal sparse control vector u_t is directed to actuators considering total cost, actuator rates, modal effects and actuator limitations, which leads to a constrained optimization problem (<ref>). This method is based on prior knowledge of control limits and CA only communicates with necessary actuators to achieve the damping requirement. The cost function of the above optimization can then be simplified to l ‖W_u u_t ‖^2_2 + ‖W_s (u_t-u_t-T_s) ‖^2_2 + λ‖u_t ‖_1 = u_t^T W^2_u u_t + (u_t-u_t-T_s)^T W^2_s (u_t-u_t-T_s) + λ‖u_t ‖_1 = u^T_t(W^2_u+W^2_s)u_t - 2u^T_t W^2_s u_t-T_s + λ‖u_t ‖_1 + const. = ‖W(u_t-u_d) ‖^2_2 + λ‖u_t ‖_1 + const. where l u_d=W^2_s (W^2_u + W^2_s)^-1 u_t-T_s , W=(W_u^2+W_s^2)^1/2 Since constant terms in the objective function will not affect the optimal solution, they can be removed and the optimization can be cast in the form of least square optimization with ℓ_1 norm regularization rCl u_tmin ‖W (u_t-u_d) ‖^2_2 + λ‖u_t ‖_1 s. t. ψB_r u_t=v_t u_min ≤u_t ≤u_max with u_d and W from (<ref>). The problem can be approximated by utilizing the first constraint in the cost function using the Lagrangian multiplier ρ and weighting function W_v. rCl ‖W ( u_t-u_d) ‖^2_2 + ρ^2 ‖W_v ( ψB_r u_t-v_t) ‖^2_2 + λ‖u_t ‖_1 = ‖[ ρW_v ψB_r; W ] u_t - [ ρW_v v_t; W u_d ] ‖^2_2 + λ‖u_t ‖_1 Finally, we obtain the following optimization problem rCl u_tmin ‖[ ρW_v ψB_r; W ] u_t - [ ρW_v v_t; W u_d ] ‖^2_2 + λ‖u_t ‖_1 s. t. u_min ≤u_t ≤u_max In the control literature, there exist other methods to distribute the control signal based on cost <cit.> or actuator limits <cit.>, but these have not considered the effects on modal system or sparsity. This technique allows us to give the highest priority to the control efforts associated with the critical inter-area modes by using the weighting function W_v and obtain the feasibility regions in modal coordinates. By decomposing the control vector u_t to positive and negative components, we introduce nonnegative variables u^+_t, u^-_t and q_t=[ u^+_t u^-_t ]^T such that rCl u_t = u^+_t - u^-_t = [ I_m -I_m ] q_t ; u^+_t, u^-_t ≥0 The ℓ_1 norm can then be modeled as ‖ u_t ‖_1 = 1̅^T q_t (with 1̅ being a vector of ones) and the ℓ_1-regularized least square problem can be transformed into a quadratic programing with simple box constraints as follow rCl q_tmin q_t^T [ 𝒜^T 𝒜 -𝒜^T 𝒜; -𝒜^T 𝒜 𝒜^T 𝒜 ] q_t + ( 2[ -𝒜^T ℬ; 𝒜^T ℬ ]^T + λ1̅^T ) q_t s. t. 0̅ ≤q_t ≤[ u_max; -u_min ] where l 𝒜= [ ρW_v ψ^T B_r; W ], ℬ= [ ρW_v v_t; W u_d ] For most problems, these quadratic programs can be solved efficiently using interior-point or active-set methods. Note the transformed problem is an optimization over 2m-dimensional vector space. § DYNAMIC MODEL OF THE WECC TEST SYSTEM A modified 286-bus WECC system is used in this study to capture the effects of redundant actuators over the inter-area modes. As shown in Fig. <ref>, this system consists of 31 synchronous generators with generation of 60.25 GW and 35 small-scale wind farms, each rated at 60 MVA and 50 MW, with total generation of 1.75 GW. Each generator is represented using a two-axis model equipped with a high-gain AVR system and a power system stabilizer (PSS1A) to damp the local oscillation modes <cit.>. All loads are assumed to be constant power and original parameters regarding the network data and operating conditions are given in <cit.>. Wind farms are represented by an aggregated model of Type 4 wind turbines. In this work, the base power of each wind farm is scaled based on the total number of wind turbines while the parameters are assumed to be constant. The equivalent circuit is shown in Fig. <ref> and further details on network and model parameters can be found in <cit.>. In this study, the damping controller is performed by adding a supplementary signal u to the reactive power control loop for reactive power modulation. We assume each wind farm is constrained to run within a specific power factor range, for example 0.9 lagging to 0.9 leading which is typical for Type 4 machines <cit.>. As a result, a hard limits of u_max=-u_min=0.4 pu are imposed on the supplementary signal of each wind farm. § NUMERICAL RESULTS Detailed studies based on a nonlinear model of the WECC system described in the previous section are performed to verify the performance of the proposed control allocation method. §.§ Linear Analysis and Design of WADC This system exhibits several low-frequency oscillation modes that are characterized in Table <ref>. Critical mode 3 with frequency of 0.564 Hz and a low damping ratio of 0.98% is of high interest and represents the inter-area mode between area 2 and 4. Based on an observability measure, speed deviation of G_10 is selected as the best candidate signal for our controller as it has the highest observability over the critical mode (details of this approach are given in <cit.>). The test system has 490 states and the order of the reduced model is chosen as n=6 to preserve the largest Hankel singular values <cit.> as shown in Fig <ref>. The WADC is designed based on the 6^th-order model to meet or exceed 6% damping over the inter-area modes. §.§ Design of Modal-Based Control Allocation The proposed CA is implemented as a user-defined model (UDM) in TSAT <cit.> and the optimization algorithm (<ref>) is performed using dynamically linked blocks (DLBs) and MATLAB with a fixed time step of T_s=0.02 s and interior-point method. The available small-scale wind farms are chosen as the set of redundant actuators as follows rCl ℛ={W_184, W_187, W_190, …, W_286 } where i^th element of vector ℛ is associated with the i^th column of matrix B_r. The weighting functions and gains are chosen as W_u:=I_35, W_s:=2 W_u, W_v:=diag(2,2,4,4,8,8), λ:=1 and ρ:=100. This choice of weighting matrix W_v gives the highest priority to the control efforts regarding the critical mode 3. Moreover, the weighting W_u can also be chosen based on the reliability of each actuator and the corresponding communication link. §.§ Nonlinear Simulations Nonlinear transient studies were performed using TSAT and Prony analysis is used to extract the damping coefficient of the inter-area oscillation based on the nonlinear response. In this study, the time frame of analysis (oscillation) is restricted to a few seconds, so it is reasonable to assume that the wind speed remains effectively constant during this period. Cases of interest include faults in both the physical system and actuators. In the physical system, a symmetrical three-phase fault is applied at bus #139, which is a severe disturbance, to stimulate the critical inter-area mode. To illustrate the benefit of sparse control allocation, three control cases were evaluated and compared during transient response. First, the system with no control is considered. Second, a WADC with fixed allocation u(t)=(ψ B_r)^† v(t) is considered based on pseudo-inverse calculation. Finally, a sparse control allocation is considered to include hard limits and actuator status in the design. Active power of the inter-area transmission line 6-27 is shown in Figs <ref>, <ref> and <ref> for the following cases * Case A: No faulty actuators and fault duration of 1 cycle; * Case B: No faulty actuators and fault duration of 6 cycles; * Case C: Faults in 70% of the actuators and fault duration of 3 cycles; It can be seen that in case A, where fault duration is short and the required control effort is less, both sparse CA and fixed allocation method can improve the damping to 7.2% compared to the open-loop damping of 0.98%. In case B, where fault duration is longer and requires extensive control efforts, the damping ratio of the fixed allocation method reduces to 2.82%. However, the sparse CA achieves a damping ratio of 5.55% as it considers the actuator limitations in control redistribution. In case C with a shorter fault duration but 70% actuator failures (either from multiple wind farms are disconnected, communication congestions, changes in wind speed), the sparse CA will again dampen the oscillations by redistributing the control signal to healthy actuators and maintain sufficient damping of 4.89% compared to 2.03% under a fixed allocation. Comparing these results, it can be seen that the proposed method enhances fault-tolerance of the WADC system. Figs. <ref> and <ref> illustrate the sparse CA outputs in case A and B, respectively. It can be seen that the control signal u is temporally sparse relative to the fixed control allocation method. Additional results for different actuator fault combinations are presented in Table <ref>. In all cases, the physical fault is assumed to be with a duration of 3 cycles. It can be observed that our proposed method tolerates various combinations of failures and maintains a higher damping ratio over the critical inter-area mode. § CONCLUSIONS This work proposes a sparse control allocation technique for fault-tolerant wide-area damping controllers and coordinated control of multiple actuators. This method leads to a modular design process where the damping controller generates the virtual control signal and the supervisory CA distributes the control efforts to the necessary actuators based on the desired control actions, actuator limits and modal effects. The proposed approach is applied to a modified 286-bus Western Electricity Coordinating Council (WECC) system with distributed small-scale wind farms. Simulation results show significant improvement in resiliency due to various system failures. IEEEtran

http://arxiv.org/abs/1701.08161v2

20170127190000

Save the Planet, Feed the Star: How Super-Earths Survive Migration and Drive Disk Accretion

astro-ph.EP

Department of Astronomy, University of California at Berkeley, Campbell Hall, Berkeley, CA 94720-3411 1NASA Sagan Fellow email: jeffrey.fung@berkeley.edu Two longstanding problems in planet formation include (1) understanding how planets survive migration, and (2) articulating the process by which protoplanetary disks disperse—and in particular how they accrete onto their central stars. We can go a long way toward solving both problems if the disk gas surrounding planets has no intrinsic diffusivity (“viscosity”). In inviscid, laminar disks, a planet readily repels gas away from its orbit. On short timescales, zero viscosity gas accumulates inside a planet's orbit to slow Type I migration by orders of magnitude. On longer timescales, multiple super-Earths (distributed between, say, ∼0.1–10 AU) can torque inviscid gas out of interplanetary space, either inward to feed their stars, or outward to be blown away in a wind. We explore this picture with 2D hydrodynamics simulations of Earths and super-Earths embedded in inviscid disks, confirming their slow/stalled migration even under gas-rich conditions, and showing that disk transport rates range up to ∼10^-7 and scale as Ṁ∝Σ M_ p^3/2, where Σ is the disk surface density and M_ p is the planet mass. Gas initially sandwiched between two planets is torqued past both into the inner and outer disks. In sum, sufficiently compact systems of super-Earths can clear their natal disk gas, in a dispersal history that may be complicated and non-steady, but which conceivably leads over Myr timescales to large gas depletions similar to those characterizing transition disks. § INTRODUCTION Protoplanetary disks have two jobs: make planets and feed their host stars. The first task is frustrated by migration: disk torques force planetary orbits to decay <cit.>, evacuating the very regions where planets are observed in abundance <cit.>. Fulfilling the second task requires a mechanism to transport away the disk's angular momentum. Magnetic torques are promising but depend on seed fields of uncertain provenance <cit.>. <cit.> proposed that the two problems are actually one: that planets themselves—if they can survive migration—can provide an effective source of disk viscosity by exciting density waves that transport angular momentum outward. <cit.> emphasized that such planets must be massive enough to open gaps and avoid Type I migration. They focused on giant Jupiter-mass planets, a demographic that is now understood to be rare <cit.>. In this paper, we turn our attention to super-Earths: bodies of mass 1–10 M_⊕ that have been discovered by Kepler to be relatively commonplace <cit.>. Can super-Earths avoid Type I migration? Yes—if their disks are sufficiently inviscid. The dependence of the Type I drift rate on disk viscosity is perhaps under-appreciated, as it is not explicitly called out in the typically quoted Type I formula (see, e.g., ). Crucially, without an intrinsic disk viscosity to smooth away the planet's perturbations to the disk's surface density, a pile-up of disk material ahead of a migrating planet exerts a “feedback” torque that slows and can even stall migration <cit.>. <cit.> calculated that for inviscid disks in which planet-driven waves dissipate by steepening into shocks, the critical planet mass above which Type I migration shuts off is: M_ cr≃ 4 (h_ p/r_ p/0.035)^3(M_∗/M_⊙) (Σ_ p r_ p^2 / M_∗/10^-3)^5/13 M_⊕ , where r and h are the disk radius and scale height, Σ is the disk gas surface density, M_∗ is the central stellar mass, and the subscript p indicates evaluation near the planet's position. <cit.> and <cit.> have performed numerical simulations supporting the analytic calculations by <cit.>, and confirming that super-Earths in low-viscosity disks migrate much more slowly (and erratically) than is predicted by Type I. A planet of mass M_ p at r_ p drives a disk mass transport rate Ṁ at distance r of Ṁ(r) = -2F_0/l r ∂φ(r)/∂ r where F_0 = Σ_ p r_ p^2 l_ pΩ_ p(M_ p/M_*)^2 (h_ p/r_ p)^-3 measures the total angular momentum carried away per time by planet-driven waves (a.k.a. the total integrated one-sided Lindblad torque), Ω is the orbital frequency, and l = Ω r^2 is the specific angular momentum. The dimensionless function φ(r) describes how waves, as they travel away from the planet, damp with distance, depositing their angular momentum to disk gas and thereby propelling material radially. From <cit.>, φ∼( M_ p/M_ thermal)^-1/2( |r_ p - r|/h_ p)^-5/4 , valid for M_ p≲ M_ thermal≡ (h_ p/r_ p)^3 M_∗ and φ≲ 1 (i.e., distances far enough from the planet that the waves are dissipating in weak shocks), and where we have ignored order-unity constants and all radial variations in h, Σ, and gas sound speed. It follows that Ṁ(r) ∼ sign(r-r_ p) Σ_ p r_ p^2 Ω_ p(M_ p/M_∗)^3/2(h_ p/r_ p)^-5/2(|r_ p-r|/h_ p)^-9/4 . At r < r_ p, Ṁ as given by Equation (<ref>) is negative (mass flows inward), and vice versa; a planet tends to repel material away from itself. To avoid minus signs, we will ignore this formal sign convention so that all our reported values for Ṁ will be positive and understood to be inward unless otherwise indicated. Note how Ṁ∝ M_ p^3/2 and not M_ p^2. Although the total Lindblad torque scales as M_ p^2 (Equation <ref>), that torque is distributed over a distance that increases with decreasing M_ p (as M_ p^-2/5, as can be seen by solving for |r_ p - r| in terms of M_ p at fixed φ in Equation <ref>). Thus at fixed distance away from the planet, Ṁ increases with M_ p with a power less than 2. Inserting M_ p = 10M_⊕ and other nominal parameters (for r < r_ p) into (<ref>) yields Ṁ(r) ∼ 10^-8 (Σ_ p r_ p^2/10^-3 M_⊙) (2π/Ω_ p/1 yr)^-1(M_ p/M_*/3 × 10^-5)^3/2(h_ p/r_ p/0.035)^-1/4 ((r_ p - r)/r_ p/0.5)^-9/4 , comparable to accretion rates measured for classical T Tauri stars <cit.>. Note how weakly Ṁ depends on h_ p/r_ p, underscoring how Ṁ does not scale simply as the total Lindblad torque in eqn:torque (which scales as (h_ p/r_ p)^-3), but depends also on the distance over which that torque is exerted, as we have described above. The above considerations indicate that with super-Earths we might have our cake (survive migration) and eat it, too (drive disk accretion). Of course, a single super-Earth is insufficient because its reach is too short (Ṁ drops as |r-r_ p|^-9/4). Multiple super-Earths are needed to shuttle the accretion flow from distances of a few AU down to the stellar radius. Reality will be non-steady and likely messy (see, e.g., Figure 4 of ), with material between adjacent planets having a fate that is not obvious: does the sandwiched gas drain inward, or does the inner planet hold back material pushed inward by the outer planet? And to what extent do super-Earths migrate with the accretion flow they drive? Here we explore these questions using fully non-linear, 2D hydrodynamical simulations of super-Earths embedded in inviscid disks. We measure the migration histories r_ p(t) and accretion rates Ṁ in simulations containing 1 or 2 super-Earths, experimenting with varying the disk surface density and the planet mass to test eqn:ana_mdot_wnum. Our numerical methods are given in Section <ref>. Results are presented in Section <ref> and placed into broader context in Section <ref>. § NUMERICAL METHOD We use the graphics processing unit (GPU) accelerated hydrodynamics code <cit.> to perform 2D simulations of disk-planet interactions. It is a Lagrangian-remap shock-capturing code that uses the piecewise parabolic method <cit.> to solve the continuity and momentum equations: DΣ/ Dt = -Σ(∇·𝐯) , D𝐯/ Dt = -1/Σ∇ p + 1/Σ∇·𝕋 - ∇Φ , where Σ is the gas surface density, 𝐯 the velocity field, p the vertically averaged gas pressure, 𝕋 the Newtonian stress tensor, and Φ the combined gravitational potential of the star and the planet(s). We use a globally isothermal equation of state: p=^2Σ with a spatially constant sound speed =0.035 v_ K, 1 AU≃ 1  km s^-1 where v_ K, 1 AU is the Keplerian velocity at 1 AU around a 1 M_⊙ star. (This corresponds to a disk temperature of 300 K assuming a mean molecular weight of 2.34.) In a polar coordinate system (radius r, azimuth ϕ) centered on the star, Φ = -GM_*/r + ∑_i=1^N_ pΦ_ p,i Φ_ p,i = -GM_ p, i/√(r^2 + r_ p, i^2 - 2rr_ p, icosϕ_i' + r_ s, i^2) + GM_ p, i rcosϕ_i'/r_ p, i^2 where G is the gravitational constant, M_∗ = 1 M_⊙ is the stellar mass, the subscript i labels each planet, N_ p is the total number of planets, M_ p is the planet mass, Φ_ p the planet's gravitational potential, r_ p the planet's radial coordinate, the smoothing length of the planet's potential, and ϕ' = ϕ-ϕ_ p the azimuthal separation from the planet. The stress tensor 𝕋 is proportional to the kinematic viscosity ν. Most of our simulations are of inviscid disks with ν=0. For our viscous disk simulations, we use ν=α h, where the Shakura-Sunyaev parameter α = 0.001, h = /Ω_ K is the local scale height, and Ω_ K = √(GM_∗/r^3) is the Keplerian orbital angular frequency. At r = 1 AU, h/r = 0.035. We set = 0.5 h, as is appropriate for 2D simulations <cit.>. A given planet feels the gravitational force from the star, the disk, and other planets. The disk force on the planet is calculated by direct summation over all mass elements in the disk, with the “background” axisymmetric component of the disk surface density subtracted off. Because the disk does not feel its own gravity at all (i.e., we ignore disk self-gravity; see equation <ref>), eliminating this axisymmetric component in the disk-planet forcing improves consistency between the motions of the planets and the disk. Planet migration should be minimally affected by this procedure, since the background component of Σ exerts no torque. Spurious forces arising from within the planet's Hill sphere are sometimes a concern if this region is under-resolved. The Hill radius, = (M_ p/3M_*)^1/3, ranges from 0.3 to 0.6 h_ p, similar to the smoothing length . We have verified that the torque generated within a radius of 0.5 from the planet is negligible, and so we do not excise the Hill sphere in force calculations. The planets' motions are integrated using a kick-drift-kick leapfrog scheme, with the drift step occurring synchronously with the hydrodynamics step; i.e., the planets' positions are linear in time within a hydrodynamics step. §.§ Initial and boundary conditions, and grid parameters ccccccccc Model Parameters Model # M_ p (M_⊕) r_ p,1 (AU) r_ p,2 (AU) Σ_0 (g cm^-2) α r_ in (AU) t_ end (years) 1 10 1 – 8.5× 10^3 10^-3 0.4 700 2 10 1 – 8.5× 10^3 0 0.4 5000 3 10 1 1.2 8.5× 10^3 0 0.4 5000 4 10 0.75 1.05 8.5 0 0.3 2000 5 3 0.75 1.05 8.5 0 0.3 2000 6 1 0.75 1.05 8.5 0 0.3 2000 t_ end is the end time of a simulation, in units where the Keplerian orbital period at 1 AU is 1 year. Also, r_ p,1 and r_ p,2 are merely the initial planet locations at t=0; the planets are completely free to migrate in the simulations. tab:setup lists the parameters used by our 6 models. The disk is initialized with a power-law surface density: Σ = Σ_0 (r/ AU)^-3/2 . We consider both gas-rich disks having Σ_0=8.5×10^3  g cm^-2 resembling the minimum-mass extrasolar nebula (), and gas-poor disks having a surface density 1000× lower. The initial velocity field is axisymmetric and Keplerian, with corrections from gas pressure: Ω = √(Ω_ K^2 + 1/rΣ d p/ d r) . One planet, whose mass is increased gradually over the first 10 yr of the simulation to the full value of M_ p (either 1, 3, or 10 M_⊕), is placed initially at r = r_ p, 1 (either 1 or 0.75 AU) and ϕ_ p, 1 = π. In two-planet models, we place a second planet of equal mass to the first at r = r_ p, 2 (either 1.2 or 1.05 AU) and ϕ_ p, 2 = π initially. Our simulation grid spans the full 2π in azimuth, and extends from an outer radius of 1.8 AU to an inner radius r_ in that equals either 0.4 or 0.3 AU depending on whether r_ p,1 = 1 AU or 0.75 AU (see tab:setup). Grid dimensions are 800 (r) × 3200 (ϕ) when r_ in = 0.4 AU, and 960 × 3200 when r_ in = 0.3 AU. Cells are spaced logarithmically in radius and uniformly in azimuth. Our choices yield a resolution of ∼18 cells per scale height h in both directions at r = 1 AU (similar to the resolution of ). Simulations at twice our standard resolution did not produce significant changes in either planet migration or disk accretion rate for the first 100 yr. We also tested our inviscid disk model without a planet, and found that the numerical noise in |Ṁ| was about 3 orders of magnitude below planet-driven disk accretion rates, corresponding to a numerical viscosity of α<10^-5. Radial boundary conditions require special care in this study. After experimenting with a few ways to measure disk accretion rates, we found that the most stable method was to track the total disk mass within a cylinder of radius 0.6 AU—a distance intermediate between the innermost planet and the inner disk boundary—while preventing mass from leaving the grid. We adopt “zero flux” boundary conditions where mass and momentum fluxes across the inner and outer disk edges are always zero. In , this is achieved by solving a special Riemann problem at the boundaries, one where no wave travels toward the simulation domain, and where the radial velocity outside the domain is always zero. This implementation conserved the total mass within the simulation domain to numerical accuracy. The accretion rate Ṁ at r = 0.6 AU is calculated by following over time the disk mass enclosed, M_ 0.6 AU(t). Because the function M_ 0.6 AU(t) fluctuates strongly, we fit independent lines to segments of data each lasting 20 yr, taking Ṁ from the best-fitting slopes. As a planet repels material away from its orbit, our boundary conditions result in gas piling up at the inner and outer boundaries. Our results can only be trusted to the extent that these boundary pile-ups do not interfere with planet migration and disk accretion. We therefore limit ourselves to studying only the first few thousand years of planet-disk interactions, before boundary effects become too large. § RESULTS We assess to what extent planets migrate in inviscid disks (<ref>), and study how planet-driven accretion rates evolve with time and depend on disk and planet masses (<ref>). For planet migration, models #1–3 demonstrate differences between viscous and inviscid disks, and between single-planet and two-planet systems. For planet-driven accretion, we vary disk and planet masses in models #3–6 to test eqn:ana_mdot_wnum. §.§ Planet migration fig:mig plots the orbital evolution of planets, each of mass 10 M_⊕, in the gas-rich disk models (#1–3). Overplotted for comparison is the trajectory expected from integrating the Type I migration rate, ṙ_ p, Type I = -2 C r_ pΩ_ p(Σ_ p r_ p^2/M_ p) (M_ p/M_*)^2 (h_ p/r_ p)^-2 , using the unperturbed surface density law in eqn:initial_sigma to evaluate Σ_ p (using the actual surface density in the viscous disk simulation #1 would give practically identical results, since gaps do not form in that model). Three-dimensional simulations suggest that C∼ 2–3 for our given disk profile <cit.>; the blue dashed curve in fig:mig uses C=2. fig:sigma displays azimuthally averaged surface density profiles at various epochs in gas-rich, inviscid disk models #2 and #3, with planet locations marked. In agreement with the simulations of <cit.> and <cit.>, the planet migration rate in our viscous disk (model #1) is similar to the Type I rate (punctuated by what appear to be episodes of even faster Type III migration; ), and much slower in inviscid disks (models #2 and #3). The initially rapid migration seen in the inviscid simulations at t ≲ 200 yr is a transient that decays after disk surface densities adjust to planetary Lindblad torques, i.e., after the surface density pile-up ahead of the planet attains a fractional amplitude on the order of unity. After this initial adjustment period, migration slows and even stalls at times, with radial positions changing by ∼10–30%, or less, over kyr timescales. In the single-planet, gas-rich simulation (black curve in fig:mig, model #2), the planet journeys slowly inward for the first ∼2000 yr and practically stops in the mean from t ≃ 2000–3500 yr, as disk gas that the planet has pushed inward to r ≃ 0.7–0.8 AU piles up (blue and green curves in the left panel of fig:sigma) and stymies further migration. At t ≃ 3500 yr, the planet experiences a sudden drop in orbital radius; we traced this drop to a close encounter between the planet and a vortex formed at its outer gap edge at r ≃ 1.1 AU (see fig:vortex). Some time after the encounter, the vortex gradually disperses, completely decaying away by the end of our simulation at 5000 yr (fig:endpoint, left panel). Similar planet-vortex interactions were found in simulations by <cit.> and <cit.>. Thereafter, the planet's migration returns to its near-zero mean pace. In our two-planet, gas-rich simulation (#3), the outer planet stalls for the first ∼3000 yr (magenta curve in fig:mig), apparently trapped at a local surface density maximum created by the inner planet whose forcing dominates: see the magenta curve in the right panel of fig:sigma, and note how similar it is to the corresponding magenta curve in the left panel for the single-planet case. Gas pushed inward by the outer planet strengthens the torque on the inner planet and forces the latter to migrate inward by ∼20% over the same time period (red curve in fig:mig); contrast this behavior with the stalling observed in the single-planet case (black curve). Eventually, at t ∼ 4000 yr, the outer planet disperses the surface density maximum in its vicinity, and proceeds to migrate slowly inward, slowing down near t∼ 5000 yr as it runs into sandwiched gas. Meanwhile, the inner planet ultimately comes to a near halt in much the same way that it does in the single-planet simulation, having run into material that has piled up just interior to its orbit. The right panel of fig:endpoint shows the final surface density distribution. We emphasize that this pile-up is physical as it is located at r ≃ 0.6 AU, away from the inner grid boundary of the simulation at r_ in = 0.4 AU. The latter location has its own separate pile-up, which does not grow to significance over the limited duration of our simulations. The same statement applies to the outer grid boundary. We note that in none of the two-planet simulations did we observe the formation of a vortex like the one seen in our single-planet simulation. This difference might be physical, and deserves attention in future studies of planet-vortex interactions. All other factors being equal, lower disk masses should lead to even slower planetary migration rates. This is confirmed in our gas-poor simulations (models #4–6) which exhibit no measurable change in planet mean radial positions. Thus the gas-poor simulations can be used to diagnose disk accretion rates without the complicating effects of planetary migration, as we discuss in the next subsection. §.§ Disk accretion fig:acc shows disk accretion rates as functions of time for all our inviscid models. As described in <ref>, the accretion rate Ṁ is measured by tracking the build up of disk mass inside r = 0.6 AU, a location interior to the planets at all times. Later, in <ref>, we track the movement of mass initially between two planets. We begin by checking whether our simulations are compatible with the analytic expectation for Ṁ given by eqn:ana_mdot_wnum. The comparison is best made at early times of the simulation, t ≲ 1000 yr, before radial surface density profiles become too distorted. For the single-planet, gas-rich model #2, eqn:ana_mdot_wnum yields Ṁ∼ 3 × 10^-8. This prediction is within a factor of ∼2 of the simulated result at early times (left panel of fig:acc). As for the corresponding two-planet model #3, we expect from (<ref>) that Ṁ should be only fractionally larger than for the single-planet case; the second planet is farther removed from where we measure Ṁ (r = 0.6 AU), and so makes only a ∼50% contribution to the accretion flow there as compared to the inner planet. This is approximately consistent with fig:acc. For our gas-poor model #4, eqn:ana_mdot_wnum predicts Ṁ∼ 1.5 × 10^-10, again within a factor of 2 of the simulated result (right panel of fig:acc). Scaling the planet mass M_ p down by a factor of 10 from models #4 to #6 should, according to eqn:ana_mdot_wnum, reduce Ṁ by a factor of 10^3/2≃ 30. By comparison, fig:acc shows a factor of ∼20 decrease between these two models; we consider this acceptable agreement with the analytic expectation. In summary, our simulations support the various functional dependencies predicted by eqn:ana_mdot_wnum to within a factor of 2. At later times, t ≳ 1000 yr, we observe time variability in Ṁ caused by the deepening of planetary gaps, and by planet migration. These variations are limited to factors of a few. The simulations easiest to interpret are models #4–6 (right panel of Figure <ref>) which have too little disk gas to drive planet migration. The initial gradual decline in Ṁ seen in model #4 is caused by the deepening of gaps opened by its 10-M_⊕ planets; over the course of 2000 yr, the gas density in the immediate vicinity of the planets decreases by a factor of ∼5 for the inner planet and by a factor of ∼3 for the outer one. Models #5 and #6 exhibit steadier accretion rates, as their planets have lower masses which are less effective at opening gaps. More complicated behavior is seen in the gas-rich simulations where planets migrate more appreciably. Comparison of Figures <ref> and <ref> reveals that increases in Ṁ can be traced to planets moving inward, either gradually, as in the first 2000 yr of models #2 and #3, or suddenly, as in the planet-vortex encounter at t ≃ 3300 yr in model #2. Decreases in Ṁ correspond to planets opening gaps upon moving to new locations. §.§.§ The fate of gas initially sandwiched between planets Disk accretion driven by planets would be impractical if material residing between planets were unable to escape. We track this sandwiched gas in simulations #4–6, each containing a pair of planets which migrate negligibly. We assign each gas parcel a “passive scalar” η that equals 1 for gas initially located between r = 0.8 and 1 AU (between the two planets), and is 0 everywhere else. Gas elements carry η as a conservative quantity. fig:eta follows the η-tagged gas by plotting ∂M_η/∂ r = ∫^2π_0ηΣ r dϕ , vs. r at various times. We find that the sandwiched gas is torqued both inward and outward, escaping in roughly equal amounts to the inside of the inner planet and to the outside of the outer planet. The opposing torques from the two planets do not in general cancel. The opposing torques from the two planets do not in general cancel, although in model #4 containing the highest mass planets, some gas does concentrate along the midline between the planets in a “shepherded” ring, resulting in less mass leaking out of the sandwiched region. Looking at models #4–6 in Figure <ref>, we see no clear trend between the rate at which sandwiched gas escapes and planet mass. There seems to be a complicated confluence of effects in the sandwiched region. Lindblad torques act to shepherd some gas while also opening gaps that reduce the local gas density; and co-orbital torques allow gas to escape via horseshoe orbits, whose libration times scale only weakly with planet mass (t_ lib∝ M_ p^-1/2; ). A detailed analysis is deferred to another paper; for now, we conclude that, at least for comparable mass planets with orbital spacings like the one we have assumed, the inner planet presents a porous barrier to material pushed inward by the outer planet. Apparently gas that is pushed by the outer planet toward the inner planet can be shuttled past the latter on horseshoe orbits (and vice versa). § SUMMARY AND DISCUSSION Using hydrodynamical simulations, we have demonstrated that super-Earths in inviscid disks can simultaneously avoid type I migration (fig:mig) and promote disk accretion (fig:acc) by driving density waves. Disk accretion rates measured from our simulations verify analytic predictions (eqn:ana_mdot_wnum) to within a factor of 2 . We observed gap opening and planet migration in inviscid disks to be modest and to introduce order-unity effects on the disk accretion rate. We also found in our two-planet simulations that material initially sandwiched between two planets leaks past both into the innermost and outermost disks (fig:eta). Our models omit a number of effects. Many of these are not overly concerning. Although our simulations are 2D, no substantive difference between 2D and 3D treatments of planet-disk interactions in viscous disks has been reported vis-à-vis gap opening <cit.> or planetary torques <cit.>. Our neglect of disk self-gravity should be an excellent approximation, as the Toomre Q-values of our disks greatly exceed unity. Our planets are not allowed to accrete gas, but super-Earths/sub-Neptunes are inferred observationally to have only modest amounts of gas—less than 10% by mass—acquired gradually over the entire disk lifetime <cit.>. More interesting frontiers to pursue include incorporating disk thermodynamics, as radiative cooling and differential heating across gap walls are thought to materially affect planet-disk interactions (e.g., ; ). Of course, extending the durations of the simulations, and including more planets with different orbital architectures, would also be welcome, for greater realism and to enable more direct connections with observations. Closer study of gap depths is warranted; we observed surface density contrasts only on the order of unity (fig:sigma), in clear deviation from scaling relations derived from viscous disks <cit.>, and surprising insofar as less viscous gas should be less effective at diffusively back-filling gaps. The shallowness of the gaps is due partly to the planets migrating and re-starting the gap-opening process at each new radial location <cit.>. But how much of it is due to the limited duration of our simulations (≤ 5000 yr), or to hydrodynamical instabilities like the Rayleigh instability <cit.>, remains to be worked out. Finally, survival against planetary migration is not guaranteed: planets with mass M_ p > M_ cr (eqn:m_cr) stall but less massive planets do not. The question is whether rocky planets can coagulate fast enough to cross the M_ cr threshold before they succumb to migration. Our results support the proposal by <cit.> that predominantly rocky planets—super-Earths and Earths—can solve, or at least help to solve, the problem of how protoplanetary gas disks ultimately disperse. Given that a single planet can push gas over a lengthscale of approximately half its orbital radius, shuttling gas from 5 AU down to 0.1 AU would require about 6 super-Earths distributed in roughly equal logarithmic intervals across this distance. Such planet multiplicities are reasonable, given the profusion of super-Earths/sub-Neptunes discovered by Kepler <cit.>. To be sure, a disk accretion flow driven by a planetary system will be unsteady, changing not only on secular, Myr-long timescales, but also on much shorter ones, with mass alternately accumulating and dispersing in interplanetary space, as we have seen in our simulations (fig:sigma). No matter how complicated the history, however, all gas must ultimately be torqued out of sufficiently compact planetary systems. It may be torqued by planets so far inward that turbulence driven by the magneto-rotational instability, activated in the innermost regions which are sufficiently thermally ionized <cit.>, takes over the job of disk accretion onto the host star. Or it may be torqued by planets so far outward that it escapes from the system altogether in a photoionized wind <cit.>. How massive a gas disk can a set of super-Earths drain? A first consideration is that the disk surface density can not be so large that the embedded planet mass M_ p < M_ cr∝Σ_ p^5/13, lest the planet migrate away. Based on the typical parameters listed in eqn:m_cr, a disk containing Σ_ pr_ p^2 ∼ 10^-3 M_∗∼ 300 M_⊕ of gas could be evacuated by ∼6 super-Earths weighing a total of ∼30 M_⊕. In such an initially gas-rich environment, we anticipate the planets would migrate to and fro by a few tens of percent in orbital distance (fig:mig). As the disk drains in the long term, whatever slow and erratic migration the planets undergo diminishes. Most of the angular momentum of interplanetary gas would be transported to the outermost disk, exterior to all the planets, either by Lindblad torques or by direct advection. The large cavities of transitional disks may have been excavated over time by families of super-Earths. Observationally, gas densities inside cavities can be suppressed relative to their values outside by two to four orders of magnitude (e.g., ; ). To reproduce these strong depletions, appeal is commonly made to giant planets in viscous α-disks that can open deep gaps <cit.>. But an alternative interpretation is that the cavities have been eroded gradually over time by much smaller mass planets in inviscid disks. So far as we have measured in our simulations, such planets open gaps having only order-unity surface density contrasts. Nevertheless, given sufficient time, they can drain interplanetary gas by orders of magnitude. Because planet-driven accretion rates Ṁ scale linearly with gas surface densities Σ, we have Σ̇∝ -Σ which implies exponential decay of the gas content. If it takes t_ e-fold∼ 10^5 yr to reduce a total disk mass of 10^-3 M_⊙ by a factor of e (this assumes a contemporaneous mass transport rate of 10^-8),[This transport can be inward or outward—it should not matter as long as the region occupied by the planets is monotonically drained of gas over time.] then it takes 7 t_ e-fold∼ 7 × 10^5 yr to reduce it by a factor of 1000. Another feature of this picture is that if super-Earths drain disk mass faster than it takes for their nascent atmospheres to cool and acquire more mass, they may be able to forestall runaway accretion <cit.>. Depleting the local disk density by a factor of ≳ 100 (relative to the minimum-mass extrasolar nebula) over timescales of ∼1 Myr suffices to keep the gas-to-solids mass fraction of super-Earths ≲ 10%, in accord with observations <cit.>. One potential problem with this scenario is that it predicts mass accretion rates to lower in proportion to disk gas densities. Although some transitional disks do have low accretion rates <cit.>, others do not, with a few having Ṁ as high as 10^-7 (e.g., ; ; ). Even so, system-to-system variations in orbital architectures, particularly in the masses of the orbiting companions, could help to resolve this problem. We have focused here on super-Earths because they are commonplace, but in principle gas giants and perhaps brown dwarfs or even low-mass stars may serve in their stead.[The case of the transitional disk HD 142527 is especially intriguing: its host star accretes at a rate of Ṁ∼ 10^-7 <cit.>, and its cavity contains a 0.2 M_⊙ companion highly inclined to the disk <cit.>.] That disk gas is intrinsically inviscid (laminar) is suggested on other, independent grounds. Large-scale asymmetries in transitional disks (e.g., ; ; ) have been interpreted as vortices (e.g., ; ), but these vortices are spawned only in low-viscosity disks. <cit.> found that the Shakura-Sunyaev viscosity parameter α needed to be 10^-4 or lower before vortices could grow from sharp density gradients. In other news, attempts to detect turbulence in the outer portions of disks using molecular line observations have so far come up empty-handed (; Flaherty et al. 2017, in preparation). And <cit.>, in a systematic analysis of disk accretion rates and masses, suggests that accretion may not proceed viscously (i.e., diffusively), but may be enabled instead by spiral density waves and/or disk winds. All these recent developments, in addition to our present work, suggest that the reason the community has not discovered a robust explanation for a non-zero α in protoplanetary disks is that none exists: that in fact such disks are for the most part inviscid, and accrete primarily by the action of gravitational torques, exerted either by disk gas itself at early times (e.g., ; ),[In self-gravitating disks, characterizing transport in terms of a non-zero α is commonly done, but mostly for convenience. Gravity is a long-range force and not naturally captured within a local theory like the one defining α.] or by planets at late times. We thank Pawel Artymowicz, Ruobing Dong, Eve Lee, Hui Li, Zhi-Yun Li, Frédéric Masset, Norm Murray, Ruth Murray-Clay, Sijme-Jan Paardekooper, Roman Rafikov, Yanqin Wu, Zhaohuan Zhu, and an anonymous referee for encouraging discussions and helpful feedback. This work was performed under contract with the Jet Propulsion Laboratory (JPL) funded by NASA through the Sagan Fellowship Program executed by the NASA Exoplanet Science Institute. EC is grateful for financial support from NASA and NSF. apj

http://arxiv.org/abs/1701.08174v2

20170127191321

Rotated Eigenstructure Analysis for Source Localization without Energy-decay Models

cs.IT

Rotated Eigenstructure Analysis for Source Localization without Energy-decay Models C. Garza ==================================================================================== Ginwidth=0.5 Ginwidth=1.0 Herein, the problem of simultaneous localization of two sources given a modest number of samples is examined. In particular, the strategy does not require knowledge of the target signatures of the sources a priori, nor does it exploit classical methods based on a particular decay rate of the energy emitted from the sources as a function of range. General structural properties of the signatures such as unimodality are exploited. The algorithm localizes targets based on the rotated eigenstructure of a reconstructed observation matrix. In particular, the optimal rotation can be found by maximizing the ratio of the dominant singular value of the observation matrix over the nuclear norm of the optimally rotated observation matrix. It is shown that this ratio has a unique local maximum leading to computationally efficient search algorithms. Moreover, analytical results are developed to show that the squared localization error decreases at a rate n^-3 for a Gaussian field with a single source, where n(log n)^2 scales proportionally to the number of samples M. § INTRODUCTION Underwater source detection and localization is an important but challenging problem. Classical range-based or energy-based source localization algorithms usually require energy-decay models and the knowledge of the environment <cit.>. However, critical environment parameters may not be available in many underwater applications, in which case, classical model-dependent methods may break down, even when the measurement snr is high. There have been some studies on source localization using nonparametric machine learning techniques, such as kernel regressions and support vector machines <cit.>. However, these methods either require a large amount of sensor data, or some implicit information of the environment, such as the choice of kernel functions. For example, determining the best kernel parameters (such as bandwidth) is very difficult given a small amount of data. This paper focuses on source detection and localization problems when only some structural properties of the energy field generated by the sources are available. Specifically, instead of requiring the knowledge of how energy decays with distance to the source, the paper aims at exploiting only the assumption that the closer to the source the higher energy received, and moreover, the energy field of the source is spatially invariant and decomposable. In fact, such a structural property is generic in many underwater applications. The prior work <cit.> studied the single source case, where an observation matrix is formed from a few energy measurements of the field in the target area, and the missing entries of the observation matrix are filled using matrix completion methods. Knowing that the matrix would be rank-1 under full and noise-free sampling of the whole area, svd is applied to extract the dominant singular vectors, and the source location is inferred from analyzing the peaks of the singular vectors. Herein, we propose to improve upon two shortcomings in <cit.>: we make rigorous an estimation/localization bound (versus focusing on the reduction of the search region) and we provide a method for localizing two sources. In the two source case, we need to tackle an additional difficulty that the svd of the observation matrix does not correspond to the signature vectors of the sources. To resolve this issue, a method of rotated eigenstructure analysis is proposed, where the observation matrix is formed by rotating the coordinate system such that the sources are aligned in a row or in a column of the matrix. We develop algorithms to first localize the central axis of the two sources, and then separate the sources on the central axis. To summarize, we derive algorithms to simultaneously localize up to two sources based on only a few power measurements in the target area without knowing any specific energy-decay model. The contributions of this paper are as follows: * We derive the location estimators with analytical results to show that the squared error decreases at a rate n^-3 for a Gaussian field with a single source, where n(log n)^2 scales proportionally to the number of samples M. * We develop a localization algorithm for the double source case based on a novel rotated eigenstructure analysis. We show that the two sources can be separated even when their aggregate power field has a single peak. The rest of the paper is organized as follows. Section <ref> gives the system model and assumptions. Section <ref> develops location estimator with performance analysis for single source case. Section <ref> proposes rotated eigenstructure analysis for double source case. Numerical results are given in Section <ref> and Section <ref> concludes this work. § SYSTEM MODEL Consider that there are K (K=1,2) sources with unknown locations 𝐬_k=(x_k^S,y_k^S)∈ℝ^2 located in a bounded area 𝒜. Suppose that the sensors can only measure the aggregate power transmitted by the sources, and is given by h(x,y)=∑_kh_k(x,y) for measurement location (x,y), where h_k(x,y)=α u(x-x_k^S)u(y-y_k^S) is the power density from source k, where α>0. The explicit form of the density function h_k(x,y) is unknown to the system, except that the characteristic function u(x) is known to have the following properties a) positive semi-definite, i.e., u(x)≥0 for all x∈ℝ b) symmetric, i.e., u(x)=u(-x) c) unimodal, i.e., u^'(x)<0 for x>0, d) smooth, i.e., |u^'(x)|<K_u for some K_u>0, and e) normalized, i.e., ∫_-∞^∞u(x)^2dx=1. Note that u(x) can be considered as the marginal power density function. Consider that M power measurements {h^(l)} are taken over distinct locations 𝐳^(l)=(x^(l),y^(l)), l=1,2,…,M, uniformly at random in the target area 𝒜. The measurements are assigned to a n_1× n_2 observation matrix 𝐇̂ as follows. First, partition the target area 𝒜 into n_1× n_2 disjoint cells 𝒢_ij, i=1,2,…,n_1 and j=1,2,…,n_2, where n_1 and n_2 are to be determined. Second, assign the power measurements h^(l) to the corresponding (i,j)th entry of 𝐇̂ as Ĥ_ij=s(𝒢_ij)h^(l) if 𝐳^(l)∈𝒢_ij,where s(𝒢_ij) measures the area of 𝒢_ij.[If multiple samples are close to each other and assigned to the same entry of 𝐇̂, the value of that entry is the average of the sample values.] Denote Ω as the set of observed entries of 𝐇̂, i.e., (i,j)∈Ω if there exists 𝐳^(l)∈𝒢_ij such that h^(l) is assigned to Ĥ_ij. For easy discussion, assume that 𝒜=[-L/2,L/2]×[-L/2,L/2], n_1=n_2=n, and 𝒢_ij are rectangles centered at (x_i,y_j), x_i=-L/2+L/2n+L/n(i-1), y_j=-L/2+L/2n+L/n(j-1), and have identical size with each other. Let 𝐇=α∑_k=1^K𝐮_k𝐯_k^T be the matrix of ideal observation, where 𝐮_k =L/N[u(x_1-x_k^S),u(x_2-x_k^S),…,u(x_n-x_k^S)]^T 𝐯_k =L/N[u(y_1-y_k^S),u(y_2-y_k^S),…,u(y_n-y_k^S)]^T for k=1,2. Thus 𝐇 has rank at most K. For (i,j)∈Ω, we have Ĥ_ij≈ H_ij, where the slight difference is due to sampling away from the centers of the cells 𝒢_ij. As a result, 𝐇̂ is a sparse and noisy observation of the low rank matrix 𝐇. An application example is illustrated in <ref>. The goal of this paper is to find the approximate locations of the sources using only the spatial invariant property (<ref>) and the four generic properties of the characteristic function u(x). Note that this problem is non-trivial. We insist on several features of the algorithm to be developed: it should be robust to structural knowledge of the signatures of the sources (as captured by g(x,y) in (<ref>)). This disallows the use of parametric regression or parameter estimation for source localization. In addition, we wish to under-sample the target area using small M. As such, maximum value entries may not represent the true locations of the sources. While not a focus of the current work, we will use matrix completion methods and the low rank property of 𝐇 as in <cit.> to cope with the under-sampled observations. § EIGENSTRUCTURE ANALYSIS FOR SINGLE SOURCE LOCALIZATION To simplify the discussion, the following mild assumptions are made.[The two assumptions are mainly to avoid discussing the effects on the boundary of 𝒜 and the high order noise term in the sampling noise model (<ref>). Straight-forward modifications can be made to handle the boundary effect in practical algorithms.] A1) The observation area 𝒜 is large enough, such that there is only negligible energy spreading outside the area 𝒜. A2) The parameter n is not too small, such that u(x_i-x_k^S)^2δ^2≈∫_x_i^x_i+1u(x-x_k^S)^2dx and u(y_i-y_k^S)^2δ^2≈∫_y_i^y_i+1u(y-y_k^S)^2dy for all i=1,2,…,n. Mathematically, the above assumptions imply that the vectors 𝐮_k and 𝐯_k have unit norm. §.§ Observation Matrix Construction We first exploit the low rank property of 𝐇 to obtain the full matrix 𝐇̂_c from the partially observed matrix 𝐇̂. Let 𝒫_Ω(𝐗) be a projection, such that the (i,j)th element of matrix 𝒫_Ω(𝐗) is [𝒫_Ω(𝐗)]_ij=X_ij if (i,j)∈Ω, and [𝒫_Ω(𝐗)]_ij=0 otherwise. The completed matrix 𝐇̂_c can be found as the unique solution to the following problem 𝐗minimize 𝐗_* subject to 𝒫_Ω(𝐗-𝐇̂)_F≤ϵ where 𝐗_* denotes the nuclear norm of 𝐗 and ϵ is a small parameter to tolerate the discrepancy between the two matrices. To choose a proper dimension n for the observation matrix 𝐇̂_c∈ℝ^n× n, we consider the results in <cit.>. It has been shown that under some mild conditions of 𝐇 (such as the strong incoherence property and small rank property), the matrix 𝐇∈ℝ^n× n can be exactly recovered with a high probability, if the dimension n satisfies Cn(log n)^2≤ M and noise-free sampling, Ĥ_ij=H_ij for (i,j)∈Ω, is performed. Here, C is a positive constant. Given this, we propose to choose n=n_c as the largest integer to satisfy n_c(log n_c)^2≤ M/C. §.§ Location Estimator Exploiting Property of Symmetry Consider the svd of the completed matrix 𝐇̂_c as 𝐇̂_c=α_1𝐮̂_1𝐯̂_1^T+∑_i=2^n_cα_i𝐮̂_i𝐯̂_i^T. We thus model the singular vectors of 𝐇̂_c as 𝐮̂_1=𝐮_1+𝐞_u and 𝐯̂_1=𝐯_1+𝐞_v. Note that the vectors 𝐮_1 and 𝐯_1 defined in (<ref>) and (<ref>), respectively, contain the source location information due to the unimodal property of u(x). However, due to the noise vectors 𝐞_u and 𝐞_v, the source location cannot be found by simply locating the peaks of 𝐮̂_1 and 𝐯̂_1. To resolve this difficulty, we exploit the symmetric property of u(x) and develop a location estimator as follows. Define a reflected correlation function as R̂(t;𝐮̂_1)=∫_-∞^∞û(x)û(-x+t)dx where û(x) is a (nonparametric) regression function from vector 𝐮̂_1. For example, û(x) can be obtained by û(x)=𝐮̂_1(i) if x=x_i, and by linear interpolation between 𝐮̂_1(i) and 𝐮̂_1(i+1) if x_i<x<x_i+1. Then the location estimator for x_1^S is given by x̂_1^S(𝐮̂_1)=1/2t∈ℝargmax R̂(t;𝐮̂_1). The location estimator for y_1^S can be obtained in a similar way. The location estimator (<ref>) exploits the fact that as 𝐮̂_1 is symmetric, the reflected correlation (<ref>), which is the correlation between 𝐮̂_1 and a reflected and shifted version of 𝐮̂_1, is maximized at the source location. Therefore, the estimator x̂_1^S(𝐮̂_1) tries to the suppress the perturbation from the noise by correlating over all the entries of 𝐮̂_1. We establish several properties for the estimator x̂_1^S(𝐮̂_1). Consider the autocorrelation for the characteristic function u(x) as τ(t)=∫_-∞^∞u(x)u(x-t)dx. Then, the following property can be derived. The autocorrelation function τ(t) is non-negative and symmetric. In addition, τ(t) is strictly decreasing in t>0. Let the dominant singular vector of 𝐇̂_c as the solution to (<ref>) be given by 𝐮̂_1=𝐮_1+𝐞_1, where 𝐮_1 is the dominant singular of 𝐇. Let 𝐞_1 be a vector with reverse elements of 𝐞_1, i.e., the jth element of 𝐞_1 equals to the last but the jth element of 𝐞_1. Let 𝐞_1^-t be a vector obtained from the t-shift of 𝐞_1, i.e., for t>0, the first t elements of 𝐞_1^-t are zeros and the remaining (n_c-t) elements of 𝐞_1^-t are identical to the first (n_c-t) elements of 𝐞_1; and for t<0, the first (n_c-t) elements of 𝐞_1^-t are identical to the last (n_c-t) elements of 𝐞_1 and the remaining t elements of 𝐞_1^-t are zeros. With such a notion, we make the following assumption on the singular vector 𝐮̂_1=𝐮_1+𝐞_1 of the completed matrix 𝐇̂_c: |𝐮_1^T𝐞_1^-t|≤ C_e|𝐮_1^T𝐞_1| for any 0≤ t≤ n_c-1, where C_e<∞ is a positive constant that only depends on the characteristic function u(x) but not n_c or M. Such an approximation is motivated by two observations. First, the entries of the vector 𝐞_1 may have roughly the same chance to take positive values or negative values because both 𝐮_1 and 𝐮_1+𝐞_1 have unit norm. Second, the magnitude of the elements in 𝐮_1 depends only on the characteristic function u(x) but not n_c or M. Although it is difficult to analytically validate the assumption (<ref>), it can be roughly confirmed by massive simulation results. As a result, we have the following theorem to characterize the estimation error of x̂_1^S. Suppose that the sampling error of 𝐇̂ from the true energy field matrix 𝐇 is bounded by 𝒫_Ω(𝐇̂-𝐇)_F≤ϵ̅ and the algorithm parameter ϵ in (<ref>) is chosen as ϵ=ϵ̅. Then, with high probability, |x̂_1^S-x_1^S|≤1/2τ^-1(1-μ_uL^6n_c(M)^-3+o(n_c(M)^-3)) where τ^-1(r) is the inverse function of r=τ(t), μ_u=C_e128u(0)^2K_u^2, and n_c(M) is the largest integer chosen such that M≥ Cn_c(log n_c)^2. The specific performance from (<ref>) depends on the characteristics of the energy field. Intuitively, if u(x) has a sharp peak (large slope of the autocorrelation function τ(t)), the localization error should be smaller. Consider a numerical example where the energy field has a Gaussian characteristic function. For a Gaussian characteristic function u(x)=(2γ/π)^1/4e^-γ x^2, there exists a constant C_μ, which only depends on the characteristic function u(x), such that with high probability, the squared estimation error is upper bounded by |x̂_1^S-x_1^S|^2+|ŷ_1^S-y_1^S|^2≤ C_μL^6n_c(M)^-3+o(n_c(M)^-3). Theorem <ref> and Corollary <ref> gives the asymptotic performance of the proposed localization algorithm without knowing the energy-decay model. For large M, the worst case squared error decays at a rate n_c(M)^-3. As a benchmark, the squared error of a naive scheme, which estimates the source location directly from the position of the measurement sample that observes the highest power, decreases as M^-1, which is equivalent to n_c(M)^-1(log n_c(M))^-2, much slower than that of the proposed algorithm. This is because, the granularity of the original observations is L/√(M). The results then confirm that by exploiting the low rank property using matrix completion and the reflected correlation technique, the proposed algorithm significantly improves the localization resolution. § ROTATED EIGENSTRUCTURE ANALYSIS FOR DOUBLE SOURCE LOCALIZATION The location estimator x̂_1^S in (<ref>) is based on the intuition that the singular vectors of 𝐇 are just the vectors 𝐮_1 and 𝐯_1, which contains the source location in their peaks. However, a similar technique cannot be applied to the two source case, because 𝐮_k and 𝐯_k may not be the singular vectors of 𝐇, as the vectors {𝐮_k} may not be orthogonal. §.§ Optimal Rotation of the Observation Matrix When there are two sources, the (ideal) observation matrix 𝐇 is not rank-1, expect for the special case where the two sources are aligned on one of the axes of the coordinate system. wlog, assume that the sources are aligned with the x-axis, where y_k^S=C for k=1,2. Consequently, we have 𝐯_1=𝐯_2, and 𝐇=α(∑_k𝐮_k)𝐯_1^T, which is rank-1. Hence, the right singular vector of 𝐇 is 𝐯_1 and, by analyzing the peak of 𝐯_1, the central axis ŷ_k^S=C can be estimated. The above observations suggest that we rotate the coordinate system such that the sources are aligned with one of the axes. Consider rotating the coordinate system by θ. The entries of 𝐇̂_c are rearranged into a new observation matrix 𝐇̂_θ, where [𝐇̂_θ]_(i,j)=[𝐇̂_c]_(p,q) in which (p,q) is the index such that (x_p^',y_q^') is the closest point in Euclidean distance to (x̅,y̅) in the original coordinate system 𝒞_0, with x̅=dcos(β+θ) and y̅=dsin(β+θ). Here β=∠(x_i,y_j) is the angle of (x_i,y_j) to the x-axis of the rotated coordinate system 𝒞_θ, and d=(x_i,y_j)_2. Note that 1≤ i,j≤ n^', where n^'≤ n_c, since the rotation of the axes induce truncation of some data samples. Let the orientation angle of the central axis of the sources wrt the x-axis in the original coordinate system 𝒞_0 be θ_0, θ_0∈[0,π). Then the desired rotation for coordinate system 𝒞_θ would be θ^*=θ_0 for θ_0<π/2, or θ^*=θ_0-π/2 for θ_0≥π/2. The desired rotation θ can be obtained as θ∈[0,π/2]maximize ρ(θ)≜λ_1(𝐇̂_θ)/∑_kλ_k(𝐇̂_θ) where λ_k(𝐀) is the kth largest singular value of 𝐀. Note that ρ(θ)≤1 for all θ∈[0,π/2] and ρ(θ^*)=1, where 𝐇̂_θ becomes a rank-1 matrix when the sources are aligned with one of the axes. The maximization problem (<ref>) is in general non-convex. An exhaustive search for the solution θ^* is computationally expensive, since for each θ, svd should be performed to obtain the singular value profile of 𝐇̂_θ. Therefore, we need to study the properties of the alignment metric ρ(θ) in order to develop efficient algorithms for the source detection. §.§ The Unimodal Property We also show that the function ρ(θ) also has the unimodal property defined as follows. A function f(x) is called unimodal in a bounded region (a,b), if there exists x_0∈[a,b], such that f^'(x)f^'(y)<0 for any a<x<x_0<y<b. The unimodality suggests that f(x) has a single peak in (a,b), and hence f(x) has a unique local maximum (or minimum). The function ρ(θ) in (<ref>) is unimodal in θ∈(θ^*-π/4,θ^*+π/4), if s·τ^'(t)>t·τ^'(s) for all 0<s<t, where τ^'(t)≜d/dtτ(t). In addition, ρ(θ) is strictly increasing over (θ^*-π/4,θ^*) and strictly decreasing over (θ^*,θ^*+π/4). The result in Theorem <ref> is powerful, since it confirms that the function ρ(θ) is unimodal within a π/2-window, and there is a unique local maximum, when the autocorrelation of the energy field characteristic function u(x) agrees with the condition (<ref>). Note that ρ(θ) is also symmetric wrt θ=θ^*. As a result, a simple bisection search algorithm can efficiently find the global optimal solution θ^* to (<ref>). An example algorithm is given in Algorithm <ref>. Note that condition (<ref>) can be satisfied by a variety of energy fields. For example, for Laplacian field u(x)=√(γ)e^-γ|x|, we have τ(t)=(1+γ t)e^-γ t, and τ^'(t)=-γ^2te^-γ t; for Gaussian field u(x)=(2γ/π)^1/4e^-γ x^2, we have τ(t)=e^-γ t^2/2, and τ^'(t)=-γ te^-γ t^2/2. In both cases, condition (<ref>) is satisfied. §.§ Source Detection In the coordinate system 𝒞_θ under optimal rotation θ=θ^* (assuming alignment on the x-axis), the left and right singular vectors of 𝐇̂_θ can be modeled as 𝐮̂_1=1/2(𝐮_1(θ^*)+𝐮_2(θ^*))+𝐞_u and 𝐯̂_1=𝐯_1(θ^*)+𝐞_v, respectively. Correspondingly, the y-coordinates of the sources can be the found using estimator (<ref>) based on reflected correlation ŷ_1^S(𝐯̂_1;θ^*)=ŷ_2^S(𝐯̂_1;θ^*)=1/2t∈ℝargmax R̂(t;𝐯̂_1). To find the x-coordinates, note that the function u_1(x)=1/2(u(x-x_1^S)+u(x-x_2^S)) is symmetric at x=1/2(x_1^S+x_2^S). Therefore, the center of the two sources can be found by ĉ=1/2t∈ℝargmax R̂(t;𝐮̂_1). In addition, after estimating ŷ_1^S, the marginal power density function u(x) can be obtained as û(y)=v̂_1(y-ŷ_1^S), where v̂_1(y) is a regression function from 𝐯̂_1 (for example, by linear interpolation among y_1,y_2,…,y_n_c). As a results, the x-coordinates of the two sources can be found using similar techniques as spread spectrum early gate synchronization <cit.>, and obtained as x̂_1^S(θ^*)=ĉ-d̂ and x̂_2^S(θ^*)=ĉ+d̂, where d̂ =d≥0argmax Q(d;𝐮̂_1,𝐯̂_1) and Q(d;𝐮̂_1,𝐯̂_1)≜1/2∫_-∞^∞û_1(x)(û(x-ĉ-d)+û(x-ĉ+d))dx. It is straight-forward to show that Q(d;𝐮̂_1,𝐯̂_1) is maximized at d^*=1/2|x_1^S-x_2^S|. As a benchmark, consider a naive scheme that estimates x_1^S and x_2^S by analyzing the peaks of 𝐮̂_1. However, such naive strategy cannot work for small source separation, because if d=1/2|x_1^S-x_2^S| is too small, the aggregate power density function ũ_1(x)=u(x-x_1^S)+u(x-x_1^S-d) would be unimodal and there is only one peak in 𝐮̂_1. As a comparison, the proposed procedure estimator from procedure (<ref>)–(<ref>) does not such a limitation. § NUMERICAL RESULTS In this section, we evaluate the performance of the proposed location estimator in both single source and double source cases. Two sources are placed in the area [-0.5,0.5]×[-0.5,0.5] uniformly and independently at random, with the restriction that the distance between the two sources is no more than 0.5.[When the two sources are far apart, the problem degenerates to two single-source-localization problems.] The power field generated by each source in an underwater environment is modeled as h_k(x,y)=e^-20(x-x_k^S)^2-20(x-y_k^S)^2, k=1,2. There are M power measurements taken in the area 𝒜=[-1,1]×[-1,1] uniformly at random. The parameter n_c of the proposed observation matrix 𝐇̂∈ℝ^n_c× n_c is chosen as the largest integer satisfying n_c(log n_c)^2≤ M/C, for C=1. As a benchmark, the proposed location estimation is compared with the naive scheme, which determines the source location directly form the position of the measurement sample that observes the highest power. In the two source case, the naive algorithm aims at detecting either one of the sources, and the corresponding localization error is computed as ℰ_ naive^2=min{𝐬̂_ naive-𝐬_1^2,𝐬̂_ naive-𝐬_2^2}. As a comparison, the localization error of the proposed algorithm is computed as ℰ^2=1/2(𝐬̂_1-𝐬_1^2+𝐬̂_2-𝐬_2^2). Fig. <ref> depicts the mse of the source location versus the number of samples M. In the single source case,the coefficient of the worst case upper bound (<ref>) is chosen as C_μ=1 to demonstrate the asymptotic decay rate of the worst case squared error bound. The decay rate of the analytic worst case error bound is roughly the same as the mse obtained from the numerical experiment. It is expected that as M increases, the two curves merge in an asymptotic way. As a benchmark, the proposed scheme requires less than half of the samples to achieve similar performance to that of the naive baseline even for small M (around 50). More importantly, it demonstrates a higher mse decay rate, where for medium M (around 200), the proposed scheme reduces the number of samples to 1/10. In the double source case, there is an error floor for the naive scheme, because the location that observes the highest power may not be either one of the source locations. As a comparison, there is no error floor in for proposed scheme as M increases. Fig. <ref> shows an example on simultaneously localizing two sources (red crosses). Although the aggregate power field has only one peak, the algorithm (black circles) is able to separate the two sources. § CONCLUSIONS This paper developed source localization algorithms from a few power measurement samples, while no specific energy-decay model is assumed. Instead, the proposed method only exploited the structural property of the power field generated by the sources. Analytical results were developed to demonstrate that the proposed algorithm decreases the localization error at a higher rate than the baseline algorithm when the number of samples increases. In addition, a rotated eigenstructure analysis technique was derived for simultaneously localizing two sources. Numerical results demonstrate the performance advantage in localizing single or double sources. § ACKNOWLEDGMENTS This research was supported, in part, by National Science Foundation under Grant NSF CNS-1213128, CCF-1410009, CPS-1446901, Grant ONR N00014-15-1-2550, and Grant AFOSR FA9550-12-1-0215. § APPENDIX §.§ Proof of Lemma <ref> τ^'(t) =d/dt∫_-∞^∞u(x)u(x-t)dx =∫_-∞^∞-u(x)u^'(x-t)dx =-∫_-∞^0u(z+t)u^'(z)dz-∫_0^∞u(z+t)u^'(z)dz =-∫_-∞^0u(z+t)u^'(z)dz+∫_0^∞u(z+t)u^'(-z)dz =-∫_-∞^0u(z+t)u^'(z)dz+∫_-∞^0u(-w+t)u^'(w)dw =-∫_-∞^0[u(z+t)-u(-z+t)]u^'(z)dz =-∫_-∞^0[u(z+t)-u(z-t)]u^'(z)dz <0 where (<ref>) is due to the change of variable z=x-t and u^'(z)=-u^'(-z), (<ref>) is to change the variable z=-w, (<ref>) exploits the fact that u(x)=u(-x), and the last inequality is due to u(z+t)-u(z-t)>0 and u^'(z)>0 for all z<0. §.§ Proof of Theorem <ref> To simplify the algebra, we only focus on the dominant terms wrt n_c as n_c goes large. §.§.§ Upper Bound of the Sampling Error For notational convenience, define u_1(x)=u(x-x_1^S) and v_1(y)=u(x-y_1^S). Consider the sampling position (x,y)∈𝒢_ij. Using a Taylor expansion, we have |h_1(x,y)-h_1(x_1,y_1)| =α|u_1(x)v_1(y)-u_1(x_1)v_1(y_1)| =α|(u_1(x_1)+u_1^'(x_1)(x-x_1)) ×(v_1(y_1)+v_1^'(y_1)(y-y_1))-u_1(x_1)v_1(y_1) +o(x-x_1)+o(y-y_1)| =α|u_1(x_1)v_1^'(y_1)(y-y_1)+v_1(y_1)u_1^'(x_1)(x-x_1)| +o(x-x_1)+o(y-y_1) ≤α u(0)K_uL/n_c+o(L/n_c) from the property u(x)≤ u(0) and |u^'(x)|≤ K_u. From (<ref>), we have |Ĥ_ij-H_ij| =(L/n_c)^2|h_1(x,y)-h_1(x_1,y_1)| ≤α u(0)K_uL^3/n_c^3+o(L^3/n_c^3). As a result, 𝒫_Ω(𝐇̂_c-𝐇)_F^2 =∑_(i,j)∈Ω|Ĥ_ij-H_ij|^2 ≤ M(α u(0)K_uL^3/n_c^3)^2≜ϵ̅^2. §.§.§ Matrix Completion with Noise and Singular Vector Perturbation When there is sampling noise, the performance of matrix completion can be evaluated by the following result. Consider that ϵ in (<ref>) is chosen such that 𝒫_Ω(𝐇̂-𝐇)_F≤ϵ=ϵ̅. Then, with high probability, δ≜𝐇̂_c-𝐇_F≤4√((2+p)n_c/p)ϵ̅+2ϵ̅ where p=M/n_c^2. As we focus on not too small n_c, which is chosen to be such that M≈ Cn_c(log n_c)^2, the bound (<ref>) can be simplified as δ ≤4√((2+Cn_c(log n_c)^2/n_c^2)n_c/Cn_c(log n_c)^2/n_c^2)ϵ+2ϵ ≈√(32/C)n_c/log n_cϵ. Let 𝐮_1 and 𝐮̂_1=𝐮_1+𝐞_1 be the dominant left singular vectors of 𝐇 and 𝐇̂_c, respectively. We exploit the following classical result from singular vector perturbation analysis. Let σ_1 and σ_2 be the first and second dominant singular values of 𝐇. Then, sin∠(𝐮_1,𝐮̂_1)≤2𝐇̂_c-𝐇_F/σ_1-σ_2=2δ/σ_1-σ_2. By exploiting Lemma <ref> for our case, we have sin∠(𝐮_1,𝐮̂_1) =√(1-|𝐮_1^T(𝐮_1+𝐞_1)|^2) =√(-2𝐮_1^T𝐞_1+|𝐮_1^T𝐞_1|^2) ≈√(2|𝐮_1^T𝐞_1|) where |·| denotes the absolute value operator, and we drop the second order term |𝐮_1^T𝐞_1|^2, since |𝐮_1^T𝐞_1| is small as we focus on large n_c(M). We also note that 𝐮_1^T𝐞_1≤0. Consider that we have chosen M≈ Cn_c(log n_c)^2, and moreover, 𝐇 is a rank-1 matrix with singular value σ_1=α. As a result, 2|𝐮_1^T𝐞_1|≈sin^2∠(𝐮_1,𝐮̂_1)≤(2δ/α)^2 ≤4/α^232/C(n_c/log n_c)^2ϵ̅^2 =128u(0)^2K_u^2L^6n_c^-3. §.§.§ Estimator based on Reflected Correlation Let e(x)=û(x)-u_1(x). Define a reflected correlation function as R(t;x_1^S)=∫_-∞^∞u(x-x_1^S)u(-x-x_1^S+t)dx. Then, it follows that R(t;x_1^S)=τ(2x_1^S-t). As a result, we have R̂(t;𝐮̂_1) =∫_-∞^∞(u_1(x)+e(x))(u_1(-x+t)+e(-x+t))dx =∫_-∞^∞u_1(x)u_1(-x+t)dx+∫_-∞^∞u_1(x)e(-x+t)dx ∫_-∞^∞e(x)u_1(-x+t)dx+∫_-∞^∞e(x)e(-x+t)dx ≈ R(t;x_1^S)+∫_-∞^∞u_1(x)e(-x+t)dx +∫_+∞^-∞e(-y+t)u_1(y)(-dy) =R(t;x_1^S)+2∫_-∞^∞u_1(x)e(-x+t)dx ≈ R(t;x_1^S)+2𝐮_1^T𝐞_1^-t where the first approximation (<ref>) is by dropping the second order term ∫_-∞^∞e(x)e(-x+t)dx, and the second approximation (<ref>) is to use the inner product 𝐮_1^T𝐞_1^-t to approximate the integral based on assumptions A1 and A2 in Section <ref>. As a result, we have R(t;x_1^S)-R̂(t;𝐮̂_1)≈-2𝐮_1^T𝐞_1^-t. Recall that t̂=2x̂_1^S maximizes R̂(t̂;𝐮̂_1) and t^*=2x_1^S maximizes R(t^*;x_1^S)=τ(2x_1^S-t^*). We have τ(0)-τ(2|x̂_1^S-x_1^S|) =R(t^*;x_1^S)-R̂(t̂;𝐮̂_1) ≈-2𝐮_1^T𝐞_1^-t ≤ C_e2|𝐮_1^T𝐞_1| ≤μ_uL^6n^-3+o(n_c^-3) where μ_u=C_e128u(0)^2K_u^2 and o(n_c^-3) is due to the fact that we keep omitting the higher order terms. Finally, we obtain τ(2|x̂_1^S-x_1^S|)=1-μ_uL^6n^-3+o(n_c^-3) and hence, |x̂_1^S-x_1^S|≤1/2τ^-1(1-μ_uL^6n_c^-3+o(n_c^-3)). §.§ Proof of Theorem <ref> We first study the singular vectors in double source case. Let 𝐮_k(θ) and 𝐯_k(θ) be the vectors defined following (<ref>) and (<ref>) in the rotated coordinate system 𝒞_θ. The svd of 𝐇_θ is given by 𝐇_θ=α_1𝐩_1𝐪_1^T+α_2𝐩_2𝐪_2^T where α_1=α/2𝐮_1+𝐮_2𝐯_1+𝐯_2 and α_2=α/2𝐮_1-𝐮_2𝐯_1-𝐯_2 are the singular values, and 𝐩_1=𝐮_1+𝐮_2/𝐮_1+𝐮_2, 𝐪_1=𝐯_1+𝐯_2/𝐯_1+𝐯_2 𝐩_2=𝐮_1-𝐮_2/𝐮_1-𝐮_2, 𝐪_2=𝐯_1-𝐯_2/𝐯_1-𝐯_2 are the corresponding singular vectors. First, 𝐇_θ =α̅(𝐮_1𝐯_1^T+𝐮_2𝐯_2^T) =α̅/2[(𝐮_1+𝐮_2)(𝐯_1+𝐯_2)^T+(𝐮_1-𝐮_2)(𝐯_1-𝐯_2)^T] =α_1𝐩_1𝐪_1^T+α_2𝐩_2𝐪_2^T Hence, these four vectors form a decomposition of 𝐇_θ. Second, we have 𝐩_1^T𝐩_2 =c(𝐮_1+𝐮_2)^T(𝐮_1-𝐮_2) =c(𝐮_1^2-𝐮_2^2) =0 where c=1/(𝐮_1+𝐮_2𝐮_1-𝐮_2). Similarly, 𝐪_1^T𝐪_2=0. In addition, all the four vectors have unit norm. As a result, (<ref>) is the svd of 𝐇_θ. Consider an arbitrary coordinate system. wlog (due to Assumption 1), assume that the first source is located at the origin, x_1^S=0 and y_1^S=0, and the second source is away from the first source with distance D and angle θ to the x-axis, x_2^S=Dcosθ and y_2^S=Dsinθ. In addition, defining u_c(x,θ)≜ u(x-Dcosθ), u_s(x,θ)≜ u(x-Dsinθ) we have 𝐮_1 =√(δ)[u(x_1),u(x_2),…,u(x_N)]^T 𝐯_1 =√(δ)[u(y_1),u(y_2),…,u(y_M)]^T 𝐮_2 =√(δ)[u_c(x_1,θ),u_c(x_2,θ),…,u_c(x_N,θ)]^T 𝐯_2 =√(δ)[u_s(y_1,θ),u_s(y_2,θ),…,u_s(y_M,θ)]^T. Based on assumption A1 and A2, we have 𝐮_k^2=(L/n)^2∑_i=1^Nu(x_i-x_k^S)^2 ≈∫_x_1^x_n-1u(x-x_k^S)^2dx ≈∫_-∞^∞u(x-x_k^S)^2dx=1 and similar integrals apply to 𝐯_k. As an equivalent statement to Theorem <ref>, we need to show that ρ(θ) is a strictly increasing function in θ∈(0,π/4). Equivalently, we should prove that the function λ_2(𝐇_θ)^2/λ_1(𝐇_θ)^2 ≈∫_-∞^∞(u(x)-u_c(x,θ))^2dx/∫_-∞^∞(u(x)+u_c(x,θ))^2dx∫_-∞^∞(u(x)-u_s(x,θ))^2dx/∫_-∞^∞(u(x)+u_s(x,θ))^2dx ≜μ(θ) is strictly increasing in θ∈(0,π/4), where the approximated integrals are obtained from (<ref>). To simplify the notation, define the integration operator ⟨·⟩ as ⟨ f⟩≜∫_-∞^∞f(x,θ)dx for a function f(x,θ). By definition, the integration operator is linear and satisfies the additive property, i.e., ⟨ af⟩=a⟨ f⟩ and ⟨ f+g⟩=⟨ f⟩+⟨ g⟩, for a constant a and a function g(x,θ). As a result, ⟨(u-u_c)^2⟩=⟨ u^2⟩+⟨ u_c^2⟩-2⟨ u· u_c⟩=2(1-⟨ u· u_c⟩), and the function μ(θ) can be written as μ(θ)=(1-⟨ u· u_c⟩)(1-⟨ u· u_s⟩)/(1+⟨ u· u_c⟩)(1+⟨ u· u_s⟩). In addition, from the properties in calculus, if f(x,θ) and ∂/∂θf(x,θ) are continuous in θ, then d/dθ⟨ f⟩ =d/dθ∫_-∞^∞f(x,θ)dx =∫_-∞^∞∂/∂θf(x,θ)dx=⟨∂/∂θf⟩. Therefore, defining u_c^'(x,θ) ≜d/dxu(x)|_x=x-Dcosθ u_s^'(x,θ) ≜d/dxu(x)|_x=x-Dsinθ we have d/dθ⟨ u· u_c⟩ =⟨ u·∂/∂θu_c(x,θ)⟩=⟨ u· u_c^'⟩ Dsinθ d/dθ⟨ u· u_s⟩ =⟨ u·∂/∂θu_s(x,θ)⟩=-⟨ u· u_s^'⟩ Dcosθ. With some algebra, the derivative of μ(θ) can be obtained as d/dθμ(θ) =η[Dcosθ⟨ u· u_s^'⟩(1-⟨ u· u_c⟩^2) -Dsinθ⟨ u· u_c^'⟩(1-⟨ u· u_s⟩^2)] =η[-t·τ^'(s)(1-τ(t)^2)+s·τ^'(t)(1-τ(s)^2)] where η=2(1+⟨ u· u_c⟩)^-2(1+⟨ u· u_s⟩)^-2, t=Dcosθ, and s=Dsinθ. Note that 0<s<t for 0<θ<π/4. Applying condition (<ref>), we have d/dθμ(θ) >η· t·τ^'(s)[(1-τ(s)^2)-(1-τ(t)^2)] =η· t·τ^'(s)(τ(t)^2-τ(s)^2) >0 since τ^'(s)<0 and τ(t)<τ(s) for 0<s<t. This confirms that μ(θ) is a strictly increasing function, and hence ρ(θ) is a strictly increasing function in θ∈(0,π/4). The results in Theorem <ref> is confirmed. IEEEtran 10 url@samestyle BecStoLi:J08 A. Beck, P. Stoica, and J. Li, “Exact and approximate solutions of source localization problems,” IEEE Trans. Signal Process., vol. 56, no. 5, pp. 1770–1778, 2008. QiXiuYua:J13 H.-D. Qi, N. Xiu, and X. Yuan, “A lagrangian dual approach to the single-source localization problem,” IEEE Trans. Signal Process., vol. 61, no. 15, pp. 3815–3826, 2013. SheHu:J05 X. Sheng and Y.-H. Hu, “Maximum likelihood multiple-source localization using acoustic energy measurements with wireless sensor networks,” IEEE Trans. Signal Process., vol. 53, no. 1, pp. 44–53, 2005. MeeMitNar:J08 C. Meesookho, U. Mitra, and S. Narayanan, “On energy-based acoustic source localization for sensor networks,” IEEE Trans. Signal Process., vol. 56, no. 1, pp. 365–377, 2008. LiuHuPan:J12 Y. Liu, Y. H. Hu, and Q. Pan, “Distributed, robust acoustic source localization in a wireless sensor network,” IEEE Trans. Signal Process., vol. 60, no. 8, pp. 4350–4359, 2012. ZisWax:C88 I. Ziskind and M. Wax, “Maximum likelihood localization of multiple sources by alternating projection,” Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, vol. 36, no. 10, pp. 1553–1560, 1988. LefRea:J17 R. Lefort, G. Real, and A. Drémeau, “Direct regressions for underwater acoustic source localization in fluctuating oceans,” Applied Acoustics, vol. 116, pp. 303–310, 2017. NguJorSin:J05 X. Nguyen, M. I. Jordan, and B. Sinopoli, “A kernel-based learning approach to ad hoc sensor network localization,” ACM Trans. on Sensor Networks, vol. 1, no. 1, pp. 134–152, 2005. JinSohWon:J10 Y. Jin, W.-S. Soh, and W.-C. Wong, “Indoor localization with channel impulse response based fingerprint and nonparametric regression,” IEEE Trans. Wireless Commun., vol. 9, no. 3, pp. 1120–1127, 2010. KimParYooKimPar:J13 W. Kim, J. Park, J. Yoo, H. J. Kim, and C. G. Park, “Target localization using ensemble support vector regression in wireless sensor networks,” IEEE Trans. on Cybernetics, vol. 43, no. 4, pp. 1189–1198, 2013. ChoMit:C15 S. Choudhary and U. Mitra, “Analysis of target detection via matrix completion,” in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing, 2015, pp. 3771–3775. ChoKumNarMit:J16 S. Choudhary, N. Kumar, S. Narayanan, and U. Mitra, “Active target localization using low-rank matrix completion and unimodal regression,” arXiv preprint arXiv:1601.07254, 2016. CanPla:J10 E. J. Candes and Y. Plan, “Matrix completion with noise,” Proceedings of the IEEE, vol. 98, no. 6, pp. 925–936, 2010. peterson:b95 R. Peterson, R. Ziemer, and D. Borth, Introduction to spread spectrum systems.1em plus 0.5em minus 0.4emEnglewood Cliffs, NJ: Prentice-Hall, 1995. vu:J11singular V. Vu, “Singular vectors under random perturbation,” Random Structures & Algorithms, vol. 39, no. 4, pp. 526–538, 2011.

http://arxiv.org/abs/1701.07818v3

20170126185300

Turaev-Viro invariants, colored Jones polynomials and volume

math.GT

*namedtheorem named[1] theoremTheorem[section] lemma[theorem]Lemma proposition[theorem]Proposition definition[theorem]Definition corollary[theorem]Corollary conjecture[theorem]Conjecture question[theorem]Question remark remark[theorem]Remark remark example[theorem]Example equationsection 16cm 24.5cm -2cm -2.0cm16cm 25.0cm -2cm -2.0cm16cm 23cm -2cm -2.0cm2mm Turaev-Viro invariants, colored Jones polynomials and volume Renaud Detcherry, Efstratia Kalfagianni [E.K. is supported by NSF Grants DMS-1404754 and DMS-1708249] and Tian Yang [T.Y. is supported by NSF Grant DMS-1405066] ========================================================================================================================================================================= We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomials of the link. As an application, we give the first examples of 3-manifolds where the “large r" asymptotics of the Turaev-Viro invariants determine the hyperbolic volume. We verify the volume conjecture of Chen and the third named author <cit.> for the figure-eight knot and the Borromean rings. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem to be predicted neither by the Kashaev-Murakami-Murakami volume conjecture and its generalizations nor by Zagier's quantum modularity conjecture. We conjecture that the asymptotics of the Turaev-Viro invariants of any link complement determine the simplicial volume of the link, and verify this conjecture for all knots with zero simplicial volume. Finally, we observe that our simplicial volume conjecture is compatible with connected summations and split unions of links. § INTRODUCTION In <cit.>, Turaev and Viro defined a family of 3-manifold invariants as state sums on triangulations of manifolds. The family is indexed by an integer r, and for each r the invariant depends on a choice of a 2r-th root of unity. In the last couple of decades these invariants have been refined and generalized in many directions and shown to be closely related to the Witten-Reshetikhin-Turaev invariants. (See <cit.> and references therein.) Despite these efforts, the relationship between the Turaev-Viro invariants and the geometric structures on 3-manifolds arising from Thurston's geometrization picture is not understood. Recently, Chen and the third named author <cit.> conjectured that, evaluated at appropriate roots of unity, the large-r asymptotics of the Turaev-Viro invariants of a complete hyperbolic 3-manifold, with finite volume, determine the hyperbolic volume of the manifold, and presented compelling experimental evidence to their conjecture. In the present paper, we focus mostly on the Turaev-Viro invariants of link complements in S^3. Our main result gives a formula of the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomials of the link. Using this formula we rigorously verify the volume conjecture of <cit.> for the figure-eight knot and Borromean rings complement. To the best of our knowledge these are first examples of this kind. Our calculations exhibit new phenomena of asymptotic behavior of the colored Jones polynomials that does not seem to be predicted by the volume conjectures <cit.> or by Zagier's quantum modularity conjecture <cit.>. §.§ Relationship between knot invariants To state our results we need to introduce some notation. For a link L⊂ S^3, let TV_r(S^3∖ L,q) denote the r-th Turaev-Viro invariant of the link complement evaluated at a root of unity q such that q^2 is primitive of degree r. Throughout this paper, we will consider the case that q=A^2, where A is either a primitive 4r-th root for any integer r or a primitive 2r-th root for any odd integer r. We use the notation 𝐢=(i_1,…,i_n) for a multi-integer of n components (an n-tuple of integers) and use the notation 1⩽𝐢⩽ m to describe all such multi-integers with 1⩽ i_k ⩽ m for each k∈{1,…, n}. Given a link L with n components, let J_L,𝐢(t) denote the 𝐢-th colored Jones polynomial of L whose k-th component is colored by i_k <cit.>. If all the components of L are colored by the same integer i, then we simply denote J_L,(i,…,i)(t) by J_L,i(t). If L is a knot, then J_L,i(t) is the usual i-th colored Jones polynomial. The polynomials are indexed so that J_L,1(t)=1 and J_L, 2(t) is the ordinary Jones polynomial, and are normalized so that J_U,i(t)=[i] =A^2i-A^-2i/A^2-A^-2 for the unknot U, where by convention t=A^4. Finally, we define η_r=A^2-A^-2/√(-2r) and η_r'=A^2-A^-2/√(-r). Before stating our main result, let us recall once again the convention that q=A^2 and t=A^4. Let L be a link in S^3 with n components. * For an integer r⩾ 3 and a primitive 4r-th root of unity A, we have TV_r(S^3∖ L,q)=η_r^21 ⩽𝐢⩽ r-1∑ | J_L, 𝐢 (t) |^2. * For an odd integer r⩾ 3 and a primitive 2r-th root of unity A, we have TV_r(S^3∖ L,q)=2^n-1(η_r')^21 ⩽𝐢⩽r-1/2∑ | J_L, 𝐢 (t)|^2. Extending an earlier result of Roberts <cit.>, Benedetti and Petronio <cit.> showed that the invariants TV_r(M,e^π i/r) of a 3-manifold M, with non-empty boundary, coincide up to a scalar with the SU(2) Witten-Reshetikhin-Turaev invariants of the double of M. The first step in our proof of Theorem <ref> is to extend this relation to the Turaev-Viro invariants and the SO(3) Reshetikhin-Turaev invariants <cit.>. See Theorem <ref>. For this we adapt the argument of <cit.> to the case that r is odd and A is a primitive 2r-th root of unity. Having this extension at hand, the proof is completed by using the properties of the SO(3) Reshetikhin-Turaev Topological Qantum Field Theory (TQFT) developed by Blanchet, Habegger, Masbaum and Vogel <cit.>. Note that for any primitive r-th root of unity with r⩾ 3, the quantities η_r and η_r' are real and non-zero. Since J_L,1(t)=1, and with the notation as in Theorem <ref>, we have the following. For any r⩾ 3, any root q=A^2 and any link L in S^3, we have TV_r(S^3∖ L,q) ⩾ H_r>0, where H_r=η_r^2 in case (1), and H_r= 2^n-1(η_r')^2 in case (2). Corollary <ref> implies that the invariants TV_r(q) do not vanish for any link in S^3. In contrast to that, the values of the colored Jones polynomials involved in the Kashaev-Murakami-Murakami volume conjecture <cit.> are known to vanish for split links and for a class of links called Whitehead chains <cit.>. Another immediate consequence of Theorem <ref> is that links with the same colored Jones polynomials have the same Turaev-Viro invariants. In particular, since the colored Jones polynomials are invariant under Conway mutations and the genus 2 mutations <cit.>, we obtain the following. For any r⩾ 3, any root q=A^2 and any link L in S^3, the invariants TV_r(S^3∖ L,q) remain unchanged under Conway mutations and the genus 2 mutations. §.§ Asymptotics of Turaev-Viro and colored Jones link invariants We are interested in the large r asymptotics of the invariants TV_r(S^3∖ L,A^2) in the case that either A=e^π i/2r for integers r⩾ 3, or A=e^π i/r for odd integers r⩾ 3. With these choices of A, we have in the former case that η_r=2sin(π/r)/√(2r), and in the latter case that η_r'=2sin(2π/r)/√(r). In <cit.>, Chen and the third named author presented experimental evidence and stated the following. <cit.> For any 3-manifold M with a complete hyperbolic structure of finite volume, we have lim_r→∞2π/rlog (TV_r(M, e^2π i/r))=Vol(M), where r runs over all odd integers. Conjecture <ref> impies that TV_r(M,e^2π i/r) grows exponentially in r. This is particularly surprising since the corresponding growth of TV_r(M, e^π i/r) is expected, and in many cases known, to be polynomial by Witten's asymptotic expansion conjecture <cit.>. For closed 3-manifolds, this polynomial growth was established by Garoufalidis <cit.>. Combining <cit.> and the results of <cit.>, one has that for every 3-manifold M with non-empty boundary, there exist constants C>0 and N such that |TV_r(M, e^π i/r)|⩽ C r^N. This together with Theorem <ref>(1) imply the following. For any link L in S^3, there exist constants C>0 and N such that for any integer r and multi-integer 𝐢 with 1⩽𝐢⩽ r-1, the value of the 𝐢-th colored Jones polynomial at t=e^2π i/r satisfies |J_L, 𝐢(e^2π i/r)|⩽ Cr^N. Hence, J_L, 𝐢(e^2π i/r) grows at most polynomially in r. As a main application of Theorem <ref>, we provide the first rigorous evidence to Conjecture <ref>. Let L be either the figure-eight knot or the Borromean rings, and let M be the complement of L in S^3. Then lim_r→ +∞2π/rlog TV_r(M, e^2π i/r)=lim_m→ +∞4π/2m+1log|J_L, m(e^4π i/2m+1)|=Vol(M), where r=2m+1 runs over all odd integers. The asymptotic behavior of the values of J_L,m(t) at t=e^2π i/m+1/2 is not predicted either by the original volume conjecture <cit.> or by its generalizations <cit.>. Theorem <ref> seems to suggest that these values grow exponentially in m with growth rate the hyperbolic volume. This is somewhat surprising because as noted in <cit.>, and also in Corollary <ref>, that for any positive integer l, J_L,m(e^2π i/m+l) grows only polynomially in m. We ask the following. Is it true that for any hyperbolic link L in S^3, we have lim_m→ +∞2π/mlog|J_L,m(e^2π i/m+1/2)|=Vol(S^3∖ L)? §.§ Knots with zero simplicial volume Recall that the simplicial volume (or Gromov norm) ||L|| of a link L is the sum of the volumes of the hyperbolic pieces in the JSJ-decomposition of the link complement, divided by the volume of the regular ideal hyperbolic tetrahedron. In particular, if the geometric decomposition has no hyperbolic pieces, then ||L||=0 <cit.>. As a natural generalization of Conjecture <ref>, one can conjecture that for every link L the asymptotics of TV_r(S^3∖ L, e^2π i/r) determines ||L||. See Conjecture <ref>. Using Theorem <ref> and the positivity of the Turaev-Viro invariants (Corollary <ref>), we have a proof of Conjecture <ref> for the knots with zero simplicial volume. Let K⊂ S^3 be a knot with simplicial volume zero. Then lim_r→∞2π/rlog TV_r(S^3∖ K, e^2π i/r) =||K||=0, where r runs over all odd integers. We also observe that, unlike the original volume conjecture that is not true for split links <cit.>, Conjecture <ref> is compatible with split unions of links, and under some assumptions is also compatible with connected summations. Since this article was first written there has been some further progress in the study of relations of the Turaev-Viro invariants and geometric decompositions of 3-manifolds: By work of Ohtsuki <cit.> Conjecture <ref> is true for closed hyperbolic 3-manifolds obtained by integral surgeries along the figure-eight knot. In <cit.>, the authors of this paper verify Conjecture <ref> for infinite families of cusped hyperbolic 3-manifolds. In <cit.>, Detcherry and Kalfagianni establish a relation between Turaev-Viro invariants and simplicial volume of 3-manifolds with empty or toroidal boundary, and proved generalizations of Theorem <ref>. In <cit.>, Detcherry proves that Conjecture <ref> is stable under certain link cabling operations. §.§ Organization The paper is organized as follows. In Subsection <ref>, we review the Reshetikhin-Turaev invariants <cit.> following the skein theoretical approach by Blanchet, Habegger, Masbaum and Vogel <cit.>. In Subsection <ref>, we recall the definition of the Turaev-Viro invariants, and consider an SO(3)-version of them that facilitates our extension of the main theorem of <cit.> in the setting needed in this paper (Theorem <ref>). The relationship between the two versions of the Turaev-Viro invariants is given in Theorem <ref> whose proof is included in the Appendix. We prove Theorem <ref> in Section <ref>, and prove Theorem <ref> and Theorem <ref> respectively in Sections <ref> and <ref>. §.§ Acknowledgement Part of this work was done while the authors were attending the conferences “Advances in Quantum and Low-Dimensional Topology 2016" at the University of Iowa, and “Knots in Hellas 2016" at the International Olympic Academy in Greece. We would like to thank the organizers of these conferences for support, hospitality, and for providing excellent working conditions. We are also grateful to Francis Bonahon, Charles Frohman, Stavros Garoufalidis and Roland van der Veen for discussions and suggestions. § PRELIMINARIES §.§ Reshetikhin-Turaev invariants and TQFTs In this subsection we review the definition and basic properties of the Reshetikhin-Turaev invariants. Our exposition follows the skein theoretical approach of Blanchet, Habegger, Masbaum and Vogel <cit.>. A framed link in an oriented 3-manifold M is a smooth embedding L of a disjoint union of finitely many thickened circles S^1× [0,ϵ], for some ϵ >0, into M. Let [A,A^-1] be the ring of Laurent polynomials in the indeterminate A. Then following <cit.>, the Kauffman bracket skein module K_A(M) of M is defined as the quotient of the free [A,A^-1]-module generated by the isotopy classes of framed links in M by the following two relations: * Kauffman Bracket Skein Relation: < g r a p h i c s > = A < g r a p h i c s > + A^-1 < g r a p h i c s > . * Framing Relation: L ∪ < g r a p h i c s > =(-A^2-A^-2) L. There is a canonical isomorphism ⟨ ⟩: K_A(S^3) →ℤ[A,A^-1] between the Kauffman bracket skein module of S^3 and ℤ[A,A^-1] viewed a module over itself. The Laurent polynomial ⟨ L ⟩∈ℤ[A,A^-1] determined by a framed link L⊂ S^3 is called the Kauffman bracket of L. The Kauffman bracket skein module K_A(T) of the solid torus T=D^2× S^1 is canonically isomorphic to the module [A,A^-1][z]. Here we consider D^2 as the unit disk in the complex plane, and call the framed link [0,ϵ]× S^1 ⊂ D^2× S^1, for some ϵ >0, the core of T. Then the isomorphism above is given by sending i parallel copies of the core of T to z^i. A framed link L in S^3 of n components defines an [A,A^-1]-multilinear map ⟨ , … , ⟩_L : K_A(T)^⊗ n→[A,A^-1], called the Kauffman multi-bracket, as follows. For monomials z^i_k∈[A,A^-1][z]≅ K_A(T), k = 1, …, n, let L(z^i_1, …, z^i_n) be the framed link in S^3 obtained by cabling the k-th component of L by i_k parallel copies of the core. Then define ⟨ z^i_1, …, z^i_n⟩_L ≐⟨ L(z^i_1, …, z^i_n) ⟩, and extend [A,A^-1]-multilinearly on the whole K_A(T). For the unknot U and any polynomial P(z)∈[A,A^-1][z], we simply denote the bracket ⟨ P(z) ⟩_U by ⟨ P(z) ⟩. The i-th Chebyshev polynomial e_i ∈[A,A^-1][z] is defined by the recurrence relations e_0=1, e_1=z, and ze_j=e_j+1+e_j-1, and satisfies ⟨ e_i ⟩ = (-1)^i[i+1]. The colored Jones polynomials of an oriented knot K in S^3 are defined using e_i as follows. Let D be a diagram of K with writhe number w(D), equipped with the blackboard framing. Then the (i+1)-st colored Jones polynomial of K is J_K,i+1(t)=((-1)^i A^i^2+2i)^w(D)⟨ e_i ⟩_D. The colored Jones polynomials for an oriented link L in S^3 is defined similarly. Let D be a diagram of L with writhe number w(D) and equipped with the blackboard framing. For a multi-integer 𝐢 = (i_1, …, i_n), let 𝐢+1 = (i_1+1,…, i_n+1). Then the (𝐢+1)-st colored Jones polynomial of L is defined by J_L, 𝐢+1(t) = ((-1)^k=1n∑i_k A^s(𝐢))^w(D)⟨ e_i_1, …, e_i_n⟩ _D, where s(𝐢)=k=1n∑(i_k^2+i_k). We note that a change of orientation on some or all the components of L changes the writhe number of D, and changes J_L,𝐢(t) only by a power of A. Therefore, for an unoriented link L and a complex number A with |A|=1, the modulus of the value of J_L,𝐢(t) at t=A^4 is well defined, and |J_L, 𝐢(t)| =|⟨ e_i_1-1, …, e_i_n-1⟩ _D|. If M is a closed oriented 3-manifold obtained by doing surgery along a framed link L in S^3, then the specialization of the Kauffman multi-bracket at roots of unity yields invariants of 3-manifolds. From now on, let A be either a primitive 4r-th root of unity for an integer r⩾ 3 or a primitive 2r-th root of unity for an odd integer r⩾ 3. To define the Reshetikhin-Turaev invariants, we need to recall some special elements of K_A(T)≅[A,A^-1][z], called the Kirby coloring, defined by ω_r=i=0r-2∑⟨ e_i ⟩ e_i for any integer r, and ω_r'=i=0m-1∑⟨ e_2i⟩ e_2i for any odd integer r=2m+1. We also for any r introduce κ_r=η_r ⟨ω_r ⟩_U_+, and for any odd r introduce κ'_r=η_r' ⟨ω_r' ⟩_U_+, where U_+ is the unknot with framing 1. Let M be a closed oriented 3-manifold obtained from S^3 by doing surgery along a framed link L with number of components n(L) and signature σ(L). * The Reshetikhin-Turaev invariants of M are defined by ⟨ M⟩_r=η_r^1+n(L) κ_r^-σ(L) ⟨ω_r, …, ω_r ⟩ _L for any integer r⩾ 3, and by ⟨ M⟩_r'=(η_r')^1+n(L) (κ_r')^-σ(L) ⟨ω_r', …, ω_r'⟩ _L for any odd integer r⩾ 3. * Let L' be a framed link in M. Then, the Reshetikhin-Turaev invariants of the pair (M,L') are defined by ⟨ M,L' ⟩_r=η_r^1+n(L) κ_r^-σ(L) ⟨ω_r, …, ω_r, 1 ⟩ _L∪ L' for any integer r⩾ 3, and by ⟨ M, L' ⟩_r'=(η_r')^1+n(L) (κ_r')^-σ(L) ⟨ω_r', …, ω_r', 1 ⟩ _L∪ L' for any odd integer r⩾ 3. * The invariants ⟨ M⟩_r and ⟨ M ⟩ _r' are called the SU(2) and SO(3) Reshetikhin-Tureav invariants of M, respectively. * For any element S in K_A(M) represented by a [A,A^-1]-linear combinations of framed links in M, one can define ⟨ M, S ⟩_r and ⟨ M, S ⟩_r' by [A,A^-1]-linear extensions. * Since S^3 is obtained by doing surgery along the empty link, we have ⟨ S^3 ⟩_r=η_r and ⟨ S^3 ⟩_r' =η_r'. Moreover, for any link L ⊂ S^3 we have ⟨ S^3 ,L ⟩_r =η_r ⟨ L ⟩, and ⟨ S^3, L ⟩_r'= η_r' ⟨ L ⟩. In <cit.>, Blanchet, Habegger, Masbaum and Vogel gave a construction of the topological quantum field theories underlying the SU(2) and SO(3) versions of the Reshetikhin-Turaev invariants. Below we will summarize the basic properties of the corresponding topological quantum field functors denoted by Z_r and Z_r', respectively. Note that for a closed oriented 3-manifold M we will use -M to denote the manifolds with the orientation reversed. <cit.> * For a closed oriented surface Σ and any integer r⩾ 3, there exists a finite dimensional ℂ-vector space Z_r(Σ) satisfying Z_r(Σ_1 ∐Σ_2) ≅ Z_r(Σ_1)⊗ Z_r(Σ_2), and, similarly, for each odd integer r⩾ 3, there exists a finite dimensional ℂ-vector space Z_r'(Σ) satisfying Z_r'(Σ_1 ∐Σ_2) ≅ Z_r'(Σ_1)⊗ Z_r'(Σ_2). * If H is a handlebody with ∂ H =Σ, then Z_r(Σ) and Z_r'(Σ) are quotients of the Kauffman bracket skein module K_A(H). * Every compact oriented 3-manifold M with ∂ M=Σ and a framed link L in M defines for any integer r a vector Z_r(M,L) in Z_r(Σ), and for any odd integer r a vector Z_r'(M,L) in Z_r'(Σ). * For any integer r, there is a sesquilinear pairing ⟨ , ⟩ on Z_r(Σ) with the following property: Given oriented 3-manifolds M_1 and M_2 with boundary Σ=∂ M_1=∂ M_2, and framed links L_1⊂ M_1 and L_2⊂ M_2, we have ⟨ M, L ⟩_r=⟨ Z_r(M_1, L_1),Z_r(M_2, L_2)⟩, where M=M_1⋃_Σ (-M_2) is the closed 3-manifold obtained by gluing M_1 and -M_2 along Σ and L= L_1∐ L_2. Similarly, for any odd integer r, there is a sesquilinear pairing ⟨ , ⟩ on Z_r'(Σ), such tor any M and L as above, ⟨ M, L ⟩_r'=⟨ Z_r'(M_1, L_1),Z_r'(M_2, L_2)⟩. For the purpose of this paper, we will only need to understand the TQFT vector spaces of the torus Z_r(T^2) and Z_r'(T^2). These vector spaces are quotients of K_A(T)≅[A,A^-1][z], hence the Chebyshev polynomials {e_i} define vectors in Z_r(T^2) and Z_r'(T^2). We have the following <cit.> * For any integer r⩾ 3, the vectors {e_0, …, e_r-2} form a Hermitian basis of Z_r(T^2). * For any odd integer r=2m+1, the vectors {e_0, …, e_m-1} form a Hermitian basis of Z_r'(T^2). * In Z_r'(T^2), we have for any i with 0⩽ i ⩽ m-1 that e_m+i=e_m-1-i. Therefore, the vectors {e_2i}_i=0,… ,m-1 also form a Hermitian basis of Z_r'(T^2). §.§ Turaev-Viro invariants In this subsection, we recall the definition and basic properties of the Turaev-Viro invariants <cit.>. The approach of <cit.> relies on quantum 6j-symbols while the definition of Kauffman and Lins <cit.> uses invariants of spin networks. The two definitions were shown to be equivalent in <cit.>. The formalism of <cit.> turns out to be more convenient to work with when using skein theoretic techniques to relate the Turaev-Viro invariants to the Reshetikhin-Turaev invariants. For an integer r⩾ 3, let I_r={0,1,…, r-2} be the set of non-negative integers less than or equal to r-2. Let q be a 2r-th root of unity such that q^2 is a primitive r-th root. For example, q=A^2, where A is either a primitive 4r-th root or for odd r a primitive 2r-th root, satisfies the condition. For i∈ I_r, define < g r a p h i c s > =(-1)^i[i+1]. A triple (i,j,k) of elements of I_r is called admissible if (1) i+j⩾ k, j+k⩾ i and k+i⩾ j, (2) i+j+k is an even, and (3) i+j+k⩽ 2(r-2). For an admissible triple (i,j,k), define < g r a p h i c s > = (-1)^-i+j+k/2[i+j-k/2]![j+k-i/2]![k+i-j/2]![i+j+k/2+1]!/[i]![j]![k]!. A 6-tuple (i,j,k,l,m,n) of elements of I_r is called admissible if the triples (i,j,k), (j,l,n), (i,m,n) and (k,l,m) are admissible. For an admissible 6-tuple (i,j,k,l,m,n), define < g r a p h i c s > = ∏_a=1^4∏_b=1^3[Q_b-T_a]!/[i]![j]![k]![l]![m]![n]!∑_z=max{T_1, T_2, T_3, T_4}^min{ Q_1,Q_2,Q_3}(-1)^z[z+1]!/∏_a=1^4[z-T_a]!∏_b=1^3[Q_b-z]!, where T_1=i+j+k/2, T_2=i+m+n/2, T_3=j+l+n/2, T_4=k+l+m/2, Q_1=i+j+l+m/2, Q_2=i+k+l+n/2, Q_3=j+k+m+n/2. The symbols < g r a p h i c s > , < g r a p h i c s > and < g r a p h i c s >, used above, are examples of spin networks: trivalent ribbon graphs with ends colored by integers. The expressions on the right hand sides of above equations give the Kauffman bracket invariant of the corresponding networks. See <cit.>. In the language of <cit.>, the second and third spin networks above are the trihedral and tetrahedral networks, denoted by θ(i,j,k) and τ(i, j,k) therein, and the corresponding invariants are the trihedral and tetrahedral coefficients, respectively. A coloring of a Euclidean tetrahedron Δ is an assignment of elements of I_r to the edges of Δ, and is admissible if the triple of elements of I_r assigned to the three edges of each face of Δ is admissible. See Figure <ref> for a geometric interpretation of tetrahedral coefficients. Let 𝒯 be a triangulation of M. If M has non-empty boundary, then we let 𝒯 be an ideal triangulation of M, i.e., a gluing of finitely many truncated Euclidean tetrahedra by affine homeomorphisms between pairs of faces. In this way, there are no vertices, and instead, the triangles coming from truncations form a triangulation of the boundary of M. By edges of an ideal triangulation, we only mean the ones coming from the edges of the tetrahedra, not the ones from the truncations. A coloring at level r of the triangulated 3-manifold (M,𝒯) is an assignment of elements of I_r to the edges of 𝒯, and is admissible if the 6-tuple assigned to the edges of each tetrahedron of 𝒯 is admissible. Let c be an admissible coloring of (M,𝒯) at level r. For each edge e of 𝒯, let |e|_c= < g r a p h i c s > . For each face f of 𝒯 with edges e_1, e_2 and e_3, let |f|_c= < g r a p h i c s > , where c_i = c(e_i). For each tetrahedra Δ in 𝒯 with vertices v_1,…,v_4, denote by e_ij the edge of Δ connecting the vertices v_i and v_j, {i,j}⊂{1,…,4}, and let |Δ|_c= < g r a p h i c s > , where c_ij=c(e_ij). Let A_r be the set of admissible colorings of (M,𝒯) at level r, and let V, E F and T respectively be the sets of (interior) vertices, edges, faces and tetrahedra in 𝒯. Then the r-th Turaev-Viro invariant is defined by TV_r(M)= η_r^2|V|∑_c∈ A_r∏_e∈ E|e|_c∏_Δ∈ T|Δ|_c/∏_f∈ E|f|_c. For an odd integer r⩾ 3, one can also consider an SO(3)-version of the Turaev-Viro invariants TV_r'(M) of M, which will relate to the SU(2) invariants TV_r(M), and to the Reshetikhin-Turaev invariants ⟨ D(M) ⟩'_r of the double of M (Theorems <ref>, <ref>). The invariant TV_r'(M) is defined as follows. Let I'_r={0,2,…, r-5, r-3} be the set of non-negative even integers less than or equal to r-2. An SO(3)-coloring of a Euclidean tetrahedron Δ is an assignment of elements of I'_r to the edges of Δ, and is admissible if the triple assigned to the three edges of each face of Δ is admissible. Let 𝒯 be a triangulation of M. An SO(3)-coloring at level r of the triangulated 3-manifold (M,𝒯) is an assignment of elements of I'_r to the edges of 𝒯, and is admissible if the 6-tuple assigned to the edges of each tetrahedron of 𝒯 is admissible. Let A'_r be the set of SO(3)-admissible colorings of (M,𝒯) at level r. Define TV_r'(M)=(η_r')^2|V|∑_c∈ A'_r∏_e∈ E|e|_c∏_Δ∈ T|Δ|_c/∏_f∈ E|f|_c. The relationship between TV_r(M) and TV'_r(M) is given by the following theorem. Let M be a 3-manifold and let b_0(M) and b_2(M) respectively be its zeroth and second ℤ_2-Betti number. * For any odd integer r⩾ 3, TV_r(M)=TV_3(M)· TV'_r(M). * (Turaev-Viro <cit.>). If ∂ M= ∅ and A=e^π i/3, then TV_3(M)=2^b_2(M)-b_0(M). * If M is connected, ∂ M≠∅ and A=e^π i/3, then TV_3(M)=2^b_2(M). In particular, TV_3(M) is nonzero. We postpone the proof of Theorem <ref> to Appendix <ref> to avoid unnecessary distractions. § THE COLORED JONES SUM FORMULA FOR TURAEV-VIRO INVARIANTS In this Section, following the argument of <cit.>, we establish a relationship between the SO(3) Turaev-Viro invariants of a 3-manifold with boundary and the SO(3) Reshetikhin-Turaev invariants of its double. See Theorem <ref>. Then, we use Theorem <ref> and results established in <cit.>, to prove Theorem <ref>. §.§ Relationship between invariants The relationship between Turaev-Viro and Witten-Reshetikhin-Turaev invariants was studied by Turaev-Walker <cit.> and Roberts <cit.> for closed 3-manifolds, and by Benedetti and Petronio <cit.> for 3-manifolds with boundary. For an oriented 3-manifold M with boundary, let -M denote M with the orientation reversed, and let D(M) denote the double of M, i.e., D(M)=M∂ M⋃(-M). We will need the following theorem of Benedetti and Petronio <cit.>. In fact <cit.> only treats the case of A=e^π i/2r, but, as we will explain below, the proof for other cases is similar. Let M be a 3-manifold with boundary. Then, TV_r(M)=η_r^-χ(M)⟨ D(M) ⟩_r for any integer r, and TV_r'(M)=(η'_r)^-χ ( M)⟨ D(M) ⟩_r' for any odd r, where χ(M) is the Euler characteristic of M. We refer to <cit.> and <cit.> for the SU(2) (r being any integer) case, and for the reader's convenience include a sketch of the proof here for the SO(3) (r being odd) case. The main difference for the SO(3) case comes from to the following lemma due to Lickorish. Let r⩾ 3 be an odd integer and let A be a primitive 2r-th root of unity. Then < g r a p h i c s > ={[ < g r a p h i c s > if i=0, r-2; ; 0 if i≠ 0, r-2 ]. I.e., the element of the i-th Temperley-Lieb algebra obtained by circling the i-th Jones-Wenzl idempotent f_i by the Kirby coloring ω_r' equals f_i when i=0 or r-2, and equals 0 otherwise. As a consequence, the usual fusion rule <cit.> should be modified to the following. Let r⩾ 3 be an odd integer. Then for a triple (i,j,k) of elements of I_r', < g r a p h i c s > ={[ (η_r')^-2/ < g r a p h i c s > < g r a p h i c s > if (i,j,k) is r-admissible,; ; 0 if (i,j,k) is not r-admissible. ]. Here the integers i,j and k being even is crucial, since it rules out the possibility that i+j+k=r-2, which by Lemma <ref> could create additional complications. This is the reason that we prefer to work with the invariant TV_r'(M) instead of TV_r(M). Note that the factor < g r a p h i c s > in the formula above is also denoted by θ(i,j,k) in <cit.> and <cit.>. Following <cit.>, we extend the “chain-mail" invariant of Roberts <cit.> to M with non-empty boundary using a handle decomposition without 3-handles. For such a handle decomposition, let d_0, d_1 and d_2 respectively be the number of 0-, 1- and 2-handles. Let ϵ_i be the attaching curves of the 2-handles and let δ_j be the meridians of the 1-handles. Thicken the curves to bands parallel to the surface of the 1-skeleton H and push the ϵ-bands slightly into H. Embed H arbitrarily into S^3 and color each of the image of the ϵ- and δ-bands by η_r'ω_r' to get an element in S_M in K_A(S^3). Then the chain-mail invariant of M is defined by CM_r(M)=(η_r')^d_0⟨ S_M ⟩, where, recall that, we use the notation ⟨ ⟩ for the Kauffman bracket. It is proved in <cit.> that CM_r(M) is independent of the choice of the handle decomposition and the embedding, hence defines an invariant of M. To prove the result we will compare the expressions of the invariant CM_r(M) obtained by considering two different handle decompositions of M. On the one hand, suppose that the handle decomposition is obtained by the dual of an ideal triangulation 𝒯 of M, namely the 2-handles come from a tubular neighborhood of the edges of 𝒯, the 1-handles come from a tubular neighborhood of the faces of 𝒯 and the 0-handles come from the complement of the 1- and 2-handles. Since each face has three edges, each δ-band encloses exactly three ϵ-bands (see <cit.>). By relation (<ref>), every η_r'ω_r' on the ϵ-band can be written as η_r'ω_r'=η_r'∑_ i=0^r-1/2-1⟨ e_i⟩ e_i=η_r'∑_ i=0^r-1/2-1⟨ e_2i⟩ e_2i. Next we apply Lemma <ref> to each δ-band. In this process the four δ-bands corresponding to each tetrahedron of 𝒯 give rise to a tetrahedral network (see also <cit.>). Then by Remark <ref> and equations preceding it, we may rewrite CM_r(M) in terms of trihedral and tetrahedral coefficients to obtain CM_r(M)=(η_r')^d_0-d_1+d_2∑_c∈ A'_r∏_e∈ E|e|_r^c∏_Δ∈ T|Δ|_r^c/∏_f∈ E|f|_r^c=(η_r')^χ(M)TV_r'(M). On the other hand, suppose that the handle decomposition is standard, namely H is a standard handlebody in S^3 with exactly one 0-handle. Then we claim that the ϵ- and the δ-bands give a surgery diagram L of D(M). The way to see it is as follows. Consider the 4-manifold W_1 obtained by attaching 1-handles along the δ-bands (see Kirby <cit.>) and 2-handles along the ϵ-bands. Then W_1 is homeomorphic to M× I and ∂ W_1 =M×{0}∪∂ M× I ∪(-M)×{1}= D(M). Now if W_2 is the 4-manifold obtained by attaching 2-handles along all the ϵ- and the δ-bands, then ∂ W_2 is the 3-manifold represented by the framed link L. Then due to the fact that ∂ W_1=∂ W_2 and Definition <ref>, we have CM_r(M)=η_r'⟨η_r'ω_r',…, η'ω' ⟩_L=(η_r')^1+n(L)⟨ω_r',…, ω_r' ⟩_L=⟨ D(M)⟩_r' (κ_r') ^σ(L). We are left to show that σ(L)=0. It follows from the fact that the linking matrix of L has the form LK(L)=[ 0 A; A^T 0 ], where the blocks come from grouping the ϵ- and the δ-bands together and A_ij=LK(ϵ_i, δ_j). Then, for any eigenvector v=(v_1,v_2) with eigenvalue λ, the vector v'=(-v_1,v_2) is an eigen-vector of eigenvalue -λ. Theorems <ref> and <ref> together with the main result of <cit.> imply that if Conjecture <ref> holds for M with totally geodesic or toroidal boundary, then it holds for D(M). §.§ Proof of Theorem <ref> We are now ready to prove Theorem <ref>. For the convenience of the reader we restate the theorem. Theorem <ref> Let L be a link in S^3 with n components. * For an integer r⩾ 3 and a primitive 4r-th root of unity A, we have TV_r(S^3∖ L,q)=η_r^21 ⩽𝐢⩽ r-1∑ | J_L, 𝐢 (t) |^2. * For an odd integer r=2m+1⩾ 3 and a primitive 2r-th root of unity A, we have TV_r(S^3∖ L,q)=2^n-1(η_r')^21 ⩽𝐢⩽ m∑ | J_L, 𝐢 (t)|^2. Here, in both cases we have t=q^2=A^4. We first consider the case that r=2m+1 is odd. For a framed link L in S^3 with n components, we let M=S^3∖ L. Since, by Theorem <ref>, we have TV_r(M)=2^n-1 TV_r'(M), from now on we will work with TV_r'(M). Since the Euler characteristic of M is zero, by Theorem <ref>, we obtain TV_r'(M)=⟨ D(M) ⟩_r'=⟨ Z_r'(M),Z_r'(M) ⟩, where Z_r(M) is a vector in Z_r(T^2)^⊗ n. Let {e_i}_i=0,…, m-1 be the basis of Z_r'(T^2) described in Theorem <ref> (2). Then the vector space Z_r(T^2)^⊗ n has a Hermitian basis given by {e_𝐢=e_i_1⊗ e_i_2… e_i_n} for all 𝐢=(i_1,i_2,…,i_n) with 0⩽𝐢⩽ m-1. We write ⟨ e_𝐢⟩_L for the multi-bracket ⟨ e_i_1,e_i_2,…,e_i_n⟩_L. Then, by relation (<ref>), to establish the desired formula in terms of the colored Jones polynomials, it is suffices to show that TV_r'(M)=(η_r')^20 ⩽𝐢⩽ m-1∑|⟨ e_𝐢⟩_L|^2. By writing Z_r'(M)=0 ⩽𝐢⩽ m-1∑λ_𝐢e_𝐢 and using equation (<ref>), we have that TV_r'(M)=0 ⩽𝐢⩽ m-1∑|λ_𝐢|^2. The computation of the coefficients λ_𝐢 of Z_r(M) relies on the TQFT properties of the invariants <cit.>. (Also compare with the argument in <cit.>). Since {e_𝐢} is a Hermitian basis of Z_r(T^2)^⊗ n, we have λ_𝐢=⟨ Z_r'(M),e_𝐢⟩. A tubular neighborhood N_L of L is a disjoint union of solid tori k=1n∐T_k. We let L(e_𝐢) be the element of K_A(N_L) obtained by cabling the component of the L in T_k using the i_k-th Chebyshev polynomial e_i_k. Then in Z_r'(T)^⊗ n, we have e_𝐢= Z_r'(N_L, L(e_𝐢)). Now by Theorem <ref> (4), since S^3=M ∪ (-N_L), we have ⟨ Z_r'(M),e_𝐢⟩= ⟨ Z_r'(M), Z_r'(N_L, L(e_𝐢)) ⟩= ⟨ M∪(-N_L ), L(e_𝐢)) ⟩_r'=⟨ S^3 , L(e_𝐢) ⟩_r'. Finally, by Remark <ref> (2), we have ⟨ S^3 , L(e_𝐢) ⟩_r'=η_r' ⟨ e_𝐢⟩_L. Therefore, we have λ_𝐢=η_r' ⟨ e_𝐢⟩_L, which finishes the proof in the case of r=2m+1. The argument of the remaining case is very similar. By Theorem <ref>, we obtain TV_r(M)=⟨ D(M) ⟩_r=⟨ Z_r(M),Z_r(M) ⟩. Working with the Hermitian basis {e_i}_i=0,… ,r-2 of Z_2r(T^2) given in Theorem <ref> (1), we have TV_r(M)=0 ⩽𝐢⩽ r-2∑|λ_𝐢|^2, where λ_𝐢=⟨ Z_r(M),e_𝐢⟩ and e_𝐢= Z_r(N_L, L(e_𝐢)). Now by Theorem <ref> (4) and Remark <ref>, one sees λ_𝐢=η_r⟨ e_𝐢⟩_L, which finishes the proof. § APPLICATIONS TO CONJECTURE <REF> In this section we use Theorem <ref> to determine the asymptotic behavior of the Turaev-Viro invariants for some hyperbolic knot and link complements. In particular, we verify Conjecture <ref> for the complement of the figure-eight knot and the Borromean rings. To the best of our knowledge these are the first calculations of this kind. §.§ The figure-eight complement The following theorem verifies Conjecture <ref> for the figure-eight knot. Let K be the figure-eight knot and let M be the complement of K in S^3. Then lim_r→ +∞2π/rlog TV_r(M, e^2π i/r)=lim_m→ +∞4π/2m+1log|J_K, m(e^4π i/2m+1)|=Vol(M), where r=2m+1 runs over all odd integers. By Theorem <ref>, and for odd r=2m+1, we have that TV_r(S^3∖ K,e^2π i/r)=(η_r')^2∑_i=1^m|J_i(K,t)|^2, where t=q^2=e^4π i/r. Notice that (η_r')^2 grows only polynomially in r. By Habiro and Le's formula <cit.>, we have J_K,i(t)=1+∑_j=1^i-1∏_k=1^j(t^i-k/2-t^-i-k/2)(t^i+k/2-t^-i+k/2), where t=A^4=e^4π i/r. For each i define the function g_i(j) by g_i(j)= ∏_k=1^j|(t^i-k/2-t^-i-k/2)(t^i+k/2-t^-i+k/2)| = ∏_k=1^j4|sin2π(i-k)/r||sin2π(i+k)/r|. Then |J_K,i(t)|⩽ 1+∑_j=1^i-1g_i(j). Now let i be such that i/r→ a∈[0,1/2] as r →∞. For each i, let j_i∈{1, …, i-1} such that g_i(j_i) achieves the maximum. We have that j_i/r converges to some j_a∈ (0,1/2) which varies continuously in a when a is close to 1/2. Then lim_r →∞1/rlog|J_K,i|⩽lim_r →∞1/rlog(1+∑_j=1^i-1g_i(j))= lim_r →∞1/rlog(g_i(j_i)), where the last term equals lim_r →∞1/r(∑_k=1^j_ilog|2sin2π(i-k)/r|+∑_k=1^j_ilog|2sin2π(i+k)/r|) = 1/2π∫_0^j_aπlog(2|sin(2π a-t)|)dt+ 1/2π∫_0^j_aπlog(2|sin(2π a+t)|)dt = -1/2π(Λ(2π (j_a- a))+Λ(2π a)) -1/2π(Λ(2π (j_a+ a))-Λ(2π a)) = -1/2π(Λ(2π (j_a- a))+Λ(2π (j_a+ a))). Here Λ denotes the Lobachevsky function. Since Λ(x) is an odd function and achieves the maximum at π/6, the last term above is less than or equal to Λ(π/6)/π=3Λ(π/3)/2π=Vol(S^3∖ K)/4π. We also notice that for i=m, i/r = m/2m+1→1/2, j_1/2=5/12 and all the inequalities above become equalities. Therefore, the term |J_K,m(t)|^2 grows the fastest, and lim_r→ +∞2π/rlog TV_r(S^3∖ K,A^2)=lim_r→ +∞2π/rlog|J_K,m(t)|^2=Vol(S^3∖ K). §.§ The Borromean rings complement In this subsection we prove the following theorem that verifies Conjecture <ref> for the 3-component Borromean rings. Let L be the 3-component Borromean rings, and let M be the complement of L in S^3. Then lim_r→ +∞2π/rlog TV_r(M, e^2π i/r)=lim_m→ +∞4π/2m+1log|J_L, m(e^4π i/2m+1)|=Vol(M), where r=2m+1 runs over all odd integers. Here, J_L, m denotes the colored Jones polynomial where all the components of L are colored by m. The proof relies on the following formula for the colored Jones polynomials of the Borromean rings given by Habiro <cit.>. Let L be the Borromean rings and k, l and n be non-negative integers. Then J_L,(k,l,n)(t)=j=0min(k,l,n)-1∑ (-1)^j [k+j]![l+j]![n+j]!/[k-j-1]![l-j-1]![n-j-1]!( [j]!/[2j+1]!)^2. Recall that in this formula [n]=t^n/2-t^-n/2/t^1/2-t^-1/2 and [n]!=[n][n-1]… [1]. From now on we specialize at t=e^4π i/r where r=2m+1. We have [n]=2sin(2nπ/r)/2sin(2π/r)={ n }/{ 1 }, where we write { j } =2sin(2jπ/r). We can rewrite formula (<ref>) as J_L,(k,l,n)(e^4iπ/r)=j=0min(k,l,n)-1∑ (-1)^j 1/{ 1 }{ k+j }!{ l+j }!{ n+j}!/{ k-j-1}!{ l-j-1}!{ n-j-1}!( { j} !/{ 2j+1} !)^2. Next we establish three lemmas needed for the proof of Theorem <ref>. For any integer j with 0<j<r, we have log(| j !|)=-r/2πΛ(2jπ/r)+O(log (r)), where O(log(r)) is uniform: there is a constant C independent of j and r, such that O(log r)≤ C log r. This result is an adaptation of the result in <cit.> for r odd. By the Euler-Mac Laurin summation formula, for any twice differentiable function f on [a,b] where a and b are integer, we have k=ab∑ f(k)=∫_a^b f(t)dt +1/2f(a)+1/2f(b)+R(a,b,f), where |R(a,b,f)|⩽3/24∫_a^b |f”(t)|dt. Applying this to log(| j !|)=k=1j∑log(2|sin(2kπ/r)|), we get [ log(| j !|) = ∫_1^jlog(2|sin(2tπ/r)+1/2(f(1)+f(j))+R(2π/r,2jπ/r,f); = r/2π∫_2π/r^2jπ/rlog(2|sin(2tπ/r)+1/2(f(1)+f(j))+R(2π/r,2jπ/r,f); = r/2π(-Λ(2jπ/r)+Λ(2π/r))+1/2(f(1)+f(j))+R(2π/r,2jπ/r,f), ] where f(t)=log(2|sin(2tπ/r)|). Since we have |rΛ(2π/r)|⩽ C' log(r) and |f(1)+f(j)|⩽ C”log(r) for constants C' and C” independent of j, and since R(1,j,f)=∫_1^j |f”(t)|dt=∫_1^j 4π^2/r^21/sin(2π t/r)^2=2π/r( (2jπ/r)-(2π/r))=O(1), we get log(| j !|)=-r/2πΛ(2jπ/r)+O(log (r)) as claimed. Lemma <ref> allows us to get an estimation of terms that appear in Habiro's sum for the multi-bracket of Borromean rings. We find that log| 1/{ 1 }{ k+j }!{ l+j }!{ n+j}!/{ k-i-1}!{ l-i-1}!{ n-i-1}!( { i} !/{ 2i+1} !)^2 | =-r/2π(f(α,θ)+f(β,θ)+f(γ,θ))+O(log(r)), where α=2kπ/r, β=2lπ/r, γ=2nπ/r and θ=2jπ/r, and f(α,θ)=Λ(α+θ)-Λ(α-θ)+2/3Λ(θ)-2/3Λ(2θ). The minimum of the function f(α,θ) is -8/3Λ(π/4)=-v_8/3. This minimum is attained for α=0 modulo π and θ=3π/4 modulo π. The critical points of f are given by the conditions Λ'(α+θ)-Λ'(α-θ)=0 Λ'(α+θ)+Λ'(α-θ)+2/3Λ'(θ)-4/3Λ'(2 θ)=0. As Λ'(x)=2log|sin(x)|, the first condition is equivalent to α+θ=±α-θπ. Thus, either θ=0 π/2 in which case f(α,θ)=0, or α=0 or π/2π. In the second case, as the Lobachevsky function has the symmetries Λ(-θ)=-Λ(θ) and Λ(θ +π/2)=1/2Λ(2θ)-Λ(θ), we get f(0,θ)=8/3Λ(θ)-2/3Λ(2θ), and f(π/2,θ)=1/3Λ(2θ)-4/3Λ(θ). We get critical points when 2Λ'(θ)=Λ(2θ) which is equivalent to (2 sin (θ))^2=2|sin (2θ)|. This happens only for θ=π/4 or 3π/4π and the minimum value is -8/3Λ(π/4), which is obtained only for α=0 π and θ=3π/4π only. If r=2m+1, we have that log(|J_L,(m,m,m)(e^4iπ/r)|)=r/2πv_8 +O(log(r)). Again, the argument is very similar to the argument of the usual volume conjecture for the Borromean ring in Theorem A.1 of <cit.>. We remark that quantum integer n admit the symmetry that m+1+i =- m-i for any integer i. Now, for k=l=n=m, Habiro's formula for the colored Jones polynomials turns into J_L,(m,m,m)(t) =j=0m-1∑ (-1)^j m ^3/ 1 ( k=1j∏ m+k m-k )^3 ( { j} !/{ 2j+1} !)^2 =j=0m-1∑ m ^3 m+j+1 / 1 m+1 ( k=1j∏ m+k )^6 ( { j} !/{ 2j+1} !)^2. Note that as n =sin (2 π n/2m+1)<0 for n ∈ m+1,m+2, …,2m , the factor m+j+1 will always be negative for 0 ⩽ m-1. Thus all terms in the sum have the same sign. Moreover, there is only a polynomial in r number of terms in the sum as m=r-1/2. Therefore, log(|J_L,(m,m,m)|) is up to O(log (r)) equal to the logarithm of the biggest term. But the term j=⌊3r/8⌋ corresponds to α=2(m-1)π/r=0+O(1/r) π and θ =2 jπ/r=3π/4+O(1/r) π, so log| m ^3/ 1 ( k=1m-1∏ m+k m-k )^3 ( { m-1} !/{ 2m-1} !)^2|=r/2πv_8+O(log(r)), and 2π/rlog |J_L,(m,m,m)|=v_8+O(log(r)/r). By Theorem <ref>, we have TV_r'(S^3∖ L,e^2π i/r)=(η_r')^21 ⩽ k,l,n⩽ m∑|J_L,(k,l,n))(e^4iπ/r)|^2. This is a sum of m^3=(r-1/2)^3 terms, the logarithm of all of which are less than r/2π(2v_8)+O(log(r)) by Lemma <ref>. Also, the term |J_L,(m,m,m)(e^4iπ/r)|^2 has logarithm r/2π(2v_8)+O(log(r)). Thus we have r→∞lim2π/rlog (TV_r'(S^3∖ L),e^2π i/r)=2v_8= (S^3∖ L). Finally we note that Theorem <ref> stated in the introduction follows by Theorems <ref> and <ref>. § TURAEV-VIRO INVARIANTS AND SIMPLICIAL VOLUME Given a link L in S^3, there is a unique up to isotopy collection T of essential embedded tori in M=S^3∖ L so that each component of M cut along T is either hyperbolic or a Seifert fibered space. This is the toroidal or JSJ-decomposition of M <cit.>. Recall that the simplicial volume (or Gromov norm) of L, denoted by ||L||, is the sum of the volumes of the hyperbolic pieces of the decomposition, divided by v_3; the volume of the regular ideal tetrahedron in the hyperbolic space. In particular, if the toroidal decomposition has no hyperbolic pieces, then we have ||L||=0. It is known <cit.> that the simplicial volume is additive under split unions and connected summations of links. That is, we have ||L_1 ⊔ L_2||= ||L_1 # L_2||= ||L_1||+ ||L_2||. We note that the connected sum for multi-component links is not uniquely defined, it depends on the components of links being connected. For every link L⊂ S^3, we have lim_r→∞2π/rlog (TV_r(S^3∖ L, e^2π i/r)) = v_3 ||L||, where r runs over all odd integers. Theorem <ref> suggests that the Turaev-Viro invariants are a better object to study for the volume conjecture for links. As remarked in <cit.>, all the Kashaev invariants of a split link are zero. As a result, the original simplicial volume conjecture <cit.> is not true for split links. On the other hand, Corollary <ref> implies that TV_r'(S^3∖ L,q)≠ 0 for any r⩾ 3 and any primitive root of unity q=A^2. Define the double of a knot complement to be the double of the complement of a tubular neighborhood of the knot. Then Theorem <ref> and the main result of <cit.> implies that if Conjecture <ref> holds for a link, then it holds for the double of its complement. In particular, by Theorem <ref>, we have Conjecture <ref> is true for the double of the figure-eight and the Borromean rings complement. Since colored Jones polynomials are multiplicative under split union of links, Theorem <ref> also implies that TV_r'(S^3∖ L,q) is up to a factor multiplicative under split union. For any odd integer r⩾ 3 and q=A^2 for a primitive 2r-th root of unity A, TV_r'(S^3∖ (L_1 ⊔ L_2),q)= (η_r')^-1 TV_r'(S^3∖ L_1,q)· TV_r'(S^3∖ L_2,q). The additivity of simplicial volume implies that if Conjecture <ref> is true for L_1 and L_2, then it is true for the split union L_1⊔ L_2. Next we discuss the behavior of the Turaev-Viro invariants under taking connected sums of links. With our normalization of the colored Jones polynomials, we have that J_L_1 # L_2,𝐢(t)= [i] J_L_1 ,𝐢_1 (t)· J_L_2,𝐢_2(t), where 𝐢_1 and 𝐢_2 are respectively the restriction of 𝐢 to L_1 and L_2, and i is the component of 𝐢 corresponding to the component of L_1#L_2 coming from the connected summation. This implies the following. Let A be a primitive 2r-th root of unity. For any odd integer r⩾ 3, q=A^2 and t=A^4, we have TV_r'(S^3∖ L_1#L_2,q)=(η_r')^2∑_1⩽𝐢⩽ m[i]^2|J_L_1,𝐢_1(t)|^2|J_L_2,𝐢_2(t)|^2, where 𝐢_1 and 𝐢_2 are respectively the restriction of 𝐢 to L_1 and L_2, and i is the component of 𝐢 corresponding to the component of L_1#L_2 coming from the connected summation. In the rest of this section, we focus on the value q=e^2 i π/r for odd r=2m +1. Notice that in this case, the quantum integers [i] for 1⩽ i ⩽ m are non-zero and their logarithms are of order O(log r). Corollary <ref> implies that r →∞lim sup2π/rlog TV_r' (S^3∖ L_1# L_2,q) ⩽r →∞lim sup2π/rlog TV_r'(S^3∖ L_1 ,q)+r →∞lim sup2π/rlog TV_r'(S^3∖ L_2, q). Moreover if we assume a positive answer to Question <ref> for L_1 and L_2, then the term |J_L_1#L_2,m(t)|^2 of the sum for L_1#L_2 satisfies r →∞lim2π/rlog |J_L_1#L_2,m(t)|^2= (S^3∖ L_1#L_2). It follows that if the answer to Question <ref> is positive, and Conjecture <ref> is true for links L_1 and L_2, then Conjecture <ref> is true for their connected sum. In particular, Theorem <ref> implies the following. Conjecture <ref> is true for any link obtained by connected sum of the figure-eight and the Borromean rings. We finish the section with the proof of Theorem <ref>, verifying Conjecture <ref> for knots of simplicial volume zero. Theorem <ref> Let K⊂ S^3 be knot with simplicial volume zero. Then, we have lim_r→∞2π/rlog (TV_r(S^3∖ K, e^2π i/r)) =||K||=0, where r runs over all odd integers. By part (2) of Theorem <ref> we have TV_r(S^3∖ K, e^2iπ/r)=(η_r')^21 ⩽ i⩽ m∑ | J_L, i (e^4iπ/r)|^2. Since J_K,1(t)=1, we have TV_r(S^3∖ K)⩾η_r'^2>0 for any knot K. Thus for r> > 0 the sum of the values of the colored Jones polynomials in (<ref>) is larger or equal to 1. On the other hand, we have η_r'≠ 0 and log (|η_r'|^2)/r→ 0 as r →∞. Therefore, r →∞lim inflog|TV_r(S^3 ∖ K)|/r⩾ 0. Now we only need to prove that for simplicial volume zero knots, we have r →∞lim suplog|TV_r(S^3 ∖ K)|/r⩽ 0. By Theorem <ref>, part (2) again, it suffices to prove that the L^1-norm ||J_K,i(t)|| of the colored Jones polynomials of any knot K of simplicial volume zero is bounded by a polynomial in i. By Gordon <cit.>, the set of knots of simplicial volume zero is generated by torus knots, and is closed under taking connected sums and cablings. Therefore, it suffices to prove that the set of knots whose colored Jones polynomials have L^1-norm growing at most polynomially contains the torus knots, and is closed under taking connected sums and cablings. From Morton's formula <cit.>, for the torus knot T_p,q, we have J_T_p,q,i(t)=t^pq(1-i^2)|k|=-i-1/2i-1/2∑t^4pqk^2-4(p+q)k+2-t^4pqk^2-4(p-q)k-2/t^2-t^-2. Each fraction in the summation can be simplified to a geometric sum of powers of t^2, and hence has L^1-norm less than 2qi+1. From this we have ||J_T_p,q,i(t)||=O(i^2). For a connected sum of knots, we recall that the L^1-norm of a Laurent polynomial is ||d ∈∑ a_d t^d||=d ∈∑|a_d|. For a Laurent polynomial R(t)=f ∈∑ c_f t^f, we let deg(R(t))=max ({ d / c_d≠ 0})-min ( { d / c_d≠ 0}). Then for two Laurent polynomials P(t)=d ∈∑ a_d t^d and Q(t)=e ∈∑ b_e t^e, we have ||PQ||=||(d ∈∑a_d t^d)(d ∈∑b_d t^d)|| ⩽ ||f ∈∑ (d+e=f∑a_d b_e)t^f|| ⩽ deg(PQ) d+e=f∑|a_d b_e| ⩽ deg(PQ)||P|| ||Q||. Since the L^1-norm of [i] grows polynomially in i, if the L^1-norms of J_K_1,i(t) and J_K_2,i(t) grow polynomially, then so does that of J_K_1#K_2,i (t)=[i] J_K_1 ,i(t)· J_ K_2 ,i(t). Finally, for the (p,q)-cabling K_p,q of a knot K, the cabling formula <cit.> says J_K_p,q,i(t)=t^pq(i^2-1)/4k=-i-1/2i-1/2∑t^-pk(qk+1)J_K,2qn+1(t), where k runs over integers if i is odd and over half-integers if i is even. It implies that if ||J_K,i(t)||=O(i^d), then ||J_K_p,q,i(t)||=O(i^d+1). By Theorem <ref> and the argument in the beginning of the proof of Theorem <ref> applied to links we obtain the following. For every link L⊂ S^3, we have r →∞lim inflog|TV_r(S^3 ∖ L)|/r⩾ 0, where r runs over all odd integers. As said earlier, there is no lower bound for the growth rate of the Kashaev invariants that holds for all links; and no such bound is known for knots as well. § THE RELATIONSHIP BETWEEN TV_R(M) AND TV_R'(M) The goal of this appendix is to prove Theorem <ref>. To this end, it will be convenient to modify the definition of the Turaev-Viro invariants given in Subsection 2 and use the formalism of quantum 6j-symbols as in <cit.>. For i∈ I_r, we let |i|=(-1)^i[i+1], and for each admissible triple (i,j,k), we let |i, j, k| =(-1)^-i+j+k/2[i+j-k/2]![j+k-i/2]![k+i-j/2]!/[i+j+k/2+1]!. Also for each admissible 6-tuple (i,j,k,l,m,n), we let |[ i j k; l m n; ]|=∑_z=max{T_1, T_2, T_3, T_4}^min{ Q_1,Q_2,Q_3}(-1)^z[z+1]!/[z-T_1]![z-T_2]![z-T_3]![z-T_4]![Q_1-z]![Q_2-z]![Q_3-z]!. Consider a triangulation 𝒯 of M, and let c be an admissible coloring of (M,𝒯) at level r. For each edge e of 𝒯, we let |e|_c=|c(e)|, and for each face f with edges e_1, e_2 and e_3, we let |f|_c=|c(e_1), c(e_2), c(e_3)|. Also for each tetrahedra Δ with edges e_ij, {i,j}⊂{1,…, 4}, we let |Δ|_c=| [ c(e_12) c(e_13) c(e_23); c(e_34) c(e_24) c(e_14); ]|. Now recall the invariants TV_r(M) and TV_r'(M) given in Definitions <ref> and <ref>, respectively. Then we have the following. * For any integer r⩾ 3, TV_r(M)= η_r^2|V|∑_c∈ A_r∏_e∈ E|e|_c∏_f∈ E|f|_c∏_Δ∈ T|Δ|_c. * For any odd integer r⩾ 3, TV_r'(M)= (η_r')^2|V|∑_c∈ A_r'∏_e∈ E|e|_c∏_f∈ E|f|_c∏_Δ∈ T|Δ|_c. The proof is a straightforward calculation. Next we establish four lemmas on which the proof of Theorem <ref> will rely. We will use the notations |i|_r, |i,j,k|_r and |[ i j k; l m n; ]|_r respectively to mean the values of |i|, |i,j,k| and |[ i j k; l m n; ]| at a primitive 2r-th root of unity A. |0|_3= |1|_3=1, |0, 0, 0|_3=|1, 1, 0|_3=1 and |[ 0 0 0; 0 0 0; ]|_3 = |[ 0 0 0; 1 1 1; ]|_3=|[ 1 1 0; 1 1 0; ]|_3=1. A direct calculation. The following lemma can be considered as a Turaev-Viro setting analogue of Theorem <ref> (3). For i∈ I_r, let i'=r-2-i. * If i∈ I_r, then i'∈ I_r. Moreover, |i'|_r=|i|_r. * If the triple (i,j,k) is admissible, then so is the triple (i', j', k). Moreover, |i',j',k|_r=|i,j,k|_r. * If the 6-tuple (i,j,k,l,m,n) is admissible, then so are the 6-tuples (i,j,k, l',m',n') and (i',j', k, l', m', n). Moreover, |[ i j k; l' m' n'; ]|_r =|[ i j k; l m n; ]|_r and |[ i' j' k; l' m' n; ]|_r =|[ i j k; l m n; ]|_r. Parts (a) (b) follow easily from the definitions. To see the first identity of (c), let T_i' and Q_j' be the sums for (i,j,k,l',m',n'), involved in the expression of the corresponding 6j-symbol. Namely, let T'_1=i+j+k/2=T_1, T'_2=j+l'+n'/2 and Q'_2=i+k+l'+n'/2, etc. For the terms in the summations defining the two 6j-symbols, let us leave T_1 alone for now, and consider the other T_i's and Q_j's. Without loss of generality we assume that, Q_3⩾ Q_2⩾ Q_1 ⩾ T_4⩾ T_3⩾ T_2. One can easily check that * Q_3-Q_1=T'_4-T'_2, Q_2-Q_1=T'_4-T'_3, Q_1-T_4=Q'_1-T'_4, T_4-T_3=Q'_2-Q'_1 and T_4-T_2=Q'_3-Q'_1, which implies * Q'_3⩾ Q'_2 ⩾ Q'_1⩾ T'_4⩾ T'_3⩾ T'_2. For z in between max{T_1,… , T_4} and min{Q_1, Q_2, Q_3}, let P(z)=(-1)^z[z+1]!/[z-T_1]![z-T_2]![z-T_3]![z-T_4]![Q_1-z]![Q_2-z]![Q_3-z]!, and similarly for z in between max{T'_1,… , T'_4} and min{Q'_1, Q'_2, Q'_3} let P'(z)=(-1)^z[z+1]!/[z-T'_1]![z-T'_2]![z-T'_3]![z-T'_4]![Q'_1-z]![Q'_2-z]![Q'_3-z]!. Then for any a∈{0, 1, …, Q_1-T_4=Q'_1-T'_4} one verifies by (1) above that P(T_4+a)=P'(Q'_1-a). There are the following three cases to consider. Case 1. T_1⩽ T_4 and T'_1 ⩽ T'_4. In this case, T_max=T_4, Q_min=Q_1, T'_max=T'_4 and Q'_min=Q'_1. By (<ref>), we have ∑_z= T_4^Q_1P(z)=∑_a=0^Q_1-T_4P(T_4+a)=∑_a=0^Q'_1-T_4P'(Q'_1-a)=∑_z= T'_4^Q'_1P'(z). Case 2. T_1 > T_4 but T'_1 < T'_4, or T_1 < T_4 but T'_1 > T'_4. By symmetry, it suffices to consider the former case. In this case T_max=T_1, Q_min=Q_1, T'_max=T'_4 and Q'_min=Q'_1, and Q'_1 - (r-2) = i + j - l -m/2 = T_1 -T_4. As a consequence Q'_1>r-2 . By (<ref>), we have ∑_z= T_1^Q_1P(z)=∑_a=T_1-T_4^Q_1-T_4P(T_4+a)=∑_a=Q'_1-(r-2)^Q'_1-T'_4P'(Q'_1-a)=∑_z= T'_4^r-2P'(z)=∑_z= T'_4^Q'_1P'(z). The last equality is because we have P'(z)=0, for z> r-2. Case 3. T_1 > T_4 and T'_1 >T'_4. In this case we have T_max=T_1, Q_min=Q_1, T'_max=T'_1 and Q'_min=Q'_1. We have Q'_1-(r-2)= i+j-l-m/2 = T_1-T_4>0, hence Q_1>r-2. Also, we have Q'_1-T'_1 = l'+m'-k /2=r-2-T_4. As a consequence, Q'_1-(r-2)=T'_1-T_4=T_1-T_4>0, and hence Q'_1>r-2. By (<ref>), we have ∑_z= T_1^Q_1P(z)=∑_z= T_1^r-2P(z)=∑_a=T_1-T_4^r-2-T_4P(T_4+a)=∑_a=Q'_1-(r-2)^Q'_1-T'_1P'(Q'_1-a)=∑_z= T'_1^r-2P'(z)=∑_z= T'_1^Q'_1P'(z). The first and the last equality are because P(z)=P'(z)=0, for z>r-2. The second identity of (c) is a consequence of the first. As an immediate consequence of the two lemmas above, we have * For all i∈ I_r, |i|_r=|0|_3|i|_r and |i'|_r=|1|_3|i|_r. * If the triple (i,j,k) is admissible, then |i,j,k|_r=|0,0,0|_3|i,j,k|_r and |i',j',k|_r=|1,1,0|_3|i,j,k|_r. * For every admissible 6-tuple (i,j,k,l,m,n) we have the following. |[ i j k; l m n; ]|_r = |[ 0 0 0; 0 0 0; ]|_3 |[ i j k; l m n; ]|_r, |[ i j k; l' m' n'; ]|_r =|[ 0 0 0; 1 1 1; ]|_3|[ i j k; l m n; ]|_r, |[ i' j' k; l' m' n; ]|_r =|[ 1 1 0; 1 1 0; ]|_3|[ i j k; l m n; ]|_r. Now we are ready to prove Theorem <ref>. For (a), we observe that there is a bijection ϕ: I_3× I'_r → I_r defined by ϕ (0, i)=i and ϕ (1, i)=i'. This induces a bijection ϕ: A_3 × A'_r → A_r. Then, by Proposition <ref>, we have TV_3(M) · TV'_r(M) = (η_3^2|V|∑_c∈ A_3∏_e∈ E|e|_c∏_f∈ F|f|_c∏_Δ∈ T|Δ|_c)(η_r'^2|V|∑_c'∈ A'_r∏_e∈ E|e|_c'∏_f∈ F|f|_c'∏_Δ∈ T|Δ|_c') =(η_3η'_r)^2|V|∑_(c,c')∈ A_3× A'_r∏_e∈ E |e|_c|e|_c'∏_f∈ F |f|_c|f|_c'∏_Δ∈ T |Δ|_c|Δ|_c' = η_r^2|V|∑_ϕ(c,c')∈ A_r∏_e∈ E |e|_ϕ(c,c')∏_f∈ F |f|_ϕ(c,c')∏_Δ∈ T |Δ|_ϕ(c,c') =TV_r(M), where the third equality comes from the fact that η_r = η_3 ·η_r' and Lemma <ref>. This finishes the proof of part (a) of the statement of the theorem. Part (b) is given in <cit.>. To deduce (c), note that by Lemma <ref> we have that TV_3(M)=∑_c∈ A_31=|A_3|. Also note that c∈ A_3 if and only if c(e_1)+c(e_2)+c(e_3) is even for the edges e_1, e_2, e_3 of a face. Now consider the handle decomposition of M dual to the ideal triangulation. Then there is a one-to-one correspondence between 3-colorings and maps c:{ 2-handles}→ℤ_2, and c∈ A_3 if and only if c is a 2-cycle; that is if and only if c∈ Z_2(M, ℤ_2). Hence we get |A_3|=(Z_2(M, ℤ_2)). Since there are no 3-handles, H_2(M,ℤ_2)≅ Z_2(M, ℤ_2). Therefore, TV_3(M)=|A_3|= (H_2(M, ℤ_2))=2^b_2(M). hamsplain Effstratia Kalfagianni Department of Mathematics, Michigan State University East Lansing, MI 48824 (kalfagia@math.msu.edu) Renaud Detcherry Department of Mathematics, Michigan State University East Lansing, MI 48824 (detcherry@math.msu.edu) Tian Yang Department of Mathematics, Texas A &M University College Station, TX 77843 ( tianyan@math.tamu.edu)

http://arxiv.org/abs/1701.07991v2

20170127100213

A Mood Value for Fair Resource Allocations

cs.NI

A Mood Value for Fair Resource Allocations Francesca Fossati1, Stefano Moretti2, Stefano Secci1 1Sorbonne Universités, UPMC Univ Paris 06, UMR 7606, LIP6, 75005 Paris, France. Email: {firstname.lastname}@upmc.fr 2 CNRS UMR7243, PSL, Université Paris-Dauphine, Paris, France. Email: stefano.moretti@lamsade.dauphine.fr. January 27, 2017 ======================================================================================================================================================================================================================================================================================== In networking and computing, resource allocation is typically addressed using classical sharing protocols as, for instance, the proportional division rule, the max-min fair allocation, or other solutions inspired by cooperative game theory. In this paper, we argue that, under awareness about the available resource and other users' demands, in a cooperative setting such classical resource allocation approaches, as well as associated notions of fairness, show important limitations. We identify in the individual satisfaction rate the key aspect of the challenge of defining a new notion of fairness and, consequently, a resource allocation algorithm more appropriate for the cooperative context. We generalize the concept of user satisfaction considering the set of admissible solutions for bankruptcy games. We adapt the Jain's fairness index to include the new user satisfaction rate. Accordingly, we propose a new allocation rule we call Mood Value. For each user it equalizes our novel game-theoretic definition of user satisfaction with respect to a distribution of the resource. We test the mood value and the new fairness index through extensive simulations showing how they better support the fairness analysis. § INTRODUCTION In communication networks and computing systems, resource allocation (in some contexts also referred to as resource scheduling, pooling, or sharing) is a phase, in a network protocol or system management stack, when a group of individual users or clients have to receive a portion of the resource in order to operate a service. Resource allocation becomes a challenging problem when the available resource is limited and not enough to fully satisfy users' demand. In such situations, resource allocation algorithms need to ensure a form of fairness. Such situations emerge in a variety of contexts, such as wireless access <cit.>, competitive routing <cit.>, transport control <cit.>. The common methodology adopted in the literature is to, on the one hand, determine allocation rules such that they satisfy desirable properties <cit.>, and, on the other hand, analyse the fairness of a given allocation through indices, the most commonly used being the Jain's index  <cit.>. Allocation rules and indices of fairness are commonly justified by some fairness criteria. For instance, among two equivalent users demanding the same amount of resource, it makes sense not to discriminate and to give to each of them the same portion of the resource. In some cases, it can be desirable to guarantee at least a minimum amount of the resource so that the maximum number of users can be served. In the networking literature, the resource allocation problem is historically solved as a single-decision maker problem in which users are possibly not aware of the other users' demands and of the total amount of available resource. It follows that the most natural and intuitive way to quantify the user satisfaction is through the proportion of the demand that is satisfied by an allocation. Large literature exists indeed in the networking area on proportional resource allocations for many practical situations, from wireless networks to transport connection management <cit.>. In this paper, we are particularly interested instead in cooperative networking contexts such that users can be aware of other users' demands and the available amount. As such, rational users shall compute their satisfaction also based on the presence of other users. In fact, such networking contexts with demand and resource availability awareness are making surface in wired and wireless network environments with an increasing level of programmability, i.e., using software-defined radio and network platforms that expose novel (northbound) interfaces to users to disseminate information and pilot network resource allocations. Our main idea is defining a new notion of user satisfaction for such interactive resource allocation situations with demand and resource awareness. Let us briefly clarify our motivation with the following allocation example. A user i asks a quantity of resource that is bigger than the resource itself (as B in Fig. <ref>). Classical fairness indices <cit.>, <cit.>, <cit.> tend to qualify the user satisfaction as maximum when i obtains exactly what it asks. In the case where i asks more than the available amount, it cannot reach the maximum satisfaction due to the fact that its demand exceeds the available resource. Instead, in demand and resource awareness conditions, it would be more reasonable that its satisfaction is maximum when it obtains all the available resource. Furthermore, if all the other users together ask a quantity of good inferior to the resource, a minimum portion of it, equal to the difference between the resource and the sum of the demands of all the others, is guaranteed to i. Under a dual reasoning, it also appears more acceptable that the minimum satisfaction of a user is reached when it receives the minimum portion of the available resource, instead of when it receives zero. If users are in complete information context the classical approach can lead to not reasonable outcomes. In this perspective, in order to better describe the user satisfaction as a function of the available resource, and to capture the interactions due to the networking context (e.g., networked users may be aware of respective demands, may ally in the formulation of their demands, etc), we propose to model the resource allocation problem as a coalitional game. Accordingly, we define a new satisfaction rate for users, able to adapt to various configurations of the demands. Furthermore, we define a new resource allocation rule, called the Mood Value, based on the idea that the most fair allocation is the one that equalizes the satisfaction of each player. Indeed, regardless of the level of satisfaction, each player is not discriminated if its satisfaction is the same than the one of all the others. Choosing this allocation, users, who have the chance to recover informations about the other users and the available resource, have the feeling to receive a fair portion of the resource. We also provide an interpretation of this approach positioning it with respect to classical traffic theory <cit.>. The paper is organized as follows. Section <ref> presents the state of the art on the topic. In Section <ref> a new satisfaction rate is proposed. In Section <ref> the mood value and a new fairness index are described. In Section <ref> we provide an interpretation of the mood value with a traffic theory methodology, i.e. as result of the maximization of an appropriate utility function. Section <ref> presents some numerical examples. Finally, Section <ref> concludes the paper. § BACKGROUND A resource allocation problem can be characterized by a pair (c, E), in which c is the vector of demands (claims) from n users (claimants) and E is the resource (estate) that should be shared between them. The set of users is N={1,...,n}. The resource allocation is a challenging problem when E is not enough to satisfy all the demands (∑_i=1^n c_i ≥ E). An allocation x ∈ℝ^n is a solution vector that satisfies three basic properties: * Non-negativity: each user should receive at least zero. * Demands boundedness: each user cannot receive more than its demand. * Efficiency: the sum of all allocations should be E. An allocation rule is a function that associates a unique allocation vector x to each (c ,E). §.§ Classical resource allocation rules Many resource allocation rules are proposed in the literature and each of them is characterized by a set of properties that justify the use of the given rule in order to find a solution of the allocation problem <cit.>. In computer networks, the most well-known rules are: the proportional rule and the weighted proportional rule <cit.>, the max-min fair allocation (MMF) <cit.>, <cit.> , and the α-fair allocation <cit.>. Each of these allocation rules, result of an optimization problem and/or an iterative algorithm, follows a fairness criterion. The weighted proportional allocation rule is based on the idea that a logarithmic utility function captures well the individual evaluation of the worth of the resource <cit.>. One way to compute it is via the maximization of ∑_i=1^n w_ilog x_i subject to demand boundness and efficiency constraints. When w_i is equal to 1 the resulting allocation is called simply proportional and when w_i is equal to c_i we obtain the allocation that actually produces allocations proportional to the demands; hence in the following, we refer to the latter rule as proportional instead of the previous (not weighted) one. The idea behind the max-min fairness (MMF) allocation is to maximize firstly the minimum allocation; secondly, the second lowest allocation, and so on <cit.>. This solution coincides with the only feasible allocation such that, if the allocation of some users is increased, the allocation of some other users with smaller or equal amount is decreased. More generally, it is possible to obtain a family of allocation rules maximizing a parametric utility function. The α-fair utility function is defined as ∑_i=1^n x_i^(1-α)/1-α <cit.>. If α→ 1 the solution of the optimization problem coincides with the weighted proportional allocation with w_i equal to 1, if α=2 with the minimum delay potential allocation, that is the allocation obtained minimizing the total potential delay ∑_i=1^n ( 1/x_i) <cit.>, and if α→∞ with the max-min fair allocation. §.§ Game theoretical allocation rules Recently game theory has been applied to communication systems in order to model network interactions. For example, in <cit.> a cooperative game model is proposed to select a fair allocation of the transmission rate in multiple access channels and in <cit.> the authors studied, using coalitional game theory, the cooperation between rational users in wireless networks. Moreover, it is possible to analyze the allocation problem as a Transferable Utility (TU) game <cit.>, which is defined as a pair (N,v), where N={1,…,n} denotes the set of players and v:2^N→ℝ is the characteristic function, (by convention, v(∅)=0). Bankruptcy games <cit.>, in particular, deal with situations where the number of claimed resource exceeds that available. A Bankruptcy game is a TU-game (N,v) in which the value of the coalition is given by v(S)= max{E-∑_i ∈ N ∖ S c_i, 0} where E ≥ 0 represents the estate to be divided and c ∈ℝ_+^N is a vector of claims satisfying the condition ∑_i ∈ N c_i > E <cit.>. The bankruptcy game is superadditive, that is: v(S ∪ T) ≥ v(S) + v(T), ∀ S, T ⊆ N | S ∩ T = ∅ it is also supermodular (or, equivalently, convex), that is: v(S ∪ T) + v(S ∩ T) ≥ v(S) + v(T) ∀ S, T ⊆ N A classical set-value solution for a TU-game is the core C(v), which is is defined as the set of allocation vectors x ∈ℝ^N for which no coalition has an incentive to leave the grand coalition N, i.e.: C(v)={ x ∈ℝ^N: ∑_i ∈ N x_i = v(N), ∑_i ∈ S x_i ≥ v(S) ∀ S ⊂ N}. A one-point solution (or simply a solution) for a class 𝒞^N of coalitional games is a function ψ: 𝒞^N →ℝ^N that assigns a payoff vector ψ(v) ∈ℝ^N to every coalitional game in the class. A well-known solution for TU-games is the Shapley value <cit.> ϕ(v) of a game (N,v), defined as the weighted mean of the players' marginal contributions over all possible coalitions and computed as follows: ϕ_i(v)= ∑_S ⊆ N: i ∈ S w_i(S) (v(S)-v(S ∖{i})), with w_i(S)=(s-1)!(n-s)!/n! where s denotes the cardinality of S ⊆ N. Another well studied solution for TU-games is the nucleolus, based on the idea of minimizing the maximum discontent <cit.>. Given a TU-game (N,v) and an allocation x ∈ℝ^N, let e(S,x)= v(S) - ∑_i ∈ S x_i be the excess of coalition S over the allocation x, and let ≤_L be the lexicographic order on ℝ. Given an imputation x, θ(x) is the vector that arranges in decreasing order the excess of the 2^n-1 non-empty coalitions over the imputation x. The nucleolus ν(v) is defined as the imputation x (i.e., ∑_i ∈ N x_i = v(N) and x_i ≥ v({i}) for each i ∈ N) such that θ(x) ≤_L θ(y) for all y imputations of the game v. Given a bankruptcy game, many other solutions can be proposed <cit.>. As already introduced in the previous section, the proportional allocation assigns to player i an allocation equal to E · c_i / ∑_i=1^n c_i. For example, it is worth mentioning the Constrained Equal Loss (CEL) allocation that divides equally the difference between the sum of the demands and E, under the constraint that no player receives a negative amount. §.§ Fairness indices The evaluation of the fairness of the allocations, used as an important system performance metric especially in networking, can be useful to discriminate among allocation rules and to evaluate the level of justice in the repartition of the resources. Jain <cit.> introduces a formula aimed at providing a quantitative measure of the fairness of a resource sharing allocation. Given an allocation problem (c,E) and an allocation x, the Jain's fairness index is: J= [∑_i=1^n (x_i/c_i) ]^2 /[ n∑_i=1^n (x_i/c_i)^2] The Jain's index is bounded between 1/n and 1 <cit.>. The maximum fairness is measured when all the users obtain the same fraction of demand and the minimum fairness is measured when it exists only one user that receives all the resource. The Jain's index has the following good properties: * Population size independence: applicable to any user set, finite or infinite. * Scale and metric independence: not affected by the scale. * Boundedness: can be expressed as a percentage. * Continuity: able to capture any change in the allocation. The index considers the proportion of demand and it gives the maximum fairness to the allocation for which all the users receive the same proportion of the demand, regardless of the type of allocation problem, it suggests to allocate the resources in a proportional way even when this allocation is not the most suitable to solve the problem. Another well-know index of fairness is the Atkinson's index <cit.>; contrary to the Jain's index, it measures the degree of inequality of a given allocation, taking value equal to 0 when the system is 100% fair in the MMF sense, and 1 when it is totally unfair. Let (c,E) be the situation of Fig. <ref> with c=(3,13,2) and E=10. The discussed allocation rules provide values in Table <ref> along with the Jain's index and 1-Atkinson's index in order to have a measure of fairness. The axiomatic theory of fairness proposed in <cit.> shows that it exists an unique family of fairness measures, which includes the Jain's and the Atkinson's indices, satisfying a set of reasonable axioms. In the rest of the paper, we consider only the Jain's index because it is the one classically used in networking applications. MMF-driven inequality indices find their most appropriate use in socio-economical contexts, because they are linked to the concept of welfare of an income distribution. Furthermore the Jain's index is based on the idea of summarizing the information about the users' satisfaction, which is close to our methodology of redefining users' satisfaction under demand and resource awareness, as discussed in the following section. § FROM DEMAND FRACTION SATISFACTION TO GAME THEORETICAL SATISFACTION In this section, we propose a game-theoretic approach to evaluate the satisfaction of a user for an allocation. §.§ User satisfaction rate A crucial issue in resource allocation is to jointly: * find the best solution in terms of a certain goal; * evaluate its fairness by referring to a fairness index. With this purpose, it is important to evaluate the individual satisfaction rates and to summarize the information given by each of them with a global fairness index. A natural way to quantify the satisfaction of a user, as proposed by Jain, is through the proportion of the demand that is satisfied by an allocation <cit.>. Given the user i with demand c_i and an allocation x_i, the Demand Fraction Satisfaction (DFS) rate of i is: DFS_i =x_i/c_i. This rate takes value between 0 and 1 since it represents the percentage of the demand that is satisfied. Unavoidably, this way to quantify the user satisfaction makes the weighted proportional allocation the fairest one since it allocates proportionally to the demand. There are, however, situations in which the common sense does not suggest to allocate in a proportional way; e.g., if there is a big gap between the demands, in order to protect the weaker users and guarantee them a minimum portion of the estate, the MMF allocation can be preferable. Furthermore, as mentioned in the introduction, the presence of other users should rationally be considered not to distort the satisfaction of each user, in case of awareness about other users' demand and the available demands. For these reasons, we aim at defining an alternative satisfaction rate such that it satisfies the following two properties we name demand relativeness and relative null satisfaction: * Demand relativeness: a user is fully satisfied when it receives its maximal right, based on the available resource; * Relative null satisfaction: a user has null satisfaction when it receives exactly its minimal right, based on other users' demands and the available resource. The minimal right for a player is the difference between the available amount and the sum of the demands of the other users (i.e., taking a worst case assumption that the others get the totality of their demand), and the maximal right is equal to the maximum available resource, i.e., c_i if c_i<E, or it is equal to E otherwise. Remembering the definition of the characteristic function of a bankruptcy game we have that: * the minimal right for player i is v(i) * the maximal right for player i is v(N)-v(N∖ i) Thus we introduce the player satisfaction (PS) rate, which satisfies the above two properties by considering the value of the bankruptcy game associated to the allocation problem. Given a bankruptcy game such that ∑_i=1^n c_i>E and an allocation x_i, the Player Satisfaction (PS) rate for i is: PS_i=x_i-min_imax_i-min_i, where: min_i=v(i), max_i=v(N)-v(N∖ i). If ∑_i=1^n c_i=E the player satisfaction rate is PS_i=1, ∀ i ∈ N. PS_i ∈ [0,1] if the allocation belongs to the core (see Proposition <ref>). Moreover it corrects DFS_i since it replaces the interval of possible values [0,c_i] for x_i with the interval [min_i,max_i]. Consequently, if for the DFS rate the maximum satisfaction for i is measured when it gets c_i and the minimum when it gets 0, with PS, i is measured to be totally satisfied when it gets max_i and totally unsatisfied when it gets min_i. Consider (c,E) of Example 1 (see Fig.<ref>) and the corresponding bankruptcy game model. It holds: Proportional allocation: DFS_2=0.555 and PS_2=0.444 MMF allocation: DFS_2=0.3846 and PS_2=0. In both cases the PS rate shows that player 2 is less satisfied than what expected with the DFS rate. This is due to the fact that the game guarantees player 2 to get at least 5. The following propositions show some interesting properties of the PS rate. If the allocation x belongs to the core of the bankruptcy game, PS_i ∈ [0,1] ∀ i ∈ N. If a solution x belongs to a core it holds: x_i≥ v(i) and x_i≤ v(N)-v(N∖ i). Thus v(i) and v(N)-v(N∖ i) are the minimum and the maximum value that an allocation in the core can take. If x_i=v(i)=min_i then PS_i=0, if x_i=v(N)-v(N∖ i)=max_i then PS_i=1. It is possible to summarize the bankruptcy regimes of the PS rate in four possible cases as in Table <ref>. Let us treat each possible cases of Table <ref>: * Case Gm: v(i)=0, c_i<E Using the definition of bankruptcy game, it holds: v(N)-v(N∖ i)= E-max{0, E-c_i}=E-E+c_i. It follows PS_i= x_i / c_i. * Case Gg: v(i)=0, c_i≥ E Using the definition of bankruptcy game, it holds: v(N)-v(N∖ i)= E-max{0, E-c_i}=E. It follows PS_i= x_i / E * Case Mm: v(i)≠0, c_i<E As in case Mg, v(N)-v(N∖ i)= E-max{0, E-c_i}=c_i. It follows PS_i= (x_i-v(i)) / (c_i-v(i)). * Case Mg: v(i)≠0, c_i≥ E As in case Gg , v(N)-v(N∖ i)= E-max{0, E-c_i}=E. It follows PS_i= (x_i-v(i)) / (E-v(i)). §.§.§ Case terminology the PS rate differentiates 4 possible cases we name Gm, Gg, Mm, Mg. If a player asks less than E we call it moderate player (m) while if it asks more than E it is a greedy player (g). In similar way, if the sum of the demand of a group of n-1 players exceeds E, that means v(i)=0, the group is a group of greedy players (G) otherwise if v(i)≠ 0 we have a group of moderate players (M). Proposition <ref> highlights that, not only there is a relation between the DFS rate and the PS rate, the satisfaction of a user should be modified when it is considered as a player inside a cooperative game. In particular, we can notice that for case Gm the PS rate coincides with the DFS one, i.e., PS_i=DFS_i; for case Gg, the user satisfaction measured with the PS rate is higher than when it is measured with the DFS rate, i.e., PS_i ≥ DFS_i; in the Mg case, we have instead that DFS_i ≥ PS_i. We can also notice that the denominator of the PS rate is always different from zero. In cases Gm and Gg this is obviously true, in case Mm the denominator is zero when ∑_i=1^n c_i=E but in this case we set PS_i=1 and in case Mg the denominator is zero when ∑_j∈ N,j≠ i c_j=0 that is impossible. Furthermore, from Proposition <ref> it follows that if an allocation, i.e. a solution of an allocation problem that satisfies efficiency, non-negativity and demand boundedness, is an imputation, then PS_i ∈ [0,1] for all the users. This holds due to the fact that for an allocation, in each of the 4 cases presented above, it is always verified that v(N)-v(N∖ i) is an upper bound for x_i. §.§ Game-theoretical interpretation To support and justify the use of the new satisfaction rate, we show an interesting game-theoretic interpretation. Gately <cit.> introduced the concept of propensity to disrupt in order to eliminate the less fair imputation inside of the core. The idea was to investigate the gain of the player from the cooperation or, instead, its propensity to leave the cooperation, and to eliminate the imputation for which the propensity to leave the coalition for some players is excessively high. The formal definition of the propensity to disrupt is given in <cit.>. For any allocation vector x, the propensity to disrupt d(x,S) of a coalition S ∈ N (S≠∅, N) is the ratio of the loss incurred by the complementary coalition N∖ S to the loss incurred by the coalition S itself if the payoff vector is abandoned. In formula, d(x,S)=x(N∖ S)-v(N∖ S)/x(S)-v(S). An equivalent definition of d(x,S) is : d(x,S)=x(S)-v(S)/x(S)-v(S)-1 where: x(S)= v(N)-v(N∖ S) <cit.>. The propensity to disrupt of a coalition S quantifies its desire to leave the coalition. When x(S)=v(S) the propensity to disrupt of S is infinite and the desire of S to leave the coalition is maximum; when x(S)>v(S) but x(S)-v(S) is small, the value of d(x,S) is very high and again S does not like the agreement; when x(S)=v(N)-v(N∖ S) the propensity to disrupt is zero and S has the propensity not to destroy the coalition; when x(S)>v(N)-v(N∖ S) the index is negative and there is an hyper-enthusiasm for such an agreement. It holds an interesting relationship between the propensity to disrupt and the player satisfaction rate. The relationship between the player satisfaction rate and the propensity to disrupt is: PS_i=1d(x,i)+1. Using the alternative definition of d(x,i) we have d(x,i)=v(N)-v(N∖ i)-v(i)/x_i-v(i)-1 but v(N)-v(N∖ i)-v(i)/x_i-v(i) is equal to 1/PS_i so d(x,i)=1/PS_i-1. It is worth noting that if d(x,i) goes to infinity, then PS_i goes to 0 and if d(x,i)=0 then PS_i = 1. This gives another interpretation of the PS rate. The higher the satisfaction is, the bigger the enthusiasm of i, for being in the coalition, is. On the contrary, the closer to zero the user satisfaction is, the higher the propensity of user i to leave the coalition is. § THE MOOD VALUE AND THE PLAYER FAIRNESS INDEX In this section, we define a new resource allocation rule we call the Mood Value. The fairness idea behind this rule is the same of the one behind the Jain's index. A repartition of a resource is fair when all the users have the same satisfaction. Furthermore, we propose a novel fairness index as a modification of the Jain's index. §.§ The Mood Value Using the defined PS rate, we can define the mood value. Given an allocation problem characterized by (c,E), the allocation x such that PS_i=PS_j ∀ i,j ∈ N is called mood value. Due to the relation between the propensity to disrupt and the player satisfaction, the fairest solution corresponds to the one in which every player has the same propensity to leave the coalition. Equalizing the propensity to disrupt of the users, this allocation equalizes the mood of each player. In particular, given a game, it exists a unique mood such that the satisfaction of each user is the same. The closer to zero the mood is, the more unsatisfied user i is; the closer to one the mood is, the more enthusiast the user i is. Let (c,E) characterize an allocation problem. It exists a unique mood m such that PS_i=m ∀ i ∈ N; it is: m=E-min /max-min where min=∑_i=1^n v(i)= ∑_i=1^n min(i) and max=∑_i=1^n [E-v(N∖ i)]=∑_i=1^n max(i). And the mood value is given by: x_i^m=v(i)+m(max(i)-min(i)). Let PS_i=m ∀ i ∈ N. It follows: x_i=m(E-v(N∖ i))+(1-m)v(i). Due to the efficiency property it holds: ∑_i=1^n m(E-v(N∖ i))+(1-m)v(i)=E. Thus (<ref>). Since x_i is the mood value iff PS_i=m ∀ i ∈ N: x_i-v(i)/E-v(N∖ i)-v(i)=m ∀ i ∈ N and (<ref>) remains proved. From (<ref>) we can notice that the mood depends only on the game setting, thus, given a bankruptcy game, we can know a priori the value of the mood that produces a fair allocation. Knowing m, on can easily calculate the mood value x_i^m. The formula (<ref>) shows that each user receives the minimum possible allocation v(i) plus a portion m of the quantity max_i-min_i. The nearer to 1 is the mood m, the greater is the happiness of each user, and the closer to the maximum the allocation is. In fact, when m is equal to 1, the player receives exactly E-v(N∖ i), that is the maximum portion of resource that it can get, being inside a bankruptcy game. The mood value owns some interesting properties. It is an allocation thus it satisfies non-negativity, demand boundedness and efficiency property; it is stable, that means it belongs to the core of the game (prop. <ref>) and it guarantees more than minimal right to each player (x_i^m>v(i)). Furthermore it satisfies the following property: if v(i)=v(j) and v(N∖ i)=v(N∖ j) then x_i^m=x_j^m. This implies the equal treatments of equals (c_i=c_j ⇒ x_i^m=x_j^m) and equal treatment of greedy claimants (given a bankruptcy game, let G be the set of greedy players, i.e. such that c_i>E: if |G|≥ 2 then x_i^m=x_j^m ∀ i,j ∈ G). This last property guarantees that even if a user has a cheating behavior its demand is bounded by the available amount of resource E. Furthermore the mood value is a strategy-proof allocation because a user has no advantages in splitting his demand. The mood value belongs to the core of (N,v). We should prove that x_S^m≥ v(S), ∀ S ⊆ N. If v(S)=0 the condition holds due to the fact that x_i^m< 0, ∀ i ∈ N. Now consider the case v(S)>0. Suppose that x_S^m < v(S)=E-∑_i ∈ N ∖ S c_i. For the efficiency property it holds E=x_S^m+x_N∖ S^m, implying x_N∖ S^m>∑_i ∈ N ∖ S c_i, which yields a contradiction with the fact that, according to the mood value solution, each user receives at most its demand. §.§.§ Mood Value Computation Complexity Differently from the other allocation solutions inspired by game theory, in order to calculate this new allocation, only the value of 2n coalitions, i.e., the ones formed by the single players and the ones containing n-1 players, is needed. The time complexity of mood value computation is dominated by the complexity of computing v(i) that is 𝒪(n). In dynamic situations, i.e. when the value of each of the n coalitions has to be updated at each slot of time, the complexity is therefore 𝒪(n^2), but it can be reduced to 𝒪(n) where v(i) pre-computation is possible. This makes the mood value the best allocation rule in terms of time complexity together with the proportional allocation: the Shapley value has a time complexity of 𝒪(n!), while iterative algorithms for the computation of MMF and CEL allocations have a 𝒪(n^2log n) time complexity; the Nucleolus computation is a NP-hard problem. In terms of spatial complexity, the mood value, proportional, MMF and CEL allocations can be considered as equivalent and in the order of 𝒪(n). Instead, the Shapley value and the Nucleolus computations have a spatial complexity of 𝒪(2^n). §.§ The Player Fairness Index Considering the observed good properties that make the Jain's index a strong fairness index, we propose its modification replacing the DFS rate of the Jain's index with the PS rate. The resulting new fairness index we propose takes value 1 when all the users have the same satisfaction, i.e., when the allocation is the mood value. Given a problem (c,E) and an allocation x, the players fairness index is: J_p=[∑_i=1^n (PS_i) ]^2 / n∑_i=1^n (PS_i)^2 The players fairness index takes value in [1/n, 1] when the allocation belongs to the core. From Proposition <ref> follows that PS_i belongs to [0,1] and that ∑_i=1^n PS_i is always not negative. The maximum fairness is measured when all the users have the same PS rate, i.e.: [∑_i=1^n (PS_i) ]^2 =( n PS_i )^2 ⇒ n∑_i=1^n (PS_i)^2=nn (PS_i)^2. Thus J_p=1. The minimum fairness is measured when ∃ ! k s.t. PS_k≠ 0 and PS_j= 0 ∀ j ≠ k. In this case: [∑_i=1^n (PS_i) ]^2 =(PS_k)^2 ⇒ n∑_i=1^n (PS_i)^2=n (PS_k)^2 ⇒ J_p=1/n For core allocations, J_p takes value in the same interval of J making possible a comparison between the two indices. Furthermore, this index maintains all the good properties of the Jain's index: the population size independence, the scale and metric independence, the boundedness and the continuity. § INTERPRETATION WITH RESPECT TO TRAFFIC THEORY In the already cited seminal works about the definition of proportional and weighted proportional allocations in network communications, network optimization models are defined using as goal the maximization of an utility function. A typical application is the bandwidth sharing between elastic applications <cit.>, i.e., protocols able to adapt the transmission rate upon detection of packet loss. In this context we show how it is possible to revisit the mood value as a value resulting of the sum of the minimum allocation and the result of a weighted proportional allocation formulation where the weights are not the original demands, but new demands re-scaled accordingly to the maximum possible allocation knowing the available resource, and the minimum allocation under the awareness of other user's demands and the available resource. More precisely, the mood value can be computed as the result of the following 4-step algorithm. Step 1: We assign to each user the minimal right v(i). Step 2: We set the new value of the estate E'=E-min= E-∑_i=1^nv(i) and the new demands c'_i = max_i-min_i. Step 3: We solve the following optimization problem xmaximize ∑_i=1^n c'_ilog x_i subject to x_i ≤ c'_i, i = 1, …, n x_i ≥ 0, i = 1, …, n ∑_i=1^n x_i=E' Step 4: The mood value coincides with the sum of the minimal right and the allocation given by step 3: x_i^m=v(i)+x_i. We should prove that the result of the optimization problem is x_i=mc_i'. The lagrangian of the problem is L(x,μ, λ) = ∑_i=1^n c'_ilog x_i-μ^T(C-Ax)-λ( E'-∑_i=1^n x_i) where the vector μ and λ are the lagrangian multipliers (or shadow prices), C is the vector of the demands [c_1', ...c_n'] and A is the identity matrix of dimension n. Then, ∂ L/∂ y_i =c'_i/y_i-μ_i-λ. The optimum is given by y_i=c'_i/μ_i+λ when μ≥ 0, Ay≤ C, ∑_i=1^n y_i=E' and μ^T(C-Ay)=0. This coincides with the case in which μ^T=0 and λ≠ 0. In fact, we have ∑_i=1^nc'_i/λ=1/λ∑_i=1^nc'_i=E' . It follows that λ=1/E'∑_i=1^nc'_i is greater or equal to 1 and y_i=c'_i/λ is less or equal to c_i', that is an admissible solution. We can now notice that λ=1/E'∑_i=1^nc'_i=max-min/E-min=1/m. It follows y_i=mc_i'. Let (c,E) be the allocation problem of Fig. <ref>. Following the algorithm we have: Step 1: v(i)=[0,5,0]. Step 2: E'=5, c'_i = [3,5,2]. Step 3: x=[1.5, 2.5, 1] Step 4: x_i^m=[1.5,7.5,1]. The algorithm shows that the mood value firstly assign the minimal right (step 1) and secondly, considering the new allocation problem resulting after the first assignment (step 2), it allocates in a proportional way the resources (step 3). The proportion of resource to allocated is the mood. We provides two ways to compute the mood value: (<ref>) and the 4-step algorithm of Section <ref>. It is clear that the computation of the mood value throught the formula (<ref>) is less complex than the one using the 4 steps algorithm. § NUMERICAL EXAMPLES We tested the mood value and the new fairness index in a few significant configurations comparing them with the classical allocations and the Jain's index. We considered two demands distributions: (i) a uniform distribution, and (ii) a Weibull distribution. The former can be considered as a baseline, while the latter a maybe more realistic one. Taking inspiration from cellular (OFDMA) resource allocation studies we emulated an indoor scenario of femtocells using the WINNER II channel model <cit.>: generating in a uniform way 10000 users around the cell station between 3 and 100 m, we associate resource blocks (RBs) to each of them with a transmit power between 1 and 100 dB; the resulting RB distribution is well fit by a Weibull distribution. We now consider a range for demand generation between 0 and 100 units and we generate from (i) a uniform distribution between 0 and 100 and (ii) from a Weibull distribution f(x) = (a/b)(x/b)^(a-1) e^- (x/b)^a for x > 0 with scale parameter a=40 and shape parameter b=1.4. It is worth noting that the Weibull distribution is quite close to the Pareto distribution (both are exponential ones), its and discrete variations (e.g., Zipf's one), for example used in in-network content caching resource allocation <cit.> . We run different instances with a ratio of E (available resource) ranging from 5% to 95% of the global demand. We first simulate 300 bankruptcy games with 3 and 5 users. Fig. <ref> show the users configuration as a function of the available resource. With 3 users (Fig. <ref>a,c), for low value of E almost all are greedy players (Gg case) due to the fact that the resource is small; increasing E the number of moderate players (Gm) increases but also some users in configuration Mg appear. In fact, increasing E some greedy players become moderate while the others remain greedy ones; some of them are greedy inside a group of greedy users (Gg), while some others greedy inside a group of moderate ones (Mg). When the available resource is higher than half of the global demand, greedy players Gg disappear and the number of moderate players increases. In particular, users Mm appear and they become the majority when the resource is large. With 5 users (Fig. <ref>b,d), we find a similar trend than with 3 users in the number of moderate players that increases when increasing E. However, Mg users are few; in fact, it holds that it can exist at most one Mg user in a game and, due to the higher number of users in the system, it is very unlikely that there exists only a player Mg in the system such that the sum of the demands of the other n-1 players exceeds E. Thus, with a number of users higher than 5, one can practically reduce the number of user cases from 4 to 3. For this reason, in order to capture all the possible scenarios, we choose a low number of user for the first round of simulations. Fig. <ref>, <ref> and <ref> show the results of the first simulations. We consider the six allocations discussed before: Proportional, Shapley, Nucleolus, Mood Value, MMF and CEL. We calculate the Jain's fairness index and the players fairness index and we plot, for each ratio of E and each index, the mean value in between the first and third quantile lines. The Jain's index is depicted with the red color and square points while the players fairness index with the blue one and round points. Due to space limit, we do not show the case of 3 users with uniform demand distribution because similar to the 3-user one with the weibull distribution. In the 3-user scenario (Fig. <ref>) it is possible to notice differences between the result obtained by the classical Jain's index and our new players fairness (PF) index. It is worth recalling that the Jain's index has value 1 when the allocation is proportional while the PF index is 1 when the allocation is the mood value. We can notice that the PF index considers the MMF allocation as a fair one when the available resource is small (high congestion), i.e., when there are many greedy users. In fact the MMF allocation and the mood value, are close: in such cases, both have the property of treating equally the greedy claimant, giving them the same portion of resource, independently of their demands. Instead, when E increases, the MMF one is not fair any longer because it satisfies more the two users with less claim while it gives the minimal right to the one with bigger claim; in fact, in such cases the mood value becomes closer to the Proportional allocation, to the Shapley value and to the Nucleolus. The similarity between the Proportional allocation and the mood value is due to the fact that the correct way to measure the satisfaction of moderate players is through the DFS rate and increasing E the number of moderate players increases. It follows that the allocation equalizing the DFS rates, i.e., the Proportional one, is close to the one equalizing the PS rates of user, i.e., the Mood Value. With 5 users (Fig. <ref> and Fig. <ref>), we can notice that, due to the fact that the group Mg is little, when E reach the 40% of the global demand the Proportional is equal to the Mood value and it shows a PF index equal to 1. Furthermore the MMF allocation show again a high (PF) fairness when the resource is little (5%), i.e., when there are many greedy players. With a second round of simulations we want to see how the Demand Fraction (DFS) and Player (PS) Satisfaction rates are distributed with an increasing number of users. Results are shown in Figure <ref> as box-plots (minimum, quantiles and maximum, without outliers), for two regimes (high congestion of a 5% E/demand ratio, and low congestion of a 95% ratio). We can notice that the value of the satisfaction is low when the resource is small (Fig.. <ref>a,b,c) while it is higher when the congestion is low (Fig. <ref>d,e,f). With 4 users, in terms of user satisfaction rate, the mood value is close to the MMF allocation when the congestion is high, and to the proportional allocation when the congestion is low. As the number of players grows, the absolute difference between allocation in terms of distribution of the satisfaction decrease. In both the congestion situations (E equal to 5% and 95% of the demand) the Shapley value is the closest allocation to the mood value in terms of PS rate. Summarizing, the simulations show that the Mood Value is able to nicely weight the nature (greedy or moderate) of users and of user groups. In particular it is close to the MMF allocation when the resource is scarce and to the proportional allocation when the resource is close to the global demand. Furthermore, it is worth noticing that with respect to classical game-theoretical allocation rules (Shapley Value, Nucleolus), the results show that the Mood Value shows a similar good behavior in terms of fairness, with the key advantage of having a much lower computation time complexity. § CONCLUSION We proposed a game-theoretical approach to analyze and solve resource allocation problems, going beyond classical approaches that do not explore the setting where users can be aware of other users' demand and the available resource. In particular, we proposed a new way of quantifying the user satisfaction and a new fairness index as enhancement of the Jain's index, describing and comparing their mathematical properties in detail. Accordingly to these new concepts, we propose a new resource allocation rule that meets the goal of providing the fairest resource allocation, we called the Mood Value, which we position with respect to game theory metrics as well the common theory of fair allocation in networks. Finally, we test our ideas via numerical simulations of representative demand distributions, showing at which extent the mood value can approach and differ from max-min-fairness, weighted proportional, constrained-equal loss, Shapley Value and Nucleolus allocations. § ACKNOWLEDGMENT This work was partially funded by the FED4PMR investissement d'avenir project. The authors would like to thank Sahar Hoteit for her suggestions, and Deep Medhi and Catherine Rosenberg for their useful feedback. 99 saad-debbah W. Saad et al. "Coalitional games for distributed collaborative spectrum sensing in cognitive radio networks." IEEE INFOCOM 2009. cath J. Ghimire, C. Rosenberg. "Revisiting scheduling in heterogeneous networks when the backhaul is limited." IEEE Journal on Selected Areas in Communications 33.10 (2015): 2039-2051. orda A. Orda, R. Rom, N. Shimkin. "Competitive routing in multiuser communication networks." IEEE/ACM Tran. on Networking (ToN) 1.5 (1993): 510-521. proutiere-server SY. Yun, A. Proutiere. "Distributed Proportional Fair Load Balancing in Heterogenous Systems." ACM SIGMETRICS Performance Evaluation Review. Vol. 43. No. 1. ACM, 2015. thomson W. Thomson. "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: an update." Math. Social Sciences 74 (2015): 41-59. jain R. Jain, DM. Chiu, WR. Hawe. A quantitative measure of fairness and discrimination for resource allocation in shared computer system. Vol. 38. Hudson, MA: Eastern Research Laboratory, Digital Equipment Corporation, 1984. atk AB. Atkinson. "On the measurement of inequality." Journal of economic theory 2.3 (1970): 244-263. alpha J. Mo, J. Walrand. "Fair end-to-end window-based congestion control." IEEE/ACM Trans. on Networking (ToN) 8.5 (2000): 556-567. prop FP. Kelly, AK. Maulloo, DKH Tan. "Rate control for communication networks: shadow prices, proportional fairness and stability." Journal of the Operational Research society 49.3 (1998): 237-252. mmf DP. Bertsekas, RG. Gallager, P. Humblet. Data networks. Vol. 2. New Jersey: Prentice-Hall International, 1992. mmf1 W. Ogryczak et al. "Fair optimization and networks: A survey." Journal of Applied Mathematics (2014). delay L. Massoulié, J. Roberts. "Bandwidth sharing: objectives and algorithms." IEEE INFOCOM 1999. cop RJ. La, V. Anantharam. "A game-theoretic look at the Gaussian multiaccess channel." DIMACS series in discrete mathematics and theoretical computer science Vol.66 (2004): 87-106. cop2 S. Mathur, L. Sankar and N. Mandayam. "Coalitions in cooperative wireless networks." IEEE Journal on Selected areas in Communications, Vol. 26 (2008): 1104-1115. net MC. Lucas-Estañ, J. Gozàlvez, J. Sanchez-Soriano. "Bankruptcy-based radio resource management for multimedia mobile networks." Trans. on Emerging Telecommunications Technologies 23.2 (2012): 186-201. nucleolus S. Hoteit, et al. "A nucleolus-based approach for resource allocation in OFDMA wireless mesh networks."IEEE Transactions on Mobile Computing 12.11 (2013): 2145-2154. bank S. Hoteit et al. "On fair network cache allocation to content providers." Computer Networks 103 (2016): 129-142. game G. Owen, Game Theory (3rd ed.), Academic Press, New York (1995) aumann RJ. Aumann, M. Maschler. Game theoretic analysis of a bankruptcy problem from the Talmud. J. of Economic Theory, 36(2), 195-213 (1985). shapley LS. Shapley. "A value for n-person games.", H Kuhn and A Tucker, eds, Contributions to the Theory of Games, Vol. 2 of Annals of Mathematics Studies, Princeton U Press., 1953 nucl D. Schmeidler. (1969). The nucleolus of a characteristic function game. SIAM Journal on applied mathematics, 17(6), 1163-1170. axiom T. Lan et al. An axiomatic theory of fairness in network resource allocation. In Proceedings of INFOCOM 2010, IEEE. Gately D. Gately. "Sharing the gains from regional cooperation: A game theoretic application to planning investment in electric power." International Economic Review (1974): 195-208. little SC. Littlechild, KG. Vaidya. "The propensity to disrupt and the disruption nucleolus of a characteristic function game." International Journal of Game Theory 5.2-3 (1976): 151-161. winner YJ. Bultitude, T. Rautiainen. "IST-4-027756 WINNER II D1. 1.2 V1. 2 WINNER II Channel Models." (2007).

http://arxiv.org/abs/1701.07616v2

20170126084704

Minimum-Distance Based Construction of Multi-Kernel Polar Codes

cs.IT

Minimum-Distance Based Construction of Multi-Kernel Polar Codes Valerio Bioglio, Frédéric Gabry, Ingmar Land, Jean-Claude Belfiore Mathematical and Algorithmic Sciences Lab France Research Center, Huawei Technologies France SASU Email: {valerio.bioglio,frederic.gabry,ingmar.land, jean.claude.belfiore}@huawei.com January 2017 =============================================================================================================================================================================================================================================================== In this paper, we propose a construction for multi-kernel polar codes based on the maximization of the minimum distance. Compared to the original construction based on density evolution, our new design shows particular advantages for short code lengths, where the polarization effect has less impact on the performance than the distances of the code. We introduce and compute the minimum-distance profile and provide a simple greedy algorithm for the code design. Compared to state-of-the-art punctured or shortened Arikan polar codes, multi-kernel polar codes with our new design show significantly improved error-rate performance. § INTRODUCTION Polar codes, introduced by Arikan in <cit.>, are a new class of channel codes which achieve capacity over various classes of channels under low encoding and decoding complexity. Also for finite-lengths, these codes show remarkable error rate performance under list decoding. Only seven years after their discovery, polar codes were recently adopted in the standardization for the control channel of the future 5G system, where the focus is on short-length codes. In their original construction, polar codes are based on the polarization effect of the Kronecker powers of the 2 × 2 kernel matrix T_2 = [ 1 0; 1 1 ]. The generator matrix of a polar code is then a sub-matrix of the transformation matrix T_2^⊗ n. Arikan conjectured in <cit.> that the polarization effect is not restricted to powers of the kernel T_2, which was verified in <cit.>, where the authors provide necessary and sufficient conditions for binary kernels T_p of size p × p, p>2, to allow for the polarization effect. Recently, polar codes based on larger kernels were proposed in <cit.>, while in <cit.> authors propose to use different kernels of the same size to construct the transformation matrix of the code. Thanks to these ideas, it is now possible to construct polar codes of any code length of the form N = p^n. However, not all code lengths can be expressed as powers of integers. To overcome this length matching problem, puncturing <cit.>, <cit.> and shortening <cit.> techniques have been proposed to construct polar codes of arbitrary lengths, at the cost of a loss in terms of polarization speed, and hence worse error rate performance. To tackle the code length problem of polar codes, a multi-kernel construction has been proposed in <cit.>. By mixing binary kernels of different sizes in the transformation matrix, codes of lengths that are not only powers of integers can be constructed. The resulting multi-kernel polar code still benefits from the polarization effect while decoded through successive cancellation <cit.>. As a result, the new multi-kernel construction largely increases the number of code lengths that can be achieved without puncturing or shortening, with comparable or even better error-rate performance. For codes based on the polarization effect, the reliability of the input positions is determined by density evolution or other techniques, and then the least reliable positions are frozen. This is the design principle of the original construction of polar codes of infinite length <cit.>, and it is similarly used for the design of multi-kernel polar codes <cit.>. Such design by reliability is appropriate for long codes under successive cancellation decoding; for short codes under list decoding <cit.>, however, design principles that give more weight to distance properties may give superior error-rate performance. Related to this is the work in <cit.>, where reliability-based design of polar codes for better channels is shown to lead to better distance properties, and ultimately to Reed-Muller codes. In this paper, we propose a construction of multi-kernel polar codes that maximizes the minimum distance. We show how to find kernels of size larger than 2 that are advantageous in our construction. Moreover, we present a simple greedy code design algorithm that maximizes the minimum distance for given kernels. Due to the special structure of the kernels of larger size and the resulting flexibility in the code design, our construction of multi-kernel polar codes leads to better distance properties and thus to superior error rate performance under successive cancellation list decoding, compared to the reliability-based construction, and also compared to shortened or punctured codes based on T_2^⊗ n. This paper is organized as follows. In Section <ref>, we review construction, encoding, and decoding of multi-kernel polar codes. In Section <ref> we describe explicitly the new distance-based design for multi-kernel polar codes. In Section <ref> we illustrate numerically the performance of the codes, and Section <ref> concludes this paper. § MULTI-KERNEL POLAR CODES In this section, we briefly review the structure, encoding and decoding of multi-kernel polar codes; for details, we refer the reader to <cit.>. Multi-kernel polar codes are a generalization of the Arikan polar codes <cit.>, simply referred to as polar codes throughout the paper, and therefore we will provide a comparison to the Arikan construction for clarity. §.§ Code Structure and Encoding Polar codes are based on the Kronecker product G_N = T_2^⊗ n, N=2^n, where T_2 denotes the 2 × 2 kernel T_2 = [ 1 0; 1 1 ] . Let us assume an information set ℐ⊂ [N], [N] = {0,1,…,N-1}, of size |ℐ| = K and a corresponding frozen set ℱ = [N] \ℐ of size |ℱ| = N-K. An (N,K) polar code of length N and dimension K is then defined by the encoder x = u G_N, mapping the input vector u ∈2^N to the codeword x ∈2^N, where u_i = 0 for i ∈ℱ, denoting the frozen bits, and u_i, i ∈ℐ, are the information bits. Multi-kernel polar codes generalize this construction by mixing binary kernels of different sizes in the Kronecker product forming the transformation matrix. Examples of such kernels, which are used in this paper, are T_3 = [ 1 1 1; 1 0 1; 0 1 1 ], T_5 = [ 1 1 1 1 1; 1 0 0 0 0; 1 0 0 1 0; 1 1 1 0 0; 0 0 1 1 1 ] . The transformation matrix of a multi-kernel polar code is given by G_N = T_p_1⊗ T_p_2⋯⊗ T_p_s , where T_p_i, i=1,2,…,s, denotes the kernel matrix of size p_i × p_i, and kernels of same size can be used multiple times, i.e., it may be that p_i = p_j for some i,j. The length of the code is N = p_1 · p_2 ⋯ p_s. Note that the ordering of the kernels in the Kronecker product is important for the frozen set design, as the Kronecker product is not commutative. An (N,K) multi-kernel polar code is defined by the transformation matrix G_N and the information set ℐ, with corresponding frozen set ℱ = [N] \ℐ. Codewords x ∈2^N are generated from the input words u ∈2^N by x = u G_N, where u_i = 0 for i ∈ℱ and u_i, i ∈ℐ, stores the information bits. In <cit.>, ℐ is generated according to the reliabilities of the positions in the input vector u = (u_0,u_1,…,u_N-1), which can be determined e.g. through density evolution <cit.>. In this case, the information set is composed by the K most reliable positions. Similar to polar codes, the Tanner graph of multi-kernel polar codes can be constructed. While the Tanner graph of polar codes consists solely of 2 × 2 blocks, each corresponding to the kernel T_2, the Tanner graph of multi-kernel polar codes consists of various blocks, corresponding to the different kernels used. The Tanner graph for the transformation matrix in (<ref>) consists of s stages. On stage i, there are N/p_i blocks, each of size p_i × p_i, corresponding to a T_p_i kernel, with p_i edges to the left and to the right. The connections and edge permutations follow from the Kronecker product <cit.>. An example is given in Fig. <ref>, with edge-permutations indicated by dotted boxes. §.§ Decoding of Multi-Kernel Polar Codes Decoding of multi-kernel polar codes is performed similarly to polar codes, using successive cancellation (SC) decoding on the Tanner graph of the code <cit.>, or enhanced SC-based decoding methods like SC list (SCL) decoding <cit.>. Log-likelihood ratios (LLRs) are passed along the Tanner graph from the right to the left, while hard decisions on decoded bits are passed from the left to the right. The major difference to decoding of polar codes is given by the computations in the blocks corresponding to the new kernels. The notation used for the p × p block corresponding to a T_p kernel is depicted in Fig. <ref>. Denote u = (u_0,u_1,…,u_p-1) the binary input vector to this block and x = (x_0,x_1,…,x_p-1) its binary output vector. Then we have the relationship u T_p = x, defining the update rule for the hard-decisions going from left to right. Further denote L_i the LLR of output bit x_i and λ_i the LLR of the input bit u_i. The general structure of the update rule for LLRs, going from right to left, is λ_i = f( L_0 , L_1 , … , L_l-1 , û_0 , û_1 , …, û_i-1 ), i.e., all LLRs L_j and only previous hard-decisions (estimates) û_j may be used for the computation, following the SC principle. The corresponding LLR calculations for T_2 from <cit.> are λ_0 = L_0 ⊞ L_1 , λ_1 = (-1)^u_0· L_0 + L_1, for T_3 depicted in (<ref>), from <cit.>, are λ_0 = L_0 ⊞ L_1 ⊞ L_2 , λ_1 = (-1)^u_0· L_0 + L_1 ⊞ L_2 , λ_2 = (-1)^u_0· L_1 + (-1)^u_0 ⊕ u_1· L_2 , and for T_5 in (<ref>), presented here for the first time, are λ_0 = L_1 ⊞ L_2 ⊞ L_4 , λ_1 = (-1)^û_0· ( L_0 ⊞ (L_2 + (L_1 ⊞ L_4)) ⊞ L_3 ) , λ_2 = (-1)^û_1 · ( L_0 ⊞ L_1 ) + ( L_3 ⊞ L_4 ) , λ_3 = (-1)^û_0 ⊕û_1 ⊕û_2· L_0 + (-1)^û_0· L_1 + ( L_2 ⊞ (L_3 + L_4) ) , λ_4 = (-1)^û_0 ⊕û_3· L_2 + (-1)^û_0 ⊕û_2· L_3 + (-1)^û_0· L_4 . The boxplus operator for two LLRs a and b may be evaluated exactly as a ⊞ b = 2 tanh^-1( tanha/2·tanhb/2) or approximately as a ⊞ b ≈ a · b ·min{|a|,|b|}; the extension to multiple LLRs is as usual. For other kernels similar LLR update equations can be derived <cit.>. In the next section, we will show that the presented kernels T_3 and T_5 permit to construct multi-kernel polar codes with good minimum distance. § DESIGN FOR MINIMUM-DISTANCE In this section, we describe how to design multi-kernel polar codes to increase the minimum distance of the code. In <cit.>, the information set ℐ is selected according to reliability. This approach, which is also commonly followed for polar codes, is optimal for SC decoding when the code length tends to infinity. For short codes, however, the distance properties are more crucial than the polarization effect. In the following, we focus on multi-kernel polar codes with transformation matrix of the form G_N = T_2^⊗ n⊗ T_p, i.e., a polar code composed with a larger kernel at the end. This is not a very limiting assumption for the following reasons. First, the proposed design is to be used for short codes, for which the use of a single larger kernel is usually sufficient. Second, the larger kernel can be the composition of multiple smaller kernels. And third, changing the order of the kernels in the Kronecker product is equivalent to a row and column permutation of G_N, and thus leads to equivalent codes. §.§ Minimum-Distance Spectrum In the following, we will determine the minimum distance d achievable by a code generated by selecting K rows of a transformation matrix G_N. More formally, we define the minimum-distance spectrum S_G_N of the transformation matrix G_N to be the mapping from dimension K to the maximal minimum distance d achievable by selecting an information set ℐ of size K, i.e., S_G_N(K) is the largest minimum distance achievable by an (N,K) multi-kernel polar code derived from the transformation matrix G_N. Finding the minimum-distance spectrum of a code is in general a complex task, which may be accomplished e.g. by an exhaustive search. Under certain constraints, however, the minimum-distance spectrum of a multi-kernel polar code can be easily calculated based on the minimum-distance spectra of its building kernels. In fact, for polar codes, S_T_2^⊗ n = ([2 1]^⊗ n) , where (x) is the vector x sorted in decreasing order, since polar codes have the same transformation matrix as Reed-Muller codes. In the following, we prove that a similar property holds for multi-kernel polar codes, allowing one to calculate the minimum-distance spectrum of the transformation matrix G_N using the Kronecker product of the spectra of the kernels composing it. 1ex0ex If G_N = T_2^⊗ n⊗ T_p, then S_G_N = (S_T_2^⊗ n⊗ S_T_p). The proposition is proved by induction on the number n of T_2 kernels employed in the transformation matrix G_N. The property obviously holds for n=0, and by inductive hypothesis we suppose that S_G_N/2 = (S_T_2^⊗ n-1⊗ S_T_p) given G_N/2 = T_2^⊗ n-1⊗ T_p. Given the transformation matrix G_N = T_2^⊗ n⊗ T_p = [ G_N/2 0; G_N/2 G_N/2 ], this matrix can be divided into two parts, an upper matrix G^U = [G_N/2 | 0] and a lower matrix G_L = [G_N/2 |G_N/2 ], for which S_G^U = S_G_N/2 and S_G^L = 2 S_G_N/2. Given V = (S_T_2^⊗ n⊗ S_T_p), the goal of the proof is to show that S_G_N = V, i.e., that for every dimension K, there exists a subset of K rows of G_N such that the span of these rows has minimum distance V(K). To do that, for every K we show how to construct a sub-matrix of G_N for which all the vectors of its span have Hamming weight not smaller than V(K). In fact, by construction, for every K there exist two integers K^U and K^L such that K^U + K^L = K, and two sub-matrices G_A^U and G_B^L, formed by K^U rows of G^U and by K^L rows of G^L respectively, such that S_G_A^U(K^U) ≥ V(K) and S_G_B^L(K^L) ≥ V(K). To end the proof, it is sufficient to use the distance property of the classical (u | u+v) construction <cit.> to verify that the code generated by G_A,B = [ G_A^U/G_B^L] has minimum distance min(S_G_A^U(K),S_G_B^L(K)) = V(K). The proposition shows how to exploit the spectra of the building kernels to evaluate the minimum-distance spectrum of the multi-kernel polar code. Moreover, the constructive nature of the proof suggests a greedy technique to build multi-kernel polar codes with optimal minimum distance. Before describing the algorithm in detail, in the following section we present kernel design principles leading to codes with good minimum distance spectra. §.§ Kernel Design For polar codes, kernels are usually designed to maximize the polarization effect on the input bits of the transformation G_N, and the information positions are then selected in reliability order. For short codes, however, the polarization effect is far less important than distances of the code, and kernels should be designed taking this aspect into account. Different kernels have different spectra, while polar codes are limited by the spectrum of the kernel T_2. Multi-kernel polar codes permit to create codes of desired minimum distance by changing the kernels composing the transformation matrix. If the kernels are designed properly, the information set can then be selected such that a large minimum distance is achieved for the desired length and dimension. As an example, consider the T_3 kernel depicted in (<ref>), introduced in <cit.>, and its minimum-distance spectrum. For the information set of size 1, one row has to be selected: in order to maximize the minimum distance, the first row, (1 1 1), is selected, giving minimum distance 3; any other row selection would result in a smaller minimum distance, namely 2. For the information set of size 2, the last two rows, (1 0 1) and (0 1 1), are selected, generating a code of minimum distance 2; any other row selection would result in a smaller minimum distance. Finally, for a code of dimension 3, all rows have to be selected, resulting in a code of minimum distance 1. T_3 thus has the minimum-distance spectrum S_T_3=(3,2,1). As opposed to that, the construction by reliability selects the last row for dimension 1, the last two rows for dimension 2, and all rows for dimension 3; this gives minimum-distance spectrum (2,2,1). The proposed T_5 kernel presents a similar behavior, with minimum-distance spectrum S_T_5=(5,3,2,1,1). §.§ Greedy Row-Selection Algorithm In the previous sections, we described how to calculate the minimum-distance spectrum of the transformation matrix of a certain class of multi-kernel polar codes. The scope of this section is to describe how to determine the actual information set that achieves this minimum distance. As for the minimum-distance spectrum itself, this may be accomplished by an exhaustive search, which in general will be very complex. The proof of Proposition <ref>, however, gives an insight on how to select rows of G_N to achieve the minimum-distance spectrum. In the following we describe a greedy algorithm able to accomplish this task; the pseudo code is provided in Algorithm <ref>. Since the algorithm is based on Proposition <ref>, it finds an optimal solution if only one kernel of size larger than 2 is used in the construction, and this kernel is the last term in the Kronecker product. The algorithm may as well be applied in the case of multiple kernels of size larger than 2, also at the end of the Kronecker product, by treating the Kronecker product of these kernels as one large kernel, for which the minimum-distance spectrum has to be determined before the algorithm is applied. Given a transformation matrix G_N = T_2^⊗ n⊗ T_p, we assume the kernel T_p to have minimum-distance spectrum S_T_p= (d_p (1), ⋯, d_p (p)), where d_p (k) is the minimum distance of the code of dimension k. The list I^k = {i^k_1,…,i^k_k} is associated to every entry d_p (k) of the spectrum, collecting the indices of the k rows of T_p giving the optimal minimum distance of the kernel. To begin with, the vector r_N = (2,1)^⊗ n⊗ S_T_p is created. This vector is an unsorted version of the minimum-distance spectrum, collecting the minimum achievable distances of each part of G_N. For a code of dimension K, at each step the algorithm adds sequentially one row index to the information set ℐ, which is initially empty. At each step, the position l, with l = 0,…,N-1, of the last largest entry in r_N is found, and r_N (l) is set to zero. After that, the value c = (l mod p)+1 and q = l-c+1 are calculated, giving the row position within the kernel and the row index in the transformation matrix where the corresponding kernel starts, respectively. In fact, since in S_T_p the distances are sorted in descending order, we know that {i^c_1+q,…,i^c_c+q}⊂ℐ. The algorithm deletes these c indices belonging to I^c, substituting them with the c+1 indices given by I^c+1; by the constructive proof of Proposition <ref>, we know that the resulting code has the desired minimum distance. Of course, if c=0, no rows of that part of the matrix are already in the information set, and therefore no information indices are deleted. In practice, at each step the information set is updated as ℐ = ℐ∖{i^c_1+q,…,i^c_c+q}∪{i^c+1_1+q,…,i^c+1_c+1+q}. The algorithm stops when ℐ includes K elements. The remaining N-K indices compose the frozen set ℱ. §.§ Construction Example To illustrate our construction, in the following we describe the minimum distance design of a multi-kernel polar code of length N = 6 depicted in Fig. <ref> with transformation matrix G_6 = T_2 ⊗T_3 = [ T_3 0; T_3 T_3 ] = [ 1 1 1 0 0 0; 1 0 1 0 0 0; 0 1 1 0 0 0; 1 1 1 1 1 1; 1 0 1 1 0 1; 0 1 1 0 1 1; ] . For the described kernel of size 3, we have that S_T_3 = (3,2,1) with ℐ_T_3(1) = {0}, ℐ_T_3(2) = {1,2} and obviously ℐ_T_3(3) = {0,1,2}. The minimum-distance spectrum is given by S_G_6 = ((2,1) ⊗ S_T_3) = (6,4,3,2,2,1); consequently, r_6 = (3,2,1,6,4,2). It is worth noticing that the minimum-distance spectrum of the reliability construction is (4,4,2,2,2,1). If a rate 1/2 code has to be designed, the positions of the K=3 information bits are needed. The information set ℐ is initially empty. At the first step, l=3, hence c=0 and q=3; since c=0, no entries of ℐ have to be deleted, and ℐ = {3}. At the second step, l=4, so c=1 and q=3; the information set is calculated as ℐ = ℐ∖{3}∪{4,5} = {4,5}. Finally, at the third step l=0, and the resulting information set is ℐ = {4,5}∪{0} = {0,4,5}. A comparison of the information sets calculated by the proposed algorithm following the distance criterion and the one resulting from the reliability order is presented in Table <ref> for various dimensions K. We observe that the proposed design always outperforms the reliability-based designs in terms of minimum distance, or performs identically when the reliability-based construction is equivalent to the minimum-distance based construction. § NUMERICAL ILLUSTRATIONS In the following, we show the performance of the proposed minimum distance construction of multi-kernel polar codes. In particular, in Figures <ref>, <ref>, <ref> and <ref>, we show the BLER performance of the codes designed according to the proposed minimum distance construction under list decoding <cit.> with list size L=8 for BPSK transmission over an additive white Gaussian noise (AWGN) channel. Our proposal, coined MK-dist in the figures, will be compared to the reliability-based design of multi-kernel polar codes proposed in <cit.>, coined MK-rel in the figures. We emphasize that MK-dist is designed according to the row selection algorithm described in the previous sections. Moreover, we add as references state-of-the-art punctured <cit.> and shortened <cit.> polar code constructions, coined polar-punct and polar-short respectively. First, in Figure <ref>, we show the performance of a code with length N = 192 and dimension K = 96. In this case, the transformation matrix is given by T_192=T_2^⊗ 6⊗ T_3, i.e., there is only one T_3 kernel at the rightmost of the Kronecker product. In this case, the proposed row selection algorithm can be run using the minimum-distance spectrum of the T_3 kernel presented before, and we observe that the minimum-distance based design outperforms all other designs. In Figure <ref> we show the performance of a code of length N = 144 and dimension K = 72. The transformation matrix is given by T_144=T_2^⊗ 4⊗ T_3^⊗ 2, i.e., there are two T_3 kernels at the rightmost of the Kronecker product. In this case, the minimum-distance spectrum for the Kronecker product kernel T_3 ⊗ T_3 has to be calculated, along with the auxiliary lists ℐ_T_3 ⊗ T_3. Potentially, two different kernels of size 3 may be used, like proposed in <cit.>, augmenting the flexibility of the minimum-distance spectrum, but this kind of optimization is out of the scope of this paper. The resulting spectrum is S_T_3 ⊗ T_3 = (9,6,4,4,3,2,2,2,1), and the multi-kernel polar codes resulting from our design still outperform all other depicted designs. In Figure <ref> we show the performance of a code of length N = 40 and dimension K = 20. The transformation matrix is given by T_40=T_2^⊗ 3⊗ T_5, i.e., there is only one T_5 kernel at the rightmost of the Kronecker product. In this case, the BLER performance of the proposed construction is again better than the one of the other constructions, though the gain is smaller than in the previous two cases where T_3 is used. Finally, in Figure <ref> we show the performance of a code of length N = 90 and dimension K = 45. In this case, the transformation mixes three different kernels, and we define T_90=T_2 ⊗ T_3^⊗ 2⊗ T_5. The BLER performance of the proposed construction, while is only able to match the performance of the shortened polar code. This shows that the proposed algorithm should be further optimized in the presence of multiple high size kernels. In conclusion, the proposed distance-based construction significantly outperforms state-of-the-art punctured and shortened polar codes for small block lengths, as well as the previously proposed reliability-based construction in <cit.>. We expect this property to hold true for short code lengths, when the polarization effect has lower importance than the distance profile in the design of the codes. Moreover, we argue that the encoder and the decoder of the proposed multi-kernel polar codes have a lower complexity compared to the encoder and the decoder of the state-of-the-art punctured polar codes, due to the larger length of the mother polar code and the reliability calculations required for these constructions <cit.>. § CONCLUSIONS In this paper, we proposed a construction for multi-kernel polar codes, introduced in <cit.>, based on the maximization of the minimum distance. While the original construction based on bit reliabilities is suitable for long codes, our new minimum-distance based construction provides significant performance gains for short codes, i.e., where the polarization effect is less important than distance properties. This gives fundamental insights for the design of multi-kernel polar codes of any length. We further introduced the minimum-distance spectrum of a transformation matrix, and we developed a greedy algorithm that finds the information set achieving this minimum distance. Simulations illustrate the competitive performance of our design for short-length codes. IEEEbib

http://arxiv.org/abs/1701.07997v1

20170127101733

Observational View of Magnetic Fields in Active Galactic Nuclei Jets

astro-ph.HE

Non Amontons-Coulomb local friction law of randomly rough contact interfaces with rubber Antoine Chateauminoisy^1 December 30, 2023 ======================================================================================== According to the currently favored picture, relativistic jets in active galactic nuclei (AGN) are launched in the vicinity of the black hole by magnetic fields extracting energy from the spinning black hole or the accretion disk. In the past decades, various models from shocks to magnetic reconnection have been proposed as the energy dissipation mechanism in the jets. This paper presents a short review on how linear polarization observations can be used to constrain the magnetic field structure in the jets of AGN, and how the observations can be used to constrain the various emission models. § INTRODUCTION Active galactic nuclei (AGN) are among the most extreme objects in the universe. They are the centers of distant galaxies hosting a supermassive black hole, billions of times the mass of the Sun. About 10% of AGN produce collimated, relativistic outflows or plasma jets that shine brightly over the entire electromagnetic spectrum from radio to gamma-ray energies. The formation and stability of these jets are not yet fully understood. It is believed that energy is extracted from the central black hole by large-scale magnetic fields (<cit.>) while rotating inner regions of the accretion disk form an outflowing wind that helps to collimate the jet (<cit.>). In this picture, the jet is launched as a magnetically dominated outflow with strong magnetic fields accelerating the flow to relativistic velocities. General relativistic magnetohydrodynamic simulations show that the efficiency of the jet generation can be very high if the black hole is threaded by a dynamically important magnetic field (<cit.>). While the simulations suggest that the jets are likely highly magnetized near the black hole, their composition and magnetic field structure in the parsec scales, where they produce most of the emission, is much less clear. A link between the magnetic fields near the black hole and the parsec-scale jets was established by <cit.> who showed that there is a correlation between the magnetic flux of the jet and the accretion disk luminosity, as predicted by the magnetically arrested disk (MAD) model (<cit.>). This result illustrates how studying the emission regions in the parsec-scale jets may help us gain knowledge about the jet formation processes. One of the main open questions is the nature of the emission region. Traditionally, the flaring behavior of AGN jets has been well explained by shock-in-jet models, especially in the radio bands (e.g., <cit.>). These models assume the jet to be kinetically dominated at parsec scales. Shock models struggle to explain the very fast variability detected at high-energy gamma-ray bands, and models involving magnetic reconnection have been invoked to explain the high-energy emission (e.g., <cit.>). In this case the jet flow should be magnetically dominated, although recent particle-in-cell simulations show that even in reconnection models, the magnetic field and particle energies can be in equipartition at the emission region (<cit.>). Another major open question is the structure of the magnetic field at the emission site. According to the simulations, the magnetic field is helical close to the black hole but whether it preserves its order in the parsec scales is debated upon. For example, in the model of <cit.>, the magnetic field is helical at the acceleration and collimation zone of the jet but gets disrupted and turbulent beyond a standing shock in the jet. Polarization observations of some jets, however, indicate that the magnetic field could be helical even on the parsec scales (e.g., <cit.>). Figure <ref> shows an example adapted from <cit.>. In this contribution, I will discuss how linear polarization observations in the radio and optical bands can be used to constrain jet emission models and the magnetic field structure of the jets. This is not intended as a comprehensive review of all possible models, but more to give some examples on how observations are used to study magnetic fields in AGN jets. § POLARIZATION OBSERVATIONS The optical and radio emission of AGN jets is synchrotron emission, which is intrinsically highly polarized. In an optically thin emission region with a uniform magnetic field, the polarization degree is up to 70% (e.g., <cit.>), although such high polarization degree values are not typically seen in AGN jets in the radio (e.g., <cit.>) or optical (e.g., <cit.>) bands. This has been taken as evidence for disordered magnetic fields. The emission in AGN jets is often described with the Stokes parameters I (for total intensity), Q and U (for linear polarization) and V (for circular polarization). I will only discuss linear polarization here. Using the Stokes parameters, the polarization degree and the electric vector position angle (EVPA) can be defined as m=(√(Q^2+U^2))/I and EVPA=1/2tan^-1(U/Q). In the simplest case, in an optically thin jet, the polarization position angle is perpendicular to the magnetic field direction and one can use the EVPA observations to infer the direction of the magnetic field. However, one should note that in relativistic jets viewed at a small angle to the line of sight of the observer, the situation is more complicated (<cit.>). Another way to obtain information about magnetic fields in blazar jets is via Faraday rotation observations. When synchrotron radiation passes through magnetized plasma, Faraday rotation of the EVPAs proportional to the line-of-sight magnetic field and electron density occurs (e.g. <cit.>). In the simplest case, the effect can be described by a linear dependence between the observed electric vector position angle (EVPA; χ_obs) and wavelength squared (λ^2), given by χ_obs = χ_0 + 0.81 ∫ n_e 𝐁·𝐝𝐥 = χ_0 + RMλ^2, where χ_0 is the intrinsic EVPA and RM is the rotation measure (in rad/m^2), related to the electron density n_e of the plasma (in cm^-3) and the magnetic field component 𝐁 (in μG) along the line of sight (in parsecs). The RM can thus be estimated by observing the EVPA at different frequencies. This will give us information on the line-of-sight component of the magnetic field. For example, if the rotation measure is positive, the magnetic field is coming towards the observer, and if negative it is going away from the observer. Therefore it was suggested that a gradient in a Faraday rotation measure transverse to the jet direction could reveal helical magnetic field structures (<cit.>). § CONSTRAINING EMISSION MODELS THROUGH POLARIMETRY Polarization observations of AGN jets in the radio and optical bands have been conducted since the 1970s (see e.g., <cit.> for a review). At the same time, a large number of theoretical models were developed to explain the total intensity and polarization observations of AGN jets (e.g, <cit.>). In the 1980s it was suggested that most of the radio variability, especially in the cm-band, are due to shocks in the relativistic jets where the magnetic field is predominantly turbulent (<cit.>). The field of polarization modeling has received a new boost since an exciting connection to high-energy gamma-ray emission was found (<cit.>). A rotation in the optical polarization angle was coincident with an ejection of a new very long baseline interferometry (VLBI) component from the core, and a very-high-energy (VHE; > 100 GeV) detection of the source BL Lac by the MAGIC telescope. A further connection between the optical polarization and gamma-ray flares was suggested in two quasars 3C 279 (<cit.>) and PKS 1510-089 (<cit.>). In both of these the optical polarization angle was seen to rotate by more than 180 degrees over a course of 20 to 50 days during which a sharp gamma-ray flare was observed. Following all these observations, a number of new models have been published in the last years (e.g., <cit.>). I will go through some of these models with the emphasis on the observations that can be used to constrain them. §.§ Turbulence in the jets Turbulent magnetic field produces stochastic variations to the polarization angle and degree (<cit.>), such as an EVPA rotation of any length, typically accompanied by a low polarization degree. The challenge in constraining these types of models is the stochastic nature of the variations so that it is not possible to fit the individual light curves (<cit.>). Instead, one should observe a large number of objects and compare the statistical properties of the variations with the turbulent models. To study the connection between high-energy emission and optical polarization in a statistical manner, the RoboPol program was initiated in 2013 (<cit.>). The RoboPol instrument is mounted on the 1.3-m telescope in Skinakas Observatory in Crete, where the observing season lasts from April until November. During its first three observing seasons, RoboPol observed about 100 AGN twice per each week in search of optical polarization angle rotations. The main goal was to study the differences in gamma-ray detected and non-detected objects, and to search for a connection between rotations and flaring behavior. RoboPol has detected 40 optical polarization angle rotations during the first three observing seasons, tripling the number of known rotations (<cit.>). Comparing the rotations with random walk simulations showed that while stochastic variability for any individual rotation could not be ruled out, it was highly unlikely that all of them would be of random walk origin. <cit.> suggested that the smoothness of the rotation could be used as an additional indicator when comparing the rotations with random walk models. They also illustrated the importance of good sampling in polarization observations by revealing how additional data changed the rotation reported by <cit.> in the quasar 3C 279. Figure <ref> shows an example of a EVPA rotation in the source 1ES 1727+502 studied in <cit.>. Based on random walk simulations, conducted in the same manner as in <cit.>, a stochastic origin for the rotation cannot be excluded. §.§ Emission in a helical field A rotation in the polarization angle could also be due to an emission feature tracing a magnetic field in the jet as suggested by <cit.> and <cit.>. In this model, the magnetic field in the jet is helical in the acceleration and collimation zone, which is probed by the optical band, and a rotation is seen when the emission feature moves along the magnetic field. The rotation may be accompanied by flaring in other bands when the emission feature reaches a standing shock in the jet. Another alternative is a shock moving down a jet with a helical field, in which case the EVPA rotation would be due to light travel time effects when parts of the shock are seen at different times (<cit.>). This model was used to successfully fit the EVPA rotation in the quasar 3C 279 (<cit.>), although one should bear in mind the caveat that with the additional data from <cit.>, the rotation was not as long as originally presented by <cit.>. Whether the magnetic field in the jet is helical, especially in regions beyond the acceleration and collimation zone, is still unclear. As explained in Sect. <ref>, observations of a Faraday rotation measure gradient across the jet could be an indication of such a field. First signatures of a helical magnetic field were observed in the quasar 3C 273 using multifrequency very long baseline array (VLBA) observations (<cit.>). Many more claims of such gradients have been made (e.g., <cit.>) but the issue has been controversial due to the limited resolution across the jets in blazars (<cit.>). In <cit.>, we performed Monte Carlo simulations to quantify the significance of the rotation measure gradients in the large sample of parsec-scale jets in the MOJAVE (Monitoring of Jets in Active galactic nuclei with VLBA Experiments) sample. Our observations confirmed the gradient in 3C 273 (see Fig. <ref>), and we also found significant gradients in three other quasars. In <cit.> we modelled the gradient of the quasar 3C 454.3 using a magnetic field with both helical and turbulent components. In the other two sources, the gradient span only over a small portion of the jet and detailed modeling was not possible. Nowadays performing simulations to quantify the significance of the gradients is a common practice (e.g., <cit.>) and the number of significant gradients has steadily increased (e.g., <cit.>). However, rotation measure gradients can also arise from changes in the density of the Faraday rotating material (e.g. <cit.>) so that detailed modeling of the gradients should be done in order to confirm that they are indeed due to helical fields. It is also unclear whether the helical field is within the jet or in a sheath layer around the jet because Faraday rotation is a propagation effect originating in the plasma outside the emission region. §.§ Shock in a jet As stated earlier, shock-in-jet models have been successful in reproducing the observed flaring especially in the radio bands. Typically, radiative transfer simulations are generated to estimate the parameters of the shocks (e.g. <cit.>) and these are then compared with observations at multiple bands (<cit.>). For example, as shown by <cit.>, the polarization degree and range of EVPA values can be used to constrain the magnetic field geometry and jet orientation, while total intensity behavior is indistinguishable in different models. Interestingly, although the magnetic field is assumed to be predominantly turbulent, in some cases an ordered field component (possibly helical) could also be present (<cit.>). This supports the findings of e.g., <cit.>, and <cit.>, who also suggest that there are both turbulent and ordered magnetic field components / deterministic variations in the jets. §.§ Magnetic reconnection Magnetic reconnection (see <cit.> for a review) is a new hot topic in the field. It was originally invoked to explain the fast high-energy variability of blazars through the jet-in-jet model (<cit.>). In the recent years, particle-in-cell simulations have evolved significantly, and it is now possible to reliably simulate the complicated structure of the reconnection layer (e.g., <cit.>). While the reconnection models can already be compared to total intensity variability of the sources (<cit.>), there are no explicit models for the polarized emission. <cit.> stated that the EVPA swing in the 2009 flare of 3C 279 was reproduced by a model that “favors magnetic energy dissipation process during the flare” because the magnetic field strength was seen to decrease when the total intensity increased. However, this model did not yet include detailed comparisons to a specific reconnection model. Considering the growing interest in the magnetic dissipation models, it is likely that in the next few years more detailed models with observational predictions will become available. §.§ Statistical trends From the previous sections, it is clear that polarization observations can be used to constrain various types of emission models and the magnetic field structure in the jets. In order to generalize the findings in individual sources into the AGN jet population, a statistical approach must be used. This is what the RoboPol program aims for by observing a sample of about 100 sources with high cadence. RoboPol has found, for example, that any class of blazars from low to high synchrotron peaking objects can show rotations (see also <cit.>). However, there seems to be a specific class of “rotators” that do so more often, and rotations are more common in the low spectral peaking objects (<cit.>). The low synchrotron peaking objects also have higher optical polarization degree than the high-peaking objects (<cit.>), a trend that has also been seen in the smaller sample studied in <cit.> and earlier in the radio band (<cit.>). These general trends can be explained with a simple, qualitative model where a shock moves down a jet, which has both helical and turbulent magnetic field components (see <cit.> for details). Any model put forward to explain the polarization behavior in individual flares, should also account for these general trends. § FUTURE DIRECTIONS One challenge in connecting the observations of magnetic fields to the theory of jet formation is that often the observations, especially in the radio bands, probe regions 10^3-10^5 gravitational radii away from the black hole. Optical observations do not suffer from this restriction, but for example, Faraday rotation with λ^2 wavelength dependence, is not typically seen at optical wavelengths. A solution can be provided by going to millimeter-band observations, as demonstrated by <cit.>. They used ALMA polarization observations to probe the Faraday rotation at the jet base of the lensed quasar PKS 1830-211, and found extremely high Faraday rotation of 10^8rad/m^2, which is two orders of magnitude higher than previous observations in other sources (e.g., <cit.>). They inferred this as a signature of a very high magnetic field at the base of the jet, in support of magnetically launched jet models. Whether similar high RM values are a common property of quasars remains to be seen with future ALMA observations. Especially interesting will be future observations using ALMA as part of the global VLBI array, which may be able to spatially resolve the regions of high magnetic fields. A major leap forward will come with the future X-ray polarization missions because the amount of polarization for different high-energy emission mechanisms in AGN jets, for example between leptonic and hadronic models, is very different (<cit.>), and X-ray polarization observations can be used to distinguish between the models. § SUMMARY Magnetic fields in AGN jets can be probed through polarization observations, which are most easily done at radio and optical bands. Modeling the polarization degree and position angle behavior can be used to test different blazar emission models, while Faraday rotation observations at radio, and now also in the millimeter bands, can be used to probe the line-of-sight magnetic field component. A statistical approach is favored when connecting the results to different AGN jet populations. [Abdo et al. 2010]abdo10 Abdo, A. A., Ackermann, M., Ajello, M. et al. 2010, Nature, 463, 18 [Aleksić et al. 2014]aleksic14 Aleksić, J., Ansoldi, S., Antonelli, L. A. et al. 2014 A&A, 569, 46 [Aller et al. 2003]aller03 Aller, M. F., Aller, H. D. & Hughes, P. A. 2003, ApJ, 586, 33 [Aller et al. 2014]aller14 Aller, M. F., Hughes, P. A., Aller, H. D. et al. 2014, ApJ, 791, 51 [Aller et al. 2016]aller16 Aller, M. F., Hughes, P. A., Aller, H. D., et al. 2016, Galaxies, 4, 35 [Angel & Stockman 1980]angel80 Angel, J. R. P. & Stockman, H. S. 1980, ARA&A, 18, 321 [Angelakis et al. 2016]angelakis16 Angelakis, E., Hovatta, T., Blinov, D. et al. 2016 MNRAS, 463, 3365 [Asada et al. 2002]asada02 Asada, K. Inoue, M., Uchida, Y., et al. 2002, PASJ, 54, L39 [Blandford 1993]blandford93 Blandford, R. 1993, in: M. Burgarella, M. Livio, & C. P. O’Dea (eds.) Astrophysical Jets, (Astrophysics and Space Science Library, Vol. 103; Cambridge: Cambridge Univ. Press), 15 [Blandford & Königl 1979]blandford79 Blandford, R. D. & Königl, A. 1979, ApJ, 232, 34 [Blandford & Payne 1982]blandford82 Blandford, R. D. & Payne, D. G. 1982, MNRAS, 199, 883 [Blandford & Znajek 1977]blandford77 Blandford, R. D. & Znajek, R. L. 1977, MNRAS, 179, 433 [Blinov et al. 2015]blinov15 Blinov, D., Pavlidou, V., Papadakis, I. et al. 2015, MNRAS, 453, 1669 [Blinov et al. 2016a]blinov16a Blinov, D., Pavlidou, V., Papadakis, I. E. et al. 2016a, MNRAS, 457, 2252 [Blinov et al. 2016b]blinov16b Blinov, D., Pavlidou, V., Papadakis, I. E. et al. 2016b, MNRAS, 462, 1775 [Burn 1966]burn66 Burn, B. J. 1966, MNRAS, 133, 67 [Gabuzda et al. 2015]gabuzda15 Gabuzda D. C., Knuettel, S. & Reardon, B. 2015 MNRAS, 450, 2441 [Gabuzda et al. 2004]gabuzda04 Gabuzda, D. C., Murray, E., & Cronin, P. 2004, MNRAS, 351, L89 [Giannios et al. 2009]giannios09 Giannios, D., Uzdensky, D. A. & Begelman, M. C. 2009, MNRAS, 359, L29 [Gómez et al. 2011]gomez11 Gómez, J.-L., Roca-Sogorb, M., Agudo, I., et al. 2011, ApJ, 733, 11 [Hovatta et al. 2012]hovatta12 Hovatta, T., Lister, M. L., Aller, M. F. et al. 2012, AJ, 144, 105 [Hovatta et al. 2016]hovatta16 Hovatta, T., Lindfors, E., Blinov, D. et al. 2016 A&A, 596, 78 [Hughes et al. 1985]hughes85 Hughes, P. A., Aller, H. D. & Aller, M. F. 1985, ApJ, 298, 301 [Hughes et al. 2011]hughes11 Hughes, P. A., Aller, M. F., & Aller, H. D. 2011, ApJ, 735, 81 [Hughes et al. 2015]hughes15 Hughes, P. A., Aller, M. F., Aller, H. D. 2015, ApJ, 799, 207 [Jermak et al. 2016]jermak16 Jermak, H., Steele, I., Lindfors, E. et al. 2016, MNRAS, 462, 4267 [Jones & O'Dell 1977]jones77 Jones, T. W. & O’Dell, S. 1977, ApJ, 215, 236 [Jones et al. 1985]jones85 Jones, T. W., Rudnick, L., Aller, H. D. et al. 1985, ApJ, 290, 627 [Kagan et al. 2015]kagan15 Kagan, D., Sironi, L. Cerutti, B. & Giannios, D. 2015, Space Science Reviews, 191, 545 [Kiehlmann et al. 2016]kiehlmann16 Kiehlmann, S., Savolainen, T., Jorstad, S. G., et al. 2016, A&A, 590, A10 [Laing 1980]laing80 Laing, R. A. 1980, MNRAS, 193, 439 [Lister & Homan 2005]lister05 Lister, M. L., & Homan, D. C. 2005, AJ, 130, 1389 [Lister et al. 2011]lister11 Lister, M. L., Aller, M. F., Aller, H. D. et al. 2011, ApJ, 742, 27 [Lyutikov et al. 2005]lyutikov05 Lyutikov, M., Pariev, V. I., & Gabuzda, D. C. 2005, MNRAS, 360, 869 [Marscher 2014]marscher14 Marscher, A. P. 2014, ApJ, 780, 87 [Marscher & Gear 1985]marscher85 Marscher, A. P. & Gear, W. K. 1985, ApJ, 298, 114 [Marscher et al. 2008]marscher08 Marscher, A. P., Jorstad, S. G., D'Arcangelo, F. D. et al. 2008, Nature, 452, 966 [Marscher et al. 2010]marscher10 Marscher, A. P., Jorstad, S. G., Larionov, V. M. et al. 2010, ApJ, 710, L126 [Martí-Vidal et al. 2015]marti-vidal15 Martí-Vidal, I., Muller, S., Vlemmings, W., Horellou, C., Aalto, S. 2015, Science, 348, 311 [Murphy & Gabuzda 2013]murphy13 Murphy E. & Gabuzda D. C. 2013, in The Inner-most Regions of Relativistic Jets and Their Magnetic Fields, EPJ Web of Conferences, Volume 61, id.07005 [Pacholczyk 1970]pacholczyk70 Pacholczyk, A. G. 1970, Radio Astophysics, Nonthermal Processes in Galactic and Extragalactic Sources (San Francisco, CA: Freeman) [Pavlidou et al. 2014]pavlidou14 Pavlidou, V., Angelakis, E., Myserlis, I. et al. 2014, MNRAS, 442, 1693 [Petropoulou et al. 2016]petropoulou16 Petropoulou, M., Giannios, D., Sironi, L. 2016, MNRAS, 462, 3325 [Plambeck et al. 2014]plambeck14 Plambeck, R. L., Bower, G. C., Rao, R. et al. 2014, ApJ, 797, 66 [Saikia & Salter 1988]saikia88 Saikia, D. & Salter, C. 1988, Ann. Rev. in Astron. & Astrophys. , 26, 93 [Sironi et al. 2015]sironi15 Sironi, L., Petropoulou, M., & Giannios, D. 2015, MNRAS, 450, 183 [Taylor & Zavala 2010]taylor10 Taylor, G. B., & Zavala, R. T. 2010, ApJ, 722, L183 [Tchekhovskoy et al. 2011]tchekhovskoy11 Tchekhovskoy, A., Narayan, R., McKinney, J. C. 2011, MNRAS, 418, L79 [Zamaninasab et al. 2014]zamaninasab14 Zamaninasab, M., Clausen-Brown, E., Savolainen, T. & Tchekhovskoy, A. 2014, Nature, 510, 126 [Zamaninasab et al. 2013]zamaninasab13 Zamaninasab, M., Savolainen, T., Clausen-Brown, E. et al. 2013, MNRAS, 436, 3341 [Zavala & Taylor 2005]zavala05 Zavala, R. T., & Taylor, G. B. 2005, ApJ, 626, L73 [Zhang & Böttcher 2013]zhang13 Zhang, H. & Böttcher, M. 2013, ApJ, 774, 18 [Zhang et al. 2014]zhang14 Zhang, H. Chen, X. & Böttcher, M. 2014, ApJ, 789, 66 [Zhang et al. 2015]zhang15 Zhang, H. Chen, X. & Böttcher, M. et al. 2015, ApJ, 804, 58

http://arxiv.org/abs/1701.08197v1

20170127212731

Towards exascale real-time RFI mitigation

astro-ph.IM

[NO \title GIVEN] [NO \author GIVEN] December 30, 2023 ====================== We describe the design and implementation of an extremely scalable real-time RFI mitigation method, based on the offline AOFlagger. All algorithms scale linearly in the number of samples. We describe how we implemented the flagger in the LOFAR real-time pipeline, on both CPUs and GPUs. Additionally, we introduce a novel simple history-based flagger that helps reduce the impact of our small window on the data. By examining an observation of a known pulsar, we demonstrate that our flagger can achieve much higher quality than a simple thresholder, even when running in real time, on a distributed system. The flagger works on visibility data, but also on raw voltages, and beam formed data. The algorithms are scale-invariant, and work on microsecond to second time scales. We are currently implementing a prototype for the time domain pipeline of the SKA central signal processor. RFI, real-time, LOFAR, SKA, CSP, TDT § INTRODUCTION Radio Frequency Interference (RFI) mitigation is extremely important to take advantage of the vastly improved bandwidth, sensitivity, and field-of-view of exascale telescopes. For current instruments, RFI mitigation is typically done offline, and in some cases (partially) manually. At the same time, it is clear that due to the high bandwidth requirements, RFI mitigation will have to be done automatically, and in real-time, for exascale instruments. In general, real-time RFI mitigation will be less precise than offline approaches. Due to memory constraints, there is much less data to work with, typically only in the order of one second or less, as opposed to the entire observation. In addition, due to memory limitations and the fact that processing is typically done in a distributed system, we can record only limited statistics of the past. Moreover, we will typically have only few frequency channels locally available at each compute node. Finally, the amount of processing that can be spent on RFI mitigation is extremely limited due to computing constraints and a limited power budget. Many existing algorithms are therefore far too expensive and not applicable. Nevertheless, there are many potential benefits as well, which include the possibility of working on higher time and frequency resolutions, before any integration is done, leading to more accurate results. Most importantly, we can remove RFI before beam forming, which combines data from all receivers. With beam forming, not only the signals, but also the RFI that is present in the data streams from the separate receivers is combined, effectively taking the union of all RFI. Thus, the RFI from any receiver will pollute all beams. Therefore, it is essential to also perform real-time RFI mitigation before the beam former, even though the data rates can be very high at this point. This is particularly important for pulsar surveys, for instance. The algorithms we use are based on earlier work by Offringa and others <cit.>. Although our techniques are generic, we describe how we implemented real-time RFI mitigation for one of the SKA pathfinders: The Low Frequency Array (LOFAR) <cit.>. The modified RFI mitigation algorithms we introduce here are extremely fast, and the computational requirements scale linearly in the number of samples and frequency channels. We evaluate the quality of the algorithms with LOFAR pulsar observations. Using the signal-to-noise ratios of the folded pulse profiles, we can qualitatively and quantitatively compare the impact of different real-time RFI mitigation algorithms. In addition to CPU versions, we have developed a prototype for Graphical Processing Units (GPUs). We present the very promising performance results of performing real-time RFI mitigation on GPUs. Finally, we are now working on incorporating our CPU and GPU codes in the SKA Central Signal Processor (CSP), in the context of the Time Domain Team (TDT) pulsar and transient pipeline. § LOF: THE LOFAR ONLINE FLAGGER For the offline RFI mitigation for the LOFAR telescope, we use the AOFlagger <cit.>. This flagger is relatively efficient, and is fast enough to be applied in modern high-resolution observatories. The AOFlagger operates on visibility data, and currently is used only in the imaging mode. Although the AOFlagger provides a selection of many different algorithms, for online use, we ported the most important and successful ones to the LOFAR real-time central processing system, that originally ran on an IBM Blue Gene/p supercomputer. We also ported the algorithms to NVIDIA GPUs using CUDA, as described in Section <ref>. We call our implementation of the real-time flagger “LOF”, short for LOFAR Online Flagger. §.§ The SumThreshold algorithm The most important algorithm we use is SumThreshold <cit.>. It performs thresholding with an exponentially increasing window size, and an increasingly sharper threshold. This way, it can detect RFI at different scales. As we will demonstrate in Section <ref>, this indeed works well in practice, and effectively removes RFI from microsecond to multiple second scales. With SumThreshold, we define the threshold for the current window as follows: threshold_I = median + stddev * factor_I * sensitivity Where the factor is: factor_I = startThreshold * p^2log(I)/I Typical values are p = 1.5, and sensitivity = 1.0. All measurements use the defaults, since we empirically found that they provide optimal results. No tuning is required. Figure <ref> shows the threshold for different iterations. Figure <ref> shows the operation of the SumThreshold algorithm. With the default values, the algorithm begins the first iteration as a simple thresholder, flagging all single samples that are more than 6 sigma away from the median. The second iteration doubles the window size to two samples, but lowers the threshold to 4.5 sigma. We typically run 7-10 iterations, depending on the resolution and size of the input. SumThreshold can be run in a one-dimensional mode, as shown in Figure <ref>. This can then be done in both time and frequency directions. Alternatively, the algorithm can operate on 2D data in one pass as well (not shown in Figure <ref>). As we will explain in Section <ref>, for real-time use, we did implement a 2D code, but we will mostly use the one dimensional version of the algorithm for performance reasons. It is important to note that the computational complexity of the SumThreshold algorithm itself is linear in the number of samples. Therefore, the algorithm is suitable for real-time use, where we are limited by the number of compute cycles we can spend. §.§ The scale-invariant rank operator The scale-invariant rank operator <cit.> makes SumThreshold more robust, by extending the ranges of flagged samples by a percentage of the size of the flagged range. A typical percentage we used is 20%. This means that all ranges of consecutive flagged samples are extended by flagging 20% more samples, before and after the original range. The algorithm can be run in both the time direction and the frequency direction. The SIR operator helps to remove RFI that slowly rises and decreases in strength, that may be otherwise undetected. The result of the SIR operator is shown in Figure <ref>. We use a version of the SIR operator implementation that has a linear computational complexity in the number of samples (the original implementation in the AOFlagger had worse computational complexity). The linear version of the algorithm is described in <cit.>. Since SIR operates on the flag masks only, and not on the actual data itself, it is extremely efficient. § INTEGRATION IN REAL-TIME PIPELINES Figure <ref> shows a high-level overview of the LOFAR real-time central signal processing pipeline. The entire pipeline is implemented in software, and is described in detail in <cit.>, including a detailed performance analysis. The LOFAR online flagger components are placed in four different places in the pipeline. Depending on the configuration and the observation type, one or more different flaggers are used. Data arrives in the form of raw voltages at the top left of the figure. Next, a number of steps are executed independently of whether we are in imaging or beam forming mode. The most important step is a polyphase filter bank that splits the broad input subbands in narrower frequency channels. Typical channel bandwidths are between 0.8 kHz and 12 kHz, with sample rates between 82 microseconds and 1.3 milliseconds. When we need extremely high time resolution (e.g, for millisecond pulsars or for the cosmic ray pipeline), we bypass the polyphase filter bank altogether. For this case we created a special high time resolution flagger. Next, the band pass of the first polyphase filter bank that runs inside the stations on FPGAs is corrected. After this bandpass correction, we inserted our pre-correlation flagger that works on the channelized raw voltage data. It is important to do this after the band pass correction, as this ensures that the sensitivity is equal across all channels. The pre-correlation version is the most important real-time flagger, especially for the beam forming modes. The beam former does a weighted addition of the data streams from the different stations, essentially taking the union of all RFI from all stations. If RFI is present at a station, this will pollute all output beams. Especially for uncorrelated RFI and long baselines, this is sub optimal. In addition, we can use our real-time flagger after the correlator, for real-time image-based transient detection, for example. The drawback of performing RFI mitigation after the correlator is that, depending on the number of baselines and the integration time, data rates can be higher than before the correlator. Finally, we have a post beam forming flagger that can potentially benefit from a better interference-to-noise ratio (INR). Moreover, depending on the number of output beams, this typically runs on lower data rates. § CHANGES FOR REAL-TIME USE To make sure the algorithms used in the AOFlagger work in a real-time context, we had to make several changes. First, depending on the input data type of the flagger, we may have to compute amplitudes first. This is the case for the high-time resolution flagger, and for the pre-correlation flagger. The post-correlation flagger runs on visibility data directly, while the post-beam forming flagger runs on Stokes I data. In pre-correlation mode, we typically integrate the data. We found it particularly useful to integrate the time direction fully for frequency flagging, and to integrate the frequency direction fully for time domain flagging. Figure <ref> shows this. This approach has two benefits: it improves the INR, while at the same time reducing the computational costs. We first flag in the frequency direction. This removes strong narrow-band RFI that would otherwise decrease the quality of the statistics used to compute the thresholds. We found that this is frequently present in the LOFAR RFI environment. This also is the reason we create the narrow approximately 1kHz channels. We have an alternative method that implements 2D flagging, while partially integrating in one or both dimensions to improve INR and to reduce compute costs. All flaggers have linear computational complexity in the number of samples, regardless of their place in the real-time pipeline. For instance, the pre-correlation flagger has a complexity of O(nrStations * nrPolarizations * nrChannels * nrTimes), the post-correlation flagger is O(nrBaselines * nrPolarizations * nrChannels), and the post beam forming flagger is O(nrBeams * nrChannels * nrTimes). §.§ Using historical information One of the most difficult problems with real-time RFI mitigation is the very limited window on the observation. Typically, we can only keep one second of data or less (e.g., a tenth of a second) in memory. Similarly, we can only keep a very limited number of frequency channels in memory. This is due to memory constraints, and partially because we process the data on a distributed system. We create parallelism by performing domain decomposition. This typically means that different frequency subbands are processed on different compute nodes. Finally, in some cases, processing a second of input data takes longer than a second. To still meet the real-time requirements, subsequent seconds are processed partially overlapping in time, by different compute nodes. All these factors severely limit our situational awareness. Our solution for this problem is the introduction of a novel history flagger that performs simple thresholding of the current data chunk, based on statistics of past chunks. Pseudo code for this history flagger is shown below. // For all channels, we do the following: // Keep a history buffer (sliding window) of // means of unflagged samples of the past seconds currentValue = meanOfUnflaggedSamples() historyMean = meanOfMeans() historyStddev = stddevOfMeans() threshold= historyMean + sensitivity * historyStddev if(currentValue < threshold) addToHistory(station, subband, currentValue) else addToHistory(station, subband, threshold) flagThisIntegrationTime() As shown in Figure <ref>, our flagger uses a history buffer that stores the means of the unflagged samples of the previous seconds of data. This buffer essentially is a sliding window over the data. We use the buffer to compute the mean and standard deviation of the previous seconds, to give us a frame of reference for the overall signal strength of the current second. This is especially important, since a strong broadband RFI event that lasts longer than our integration time can otherwise not be detected. §.§ Statistics in a real-time pipeline The quality of the statistics that we compute and keep is important, especially since the window on the data is so small. We use only very basic statistics, such as (winsorized) means, medians, standard deviations, and the MAD (Median Absolute Deviation). To compute the medians, which can be expensive, we use an efficient O(N) implementation (note that sorting the data already is O(NlogN) at best. More complications result from using a distributed platform: statistics are often in the wrong place, at the wrong time. Moreover, we cannot compute running statistics, since a second of data takes more than a second to compute. Finally, our real-time pipeline uses complex communication patterns due to scheduling, and asynchronous communication for better performance. Together, this means that even computing basic statistics is quite complex in practice. An important consideration for the history flagger is presented by the space requirements of the statistics of the past we want to keep in the history buffer. Let us use LOFAR numbers as an example. For the pre-correlation flagger, we need stations * subbands * channels * 32 bits = 64 * 248 * 256 * 4 = 15.5 MByte per second. If we want to keep 5 minutes of history in the buffer, we already need 300 samples, leading to a storage requirement of 4.5 GBytes. After the correlator, requirements are even higher: baselines * subbands * channels * 32 bits = 2080 * 248 * 256 * 4 = 504 MByte per second, which, even if we want to store only 5 minutes, already leads to 148 Gbytes of statistics data. Therefore, in practice, even keeping these very limited statistics of only a few minutes of the past already is extremely difficult, and down-sampling may be needed. § EVALUATION In this section, we will present a qualitative and quantitative evaluation of the real-time flagger, using a LOFAR pulsar observation. We use the pulsar pipeline, because it allows for a quantitative comparison: we perform dedispersion and folding to create a folded pulse profile. Next, we compute the SNR of the pulse profile as a measure of quality. Better RFI mitigation directly leads to a higher SNR. §.§ A pulsar observation We performed an observation of pulsar B1919+21, which has a period of 1.3373 s, a pulse width 0.04 s, and a dispersion measure (DM) of 12.455. We observed at 138.0–145.2 MHz (32 subbands) with 5 stations: CS005, CS006, RS205, RS406, and UK608. We deliberately chose a number of core stations, where RFI is expected to be correlated, a remote station in the Netherlands, where the RFI environment is known to be particularly bad, and an international station in the UK, to guarantee we also have uncorrelated RFI. We used a special LOFAR mode that allowed us to store the raw UDP network packets, before the data even enters the correlator. This allows us to replay the entire real-time pipeline in an offline mode, enabling comparisons between different flagging algorithms, parameter settings and even observation modes (e.g., imaging or beam forming). For the beam forming mode that we use for the pulsar pipeline, we split the frequency subbands into 16 channels (12 KHz / 82 μs). Figure <ref> shows two waterfall plots. The left side is the original observation with RFI present; on the right the improvement of our LOF compared to a simple thresholding scheme, where we manually determined the optimal threshold. There clearly is a lot of residual RFI that is not removed by the thresholder, that is removed by LOF. Figure <ref> shows Stokes I data of an output beam; Figure <ref> shows the same, but zoomed in. The top panels are without RFI mitigation, the bottom panels with LOF. Almost all RFI is removed. The data is not de-dispersed, but the pulsar signal is so strong that it is clearly visible in the data. Note that the pulses are not flagged away by our mitigation algorithms. In Figure <ref> (left side), we show the folded pulse profiles, without RFI mitigation, with a simple thresholder, and with LOF. Without RFI mitigation, the pulsar signal is completely below the noise floor, and cannot be detected. With the thresholder, the pulse is visible, but there also are false positives, caused by strong RFI events that were not removed. With LOF, there is one clear peak with the right shape, with a good SNR. This can be seen better in the right side of Figure <ref>, which shows the same data, but zoomed in, with the non-flagged line removed. Figure <ref> shows that LOF flags 2.9% of the data in the observation, significantly more than the simple thresholder that only flags 1.7%. In the bottom left graph, we show that LOF flags away about 15% of the total signal power, only slightly more than the simple thresholder. The bottom right graph shows the SNR of the folded pulse profile. LOF is significantly better than the simple thresholder, and almost as high as performing offline RFI mitigation with Presto's rfifind <cit.>. § TOWARDS EXASCALE The LOFAR online processor needs hundreds of teraflops of computational power. Future instruments such as the Square Kilometre Array (SKA) will be much more sensitive, and will require orders of magnitude more processing <cit.>. Therefore, we were careful to make sure that all algorithms used in our real-time flagger have a linear computational complexity, allowing excellent scaling. In addition, we investigated the use of modern processing architectures, such as GPUs. For LOFAR, we already switched from an IBM Blue Gene/p system to a GPU cluster. For the central signal processor of the SKA, a combination of FPGAs and GPUs are likely. In this section, we present a prototype GPU version of the LOF. §.§ GPU implementation With Linus Schoemaker <cit.>, we worked on the GPU port of LOF. Due to the limited space, we will only describe the most important differences with the normal parallel CPU version here. The normal LOF exploits parallelism by scheduling different subbands to different compute nodes. Inside the compute nodes we use C++ and OpenMP for multi-threading, and asynchronous MPI messages for inter-node communication. To avoid synchronization and parallelization overhead, we make the parallelism as coarse grained as possible. This means that each thread handles all data for one subband. GPUs, in contrast, can efficiently exploit extremely fine grained parallelism. Hundreds of thousands of threads can work in parallel on a single GPU, without overhead. In fact, in our implementation, we create one GPU thread per sample. We exploit data-reuse by using the shared memory that is available inside the GPU's streaming multiprocessors. The performance results are shown in figure <ref>. The top two lines (red and green) show the run time without doing any data integration, running SumThreshold on the full input data rate. For the red line, we also run the SIR operator; green is without. The bottom two lines show performance if we fully integrate and run SumThreshold two times, once in the frequency direction, and once in the time direction, as described in Section <ref>. In all cases, we achieve linear Scalability. The GPU performs so well, that we can handle all LOFAR stations on single GPU in real time. §.§ Integration in SKA time domain pipeline We are currently working on incorporating our CPU and GPU codes in the SKA Central Signal Processor (CSP), in the context of the Time Domain Team (TDT) pulsar and transient pipeline. The initial stages of the CSP, including the beam former, will likely use FPGAs. After the beam former, GPUs are the most likely candidate for further processing. Therefore, we will perform post-beamforming flagging using our algorithms on GPUs. For an FPGA implementation, more research is needed. § CONCLUSIONS AND FUTURE WORK We have demonstrated that our online flagger can achieve much higher quality than simple thresholding, in real time, even on a distributed system. The SumThreshold algorithm was originally used mostly on visibility data. In this paper, we demonstrated that the algorithm also works well on raw voltages, pre-correlation data, and post-beam forming data. Therefore, we have one robust algorithm for extremely different scales, from microseconds to multiple seconds. The algorithms are scalable and have linear computational complexity, adding little overhead to existing pipelines. One complication is that we have an extremely limited view on our data, and therefore need a history flagger. Due to the high data rates, we have to be flexible in storage requirements, even for statistics. We are currently working on commissioning of the GPU code for LOFAR. Moreover, we are constructing a performance model that can extrapolate Scalability towards SKA sizes. This model includes power dissipation as well, since this will be an important bottleneck for the SKA. We plan to use the Dome ExaBounds tool for this analysis <cit.>. All code used in this paper is available as open source: <https://github.com/NLeSC/eAstroViz>. IEEEbib

http://arxiv.org/abs/1701.07647v1

20170126104540

Direct photon yield in pp and in Pb-Pb collisions measured with the ALICE experiment

nucl-ex

Utrecht University, Princetonplein 5, Utrecht, the Netherlands [E-mail:]davide.francesco.lodato@cern.ch The measurement of direct photon production in Pb-Pb collisions at √(s_NN) = 2.76 TeV and in pp collisions at √(s)= 7 TeV with the ALICE experiment is presented. In Pb-Pb collisions a clear direct photon signal below 3 GeV/c is observed only for the 0-20 % most central collisions. No excess is observed for semi-central and peripheral Pb-Pb collisions and in pp collisions. Furthermore, in pp collision the measurement of direct photon production in the range 10 ≤ p_T≤ 60 GeV/c is reported. The analysis is performed on the EMCal-triggered data taken in 2011. The two main sources of background, namely photons from fragmentation processes and decay photons, have been subtracted from the inclusive photon spectrum by means of an isolation technique combined with the study of the transverse dispersion of electromagnetic showers. The measurement is in agreement both with NLO pQCD calculations and with those performed by the ATLAS and the CMS collaborations, extending the isolated photon spectrum investigated at LHC towards lower values of p_T. Direct PhotonsThermal RadiationElectroweak Probes QGPIsolated Photons § INTRODUCTION In pp collisions the measurement of direct photon production can be used to test both the pQCD calculations and the binary scaling behaviour of the initial hard scattering. At high p_T, pQCD processes like quark-gluon Compton scattering and quark-antiquark annihilation are the main contribution to direct photon production, and allow us to probe directly the gluons within hadrons. However, photons are also produced in jet fragmentation processes, in which part of the information about the hard scattering dynamics is lost. Reducing the contribution of photons from the latter source via isolation techniques helps to better constrain the gluon parton distribution function <cit.>. In nucleus-nucleus collisions direct photons are produced at every stage of the collision and therefore are sensitive to the different phases of the medium evolution. The low-p_T component of the direct photon spectrum is dominated by thermal production in the quark-gluon plasma and in the hadron-gas phase, giving us access to information on the temperature of the hot and dense medium in which direct photons are produced. For p_T greater than 5 GeV/c, direct photons are mainly produced in hard partonic scattering processes in the early stage of the collision, leaving the strongly interacting medium unscathed and provide access to information about the initial dynamics. § LOW-P_T DIRECT PHOTON MEASUREMENT IN PP AND PB-PB COLLISIONS: R_Γ The inclusive photon yield is measured both directly via the calorimetric method with the ALICE PHOton Spectrometer (PHOS) and via the reconstruction of photons via Photon Conversion Method (PCM). In the latter case a secondary vertex finder is used to pair electron positron tracks with a large impact parameter. Several selection criteria, like constraints on the opening angle and on the reconstructed invariant mass, are applied in order to optimize the signal to background ratio. PHOS is a highly granulated leadt ungstate (PbWO_4) homogeneous calorimeter with a coverage of Δφ = 60^∘ and |η|< 0.12. The analysis performed via PCM makes use of the (η-φ) coverage of the ALICE Time Projection Chamber <cit.>, respectively |Δη| < 0.9 and full azimuthal coverage . The same strategy is used to analyse the data taken both in Pb-Pb at √(s_NN) = 2.76 TeV and pp collisions at √(s_NN) = 7 TeV. The Pb-Pb analysis is performed in three bins of centrality (central: 0-20%, semi-central: 20-40% and peripheral: 40-80 %). The direct photon signal is obtained by subtracting the contribution of decay photons from the inclusive photon spectrum: γ_direct =γ_inc - γ_decay = ( 1 - γ_decay/γ_inc) ·γ_inc, where the main sources of decay photons are neutral meson (mainly π^0 and η via their 2-photon decay channel). The raw photon spectrum is corrected for purity, reconstruction efficiency and, in case of the PCM analysis, for the conversion probability in the detector material. In pp collisions the decay photon spectrum is calculated from a parametrization of the measured yield of π^0 and η mesons, reconstructed via the 2-photon decay channel; in Pb-Pb the parametrization is available only for π^0. In both cases, additional sources of decay photons yields from other mesons like η('), ω, ϕ and ρ_0 are computed via m_T-scaling and included into the cocktail calculations. The decay photon contribution was subtracted by calculating the so-called double ratio R_γ = (γ_inc/ π^0_param)/(γ_decay/π^0_param). By doing so, some uncertainties of the measurement cancel exactly. In pp collisions no direct photon signal is found <cit.>. The results of the measurements of R_γ in Pb-Pb collisions are presented in Fig. <ref> for different centralities. For comparison, the NLO pQCD calculations for pp collisions and scaled by N_coll are overlayed. The results obtained for non-central collisions are found to be in agreement with the theoretical predictions; the direct photon signal increases with the centrality of the collision. The shown R_γ is obtained by combining the PHOS and PCM measurements, for which the two analysis reach an agreement of 0.4 standard deviations. More details on the analyses and their comparison and combination can be found in <cit.>. In Fig. <ref>, the direct photon p_T spectrum computed for the most central collisions is presented. In the range 1 < p_T < 2 GeV/c, a fit with an exponential function is performed to extract the effective temperature of the medium, T_eff from the inverse slope parameter, which is estimated to be T_eff = 304 ± 11^stat± 40 ^syst MeV. As a comparison, results from PHENIX for the same measurement at √(s_NN)= 0.2 TeV are also shown. § DIRECT PHOTON AT HIGH P_T: ISOLATION TECHNIQUE FOR CONTAMINATION ESTIMATE The analysis of direct photon production at high p_T has been carried out in pp collisions at √(s)= 7 TeV by analysing the EMCal L0-triggered data collected by the ALICE experiment in 2011. The photons are measured directly via the calorimetric method with EMCal <cit.> while charged particles are reconstructed by the full ALICE tracking system. The EMCal is a Pb-scintillator sampling calorimeter with a granularity of Δη×Δφ = 0.0143×0.0143 and a total coverage of Δφ = 100^∘ and |η|< 0.7. The EMCal-L0 trigger correlates the energy deposited in the V0 detectors, placed at 2.8 < η < 5.1 and -3.7 < η < -1.7 on the A and C side respectively, with the energy measured by the EMCal detector and it is used to select events with a large energy deposition (> 5 GeV) in a 4×4 EMCal towers patch. This allows for a reduction of the data volume and of the detector dead time, enhancing, as a consequence, the integrated luminosity recorded. Particles will deposit their energy in multiple towers (or cells) of the calorimeter. A clustering algorithm identifies cells (called seeds) with energy deposit E_seed≥300 MeV, aggregating its neighbouring cells as long as E_cell≥ 100 MeV. In order to select direct photon candidates, various selection criteria are applied. Clusters produced by charged particles are rejected on the base of a track-proximity criterion. The production of direct prompt photons at high p_T is mainly due to quark-gluon Compton scattering and quark-antiquark annihilation. Neutral pions represent the main source of background via the 2-photon decay process. In the range 10 < p_T < 60 GeV/c, the two photons will likely be reconstructed as single elongated cluster. Direct photon cluster selection is based on the study of the width of the energy distribution σ^2_long = 0.5 × (d_ηη + d_φφ) + √(0.25 ×(d_ηη - d_φφ)^2 + d_ηφ^2), along the major axis of the reconstructed cluster. The dispersions d_ij are computed with a logarithmic weighting as in <cit.>. In Figure <ref> the dependence of the distribution of the σ^2_long parameter on the energy of the cluster is presented. Two main regions can be identified: single photon clusters dominate at σ^2_long≤ 0.3 while clusters from π^0 decay photons are clearly visible for E ≤ 20 GeV/c. Isolation techniques can help to distinguish direct photons both from those produced in fragmentation processes and further reduces the decay photon background. The advantage of applying an isolation criterion is discussed in <cit.>. The isolation criterion used in this analysis is based on the measurement of the total activity in a cone of radius R=0.4 around the selected photon candidates; the condition for which a cluster is considered isolated is: E_T^cone = Σ(E_T^clust + p_T^tracks) < 2 c. Only clusters whose cone is fully contained within the EMCal acceptance are selected, reducing the η-φ acceptance of the analysis to |η|<0.27 and 1.8 <φ < 2.7. By making use of both the σ^2_long and the E_T^cone distributions it is possible to estimate the contamination in our sample of isolated photon-like clusters by means of a double sideband method. The validity of this method is related to the fact that the E_T^cone distribution is independent of σ^2_long, for the background. Studies performed on MonteCarlo samples have shown the presence of cross-talk between the σ^2_long and E_T^cone variables. We can compute then a factor α_MC(p_T) and use it to correct the estimated contamination in data, which is then subtracted from the raw isolated photon spectrum. The remaining spectrum is corrected for reconstruction and identification efficiency. The differential cross section density is computed by scaling the corrected spectrum for trigger efficiency and total inelastic cross section measured by the ALICE collaboration <cit.>. The analysis has been repeated by varying different selection criteria in order to estimate the uncertainty related to various choices of cuts in the analysis. The main source of uncertainties is due to discrepancies in the modelling of the transverse shower shape distribution in MC with respect to data. Fig. <ref> shows the comparison between the measured cross section and the theoretical calculation from JETPHOX <cit.>; a reasonable agreement is found in the whole p_T range investigated. This result extends the range investigated by the ATLAS <cit.> and the CMS collaborations <cit.> down to 10 c . § SUMMARY In these proceedings we presented a summary of the direct photon measurements in pp and Pb-Pb collisions performed by the ALICE collaboration. In pp collision the direct photon yield is measured in the p_T range [10-60] c using the calorimetric method (EMCal) complemented by means of isolation techniques. All results obtained in pp collisions are in good agreement with NLO pQCD calculations. For lower values of the transverse momentum both the calorimetric method (PHOS) and the Photon Conversion Method are used. The signal has been studied by means of the double ratio R_γ <cit.>, and no direct photon signal has been found at low p_T in pp collisions. In Pb-Pb collisions, the same low p_T part (up to 14 GeV/c) of the direct photon spectrum has been investigated. The measurement has been carried out with PHOS and PCM using the same analysis strategy applied to pp collisions. The Pb-Pb analysis is performed in three centrality classes. An excess in the R_γ is found only for the most central (0-20%) collisions. A fit of the transverse momentum spectrum with an exponential function gives an inverse slope from which the effective temperature of the medium is estimated to be T_eff = 304 ± 11^stat± 40 ^syst. * elsarticle-num

http://arxiv.org/abs/1701.07736v2

20170126151720

Information-geometrical characterization of statistical models which are statistically equivalent to probability simplexes

cs.IT

op-tical net-works semi-conduc-tor

http://arxiv.org/abs/1701.07580v2

20170126051908

Kubo formulae for the shear and bulk viscosity relaxation times and the scalar field theory shear $τ_π$ calculation

nucl-th

red

http://arxiv.org/abs/1701.07730v1

20170126145531

Alpha Fair Coded Caching

cs.IT

Alpha Fair Coded Caching Apostolos Destounis^1, Mari Kobayashi^2, Georgios Paschos ^1, Asma Ghorbel^2 ^1 France Research Center, Huawei Technologies Co. Ltd., email: firstname.lastname@huawei.com ^2Centrale-Supélec, France, email: firstname.lastname@centralesupelec.fr ====================================================================================================================================================================================================================================================================================== The performance of existing coded caching schemes is sensitive to worst channel quality, a problem which is exacerbated when communicating over fading channels. In this paper we address this limitation in the following manner: in short-term, we allow transmissions to subsets of users with good channel quality, avoiding users with fades, while in long-term we ensure fairness across the different users. Our online scheme combines (i) joint scheduling and power control for the broadcast channel with fading, and (ii) congestion control for ensuring the optimal long-term average performance. We restrict the caching operations to the decentralized scheme of <cit.>, and subject to this restriction we prove that our scheme has near-optimal overall performance with respect to the convex alpha-fairness coded caching optimization. By tuning the coefficient alpha, the operator can differentiate user performance with respect to video delivery rates achievable by coded caching. We demonstrate via simulations our scheme's superiority over legacy coded caching and unicast opportunistic scheduling, which are identified as special cases of our general framework. Broadcast channel, coded caching, fairness, Lyapunov optimization. § INTRODUCTION A key challenge for the future wireless networks is the increasing video traffic demand, which reached 70% of total mobile IP traffic in 2015 <cit.>. Classical downlink systems cannot meet this demand since they have limited resource blocks, and therefore as the number of simultaneous video transfers K increases, the per-video throughput vanishes as 1/K. Recently it was shown that scalable per-video throughput can be achieved if the communications are synergistically designed with caching at the receivers. Indeed, the recent breakthrough of coded caching <cit.> has inspired a rethinking of wireless downlink. Different video sub-files are cached at the receivers, and video requests are served by coded multicasts. By careful selection of sub-file caching and exploitation of the broadcast wireless channel, the transmitted signal is simultaneously useful for decoding at users with different video requests. Although this scheme–theoretically proved to scale well–can potentially resolve the future downlink bottleneck, several limitations hinder its applicability in practical systems <cit.>. In this work, we take a closer look to the limitations that arise from the fact that coded caching was originally designed for a symmetric error-free shared link. If instead we consider a realistic model for the wireless channel, we observe that a naive application of coded caching faces a short-term limitation: since the channel qualities of the users fluctuate over time and our transmissions need to reach all users, the transmissions need to be designed for the worst channel quality. This is in stark contrast with standard downlink techniques, like opportunistic scheduling <cit.>, which serve the user with the best instantaneous channel quality. Thus, a first challenge is to discover a way to allow coded caching technique to opportunistically exploit the fading of the wireless channel. Apart from the fast fading consideration, there is also a long-term limitation due to the network topology. The user locations might vary, which leads to consistently poor channel quality for the ill-positioned users. The classical coded caching scheme is designed to deliver equal video shares to all users, which leads to ill-positioned users consuming most of the air time and hence driving the overall system performance to low efficiency. In the literature, this problem has been resolved by the use of fairness among user throughputs <cit.>. By allowing poorly located users to receive less throughput than others, precious airtime is saved and the overall system performance is greatly increased. Since the sum throughput rate and equalitarian fairness are typically the two extreme cases, past works have proposed the use of alpha-fairness <cit.> which allows to select the coefficient α and drive the system to any desirable tradeoff point in between of the two extremes. Previously, the alpha-fair objectives have been studied in the context of (i) multiple user activations <cit.>, (ii) multiple antennas <cit.> and (iii) broadcast channels <cit.>. However, here the fairness problem is further complicated by the interplay between scheduling and the coded caching operation. In particular, we wish to shed light into the following questions: what is the right user grouping and how we should design the codewords to achieve our fairness objective while adapting to changing channel quality? To address these questions, we study the content delivery over a realistic block-fading broadcast channel, where the channel quality varies across users and time. In this setting, we design a scheme that decouples transmissions from coding. In the transmission side, we select the multicast user set dynamically depending on the instantaneous channel quality and user urgency captured by queue lengths. In the coding side, we adapt the codeword construction of <cit.> depending on how fast the transmission side serves each user set. Combining with an appropriate congestion controller, we show that this approach yields our alpha-fair objective. More specifically, our approaches and contributions are summarized below: 1) We impose a novel queueing structure which decomposes the channel scheduling from the codeword construction. Although it is clear that the codeword construction needs to be adaptive to channel variation, our scheme ensures this through our backpressure that connects the user queues and the codeword queues. Hence, we are able to show that this decomposition is without loss of optimality. 2) We then provide an online policy consisting of (i) admission control of new files into the system; (ii) combination of files to perform coded caching; (iii) scheduling and power control of codeword transmissions to subset of users on the wireless channel. We prove that the long-term video delivery rate vector achieved by our scheme is a near optimal solution to the alpha-fair optimization problem under the specific coded caching scheme <cit.>. 3) Through numerical examples, we demonstrate the superiority of our approach versus (a) opportunistic scheduling with unicast transmissions and classical network caching (storing a fraction of each video), (b) standard coded caching based on transmitting-to-all. §.§ Related work Since coded caching was first proposed <cit.> and its potential was recognized by the community, substantial efforts have been devoted to quantify the gain in realistic scenarios, including decentralized placement <cit.>, non-uniform popularities <cit.>, and device-to-device (D2D) networks <cit.>. A number of recent works replace the original perfect shared link with wireless channels <cit.>. Commonly in the works with wireless channels, the performance of coded caching is limited by the user in the worst channel condition because the wireless multicast capacity is determined by the worst user <cit.>. This limitation of coded caching has been recently highlighted in <cit.>, while similar conclusions and some directions are given in <cit.>. Our work is the first to addresses this aspect by jointly designing the transmissions over the broadcast channel and scheduling appropriate subsets of users. Most past works deal with offline caching in the sense that both cache placement and delivery phases are performed once and do not capture the random and asynchronous nature of video traffic. The papers <cit.> addressed partly the online nature by studying cache eviction strategies, and delay aspects. In this paper, we explore a different online aspect. Requests for video files arrive in an online fashion, and transmissions are scheduled over time-varying wireless channels. Online transmission scheduling over wireless channels has been extensively studied in the context of opportunistic scheduling <cit.> and network utility maximization <cit.>. Prior works emphasize two fundamental aspects: (a) the balancing of user rates according to fairness and efficiency considerations, and (b) the opportunistic exploitation of the time-varying fading channels. Related to our work are the studies of wireless downlink with broadcast degraded channels; <cit.> gives a maxweight-type of policy and <cit.> provides a throughput optimal policy based on a fluid limit analysis. Our work is the first to our knowledge that studies coded caching in this setting. The new element in our study is the joint consideration of user scheduling with codeword construction for the coded caching delivery phase. § SYSTEM MODEL AND PROBLEM FORMULATION We study a wireless downlink consisting of a base station and K users. The users are interested in downloading files over the wireless channel. §.§ Fair file delivery The performance metric is the time average delivery rate of files to user k, denoted by r_k. Hence our objective is expressed with respect to the vector of delivery rates . We are interested in the fair file delivery problem: r^* = max_r∈Λ∑_k=1^Kg(r_k), where Λ denotes the set of all feasible delivery rate vectors–clarified in the following subsection–and the utility function corresponds to the alpha fair family of concave functions obtained by choosing: g(x) = (d+x)^1-α/1-α, α≠ 1 log(1+x/d), α = 1 for some arbitrarily small d>0 (used to extend the domain of the functions to x=0). Tuning the value of α changes the shape of the utility function and consequently drives the system performance r^* to different points: (i) α=0 yields max sum delivery rate, (ii) α→∞ yields max-min delivery rate <cit.>, (iii) α = 1 yields proportionally fair delivery rate <cit.>. Choosing α∈ (0,1) leads to a tradeoff between max sum and proportionally fair delivery rates. The optimization (<ref>) is designed to allow us tweak the performance of the system; we highlight its importance by an example. Suppose that for a 2-user system Λ is given by the convex set shown on figure <ref>. Different boundary points are obtained as solutions to (<ref>). If we choose α=0, the system is operated at the point that maximizes the sum r_1+r_2. The choice α→∞ leads to the maximum r such that r_1=r_2=r, while α=1 maximizes the sum of logarithms. The operation point A is obtained when we always broadcast to all users at the weakest user rate and use <cit.> for coded caching transmissions. Note that this results in a significant loss of efficiency due to the variations of the fading channel, and consequently A lies in the interior of Λ. We may infer that the point α→∞ is obtained by avoiding transmissions to users with instantaneous poor channel quality but still balancing their throughputs in the long run. §.§ Transmission model To analyze the set of feasible rate vectors Λ we need to zoom in the detailed model of transmissions. Caching model. There are N equally popular files W_1, …, W_N, each F bits long. The files are available to the base station. User k is equipped with cache memory Z_k of MF bits, where M∈ [0,N]. Caching placement is performed during off-peak hour, and the goal is to fill the caches up to the memory constraint with selected bits. To this end, we need to select K caching functions ϕ_k: _2^NF→_2^MF which map the files W_1, …, W_N into the cache contents Z_k ϕ_k(W_1,…, W_N),   ∀ k=1,…,K. The caching functions can be used to cache a few entire files, or a small fraction from each file, or even coded combinations of subfiles <cit.>. It is important to note that the caching functions are selected once, without knowledge of future requests, and are fixed throughout our system operation.[A reasonable extension is to enable infrequent updates of the caching placement phase.] Downlink channel model. We consider a standard block-fading broadcast channel, such that the channel state remains constant over a slot of T_ slot channel uses and changes from one slot to another in an i.i.d. manner. The channel output of user k in any channel use of slot t is given by _k(t)=√(h_k(t))(t)+_k(t), where the channel input ∈^T_ slot is subject to the power constraint [‖‖^2] ≤ PT_ slot; _k(t)∼_(0, _T_ slot) are additive white Gaussian noises with covariance matrix identity of size T_ slot, assumed independent of each other; {h_k(t)∈} are channel fading coefficients ∼β_k^2exp(1) independently distributed across time and users, with β_k denoting the path-loss parameter of user k. Encoding and transmissions. The transmissions aim to contribute information towards the delivery of a specific vector of file requests (t), where d_k(t)∈{1, …, N} denotes the index of the requested file by user k in slot t. Here N is the video library size, typically in the order of 10K. The requests are generated randomly, and whenever a file is delivered to user k, the next request of this user will be for another randomly selected file. At each time slot, the base station observes the channel state (t)=(h_1(t), …,h_K(t)) and the request vector up to t, ^t, constructs a transmit symbol using the encoding function f_t:{1,..,N}^Kt×^K→^T_ slot. (t) = f_t ( ^t,(t)), Finally, it transmits a codeword (t) for the T_ slot channel uses over the fading broadcast channel in slot t . The encoding function may be chosen at each slot to contribute information to a selected subset of users 𝒥(t)⊆{1,…,K}. This allows several possibilities, e.g. to send more information to a small set of users with good instantaneous channel qualities, or less information to a large set that includes users with poor quality. Decoding. At slot t, each user k observes the local cache contents Z_k and the sequence of channel outputs so far y_k(τ),  τ=1,…,t and employs a decoding function ξ_k to determine the decoded files. Let D_k(t) denote the number of files decoded by user k after t slots. The decoding function ξ_k is a mapping ξ_k: ^T_slott×^Kt×_2^FM×{1,..,N}^Kt→_2^FD_k(t). The decoded files of user k at slot t are given by ξ_k(y_k^T_ slott, Z_k, ^t, ^t), and depend on the channel outputs and states up to t, the local cache contents, and the requested files of all users up to t. A file is incorrectly decoded if it does not belongs to the set of requested files. The number of incorrectly decoded files are then given by |∪_t{ξ_k(t)}∖ d_k^t| and the number of correctly decoded files at time t is: C_k(t)= D_k(t)- |∪_t{ξ_k(t)}∖ d_k^t| A rate vector =(r_1,…, r_K) is said to be feasible ∈Λ if there exist functions ([ϕ_k], [f_t], [ξ_k]) such that: r_k=lim sup_t→∞C_k(t)/t, where the rate is measured in file/slot. In contrast to past works which study the performance of one-shot coded caching <cit.>, our rate metric measures the ability of the system to continuously deliver files to users. §.§ Code-constrained rate region Finding the optimal policy is very complex. In this paper, we restrict the problem to specific class of policies given by the following mild assumptions: The admissible policies have the following characteristics: * The caching placement and delivery follow the decentralized scheme <cit.>. * The users request distinct files, i.e., the ids of the requested files of any two users are different. Since we restrict our action space, the delivery rate feasibility region, Λ^CC, of the class of policies Π^CC is smaller than the one for the original problem Λ. However, these restrictions allow us to come up with a concrete solution approach. Note that the optimal cache and transmission design policy is already a very hard problem even in the simple case of broadcast transmissions with a fixed common rate, and the method in <cit.>, <cit.> are practical approaches with good performance. In addition, looking at demand IDs when combining files would be very complex and, because of the big library sizes, is not expected to bring substantial gains (it is improbable that two users will make request for the same file in close time instances). § OFFLINE CODED CACHING In this section we briefly review decentralized coded caching, first proposed in <cit.>, and used by all admissible policies Π^CC. We set m=M/N the normalized memory size. Under the memory constraint of MF bits, each user k independently caches a subset of mF bits of file i, chosen uniformly at random for i=1,…, N. By letting W_i| denote the sub-file of W_i stored exclusively in the cache memories of the user set , the cache memory Z_k of user k after decentralized placement is given by Z_k ={ W_i : ∀⊆[K], ∀∋ k , ∀ i =1,…, N }. The size of each sub-file measured in bits is given by |W_i |= m^||(1-m)^K-|| as F→∞. The above completely determine the caching functions. Once the requests of all users are revealed, the offline scheme proceeds to the delivery of the requested files (delivery phase). Assuming that user k requests file k, i.e. d_k =k, the server generates and conveys the following codeword simultaneously useful to the subset of users : V_=⊕_k∈W_k|∖{k}, where ⊕ denotes the bit-wise XOR operation. The main idea here is to create a codeword useful to a subset of users by exploiting the receiver side information established during the placement phase. It is worth noticing that the coded delivery with XORs significantly reduces the number of transmissions. Compared to uncoded delivery, where the sub-files are sent sequentially and the number of transmissions are equal to ||×|W_k|∖{k}|, the coded delivery requires the transmission of |W_k|∖{k}|, yielding a reduction of a factor ||. In a practical case of N>K, it has been proved that decentralized coded caching achieves the total number of transmissions, measured in the number of files, given by <cit.> T_ tot(K,m) = 1/m(1-m) {1-(1-m) ^K }. On the other hand, in uncoded delivery, the number of transmissions is given by K(1-m) since it exploits only local caching gain at each user. For a system with K=30 users and normalized memory of m=1/3, the minimum transmissions required by uncoded delivery is 20 and that of decentralized coded caching is 2, yielding a gain of factor 10. In order to further illustrate the placement and delivery of decentralized coded caching, we provide an three-user example. For the case of K=3 users in Fig.<ref>, let us assume that user 1, 2, 3, requests file A, B, C, respectively. After the placement phase, a given file A will be partitioned into 8 subfiles. Codewords to be sent are the following * A_∅, B_∅ and C_∅ to user 1, 2 and 3 respectively. * A_ 2⊕ B_ 1 is intended to users {1,2}. Once received, user 1 decodes A_ 2 by combining the received codeword with B_ 1 given in its cache. Similarly user 2 decodes B_ 1. The same approach holds for codeword B_ 3⊕ C_ 2 to users {2,3} and codeword A_ 3⊕ C_ 1 to users {1,3} * A_ 23⊕ B_ 13⊕ C_ 12 is intended users 1,2,3. User 1 can decode A_ 23 by combining the received codeword with {B_ 13,C_ 12} given in its cache. The same approach is used for user 2, 3 to decode B_ 13, C_ 12 respectively. § BROADCASTING PRIVATE AND COMMON MESSAGES In this section, we address the question on how the transmitter shall convey private and multiple common messages, each intended to a subset of users, while opportunistically exploiting the underlying wireless channel. We start by remarking that the channel in (<ref>) for a given channel realization corresponds to the Gaussian degraded broadcast channel. Without loss of generality, let us assume h_1≥…≥ h_K so that the following Markov chain holds. X ↔ Y_1 ↔…↔ Y_K. The capacity region of the degraded broadcast channel for K private messages and a common message is well-known <cit.>. In this section, we consider a more general setup where the transmitter wishes to convey 2^K-1 mutually independent messages, denoted by {M_}, where M_ denotes the message intended to the users in subset ⊆{1,…, K}. Each user k must decode all messages {M_} for ∋ k. By letting R_ denote the multicast rate of the message M_, we say that the rate-tuple ∈_+^2^K-1 is achievable if there exists encoding and decoding functions which ensure the reliability and the rate condition. The capacity region is defined as the supremum of the achievable rate-tuple, where the rate is measured in bit/channel use. The capacity region Γ() of a K-user degraded Gaussian broadcast channel with fading gains h_1 ≥…≥ h_K and 2^K-1 independent messages {M_} is given by R_1 ≤log(1+ h_1α_1 P) ∑_⊆{1,…, k}: k∈ R_ ≤log1+ h_k ∑_j=1^kα_j P/1+ h_k∑_j=1^k-1α_j P k=2, …, K for non-negative variables {α_k} such that ∑_k=1^K α_k ≤ 1. Please refer to Appendix <ref> for the proof. The achievability builds on superposition coding at the transmitter and successive interference cancellation at receivers. For K=3, the transmit signal is simply given by x = x_1 + x_2 + x_3 + x_12 + x_13+ x_123 where {x_} are mutually independent Gaussian distributed random variables satisfying the power constraint and x_ denotes the signal corresponding to the message M_ intended to the subset ⊆{1,2,3}. User 3 (the weakest user) decodes M̃_3 ={M_3, M_13, M_23, M_123} by treating all the other messages as noise. User 2 decodes first the messages M̃_3 and then jointly decodes M̃_2 ={M_2, M_12}. Finally, user 1 (the strongest user) successively decodes M̃_3, M̃_2 and, finally, M_1. Later in our online coded caching scheme we will need the capacity region Γ(), and more specifically, we will need to characterize its boundary. To this end, it suffices to consider the weighted sum rate maximization: max_∈Γ()∑_: ⊆{1,…, K}θ_ r_. We first simplify the problem using the following theorem. The weighted sum rate maximization with 2^K-1 variables in (<ref>) reduces to a simpler problem with K variables, given by f() = ∑_k=1^K θ̃_k log1+h_k ∑_j=1^kα_j P/1+ h_k∑_j=1^k-1α_j P. where θ̃_k denotes the largest weight for user k θ̃_k=max_: k∈⊆{1,…, k}θ_. The proof builds on the simple structure of the capacity region. We first remark that for a given power allocation of other users, user k sees 2^k-1 messages {W_} for all such that k∈⊆{1,…, k} with the equal channel gain. For a given set of {α_j}_j=1^k-1, the capacity region of these messages is a simple hyperplane characterized by 2^k-1 vertices C_k _i for i=1, …, 2^k-1, where C_k is the sum rate of user k in the RHS of (<ref>) and _i is a vector with one for the i-th entry and zero for the others. Therefore, the weighted sum rate seen is maximized for user k by selecting the vertex corresponding to the largest weight, denoted by θ̃. This holds for any k. We provide an efficient algorithm to solve this power allocation problem as a special case of the parallel Gaussian broadcast channel studied in <cit.>. Following <cit.>, we define the rate utility function for user k given by u_k(z)= θ̃_k/1/h_k+z-λ where λ is a Lagrangian multiplier. The optimal solution corresponds to selecting the user with the maximum rate utility at each z and the resulting power allocation for user k is α^*_k = { z: [max_j u_j(z) ]_+ = u_k(z) }/P with λ satisfying P= [ max_k θ̃_k/λ -1/h_k]_+. § PROPOSED ONLINE DELIVERY SCHEME This section presents first the queued delivery network and its feasible rate region of arrival rates, then describes the proposed control policy. §.§ Solution plan At each time slot t, the controller admits a_k(t) files to be delivered to user k, and hence a_k(t) is a control variable.[We note that random file arrivals can be directly captured with the addition of an extra queue <cit.>, which we avoid to simplify exposition.] As our model dictates, the succession of requested files for user k is determined uniformly at random. Queueing model. The base station organizes the information into the following types of queues: * User queues to store admitted files, one for each user. The buffer size of queue k is denoted by S_k(t) and expressed in number of files. * Codeword queues to store codewords to be multicast. There is one codeword queue for each subset of users ⊆{1, …, K}. The size of codeword queue is denoted by Q_(t) and expressed in bits. A queueing policy π performs the following operations: (i) decides how many files to admit into the user queues S_k(t) in the form of (a_k(t)) variables, (ii) then it decides how to combine together files from different user queues to be encoded into the form of multiple codewords which represent the required broadcast transmissions for the reception of this file–these codewords are stored in the appropriate codeword queues Q_(t), (iii) and last it decides the encoding function f_t. (ii) and (iii) are further clarified in the next section. A queue S(t) is said to be (strongly) stable if lim sup_T→∞1/T∑_t=0^T-1𝔼[S(t)] < ∞. A queueing system is said to be stable if all its queues are stable. Moreover, the stability region of a system is the set of all arrival rates such that the system is stable. The above definition implies that the average delay of each job in the queue is finite. In our problem, if we develop a policy that keeps user queues (t) stable, then all admitted files will, at some point, be combined into codewords. If in addition codeword queues (t) are stable, then all generated codewords will reach their destinations, meaning that all receivers will be able to decode the admitted files that they requested. The region of all feasible delivery rates Λ is the same as the stability region of the system (i.e. the set of all demand arrival rates for which there exists a policy that stabilizes the queueing system). Let a_k = lim sup_t→∞1/t∑_t=0^t-1𝔼[a_k(t)], denote the time average number of admitted files for user k. Lemma <ref> implies the following Corollary. Solving (<ref>) is equivalent to finding a policy π such that a^π = ∑_k=1^Kg_k(a_k) s.t. the system is stable. See Appendix <ref> §.§ Feasible Region Contrary to the offline coded caching in <cit.>, we propose an online delivery scheme consisting of the following three blocks. Each block is operated at each slot. * Admission control: At the beginning of each slot, the controller decides how many requests for each user, a_k(t) should be pulled into the system from the infinite reservoir. * Routing: The cumulative accepted files for user k are stored in the admitted demand queue whose size is given by S_k(t) for k=1,…, K. The server decides the combinations of files to perform coded caching. The decision at slot t for a subset of users ⊆{1,..,K}, denoted by σ_(t)∈{0,1,…,σ_max}, refers to the number of combined requests for this subset of users. It is worth noticing that offline coded caching lets σ_ = 1 for ={1,…, K} and zero for all the other subsets. The size of the queue S_k evolves as: S_k(t+1) = [S_k(t) - ∑_: k∈σ_(t)]^+ + a_k(t) If σ_(t)> 0, the server creates codewords by applying offline coded caching explained in Section [] for this subset of users as a function of the cache contents {Z_j: j∈}. * Scheduling: The codewords intended to the subset of users are stored in codeword queue whose size is given by Q_(t) for ⊆{1,…, K}. Given the instantaneous channel realization (t) and the queue state {Q_(t)}, the server performs scheduling and rate allocation. Namely, at slot t, it determines the number μ_(t) of bits per channel use to be transmitted for the users in subset . By letting b_, denote the number of bits generated for codeword queue ⊆ when offline coded caching is performed to the users in , codeword queue evolves as Q_(t+1) = [Q_(t) - T_ slotμ_(t)]^+ + ∑_:⊆b_,σ_(t) where b_,=m^||(1-m)^||-||-1. In order determine our proposed policy, namely the set of decisions {(t), (t),(t)} at each slot t, we first characterize the feasible region Λ as a set of arrival rates . We let π_𝐡 denote the probability that the channel state at slot t is ∈ where is the set of all possible channel states. We let Γ() denote the capacity region for a fixed channel state 𝐡. Then we have the following A demand rate vector is feasible, i.e. ∈Λ^CC, if and only if there exist ∈∑_𝐡∈ℋπ_𝐡Γ(𝐡), σ̅_∈[0,σ_max], ∀⊆{1,…, K} such that: ∑_: k∈σ̅_≥a̅_k, ∀ k =1,…,K T_slotμ_≥∑_: ⊆ b_, σ̅_, ∀∈ 2^ . Constraint (<ref>) says that the service rate at which admitted demands are combined to form codewords is greater than the arrival rate, while (<ref>) implies that the long-term average transmission rate μ_ for the subset of users should be higher than the rate at which bits of generated codewords for this group arrive. In terms of the queueing system defined, these constraints impose that the service rates of each queue should be greater than their arrival rates, thus rendering them stable. Theorem <ref> implies that the set of feasible average delivery rates is a convex set. §.§ Admission Control and Routing In order to perform the utility maximization (<ref>), we need to introduce one more set of queues. These queues are virtual, in the sense that they do not hold actual file demands or bits, but are merely counters to drive the control policy. Each user k is associated with a queue U_k(t) which evolves as follows: U_k(t+1) = [U_k(t) - a_k(t)]^+ + γ_k(t) where γ_k(t) represents the arrival process to the virtual queue and is given by γ_k(t) = max_0≤ x≤γ_k,max[Vg_k(x) - U_k(t)x] In the above, V>0 is a parameter that controls the utility-delay tradeoff achieved by the algorithm (see Theorem <ref>). The general intuition here is as follows: Observe that the number a_k(t) of admitted demands is the service rate for the virtual queues U_k(t). The control algorithm actually seeks to optimize the time average of the virtual arrivals γ_k(t). However, since U_k(t) is stable, its service rate, which is the actual admission rate, will be greater than the rate of the virtual arrivals, therefore giving the same optimizer. Stability of all other queues will guarantee that admitted files will be actually delivered to the users. We present our on-off policy for admission control and routing. For every user k, admission control chooses a_k(t) demands given by a_k(t) = γ_k, max{U_k(t) ≥ S_k(t) } For every subset ⊆{1,…, K}, routing combines σ_(t) demands of users in given by σ_(t) = σ_max{∑_k∈S_k(t) > ∑_: ⊆b_, /F^2 Q_(t) } . §.§ Scheduling and Transmission In order to stabilize all codeword queues, the scheduling and resource allocation explicitly solve the following weighted sum rate maximization at each slot t where the weight of the subset corresponds to the queue length of Q_ (t) = max_∈Γ((t))∑_⊆{1, …, K} Q_(t)r_ . We propose to apply the power allocation algorithm in Section <ref> to solve the above problem by sorting users in a decreasing order of channel gains and treating Q_(t) as θ_. In adition, we assume that the number of channel uses in one coherence block is large enough such that the decoding error from choosing channel codes with rate (t) is very small. In this case, no feedback from the receivers is given. §.§ Example We conclude this section by providing an example of our proposed online delivery network for K=3 users as illustrated in Fig. <ref>. At slot t the server decides to combine W_1 requested by user 2 with W_8 requested by user 2 and to process W_4 requested by user 1 uncoded. Therefore σ_{1,2}(t)=σ_{1}(t)=1 and σ_(t)=0 otherwise. Given this codeword construction, codeword queues have inputs as described in Table I. In addition, data from queues Q_{2}(t), Q_{2.3}(t) are transmitted. § PERFORMANCE ANALYSIS In thi section, we present the main result of the paper, that our proposed online algorithm leads to close to optimal performance for all policies in the class Π^CC: Let r̅^π_k the mean time-average delivery rate for user k achieved by the proposed policy. Then ∑_k=1^Kg_k(r̅^π_k) ≥max_∈Λ^CC∑_k=1^Kg_k(r̅_k) - 1/V lim sup_T→∞1/T∑_t=0^T-1𝔼{Q̂(t)} = V, where Q̂(t) is the sum of all queue lengths at the beginning of time slot t, thus a measure of the mean delay of file delivery. The above theorem states that, by tuning the constant V, the utility resulting from our online policy can be arbitrarily close to the optimal one, where there is a tradeoff between the guaranteed optimality gap 1/V and the upper bound on the total buffer length V. For proving the Theorem, we use the Lyapunov function L(t) = 1/2(∑_k=1^KU_k^2(t) + S_k^2(t) + ∑_∈ 2^1/F^2Q_^2(t)) and specifically the related drift-plus-penalty quantity, defined as: 𝔼{L(t+1) - L(t)| 𝐒(t), 𝐐(t), 𝐔(t)} - V𝔼{∑_k=1^Kg(γ_k(t))|𝐒(t), 𝐐(t), 𝐔(t) }. The proposed algorithm is such that it minimizes (a bound on) this quantity. The main idea is to use this fact in order to compare the evolution of the drift-plus-penalty under our policy and two "static" policies, that is policies that take random actions (admissions, demand combinations and wireless transmissions), drawn from a specific distribution, based only on the channel realizations (and knowledge of the channel statistics). We can prove from Theorem 4 that these policies can attain every feasible delivery rate. The first static policy is one such that it achieves the stability of the system for an arrival rate vector ' such that '+δ∈∂Λ^CC. Comparing with our policy, we deduce strong stability of all queues and the bounds on the queue lengths by using a Foster-Lyapunov type of criterion. In order to prove near-optimality, we consider a static policy that admits file requests at rates ^* = max_∑_kg_k(a_k) and keeps the queues stable in a weaker sense (since the arrival rate is now in the boundary Λ^CC). By comparing the drift-plus-penalty quantities and using telescopic sums and Jensen's inequality on the time average utilities, we obtain the near-optimality of out proposed policy. The full proof is in Appendix <ref>. § NUMERICAL EXAMPLES In this section, we compare our proposed delivery scheme with the following two other schemes, all building on decentralized cache placement in (<ref>) and (<ref>). * Unicast opportunistic scheduling: for any request, the server sends the remaining (1-m)F bits to the corresponding user without combining any files. Here we only exploit the local caching gain. In each slot the serve sends with full power to user k^*(t)=max_klog( 1+h_k(t)P) / T_k(t) ^α, where T_k(t)=∑_1≤τ≤ t-1μ_k(τ)/(t-1) is the empirical average rate for user k up to slot t. * Standard coded caching: we use decentralized coded caching among all K users. For the delivery, non-opportunistic TDMA transmission is used. The server sends sequentially codewords V_ to the subset of users at the weakest user rate among : μ_(t)=log( 1+Pmin_k∈(h_k(t))) . Once the server has sent codewords {V_}_∅≠⊆{1,..,K}, every user is able to decode one file. Then the process is repeated for all the demands. We consider the system with normalized memory of m=0.6, power constraint P=10dB, file size F=10^3 bits and number of channel uses per slot T_ slot=10^2. We divide users into two classes of K/2 users each: strong users with β_k=1 and weak users with β_k=0.2. We compare the three algorithms for the cases where the objective of the system is sum rate maximization (α=0) and proportional fairness (α=1). The results are depicted in Fig. <ref> and  <ref>, respectively. Regarding the sum rate objective, standard coded caching performs very poorly, indicative of the adverse effect of users with bad channel quality. It is notable that our proposed scheme outperforms the unicast opportunistic scheme, which maximizes the sum rate if only private information packets are to be conveyed. The relative merit of our scheme increases as the number of users grows. This can be attributed to the fact that our scheme can exploit any available multicast opportunities. Our result here implies that, in realistic wireless systems, coded caching can indeed provide a significant throughput increase when an appropriate joint design of routing and opportunistic transmission is used. Regarding the proportional fair objective, we can see that the average sum utility increases with a system dimension for three schemes although our proposed scheme provides a gain compared to the two others. § CONCLUSIONS We provided an algorithm to solve the problem of ensuring fairness in the long term delivery rates in wireless systems employing decentralized coded caching. Our results imply that appropriately combining the opportunism arising from the fading channels with the multicasting opportunities that arise from coded caching can mitigate the harmful impact of users with bad channel conditions in standard coded caching schemes and provide significant increase in the performance of the system. § PROOF OF THEOREM <REF> §.§ Converse We provide the converse proof for K=3 and the general case K>3 follows readily. Notice that the channel output of user k in (<ref>) for n channel use can be equivalently written as _k=+_k, where _k=_k/√((h_k))∼_(0, N_k_n) for N_k=1/h_k and _n identity matrix of size n. Since N_1≤ N_2≤ N_3, we set M̃_k=∪_k∈⊆[k] M_ the message that must be decoded by user k (user k decodes all bits that user k'≥ k decodes) at rate R̃_k. More explicitly, M̃_1={M_1}, M̃_2={M_2,M_12}, M̃_3={M_3,M_13,M_23,M_123}. By Fano's inequality, we have nH(M̃_1) ≤ I(M̃_1;Y_1M̃_2,M̃_3) nH(M̃_2) ≤ I(M̃_2;Y_2M̃_3) nH(M̃_3) ≤ I(M̃_3;Y_3). Consider I(M̃_3;Y_3)=H(Y_3)-H(Y_3M̃_3). Since nlog( 2π eN_3)=H(Y_3M̃_3,X) ≤ H(Y_3M̃_3)≤ H(Y_3)≤ nlog( 2π e(P+N_3)), there exist 0≤α_3≤ 1 such that H(Y_3M̃_3)=nlog( 2π e((1-α_3)P+N_3)). Using (<ref>) and (<ref>) we obtain I(M̃_3;Y_3) = H(Y_3)-H(Y_3M̃_3) ≤ nlog( 2π e(P+N_3))-nlog( 2π e((1-α_3)P+N_3)) =nlog(N_3+P/N_3+(1-α_3)P). Next consider I(M̃_2;Y_2M̃_3)=H(Y_2M̃_3)-H(Y_2M̃_2,M̃_3). Using the conditional entropy power inequality in <cit.> , we have H(Y_3M̃_3) =H(Y_2+n_3-n_2M̃_3) ≥ nlog(2^2H(Y_2M̃_3)/n+2^2H(n_3-n_2M̃_3)/n) = nlog(2^2H(Y_2M̃_3)/n+2π e(N_3-N_2)) (<ref>) and (<ref>) imply nlog( 2π e((1-α_3)P+N_3)) ≥ nlog(2^2H(Y_2M̃_3)/n+2π e(N_3-N_2)) equivalent to H(Y_2M̃_3) ≤ nlog(2π e((1-α_3)P+N_2)). Since nlog( 2π eN_2)=H(Y_2M̃_2,M̃_3,X)≤ H(Y_2M̃_2,M̃_3)≤ H(Y_2M̃_3), there exists α_2 such that 0≤1-α_2-α_3≤ 1-α_3 and H(Y_2M̃_2,M̃_3)=nlog(2π e((1-α_2-α_3)P+N_2)). Using (<ref>), (<ref>) and (<ref>) it follows I(M̃_2,M̃_3;Y_2) =H(Y_2M̃_3)-H(Y_2M̃_2,M̃_3) ≤ nlog(2π e((1-α_3)P+N_2)) -nlog(2π e((1-α_2-α_3)P+N_2)) =nlog(N_2+(1-α_3)P/(1-α_2-α_3)P+N_2). Last we consider I(M̃_1;Y_1M̃_2,M̃_3) =H(Y_1M̃_2,M̃_3)-H(Y_1M̃_1, M̃_2,M̃_3) =H(Y_1M̃_2,M̃_3)-H(Y_1M̃_1, M̃_2,M̃_3, X) =H(Y_1M̃_2,M̃_3)-H(Y_1 X) =H(Y_1M̃_2,M̃_3)-nlog( 2π eN_1) Using the conditional entropy power inequality in <cit.> , we have H(Y_2M̃_2, M̃_3) =H(Y_1+n_2-n_1M̃_2,M̃_3) ≥ nlog(2^2H(Y_1M̃_2,M̃_3)/n+2^2H(n_2-n_1M̃_2,M̃_3)/n) = nlog(2^2H(Y_1M̃_2,M̃_3)/n+2π e(N_2-N_1)) (<ref>) and (<ref>) imply nlog(2π e((1-α_2-α_3)P+N_2)) ≥ nlog(2^2H(Y_1M̃_2,M̃_3)/n+2π e(N_2-N_1)) equivalent to H(Y_1M̃_2,M̃_3) ≤ nlog(2π e((1-α_2-α_3)P+N_1)). Let α_1=1-α_2-α_3. Combining the last inequality with (<ref>) we obtain I(M̃_1;Y_1M̃_2,M̃_3)≤ nlog( N_1+α_1P/N_1). From (<ref>), (<ref>), (<ref>) and (<ref>), it readily follows that ∃ 0≤α_1,α_2,α_3≤1 such that α_1+α_2+α_3=1 and H(M̃_1) ≤log(1+ α_1P/N_1), H(M̃_2) ≤log(1+α_2P/N_2+α_1P), H(M̃_3) ≤log(1+α_3P/N_3+(α_1+α_2)P). By replacing H(M̃_k) with ∑_k∈⊆[k]R_ and N_k with 1/h_k we obtain the result R_1 ≤log( 1+h_1α_1P) R_2+R_12 ≤log(1+h_2(α_1+α_2)P/1+h_2α_1P) R_3+R_13+R_23+R_123 ≤log(1+h_3P/1+h_3(α_1+α_2)P), §.§ Achievability Superposition coding achieves the upper bound. For 1≤ k≤3, generate random sequences u^n_k(m_k), m_k∈[1:2^nR̃_k] each i.i.d. _(0, α_kP). To transmit a triple message (m_1,m_2,m_3) the encoder set X=u^n_1(m_1)+u^n_2(m_2)+u^n_3(m_3). For decoding: * Receiver 3 recover m_3 from Y_3=u^n_3(m_3)+( u^n_1(m_1)+u^n_2(m_2)+n_3) by considering u^n_1(m_1)+u^n_2(m_2) as noise. The probability of error tends to zero as n→∞ if R̃_3≤log( 1+α_3P/N_3+(α_1+α_2)P). * Receiver 2 uses successive cancellation. First, it decodes m_3 from Y_2=u^n_3(m_3)+( u^n_1(m_1)+u^n_2(m_2)+n_2) by considering u^n_1(m_1)+u^n_2(m_2) as noise. The probability of error tends to zero as n→∞ if R̃_3≤log(1+ α_3P/N_2+(α_1+α_2)P). Since N_2≤ N_3 and R̃_3≤log(1+α_3P/N_3+(α_1+α_2)P), the later condition is satisfied. Second, it subtracts off u^n_3(m_3) and recover u^n_2(m_2) from Ỹ2=u^n_2(m_2)+( u^n_1(m_1)+n_2) by treating u^n_1(m_1) as noise. The probability of error tends to zero as n→∞ if R̃_2≤log(1+α_2P/N_2+α_1P). * Receiver 1 uses successive cancellation twice. First, it decodes m_3 from Y_1=u^n_3(m_3)+( u^n_1(m_1)+u^n_2(m_2)+n_1) by considering u^n_1(m_1)+u^n_2(m_2) as noise. The probability of error tends to zero as n→∞ if R̃_3≤log(1+α_3P/N_1+(α_1+α_2)P). Since N_1≤ N_3 and R̃_3≤log(1+α_3P/N_3+(α_1+α_2)P), the later condition is satisfied. Second, it subtracts off u^n_3(m_3) and decodes u^n_2(m_2) by treating u^n_1(m_1) as noise. The probability of error tends to zero as n→∞ if R̃_2≤log(1+α_2P/N_1+α_1P). Since N_1≤ N_2 and R̃_2≤log(1+α_2P/N_2+α_1P), the later condition is satisfied. Last, it subtracts off u^n_2(m_2) and recover u^n_1(m_1). The probability of error tends to zero as n→∞ if R̃_1≤log(1+α_1P/N_1). § PROOF OF LEMMA <REF> Denote A_k(t) the number of files that have been admitted to the system for user k up to slot t. Also, note that due to our restriction on the class of policies Π^CC and our assumption about long enough blocklengths, there are no errors in decoding the files, therefore the number of files correctly decoded for user k till slot t is D_k(t). Since D_k(t)≤ A_k(t), ∀ t≥ 0, ∀ k=1,..,K, if suffices to show that for every arrival rate vector ∈Λ^CC, there exists a policy in Π^CC for which the delivery rate vector is =. We shall deal only with the interior of Λ^CC (arrival rates at the boundaries of stability region are exceptional cases). Take any arrival rate vector ∈ Int(Λ^CC). From <cit.> it follows that for any there exists a randomized demand combination and transmission policy π^RAND, the probabilities of which depending only on the channel state realization each slot, for which the system is strongly stable. In addition, any arrival rate vector can be constructed via a randomized admission policy. Since the channels are i.i.d. random with a finite state space and queues are measured in files and bits, the system now evolves as a discrete time Markov chain (𝐒(t), 𝐐(t), 𝐇(t)), which can be checked that is aperiodic, irreducible ad with a single communicating class. In that case, strong stability means that the Markov chain is ergodic with finite mean. Further, this means that the system reaches to the set of states where all queues are zero infinitely often. Let T[n] be the number of timeslots between the n-th and (n+1)-th visit to this set (we make the convention that T[0] is the time slot that this state is reached for the first time). In addition, let Â_k[n], D̂_k[n] be the number of demands that arrived and were delivered in this frame, respectively. Then, since within this frame the queues start and end empty, we have [Â_k[n] = D̂_k[n], ∀ n, ∀ k. In addition since the Markov chain is ergodic, a̅_k = lim_t→∞A(t)/t = lim_N→∞∑_n=0^NÂ_k[n]/∑_n=0^NT[n] and r̅_k = lim_t→∞D(t)/t = lim_N→∞∑_n=0^ND̂_k[n]/∑_n=0^NT[n] Combining the three expressions, = thus the result follows. § PROOF OF THEOREM <REF> From Lemma <ref> and Corollary <ref>, it suffices to prove that under the online policy the queues are strongly stable and the resulting time average admission rates maximize the desired utility function subject to minimum rate constraints. We first look at policies that take random decisions based only on the channel realizations. Since the feasibility region Λ^CC is a convex set (see Theorem ), any point in it can be achieved by properly time-sharing over the possible control decisions. We focus on two such policies, one that achieves the optimal utility and another on that achieves (i.e. admits and stabilizes the system for that) a rate vector in th δ- interior of Λ^CC. We then have the following Lemmas: Define a policy π^*∈Π^CC that in each slot where the channel states are 𝐡 works as follows: (i) it pulls random user demands with mean a̅_k^*, and it gives the virtual queues arrivals with mean γ̅_k = a̅_k^* as well (ii) the number of combinations for subset is a random variable with mean σ̅^*_ and uniformly bounded by σ_max, (iii) selects one out of K+1 suitably defined rate vectors μ^𝐥∈Γ(), l=1,..,K+1 with probability ψ_l,. The parameters above are selected such that they solve the following problem: max_ ∑_k=1^Kg_k(a̅_k^*) s.t. ∑_: k∈σ̅_≥a̅_k^*, ∀ k∈{1,..,K} ∑_: ⊆b_, σ̅_≥∑_π_∑_l=1^K+1ψ_l, μ^l_(), ∀∈ 2^ Then, π^* results in the optimal delivery rate vector (when all possible policies are restricted to set Π). Define a policy π^δ∈Π^CC that in each slot where the channel states are 𝐡 works as follows: (i) it pulls random user demands with mean a̅_k^δ such that (+δ)∈Π^CC, and gives the virtual queues random arrivals with mean γ̅_k ≤ + ϵ' for some ϵ'>0 (ii) the number of combinations for subset is a random variable with mean σ̅^δ_ and uniformly bounded by σ_max, (iii) selects one out of K+1 suitably defined rate vectors μ^𝐥∈Γ(), l=1,..,K+1 with probability ψ^δ_l,. The parameters above are selected such that: ∑_: k∈σ̅^̅δ̅_≥ϵ + a̅_k^δ, ∀ k∈{1,..,K} ∑_: ⊆b_, σ̅^δ_≥ϵ + ∑_π_∑_l=1^K+1ψ'_l, μ^l_(), ∀∈ 2^ for some appropriate ϵ < δ. Then, the system under π^δ has mean incoming rates of ^δ and is strongly stable. The proof of the performance of our proposed policy is based on applying Lyapunov optimization theory <cit.> with the following as Lyapunov function L()= L(𝐒, 𝐐, 𝐔) = 1/2(∑_k=1^KU_k^2(t) + S_k^2(t) + ∑_∈ 2^Q_^2(t)/F^2) . Defining its drift as Δ L(𝐙) = 𝔼{L(𝐙(t+1))-L(𝐙(t))|𝐙(t)=𝐙} , using the queue evolution equations and the fact that ([x]^+)^2≤ x^2, we have Δ L(𝐙(t)) ≤ B + ∑_∈ 2^Q_(t)/F^2𝔼{∑_: ⊇b_,σ_(t) - μ_(t)} + ∑_k=1^KS_k(t)𝔼{a_k(t) - ∑_: k∈σ_(t)} + ∑_k=1^KU_k(t)𝔼{γ_k(t) - a_k(t)} , where B<∞ is a constant that depends only on the parameters of the system. Adding the quantity -V∑_k=1^K𝔼{g_k(γ_k(t))} to both hands of (<ref>) and rearranging the right hand side, we have Δ L(𝐙(t)) - V∑_k=1^K𝔼{g_k(γ_k(t))}≤ B +∑_k=1^K𝔼{-Vg_k(γ_k(t)) + γ_k(t)U_k(t))} +∑_𝔼{σ_(t)}(∑_:⊆Q_(t)/F^2b_, - ∑_k∈S_k(t)) + ∑_k=1^K(S_k(t) - U_k(t))𝔼{a_k(t)} -∑_Q_(t)/F^2𝔼{μ_(t)} Now observe that the control algorithm minimizes right hand side of (<ref>) given the channel state (t) (for any channel state). Therefore, taking expectations over the channel state distributions, for every vectors ∈ [1,γ_max]^K, γ̅∈ [1,γ_max]^K, σ̅∈ Conv({0,..,σ_max}^M), μ̅∈∑_∈ℋπ_Γ() it holds that Δ L(𝐙(t)) - V∑_k=1^K𝔼{g_k(γ^π_k(t))}≤ B - V∑_k=1^Kg_k(γ̅_k) + ∑_k=1^KU_k(t)(γ̅_k - a̅_k) +∑_k=1^KS_k(t)(a̅_k -∑_ :k∈σ̅_) + ∑_Q_(t)/F^2(∑_ :⊆b_,σ̅_ - μ̅_) We will use (<ref>) to compare our policy with the static policies defined in Lemmas <ref>, <ref>. More specifically, replacing the time averages we get from the static stabilizing policy π^δ of Lemma <ref> for some δ>0, we get that thre exist ϵ >0 such that Δ L(𝐙(t)) ≤ B + V∑_k=1^K𝔼{g_k(a^π_k(t))}- V∑_k=1^Kg_k(a̅_k^δ) -ϵ(∑_k=1^K S_k(t) + ∑_mQ_(t)/F^2) ϵ'∑_k=1^KU_k(t) Since a_k(t)≤γ_max∀ t, it follows that g_k(a̅_k^δ)<g_k(γ_max), therefore, we have from the Foster-Lyapunov criterion that the system (𝐒(t), 𝐐(t), 𝐔(t)) has a unique stationary probability distribution, under which the mean queue lengths are finite [For the utility-related virtual queues, note that if g_k'(0)<∞, then Y_k(t)<Vg'_k(0)+γ_k,max, i.e. their length is deterministically bounded]. Therefore the queues are strongly stable under our proposed policy. We now proceed to proving the utility-delay tradeoff. Proof of near optimal utility: Here we compare π with the static optimal policy π^* from Lemma <ref>. Since π^* takes decisions irrespectively of the queue lengths, we can replace quantities 𝐚̅, σ̅, μ̅ with the time averages corresponding to π^*, i.e. 𝐚̅^*, σ̅^*, μ̅^*. We thus have: V∑_k=1^K𝔼{g_k(γ^π_k(t))}≥ V ∑_k=1^Kg_k(a̅^*_k) - B + 𝔼{Δ L (𝐙(t))} Taking expectations over 𝐙(t) for both sides and summing the inequalities for t=0,1,..,T-1 we get 1/T∑_t=1^T-1∑_k=1^K𝔼{g_k(γ^π_k(t))}≥∑_k=1^K g_k(a̅^*_k) - B/V - 𝔼{L(𝐙(0))}/VT + 𝔼{L(𝐙(T))}/VT Assuming 𝔼{L(𝐙(0))} < ∞ (this assumption is standard in this line of work, for example it holds if the system starts empty), taking the limit as T goes to infinity gives lim_T→∞1/T∑_t=1^T-1∑_k=1^K𝔼{g_k(γ^π_k(t))}≥∑_k=1^Kg_k(a̅^*_k) - B/V In addition, since g_k(x) are concave, Jensen's inequality implies ∑_k=1^Kg_k(γ̅_l^π) =∑_k=1^Kg_k(lim_T→∞1/T∑_t=0^T𝔼{γ_k^π(t)}) ≥lim_T→∞1/T∑_t=1^T-1∑_k=1^K𝔼{g_k(γ^π_k(t))} ≥∑_k=1^Kg_k(a̅^*_k) - B/V . Proving the near optimality of the online policy follows from the above and the fact that a̅_k^π > γ̅_k^π (since the virtual queues U_k(t) are strongly stable). 10 url@samestyle cisco15 “White paper: Cisco VNI Forecast and Methodology, 2015-2020”, Tech. Report, 2015. maddah2013fundamental M. Maddah-Ali and U. Niesen, “Fundamental Limits of Caching,” IEEE Trans. Inf. Theory, vol. 60, no. 5, pp. 2856–2867, 2014. misconceptions G. S. Paschos, E. Bastug, I. Land, G. Caire, and M. Debbah, “Wireless caching: technical misconceptions and business barriers”, IEEE Communications Magazine, 2016. ji2013fundamental M. Ji, G. Caire, A. Molisch, “Fundamental Limits of Distributed Caching in D2D Wireless Networks” , arXiv/1304.5856, 2013. ji2015order M. Ji, A. Tulino, J. Llorca, and G. Caire, “Order-Optimal Rate of Caching and Coded Multicasting with Random Demands”, arXiv:1502.03124, 2015. maddah2013decentralized M. Maddah-Ali and U. Niesen, “Decentralized Coded Caching Attains Order-Optimal Memory-Rate Tradeoff”, IEEE/ACM Trans. Netw., vol. 23, no. 4, pp. 1029–1040, 2015. niesen2013coded M. Maddah-Ali and U. Niesen, “Coded Caching with Nonuniform Demands”,in IEEE INFOCOM Workshops, 2014. pedarsani2016online R. Pedarsani, M. Maddah-Ali, and U. Niesen, “Online Coded Caching," in IEEE/ACM Trans. Netw., vol. 24, no. 2, pp. 836-845, 2016. zhang2016wireless J. Zhang, and P. Elia, "Wireless Coded Caching: a Topological Perspective". arXiv:1606.08253, 2016. bidokhti2016noisy S. S. Bidokhti, M. Wigger, and R. Timo, “Noisy Broadcast Networks with Receiver Caching”, arXiv preprint arXiv:1605.02317, 2016. NgoAllerton2016 K-H. Ngo, S. Yang, and M. Kobayashi, “Cache-Aided Content Delivery in MIMO Channels”, in Proc. Allerton, IL, USA, 2016. niesen2015coded U. Niesen M. Maddah-Ali “Coded Caching for Delay-Sensitive Content”, in IEEE ICC, pp. 5559-5564, 2015. el2011network A. El Gamal and Y. H. Kim, “Network Information Theory”, Cambridge university press, 2011. neely10 M. Neely, “Stochastic Network Optimization with Application to Communication and Queueing Systems”, Morgan & Claypool, 2010. mowalrand J. Mo and J. Walrand, “Fair end-to-end window-based congestion control”, IEEE/ACM Trans. Netw. , Vol. 8, No. 5, Oct. 2000. pfscheduling F. Kelly, “Charging and Rate Control for Elastic Traffic”, European Transactions on Telecommunications, 1997. stolyar A. L. Stolyar, “On the asymptotic optimality of the gradient scheduling algorithm for multiuser throughput allocation." Operations Research Vol. 53 No. 1, 2005. knopp R. Knopp and P. A. Humblet, “Information capacity and power control in single-cell multiuser communications," in IEEE ICC, Seattle, WA, 1995. caire_fairnessMIMO H. Shirani-Mehr, G. Caire and M. J. Neely, “MIMO Downlink Scheduling with Non-Perfect Channel State Knowledge," in IEEE Trans. Commun. , vol. 58, no. 7, pp. 2055-2066, July 2010. caire_fairnessBC G. Caire, R. R. Muller and R. Knopp, “Hard Fairness Versus Proportional Fairness in Wireless Communications: The Single-Cell Case," in IEEE Trans. Inf. Theory, vol. 53, no. 4, pp. 1366-1385, April 2007. Seong06 K. Seong, R. Narasimhan, and J. Cioffi, “Queue Proportional Scheduling via Geometric Programming in Fading Broadcast Channels”, in IEEE JSAC, vol. 24, no. 8, Aug 2006. Eryilmaz01 A. Eryilmaz, and R. Srikant, and J. R. Perkins, “Throughput-optimal Scheduling for Broadcast Channels”, in Proc. ITCom, Denver, CO, August 2001. TseOptimal D. Tse, “Optimal Power Allocation over Parallel Gaussian Broadcast Channels”, unpublished Li05 M. Neely, E. Modiano, and C.-P. Li, “Fairness and Optimal Stochastic Control for Heterogeneous Networks”, IEEE/ACM Trans. Netw., 2005.

http://arxiv.org/abs/1701.07782v1

20170126172706

Beyond the Standard Model: Charting Fundamental Interactions via Lattice Simulations

hep-lat

§ INTRODUCTION In 2012 the ATLAS and CMS experiments announced the discovery of a new particle, the Higgs boson, almost fifty years after its existence was first postulated. Since then, the LHC experiments are providing precise measurements of the properties of the newly discovered particle with the aim to establish its true nature, while at the same time searching for hints of new beyond the SM physics. While no signs of new physics have yet been found, the properties of the Higgs particle are being explored in great detail. The favored quantum numbers for the new particle are the ones predicted by the SM, i.e. a CP-even, spin-0 scalar state. The mass of the Higgs boson is measured to be: m_H=125.09± 0.21 (stat)±0.11 (syst) GeV. The couplings of the new state to other SM fermions and vector bosons are also being measured precisely <cit.>. Several tests of the consistency of the Higgs couplings with the SM can be performed, based on different assumptions, showing no significant deviations from the predictions of the SM. fig:higgscouplings from <cit.> shows an example of such analysis which assumes all couplings to fermions and weak vector bosons rescaled by a universal factors κ_F and κ_V respectively. Given the current experimental evidence of a light scalar particle with couplings compatible with the SM, it is tempting to conclude that its last missing piece has been found and the SM is valid up to the Plank scale. While attractive, there are several theoretical and experimental reasons which indicate the SM is incomplete. On the experimental side, the SM does not account for the neutrino masses and the existence of dark matter nor does it provide an explanation for the origin of the matter-antimatter asymmetry of the Universe. In addition some tensions exist between SM predictions and a few experimental measures such as the anomalous magnetic moment of the muon or the proton radius. From a more theoretical perspective, the Higgs sector of SM is by many regarded as a convenient parametrization rather than a fundamental theory of electroweak symmetry breaking. Moreover, the elementary scalar nature of the Higgs field comes with an intrinsic instability of its mass under radiative corrections, giving rise to the hierarchy problem. The two most solid proposals for resolving the Higgs naturalness problem are supersymmetry and compositeness. Supersymmetric models allow to maintain the perturbativity of the SM, at the expense of doubling the field content of the SM. Supersymmetric models are the main target for searches of new physics by the LHC experiments, but no evidence has been found for the existence of supersymmetric particles yet. The second possibility is that the Higgs boson is a composite bound state of a new strong force. This solves the naturalness problem of the Higgs mass in the same way as the mass of the pion is natural in QCD: the radiative corrections to the pion mass are screened at scales of the order of Λ_QCD as, at those scales, the pion “dissolves” into its UV degrees of freedom, quarks and gluons. The composite nature of the Higgs is a more economical solution in terms of new fields, but it requires the presence of a new strongly-coupled sector in the model. Here I will focus on this second possibility and, in particular, models of pseudo-Nambu-Goldstone boson (pNGB) Higgs or walking Technicolor (WTC) models. Lattice studies provide a systematic and non-perturbative method to understand the new strong dynamics needed for composite Higgs models and can therefore provide valuable input for model building and the experimental searches at the LHC. I will summarize the theoretical framework for composite Higgs models in sect:composite, and review the recent efforts of the lattice community presented at this conference in sect:cwindow (conformal window and the search for infrared conformal models with large anomalous dimensions), sect:scalars (searches for walking Technicolor models with light scalars) and sect:pngbhiggs (non-perturbative studies of pNBG models). Although composite Higgs models are the primary focus of the lattice BSM community, many other topics have also been presented which I will not discuss here, such as supersymmetry <cit.>, extradimensions <cit.>, gauge/gravity duality <cit.> and asymptotically safe gauge-Yukawa theories <cit.>. For a review of recent results for dark matter models from lattice simulations see <cit.>. § COMPOSITE HIGGS MODELS Within the SM, electroweak symmetry breaking (EWSB) is assumed but it has no dynamical origin. The composite Higgs idea is an interesting realization of EWSB via a new strong dynamics. Composite Higgs models are built starting from the SM by removing the elementary Higgs field and replacing it with a new strongly interacting sector. This new strong sector is required to break electroweak symmetry dynamically, to give the correct mass to the W and Z bosons and to feature a composite scalar Higgs particle that mimics the SM one. If we require the new strong sector to be UV complete, the simplest realizations are based on non-abelian gauge theories with only fermionic matter. This new strong sector will possess a full spectrum of resonances, some of which could also be relevant in the context of dark matter models. The scale of the new strong interaction is set at the electroweak scale by the requirement that the W and Z bosons should acquire the experimentally observed mass. In order to generate masses for the SM fermions, additional interactions are required in any realistic model. These new interactions can be included in the model as effective operators of the form of four-fermion operators, stemming from some unspecified UV dynamics at a much high energy scale. A cartoon of the structure of a generic composite Higgs model is sketched in fig:chmodel, where L_SM-Higgs is the Lagrangian describing the SM without the Higgs field, L_SD the new strong sector, and L_int is the effective Lagrangian describing the interactions required to generate SM fermion masses. At the electroweak scale, the composite Higgs model is constrained by experimental data to closely resemble the SM, but it will feature additional new resonance states, which are composites from the new strong sector. To avoid conflict with electroweak precision data, it is necessary that a mass gap separates the Higgs resonance from the other resonances of the strong sector. It is worth to stress that in lattice simulations only the new strong sector in isolation is studied, while in any realistic model the SM and other additional interactions will affect the dynamics of the new strong sector. In particular the composite Higgs properties are expected to be affected significantly by these interactions (see below), while heavier resonances should be less affected. When comparing lattice simulations to experiments, it is therefore important to consider only observables which only depend on the strong sector or to estimate the corrections from the other interactions. The two most interesting limits of composite Higgs models are Walking Technicolor and pNBG Higgs models. In such models the little hierarchy between the Higgs mass and the other resonances of the strong model is explained by an extra approximate global symmetry and the Higgs boson being the associated pNBG. This extra symmetry is a global flavor symmetry in the case of pNBG Higgs models, or an approximate scale invariance symmetry in the case of WTC. The main features of WTC and pNBG models are summarized below. §.§ Walking Technicolor A time-honoured idea for dynamical electroweak symmetry breaking is walking techinicolor <cit.>. This idea was recently revived by extending the original framework by considering an enlarged theory space in the choice of the gauge group, number of flavors, and fermion representation <cit.> which triggered considerable new interest in the lattice community, as non-perturbative methods are required to make precise statements about the properties of these strongly coupled models. In recent years, most of the effort of the lattice community in BSM physics has been devoted to the study of the so-called “conformal window”, motivated by WTC models. In fact a number of four dimensional UV complete models are readily available to explore. I show in fig:chart a chart of the “theory space” with the models of strong dynamics presented at the Lattice 2016 conference. The exploration of novel strong dynamics, different than QCD, is a major ongoing effort of the BSM lattice community, and many results are available for a number of interesting models. Technicolor models feature a new strong sector typically based on a SU(N) gauge group, but Sp(2N), SO(N) and even exceptional simple Lie groups have also been considered <cit.>, and n_f massless techni-fermions in a given representation of the gauge group[WTC models with fermions in two different representations of the gauge group also exist <cit.> but they will not be addressed here.]. It is required that the new strong dynamics features dynamical symmetry breaking leading to at least three Nambu-Goldstone boson and the formation of a techni-fermion condensate. When electroweak interactions are introduced, the techni-fermion condensate breaks electroweak symmetry and the W and Z bosons acquire the correct mass, provided that the scale of the new strong force is chosen so that the techni-pion decay constant is equal to the electroweak vev, F_π≃ 246 GeV. A minimal realization is obtained considering n_f=2 techni-fermions in a complex representation of the gauge group, such as the fundamental of SU(3) leading to a scaled-up version of QCD, whose pattern of spontaneous chiral symmetry breaking is SU(2)×SU(2)/SU(2) which as the minimal number of NG bosons required for a TC model. For a larger number of techni-fermions, or for real or pseudoreal representations, TC models will in general lead to additional NG bosons in the physical spectrum. While TC models provide a natural explanation of the EW scale, there are severe constraints for a realistic TC model. The TC composite Higgs particle is identified as the lightest scalar excitation of the condensate, which has to be a narrow, light resonance (unlike the equivalent f_0(500) state in QCD). Moreover the couplings between the composite Higgs particle and the SM fermions must be SM-like. Electroweak precision data, such as the Peskin-Takeuchi S and T parameters impose strict constraints on the model, which are not easy to satisfy. Finally to generate SM fermion masses, additional interactions are needed in the form of extended TC interactions (ETC), which generically also generate flavor changing neutral current (FCNC) among SM particles which are experimentally very small. The “walking” TC idea was introduced to alleviate the problems of TC. Let's start by considering the problem of SM fermion mass generation. ETC interactions at some large UV scale Λ_ETC will produce effective couplings among the techni-fermions and the SM fermions of three kinds: A_abQT^a QψT^b ψ/Λ_ETC^2+B_abQT^a QQT^b Q/Λ_ETC^2+C_abψT^a ψψT^b ψ/Λ_ETC^2 , where we have schematically indicated with Q a generic techni-fermion, ψ a SM fermion, T the ETC gauge group generators and A, B, C are adimensional coefficients of order assumed generically of order 1. The last term generates FCNC, and the experimental constraints from BB̅ and KK̅ mixings can be used to set a lower limit on Λ_ETC. Assuming no additional structure for the ETC interactions and coefficients C≃ 1 one obtains a limit Λ_ETC≳ 10^3 TeV for the second generation of SM quarks<cit.>. After techni-fermions have condensed, the A terms in eq:etc lead to mass terms for the SM fermions: m_q≃1/Λ_ETC^2⟨Q̅ Q⟩_ETC . From this expression it is clear that in order to generate large hierarchies in the quark masses, the generation of the four-fermion operators for different SM flavors cannot just happen at one single ETC scale. Rather one should assume a different scale Λ_ETC for each SM quark family. From the FCNC constraint above, the charm and strange quark mass should be generated at a scale of 10^3 TeV, if no further suppression mechanism is present in the ETC model. The techni-quark condensate in eq:qmass is evaluated at the ETC scale and it is related to the condensate at the TC scale by the renormalization group equation: ⟨Q̅ Q⟩_ETC=⟨Q̅ Q⟩_TCexp(∫_Λ_TC^Λ_ETCγ(μ)dln(μ)), where γ is the anomalous dimension of the techni-quark mass operator. Assuming that the TC model is asymptotically free just above the TC ∼ 1 TeV, i.e. that γ∼ 0 above Λ_TC, and estimating by simple dimensional analysis ⟨Q̅ Q⟩_TC∼Λ_TC^3, one obtains m_q∼Λ_TC(Λ_TC/Λ_ETC)^2. This results in a mass of ∼ 1 MeV for the second generation of SM quarks, which is clearly too small. To resolve this tension the mechanism of walking TC was proposed. Instead of assuming the TC dynamics to be QCD-like as done above, one requires the model have an approximate conformal symmetry, so that the anomalous dimension γ(μ) remains almost constant between the TC and ETC scales and equal to γ^*. Under this assumption the SM fermion mass can be estimated as m_q∼Λ_TC(Λ_TC/Λ_ETC)^2-γ^*. For large values of γ^*∼1 this results in a considerable enhancement of the generated SM fermion mass ∼ 1 GeV. A large γ^*∼1 requires a strongly coupled (near) conformal dynamics. Whether or not this can be realized is a very interesting question which can in principle be answered by lattice simulations. I will discuss below in sect:cwindow the current status for lattice searches of IR conformal models with large mass anomalous dimensions. The B terms in eq:etc, four-fermion interactions among techni-quarks, will affect the strong TC dynamics[One could also use an infrared conformal TC model for the new strong sector, in which case four-fermions interactions could change the anomalous dimensions and also generate walking <cit.>], and they will e.g. generate masses for the techni-pions. Besides SM fermion mass generation, the Peskin-Takeuchi S-parameter also imposes a strict constraint on the TC dynamics. It has been suggested that the value of the S-parameter, normalized to the number of new electroweak doublets, is reduced in near-conformal models of walking TC <cit.>. Evidence from non-perturbative lattice simulations is still rather limited, see <cit.> for a recent result for the SU(3) gauge group with N_f=2, 6, 8 flavors. This preliminary evidence favors the expected reduction of the S-parameter for walking TC models, although a precise determination is hindered by difficulties related to the chiral extrapolation in near-conformal models. Walking TC models are also challenged by the requirement of a light composite Higgs scalar, with the correct couplings to the SM fermions. In TC models the composite Higgs is the lightest isospin-0 scalar composite of techni-quarks. In a strongly coupled model this composite state will also contain a techni-glue component. The observation that the QCD analogue, the f_0(500) resonance, is a quite broad resonance, has driven the belief that light composite states cannot exist in a TC model. However this naive expectation does not take into account that any realistic WTC model should: 1) to take into account interactions with the SM particles; and 2) feature a near conformal dynamics quite different than QCD. Interactions with SM particles will change the mass of the composite Higgs state. For a realistic model, assuming SM-like couplings to gauge bosons and SM fermions, the corrections are large and dominated by the negative top quark loop contribution<cit.>, so that a mass as large as ∼1 TeV for the TC Higgs in isolation is not excluded. If the mass is reduced significantly by these interactions, the composite Higgs state would become narrow due to kinematics. The assumption of SM-like coupling is a delicate issue and it is unclear that it should hold for a generic WTC model. In <cit.> it was argued that couplings of the TC Higgs to the W and Z bosons are SM-like by comparison with the σππ effective coupling in QCD. A more detailed analysis in <cit.> has confirmed this expectation and found the effective coupling in QCD to be in surprisingly good agreement with the corresponding SM coupling. Couplings to SM fermions are more model dependent. Nonetheless if the SM fermion mass is generated via ETC interactions, the same four-fermion interactions will generate effective Yukawa couplings for the composite TC Higgs, which are then expected to be proportional to the SM fermion masses <cit.>. Another interesting possibility has been considered which leads to a light composite Higgs with correct couplings to the SM particles, namely the possibility of a dilaton-Higgs <cit.>. A dilaton-Higgs would be light, as the pNGB associated to an approximate scale invariance of the model, and with SM-like couplings, as the dilatonic states couples to the trace of the energy-momentum tensor, if the scale at which scale invariance is broken is the electroweak scale. Such a dilaton-Higgs would be difficult to distinguish from the SM Higgs <cit.>. If a near-conformal WTC model can produce such a light dilatonic scalar particle is still an open question, which lattice simulations are trying to address. In recent years the lattice community has provided evidence for the existence of light composite scalar states in strongly coupled gauge theory, pointing against the common belief that such states cannot exist. I will summarize below in sect:scalars the status of lattice searches for light composite scalars. §.§ pNGB Composite Higgs pNGB composite Higgs models interpolate between TC models and the SM with a fundamental Higgs. As in TC, pNGB composite Higgs models start by assuming a new strongly interacting sector with a global symmetry group G_F that is spontaneously broken to a subgroup H_F. Unlike in TC, the Higgs particle is identified with a Nambu-Goldstone boson<cit.> of the new strongly interacting sector. This naturally explains why the composite Higgs is light, and, since based on a similar mechanism, one can easily recover the correct SM-like coupling between the composite Higgs and the electroweak gauge bosons. For a viable realization of this scenario, as long as only electroweak symmetry is concerned, the pattern of symmetry breaking G_F/H_F should be such that the custodial symmetry of the SM is preserved, i.e. H_F⊃ G_ cust=SU(2)_L×SU(2)_R, and that one of the Nambu-Goldstone bosons has the correct quantum numbers for the Higgs particle, i.e. belong to the irrep (2,2) of G_ cust. If we consider UV-complete models in four dimensions featuring fermionic matter, the three minimal cosets are SU(4)×SU(4)/SU(4) for fermions in a complex representation of the gauge group, SU(4)/Sp(4) for fermions in a pseudoreal representation and SU(5)/SO(5) for fermions in a real representation. Models in all three of the minimal cosets have been considered at this Lattice conference. Models based on the pattern SU(4)/Sp(4) contain only five NGB, i.e. the three required to generate masses for the W and Z bosons, one composite Higgs scalar, and one additional NGB. This pattern of symmetry breaking is realized by a SU(2) technicolor gauge group with N_f=2 Dirac fermions in the fundamental representation, which therefore is the minimal realization of a UV-complete pNGB composite Higgs model[The SU(4)/Sp(4) is equivalent to the SO(6)/SO(5) coset, sometimes called next to minimal coset, which has been studied in detail via an effective sigma model description. The so-called minimal coset SO(5)/SO(4) lacks a four dimensional UV completion.]. The familiar-looking SU(4)×SU(4)/SU(4) coset can be realized for an SU(3) gauge group and four Dirac fermions in the fundamental representation, while the SU(5)/SO(5) coset can emerge from an SU(4) gauge group with five Majorana fermions in the two-index antisymmetric representation. Electroweak interactions will break the global symmetry of the strong sector and generate a mass for the composite Higgs, in a similar way as electromagnetic interactions generate a mass for charged pions in massless QCD. This potential however will not generate a for the composite Higgs field so that the minimum of the effective potential occurs at ⟨ h⟩=0 and electroweak symmetry remains unbroken. To break electroweak symmetry interactions with SM fermions are needed, the dominant contribution coming from the top quark. Possibly other sources of explicit symmetry breaking for the global flavor symmetry of the new strong sector can be considered. Interaction with the SM top quark will change the composite Higgs mass and generate a non-zero ⟨ h⟩ and break electroweak symmetry. The breaking can be parametrized by an angle θ so that: v/F_π=sinθ=sin⟨ h⟩/F_π, where v=246 GeV and F_π is the NGB decay constant of the strong sector. Other composite strong resonances are expected at a scale ∼ 4π F_π in a QCD-like dynamics. Therefore, while interactions with gauge bosons W and Z tend to align the vacuum angle θ towards zero, i.e. the NGB composite Higgs direction, SM fermions interactions have the opposite effect of pushing towards large alignment angles in the Technicolor direction θ≃π/2. In the limit θ→ 0, the model naturally has couplings to the electroweak gauge boson identical to the ones of the SM: g_VVh=g_VVh^SMcosθ and g_VVhh=g_VVhh^SMcos 2θ. Realistic models require a light Higgs mass and SM-like couplings with the EW bosons, which implies rather small θ angles. There is therefore a degree of fine tuning required between EW gauge bosons and SM fermions contributions, which are completely different in nature. Coupling to SM fermions can be introduced either as in ETC, i.e. quadratic in the SM fermions 𝒪_Sψψ/Λ_UV^d-1, or in the partial compositeness way, i.e. linearly 𝒪_F ψ/Λ_UV^d-5/2, where d is the dimension of the composite scalar or fermionic operator 𝒪. As before, these interactions will generically also produce FCNC. To evade experimental constraints one then consider models where Λ_UV can be pushed to a high scale, while still generating the required mass. In the extreme case of vanishing exponent for Λ_UV, one can consider the case of a fermion bilinear for 𝒪_S as in ETC which implies an anomalous dimension of γ=2 and a three-quark "baryon" fermionic operator for 𝒪_F also implying γ=2 for its anomalous dimension[Both values are within the unitarity limit for conformal field theories, the limit for a scalar being 2 and for a fermion being 3.]. Therefore, as in TC, large anomalous dimensions are naturally advocated in models of partial compositeness. Whether such models of strong dynamics exists or not is still an open question. Recently it was realized that such large anomalous dimensions for baryon operators in the simplest realization of partial compositeness, featuring only fermions in one representation of the gauge group, are very unlikely <cit.>. Couplings of the composite Higgs to SM fermions are model dependent, however one can generically recover MS-like couplings in the limit θ→ 0: g_hf̅f=g_hf̅f^SM(1 + c θ^2 + …). The S-parameter are also model dependent but it can naively be estimated, via the so-called "zertoh" Weinberg sum rule, to be S∝ (v/m_ρ)^2∝sin^2θ, which shows that in the limit of small θ the contributions to the S-parameter are small, as the resonances from the strong sector become heavy. The experimental constraint on the S-parameter poses one of the most stringent limit on the size of θ≲ 0.2-0.3. Models of partial compositeness require the composite fermions operators, which are the partners of SM quarks, to carry QCD color. This in turn implies that the global symmetry of the new strong sector should be enlarged to allow the embedding of SU(3)_c. Since models with partners for all the SM fermions are difficult to construct, a common approach is to use partial compositeness for the top sector only. A classification of possible models featuring only fermionic matter in two different representation of the gauge group was given in <cit.>, while a solution based on SU(3) hypercolor group and n_f>6 fermions in the fundamental representation can be found in <cit.>. Among models with fermions in two different representations, an interesting one, obtained by extending the SU(5)/SO(5) coset, is based an SU(4) gauge group with five Majorana fermions in the two-index antisymmetric representation plus three Dirac fermions in the fundamental representation <cit.>. Finally, recently models of partial compositeness based on a strong sector with both fermions and (hyper)colored scalars have been proposed <cit.>, in which a mass for all SM fermions can be generated without requiring large anomalous dimensions. These can be seen as effective description for any partial compositeness model with large anomalous dimensions and in <cit.> it was shown that such models can be well-defined with small quartic scalar couplings up to the Plank scale. §.§ Non-perturbative questions for the lattice Lattice gauge theory simulations are a unique tools to understand strong dynamics, and they can be used to address a number of relevant questions about the non-perturbative dynamics of the strongly interacting composite Higgs models, such as: * Where is the exact location of the conformal window (see fig:chart)? i.e. which asymtotically free models have an IR fixed point? * For models inside the conformal window, what is the (fixed point) anomalous dimension of the mass or the anomalous dimension of baryonic operators? Is there any model with large anomalous dimensions? * For models outside the conformal window, can composite scalar states analogue to the σ resonance be light and narrow in a strongly coupled dynamics? If so, is such state a dilaton? What are the couplings of such composite light scalar to NGBs? * More generally, how does the spectrum of a strongly coupled model changes with the number of fermion field or when changing the number of colors or fermion representation? e.g. what is the ratio m_ρ/F_π or m_σ/F_π ? * How big is the S-parameter for models just below the conformal window? It is the primary goal of the lattice BSM effort to answer such questions, and a number of interesting results for the models in fig:chart is already available. I will summarize below the new lattice results presented at this conference. § CONFORMAL WINDOW AND ANOMALOUS DIMENSIONS The precise location of the conformal window is a primary objective of lattice BSM studies. The focus has been on models based on SU(2) and SU(3) gauge groups, with many fermions in the fundamental representation, but also an interesting update for models based on SU(2) with adjoint fermions has been presented at this conference. I summarize below the main new results which show that models with large anomalous dimensions inside the CW have not yet been found despite the large number of models investigated so far. §.§ SU(3) with fundamental fermions The exact size of the conformal window in this case is still debated. There is a general consensus that n_f=6 lies outside the conformal window and most groups agree that n_f=8 is outside the CW[See however e.g. <cit.> for an alternative point of view.] (studies on the spectrum of the n_f=8 model will be reported below in sect:scalars). New results have been presented for the cases of 10 and 12 fermions. The n_f=12 case has been studied extensively by several groups by a variety of different methods <cit.>. While many of these studies indicate that the model is inside the conformal window and the anomalous dimension of the mass is small γ^*∼ 0.2-0.3, new evidence was presented in <cit.>, and reported at this conference, against the existence of the IR fixed point at the location reported by previous lattice studies. This new study is a high-precision measurement of the non-perturbative discrete β-function of the model in the gradient flow coupling and finite volume scheme <cit.>. The scheme was chosen to match, in the continuum, the one used in previous studies of the n_f=12 model so that the results can be easily compared. By using much larger lattice volumes up to L/a=56, and a precise tuning of the bare coupling, the quality of the continuum extrapolation was dramatically improved over previous state-of-the-art determinations. In fig:nogradi we show an example of continuum extrapolation (left panel) and the final result of <cit.> (right panel) for the discrete β-function. The new results are clearly incompatible with the existence of an IR fixed point where previously reported. The possibility that the n_f=12 model is IR conformal with a fixed point at a stronger coupling is also clearly not excluded. This study proves the necessity of using very large lattice volumes in the vicinity of a candidate IR fixed point. In the case of the determination of a non-perturbative β-function via a step-scaling procedure, the large lattice volume is then required to reach the correct continuum limit and avoid lattice artifacts. For the n_f=10 model, an update of <cit.> was presented at this conference <cit.> which includes a larger L/a=32 lattice volume. The study measure the β-function of the model in the finite volume gradient flow coupling scheme, similar to the n_f=12 studies above, but it uses an optimal domain-wall fermion action <cit.>. The use of this particular action with good chiral properties seems to allow the use of rather small lattices in a leading order continuum extrapolation with only linear (a/L)^2 terms. Using then four steps, the continuum extrapolation is rather well constrained as shown in fig:chiu. T he result clearly points to the existence of an IR fixed point, although a full estimate of systematic errors due to e.g. different choice of discretization of the gradient flow observable, to the choice of interpolation function used for the coupling, or to the choice of continuum extrapolation has not yet been carried out. If confirmed, this results will also settle the n_f=12 case, as it would be very difficult to imagine it is not IR conformal if the n_f=10 model is. §.§ SU(2) with fundamental fermions In the case of two colors, the location of the CW seems more established, as there is agreement among various groups that models with n_f≥ 8 lie inside the conformal window whereas models with n_f≤ 4 lie outside <cit.>. The case n_f=6 is more debated and still not settled <cit.>. A new study was presented at this conference <cit.> of the non-perturbative β-function in the SF gradient flow coupling scheme with an HEX-smeared Wilson-clover action which allows to reach rather strong couplings. The authors perform a careful study of the systematic errors involved and conclude in favor of the existence of an IR fixed point at a rather large value of the coupling g^2∼ 15, see fig:hel for an example of their final result. The same authors also investigate the mass anomalous dimension finding a value at the fixed point of γ∼ 0.3. Given the experience with the SU(3) n_f=12 case, large volumes will be required to check these results in the vicinity of the observed fixed point and to confirm its existence. In fact this is the region where the raw data shows the most sensitivity to volume, as expected, and can therefore affect the continuum extrapolation in a step-scaling analysis for the extraction of the continuum β-function. §.§ SU(2) with adjoint fermions Previous effort focused mainly on the model with n_f=2 adjoint Dirac fermions <cit.>. The model was found to be IR conformal with an anomalous dimension of the mass[Different methods for the determination of γ do not fully agree with each other, hinting to residual systematic errors not fully under control.] γ∼ 0.2-0.4. A previous study of n_f=1 Dirac fermion[The global symmetry in this case is SU(2) which is too small for a model for dynamical EW symmetry breaking.] <cit.> presented evidence hinting at the model being inside the conformal window with a rather large mass anomalous dimension γ∼ 1. At this lattice conference this study was extended to cover the full range of allowable number of adjoint fermions n_f=1/2, 1, 3/2, 2 <cit.>, where half-integer numbers of Dirac fermions represent odd numbers of Weyl fermions. The case of n_f=1/2 massless fermion corresponds to N=1 supersymmetric Yang-Mills theory. In particular, the authors of <cit.> presents new results for the n_f=2, 3/2 models. The measured observables include the masses of triplet and singlet mesons, the 0^++ glueball mass, the mass of the lightest composite spin-1/2 fermion-gluon state, and the mass anomalous dimensions at two different values of the lattice spacing. By studying ratios of masses for different states as a function of the bare quark mass in the chiral limit, evidence was presented that the n_f=2, 3/2 models lie inside the conformal window. The mass anomalous dimension has been determined both from the spectral quantities and from the mode number of the Dirac operator. The study confirmed the presence of very large finite size effects when approaching the chiral limit, as pointed out in <cit.>, which makes numerical simulation close to the chiral limit extremely expensive. By comparing their results at two different values of the lattice spacing, a residual dependence on the lattice cutoff was observed, in particular for the measured value of the mass anomalous dimension which seems to decrease significantly at smaller lattice spacing. For example at n_f=1 the (preliminary) value of γ drops to 0.75 on the finer lattice spacing used in this work. Such residual dependence is observed for all models inside the CW, i.e. n_f=1,3/2, 2, and it might be due to residual finite size effects, which drive the system away from the IR conformal fixed point. More studies are needed to determine the precise value of γ for these models. § LIGHT COMPOSITE SCALARS Several interesting examples of strongly coupled models with light scalars have been presented at this conference. These are based on SU(3) gauge group with either n_f=4+8 or n_f=8 fundamental fermions, or with n_f=2 fermions in the 2-index symmetric representation (sextet). §.§ SU(3) with n_f=4l+8h fundamental fermions Based on the idea that walking can be "generated" by continuously deforming a model with IR fixed point, one can introduce a small fermion mass to study the spectrum of the model close to an IR fixed point <cit.> in the "walking" regime. The model considered here is build on the assumption that the SU(3) model with n_f=12 fundamental fermion has an IR fixed point (see sect:su3f). One could then introduce a mass m_h for eight of the twelve fermions so that in the limit of large m_h the model is reduced to the n_f=4 model which is then similar to ordinary QCD. For intermediate masses m_h, corresponding to a walking regime, the spectrum of the model can be studied. In actual lattice simulations, it is also necessary to introduce a mass m_l for the four light fermion, so that an extrapolation to zero m_l is required for each m_h. Although not a realistic model of dynamical EW symmetry breaking, this model can be considered as a prototype for models of walking TC and for models of pNGB composite Higgs <cit.>, in the latter case based on the coset SU(4)×SU(4)/SU(4)[As noted in <cit.>, in this case a simple realistic model which takes into account partial compositeness effectively reduces the coset to SU(4)/Sp(4).]. Extending their previous work <cit.>, new results for the spectrum and scaling properties of this model were presented at this conference <cit.>. The spectrum for the light-light mesons as a function of the heavy m_h and light fermion mass m_l is shown in fig:rebbi, where meson masses are normalized by the value of the pseudoscalar decay constant F_π(m_l,m_h). In the chiral limit m_l→ 0 the ratios M_H/F_π seem to depend only weakly on the value of m_h, with the exception of the scalar 0^++ state which seems to become much lighter as m_h is reduced, i.e. when entering the "walking regime" of the model. In fact for a m_h≤ 0.06 the scalar state becomes degenerate, within errors, with would be NGBs of the model, which is a common feature observed in numerical simulations of models with light scalar states. This feature makes it difficult to extrapolate the numerical data for the NBGs and scalar sector to the chiral limit, as the ordinary chiral perturbation theory is not applicable in the regime where numerical simulations can be performed, and one should consider how to include light scalar states in the effective description of the model (see also sect:nf8 below). It is therefore still unclear precisely how light such scalar states are. The light-heavy and heavy-heavy meson states were also investigated in <cit.>. Assuming its existence, close enough to the IR fixed point for the n_f=12 model at m_l=m_h=0, hyperscaling relations hold for the masses of hadrons and decay constants, and their ratios. In <cit.> evidence is provided for hyperscaling, which supports the hypothesis of an IR fixed point in the n_f=12 system. §.§ SU(3) with n_f=8 fundamental fermions The results from the previous section point to the possibility of light scalars for walking models which lie just below the onset of the CW. As a direct test of this hypothesis, different groups have been investigating the spectrum of the SU(3) model with n_f=8 fundamental fermions <cit.>. At this conference both the LatKMI and the LSD collaboration have presented new results for the spectrum of this model. The LatKMI collaboration presented new results for spectrum of the n_f=8 and n_f=4 models <cit.>, shown in fig:aoki, in particular for the flavor singlet mesons σ and η'. In the n_f=8 model, the σ meson appears to be degenerate with the pion over the whole range of fermion masses explored, while for the n_f=4 case it remains heavier than pion at the most chiral point investigated. This hints at the possibility of a much lighter σ resonance than in QCD. However if one considers the ratio m_σ/m_ρ, or the ratio m_σ/F_π, at the most chiral point in fig:aoki, this is unchanged between the n_f=4 and 8 models, and it is also similar to the QCD case. At face value, this constrasts our previous indication and it would indicate that the σ resonance does not become lighter respect to the strong scale at the onset of the conformal window. Similar results were presented by the LSD collaboration <cit.> at a smaller value of the pion mass. The mass spectrum of the model, as obtained from the LSD collaboration, is shown in the left panel of fig:fleming, while in the right panel shows the ratios m_H/F_π. Very large volumes up to 64^3× 128 were required in order to keep systematic errors under control. In agreement with the results from the LatKMI collaboration, the mass of the σ resonance is found to be degenerate with the would be NG boson of the model and the ratios m_H/F_π are very similar to the QCD values, indicating only a very weak dependence of the ratios on the number of flavors n_f. From the right panel of fig:fleming one should notice that both the would be NG boson and the light scalar resonance mass in units of F_π(m_q) show only a very weak dependence on the fermion mass m_q in the region explored. If the n_f=8 is not inside the conformal window, then a sharp decrease of m_π/F_π is expected close to the chiral limit, while the σ resonance should remain massive. It is therefore crucial to be able to extrapolate the current results to the chiral limit to establish if the scalar σ resonance becomes much lighter for walking models. The difficulty stems from the fact that the usual chiral perturbation theory cannot be trusted in the presence of a scalar state as light as the pion. One attempt to develop a more appropriate effective description which takes into account the presence of a light scalar states was prensented at this conference <cit.> based on previous results by the same authors in <cit.>. In this new effective model, the authors consider the case of dilaton scalar state and develop a systematic expansion in the three small parameters m_q, 1/N, and n̂_f-n̂_f^*, assumming that scale invariance is recovered in the limit m_q→0, large-N Veneziano limit, and n̂_f/n̂_f^*→1^-, where n̂_f=n_f/N and n̂_f^* is the critical number of flavors, in the Veneziano large-N limit, for which the conformal window opens. Results for m_π, m_σ and the fermion condensate were obtained at leading order, that predict a distinctive behavior as a function of the fermion mass which can in principle be compared to lattice data and it could therefore be a useful analytic tool to test the dilaton-Higgs scenario in walking TC models. However one must bear in mind that the validity of the effective theory is limited to small values of the three expansion paramters. Since a naive estimate indicates a value of n̂_f^*∼ 4, in the present case of n_f=8 this corresponds to n̂_f^*-n̂_f≃ 1.3, which might be too large for the expansion to hold. Moreover, given the use of the Veneziano limit, the effective model cannot be used without modifications for the case of two-index representations. Other approaches to an effective description of the π-σ system also exist such as <cit.>. §.§ SU(3) with n_f=2 sextet fermions Another very interesting walking TC model is based on SU(3) with n_f=2 fermions in the two index symmetric representation (the sextet representation). The model has the three NGBs, i.e. the minimal number required for a TC model, but from the higher dimensional representation of the fermions, one expects the model to be walking <cit.>, with possible light scalar states. The spectrum of this sextet model has been studied in detail on the lattice with staggered fermions <cit.> and more recently an investigation with Wilson fermions also began <cit.>. Updates for both lattice formulations have been presented at this conference <cit.>. I report in fig:kuti the spectrum of the model as obtained by staggered fermions lattice simulations, at two different lattice spacings <cit.>. The qualitative features of the spectrum are similar to the other walking models described above. The spectrum features a light σ resonance over the entire range of light quark masses explored, in fact lighter that the would be NGB, which should eventually become massless in the chiral limit for a chirally broken model. There is still some controversy about the sextet model being inside or outside the conformal window. From an eye inspection of fig:kuti, one might be tempted to conclude that the model is in fact IR conformal and inside the conformal window, however a more detailed analysis has led the authors of <cit.> to conclude that this is not the case as the data do not follow the hyperscaling predictions. On the other hand the data seems to fits well with the prediction of rooted staggered chiral perturbation theory, even if the use of such effective model is not justified, given the presence of the light scalar state. As probing the model at even lighter masses in the p-regime would be prohibitively expensive, the authors of <cit.> are moving to use more sophisticated analysis methods involving the cross-over regime from the p to the ϵ-regime, the use of random matrix theory and the use of effective models which take into account the light scalar particle in the spectrum. The spectrum of the sextet model is also being investigated with Wilson fermions. Similarly to the staggered fermion case, the mass spectrum can be fitted to a prediction from Wilson chiral perturbation theory, although in this case one cannot also exclude the possibility of hyperscaling from an IR fixed point <cit.>. There is however a striking difference in the between the spectra from staggered and Wilson fermions: in the latter case, the vector resonance never appears to become much heavier that the pion in the weak coupling phase. To better understand the behavior of the model, a new study of the full phase structure of the lattice model with Wilson fermions was presented at this conference <cit.>, see fig:hansen. A phase at strong coupling was identified, separated from the weak coupling phase by a crossover. At strong coupling there is a first order transition as the fermion mass is reduced, which becomes a continuous transition in the weak coupling phase corresponding to the line of vanishing PCAC mass. The behavior of several quantities, which include the mass spectrum and the scale-setting observables w_0 and t_0, was studied both in the weak coupling and strong coupling phase. The results show a sharp change in the qualitative behavior of the measured quantities, see e.g. the right panel of fig:hansen. While at strong coupling, the observations are compatible with a chirally broken model as expected, data in the weak coupling phase do not show any clear indications of spontaneous chiral symmetry breaking. The question of the infrared conformality of this model will require the use of more data at weak coupling on the spectrum of the model, at several lattice spacings, to show consistently the presence or not of a critical behavior in the chiral limit. § PNGB HIGGS MODELS Several interesting pNGB Higgs models have been considered at this conference, which cover the three minimal cosets discussed in sect:ngh. The case of SU(4)×SU(4)/SU(4) can be realized with fundamental fermions of SU(3), and it has been discussed above. Here we discuss the contributions related to the other two cases. §.§ SU(4) with sextet and fundamental fermions The case of the SU(5)/SO(5) coset can be realized with five Majorana fermions in two-index antisymmetric (sextet) represetation of SU(4). This coset has also been suggested as the base for a model of partial compositeness with three additional Dirac fundamental fermions <cit.>. The odd number of fermions makes this model harder to study via lattice simulations. As a first step, the case of SU(4) with four Majorana in the two-index antisymmetric represetation and the case of SU(4) with two fundamental fermions plus quenched two-index antisymmetric fermions were studied <cit.>. For the latter case, the spectrum of the model was presented in <cit.>, which is shown in fig:jay. It is possible to have baryons which are composite of fermions in both representations, and the quenched study shows that these can become lighter than baryons made of fermion in one representation only. This could be interesting for model building of light top-partners. However this result in the quenched approximation is not conclusive and the full spectrum of the model should be considered with dynamical fermions. In <cit.> the radiative contributions from electroweak gauge bosons to the composite Higgs potential of the SU(4) model with four Majorana fermions in the sextet representation were considered. This calculation is similar to the electromagnetic contribution to the masses of the pions in QCD. Electroweak gauge boson generate a potential for the composite Higgs h of the form: V(h)=C_LR (3g^2+g'^2)(h/F_π)^2+𝒪(h^4) where the positive constant C_LR can be computed from a vacuum polarization function Π_LR. A test of the feasibility of the measure of C_LR on the lattice was presented in <cit.> with two different approaches found to be in good agreement. In the future this approach will be extended to the full model with fermions in two different representations. One should remember that potential generated by EW gauge bosons will not generate a vev for the higgs field and will not break electroweak symmetry, as C_LR>0. For this to occur, radiative corrections from the interactions with SM fermions, in particular the top quark, should be included. As discussed above in sect:ngh, for the composite pNGB Higgs scenario to be viable the Higgs vev should be small compared to the strong scale F_π, implying a small alignement angle θ, which requires some subtle cancellation in the effective Higgs potential between the electroweak contibutions and the SM fermion sector. §.§ SU(2) with n_f=2 fundamental fermions This model is the minimal realization of a composite pNGB model, requiring only two fundamental fermions of SU(2). The model can be used as the building block for a fully realistic composite Higgs model <cit.> compatible with the experimental constraints if, roughly, θ<0.2 (see <cit.> for details). The model can also be used to build a model of partial compositeness for all SM fermions <cit.>, which features hyper-colored scalars and it is free from Landau poles up to the Plank scale. In a different context, the same model has also been considered as a model of composite dark matter. Lattice simulations of this model are straightforward, requiring only two colors, two Dirac fermions and no high dimensional representations. The challenge is then to perform all the necessary extrapolations, i.e infinite volume, zero quark mass and continuum limit for the spectrum of the model, as this has not been done for any other BSM model studied so far. In <cit.> an update for the spectrum of the model was presented, based on <cit.>. Lattice simulations were performed at four different lattice spacings, for a number of quark masses at each lattice spacing, while keeping large enough volumes to reduce finite-volume effects as much as possible. The scale was set by using the w_0 observable <cit.> and the RI-MOM scheme <cit.> was used to measure the required non-perturbative renomalization costants. Continuum extrapolated results were obtained for F_π, the fermion condensate, and the lightest spin one and zero resonances, analogue to the QCD ρ, a_1, σ, η', a_0 resonances. A combined chiral and continuum extrapolation was used to extract the physically interested quantities. The left panel of fig:drach shows an example of such extrapolation for the ρ meson. The final spectrum for the model is summarized in the right panel of fig:drach, in units of F_π and compared to the QCD spectrum. Taken at face value, these results indicate a spectrum which is quite different from the QCD one, featuring heavier resonances which are beyond the present LHC constraints, even in the Technicolor limit of θ=π/2. In the pNGB limit, for sinθ<0.2, these resonances seem beyond the reach of LHC experiments. These results are still affected by large systematic errors, as shown in fig:drach, mainly due to the chiral and continuum extrapolations required to obtain phenomenological predictions. The accuracy of these results will be increased in the future. § CONCLUSIONS The lattice community is actively investigating interesting models for BSM physics. These non-perturbative studies complement the phenomenological approach by providing valuable information on the strongly coupled dynamics. The numerical studies of many such models present great numerical and theoretical challenges, most crucially stemming from the near conformal nature of the model requiring very large volumes and the presence of light scalars making the chiral extrapolation difficult. Nonetheless thanks to the continuous numerical effort and the development of new techniques and tools significant progress has been made in the last few years. Many results are already available for models based on the gauge groups SU(2) or SU(3) with fermions in the fundamental or higher representations. A great effort is undergoing to determine the precise location of the conformal window for two or three color models, and only a few borderline cases remaining elusive. Interesting models featuring light composite scalar states are being investigated in detail and many results for the spectrum of these WTC or pNGB Higgs models are already available. I wish to thank organizers of the Lattice 2016 conference for the kind hospitality. The work of CP is supported by the Danish National Research Foundation under grant number DNRF90 and by a Lundbeck Foundation Fellowship grant. 99 Khachatryan:2016vau G. Aad et al. [ATLAS and CMS Collaborations], JHEP 1608 (2016) 045 doi:10.1007/JHEP08(2016)045 [arXiv:1606.02266 [hep-ex]]. Kamata:2016 S. Kamata, PoS(LATTICE2016)210 giedt:2016 J. Giedt, S. Catterall, P. Damgaard and D. Schaich, PoS(LATTICE2016)209 schaich:2016 D. Schaich, PoS(LATTICE2016)221 august:2016 D. August, B. Wellegehausen and A. Wipf, PoS(LATTICE2016)234 joseph:2016 A. Joseph, PoS(LATTICE2016)220 giudice:2016 S. Ali, G. Bergner, H. Gerber, P. Giudice, I. Montvay, G. Münster and S. Piemonte, PoS(LATTICE2016)222 alberti:2016 M. Alberti, N. Irges, F. Knechtli and G. Moir, PoS(LATTICE2016)215 bennett:2016 E. Bennett, PoS(LATTICE2016)232 forini:2016 V. Forini, PoS(LATTICE2016)206 berkowitz:2016 E. Berkowitz, PoS(LATTICE2016)238 buy:2016 T. Buyukbese, PoS(LATTICE2016)233 bond:2016 A. Bond, PoS(LATTICE2016)208 rinaldi:2016 E. Rinaldi PoS(LATTICE2016)008 Weinberg:1979bn S. Weinberg, Phys. Rev. D 19 (1979) 1277. doi:10.1103/PhysRevD.19.1277 Susskind:1978ms L. Susskind, Phys. Rev. D 20 (1979) 2619. doi:10.1103/PhysRevD.20.2619 Dimopoulos:1979es S. Dimopoulos and L. Susskind, Nucl. Phys. B 155 (1979) 237. doi:10.1016/0550-3213(79)90364-X Eichten:1979ah E. Eichten and K. D. Lane, Phys. Lett. 90B (1980) 125. doi:10.1016/0370-2693(80)90065-9 Farhi:1980xs E. Farhi and L. Susskind, Phys. Rept. 74 (1981) 277. doi:10.1016/0370-1573(81)90173-3 Holdom:1984sk B. Holdom, Phys. Lett. 150B (1985) 301. doi:10.1016/0370-2693(85)91015-9 Yamawaki:1985zg K. Yamawaki, M. Bando and K. i. Matumoto, Phys. Rev. Lett. 56 (1986) 1335. doi:10.1103/PhysRevLett.56.1335 Bando:1986bg M. Bando, K. i. Matumoto and K. Yamawaki, Phys. Lett. B 178 (1986) 308. doi:10.1016/0370-2693(86)91516-9 Appelquist:1987fc T. Appelquist and L. C. R. Wijewardhana, Phys. Rev. D 36 (1987) 568. doi:10.1103/PhysRevD.36.568 Caswell:1974gg W. E. Caswell, Phys. Rev. Lett. 33 (1974) 244. doi:10.1103/PhysRevLett.33.244 Banks:1981nn T. Banks and A. Zaks, Nucl. Phys. B 196 (1982) 189. doi:10.1016/0550-3213(82)90035-9 Sannino:2004qp F. Sannino and K. Tuominen, Phys. Rev. D 71 (2005) 051901 doi:10.1103/PhysRevD.71.051901 [hep-ph/0405209]. Hong:2004td D. K. Hong, S. D. H. Hsu and F. Sannino, Phys. Lett. B 597 (2004) 89 doi:10.1016/j.physletb.2004.07.007 [hep-ph/0406200]. Pica:2010xq C. Pica and F. Sannino, Phys. Rev. D 83 (2011) 035013 doi:10.1103/PhysRevD.83.035013 [arXiv:1011.5917 [hep-ph]]. Sannino:2009aw F. Sannino, Phys. Rev. D 79 (2009) 096007 doi:10.1103/PhysRevD.79.096007 [arXiv:0902.3494 [hep-ph]]. Mojaza:2012zd M. Mojaza, C. Pica, T. A. Ryttov and F. Sannino, Phys. Rev. D 86 (2012) 076012 doi:10.1103/PhysRevD.86.076012 [arXiv:1206.2652 [hep-ph]]. Ryttov:2008xe T. A. Ryttov and F. Sannino, Phys. Rev. D 78 (2008) 115010 doi:10.1103/PhysRevD.78.115010 [arXiv:0809.0713 [hep-ph]]. Ryttov:2009yw T. A. Ryttov and F. Sannino, Int. J. Mod. Phys. A 25 (2010) 4603 doi:10.1142/S0217751X10050391 [arXiv:0906.0307 [hep-ph]]. Bona:2007vi M. Bona et al. [UTfit Collaboration], JHEP 0803 (2008) 049 doi:10.1088/1126-6708/2008/03/049 [arXiv:0707.0636 [hep-ph]]. Chivukula:2010tn R. S. Chivukula and E. H. Simmons, Phys. Rev. D 82 (2010) 033014 doi:10.1103/PhysRevD.82.033014 [arXiv:1005.5727 [hep-lat]]. Andersen:2011yj J. R. Andersen et al., Eur. Phys. J. Plus 126 (2011) 81 doi:10.1140/epjp/i2011-11081-1 [arXiv:1104.1255 [hep-ph]]. Fukano:2010yv H. S. Fukano and F. Sannino, Phys. Rev. D 82 (2010) 035021 doi:10.1103/PhysRevD.82.035021 [arXiv:1005.3340 [hep-ph]]. Rantaharju:2016zjj J. Rantaharju, V. Drach, C. Pica and F. Sannino, arXiv:1610.10074 [hep-lat]. Appelquist:1998xf T. Appelquist and F. Sannino, Phys. Rev. D 59 (1999) 067702 doi:10.1103/PhysRevD.59.067702 [hep-ph/9806409]. Sannino:2010ca F. Sannino, Phys. Rev. D 82 (2010) 081701 doi:10.1103/PhysRevD.82.081701 [arXiv:1006.0207 [hep-lat]]. DiChiara:2010xb S. Di Chiara, C. Pica and F. Sannino, Phys. Lett. B 700 (2011) no.3-4, 229 doi:10.1016/j.physletb.2011.05.008 [arXiv:1008.1267 [hep-ph]]. Appelquist:2014zsa T. Appelquist et al. [LSD Collaboration], Phys. Rev. D 90 (2014) no.11, 114502 doi:10.1103/PhysRevD.90.114502 [arXiv:1405.4752 [hep-lat]]. Foadi:2012bb R. Foadi, M. T. Frandsen and F. Sannino, Phys. Rev. D 87 (2013) no.9, 095001 doi:10.1103/PhysRevD.87.095001 [arXiv:1211.1083 [hep-ph]]. Belyaev:2013ida A. Belyaev, M. S. Brown, R. Foadi and M. T. Frandsen, Phys. Rev. D 90 (2014) 035012 doi:10.1103/PhysRevD.90.035012 [arXiv:1309.2097 [hep-ph]]. Dietrich:2005jn D. D. Dietrich, F. Sannino and K. Tuominen, Phys. Rev. D 72 (2005) 055001 doi:10.1103/PhysRevD.72.055001 [hep-ph/0505059]. Goldberger:2008zz W. D. Goldberger, B. Grinstein and W. Skiba, Phys. Rev. Lett. 100 (2008) 111802 doi:10.1103/PhysRevLett.100.111802 [arXiv:0708.1463 [hep-ph]]. Bellazzini:2012vz B. Bellazzini, C. Csaki, J. Hubisz, J. Serra and J. Terning, Eur. Phys. J. C 73 (2013) no.2, 2333 doi:10.1140/epjc/s10052-013-2333-x [arXiv:1209.3299 [hep-ph]]. Kaplan:1983fs D. B. Kaplan and H. Georgi, Phys. Lett. 136B (1984) 183. doi:10.1016/0370-2693(84)91177-8 Kaplan:1983sm D. B. Kaplan, H. Georgi and S. Dimopoulos, Phys. Lett. 136B (1984) 187. doi:10.1016/0370-2693(84)91178-X Banks:1984gj T. Banks, Nucl. Phys. B 243 (1984) 125. doi:10.1016/0550-3213(84)90389-4 Georgi:1984ef H. Georgi, D. B. Kaplan and P. Galison, Phys. Lett. 143B (1984) 152. doi:10.1016/0370-2693(84)90823-2 Georgi:1984af H. Georgi and D. B. Kaplan, Phys. Lett. 145B (1984) 216. doi:10.1016/0370-2693(84)90341-1 Dugan:1984hq M. J. Dugan, H. Georgi and D. B. Kaplan, Nucl. Phys. B 254 (1985) 299. doi:10.1016/0550-3213(85)90221-4 Pica:2016rmv C. Pica and F. Sannino, Phys. Rev. D 94 (2016) no.7, 071702 doi:10.1103/PhysRevD.94.071702 [arXiv:1604.02572 [hep-ph]]. Ferretti:2013kya G. Ferretti and D. Karateev, JHEP 1403 (2014) 077 doi:10.1007/JHEP03(2014)077 [arXiv:1312.5330 [hep-ph]]. Ferretti:2016upr G. Ferretti, JHEP 1606 (2016) 107 doi:10.1007/JHEP06(2016)107 [arXiv:1604.06467 [hep-ph]]. Vecchi:2015fma L. Vecchi, arXiv:1506.00623 [hep-ph]. Sannino:2016sfx F. Sannino, A. Strumia, A. Tesi and E. Vigiani, JHEP 1611 (2016) 029 doi:10.1007/JHEP11(2016)029 [arXiv:1607.01659 [hep-ph]]. daSilva:2015vna T. Nunes da Silva, E. Pallante and L. Robroek, arXiv:1506.06396 [hep-th]. Fodor:2012td Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. H. Wong, JHEP 1211 (2012) 007 doi:10.1007/JHEP11(2012)007 [arXiv:1208.1051 [hep-lat]]. Appelquist:2009ty T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. D 79 (2009) 076010 doi:10.1103/PhysRevD.79.076010 [arXiv:0901.3766 [hep-ph]]. Appelquist:2007hu T. Appelquist, G. T. Fleming and E. T. Neil, Phys. Rev. Lett. 100 (2008) 171607 Erratum: [Phys. Rev. Lett. 102 (2009) 149902] doi:10.1103/PhysRevLett.100.171607 [arXiv:0712.0609 [hep-ph]]. Lin:2012iw C.-J. D. Lin, K. Ogawa, H. Ohki and E. Shintani, JHEP 1208 (2012) 096 doi:10.1007/JHEP08(2012)096 [arXiv:1205.6076 [hep-lat]]. Fodor:2011tu Z. Fodor et al., Phys. Lett. B 703 (2011) 348 doi:10.1016/j.physletb.2011.07.037 [arXiv:1104.3124 [hep-lat]]. Appelquist:2011dp T. Appelquist, G. T. Fleming, M. F. Lin, E. T. Neil and D. A. Schaich, Phys. Rev. D 84 (2011) 054501 doi:10.1103/PhysRevD.84.054501 [arXiv:1106.2148 [hep-lat]]. DeGrand:2011cu T. DeGrand, Phys. Rev. D 84 (2011) 116901 doi:10.1103/PhysRevD.84.116901 [arXiv:1109.1237 [hep-lat]]. Aoki:2012eq Y. Aoki et al., Phys. Rev. D 86 (2012) 054506 doi:10.1103/PhysRevD.86.059903, 10.1103/PhysRevD.86.054506 [arXiv:1207.3060 [hep-lat]]. Cheng:2013eu A. Cheng, A. Hasenfratz, G. Petropoulos and D. Schaich, JHEP 1307 (2013) 061 doi:10.1007/JHEP07(2013)061 [arXiv:1301.1355 [hep-lat]]. Cheng:2013xha A. Cheng, A. Hasenfratz, Y. Liu, G. Petropoulos and D. Schaich, Phys. Rev. D 90 (2014) no.1, 014509 doi:10.1103/PhysRevD.90.014509 [arXiv:1401.0195 [hep-lat]]. Aoki:2013zsa Y. Aoki et al. [LatKMI Collaboration], Phys. Rev. Lett. 111 (2013) no.16, 162001 doi:10.1103/PhysRevLett.111.162001 [arXiv:1305.6006 [hep-lat]]. Lombardo:2014pda M. P. Lombardo, K. Miura, T. J. Nunes da Silva and E. Pallante, JHEP 1412 (2014) 183 doi:10.1007/JHEP12(2014)183 [arXiv:1410.0298 [hep-lat]]. Hasenfratz:2011xn A. Hasenfratz, Phys. Rev. Lett. 108 (2012) 061601 doi:10.1103/PhysRevLett.108.061601 [arXiv:1106.5293 [hep-lat]]. Cheng:2014jba A. Cheng, A. Hasenfratz, Y. Liu, G. Petropoulos and D. Schaich, JHEP 1405 (2014) 137 doi:10.1007/JHEP05(2014)137 [arXiv:1404.0984 [hep-lat]]. Fodor:2016zil Z. Fodor, K. Holland, J. Kuti, S. Mondal, D. Nogradi and C. H. Wong, Phys. Rev. D 94 (2016) no.9, 091501 doi:10.1103/PhysRevD.94.091501 [arXiv:1607.06121 [hep-lat]]. chiu:2016 T.W. Chiu, PoS(LATTICE2016)228 Chiu:2016uui T. W. Chiu, arXiv:1603.08854 [hep-lat]. Chiu:2015sea T. W. Chiu, Phys. Lett. B 744 (2015) 95 doi:10.1016/j.physletb.2015.03.036 [arXiv:1503.01750 [hep-lat]]. Leino:2016njf V. Leino, T. Rantalaiho, K. Rummukainen, J. M. Suorsa, K. Tuominen and S. Tähtinen, arXiv:1610.09989 [hep-lat]. Suorsa:2016jsf J. M. Suorsa, V. Leino, J. Rantaharju, T. Rantalaiho, K. Rummukainen, K. Tuominen and S. Tähtinen, arXiv:1611.02022 [hep-lat]. Karavirta:2011zg T. Karavirta, J. Rantaharju, K. Rummukainen and K. Tuominen, JHEP 1205 (2012) 003 doi:10.1007/JHEP05(2012)003 [arXiv:1111.4104 [hep-lat]]. Ohki:2010sr H. Ohki et al., PoS LATTICE 2010 (2010) 066 [arXiv:1011.0373 [hep-lat]]. Rantaharju:2014ila J. Rantaharju, T. Karavirta, V. Leino, T. Rantalaiho, K. Rummukainen and K. Tuominen, PoS LATTICE 2014 (2014) 258 [arXiv:1411.4879 [hep-lat]]. Lewis:2011zb R. Lewis, C. Pica and F. Sannino, Phys. Rev. D 85 (2012) 014504 doi:10.1103/PhysRevD.85.014504 [arXiv:1109.3513 [hep-ph]]. Hietanen:2013fya A. Hietanen, R. Lewis, C. Pica and F. Sannino, JHEP 1412 (2014) 130 doi:10.1007/JHEP12(2014)130 [arXiv:1308.4130 [hep-ph]]. Hietanen:2014xca A. Hietanen, R. Lewis, C. Pica and F. Sannino, JHEP 1407 (2014) 116 doi:10.1007/JHEP07(2014)116 [arXiv:1404.2794 [hep-lat]]. Arthur:2016dir R. Arthur, V. Drach, M. Hansen, A. Hietanen, C. Pica and F. Sannino, Phys. Rev. D 94 (2016) no.9, 094507 doi:10.1103/PhysRevD.94.094507 [arXiv:1602.06559 [hep-lat]]. Bursa:2010xn F. Bursa, L. Del Debbio, L. Keegan, C. Pica and T. Pickup, Phys. Lett. B 696 (2011) 374 doi:10.1016/j.physletb.2010.12.050 [arXiv:1007.3067 [hep-ph]]. Hayakawa:2013yfa M. Hayakawa, K.-I. Ishikawa, S. Takeda and N. Yamada, Phys. Rev. D 88 (2013) no.9, 094504 doi:10.1103/PhysRevD.88.094504 [arXiv:1307.6997 [hep-lat]]. Appelquist:2013pqa T. Appelquist et al., Phys. Rev. Lett. 112 (2014) no.11, 111601 doi:10.1103/PhysRevLett.112.111601 [arXiv:1311.4889 [hep-ph]]. Catterall:2007yx S. Catterall and F. Sannino, Phys. Rev. D 76 (2007) 034504 doi:10.1103/PhysRevD.76.034504 [arXiv:0705.1664 [hep-lat]]. Hietanen:2008mr A. J. Hietanen, J. Rantaharju, K. Rummukainen and K. Tuominen, JHEP 0905 (2009) 025 doi:10.1088/1126-6708/2009/05/025 [arXiv:0812.1467 [hep-lat]]. DelDebbio:2008zf L. Del Debbio, A. Patella and C. Pica, Phys. Rev. D 81 (2010) 094503 doi:10.1103/PhysRevD.81.094503 [arXiv:0805.2058 [hep-lat]]. DelDebbio:2009fd L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago, Phys. Rev. D 80 (2009) 074507 doi:10.1103/PhysRevD.80.074507 [arXiv:0907.3896 [hep-lat]]. Catterall:2009sb S. Catterall, J. Giedt, F. Sannino and J. Schneible, arXiv:0910.4387 [hep-lat]. DelDebbio:2010hx L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago, Phys. Rev. D 82 (2010) 014510 doi:10.1103/PhysRevD.82.014510 [arXiv:1004.3206 [hep-lat]]. DelDebbio:2010hu L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago, Phys. Rev. D 82 (2010) 014509 doi:10.1103/PhysRevD.82.014509 [arXiv:1004.3197 [hep-lat]]. Bursa:2011ru F. Bursa et al., Phys. Rev. D 84 (2011) 034506 doi:10.1103/PhysRevD.84.034506 [arXiv:1104.4301 [hep-lat]]. DelDebbio:2015byq L. Del Debbio, B. Lucini, A. Patella, C. Pica and A. Rago, Phys. Rev. D 93 (2016) no.5, 054505 doi:10.1103/PhysRevD.93.054505 [arXiv:1512.08242 [hep-lat]]. Athenodorou:2014eua A. Athenodorou, E. Bennett, G. Bergner and B. Lucini, Phys. Rev. D 91 (2015) no.11, 114508 doi:10.1103/PhysRevD.91.114508 [arXiv:1412.5994 [hep-lat]]. bergner:2016 G. Bergner, PoS(LATTICE2016)237 Brower:2015owo R. C. Brower, A. Hasenfratz, C. Rebbi, E. Weinberg and O. Witzel, Phys. Rev. D 93 (2016) no.7, 075028 doi:10.1103/PhysRevD.93.075028 [arXiv:1512.02576 [hep-ph]]. Hasenfratz:2016uar A. Hasenfratz, C. Rebbi and O. Witzel, arXiv:1611.07427 [hep-lat]. rebbi:2016 A. Hasenfratz, C. Rebbi and O. Witzel, PoS(LATTICE2016)235 aoki:2016 Y. Aoki, PoS(LATTICE2016)230 Aoki:2016wnc Y. Aoki et al. [LatKMI Collaboration], arXiv:1610.07011 [hep-lat]. Aoki:2013xza Y. Aoki et al. [LatKMI Collaboration], Phys. Rev. D 87 (2013) no.9, 094511 doi:10.1103/PhysRevD.87.094511 [arXiv:1302.6859 [hep-lat]]. Aoki:2014oha Y. Aoki et al. [LatKMI Collaboration], Phys. Rev. D 89 (2014) 111502 doi:10.1103/PhysRevD.89.111502 [arXiv:1403.5000 [hep-lat]]. Appelquist:2016viq T. Appelquist et al., Phys. Rev. D 93 (2016) no.11, 114514 doi:10.1103/PhysRevD.93.114514 [arXiv:1601.04027 [hep-lat]]. fleming:2016 G.T. Fleming, PoS(LATTICE2016)241 gasbarro:2016 A. Gasbarro, PoS(LATTICE2016)242 Golterman:2016eth M. Golterman and Y. Shamir, arXiv:1610.01752 [hep-ph]. Golterman:2016lsd M. Golterman and Y. Shamir, Phys. Rev. D 94 (2016) no.5, 054502 doi:10.1103/PhysRevD.94.054502 [arXiv:1603.04575 [hep-ph]]. Soto:2011ap J. Soto, P. Talavera and J. Tarrus, Nucl. Phys. B 866 (2013) 270 doi:10.1016/j.nuclphysb.2012.09.005 [arXiv:1110.6156 [hep-ph]]. Matsuzaki:2013eva S. Matsuzaki and K. Yamawaki, Phys. Rev. Lett. 113 (2014) no.8, 082002 doi:10.1103/PhysRevLett.113.082002 [arXiv:1311.3784 [hep-lat]]. Hansen:2016fri M. Hansen, K. Langaeble and F. Sannino, arXiv:1610.02904 [hep-ph]. Fodor:2016pls Z. Fodor, K. Holland, J. Kuti, S. Mondal, D. Nogradi and C. H. Wong, PoS LATTICE 2015 (2016) 219 [arXiv:1605.08750 [hep-lat]]. Fodor:2012ty Z. Fodor, K. Holland, J. Kuti, D. Nogradi, C. Schroeder and C. H. Wong, Phys. Lett. B 718 (2012) 657 doi:10.1016/j.physletb.2012.10.079 [arXiv:1209.0391 [hep-lat]]. Fodor:2014pqa Z. Fodor, K. Holland, J. Kuti, D. Nogradi and C. H. Wong, PoS LATTICE 2013 (2014) 062 [arXiv:1401.2176 [hep-lat]]. Fodor:2016wal Z. Fodor, K. Holland, J. Kuti, S. Mondal, D. Nogradi and C. H. Wong, Phys. Rev. D 94 (2016) no.1, 014503 doi:10.1103/PhysRevD.94.014503 [arXiv:1601.03302 [hep-lat]]. Drach:2015sua V. Drach, M. Hansen, A. Hietanen, C. Pica and F. Sannino, PoS LATTICE 2015 (2016) 223 [arXiv:1508.04213 [hep-lat]]. kuti:2016 J. Kuti PoS(LATTICE2016)239 wong:2016 C.H. Wong PoS(LATTICE2016)224 Hansen:2016sxp M. Hansen and C. Pica, PoS LATTICE2016 213 [arXiv:1610.08072 [hep-lat]]. DeGrand:2016mxr T. A. DeGrand, D. Hackett, W. I. Jay, E. T. Neil, Y. Shamir and B. Svetitsky, arXiv:1610.06465 [hep-lat]. DeGrand:2016het T. A. DeGrand, M. Golterman, W. I. Jay, E. T. Neil, Y. Shamir and B. Svetitsky, arXiv:1610.06121 [hep-lat]. Cacciapaglia:2014uja G. Cacciapaglia and F. Sannino, JHEP 1404 (2014) 111 doi:10.1007/JHEP04(2014)111 [arXiv:1402.0233 [hep-ph]]. Arbey:2015exa A. Arbey, G. Cacciapaglia, H. Cai, A. Deandrea, S. Le Corre and F. Sannino, arXiv:1502.04718 [hep-ph]. Arthur:2016ozw R. Arthur, V. Drach, A. Hietanen, C. Pica and F. Sannino, arXiv:1607.06654 [hep-lat]. Drach:2016 V. Drach, T. Janoswki, C. Pica and F. Saninno PoS(LATTICE2016)229 Janowski:2016hsg T. Janowski, V. Drach and C. Pica, arXiv:1611.09195 [hep-lat]. Borsanyi:2012zs S. Borsanyi et al., JHEP 1209 (2012) 010 doi:10.1007/JHEP09(2012)010 [arXiv:1203.4469 [hep-lat]]. Martinelli:1994ty G. Martinelli, C. Pittori, C. T. Sachrajda, M. Testa and A. Vladikas, Nucl. Phys. B 445 (1995) 81 doi:10.1016/0550-3213(95)00126-D [hep-lat/9411010].

http://arxiv.org/abs/1701.08005v1

20170127104339

On the Degrees-of-Freedom of the MIMO Three-Way Channel with Intermittent Connectivity

cs.IT

On the Degrees-of-Freedom of the MIMO Three-Way Channel with Intermittent Connectivity Anas Chaaban, Aydin Sezgin, and Mohamed-Slim Alouini A. Chaaban and M.-S. Alouini are with the Division of Computer, Electrical, and Mathematical Sciences and Engineering (CEMSE) at King Abdullah University of Science and Technology (KAUST), Thuwal, Saudi Arabia. Email: {anas.chaaban,slim.alouini}@kaust.edu.sa. A. Sezgin is with the Institute of Digital Communication Systems at the Ruhr-Universität Bochum, Bochum, Germany. Email: aydin.sezgin@rub.de. December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ The degrees-of-freedom (DoF) of the multi-antenna three-way channel (3WC) with an intermittent node is studied. Special attention is given to the impact of adaptation. A nonadaptive transmission scheme based on interference alignment, zero-forcing, and erasure-channel treatment is proposed, and its corresponding DoF region is derived. Then, it is shown that this scheme achieves the sum-DoF of the intermittent channel, in addition to the DoF region of the nonintermittent one. Thus, adaptation is not necessary from those perspectives. To the contrary, it is shown that adaptation is necessary for achieving the DoF region of the intermittent case. This is shown by deriving an outer bound for the intermittent channel with nonadaptive encoding, and giving a counterexample of an adaptive scheme which achieves DoF tuples outside this bound. This highlights the importance of cooperation in this intermittent network. § INTRODUCTION Multi-way, full-duplex, and device-to-device (D2D) communications are important techniques that are expected to gain more prominence in future communication systems. Multi-way communication refers to communication between multiple nodes each acting as a source, a destination, and possibly a relay. Full-duplex operation is defined as when these three functionalities take place over the same time/frequency resources, and D2D communication refers to direct communication between users without or with limited base-station intervention. Those techniques attracted and continue to attract research attention <cit.>. Consider a setup where three nodes (D2D users e.g.) communicate with each other in a multi-way fashion. This setup can be modeled as a three-way channel (3WC), an extension of Shannon's two-way channel <cit.> which has been studied in <cit.>. Therein, it is assumed that the nodes are always connected. This assumption is not always valid in practice since a node might have intermittent connectivity, e.g. due to shadowing, or if a D2D node causes strong interference to a cellular user, in which case it is not permitted to use its band <cit.>. The impact of intermittency on the performance of various networks was studied in <cit.> for instance. In this paper, we study the impact of intermittency on the multiple-input multiple-output (MIMO) 3WC. We consider a full-duplex MIMO 3WC with full message-exchange, where each node has an independent message to each of the other two nodes. The permanent nodes have only causal knowledge of the availability of the intermittent node, which can be obtained by estimating its activity from the received signals. For this model, we study the degrees-of-freedom (DoF), i.e., the capacity scaling versus signal-to-noise ratio (SNR) in a dB scale. We pay particular attention to the necessity or the lack thereof, of adaptive encoding where the transmit signal of each node is allowed to depend on its previously received signals. This issue has been studied for various channels earlier <cit.>. First, we devise a nonadaptive scheme based on interference alignment and zero-forcing, where intermittency is treated as an erasure channel, and we derive its achievable DoF region. Then, we derive DoF upper bounds that prove that this scheme achieves the sum-DoF of the channel. It follows that as far as the sum-DoF is concerned, adaptation is not necessary. This scheme also achieves the DoF region of the channel without intermittency, and hence, for the nonintermittent channel, adaptation is not necessary for achieving the DoF region. After showing the unnecessity of adaptation in those two cases, we prove that adaptation is necessary to achieve the DoF region of the intermittent channel. To show this, we derive a DoF outer bound that holds under nonadaptive encoding, and provide an adaptive scheme that achieves rates that violate this outer bound. This proves that adaptation enlarges the DoF region in the intermittent 3WC. Throughout the paper, we use x_i^n for some i to denote (x_i,1,…,x_i,n). The N× N identity matrix is denoted _N. We write ∼𝒞𝒩(0,Q) to indicate that is a complex Gaussian random variable with zero mean and covariance matrix Q. We write x^+ to denote max{0,x} for some x∈ℝ, _i to denote the ℓ_i-norm of , and ^̋†, ^̋H, and span()̋ to denote the pseudo-inverse, the Hermitian transpose, and the the subspace spanned by the columns of $̋. § SYSTEM MODEL Consider a system where three MIMO full-duplex nodes communicate in a multi-way manner using the same medium, with one of the nodes being intermittently available (Fig. <ref>). For some transmission durationn∈ℕ(in symbols), lets^ndenote the intermittency of node 1, where forℓ∈{1,…,n},s_ℓ=1means that node 1 is available, ands_ℓ=0otherwise. The states^nis a sequence of independent and identically distributed (i.i.d) Bernoulli random variablesS_ℓwith probabilityℙ(s_ℓ=1)=τandℙ(s_ℓ=0)=1-τ≜τ̅. This sequence is known at node 1. However, knowledge ofs_ℓis only available causally at nodei∈{2,3}, i.e., nodeidoes not knows_ℓat the beginning of theℓ-th transmission, and can only obtain it after receiving theℓ-th received signal from which the activity of node 1 can be detected with certainty. Nodei∈{1,2,3}is equipped withM_itransmit and receive antennas. Its transmit signal at time indexℓis represented by_i,ℓ; a realization of a random vector_i,ℓ∈ℂ^M_i×1that satisfies a power constraint[Any power discrepancy is absorbed into the channel gains.]∑_ℓ=1^n𝔼[_i,ℓ_2^2]≤nP. Clearly_1,ℓ=0ifs_ℓ=0. The received signals are _1,ℓ =_̋21_2,ℓ+_̋31_3,ℓ+_1,ℓ, if s_ℓ=1 _2,ℓ =_̋12_1,ℓ+_̋32_3,ℓ+_2,ℓ, _3,ℓ =_̋13_1,ℓ+_̋23_2,ℓ+_3,ℓ, and_1,ℓ=0ifs_ℓ=0, where_̋ji∈ℂ^M_j×M_iand_̋ki∈ℂ^M_k×M_irepresent the channel matrices from nodesjandkto nodei, respectively, and_i,ℓis a realization of_i,ℓ∼𝒞𝒩(0,σ^2_M_i), i.i.d. with respect toℓ. We denoteP/σ^2byρand call it SNR throughout the paper. We assume without loss of generality thatM_2≥M_3. We also assume thatM_1≥M_2. The channel matrices are generated randomly from a continuous distribution, held constant throughout the transmission, and are known globally. The message sets, encoding, and decoding, and achievability are defined in the standard Shannon sense <cit.>. The encoder at nodei,ℰ_i,ℓ, can be either adaptive where dependence of_i,ℓon_i^ℓ-1is allowed, or restricted (nonadaptive) where it is not. These possibilities are shown in Table <ref>. The DoF region is the set of achievable DoF tuples=̣(d_12,d_13,d_21,d_23,d_31,d_32)∈ℝ_+^6defined as in <cit.>. Roughly speaking, if a rate tuple (function ofρ) (ρ) =(R_12(ρ),R_13(ρ),R_21(ρ),R_23(ρ),R_31(ρ),R_32(ρ)) whereR_ij(ρ)is the rate of the message from nodeito nodej, is achievable, then the DoF tuple$̣ with d_ij=limsup_ρ→∞R_ij(ρ)/log(ρ) is achievable. We denote the DoF region under restricted encoding and adaptive encoding for a given τ by 𝒟_ r,τ and 𝒟_ a,τ, respectively, and we define the sum-DoF as d_ r,τ=max_∈̣𝒟_ r,τ_1 and d_ a,τ=max_∈̣𝒟_ a,τ_1. Next, we describe a restricted transmission scheme, and we derive its achievable DoF region. § RESTRICTED ENCODING TRANSMISSION SCHEME In this section, we prove the following theorem. The DoF region of the 3WC satisfies 𝒟_ r,τ^[ in]⊆𝒟_ r,τ⊆𝒟_ a,τ, where the achievable inner bound 𝒟_ r,τ^[ in] is the set of ∈̣ℝ_+^6 satisfying the following for i,j∈{2,3}, i≠ j: d_1i+d_1j+τ d_ij ≤τ M_1, d_31+τ d_32 ≤τ M_3, d_i1+d_j1+τ d_ij ≤τ M_1, d_13+τ d_23 ≤τ M_3, d_i1+d_1j+τ d_ij ≤τ M_2. The inclusion of 𝒟_ r,τ in 𝒟_ a,τ is obvious. The achievability of 𝒟_ r,τ^[ in] is proved in the rest of this section. Note that the factor τ in the inequalities above imposes a larger penalty on the streams going through the intermittent links. If we interpret τ M_3 in d_31+τ d_32≤τ M_3 as available resources, then increasing d_32 by 1 `eats' τ units of resources, while increasing d_31 by 1 `eats' 1 unit of resources. Thus, transmission between nodes 2 and 3 is `cheaper' by a factor of τ, no matter how large M_1 is as we shall see later. Next, we prove the achievability of 𝒟_ r,τ^[ in]. §.§.§ Encoding Each node splits its message w_ij into w_ij^[1] and w_ij^[2] to be sent using zero-forcing and interference alignment, respectively. Encoding proceeds as follows. Since node 1 is available for a fraction of time, say m=s^n_0 out of the n transmissions, it encodes w_12^[q] and w_13^[q], q∈{1,2}, into codewords _12^[q]m and _13^[q]m with i.i.d. 𝒞𝒩(0,p_1_a_12^[q]) and 𝒞𝒩(0,p_1_a_13^[q]) symbols, respectively. Here a_ij^[q] is the vector length, and p_1 is the power of each component of _12^[q]m and _13^[q]m. Then, those codewords are extended to length n codewords _12^[q]n and _13^[q]n by inserting zeros where s_ℓ=0 (note that s^n is known at node 1). Now, nodes 2 and 3 are available all the time, but they do not have apriori knowledge of s_ℓ. Thus, these nodes use standard random Gaussian codebooks to encode their messages, and treat the channel to node 1 as an erasure channel with erasure probability τ̅. Node 2 encodes w_21^[q] and w_23^[q], q∈{1,2}, into codewords _21^[q]n and _23^[q]n with i.i.d. 𝒞𝒩(0,p_2_a_21^[q]) and 𝒞𝒩(0,p_2_a_23^[q]) symbols, respectively. Similarly, node 3 encodes w_31^[q] and w_32^[q], q∈{1,2}, into codewords _31^[q]n and _32^[q]n with i.i.d. 𝒞𝒩(0,p_3_a_31^[q]) and 𝒞𝒩(0,p_3_a_32^[q]) symbols, respectively. To satisfy the power constraint, the powers are chosen as p_1 =(a_12^[1]+a_12^[2]+a_13^[1]+a_13^[2])^-1m^-1nP, p_i =(a_i1^[1]+a_i1^[2]+a_ij^[1]+a_ij^[2])^-1P, i,j∈{2,3}, i≠ j. This encoding is restricted since it uses neither s^ℓ-1 at nodes 2 and 3, nor _i^ℓ-1 at nodes 1, 2, and 3. §.§.§ Transmission At time ℓ, node i sends _i,ℓ =∑_q=1^2[_ij^[q]_ij,ℓ^[q]+_ik^[q]_ik,ℓ^[q]], where j,k∈{1,2,3}∖{i}, j≠ k, and _ij^[q]∈ℂ^M_i× a_ij^[q] is a beamforming matrix. Zero-forcing is achieved by choosing the matrices _ij^[1] so that _̋ik_ij^[1]=0 for distinct i,j,k∈{1,2,3}. These matrices exist if (M_i-M_k)^+≥ a_ij^[1], ensuring that node i has enough antennas to send a_ij^[1] streams to node j without interfering with node k. To avoid any overlap of the transmit signals in the transmit signal space, we require ∑_q=1^2 (a_ij^[q]+a_ik^[q])≤ M_i. §.§.§ Decoding Node 1 receives _1,ℓ=0 if s_ℓ=0 and _1,ℓ =∑_j=2^3∑_q=1^2_̋j1_j1^[q]_j1,ℓ^[q]+_23[[ _23ℓ^[2]; _32,ℓ^[2] ]]+_1,ℓ otherwise, where _23=[_̋21_23^[2], _̋31_32^[2]]. This signal consists of four desired signals plus interference. To decode a desired signals, say _21^[1]n, node 1 zero-forces the remaining signals by multiplying _1^n by a post-coder _21^[1]∈ℂ^a_21^[1]× M_1 satisfying _21^[1]_21^[1]H=_a_21^[1] and _21^[1][_̋21_21^[2], _̋31_31^[1], _̋31_31^[2], _23] =0, rank(_21^[1]_̋21_21^[1]) =a_21^[1]. After post-coding, node 1 is left with the signal _21,ℓ=_21^[1]_̋21_21^[1]_21,ℓ^[1]+_21^[1]_1,ℓ if s_ℓ=1 and _21,ℓ=0 otherwise. This is an erasure channel over which the rate I(_21,ℓ^[1];_21,ℓ,s_ℓ)= I(_21,ℓ^[1];_21,ℓ|s_ℓ)=τ I(_21,ℓ^[1];_21,ℓ|s_ℓ=1) is achievable from node 2 to node 1 for n large.[Recall that s^n is known at the decoding stage.] Since we used Gaussian i.i.d. codes, this rate is τlog|_a_21^[1]+p_2/σ^2_21^[1]_̋21_21^[1]_21^[1]H_̋21^H_21^[1]H|, leading to a DoF of τ a_21^[1] as long as (<ref>) is satisfied (cf. (<ref>)). A similar procedure can be applied for decoding _21^[2]n, _31^[1]n, and _31^[2]n, to achieve DoF of τ a_21^[2], τ a_31^[1], τ a_31^[2]. The existence of the post-coders _i1^[q] which allow this procedure is guaranteed as long as the columns of [_̋21_21^[1], _̋21_21^[2], _̋31_31^[1], _̋31_31^[2], _23] are linearly independent. Let a̅_23^[2] be the dimension of span(_̋21_23^[2])∩ span(_̋31_32^[2]). Then, span(_23) has a_23^[2]+a_32^[2]-a̅_23^[2] dimensions, and the above linear independence is possible if we choose ∑_q=1^2(a_21^[q]+a_31^[q])+a_23^[2]+a_32^[2]-a̅_23^[2]≤ M_1. To minimize the impact of interference, we choose _ij^[q] so that a̅_23^[2] is maximized. This can not be chosen arbitrarily large, as it has to be smaller than each of a_23^[2] and a_32^[2], and also smaller than the dimension of span(_̋21)∩ span(_̋31), which is (M_2+M_3-M_1)^+ almost surely. Thus, min{a_23^[2],a_32^[2],(M_2+M_3-M_1)^+}≥a̅_23^[2]. The same arguments can be applied at nodes 2 and 3, for decoding their desired signals. This achieves τ a_12^[1], τ a_12^[2], a_32^[1], and a_32^[2] DoF at node 2, and τ a_13^[1], τ a_13^[2], a_23^[1], and a_23^[2] DoF at node 3, leading to similar constraints as (<ref>) and (<ref>). §.§.§ Achievable DoF Region The constraints can be combined as follows (M_i-M_k)^+ ≥ a_ij^[1], ∑_q=1^2 (a_ij^[q]+a_ik^[q]) ≤ M_i, min{a_ij^[2],a_ji^[2],(M_i+M_j-M_k)^+} ≥a̅_ij^[2], ∑_q=1^2(a_ji^[q]+a_ki^[q])+a_jk^[2]+a_kj^[q]-a̅_jk^[2] ≤ M_i. for distinct i,j,k∈{1,2,3}, where a̅_12^[2] and a̅_13^[2] are the dimensions of span(_̋13_12^[2])∩ span(_̋23_21^[2]) and span(_̋12_13^[2])∩ span(_̋32_31^[2]), respectively. By adding the achievable DoF per stream, we obtain d_ij (e.g. d_21=τ a_21^[1]+τ a_21^[2]). Substituting d_ij in (<ref>)–(<ref>), using M_1≥ M_2≥ M_3 and Fourier Motzkin's elimination leads to the DoF region in Theorem <ref>. Details are omitted due to space limitations. Next, we study the optimality of this scheme. § OPTIMALITY DISCUSSION §.§ Sum-DoF We first consider the sum-DoF of the channel, and start by presenting the following DoF upper bounds. The following must be satisfied by any DoF tuple ∈̣𝒟_ a,τ (and hence also ∈̣𝒟_ r,τ): d_13+d_23+d_21 ≤τ M_2+τ̅M_3, d_31+d_32+d_12 ≤τ M_2+τ̅M_3. For brevity, we denote (W_ij,W_ik) by _i, and use ϵ_1n, ϵ_2n, and ϵ_3n to denote quantities that vanish as n→∞. Let _23 be an (M_2-M_3)× M_2 matrix so that _23≜[_̋23^T, _23^T]^T has full rank M_2. Such a matrix exists almost surely. Also, let _3,ℓ be defined as _23_2,ℓ+_3,ℓ if S_ℓ=1 and 0 otherwise, where _3,ℓ∼𝒞𝒩(0,σ_3^2_M_2-M_3), and define _3,ℓ=[_3,ℓ^T, _3,ℓ^T]^T. Now, consider any code for the 3WC, and let us establish a bound on R_13+R_23+R_21.[We write R_ij(ρ) simply as R_ij for brevity.] We give (_3^n,W_12) and (_3^n,_3,W_23) as side information to nodes 3 and 1, respectively. By Fano's inequality, we have n(R_13+R_23-ϵ_1n) ≤ I(W_13,W_23;_3^n,S^n,_3,W_12), n(R_21-ϵ_2n) ≤ I(W_21;_1^n,_3^n,S^n,_1,_3,W_23). Recall that each node can estimate S^n with certainty from the received signals as assumed in the system model. Using the chain rule, the independence of the messages of each other and of S^n, and combining the two bounds yields n(R_13+R_23+R_21-ϵ_3n) ≤ I(_2,W_13;_3^n|S^n,_3,W_12) +I(W_21;_1^n|_3^n,S^n,_1,_3,W_23). The second term in this bound is equal to ∑_ℓ=1^n I(W_21;_1,ℓ|_1^ℓ-1,_3^n,S^n,_1,_3,W_23), which is no(log(ρ)),[lim_ρ→∞o(log(ρ))/log(ρ)=0.] since given _1, _3, _1^ℓ-1, _3^n, and S^n, and using the adaptive encoder, we can construct a noisy version of _1,ℓ for all ℓ with S_ℓ=1 given by _̋21_23^-1[[ _3,ℓ-_̋13_1,ℓ; _3,ℓ ]]+_̋31_3,ℓ. On the other hand, using standard steps I(_2,W_13;_3^n|S^n,_3,W_12) ≤∑_ℓ=1^n I(_1,ℓ,_2,ℓ;_3,ℓ|S_ℓ) = ∑_ℓ=1^n τ I(_1,ℓ,_2,ℓ;_3,ℓ|S_ℓ=1)+τ̅ I(_2,ℓ;_3,ℓ|S_ℓ=0) ≤ n(τ M_2+τ̅M_3)log(ρ)+no(log(ρ)), since the first and second terms represent (M_1+M_2)× M_2 and M_2× M_3 MIMO channels with M_2 and M_3 DoF almost surely (M_3≤ M_2), respectively. Combining terms, dividing by n and letting n→∞, this yields the bound R_13+R_23+R_21≤ (τ M_2+τ̅M_3)log(ρ)+o(log(ρ)), which consequently leads to the first DoF bound. The second is obtained similarly by giving (_2^n,_2,W_32) and W_13 as side information to nodes 1 and 2, respectively. Based on Lemma <ref>, we can state the following theorem. The sum-DoF of the intermittent 3WC is given by d_ r,τ=d_ a,τ=2τ M_2+2τ̅M_3. Achievability follows from Theorem <ref> by using the simplex method <cit.> to maximize the sum-DoF subject to the DoF constraints. In particular, it follows by setting a_12^[1]=a_21^[1]=M_2-M_3, a_23^[2]=a_32^[2]=M_3, and a_12^[2]=a_21^[2]=a_23^[1]=a_32^[1]=d_13=d_31=0 in the scheme described in Sec. <ref>. The converse follows by adding the DoF bounds in Lemma <ref>. This agrees with intuition. To maximize the sum-DoF, one should capitalize on the stable links between nodes 2 and 3, and use any remaining resources for communicating with the intermittent node 1. This theorem proves that adaptation is not necessary for achieving the sum-DoF of the intermittent 3WC. The same does not hold true from a DoF region perspective as we shall see next. §.§ DoF Region In this section, we show that adaptation is necessary for achieving the DoF region of the intermittent 3WC. This result is particularly interesting in light of the following statement. The scheme in Sec. <ref> achieves the DoF region of the nonintermittent 3WC (τ=1) given by 𝒟_ a,1=𝒟_ r,1^[ in]. The proof is based on upper bounds in <cit.>, and is omitted for lack of space. Therefore, from a DoF-region point-of-view, adaptation is not necessary in the nonintermittent case.[Adaptation is still necessary from an achievable rate point-of-view <cit.>, but the gain of adaptation does not scale with ρ.] Interestingly, the same is not true in the intermittent case. To prove this, first we need a DoF outer bound for the restricted intermittent 3WC, and second, we need an adaptive scheme which achieves DoF tuples outside this outer bound. The first step is tackled in the following lemma. Under restricted encoding, we have 𝒟_ r,τ⊂𝒟_ r,τ^[ out] defined as the set of ∈̣ℝ_+^6 satisfying d_31+τ d_32≤τ M_3. Let _2,ℓ=S_ℓ_2,ℓ and let us give (_2^n,_2) to node 1 as side information. From Fano's inequality, we have n(R_31-ϵ_1n) ≤ I(W_31;_1^n,_2^n|S^n,_1,_2) = I(W_31;_2^n|S^n,_1,_2) + I(W_31;_1^n|S^n,_1,_2,_2^n). Given _2^n, _1, _2, and S^n, we can construct a noisy version of _1^n for S_ℓ=1 given by _̋31_̋32^†(_2,ℓ-_̋12_1,ℓ)+_̋21_2,ℓ, where _̋32^† exists almost surely. Thus, I(W_31;_1^n|S^n,_1,_2,_2^n)=nτ o(log(ρ)), and hence n(R_31-ϵ_1n-τ o(log(ρ))) ≤ I(W_31;_2^n|S^n,_1,_2) = τ∑_ℓ=1^n I(W_31;_2,ℓ|S^n,_1,_2,_2^ℓ-1,S_ℓ=1). On the other hand, giving (_1,W_31) to node 3 as side information and using Fano's inequality, we have n(τ R_32-ϵ_2n) ≤τ I(W_32;_2^n|S^n,_1,_2,W_31) = τ∑_ℓ=1^n [h(_2,ℓ|S^n,_1,_2,W_31,_2^ℓ-1) -h(_2,ℓ|S^n,_1,_2,_3,_2^ℓ-1)]. Since conditioning does not increase entropy, the first term is upper bounded by h(_2,ℓ|S^n,_1,_2,W_31,_2^ℓ-1). Moreover, since restricted encoding can be used to generate _1^n and _3^n from _1 and _3, the second entropy term is equal to h(_2,ℓ)=h(_2,ℓ|S^n,_1,_2,_3,_2^ℓ-1). Thus, n(τ R_32-ϵ_2n) ≤τ∑_ℓ=1^n I(W_32;_2,ℓ|S^n,_1,_2,W_31,_2^ℓ-1) = τ∑_ℓ=1^n I(W_32;_2,ℓ|S^n,_1,_2,W_31,_2^ℓ-1,S_ℓ=1), since for a given ℓ, this mutual information is equal to I(W_32;_̋32_3,ℓ+_2,ℓ|S^n,_1,_2,W_31,_2^ℓ-1) independent of the state S_ℓ. Combining the two bounds yields n(R_31+τ R_32-ϵ_3n-τ o(log(ρ))) ≤τ∑_ℓ=1^n I(_3;_2,ℓ|S^n,_1,_2,_2^ℓ-1,S_ℓ=1) ≤ nτ M_3log(ρ)+nτ o(log(ρ)) which follows using similar steps as in the proof of Lemma <ref>. This leads to the desired result. Despite its simplicity, Lemma <ref> suffices for our purpose. Based on this lemma, the following theorem proves the necessity of adaptation in the intermittent case. For an intermittent 3WC with M_1>M_3, 𝒟_ a,τ⊄𝒟_ r,τ^[ out], and hence adaptation is necessary. It suffices to show that ∃∉̣𝒟_ r,τ^[ out] which is achievable using an adaptive scheme. To is end, suppose that only node 3 has a message to node 1, while node 2 acts as a relay to support node 3 which employs block-Markov encoding. Consider B transmission blocks, each consisting of n channel uses, and let a_2,a_3∈[0,1], be chosen so that a_2M_2,a_3M_3∈ℕ. In block 1, node 3 encodes a message w_31(1) to a codeword _31^n with _31,ℓ∈ℂ^a_3M_3, and sends it to node 1 using a_3M_3 antennas. Node 1 receives only m symbols corresponding to s_1,ℓ=1 where s_1^n is the state sequence in this block, with m≤ n and m/n≈τ as n grows. However, node 2 receives all symbols, and thus, obtains n-m codeword symbols from node 3 that have not been received by node 1. In block 2, node 3 sends w_31(2) similar to block 1, while nodes 2 cooperates with node 3. It does so by multiplying the received signal in block 1 by _̋32^† to obtain a noisy version of _31^n given by _31^n=_31^n+_2^n where _2,ℓ consists of a_3M_3 components of _̋32^†_2,ℓ, and then constructing _̌31^m out of _31,ℓ with ℓ∈{t∈{1,…,n}|s_1,t=0}, where _̌31,ℓ∈ℂ^a_2M_2. Then, it sends a new symbol of _̌31^m to node 1 in transmission ℓ if s_2,ℓ-1=1, and repeats the previously transmitted symbol otherwise. This construction requires ma_2M_2≤ (n-m)a_3M_3. The signal _̌31^m is sent from node 2 so that it is received linearly independent of _31^n at node 1. Thus, node 1 receives a total of m a_2M_2+m a_3M_3 symbols in this block if a_2M_2+a_3M_3≤ M_1. At the end of this block, node 1 is able to decode w_31(1) by combining its received signals from blocks 1 and 2. The same is repeated over blocks 3,…,B-1. In block B, only node 2 is active and delivers m a_2M_2 symbols to node 1. The achievable DoF is the ratio of the total number of delivered symbols to the total number of transmissions, i.e., d_31=(B-1)(m a_3M_3+m a_2M_2)/nB≈τ a_3M_3+τ a_2M_2, for large n and B. The constraints from above are 0≤ a_2,a_3 ≤ 1, a_2M_2,a_3M_3 ∈ℕ, a_2M_2+a_3M_3 ≤ M_1, τ a_2M_2-τ̅a_3M_3 ≤0. Now, we need to maximize d_31 with respect to a_2 and a_3 subject to these constraints. A feasible solution can be obtained as follows. First, we ignore the second constraint, which leads to a linear program which can be solved using the simplex method <cit.>. Solving the linear program leads to a_3^*=1 and a_2^*=min{M_1-M_3/M_2,1,τ̅M_3/τ M_2}. Then, we round a_2^*M_2 and a_3^*M_3 down to the nearest integer to obtain a_3=1, and a_2M_2=min{M_1-M_3,M_2,⌊τ̅M_3/τ⌋}. This leads to the achievability of min{τ M_1,τ M_2+τ M_3, τ M_3+τ⌊τ̅M_3/τ⌋}≜ d_31, a. Thus, the DoF tuple _̣ a=(0,0,0,0,d_31, a,0)∉𝒟_ r,τ^[ out] is achievable, which proves the desired result. This theorem proves the necessity of adaptation in the intermittent case, where cooperation between nodes 2 and 3 becomes necessary to achieve higher DoF. § CONCLUSION In this paper, we have investigated the impact of intermittency on the DoF region of the MIMO three-way channel. We have seen that adaptive encoding, can be either necessary or not, depending on the performance criterion. As far as the sum-DoF is concerned, adaptive encoding is not necessary, and the optimal sum-DoF can be achieved with restricted (nonadaptive) encoding. Since the sum-DoF might be unfair in terms of per-user or per-stream DoF, the DoF region is of high importance. In this case, we have shown that adaptation is in fact necessary, and that collaboration between nodes using adaptive encoding enlarges the DoF region beyond what can be achieved using adaptive encoding. 10 url@samestyle ChaabanSezgin_FnT A. Chaaban and A. Sezgin, “Multi-way communications: An information theoretic perspective,” Found. Trends Commun. Inf. Theory, vol. 12, no. 3-4, pp. 185–371, 2015. SongDevroyeShaoNgo Y. Song, N. Devroye, H.-R. Shao, and C. Ngo, “Lattice coding for the two-way two-relay channel,” submitted to IEEE Trans. Inf. Theory, arXiv:1212.1198 [cs.IT], Dec. 2012. ChengDevroyeLiu Z. Cheng, N. Devroye, and T. Liu, “The Degrees of Freedom of full-duplex bidirectional interference networks with and without a MIMO relay,” IEEE Trans. Wireless Commun., vol. 15, no. 4, pp. 2912–2924, April 2016. SabharwalSchniterGuo A. Sabharwal, P. Schniter, D. Guo, D. W. Bliss, S. Rangarajan, and R. Wichman, “In-band full-duplex wireless: Challenges and opportunities,” IEEE J. Sel. Areas Commun., vol. 32, no. 9, pp. 1637–1652, Sep. 2014. TehraniUysalYanikomeroglu M. N. Tehrani, M. Uysal, and H. Yanikomeroglu, “Device-to-device communications in 5G cellular netowrks: Challenges, solutions, and future directions,” IEEE Commun. Mag., pp. 86–92, May 2014. Shannon_TWC C. E. Shannon, “Two-way communication channels,” in Proc. of 4th Berkeley Symp. Math. Stat. Probab., vol. 1, 1961, pp. 611–644. ChaabanMaierSezginMathar A. Chaaban, H. Maier, A. Sezgin, and R. Mathar, “Three-way channels with multiple unicast sessions: Capacity approximation via network transformation,” IEEE Trans. Inf. Theory, vol. 62, no. 12, pp. 7086–7102, Dec. 2016. Ong L. Ong, “Capacity results for two classes of three-way channels,” in Proc. IEEE Int. Symp. Commun. Inf. Tech. (ISCIT), Gold Coast, QLD, Oct. 2012. ElmahdyKeyiMohassebElBatt A. M. Elmahdy, A. El-Keyi, Y. Mohasseb, T. ElBatt, M. Nafie, and K. G. Seddik, “Asymmetric degrees of freedom of the full-duplex MIMO 3-way channel,” in Proc. IEEE Inf. Theory Workshop, Sep. 2016, pp. 469–473. mach2015band P. Mach, Z. Becvar, and T. Vanek, “In-band device-to-device communication in OFDMA cellular networks: A survey and challenges,” Commun. Surveys Tuts, vol. 17, no. 4, pp. 1885–1922, Nov. 2015. KarakusWangDiggavi C. Karakus, I. H. Wang, and S. Diggavi, “Gaussian interference channel with intermittent feedback,” IEEE Trans. Inf. Theory, vol. 61, no. 9, pp. 4663–4699, Sep. 2015. WangSuhDiggaviViswanath I. H. Wang, C. Suh, S. Diggavi, and P. Viswanath, “Bursty interference channel with feedback,” in Proc. IEEE Int. Symp. Inf. Theory, July 2013, pp. 21–25. VahidMaddahAliAvestimehr A. Vahid, M. A. Maddah-Ali, and A. S. Avestimehr, “Binary fading interference channel with no CSIT,” in Proc. IEEE Int. Symp. Inf. Theory, June 2014, pp. 666–670. YehWang S. Y. Yeh and I. H. Wang, “Degrees of freedom of the bursty MIMO X channel without feedback,” in Proc. IEEE Int. Symp. Inf. Theory, July 2016, pp. 1312–1316. Varshney L. R. Varshney, “Two way communication over exponential family type channels,” in Proc. IEEE Int. Symp. Inf. Theory, July 2013, pp. 2795–2799. ChengDevroye Z. Cheng and N. Devroye, “Two-way networks: When adaptation is useless,” IEEE Trans. Inf. Theory, vol. 60, no. 3, pp. 1793–1813, March 2014. SongAlajajiLinder L. Song, F. Alajaji, and T. Linder, “Adaptation is useless for two discrete additive-noise two-way channels,” in Proc. IEEE Int. Symp. Inf. Theory, July 2016, pp. 1854–1858. CoverThomas T. Cover and J. Thomas, Elements of Information Theory (Second Edition).1em plus 0.5em minus 0.4emJohn Wiley and Sons, Inc., 2006. JafarShamai_XChannel S. A. Jafar and S. Shamai, “Degrees of freedom region for the MIMO X channel,” IEEE Trans. Inf. Theory, vol. 54, no. 1, pp. 151–170, Jan. 2008. MatousekGartner J. Matous̆ek and B. Gärtner, Understanding and using linear programming.1em plus 0.5em minus 0.4emSpringer, 2006. MaierChaabanMathar H. Maier, A. Chaaban, and R. Mathar, “Degrees of Freedom of the MIMO 3-way channel,” in Proc. Int. Symp. Inf. Theory Appl., Melbourne, Australia, Oct. 2014, pp. 625–629.

http://arxiv.org/abs/1701.07659v1

20170126112430

Ab initio calculations for non-strange and strange few-baryon systems

nucl-th

The EUSO@TurLab Project for the JEM-EUSO Collaboration December 30, 2023 ============================================= Concerning the non-strange particle systems the low-energy excitation spectra of the three- and four-body helium isotopes are studied. Objects of the study are the astrophysical S-factor S_12 of the radiative proton deuteron capture d(p,γ)^3He and the width of the ^4He isoscalar monopole resonance. Both observables are calculated using the Lorentz integral transform (LIT) method. The LIT equations are solved via expansions of the LIT states on a specifically modified hyperspherical harmonics (HH) basis. It is illustrated that at low energies such a modification allows to work with much higher LIT resolutions than with an unmodified HH basis. It is discussed that this opens up the possibility to determine astrophysical S-factors as well as the width of low-lying resonances with the LIT method. In the sector of strange baryon systems binding energies of the hypernucleus ^3_ΛH are calculated using a nonsymmetrized HH basis. The results are compared with those calculated by various other groups with different methods. For all the considered non-strange and strange baryon systems it is shown that high-precision results are obtained. The EUSO@TurLab Project for the JEM-EUSO Collaboration December 30, 2023 ============================================= § INTRODUCTION Non-strange and strange few-baryon systems are particularly interesting particle systems in the hadronic sector. On the one hand they serve for parametrization and test of potential models for nucleon-nucleon (NN), nuc-leon-hyperon (NY), hyperon-hyperon (YY) interactions and the various analogous three-body interactions. On the other hand they play an important role in testing the quality of ab initio methods, for example in benchmark calculations. In the present work the second of these two aspects is of relevance. In fact our aim is to test the quality of two different ab initio techniques for specific physical questions as is explained in greater detail in the following. One of the tested ab initio methods is the LIT. The LIT approach is well-established <cit.> and allows to determine observables involving the many-body continuum without the necessity to calculate continuum wave functions. In the present work it is investigated to what extent specific features in the low-energy electromagnetic response of nuclei can be determined with the LIT method. We consider two examples: (i) the width of the ^4He isoscalar monopole resonance 0^+ and (ii) the threshold cross section in ^3He photodisintegration. The inverse reaction of the latter, the radiative proton-deuteron capture, is of relevance for the nucleosynthesis and usually parametrized via the astrophysical S-factor S_12. As explained in section <ref> the crucial point for an exact description of the observables mentioned above lies in the question whether a sufficiently high density of LIT states can be obtained in the low-energy region. In a rather recent LIT calculation based on HH expansions <cit.>, where the ^4He inelastic isoscalar monopole response function was computed with realistic nuclear forces, this aim could not be achieved even though the HH basis was quite large. Therefore it was not possible to determine the width of the 0^+ resonance in this calculation. In <cit.> it was then shown that the problem is due to the employed HH basis and that a somewhat modified many-body basis solves the problem of a too low low-energy density of LIT states. The modification consists in using for an A-body system, instead of an A-body HH basis, an (A-1)-body HH basis times a basis set for the relative motion of the A-th particle with respect to the center of mass of the (A-1)-body system. For the strange particle systems responses to external probes have not yet been determined in experiment. In fact the knowledge of such systems is still rather scarce. One source of experimental information are binding energies of hypernuclei. One of the future aims of our group are ab initio calculations of such binding energies with realistic forces. However, first we want to test the reliability and the precision of the ab initio approach chosen by us. Different from the basis systems, which is used for the above mentioned LIT calculations, where the various HH basis states have a well-defined permutational symmetry, we take for the bound-state calculations of strange few-body baryon systems a nonsymmetrized HH (NSHH) basis. The results are then compared with results coming from other ab initio approaches. We switch from a symmetrized to a nonsymmetrized HH basis because we are confident that calculations for A-body baryon systems with A≥6 can be carried out with less computational effort. Since in the present work it is the aim to test the precision of various theoretical ab initio approaches we do not employ realistic interaction models, but use instead simpler potential models, which will be defined in the following sections. This work is organized as follows. In section <ref> the LIT method and the used many-body basis systems are briefly described. Furthermore, the LIT results for the above mentioned S-factor S_12 as well as for the 0^+ resonance of ^4He are discussed. In both cases results with the HH and the new basis are compared. In section <ref> it is described how nonsymmetrized basis systems can nonetheless be used to determine ground states of systems which obey a specific permutational symmetry. Subsequently the results for binding and Λ separation energies of ^3_ΛH are illustrated in comparison to results from other authors with different ab initio few-body techniques. Finally, in section 4 a summary is given. § THE LIT METHOD Nuclear cross sections of inclusive reactions with electromagnetic probes are expressed in terms of inclusive response functions, which contain the information about the dynamics of the nucleus under investigation. Inclusive response functions are in general of the following form R(ω) = ∫ df |⟨ f| Ô | 0⟩|^2 δ(E_f - E_0 - ω) , where |0 ⟩ and |f⟩ are nuclear ground and final states, E_0 and E_f are the corresponding eigenenergies and ω is the energy of the exchanged real (photoabsorption) or virtual photon (electron scattering). Finally, Ô denotes the operator inducing the reaction. A calculation of R(ω) can be become very difficult or even impossible for cases where |f⟩ is a many-body continuum state. However an explicit calculation of |f⟩ can be avoided by the use of the LIT, which is an integral transform defined as follows L(σ) = ∫ dω R(ω)/(ω-σ_R)^2 + σ_I^2 with σ = σ_R + i σ_I. Due to the variable width of 2 σ_I of the Lorentzian kernel the LIT is an integral transform with a controlled resolution. But it is important to realize that in a given calculation one cannot simply increase the resolution by choosing smaller and smaller σ_I values. In fact one has to make sure that the precision of the LIT calculation allows the choice of a smaller σ_I value. How this can be achieved becomes clearer in the discussion that follows next. Since the aim is to determine the response function without the knowledge of the continuum wave function it is useless to calculate the LIT via its definition of eq. (<ref>). Fortunately, the LIT can be determined in an alternative way, namely by solving an equation, the LIT equation, given by (Ĥ-E_0-σ) |Ψ̃(σ) = Ô | 0⟩ , where Ĥ is the nuclear Hamiltonian. The important feature of the solution Ψ̃(σ) is that it is a localized function. Therefore one can compute Ψ̃(σ) using bound-state methods. After having determined Ψ̃(σ) one calculates the LIT from the following expression L(σ) = ⟨Ψ̃(σ) | Ψ̃(σ) ⟩ . In order to obtain the response function R(ω) one has to invert the LIT. Details about inversion methods are described in <cit.>. As already mentioned in the introduction an expansion on a complete many-body basis is used for the solution of the LIT equation (<ref>). First, the Hamiltonian matrix for such a basis is determined, then, in a subsequent diagonalization of this matrix, N eigenvalues E_n and eigenstates ϕ_n (LIT states) (n = 1,2,...,N) are obtained, where N is the dimension of the basis. The LIT can then be expressed in terms of the energy eigenvalues and the LIT states. One obtains L(σ) = ∑_n=1^N S_n/(σ_R-(E_n-E_0))^2 + σ_I^2 . with S_n = |⟨ϕ_n| Ô | 0 ⟩ |^2 . Coming back to the question which resolution or in other words which value of σ_I can be sustained in a given LIT calculation it is already pointed out in the introduction that the density of LIT states plays a crucial rule. In fact the higher the density of LIT states the higher is also the resolution. This point will be better illustrated in section <ref>. §.§ Many-body basis systems In most of the LIT applications the HH basis has been used for the expansions of ground-state wave functions and LIT states of the considered A-body system. This is mainly attributed to its property of being a complete A-body basis for localized states. An HH basis has the following form HH_[K]n(Ω_A,ρ_A) = Y_[K](Ω_A) R_n(ρ_A) . It consists of an hyperangular part Y_[K](Ω_A) and a hyperradial part R_n(ρ_A), where Ω_A is a set of 3A-4 hyperangles, ρ_A denotes the hyperradius and [K] stands for a set of hyperspherical quantum numbers. For the hyperradial basis functions Laguerre polynomials L_n^(β)(ρ_A) times an exponential factor exp(-ρ_A/2b) are used, where β and b are a free parameters. In addition one can also introduce as a multiplicative factor NN short-range correlation functions of Jastrow type, which can be purely central or can also become spin and/or isospin dependent (see e.g. <cit.>). Usually the hyperangular states Y_[K] are constructed with a well-defined permutational symmetry. With a complementary permutational symmetry of the spin-isospin part of the nuclear wave function one then obtains an antisymmetric basis. In case of A=2 the hyperangular basis functions reduce to the well-known spherical harmonics Y_lm. A detailed description of the HH expansion technique is given e.g. in <cit.>. In the LIT applications of the present work a new basis Φ_[K]nn'l is employed in addition. It consists of a separation of the A-body basis in a (A-1)-part with HH basis functions R_n(ρ_A-1) Y_[K](Ω_A-1) and a single-particle part with basis functions R_n'^(2)(r_A') Y_lm(Ω_r_A'): Φ_[K]nn'l = Y_[K](Ω_A-1) R_n(ρ_A-1) R_n'^(2)(r_A') Y_l(m)(Ω_r_A') with r'_A = r_A - R_cm^(A-1), where r_A and R_cm^(A-1) is the position of the A-th particle and the center of mass of the (A-1)-particle system, respectively. For R_n'^(2)(r'_A) a similar expansion as for the hyperradial part is taken, namely a Laguerre polynomial L_n'^(2)(r_A') times an exponential factor exp(-r'_A/2b_A). Also in this case one may use the NN correlation functions, discussed above, in addition. Of course, as for the HH basis given in eq. (<ref>), one has to multiply the basis functions of eq. (<ref>) with appropriate A-body spin-isospin wave functions and then one has to care for an antisymmetric state by making a proper antisymmetrization of the basis states. §.§ Photodisintegration of ^3He and astrophysical S-factor S_12 1cm As pointed out in the introduction, in this work the S-factor of the reaction d(p,γ)^3He is determined via the inverse reaction, the ^3He photodisintegration, then time reversal invariance is applied to obtain S_12. We take the unretarded dipole approximation for the calculation of the ^3He photodisintegration cross section, which is given by σ_ E1(ω) = 4 π^2 αω R_ E1(ω) , where α is the fine structure constant and R_ E1(ω) denotes the dipole response function. In this case the components Ô and |0⟩ of eq. (<ref>) become equal to D_z, the third component of the nuclear dipole operator D, and the ^3He ground-state wave function, respectively. In order to determine the S-factor S_12 one only needs to take into account the low-energy ^3He photoabsorption cross section, which is exclusively due to the two-body breakup channel ^3He+ γ→ p + d. Since the pd channel has isospin T=1/2, only the T=1/2 channel is considered for the LIT equation (<ref>). As NN potential the MT-I/III potential <cit.> is employed. To speed up the convergence of the expansions, both with HH basis and new basis, the already mentioned central NN short-range correlation functions are taken into account in addition. First LIT results with a three-body HH basis are considered. As mentioned in section <ref> the hyperradial basis functions contain Laguerre polynomials L_n^(β), here β=5 is taken. In fig. 1 the LIT for the case with 30 hyperangular and 31 hyperradial states (b = 0.3 fm) is shown. One sees that a smooth LIT is obtained with σ_I =20 MeV, while with σ_I = 2.5 MeV the contributions of single LIT states becomes visible at higher energies. Such contributions due to single LIT states become even the dominant feature for σ_I = 0.5 MeV. This is a clear sign that the LIT-state density is too low to support a resolution with a σ_I value of 0.5 MeV. In fact the resolution of strength encoded in the LIT depends on the relative distance Δ E of two neighbouring LIT states. Structures with a width smaller than Δ E can hardly be resolved by an inversion of the LIT. In other words the higher the density of LIT states the finer the details that can be resolved inverting the LIT. 1cm In order to enhance the LIT-state density one can increase the number of HH basis states taking more hyperangular and/or hyperradial states. In addition one can use a larger spatial extension of the basis by taking a greater value for the hyperradial parameter b, which then leads to a shift of LIT states towards lower energies. The resulting effects on the LIT are discussed in greater detail in <cit.>. Here, in fig. 2, we only compare the LITs of fig. 1 at low energies with those obtained with an HH basis of 40 hyperangular and 76 hyperradial states with b = 1 fm. From the LIT results with σ_I = 0.01 MeV it is readily seen that the density of LIT states becomes much larger with the increased HH basis. Accordingly one finds for the lower resolutions of σ_I equal to 0.1 and 0.5 MeV much smoother LIT results with the increased HH basis. However, one also notes a very important point: even in fig. 2b there is not a single LIT state below the three-body breakup threshold at about 8 MeV (^3He binding energy with MT potential). Thus the information about the response function is only rather scarce in the energy range between the two-body breakup threshold at about 5.8 MeV and the three-body breakup threshold. Also with regard to the results following in section <ref> and to those of <cit.> one may conclude that for an HH basis a systematic increase of the LIT-state density cannot easily be achieved in an energy region where only two-body breakup channels are open. Therefore the HH basis is not very suitable to obtain precise results for such cases, as for example the present one of low-energy ^3He photodisintegration. 1cm Now we turn to the results with the new basis described in section <ref>. The (A-1)-basis corresponds in the present three-body case to a two-body basis with basis states Y_lm(Ω_r) L_n'^(2)(r) exp(-r/b_2) with r = r_2 - r_1, where r_i is the position of the i-th particle. A basis is used with 25 and 80 radial states for the two-body and single-particle basis, respectively (b_2=0.75 fm, b_3=0.5 fm). Since in the present case only the low-energy part of the response function is relevant, it is sufficient to take into account only s-states for the two-body basis. In fig. 3 LITs resulting from the increased HH basis of fig. 2b and those obtained with the new basis are shown. Figure 3a illustrates that both results are very similar for σ_I=20 MeV, while with σ_I=10 MeV one finds some differences in the region of the maximum. For a much smaller σ_I of 0.01 MeV, shown in fig. 3b, strong differences become evident. In fact, only with the new basis LIT states are present right above the two-body breakup threshold at about 5.8 MeV. Moreover these states have a rather high density. More details of the LIT calculation with the new basis, like for example the convergence behaviour with respect to the two-body and single-particle basis systems, are discussed in <cit.>. The results presented in fig. 3 show that the use of a proper many-body basis can become important for specific questions. There are two conditions which should be considered: (i) is the density of LIT states sufficiently high in order to extract specific structures in the response function and (ii) is the LIT-state density sufficiently regular in order to work with a single σ_I value. If the second condition is not fulfilled one should take in a region of lower LIT-state density a different, more suitable, value for σ_I, otherwise one risks to misplace strength in the inversion. In fact in <cit.> quite a number of different σ_I values were used in order to take into account a lower LIT-state density with growing energy. For the LIT with the present HH basis one can conclude that the completely missing LIT states in the two-body breakup region do not only prevent to resolve the correct threshold behaviour of R_ E1(ω), but that one would also obtain an overestimation of the peak height of the response function if for the inversion one uses a LIT with a σ_I value much smaller than 20 MeV. As illustrated in <cit.> (see fig. 7 therein) one can still obtain a rather reasonable inversion result with σ_I=20 MeV using the standard inversion method, described in <cit.>, where the correct threshold behaviour of the response function is implemented. Concerning the LIT results with the new basis one can certainly say that the low-energy density of LIT states is quite high and that the pattern is very regular. Thus one may expect that the low-energy response, and thus the astrophysical S-factor S_12, can be determined very precisely by the inversion. In fig. 4 we show the result for S_12 obtained with the new basis in comparison to a calculation of S_12 with explicit wave functions for the proton-deuteron continuum states (for details of the continuum state calculation see <cit.>). As anticipated one observes an excellent agreement between both results. Note that the figure also contains an error estimate due to the LIT inversion, also here we refer to <cit.> for a more detailed description of the error estimate. 1.1cm §.§ The ^4He isoscalar monopole resonance The isoscalar monopole resonance 0^+ of the α-particle leaves a strong signal in inclusive inelastic electron scattering experiments <cit.>. The corresponding transition form factor was studied in a LIT calculation, where an HH basis and modern realistic forces were used <cit.>. A rather strong potential model dependence was found, but the experimental data were overestimated quite a bit. The present work, however, is not devoted to determine the strength of the transition form factor, but rather to a different aspect of the resonance, namely its rather small width of 270(50) keV as determined in the ^4He(e,e') experiments mentioned above. In <cit.> this question could not be addressed because the density of LIT states was not sufficiently high in the region of the 0^+ resonance, which is located closely above the lowest ^4He two-body breakup threshold. The isoscalar monopole response function R_ C0(q,ω) depends on energy transfer ω and momentum transfer q mediated in electron scattering by the exchanged virtual photon. Thus the corresponding transition operator Ô of eq. (<ref>) becomes q-dependent: Ô(q) =G_E^s(q^2)/2∑_i=1^A j_0(q r_i) . In the equation above G_E^s(q^2) is the nucleon isoscalar electric form factor, r_i is the position of nucleon i, and j_0 is the spherical Bessel function of 0^th order. For the present study the LIT of R_ C0(q,ω) is taken at q=300 MeV/c, a q value, which lies in the momentum transfer range of maximal strength of the 0^+ transition form factor. Here we consider results with an HH basis for the four-body system and in addition the new basis as described in section <ref> (three-body HH basis plus single-particle basis). For details of the used basis states we refer to <cit.>. As NN potential model the central TN potential is taken, it has been used in the very first LIT applications for the α-particle (see for example <cit.>). Like in the previous case in section <ref> central NN correlation functions are used in order to accelerate the convergence of HH and new basis. In fig. 5 the LIT results for the response function are shown. For the HH basis in fig. 5a one sees a very similar picture as in fig. 2a. There is only one essential difference, in fig. 2a there are no LIT states below the many-body breakup threshold, whereas in fig. 5a one finds just one LIT state below the many-body breakup threshold at about 30 MeV (note ^4He binding energy with TN potential is 31.4 MeV). The isolated low-energy LIT state for the ^4He case is due to the 0^+ resonance. It is evident that with a single LIT state it is impossible to determine a resonance width. In the already mentioned LIT calculation for R_ C0(q,ω) with realistic nuclear forces of <cit.> the situation was somewhat better, but the LIT-state density could not be systematically improved in order to have sufficient information to compute the 0^+ resonance width. As illustrated in fig. 5b the situation is much better in case of the new basis. It is interesting to study the results with the various σ_I values. With a low resolution of σ_I= 5 MeV one does not realize that there is a resonance (note the logarithmic scale). If one increases the resolution using smaller σ_I values the resonance becomes more and more distinct from the background. For the highest resolution of σ_I= 0.01 MeV one sees that there are various LIT states in the region of the resonance at about 26 MeV. In fact in <cit.> it was possible to determine the width using besides the results of fig. 5b also LIT results with a different basis size (for details see <cit.>). The obtained width of 180(70) keV agrees quite well with the experimental value of 270(50) keV. 0.75cm § STRANGE BARYON SYSTEMS In the benchmark calculation of <cit.> bound baryon systems from three up to five particles are considered, where one of the baryons is the Λ hyperon which has strangeness S=-1. As already mentioned in the introduction the experimental information about the YN interaction is still rather scarce. On the other hand, our present aim is not yet a realistic calculation of the binding energy of hypernuclei, but rather a check of the precision of the ab initio method used by us. Therefore in <cit.> calculations with non-fully realistic interactions models are made. Before coming to some of these results in section <ref>, first, a short description of the ab initio method of our choice is given in the following section. §.§ The nonsymmetrized hyperspherical harmonics (NSHH) expansion The NSHH expansion relies on the HH expansion given in eq. (<ref>), but the hyperspherical functions Y_[K](Ω_A) are not constructed with any permutational symmetry. Also the spin-isospin part of a hypernuclear basis state is taken without imposing a permutational symmetry. On the other hand it is clear that a hypernuclear wave function has to be antisymmetric under the exchange of two identical fermions. At this point it is helpful to consider the Casimir operator Ĉ of the particle system. Taking an A-body baryon system with N nucleons (n=1,2,...,N) and L Λ hyperons (n=N+1,N+2,...,N+L=A) one has Ĉ = Ĉ_N + Ĉ_Λ = ∑_j>i=1^N P̂_ij + ∑_j>i=N+1^A P̂_ij , where the operator P̂_ij exchanges particles i and j. The eigenvalues λ_[I] of the Casimir operator depend on the specific permutational symmetry of the eigen functions. In our case with just one Λ particle only the permutational symmetry of the nucleons is relevant. One has the lowest eigenvalue for the antisymmetric case (λ_[A]=-N(N-1)/2) and the highest for the symmetric case (λ_[S]=+N(N-1)/2). Thus diagonalizing the Hamiltonian matrix for an NSHH basis, one can find out the symmetry of a given eigenstate by calculating the corresponding eigenvalue λ_[I] applying the Casimir operator. This is the strategy which has been put forward in <cit.>. If, however, the NSHH basis is very large and one wants to find the lowest state being antisymmetric for the nucleonic part, which in general is not the lowest energy state, it is more convenient concerning the computational resources to apply the strategy of <cit.>. In fact in case of large basis systems it is not advisable to perform a complete diagonalization of H. It is much better to use the Lanczos technique, which saves computational resources and leads to a fast determination of the lowest energy state. In order to bring the lowest antisymmetric state for the nucleonic part to the lowest overall state the following fictitious Hamiltonian Ĥ' has been introduced in <cit.>: Ĥ' = Ĥ + γĈ_N . Thus, for a sufficiently large γ, such a lowest antisymmetric state will become the absolute ground state of the particle system with ground-state energy E_0. Therefore, taking Ĥ' instead of Ĥ, one can apply the Lanczos technique to find the proper ground state for an A-baryon system with A-1 nucleons and one Λ hyperon. In order to have the correct bound-state energy one needs to correct E_0 only by γ N(N-1)/2. §.§ The hypernucleus ^3_ΛH Here the benchmark results for one of the strange baryon systems discussed in <cit.> are illustrated, namely those for ^3_ΛH. Two different potential sets were used in these calculations: (i) the AV4' NN potential <cit.> together with the Bodmer-Usmani YN potential <cit.> and (ii) the AV8' NN potential <cit.> together with a parametrization <cit.> of the meson-theoretical NSC97f YN potential <cit.>. In order to accelerate the NSHH expansion an effective interaction is used as described in <cit.>. Besides the NSHH results new results with the auxiliary field diffusion Monte Carlo (AFDMC) technique <cit.> have been obtained. Also results due to the Faddeev approach (FY) and due to the Gaussian expansion method <cit.> (GEM) are included in <cit.>. In table 1 the binding energy and the Λ separation energy of ^3_ΛH are listed for the two different potential models defined above. One observes a rather good agreement between the various methods. Only the AFDMC results are a bit different, but this is not a real surprise since the AFDMC is an ab initio method more suitable for systems with more than three particles and with some preference for closed shell nuclei. In fact further calculations discussed in <cit.> show that the comparison of AFDMC results with those of the other ab initio methods become decisively better for ^4_ΛH and ^5_ΛHe. With the results of table 1 and the further ones given in <cit.> one can conclude that the NSHH method is very well suited to give precise results for observables of hypernuclei. A further benchmark with the AFDMC method, where also three-body interactions are taken into account, will be published in the near future. Thus, more ambitious calculations with more realistic interaction models can be tackled with the NSHH method in future. § SUMMARY The purpose of this work is twofold. Firstly, it is a check of the applicability of the LIT method for a precise determination of specific details in nuclear low-energy cross sections that are induced by external electromagnetic probes. To this end the reactions ^3He(γ) and ^4He(e,e') have been considered. The ^3He photodisintegration has been calculated in order to obtain the astrophysical S-factor S_12 of its inverse reaction, i.e. d(p,γ)^3He, by applying time reversal invariance. Thus the actual aim has been a precise determination of S_12 via the LIT method. Comparing to results of a calculation with explicit proton-deuteron continuum wave functions it has been shown that the LIT leads to excellent results for S_12. The calculation has been carried out with a simple central NN potential, but the real importance of the calculation does not lie in a realistic calculation of S_12, more important is the fact that the LIT method could serve to calculate astrophysical S-factors of reactions involving more than three nucleons. Also the inclusive electrodisintegration of ^4He has been computed here with a central NN interaction. Again, the essential aim has not been to obtain realistic results, but to test the possibility to determine the width of a narrow resonance with the LIT method. In fact the ^4He continuum exhibits a rather narrow resonance, the so-called isoscalar monopole resonance 0^+. Therefore the LIT calculation has been performed for the ^4He isoscalar monopole response function R_ C0(q,ω). It has been shown that the resonance width can be determined with the LIT method and that a value of 180(70) keV is obtained. This agrees quite well with the experimental result of 270(50) keV. Thus, there is cause for hope that the 0^+ resonance width can be determined with the LIT method also for the case of a realistic nuclear force. It has been pointed out that the decisive point for precise determinations of S-factor S_12 and of ^4He 0^+ resonance width is a sufficiently high density of LIT states. Unfortunately, this seems to be very difficult to achieve if one uses as A-body basis an HH A-body basis. It is much better to take an HH hybrid basis consisting of an (A-1)-body HH basis and an additional single-particle basis for the A-th nucleon. It has been illustrated that with such a basis it is possible to systematically increase the density of LIT states in the two-body breakup region at low-energies. The second purpose of the present work has been a benchmark calculation for ^3_ΛH. The ab initio method of our choice, expansion of the hypernuclear ground state on a nonsymmetrized HH basis, has been discussed, in particular, how such a basis can serve to obtain a ground state with a proper permutational symmetry. Two different potential model sets have been employed for NN and NΛ interactions. In comparison to results from other ab initio approaches it has been found that the present calculation leads to reliable results for the ^3_ΛH binding energy and the corresponding Λ separation energy. Finally, it should be mentioned that more details of the various calculations are given in <cit.>. 9 EfL07 Efros V D, Leidemann W, Orlandini G and Barnea N 2007 J. Phys. G 34 R459 BaB13 Bacca S, Barnea N, Leidemann W and Orlandini G 2013 Phys. Rev. Lett. 110 042503 Lei15 Leidemann W 2015 Phys. Rev. C 91 054001 Lei08 Leidemann W 2008 Few-Body Syst. 42 139 EfL00 Efros V D, Leidemann W, Orlandini G and Tomusiak E L 2000 Phys. Lett. B 484 223 MaT69 Malfliet R A and Tjon J A 1969 Nucl. Phys. A 127 161 DeE17 Deflorian S, Efros V D and Leidemann W 2017 Few-Body Syst. 58 3 Wa70 Walcher Th 1970 Phys. Lett. B 31 442 Fr65 Frosch R F, Rand R E, Yearian M R, Crannell H and Suelzle L R 1965 Phys. Lett. 19, 155; Frosch R F, Rand R E, Crannell H, Mccarthy J S and L R Suelzle L R 1968 Nucl. Phys. A 110 657 Ko83 Köbschall G, Ottermann C, Maurer K, Röhrich K, Schmitt Ch and Walther V H 1983 Nucl. Phys. A 405 648 FeB17 Ferrari Ruffino F, Barnea N, Deflorian S, Leidemann W, Lonardoni D, Orlandini G and Pederiva F 2017 arXiv: 1701.06399 GaK11 Gattobigio M, Kievsky A and Viviani M 2011 Phys. Rev. C 83 024001 DeB13 Deflorian S, Barnea N, Leidemann W and Orlandini G 2013 Few-Body Syst. 54 1879 WiP02 Wiringa R B and Pieper S C 2002 Phys. Rev. Lett. 89 182501 BoU88 Bodmer A R and Usmani Q N 1988 Nucl. Phys. A 477 621; Usmani A A and Khanna F C 2008 J. Phys. G 35 025105 HiO14 Hiyama E, Ohnishi S, Gibson B F and Rijken Th A 2014 Phys. Rev. C 89 061302 RiS99 Rijken Th A, Stoks V G J and Yamamoto Y 1999 Phys. Rev. C 59 21 BaL00 Barnea N, Leidemann W and Orlandini G 2000 Phys. Rev. C 61 054001 LoG13 Lonardoni D, Gandolfi S, Pederiva F 2013 Phys. Rev. C 87 041303R; Lonardoni D, Gandolfi S, Pederiva F 2014 Phys. Rev. C 89 014314

http://arxiv.org/abs/1701.07441v1

20170125190011

Cosmic ray feedback heating of the intracluster medium

astro-ph.HE

^1Department of Astronomy, University of Michigan, 1085 S University Ave, 311 West Hall, Ann Arbor, MI 48109 ^2Department of Astronomy, University of Maryland, College Park, MD 20742 3Einstein Fellow mateuszr@umich.edu (MR), hsyang@astro.umd.edu (KY), chris@astro.umd.edu (CR) Self-regulating active galactic nuclei (AGN) feedback in the cool cores of galaxy clusters plays central role in solving the decades-old cooling flow problem. While there is consensus that AGN provide most if not all of the energy needed to offset radiative losses in the intracluster medium (ICM) and prevent catastrophically large star formation rates, one major problem remains unsolved – how is the AGN energy thermalized in the ICM and what are the effective black hole feeding rates in realistic systems? We perform a suite of three-dimensional magneto-hydrodynamical (MHD) adaptive mesh refinement simulations of AGN feedback in a cool core cluster including cosmic ray (CR) physics. CRs are supplied to the ICM via collimated AGN jets and subsequently disperse in the magnetized ICM via streaming, and interact with the ICM via hadronic, Coulomb, and streaming instability heating. We find that CR transport is an essential model ingredient needed for AGN feedback to self-regulate, at least within the context of the physical model considered here. When CR streaming is neglected, the suppression of CR mixing with the ICM by magnetic fields significantly reduces ICM heating, which leads to cooling catastrophes. In the opposite case, when CR streaming is included, CRs come into contact with the ambient ICM and efficiently heat it, which results in globally stable atmospheres. Moreover, the dynamical state and intermittency of the central AGN are dramatically altered when CR streaming is present – while the AGN is never in a completely off-state, it is more variable, and the atmosphere goes through cycles characterized by low gas velocity dispersion interspersed with more violent episodes. We find that CR streaming heating dominates over the heating due to Coulomb and hadronic processes. Importantly, in simulations that include CR streaming, CR pressure support in the central 100 kpc is very low and does not demonstrably violate observational constraints. On the contrary, when CR streaming is neglected, CR energy is not spent on the ICM heating and CR pressure builds up to the level that is in disagreement with the data. Overall, our models demonstrate that CR heating is a viable channel for the thermalization of AGN energy in clusters, and likely also in elliptical galaxies, and that CRs play an important role in determining AGN intermittency and the dynamical state of cool core atmospheres. Cosmic ray feedback heating of the intracluster medium Mateusz Ruszkowski^1,2, H.-Y. Karen Yang^2,3, and Christopher S. Reynolds^2 =============================================================================== § INTRODUCTION One of the long-standing puzzles in modeling of galaxy clusters is the “cooling-flow problem” <cit.> – clusters with short central radiative cooling times, i.e., cool-core clusters, are predicted to host massive inflows of gas and to harbor large amounts of cold gas and stars near their centers, significantly in excess of what is observed. Various heating mechanisms of the ICM in cool cores have been proposed in order to prevent or reduce these inflows, among which AGN feedback is the most promising one <cit.>. These mechanisms include heating by dynamical friction acting on substructure (e.g., <cit.>), conduction of heat from the outer hot layers of cool cores to their centers (e.g., <cit.>, <cit.>, <cit.>), precipitation-driven AGN feedback (e.g., <cit.>), conduction and AGN feedback (e.g., <cit.>), dissipation of AGN-induced sound waves and weak shocks (e.g., <cit.>), and cosmic ray heating (e.g., <cit.>). Strong argument in favor of the AGN mechanism comes from the prevalence of AGN jet-inflated radio bubbles in cool-core clusters and the correlation between the estimated jet power and central cooling luminosity. Despite the observational evidence supporting AGN feedback, numerical modeling of AGN accretion and feedback suffers from large uncertainties rooted in the huge separation of scales between the size of supermassive black hole accretion disks and that of clusters. Another major unsolved problem in modeling AGN feedback concerns the issue of thermalization of the AGN jet energy in the ICM. Detailed understanding of this process is needed to discover how the supermassive black hole feedback and feeding really work in realistic systems. In recent years hydrodynamic simulations made substantial progress in terms of understanding AGN accretion and feedback processes in clusters. Earlier simulations that include Bondi accretion of hot gas and injection of thermal energy demonstrated that supermassive black hole feedback can be self-regulated (e.g., <cit.>). More recently, motivated by multiple theoretical and observational studies that focus on the role of thermal instability in the ICM in feeding the central supermassive black hole (e.g., <cit.>), simulations including cold-gas accretion and momentum-driven feedback have successfully reproduced the positive temperature gradients and properties of cold gas within the cool cores <cit.>. These kinds of simulations provided valuable insights into the mysteries of how the AGN energy is transformed into heat and how the heat is distributed radially and isotropically throughout the cool core. Specifically, <cit.> and <cit.> showed that mixing with ultra-hot thermal gas within bubbles and shock heating are the dominant heating mechanisms. Moreover, <cit.> showed that a gentle circulation flow on billion-year timescale is responsible for partially compensating cooling and transporting the heat provided by the AGN in an isotropic manner. Despite these successes, fundamental and important physical processes are not captured in purely hydrodynamic models. One of the assumptions of the above-mentioned hydrodynamic models is that, because the injected kinetic energy is quickly turned into thermal energy by shocks during the initial inflation phase, the bubbles are filled with ultra-hot thermal gas. In reality, the composition of radio bubbles is still largely unknown. Observational estimates generally show that the pressure contributed by radio-emitting CR electrons plus magnetic pressure is small compared to the ambient pressure, suggesting that the bubbles are dominated by either non-radiating CR particles or ultra-hot thermal gas <cit.>. While momentum-driven jet models often produce radially elongated bubbles, CR-dominated light jets can naturally inflate fat bubbles like those observed at the center of Perseus <cit.>. Both types of bubble shapes appear to exist in observed cool cores, suggesting that the bubbles could have a range of different compositions <cit.>. In terms of heating the ICM, CR-dominated bubbles are expected to behave qualitatively differently from hydrodynamic bubbles. First, they expand with an effective adiabatic index of 4/3 instead of 5/3. Second, while mixing is a primary heating mechanism for hydrodynamic bubbles, CR bubbles contain less thermal energy that could be accessed by the ICM via mixing. Also, the level of mixing and the distance bubbles could travel before getting disrupted by instabilities depend on a number of factors, such as the smaller amount of momentum they carry, their lower density, CR diffusion along the magnetic field, and the topology of the magnetic field in the ICM <cit.>. Third, the surrounding ICM partially mixed with the CR bubbles is more buoyant and could result in a significant outward mass transfer. In fact, <cit.> showed that this has a net cooling effect on the gas as the ICM displaced by the CR bubbles expands. Therefore, it is unclear how the heating occurs and how self-regulation can be established in cases where CRs dominate the bubble energy content. Some recent works on CR bubbles focused on 2D simulations; however, 3D simulations are required in order to accurately capture the properties of mixing. CRs can scatter on either magnetic field irregularities generated by externally driven turbulence or by self-excited Alfvén waves via the CR streaming instability. In the latter case CRs stream down their pressure gradients along magnetic field lines at (or above) the Alfvén speed. In this case, CRs experience an effective drag force that heats the gas <cit.>. This Alfvén wave heating was proposed as a viable mechanism to offset radiative cooling <cit.>. However, so far only spherically symmetric 1D models of Alfvén wave heating were explored in the literature. In this paper we study the ICM heating by CR-dominated bubbles using 3D MHD simulations including CR advection, streaming, Alfvén wave heating due to streaming and CR heating due to hadronic interactions between CRs and the thermal ICM. We demonstrate that CR transport by streaming is essential for constructing self-regulating feedback loop models, at least within the context of the physical model considered here. We show that CR contribution to the heating budget can be very important and that heating due to streaming can dominate over the hadronic and Coulomb heating. We also show that the simulations that include CR heating result in more intermittent AGN feedback. The paper is organized as follows. In Section 2 we describe basic physics relevant to CR heating of the ICM and the numerical techniques employed in our work. In Section 3 we present our main results. Summary and Conclusions are presented in Section 4. § METHODS §.§ Initial and boundary conditions and the jet feedback model The gravitational potential and initial conditions for the temperature and density distributions of the gas resemble those adopted by <cit.>. In brief, the cluster atmosphere is initially close to hydrostatic equilibrium and its density profile is similar to that corresponding to the Perseus cluster. We include tangled magnetic fields that are generated using the method similar to that described in <cit.>. We assume that in Fourier space the field has the following form B∝ k^-11/6exp[-(k/k_ in)^4] where k_ in=10^2(2π/L), where L=1Mpc is the size of the computational domain. We perform an inverse Fourier transform to generate real-space magnetic fields and, following <cit.>, we rescale the field such that B∝ρ_o^0.3, where ρ_o is the ICM density. This ensures that the magnetic pressure is approximately proportional to the gas pressure. In order to generate divergence-free field, we Fourier transform the field and perform divergence cleaning as in <cit.>. This procedure is repeated until a divergence-free field proportional to ρ_o^0.3 is obtained. The final field is normalized such that plasma β∼10^2. We also impose small isobaric perturbations δρ/ρ on top of the average gas density profiles. Following <cit.>, these fluctuations are approximately characterized by white noise spectrum with the amplitude of 0.1. The resulting ICM gas density distribution is given by ρ=ρ_omax(0.8,1+δρ/ρ). We use adaptive mesh refinement to refine the domain up to the maximum resolution of 1.95 kpc. Refinement is triggered by temperature gradients. We employ diode boundary conditions (the gas is only allowed to flow out of the domain; code variables have vanishing gradient at the boundary) but note that the choice of boundary conditions is not critical as the domain is much larger than the size of the central parts of the cool core. The black hole feedback model adopted here is based on the “chaotic cold accretion” model <cit.> and closely follows that used by <cit.>. In this model the cooling gas is removed from the hot phase of the ICM when its temperature drops below T=5×10^5K. The cold gas is then converted to passive particles that follow the fluid and are allowed to accrete onto the central black hole triggering feedback. The AGN energy is supplied back to the ICM via bipolar precessing jets. Compared to the feedback model used by <cit.>, the main difference is that here we also include MHD and CR physics and consequently the energy injected by the AGN jets is supplied in kinetic and CR form. We consider jets dominated by the CR component and assume that a fraction of f_ cr=0.9 of the energy of the jet fluid is in the form of CRs. Other model parameters are: jet mass loading factor η=1, feedback efficiency ϵ=3× 10^-4, accretion timescale t_ ff=5 Myr, accretion radius r_ accre=5.85 kpc, precession period of the jet t_ prec=10 Myr, and precession angle of 15^o. The feedback energy is injected in a cylinder of 5 kpc in radius and 4 kpc in height. We refer the reader to <cit.> and references provided therein for definitions of these quantities and further details. §.§ Model equations We solve the MHD equations including CR advection, dynamical coupling between CR and the thermal gas, CR streaming along the magnetic field lines and the associated heating of the gas by CR, heating of the ICM by Coulomb and hadronic interactions, and radiative cooling ∂ρ/∂ t + · (ρ u_g) = ρ̇_ j, ∂ρ u_g/∂ t + ∇·( ρ u_g u_g- B B/4π) + p_ tot = ρ g + ṗ_ j, ∂ B/∂ t - × ( u_g× B) = 0, ∂ e/∂ t + ·[ (e+p_ tot) u_g - B( B· u_g)/4π] = ρ u_g· g -∇· F_ c - C + H_ c+ H_ j, ∂ e_ c/∂ t + · (e_ c u_g) = -p_ c· u_g -∇· F_ c + C_ c+ H_ j, where ρ is the gas density, u_g is the gas velocity, B is the magnetic field, g is the gravitational field, ρ̇_ j is the rate of injection of thermal gas via jet, ṗ_j is the rate of momentum injection associated with the AGN, e_ c is the specific CR energy density, and e=0.5ρ u_g^2 + e_ g + e_ c + B^2/8π is the total energy density, C is the radiative cooling energy loss rate per unit volume, F_ c is the CR flux due to streaming relative to the gas, H_ c is the rate of change of total specific energy due to streaming instability heating of the gas and Coulomb and hadronic CR losses, C_ c is the CR cooling rate due to the streaming instability, Coulomb, and hadronic CR losses, and H_j represents heating due to the AGN. The total pressure is p_ tot = (γ_g -1)e_ g + (γ_c -1) e_ c + B^2/8π, where e_ g and e_ c are the specific thermal energy density of the gas, γ_g=5/3 is the adiabatic index for ideal gas, and γ_c=4/3 is the effective adiabatic index of CR fluid. Radiative cooling is included using the Sutherland & Dopita cooling function <cit.>. In order to speed up the computations we employ the sybcycling method <cit.> when the local cooling time becomes shorter than the hydrodynamical timestep. We solve the above equations using the adaptive mesh refinement MHD code FLASH4.2 <cit.>. We employ the directionally unsplit staggered mesh solver <cit.>. This solver is based on a finite-volume, high-order Godunov scheme and utilizes a constrained transport method to enforce divergence-free magnetic fields. We use third order MHD scheme and HLLD Riemann solver. §.§ Cosmic ray physics We include the heating of the ICM by CRs and transport of CRs with respect to the gas. Details of the CR physics module can be found in <cit.> and <cit.>, where we discuss simulations the Fermi bubbles and CR-driven galactic winds, respectively. We now summarize key CR physics processes described in that paper and discuss extensions of the CR module specific to the modeling of the ICM presented here. §.§.§ Streaming of cosmic rays Propagation of CRs in the magnetized ICM can be described in the framework of the self-confinement model. In this picture, CR scatter on waves excited by the streaming instability <cit.>. In a state of marginally stable anisotropy, the CRs stream at the Alfvén speed down their pressure gradients. However, the waves excited by the streaming instability can be damped by various mechanisms, e.g., by turbulent or Landau damping. When this happens, CRs can stream at speeds exceeding the Alfvén speed. The effective streaming speed increases with the strength of the damping mechanism. The streaming flux is given by F_ cr = (e_ cr + p_ cr) u_s, where u_s = - sgn(b̂·∇ e_cr)f u_A is the streaming velocity, u_A is the Alfvén velocity, and f is the streaming speed boost factor. As demonstrated by <cit.>, the effective streaming speed in the ICM can significantly exceed the Alfvén speed in the cluster outskirts. For conditions representative of the cluster cool cores, damping mechanisms can lead to moderately super-Alfvénic speeds for the following reasons. <cit.> consider turbulent and non-linear Landau damping mechanisms. In the turbulent damping case, the effective streaming speed is u_s=u_A(1 + 0.08B_10μ G^1/2n_i,-2^1/2/L_ mhd, 10^1/2 n_c,-9γ_3^n-3.510^2(n-4.6)), where n_i,-2=n_i/(10^-2 cm^-3) is the ion number density, n_c,-9=n_c/(10^-9 cm^-3) is the CR number density, L_ mhd, 10=L_ mhd/(10 kpc) is the lengthscale at which turbulence is driven at the Alfvén speed u_A, γ_3=γ/3 is the average CR Lorentz factor, and n>4 is the slope of the CR distribution function in momentum (approximately n=4.6). In the non-linear Landau damping case the effective streaming speed is u_s=u_A(1 + 0.03n_i,-2^3/4T_5 keV^1/410^n-4.6γ_3^(n-3)/2/B_10μ GL_ cr,10^1/2 n_c,-9^1/2), where L_ cr, 10=L_ cr/(10 kpc) is the characteristic lengthscale of the fluctuations in the CR distribution and T_5 keV=T/(5 keV) is the ICM temperature. For the conditions representative of cool cores, in both of these cases, CR streaming is not typically super-Alfvénic. However, the damping rate Γ may be further boosted by linear Landau damping leading to Γ_ Landau/Γ_ turb∼β^1/2, where β is the plasma β∼ 10^2 parameter in the ICM (Zweibel, in prep.). When this process is included, the second term in Eq. (7) needs to be multiplied by β^1/2. For plausible cool core parameters, the CR number density is n_c=3× 10^-9n-4/n-3q_-2n_i,-2T_5 keVE_ min, GeV^-1, where q is the ratio of CR pressure to the ICM pressure and E_ min, GeV is the low-energy cutoff in CR momentum distribution. Given the uncertainty in β, L_ mhd, and n_c, it is plausible that the effective CR streaming speed could be moderately super-Alfvénic, i.e., boosted by a factor of order unity beyond the Alfvén speed. Therefore, in addition to Alfvénic streaming we also consider super-Alfvénic streaming for f=4 in order to bracket our solutions. CR streaming is incorporated using the method of <cit.>. Because the term -∇· F_ cr varies infinitely fast due to the discontinuity in the streaming flux near CR energy local extrema, it leads to a prohibitively small simulation timestep. In order to remove the singularity and speed up computations, we regularize the streaming flux by F_ c = -(e_ c+p_ c) u_A tanh(h_ cb̂·∇ e_ c/e_ c), where h_c is a free (regularization) parameter. In the calculations presented in this paper we adopt h_c=100 kpc. §.§.§ ICM heating by cosmic rays As the CRs stream, they also experience an effective drag force. Consequently, CRs lose energy and the gas is heated due to the Alfvén wave heating at the rate of H_ cr, stream = - u_A·∇ p_ cr. In addition to the heating of the ICM associated with the streaming instability, CRs also heat the gas via Coulomb and hadronic interactions. We approximate the effects of CR cooling due to Coulomb and hadronic losses due to pion production via <cit.> C_ cr, c=-4.93× 10^-19n-4/n-3e_cρ/E_ minρ/μ_em_p erg cm^-3 s^-1 and due to the hadronic losses via C_ cr, h=-8.56× 10^-19n-4/n-3e_cρ/E_ minρ/μ_pm_p erg cm^-3 s^-1, where E_ min=1 GeV is the minimum energy of CRs, μ_e and μ_p are the mean molecular weights per electron and proton, respectively. In the simulations we assume n=4.5 and mean proton Lorentz γ = 3. While all of the CR energy loss due to Coulomb collisions is transferred to the gas, only ∼1/6 of the CR energy loss due to pion production is used to heat the gas and the remainder is removed as gamma ray emission and neutrinos. Consequently, the rate of change of the total specific energy density of the gas, that includes the thermal and CR specific energy densities, is H_ cr=(5/6)C_ cr,h/ρ<0 and the CR specific energy density loss rate is C_ c=(C_ cr,c+C_ cr,h)/ρ. § RESULTS The list of the performed runs is shown in Table 1. Figure 1 presents cross sections through the cluster center showing the distribution of the specific CR energy density. From left to right these slices correspond to the following cases: (i) hadronic and Coulomb heating but no transport processes (CHT0), (ii) CR streaming and streaming heating (ST1), (iii) CR streaming and heating due to streaming, hadronic and Coulomb processes (SCHT1), and (iv) same as the last panel but for super-Alfvénic streaming (SCHT4). All snapshots were taken at 3 Gyr. This figure demonstrates that CR transport processes affect the morphology of the radio emitting plasma and effectively redistribute CRs. The redistribution of CR energy is efficient despite the fact the jet is pointed in approximately constant direction. As expected, the widening of the CR distribution is most significant when the CR transport is the fastest, i.e., super-Alfvénic. Note that these results also imply that the dynamical state of the atmosphere does depend on whether CR transport is included. Despite the fact that all snapshots were taken at the same time, the case where the CR streaming is neglected corresponds to the most perturbed atmosphere at the center of the cool core, while in all cases that include streaming, the ICM is relatively less disturbed and calmer at this particular time. As described in detail below, in the simulations including CR streaming the ICM generally exhibits larger variations due to more intermittent AGN feedback. This means that the atmosphere can experience both the periods of relative calm and more stormy conditions. Recent Perseus data from Hitomi is consistent with relatively low level of turbulence in this cluster <cit.>. It is plausible that the dynamical state of the Perseus cluster currently corresponds to relatively low-turbulence state captured in Figure 1 in cases including transport processes (see also <cit.>). Alternatively, turbulent motions in the cluster atmosphere could be reduced due to viscosity. We also point out that the iron line shifts corresponding to large gas velocities induced by the AGN at the center of the cool core may be partially diluted by slower moving gas away from the center. This may give an impression of relative calm in the ICM even if fast gas motions are present. This dilution effect has been seen in mock Hitomi simulations that show line shifts consistent with the data (Morsony, priv. comm.). We defer to a future publication the study of the iron emission line profiles and observational predictions for the planned Hitomi replacement and the X-ray Surveyor missions. As expected, the dispersal of CRs throughout the core is more pronounced at later times since the onset of feedback and when the speed of CR transport is faster. Interestingly, observations of M87 with LOFAR reveal a sharp radio emission boundary that does not seem to depend sensitively on radio frequency <cit.>, i.e., it appears that the boundary corresponds to the physical extent of CRs. At late times no such boundary is seen in the simulations. However, such boundary in the spatial distribution of CRs could be explained by large-scale sloshing motions that order magnetic fields on large scales and prevent the leakage of CRs to large distances by suppressing cross-field CR transport. Simulations of <cit.> show that sloshing motions induced by substructure in the cluster can generate tangential magnetic fields. Such fields could slow down radial transport of CRs away from the core. Alternatively, weaker or less collimated AGN feedback could prevent the bubbles from overshooting the critical radius at which their internal entropy equals that of the ambient ICM. In such a case, we would expect CR to exist predominantly within such critical radius. We defer exploration of these possibilities to a future publication and point out that there exist counter-examples to the morphological appearance of M87. In Abell 262 <cit.> and A2597 <cit.> the radio emission at lower frequencies extends to larger distances from the cluster center. The pressure support due to CRs is quantified in Figure 2. Pressure support is defined as the ratio of the pressure provided by CRs to the sum of the thermal and CR pressures. In order to exclude CR-filled bubbles that are cooling very inefficiently, this quantity is set to 10^-2 whenever the local cooling time exceeds the Hubble time. All panels show the evolution of the profiles of the pressure support. Dark lines corresponds to 50% of CR contribution to the total pressure support. In the case excluding CR transport (left panel), CR interaction with the ambient medium is inhibited. This is caused by the presence of the magnetic fields that slow down the mixing process and the fact that CRs are simply advected with the gas and do not stream with respect to the location of the fluid injected by the AGN. Consequently, even though hadronic and Coulomb heating processes are included, the CR heating of the ambient ICM is ineffective because CRs do not easily come in contact with the thermal ICM. This means that the cooling catastrophe can easily develop, which leads to large mass accretion rates onto the central supermassive black hole. As a result of this accretion the black hole feedback increases and more CRs are injected into the ICM. This is a runaway process in which CRs account for progressively larger fraction of the total pressure support. At the end of the simulation the CR pressure support in ∼50 kpc is dominant and thus it is inconsistent with observational constraints <cit.>. The remaining three panels illustrate that the role of transport processes is essential for removing this tension with observations. The second panel shows that including CR streaming and associated with it streaming heating dramatically reduces CR contribution to the pressure support. This reduction in CR pressure occurs because CRs can now come into contact with the thermal ICM and heat it, thus reducing the CR energy density and associated with it CR pressure. Similarly, CR pressures are further reduced when, in addition to the processes included in the second panel, we also include CR hadronic and Coulomb losses. These two processes further drain the energy from CRs and heat the thermal gas. Finally, the last panel demonstrates the consequences of including faster (super-Alfvénic) streaming. As expected, this further reduces CR pressure support. Note that this boost in the CR streaming speed only affects the rate of CR transport rather than the Alfvén wave heating. In all cases but the one shown in the leftmost panel, the CR pressure support is very small. We also performed a run without streaming instability heating but including transport by streaming and heating by Coulomb and hadronic processes (CHT1; not shown). While this run is unphysical, it helps us to better understand the role of CR transport. In this run, the values of CR pressure support (and typical variability timescales of CR pressure support; see below for more detailed discussion of variability) are similar to those seen in the three right panels in Figure 2. This experiment shows that CR transport is essential for preventing cooling cathastrope. In all three cases that include transport processes (panels 2 to 4 in Figure 2) there is a significant variation in the CR pressure support over time. This is a consequence of self regulation of the AGN feedback that was absent from the non-streaming case (panel 1 in Figure 2) where a global runaway cooling instability dominated the evolution of the ICM. This self-regulating behavior of the atmosphere is reflected in Figure 3 which shows AGN jet power as a function of time. In all four cases but the one shown in the first panel, the black hole feedback is highly variable. Note that despite the large variability, the AGN jet never completely switches off. While predicting detailed observational gamma-ray and radio signatures based on these simulations is beyond the scope of this paper, we point out that typical levels of CR pressure support that we find in simulations including CR transport are generally consistent with the data. Based on one-dimensional models that include heating by thermal conduction and CRs, <cit.> argue that in those cool core clusters that do not host radio mini halos, AGN activity and CR heating are the strongest, and that CRs can provide adequate level of heating without violating observational radio and gamma-ray constraints. They further argue that primary and secondary CR electron radio emission associated with the AGN outbursts could be difficult to detect due to the small physical extent of the radio emission in this case and the large flux dynamical range of the AGN jet and the halo. This picture is likely to be consistent with the elevated CR pressure support during AGN outbursts that is seen in Figure 2 (e.g., near ∼3 Gyr in the rightmost panel). In <cit.> typical values of CR-to-thermal pressure are on the order of 0.1 and vary substantially from object to object and thus presumably depend on the cluster dynamical state. Interestingly, <cit.> shows that in the Virgo cluster, in the absence of thermal conduction, adequate CR heating rate can be supplied when CR fraction is around 0.3 while not violating observational data. Levels of CR pressure support that we observe in our simulations during outbursts are comparable to those suggested by <cit.> and could presumably be reduced further if we included thermal conduction. In the case of cool cores that are associated with radio mini halos, <cit.> predict that the amount of CR pressure support needed to stably heat the cool core exceeds observational limits and suggest that such objects are expected to be dominated by radiative cooling. This situation could correspond to the periods in between the outbursts seen in Figure 2. Thus, the general properties of our simulations, and in particular the presence of the feedback loop and two classes of cool cores, are broadly consistent with the picture based on the above one-dimensional models. We also note that the simulations that do not include CR transport processes (left panel in Figure 2), do not show intermittent AGN activity and would therefore not be able to account for the transitions between cool cores with and without radio mini halos. Finally, we note that here we focus on general trends and defer to future publication the study of the parameter space of the models that meet observational constraints in detail. The evolution of the X-ray luminosity within the central 100 kpc is shown in Figure 4. Green line corresponds to bolometric brehmsstrahlung luminosity and black line to the X-ray emission integrated in the 0.5 to 10 keV range. As the X-ray emission is dominated by the densest central region of the cool core, an increase in the X-ray luminosity implies larger accretion of gas onto the central supermassive black hole. This boost in the accretion rate consequently implies stronger AGN feedback and this is why peaks in the X-ray luminosity closely correlate with the times when the jet power increases (see Figure 3). This cyclic behavior of the X-ray luminosity is evident in the cases including CR streaming. The evolution of the profiles of the ratio of heating to radiative cooling is shown in Figure 5. As in the case of the profiles of the CR pressure support shown in Figure 2, in order to exclude regions that are cooling very inefficiently, the heating-to-cooling ratio is set to 10^-2 whenever the local cooling time exceeds the Hubble time. From left to right, top row corresponds to the heating due to streaming in the case with: (i) streaming heating (ST1), (ii) streaming heating and hadronic and Coulomb heating (SCHT1), (iii) super-Alfvénic streaming heating and hadronic and Coulomb heating (SCHT4). Bottom row shows the ratio of combined Coulomb and hadronic heating to radiative cooling. Shown from left to right in the bottom row are the following cases: (i) Coulomb and hadronic heating without CR streaming transport (CHT0), (ii) Coulomb and hadronic heating with CR streaming transport (SCHT1), (iii) same as (ii) but for super-Alfvénic CR streaming transport (SCHT4). Let us begin discussing Figure 5 by focusing on the bottom left panel. This panel shows the ratio of the combined heating due to Coulomb and hadronic interactions to radiative cooling without including CR transport effects. This panel mirrors what is shown in the left panel in Figure 2 in the sense that the regions characterized by high heating-to-cooling ratios increase in size over time just as the regions occupied by high CR fraction grow with time. This significant heating is a direct consequence of the accumulation of large amounts of CRs in the cluster center. The accumulation of CRs is caused by increased AGN energy injection. However, because the mixing of CRs with the ICM is inefficient in this case, bulk of the ICM begins to overcool, which in turn leads to the stronger AGN feedback and associated with it CR injection. This particular case is ultimately unsuccessful because the CR heating does not couple well to the bulk of the ambient ICM. This is also consistent with the evolution of the jet power shown in Figure 3. By comparing the leftmost panel in Figure 3 that corresponds to the case without CR streaming to the jet power evolution in the cases that do include streaming (panels 2 through 4 in Figure 3), one can see that the integrated jet power, and thus the amount of CRs that accumulate in the cluster core, is the largest in the non-streaming case. When CR transport is neglected, the coupling of CRs to the gas is very weak. Consequently, gas accretion is unopposed, jet is constantly turned on, but its energy is not used efficiently to offset radiative losses in the ICM. Thus, accretion proceeds uninterrupted, and the AGN is not intermittent. The non-streaming case is deceptively similar to the cases considered by <cit.>, who simulated AGN feedback using hydrodynamical simulations. The main differences between the non-streaming case presented here and their simulations is that (i) in their model AGN jets inflate bubbles dominated by thermal energy whereas in our case the injection is dominated by CRs, and (ii) we include magnetic fields. Even though hadronic and Coulomb interactions are included in the non-streaming case, mixing of the AGN fluid with the ambient ICM is inhibited by magnetic fields and so the coupling of the AGN fluid to the ambient thermal gas is suppressed. This suppression is absent from <cit.> simulations and the heating of the ambient ICM can proceed via mixing with the thermal AGN jet fluid. This interpretation is also consistent with the results of <cit.> who do include CRs but neglect magnetic fields. In their case cooling catastrophe is prevented most likely as a result of more efficient mixing of the AGN fluid containing CRs and subsequent interactions of CRs with the ambient ICM via processes other than streaming heating. The fact that the non-streaming case fails to self-regulate also implies that other heating mechanisms such as dissipation of turbulence or weak shocks, though present, are not the dominant sources of heating of the ICM. Instead, CR heating through interaction between the CRs and the ICM is essential for reaching a global thermal balance. This conclusion is analogous to that of <cit.> who point out a similar hierarchy of heating sources, but that the role of CR heating in our simulations is replaced by mixing of the ultra-hot gas within the bubbles with the ambient ICM in the hydrodynamic case. We point out that the increase of the ICM entropy in cool cores may be dominated by CR heating rather than by, for example, turbulent dissipation. After the ICM has come into contact with CRs and experienced localized heating, it can expand locally. Such generated gas motions could eventually decay via turbulent dissipation. However, the primary heating mechanism in this case would be the CR heating rather then “secondary” turbulent dissipation. However, we also note that the framework we are using does not allow for the dissipation of sound waves by conductive and viscous processes. While these processes are likely to play an important role too (see, e.g., <cit.>), including these processes is beyond the scope of this paper. Typical patterns in the evolution of the heating-to-cooling ratios shown in Figure 5 are dramatically different when CR streaming is included, i.e., in all other panels except for the bottom left panel. It is evident that including streaming increases temporal variability in the CR heating profiles. This variable behavior also mirrors what is seen in Figure 2 showing the evolution of the CR pressure support. In particular, the top left panel in Figure 5, that includes CR streaming and associated with it streaming heating, shows that the source is highly intermittent and that CR heating no longer systematically increases over time. Importantly, each significant AGN outburst results in CR heating rates being comparable to radiative cooling. Similar conclusion can be drawn from the top middle panel that corresponds to the cases that also includes hadronic and Coulomb heating. It also applies to its analog shown in the top right panel that corresponds to super-Alfvénic streaming though the heating rates are somewhat reduced due to (i) accelerated transport of CRs away from the center of the cool core and (ii) the fact that the heating rate depends on the gradient of CR distribution that is somewhat flatter in this case due to smoother CR distribution. We can also compare the contributions of CR streaming heating and the combined Coulomb and hadronic losses to the total heating budget by comparing top and bottom panels in the middle and right columns. Top panels show the contribution from the CR streaming case while the bottom ones that due to the sum of Coulomb and hadronic heating. Interestingly, it is the CR streaming heating that dominates in all cases. In Figure 6 we show profiles of temperature, entropy normalized to the initial entropy distribution, emission-weighted temperature, emission-weighted density, and emission-weighted entropy (from top to bottom, respectively; ordering of columns is the same as in Fig. 1; weighting is computed using X-ray band extending from 0.5 to 10 keV). Color-coded lines correspond to different times. There is significant qualitative difference between the evolution of the temperature profiles in the non-streaming case (upper left panel) and all other cases. In the non-streaming case, the temperature systematically decreases over time due to the development of global thermal instability which origin, as mentioned above, can be traced back to inefficient mixing of CRs with the thermal ICM and thus inefficient heating of the bulk of the ICM. In all other cases but this one, the cluster atmosphere exhibits temperature variations but profiles vary around an average profile that does not exhibit very low temperatures. Similar trends are seen in the second row that shows profiles of the entropy profile normalized to the initial entropy distribution. Only in the non-streaming case does the gas entropy systematically decrease down to very low values. This demonstrates that the case without CR transport is unsuccessful. Very low gas temperatures and entropies would lead to significant line emission and star formation in excess of what is observed in cool cores. The third row shows X-ray emission-weighted temperature profiles. Unlike the temperature distributions shown in the first row, the emission-weighted ones do not show occasional very large departures from the mean profile, and in particular they do not exhibit centrally inverted temperature slopes, which is consistent with observations. Similarly, the emission-weighted gas density distributions shown in the fourth row are well-behaved. As a side comment, note that the simulations by construction start from a state that is out of thermodynamical equilibrium. This means that we do expect larger temperature variations compared to what one could have predicted starting from hydrostatic and thermal equilibrium in the initial state. Finally, the last row shows emission-weighted entropy profiles and demonstrates that the AGN feedback is gentle enough to preserve the positive entropy gradient in agreement with observations. § SUMMARY AND CONCLUSIONS We presented simulations of AGN feedback in cluster cool cores including the effects of CRs. Specifically, our simulations include CR injection by AGN jets, CR streaming along the magnetic field lines, radiative cooling, CR heating of the ICM via CR streaming instability, Coulomb interactions and hadronic processes. Our conclusions can be summarized as follows. * We presented a numerical proof of concept that CRs supplied to the ICM via an AGN jet can efficiently heat the ICM in a self-regulating fashion. This mode of heating does not demonstrably violate observational constraints as only a low level of CR pressure support is needed to offset radiative cooling during the feedback cycle. * The emission-weighted temperature and entropy profiles predicted by this model are broadly consistent with the data. * CR streaming is an essential ingredient of the model. When CR streaming is neglected, the CRs inside the AGN-inflated bubbles do not efficiently interact with the ambient thermal ICM, which leads to inefficient coupling of the AGN energy to the ICM, global cooling catastrophe, and excessive accumulation of CRs in the center of the cool core. On the other hand, when streaming is included, CRs mix efficiently with the thermal ICM and transfer their energy to the gas via CR streaming heating and Coulomb and hadronic interactions. * In the simulations that include CR streaming, the AGN jet and the X-ray luminosity of the cool core are intermittent. When CR transport is neglected, feedback loop is broken, AGN power is relatively weakly variable and is not used efficiently to offset cooling. * When CR streaming heating and Coulomb and hadronic heating processes are all included, it is the CR streaming heating that dominates over other CR heating mechanisms. M.R. thanks Department of Astronomy at the University of Maryland for hospitality during his sabbatical stay. M.R. is grateful for the hospitality of the Harvard-Smithsonian Center for Astrophysics and the Astronomy Department at the University of Wisconsin–Madison, which was made possible in part by a generous gift from Julie and Jeff Diermeier. We thank Ellen Zweibel for very useful discussions, and specifically for highlighting the role of Landau damping. MR thanks Brian McNamara, Aneta Siemiginowska, Ralph Kraft, Christine Jones, Bill Forman, Reinout van Weeren, and Brian Morsony for very helpful conversations. H.Y.K.Y. acknowledges support by NASA through Einstein Postdoctoral Fellowship grant number PF4-150129 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA. The software used in this work was in part developed by the DOE NNSA-ASC OASCR Flash Center at the University of Chicago. M.R. acknowledges NASA grant NASA ATP 12-ATP12-0017. C.S.R. thanks for the support from the US National Science Foundation under grant AST 1333514. Simulations were performed on the Pleiades machine at NASA Ames. Data analysis presented in this paper was performed with the publicly available yt visualization software <cit.>. We are grateful to the yt development team and the yt community for their support.

http://arxiv.org/abs/1701.08657v2

20170126032214

ABC of ladder operators for rationally extended quantum harmonic oscillator systems

math-ph

30003000 18pt0pt 0pt0pt 0pt0pt 32pt35by 460pt addtoresetequationsection.equation =-2.0truecm 3000 3000 18pt 0pt 0pt 60pt 0pt 0pt 32pt 36 by 470pt addtoresetequationsection.equation 𝒞𝒲𝔔ℋ𝒫𝒦𝒟ℒRQSN1𝔸𝔹ℤℝ𝕎ℙ𝕋ℂℍ̋0 grx⃗y⃗P⃗p⃗∇⃗c⃗A⃗P⃗b⃗ #1 #1 [ 𝔰𝔬σπsncndnnsncndcdsdsechℋ𝒜𝒫ℒ𝒮𝒬𝒟𝒱𝒲#1red#1#1blue#1#1green#1 ABC of ladder operators for rationally extended quantum harmonic oscillator systems José F. Cariñena^a and Mikhail S. Plyushchay^b [8pt] ^aDepartamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain ^bDepartamento de Física, Universidad de Santiago de Chile, Casilla 307, Santiago 2, Chile [4pt] E-mails: jfc@unizar.es, mikhail.plyushchay@usach.cl ======================================================================================================================================================================================================================================================================================================= The problem of construction of ladder operators for rationally extended quantum harmonic oscillator (REQHO) systems of a general form is investigated in the light of existence of different schemes of the Darboux-Crum-Krein-Adler transformations by which such systems can be generated from the quantum harmonic oscillator. Any REQHO system is characterized by the number of separated states in its spectrum, the number of `valence bands' in which the separated states are organized, and by the total number of the missing energy levels and their position. All these peculiarities of a REQHO system are shown to be detected and reflected by a trinity (𝒜^±, ℬ^±, 𝒞^±) of the basic (primary) lowering and raising ladder operators related between themselves by certain algebraic identities with coefficients polynomially-dependent on the Hamiltonian. We show that all the secondary, higher-order ladder operators are obtainable by a composition of the basic ladder operators of the trinity which form the set of the spectrum-generating operators. Each trinity, in turn, can be constructed from the intertwining operators of the two complementary minimal schemes of the Darboux-Crum-Krein-Adler transformations. .5cm § INTRODUCTION There are two basic exactly solvable quantum mechanical systems which reveal themselves directly or indirectly in association with other systems and play a fundamental role in many physical theories and applications. One of them is a free particle characterized by a continuous spectrum. The other one is a harmonic oscillator with its infinite equidistant discrete spectrum of the bound states. Free particle is essential, in particular, for understanding the properties of the soliton solutions in integrable systems. Quantum reflectionless potentials represent a snapshot of soliton solutions to the classical KdV equation. Initially, reflectionless potentials were obtained with the help of the method of the inverse scattering transform in solving the problem of theoretical construction of a solid dielectric medium that is perfectly transparent to electromagnetic radiation <cit.>. This important class of the systems can also be generated from the quantum free particle by means of Darboux transformations (DTs) and their generalization in the form of Darboux-Crum transformations (DCTs) <cit.>. Notice here that reflectionless systems appear for instance in the Gross-Neveu model <cit.> in the context of the hadron physics <cit.> and the physics of conducting polymers <cit.>. Any quantum reflectionless system with n bound states in its spectrum is characterized by the presence of a nontrivial Lax-Novikov integral that is a differential operator of order 2n+1 having a structure of a Darboux-dressed momentum operator of the free particle system. It is this integral of motion that distinguishes the states in the doubly degenerate continuous part of the spectrum of a reflectionless system and detects all the non-degenerate bound states as well as a state at the very edge of the continuous part of the spectrum by annihilating them <cit.>. It is the same operator that plays a fundamental role in the theory of nonlinear integrable systems <cit.>. The peculiarity of reflectionless systems also reveals itself in the nature of the quantum mechanical supersymmetry associated with them. Instead of a usual linear or non-linear 𝒩=2 supersymmetric structure which appears in an extended system composed from a pair of quantum mechanical systems related by a DT or DCT, the extended system composed from a pair of reflectionless systems is described by the exotic nonlinear 𝒩=4 supersymmetry generated by two pairs of supercharges alongside with the two bosonic integrals of motion constructed from the Lax-Novikov integrals of the subsystems <cit.>. The appearance of the exotic nonlinear supersymmetric structure associated with reflectionless systems is traced to the fact that any reflectionless Schrödinger system can be related to a free particle system not only by one but by two different DCTs due to the presence of the momentum operator in the quantum free particle system in the form of its integral of motion [A general picture is more complicated, however. In the case of a coincidence of some or all discrete energy levels of the two subsystems, the supersymmetric reflectionless partners can be related directly by a DCT of a lower order, without a necessity to construct a chain of Darboux transformations via a free particle system. This happens due to the opening of a kind of a `direct tunnelling channel' that can appropriately be understood from the standpoint of the picture of soliton scattering, see <cit.>.]. Some time ago, there has been discovered a new very broad class of exactly solvable systems which represent certain rational extensions of the quantum harmonic oscillator <cit.>. The eigenstates of rationally extended quantum harmonic oscillator (REQHO) systems are given in terms of exceptional Hermite polynomials, and can be obtained from the quantum harmonic oscillator (QHO) system by an appropriate DCT, or its further generalization in the form of the Darboux-Crum-Krein-Adler transformation <cit.> [Exceptional Jacobi and Laguerre polynomials <cit.> can be associated with Darboux-Crum transformed free particle on finite interval (particle in infinite potential well) and isotonic oscillator, respectively; the isotonic oscillator, in turn, can be related to the QHO by a singular Darboux transformation.]. In what follows we shall refer to any generalized Darboux transformation of the QHO with intertwining operators to be higher order differential operators as the Darboux-Crum-Krein-Adler transformation (DCKAT). Instead of a continuous spectrum of the free particle, the QHO is characterized by an infinite discrete spectrum of bound states. In spite of such a radical difference, the QHO is also a very peculiar system because its discrete spectrum is equidistant. As a consequence, instead of the Hermitian momentum operator integral that encodes reflectionless nature of the free particle, the QHO possesses a pair of Hermitian conjugate ladder operators which are spectrum-generating operators. Similarly to the relation between reflectionless and free particle systems, a given REQHO system can be obtained from the QHO by different DCKATs. Then one can expect by analogy with the pairs of reflectionless systems related by DCTs and the exotic supersymmetric structure associated with them that the REQHO systems should be characterized by some special properties. Particularly, it seems to be natural to expect the appearance of peculiarities related to the ladder operators for such a family of quantum systems. It is worth to mention here that finite-gap systems, a limit case of which corresponds to reflectionless systems, and the QHO are generated by the periodic Darboux chains <cit.>. The last construction also produces Painlevé equations <cit.>, that are intimately related with isomonodromic deformations of linear systems and integrability properties of nonlinear systems in partial derivatives <cit.>. The REQHO systems are isomonodromic deformations of the QHO <cit.>. Some investigations on ladder operators in REQHO systems have already been realized in <cit.>. This has been done, however, for some particular examples of the REQHO systems or for some particular families of such systems, while the problem of construction of ladder operators and investigation of their properties for REQHO systems of a general form remains open. Note also that in the indicated works only some special aspects related to the ladder operators of the REQHO systems were studied. In particular, in a recent paper <cit.> we have considered the problem of construction of the ladder operators for the simplest case of the REQHO by exploiting the simplest DT which relates the system to the QHO. We also investigated there the discrete chains related with the obtained ladder operators given by a pair of Hermitian conjugate third order differential operators to be fermionic generators of the polynomially deformed bosonized 𝔬𝔰𝔭(1|2) superalgebra. In the present article we investigate the problem of construction of ladder operators for REQHO systems of a general form in the light of existence of different DCKATs by which any such a system can be related to and generated from the QHO. In this point there shows up a similarity of the REQHO systems with reflectionless systems related with a free particle. We show that for any REQHO, there exists a trinity of the basic (primary) pairs of the lowering and raising ladder operators. This trinity is proved to form the set of the spectrum-generating operators which detects and reflects all the peculiar properties of a given REQHO system. The paper is organized as follows. In the next Section we briefly review the properties of the DTs and their generalization in the form of the DCKATs. In Section <ref> we discuss general schemes of the DCKATs for generation of the REQHO systems from the QHO by using different sets of physical and non-physical eigenstates of the latter. Any REQHO system can be characterized by the total number of separated states in the low part of its spectrum, by the number of the gapless `valence bands' in which the eigenvalues of separated states are organized, and by the total number of missing energy levels and their position in the low part of the spectrum. In Section <ref> we consider in detail the problem of construction of different ladder operators for the simplest REQHO system with one separated state and two missing energy levels in a unique gap that separates it from the equidistant infinite part of the spectrum. We investigate there the general properties and relations between the basic (primary) and secondary, higher-order ladder operators. In Section <ref> we consider two more particular examples of the REQHO systems, one of which corresponds to a generalization of the system from Section <ref>. Another example corresponds to a REQHO system with one valence band composed from the two energy levels separated from the equidistant infinite part of the spectrum by the gap of two missing energy levels. The results and observations obtained in Sections <ref> and <ref> are generalized then in Section <ref> for the case of the REQHO systems of a general form. Section <ref> is devoted to a summary of the obtained results, where we also indicate some problems that could be interesting for further investigation. § DARBOUX-CRUM-KREIN-ADLER TRANSFORMATIONS Let ψ_*(x) be a nodeless physical or non-physical real eigenfunction of a Hamiltonian operator H=-d^2/dx^2+V(x) with eigenvalue E_*, Hψ_*=E_*ψ_*. We assume that potential V(x) is a non-singular real function on all the real line . Define the first order differential operators A≡ψ_*d/dx1/ψ_*=d/dx-𝒲 , A^†=-1/ψ_*d/dxψ_* , where 𝒲=ψ'_*/ψ_*, ψ'_*=dψ_*/dx. They factorize the shifted Hamiltonian, H-E_*=A^† A, whose potential is given by V=𝒲^2+𝒲'+E_* in terms of the superpotential 𝒲 and factorization energy E_*. The product with permuted operators A and A^† defines a supersymmetric partner Hamiltonian, AA^†≡H̆-E_*, H̆=-d^2/dx^2+V_*, for which V̆=𝒲^2-𝒲'+E_*. The relation of the potential V̆(x) of the superpartner H̆ to the potential V(x) of the system H can be rewritten in a more convenient form for a further generalization, V̆=V-2(lnψ_*)” . From factorization relations it follows that the operators A and A^† intertwine the partner Hamiltonians, AH=H̆A , A^†H̆=H A^† . As a consequence, A and A^† mutually map the eigenstates of the superpartners. Namely, if ψ(x;E) is a physical or non-physical eigenstate of H of eigenvalue E≠ E_*, Hψ(x;E)=Eψ(x;E), then Ψ(x;E)=Aψ(x;E) is an eigenstate of H̆ of the same eigenvalue and of the same physical or non-physical nature. Vice versa, if Ψ(x;E) is an eigenstate of H̆ of eigenvalue E≠ E_*, then A^†Ψ(x;E) is an eigenstate of H of the same eigenvalue and of the same nature. If ψ(x) is a solution of the second order differential equation Hψ(x)=Eψ(x) for arbitrary value of E, a second, linearly independent solution of this equation is ψ(x)≡ψ(x) ∫^x dξ/(ψ(ξ))^2 . If ψ(x) is a normalizable on function, then ψ(x) is not normalizable, and vice versa. The properties of A and A^† as the operators that mutually map the corresponding eigenstates ψ(x) of H and Ψ(x) of H̆ of eigenvalue E≠ E_* are valid also for the associated eigenstates ψ(x) and Ψ(x). The case E=E_* in this context is different. The eigenstate ψ_* of H of eigenvalue E=E_* constitutes the kernel of the operator A, Aψ_*=0. The same is valid for the state Ψ_*(x)≡ 1/ψ_*(x), which constitutes the kernel of A^†, A^†Ψ_*=0, and is the eigenstate of H̆ of the same eigenvalue E=E_*. In this special case of E=E_*, the operator A transforms the state ψ_* into the state Ψ_*=1/ψ_*∈ (A^†): Aψ_*(x)=(ψ_*d/dx1/ψ_*)ψ_*(x)= ψ_*(x)d/dx∫^x dξ/(ψ_*(ξ))^2= 1/ψ_*(x) . Analogously, the state Ψ_*=(1/ψ_*) is transformed by A^† into ψ_* being a kernel of A. More details can be found in <cit.> where the same notation is used. Let us stress that for our constructions below it is important that if ψ_* is a normalizable state, then the state ψ_* is not normalizable, and vice versa, and that the same property is valid for the pair of the states Ψ_*=1/ψ_* and Ψ_*. Before the discussion of the concrete quantum systems we, however, completely neglect the questions of normalizability of the corresponding wave functions. Furthermore, here and in what follows we do not preoccupy about normalization of the states, and specify wave functions modulo a constant multiplication factor. The DT construction can be generalized for the case of the DCKAT. The latter is generated on the basis of several seed eigenstates ψ_i_1, ψ_i_2,…, ψ_i_n of H of different eigenvalues E_i_k, Hψ_i_k=E_i_kψ_i_k, k=1,…, n, with E_i_k≠ E_i_k' for i_k≠ i_k'. To get a nonsingular partner system H_n, these states should be such that their Wronskian _n(x)≡(ψ_i_1(x), ψ_i_2(x),…, ψ_i_n(x))=||ℱ(x)||, ℱ_ij=d^i-1/dx^i-1ψ_j, i=1,…, n, j=i_1,…, i_n, is a nodeless function. The partner system H_n is then given by the potential V_n(x)=V(x)-2(ln_n(x))”. The eigenstates ψ(x;E) of H are mapped into the eigenstates Ψ(x;E) of H_n of the same eigenvalue E via the relation Ψ(x;E)=(ψ_i_1(x),…,ψ_i_n(x),ψ(x;E))/_n(x) . In the case of n=1, Eq. (<ref>) reduces to Eq. (<ref>) while (<ref>) reduces to the relation corresponding to the case of the DT presented in the form Ψ(x;E)=A_1ψ(x;E), with ψ_i_1=ψ_*, A_1=A and H_1=H̆. Furthermore, for n>1 Eq. (<ref>) can be presented in a form that generalizes the indicated DT's formula. For this we iteratively define two sequences of related differential operators A_m, m=1,…, and _m, m=0,1,…, as follows: _0≡ 1, A_m=(_m-1ψ_i_m)d/dx1/(_m-1ψ_i_m) , m=1,2,…, _n=A_n A_n-1… A_1 . Let us denote H_0≡ H and define H_m-1=A_m^† A_m+E_i_m. We have A_m A_m^†=H_m-E_i_m, and obtain a generalization of the intertwining relations (<ref>), A_m H_m-1=H_m A_m, A_m^† H_m=H_m-1 A_m^†. Then we can present (<ref>) in the equivalent form Ψ(x;E)=_nψ(x;E) . The n-th order differential operators _n and _n^† intertwine the partner systems H_0 and H_n, _n H_0=H_n_n , _n^† H_n=H_0_n^† . The products of operators _n and _n^† turn out to be polynomials in the corresponding Hamiltonian operators with roots equal to the energies of factorization: _n^†_n=∏_k=1^n (H_0-E_i_k) , _n _n^† =∏_k=1^n (H_n-E_i_k) . For non-singular partners H_0 and H_n, some or all of the intermediate second-order differential operators H_m with m=1,…,n-1 can be singular. If we change the order of the seed states, the Wronskian is left invariant modulo a possible multiplication by (-1), that does not change Eqs. (<ref>) and (<ref>). Differential operator _n is not changed either, and from (<ref>) one can conclude that the kernel of _n is spanned by the complete set of the seed states, (_n)=span {ψ_i_1,…,ψ_i_n}. However, under permutation in the order of the seed eigenstates the first order differential operators entering into the factorized form (<ref>) of _n are changed. As a result, the nature of some or of all of the intermediate systems H_m can be changed from a non-singular (singular) to a singular (non-singular). § GENERATION OF THE REQHO SYSTEMS FROM THE QHO Let us turn now to the specific example of the QHO system given by the Hamiltonian operator H_osc=-d^2/dx^2+x^2. Its bound eigenstates are described by (not normalized here) wave functions ψ_n(x)=H_n(x)e^-x^2/2 which correspond to eigenvalues E_n=2n+1, n=0,1,…, where H_n(x) are Hermite polynomials. The change of variable x→ i x generates a change of the sign of the Hamiltonian, H_osc→ -H_osc, and so, transforms physical eigenstates ψ_n(x) into non-physical eigenstates ψ^-_n(x)=ℋ_n(x)e^x^2/2 of H_osc of eigenvalues E_n^-=-(2n+1), n=0,1,…, where ℋ_n(x)=H_n(i x). Unlike H_n(x), the polynomials ℋ_n(x) with even index have no real zeros while the unique real zero of these polynomials with odd index is at x=0. More details can be found in <cit.>. The choice of the ground state ψ_0=e^-x^2/2 as a seed eigenfunction ψ_* for the DT in (<ref>) generates the first order differential operators A_1=d/dx+x≡ a^- , A^†_1=-d/dx+x≡ a^+ . They factorize the shifted Hamiltonian, a^+a^-=H_osc-1, and satisfy the commutation relation [a^-,a^+]=2 . As a consequence we have [H_osc,a^±]=± 2 a^± . The two relations in (<ref>) can be rewritten equivalently in the form a^-H_osc=(H_osc+2)a^- , a^+(H_osc+2)=H_osc a^+ , and mean that a^- and a^+ are the ladder lowering and raising operators of the QHO. Relations in (<ref>) can also be interpreted as that a^- and a^+ intertwine the QHO system H_osc=H_0 with the SUSY-partner system H_1=H_osc+2 which is just the shifted QHO. Since a^-ψ_0=0, from the point of view of the DT one can consider the shifted system H_1 as the QHO with the removed ground state. But since a^-ψ_n=ψ_n-1, the partner system H_1 is the same QHO with the Hamiltonian shifted for +2. In this DT picture, the ground state ψ_0 of H_osc is created by application of A_1^†=a^+ to the function ψ^-_0=(1/ψ_0) that is a non-physical eigenstate of H_1 of eigenvalue E=1. The DCKAT generated on the basis of the set of the seed eigenstates ψ_0,…,ψ_n-1,ψ_n, n=1,…, produces the intertwining operators _n+1^-=(a^-)^n+1 and ^+_n+1=_n+1^†=(a^+)^n+1, which intertwine the QHO with the SUSY-partner system H_n+1=H_osc+2(n+1), _n+1^-H_osc=H_n+1_n+1^-, _n+1^+ H_n+1=H_osc_n+1^+ . The case n=0 here reproduces the relations of the DT generated by the choice ψ_*=ψ_0. Note that in the case of n=1 the permutation of the seed states in the DCKAT construction, (ψ_0,ψ_1)→ (ψ_1,ψ_0), gives rise to a singular operator A_1=ψ_1d/dx1/ψ_1= d/dx+x-1/x≡ a^-_iso . This operator acting on the second chosen seed eigenstate ψ_0 gives a function -1/xe^-x^2/2=-1/ψ_1^-, which according to (<ref>) generates the second factorization operator A_2=d/dx+x+1/x that also is singular. The product of these two singular operators gives the same second-order non-singular intertwining operator [For the discussion of related phenomena in a context of the quantum second-order supersymmetry anomaly and coupling-constant metamorphosis see ref. <cit.>] as the scheme with the (ψ_0,ψ_1) pair: A_2A_1=(a^-)^2=_2^-. The intermediate Hamiltonian H_1 in this case is a singular at x=0 operator corresponding to the (shifted) quantum isotonic oscillator, A_1^† A_1=H_osc-3, A_1 A_1^†=H_1-3, which is given by the potential V_1(x)=x^2+2/x^2+2. We also have here the relations A_2^† A_2=H_1-1, A_2A^†_2=H_osc+3. The choice of a nodeless non-physical state ψ_*=ψ^-_0 corresponding to factorization energy E_*=-1 in the DT construction gives A_1=d/dx-x=-a^+ , A_1^†=-a^- . We obtain the same ladder operators of the QHO, but now they will intertwine the QHO with the SUSY-partner system H_1=H_osc-2, a^+H_osc=(H_osc-2)a^+ , a^-(H_osc-2)=H_osc a^- . Here the action of A_1 on the non-physical eigenstate ψ^-_0=(1/ψ_0) of H_0=H_osc of eigenvalue E=-1 produces a physical ground state ψ_0 for the partner system H_1. In the case of the DCKAT generated on the basis of the seed eigenstates ψ_0^-,…,ψ_n-1^-,ψ_n^-, n=1,…, we obtain a partner system with the added n+1 bound states in the lower part of the spectrum of the QHO which is described by the shifted QHO Hamiltonian H_n+1=H_osc-2(n+1). The described DCKATs based on the choice of the seed eigenstates ψ_0,…,ψ_n or ψ^-_0,…,ψ^-_n reflect the property of the special shape invariance of the QHO system. 0.1cm Before we proceed further, let us summarize briefly the main features of the non-singular DT and DCKAT schemes based on other choices of the sets of physical, ψ_n, and non-physical, ψ^-_n, eigenstates of the QHO as the seed eigenstates <cit.>. This will generalize the preceding discussion and will allow us to generate the REQHO systems. 0.1cm Consider the DCKAT generated on the basis of the physical eigenstates ψ_i_1, ψ_i_2, …, ψ_i_n, and non-physical eigenstates ψ_j_1^-, ψ_j_2^-, …, ψ_j_l^-, of the QHO. The peculiarity of the physical and non-physical eigenstates in both families is that they have a form of polynomials multiplied by exponential functions e^-x^2/2 and e^x^2/2, respectively. As a result we obtain a quantum system described by a potential to be a rational function. In order a new quantum system generated by the DCKAT be non-singular, the complete set of the seed states has to be composed from the blocks of the states (ψ_0^-,…,ψ_j_1-1^-,ψ_j_1^-), (ψ_j_2^-,…, ψ_j_2+l_2^-),…,(ψ_j_r^-,…, ψ_j_r+l_r^-), (ψ_0,…,ψ_i_1-1,ψ_i_1), (ψ_i_2,…, ψ_i_2+2m^+_2+1),…,(ψ_i_s,…, ψ_i_s+2m^+_s+1), where j_2=j_1+2m^-_1+3, j_k+1= j_k+2m^-_k+3, i_1< i_2, i_k+2m^+_k+1<i_k+1, and m^+_k, m^-_k and l_k can take values 0,1,…. The total number of such blocks can be arbitrary and blocks that include the states ψ_0 and ψ^-_0 can be absent. Up to a possible constant shift, the generated system will have the gapped spectrum of the QHO with the deleted levels appearing at the positions of energies corresponding to physical states ψ_i in these blocks, and with new, added energy levels appearing at the positions of energies of non-physical eigenstates ψ^-_j. In other words, the inclusion of physical states ψ_i into the generating set of the seed states eliminates the energy levels, while the inclusion of non-physical states ψ^-_j introduces corresponding additional energy levels into the spectrum. Each gap in the spectrum of the resulting system contains an even number of the missing energy levels. As in the simplest examples we considered above with the partner system to be the same but the shifted QHO, the same REQHO system can be produced by DCKATs based on different choices of the sets of the seed eigenstates. Different choices of the seed states generate different intertwining operators which relate REQHO system with the QHO. As a consequence, as we shall see, there exist different ladder operators for the same REQHO, which possess different properties. Below we first consider some simple concrete examples of the REQHO systems. This will allow us to investigate in detail the families of the DCKATs associated with a given REQHO system, to identify different ladder operators, and to study their properties as well as to establish the relations between them. Then the results will be developed for the case of the REQHO systems of a general form. § SIMPLEST REQHO SYSTEM AND ITS LADDER OPERATORS A simplest REQHO system can be produced by taking a nodeless non-physical eigenstate of energy E=-5 of H_osc, ψ^-_2=(1+2x^2)e^x^2/2 . The first order differential operators A^-≡ψ_2^-d/dx1/ψ^-_2=d/dx-x-4x/2x^2+1 , A^+=(A^-)^† factorize the shifted QHO Hamiltonian A^+A^-=-d^2/dx^2+x^2+5=H_osc+5≡ H . Their permuted product generates a simplest REQHO system, A^-A^+=-d^2/dx^2+x^2+3 +82x^2-1/(2x^2+1)^2≡H̆ . We have the intertwining relations A^-H=H̆A^- , A^+H̆=H A^+ , from which it follows that the systems H and H̆ are almost isospectral, and operators (<ref>) provide a map between the eigenstates of the QHO and the REQHO systems. The excited eigenstates of H̆ are the bound states Ψ_n(x)=A^-ψ_n(x) , E_n=6+2(n-1) , n=1,2,…, where ψ_n(x) are the QHO eigenstates (<ref>). In correspondence with the general relation (<ref>), the ground state and its energy are Ψ_0=A^-ψ^-_2=1/ψ^-_2 , E_0=0 . In correspondence with a general picture described above, this state constitutes the kernel of the operator A^+. The ladder operators for H̆ can be constructed by the Darboux-dressing of the ladder operators a^± of the QHO, 𝒜^±=A^-a^± A^+ . We have [H̆,𝒜^±]=± 2𝒜^± , and 𝒜^+𝒜^-=H̆(H̆-2)(H̆-6) , 𝒜^-𝒜^+=H̆(H̆+2)(H̆-4). Due to the last factor in (<ref>) and in correspondence with relations (<ref>), the ground-state of the REQHO of zero energy, Ψ_0=1/ψ_2^-, is annihilated by both ladder operators 𝒜^- and 𝒜^+. Another peculiarity is that the kernel of the lowering operator 𝒜^- also contains the first excited physical state Ψ_1=A^-ψ_0 of energy E=6, and the non-physical state A^-ψ^-_1 which is the eigenstate of H̆ of the eigenvalue E=2. The three-dimensional kernel of the lowering ladder operator 𝒜^- is therefore (𝒜^-)=span {Ψ_0,A^-ψ^-_1, Ψ_1} . Besides the ground state Ψ_0, the kernel of 𝒜^+ includes the two states A^-ψ^-_3 and A^-ψ^-_0, which are non-physical eigenstates of H̆ of the eigenvalues -2 and 4, (𝒜^+)=span {Ψ_0,A^-ψ^-_3, A^-ψ_0^-} . We denote (α_1) this scheme based on the DT with generating function ψ^-_2, (α_1)={ψ^-_2}. Up to a global shift, the same REQHO system can also be produced by means of any of the DCKAT schemes (α_2)={ψ^-_0,ψ^-_3} , …, (α_n+1)={ψ^-_0,ψ^-_1,…, ψ^-_n-1,ψ^-_n+2} . The presence of the first n states ψ^-_0,ψ^-_1,…, ψ^-_n-1 in the scheme (α_n+1) gives rise to the addition of the corresponding energy levels into the spectrum of the QHO while the inclusion of the state ψ^-_n+2 results finally in the generation of the shifted gapped REQHO system H_n+1=H̆-2n. The intertwining operators between H defined by (<ref>) and H_n+1 in this case are _n+1=A^-(a^+)^n , _n+1^†=(a^-)^nA^+ , where A^- and A^+ are given by Eq. (<ref>). Combining the intertwining operators of the schemes (α_n+1) and (α_1), we can construct the higher-order ladder operators A^-_n+1^† =A^-(a^-)^nA^+≡𝒜_n^- , _n+1A^+=A^-(a^+)^nA^+=(𝒜_n^-)^†≡𝒜_n^+ , [H̆, 𝒜_n^±]=± 2n 𝒜_n^±, where 𝒜^±_1=𝒜^±. This in particular means that the third-order differential operators (<ref>), which have the nature of the Darboux-dressed QHO operators a^±, can also be considered as the ladder operators generated via a composition of the intertwining operators corresponding to the schemes (α_1) and (α_2), 𝒜^-=A^-(a^-A^+), 𝒜^+=(A^-a^+)A^+. The use of the scheme (α_n) with n>2 instead of (α_2) in such a composition provides us therefore with the (α_1)-Darboux-dressed form (<ref>) of the higher-order ladder operators (a^±)^n of the QHO. The following relations between 𝒜^- and 𝒜^-_n are valid, (𝒜^-)^n=∏_j=1^n-1(H̆+2j)·𝒜^-_n= 𝒜^-_n·∏_j=1^n-1(H̆-2j) , and analogous relations are obtained from them for 𝒜^+ and 𝒜^+_n by the Hermitian conjugation. In a more general case the composition of the intertwining operators of the schemes (α_n) and (α_m) with n>m generates the higher-order ladder operators 𝒜^±_n-m, _m+1_n+1^†=∏_j=1^m(H̆-4-2j)·𝒜_n-m^- , _n+1_m+1^†= 𝒜_n-m^+·∏_j=1^m(H̆-4-2j) . The REQHO system (<ref> ) can also be generated via the DCKAT based on the physical eigenstates ψ_1 and ψ_2. We denote this scheme, which eliminates two neighbour energy levels E=8 and E=10 in the spectrum of the shifted QHO (<ref>), as (β_2): (β_2)={ψ_1,ψ_2} . We denote ^±_2 the second-order intertwining operators _2 and _2^† constructed on the basis of these two states according to (<ref>), _2^-≡ A^-_isoa^-_iso , _2^+≡ (_2^-)^†=a^+_isoA^+_iso . The operator a^-_iso is defined in Eq. (<ref>), and a^+_iso=(a^-_iso)^†. The result of the action of the operator a^-_iso on the QHO's eigenstate ψ_2 can be presented in terms of its physical and non-physical eigenstates in the form a^-_isoψ_2=-ψ_0ψ^-_2/ψ^-_1≡ϕ. The first-order differential operators A^±_iso are generated by the function ϕ(x), according to (<ref>), A^-_iso=ϕ(x)d/dx1/ϕ(x)=d/dx +x+1/x-4x/1+2x^2 , A^+_iso=(A^-_iso)^† . We have a^+_isoa^-_iso=H-8 , a^-_isoa^+_iso=H_iso , where H_iso=-d^2/dx^2+x^2+2/x^2-1 is the shifted isotonic oscillator to be singular at x=0, and H corresponds to the shifted Hamiltonian of the QHO defined in (<ref>). We also have the relations A^+_isoA^-_iso=H_iso-2 , A^-_isoA^+_iso=H̆-4 , where H̆ is the Hamiltonian of the REQHO system defined in Eq. (<ref>). From (<ref>) and (<ref>) we find _2^- H=(H̆+6)_2^- , _2^+ (H̆+6)=H_2^+ , and _2^+ _2^-=(H-8)(H-10) , _2^-_2^+=(H̆-2)(H̆-4) . By the construction, (_2^-)=span {ψ_1,ψ_2}. One can also see that (_2^+)=span {A^-ψ^-_0,A^-ψ^-_1}= span {^-_2ψ_1, ^-_2ψ_2}= span {ψ_1/_2,ψ_2/_2}, where _2(x)=(ψ_1,ψ_2)(x)=-ψ^-_2(x) e^-3/2x^2. 0.1cm The DCKATs corresponding to the (α_1)- and (β_2)-schemes are in some sense complementary. The (α_1)-scheme introduces effectively a new energy level into the spectrum of the QHO below its ground-state energy at the distance equal to the tripled distance between equidistant energy levels. The (β_2)-scheme makes a similar job but by deleting the first two excited energy levels in the spectrum of the QHO. Since the Wronskian in the DCKAT in the latter scheme includes the additional exponential factor e^-3/2x^2 in comparison with the structure of the non-physical eigenstate ψ^-_2, this produces the additional constant shift +6 in the potential generated by means of relation (<ref>) that is reflected in Eq. (<ref>), cf. Eq. (<ref>). From Eq. (<ref>) it follows that if ψ(x;E) is an eigenstate of H of energy E, then ^-_2ψ(x;E) is an eigenstate of H̆ of energy (E-6). As a consequence, all the spectrum of the system generated by the (β_2)-scheme will be shifted for -6 in comparison with the spectrum of the REHQO system (<ref>) produced via the (α_1)-scheme. In correspondence with this picture, the ground-state Ψ_0 of H̆ can alternatively be constructed from the QHO ground-state ψ_0, Ψ_0=^-_2ψ_0, cf. (<ref>). The excited states Ψ_n+1 of energy E_n+1=6+6n with n=0,… can be presented in the alternative to (<ref>) form Ψ_n+1=^-_2ψ_n+3. All this gives a possibility for the construction of another pair of ladder operators for the REQHO system (<ref>) with the properties rather different to those of the ladder operators we obtained by using only the (α)-schemes. For this we take now the composition of the intertwining operators of the (α_1)- and (β_2)- schemes to construct the operators 𝒞^-=_2^-A^+ , 𝒞^+=A^-^+_2 . Instead of (<ref>) and (<ref>), they satisfy the relations [H̆,𝒞^±]=± 6 𝒞^± , and 𝒞^+𝒞^-=H̆(H̆-8)(H̆-10) , 𝒞^-𝒞^+=(H̆+6)(H̆-2)(H̆-4) . Like 𝒜^±, these are third-order differential operators of the nature of ladder operators. However, acting on eigenstates of the REQHO system H̆, they change the energies not in 2 but in 6. In this aspect they are somewhat similar to the higher-order ladder operators 𝒜^±_3, which are fifth-order differential operators discussed above. The essential difference of 𝒞^- from 𝒜^-_n and in particular 𝒜^- is that in correspondence with the first relation from (<ref>), the kernel of 𝒞^- is composed only from physical eigenstates of H̆, (𝒞^-)=span {Ψ_0,Ψ_2,Ψ_3} . The energies 0, 8 and 10 of these states are the roots of the third-degree polynomial in the first identity in (<ref>). Also, unlike 𝒜^+, the kernel of the raising operator 𝒞^+ is composed only from the non-physical eigenstates of H̆, (𝒞^+)=span {A^-ψ^-_5,A^-ψ^-_1,A^-ψ^-_0}= span {^-_2ψ^-_3,^-_2ψ_1,^-_2ψ_2}. In correspondence with the second relation in (<ref>), the eigenvalues of the states in (<ref>) are -6, 2 and 4. From the point of view of the structure of the kernels and commutation relations (<ref>), the ladder operators 𝒞^± are similar to the third-order differential operators (a^±)^3 in the QHO system. However, unlike 𝒞^-, the operator (a^-)^3 annihilates the three lowest physical eigenstates of the QHO of the three subsequent values of energy. The first exited state Ψ_1 of the REQHO system of energy E=6 does not belong to the kernel of 𝒞^- and is annihilated by (𝒞^-)^2: 𝒞^-Ψ_1=Ψ_0, (𝒞^-)^2Ψ_1=0. The following relations can be established by comparing the kernels of the operators on both sides of the equalities, A^+^-_2=-(a^-)^3 , ^+_2A^-=-(a^+)^3 . These and their analogous relations for other REQHO systems will play important role in what follows. From them one can find in particular the operator identities a^-^+_2=-(a^+)^2A^+, (a^-)^2^+_2=-a^+A^+(H̆-2), as well as the Hermitian conjugate ones. One can introduce additionally the operators a^± inside the factorized structure of the operators 𝒞^±. In this way one can construct the operators 𝒞^-_n+1=_2^-(a^-)^nA^+ , 𝒞^+_n+1=A^-(a^+)^n^+_2 , n=0,… , with the implied identification 𝒞^±_1=𝒞^± for n=0. They satisfy the relation [H̆,𝒞^±_n+1]= ± (6+2n)𝒞^±_n+1. These operators can be treated either as the QHO operators (a^±)^n dressed by the intertwining generators of the (α_1) and (β_2) schemes, or as the operators produced by intertwining operators (<ref>) from the (α_n+1) scheme and those from the same (β_2) scheme. The kernel of 𝒞^-_n+1 is composed only by the physical eigenstates of H̆, while the kernel of 𝒞^+_n+1 is spanned only by its non-physical eigenstates. For instance, (𝒞^-_2)=span {Ψ_0,Ψ_1,Ψ_3,Ψ_4}. With the help of identities (<ref>) we also find that (𝒞^±)^n=(-1)^n+1𝒞^±_3(n-1)+1, n=1,… . Analogously to 𝒜^±, we also introduce the operators ℬ^±=^-_2 a^±^+_2 . Unlike the third-order ladder operators 𝒜^± and 𝒞^±, the ℬ^± are fifth-order differential operators. They satisfy the relations [H̆,ℬ^±]=± 2ℬ^± and ℬ^+ℬ^-= H̆(H̆-2)(H̆-6)(H̆-4)^2 , ℬ^-ℬ^+= H̆(H̆+2)(H̆-4)(H̆-2)^2 . The kernel of ℬ^- involves two physical and three non-physical eigenstates of H̆, (ℬ^-)=span {Ψ_0,Ψ_1,A^-ψ^-_0,A^-ψ^-_1,A^-ψ^-_0} . The eigenvalue E=4 of the non-physical eigenstates A^-ψ^-_0 and A^-ψ^-_0 in the kernel of ℬ^- corresponds to the double root of the last factor in the first relation in (<ref>). The kernel of the increasing ladder operator ℬ^+ includes only one physical eigenstate, (ℬ^+)=span {Ψ_0,A^-ψ^-_0,A^-ψ^-_1,A^-ψ^-_3,A^-ψ^-_1} . The eigenvalue E=2 of the non-physical eigenstates A^-ψ^-_1 and A^-ψ^-_1 in the kernel of the increasing ladder operator ℬ^+ corresponds to the double root of the last factor in the second relation in (<ref>). By analogy with (<ref>), one can consider the higher-order ladder operators ℬ^±_n=^-_2(a^±)^n^-_2 , [H̆,ℬ^±_n]=± 2n ℬ^±_n, with the identification ℬ^±_1=ℬ^±. The lowering operator ℬ^- can be related to the ladder operators 𝒜^- and 𝒞^- via the identities ℬ^-=𝒜^-(H̆-4) , ℬ^-_2=(𝒜^-)^2 , ℬ^-_3 =-𝒞^-(H̆-2)(H̆-4) , ℬ^-_4=-𝒞^-_2(H̆-2)(H̆-4), ℬ^-_5=-𝒞^-(𝒜^-)^2, ℬ^-_6=(𝒞^-)^2(H̆-2)(H̆-4), etc. The increasing operator 𝒞^+ is related to 𝒜^+ and 𝒞^+ via the conjugate identities. Similarly to (<ref>), for degrees n>1 of ℬ^- we have (ℬ^-)^n=∏_j=1^n-1(H̆-2+2j) (H̆-4+2j)·ℬ^-_n , and an analogous relation for (ℬ^+)^n. A generalization of the (β_2)-scheme corresponds to the family of the DCKAT schemes (β_3)={ψ_0,ψ_2,ψ_3} , …, (β_n+2)={ψ_0,…,ψ_n-1,ψ_n+1,ψ_n+2} . In the case of the scheme (β_n+2), the intertwining operators constructed according to the prescription (<ref>) are _n+2^-≡^-_2(a^-)^n and ^+_n+2=(a^+)^n_2^+. The schemes (β_n+2) do not give anything new but allow us to re-interprete the already discussed higher-order ladder operators 𝒞^±_n+1 as those produced via the composition of the (β_n+2) and (α_1) schemes, 𝒞^-_n+1=^-_n+2A^+, 𝒞^+_n+1=A^-^+_n+2. Analogously, ℬ^-_n+1=^-_n+2^+_2, ℬ^+_n+1=^-_2^+_n+2. Besides the two infinite families (α_n), n=1,…, and (β_n), n=2,…, which involve as the seed eigenfunctions either only non-physical or only physical eigenstates of the QHO, there are two additional, `intermediate' schemes which simultaneously include eigenstates of both types. These are the schemes (γ_2)={ψ_0,ψ^-_1} , (γ_3)={ψ_0, ψ_1,ψ^-_0} . The intertwining operators in the case of the scheme (γ_2) are the second-order differential operators _2=A^-a^- and _2^†=a^+A^+, while in the scheme (γ_3), the intertwining operators are the third-order differential operators _3=A^-(a^-)^2 and _3^†=(a^+)^2A^+. These operators have a structure similar to that of the intertwining operators in the family of the schemes (α_n). With their help we do not obtain anything essentially new for the construction of the ladder operators for the REQHO system H̆ in comparison with the already discussed structures. Indeed, employing the superposition of the intertwining operators from the (β_2)-scheme and either (γ_2)- or (γ_3)- schemes, one can construct the ladder operators 𝒞^-_-n≡^-_2(a^+)^n A^+ , 𝒞^+_-n≡ A^- (a^-)^n ^+_2 , n=1,2 . Here n=1 and n=2 correspond, respectively, to the (γ_2)- and (γ_3)- schemes. Operators 𝒞^±_-1 are fourth-order differential operators, while 𝒞^±_-2 are fifth-order differential operators. They, however, are not independent but can be expressed in terms of the already constructed intertwining operators. Namely, we have, in particular, 𝒞^±_-1=-𝒜^±_2 , 𝒞^-_-2=-(H̆-2)𝒜^- , 𝒞^+_-2=-(H̆-4)𝒜^+ . These relations can be established by comparing the kernels of 𝒞^-_-1 and 𝒞^-_-2, (𝒞^-_-1)=span {Ψ_0,A^-ψ^-_0,Ψ_1,Ψ_2}, (𝒞^-_-2)=span {Ψ_0,A^-ψ^-_1, A^-ψ^-_0, A^-ψ^-_0, Ψ_1}, with the kernels of 𝒜^-_2 and of the operator (H̆-2)𝒜^-=𝒜^-(H̆-4), respectively, and by comparison of the signs before the leading derivative terms in the corresponding pairs of operators. Due to the identities (<ref>), a generalization of the operators (<ref>) for n>2 does not give us anything new since 𝒞^±_-3=-H̆(H̆-2) (H̆-4). This last relation as well as relations (<ref>) can also be obtained by employing the identities (<ref>). We also have the operator identities 𝒜^+𝒞^+_n=(H̆-2)𝒞^+_n+1 , 𝒜^-𝒞^-_n=-𝒜^-_n+3 , n=1,…, 𝒜^+𝒞^-=-(H̆-6)𝒜^-_2 , 𝒜^-𝒞^+=-(H̆+2)𝒜^+_2 , as well as the Hermitian conjugate relations. Let us look in more detail at the already mentioned similarity between the operators 𝒞^± and 𝒜_3^±. Using the first identity from (<ref>) and factorization relation (<ref>), we obtain 𝒜^-_3=A^-(a^-)^3A^+ =-H̆𝒞^- and 𝒜^+_3=-(H̆-6)𝒞^+. These relations are similar to those in (<ref>). Employing Eq. (<ref>), we find that (𝒜^-)^3=-H̆ (H̆+2)(H̆+4)𝒞^-= -𝒞^-(H̆-6) (H̆-4)(H̆-2) . This relation from the point of view of the kernels of the involved operators corresponds to the following picture. The kernel of the operator 𝒜^- is formed by the two physical eigenstates Ψ_0 and Ψ_1 and by one non-physical eigenstate A^-ψ^-_1. We have also the relations <cit.>𝒜^-(A^-ψ^-_1)=Ψ_0, 𝒜^-(A^-ψ^-_0)=A^-ψ^-_1, 𝒜^-Ψ_2=Ψ_1, 𝒜^-(A^-ψ^-_0)=A^-ψ^-_1, 𝒜^-(A^-ψ_0)=A^-ψ^-_0, 𝒜^-Ψ_3=Ψ_2. As a result we obtain (𝒜^-)^3=span {Ψ_0, A^-ψ^-_1, Ψ_1, A^-ψ^-_1, A^-ψ^-_0, Ψ_2, A^-ψ^-_0, A^-ψ_0,Ψ_3} . The pairs of states (A^-ψ^-_1, A^-ψ^-_1), (A^-ψ^-_0, ψ^-_0) and (Ψ_1=A^-ψ_0, A^-ψ_0) constitute, respectively, the kernels of the factors (H-2), (H-4) and (H-6) in (<ref>). The remaining three physical eigenstates Ψ_0, Ψ_2 and Ψ_3 in (<ref>) correspond to the kernel of the operator 𝒞^-. According to Eq. (<ref>) and its conjugate version, the ladder operators 𝒞^± can be generated by 𝒜^±. Then with taking into account Eq. (<ref>) and all the described relations, we conclude that in the case of the simplest REQHO system given by the Hamiltonian H̆ defined in (<ref>) all the set of the ladder operators can be obtained, in principle, from the compositions of the ladder operators ℬ^+ and ℬ^-. In conclusion of this section let us note, however, that in comparison with the QHO picture, the peculiarity of the system (<ref>) in particular is that its ground-state Ψ_0 cannot be achieved from physical states by action of the lowering operators 𝒜^- and ℬ^- which are differential operators of orders 3 and 5. Like the first-order differential operator a^- in the QHO, the ladder operators 𝒜^- and ℬ^- decrease the energy values of H̆ in 2, but they produce the ground-state by acting on the non-physical eigenstate A^-ψ^-_1 of the eigenvalue E=2. The ground-state Ψ_0 of zero energy can be achieved, however, by application of the lowering operator 𝒞^-, which is a third-order differential operator, to the physical eigenstate Ψ_1 with eigenvalue E=6. We also notice here that the ladder operators (<ref>) for the REQHO system of the simplest form (<ref>) were constructed (without employing the Darboux-dressing procedure) in <cit.> where this model was introduced and investigated for the first time. Later these ladder operators were constructed, particularly, in <cit.>, <cit.> and recently in <cit.> via the Darboux-dressing prescription based on the non-physical seed state we used here. The ladder operators (<ref>) we constructed on the basis of the (α_1)- and (β_2)- schemes by employing the analogy with reflectionless quantum systems <cit.> where the corresponding Lax-Novikov integrals of motion can be generated either by Darboux-dressing of the free particle momentum operator or by `gluing' two different intertwining operators that act in the opposite directions. The same last mentioned method also allows ones to generate the Lax-Novikov integrals for periodic finite-gap systems where the Darboux-dressing mechanism can not be applied, see <cit.>. In reflectionless and finite-gap systems, however, the two glued intertwiners always are differential operators of the `opposite', even and odd, differential orders, but both intertwine the two corresponding partner systems without additional relative displacement. It is because of the relative displacement in intertwining relations (<ref>) and (<ref>) that here we obtain the ladder operators 𝒞^± while in reflectionless and finite-gap systems analogous procedure generates the integrals of motion. Within the same framework we used here and based on employing the two schemes with physical and non-physical seed states of the quantum harmonic oscillator, the ladder operators (<ref>) were introduced earlier in <cit.> (but without exploiting the indicated analogy with generation of the Lax-Novikov integrals) and derived later in <cit.> for more general families related to multi indexed exceptionnal orthogonal polynomials. By another method such ladder operators were introduced for the system (<ref>) even earlier in <cit.>. The ladder operators (<ref>) constructed here by the Darboux-dressing procedure based on physical seed states seems were not discussed earlier in the literature. With subsequent analysis we shall see that the trinity of the basic ladder operators (𝒜^±, ℬ^±, 𝒞^±) admits a natural generalization for the case of REQHO systems of a general form, and that each pair of the conjugate lowering and raising ladder operators detects and reflects some specific properties of a corresponding quantum system. § TWO FURTHER EXAMPLES OF THE REQHO SYSTEMS A REQHO system generated by the DT based on the non-physical state ψ^-_2n with n>1 is similar to the considered REQHO system generated by the DT based on ψ^-_2. In this case the gap in the spectrum of the REQHO corresponds, up to a global shift, to the missing 2n energy levels with E=3,…,4n+1 in the spectrum of the QHO. We also have here the two infinite families of the DCKAT schemes of the structures which generalize those of the case n=1. For instance, in the case of n=2, we have ψ^-_4=(4x^4+12x^2+3)e^x^2/2, and the two infinite families of the schemes are (α_1)={ψ^-_4} , (α_2)={ψ^-_0,ψ^-_5} , (α_n+1)={ψ^-_0,…,ψ^-_n-1,ψ^-_n+4} , and (β_4)={ψ_1,ψ_2,ψ_3,ψ_4} , (β_n+4)={ψ_0,…,ψ_n-1, ψ_n+1,ψ_n+2,ψ_n+3,ψ_n+4} . In addition, we have the `intermediate' schemes whose sets of seed states include both physical and non-physical eigenstates of the QHO. These are (γ_2)={ψ_0,ψ^-_3} , (γ_3)={ψ_0,ψ_1,ψ^-_2} , (γ_4)={ψ_0,ψ_1,ψ_2,ψ^-_1} , (γ_5)={ψ_0,ψ_1,ψ_2,ψ_3,ψ^-_0} . The scheme (α_1) generates the intertwining operators A^-=ψ^-_4d/dx1/ψ^-_4, and A^+=(A^-)^†. They allow us to construct ladder operators that are third-order differential operators, the Darboux-dressed ladder operators of the QHO, 𝒜^±= A^-a^± A^+. They satisfy the relations of the form (<ref>), [H̆,𝒜^±]=± 2𝒜^±, with H̆≡ A^-A^+= -d^2/dx^2+x^2+7+324x^6+4x^4+3x^2-6/(4x^4+12x^2+3)^2 , and similarly to the already considered case, here both ladder operators 𝒜^± annihilate the ground state of H̆ which is Ψ_0=1/ψ_4^-=A^-ψ^-_4. Besides the ground state Ψ_0 of energy E=0, the kernel of 𝒜^- contains, the first excited state Ψ_1=A^-ψ_0 of energy E=10 and one non-physical eigenstate A^-ψ^-_3 of energy E=2. The kernel of 𝒜^+ contains besides the ground state Ψ_0 also two non-physical eigenstates A^-ψ^-_0 and A^-ψ^-_5 of H̆ of eigenvalues E= 8 and E=-2. The intertwining operators corresponding to the DCKAT scheme (β_4)={ψ_1,ψ_2,ψ_3,ψ_4} are the fourth order differential operators constructed in accordance with Eq. (<ref>), which by analogy with the already considered case we denote here as ^-_4 and ^+_4=(^-_4)^†. Then we define another pair of ladder operators via a composition of the intertwining operators of this (β_4)-scheme and of the (α_1)-scheme, 𝒞^-=^-_4 A^+, 𝒞^+= A^-^+_4. Unlike the previously discussed case of the REQHO system (<ref>), these are fifth-order differential operators, which satisfy the relations [H̆,𝒞±]= ± 10𝒞^±. The ladder operator 𝒞^- annihilates five physical eigenstates of H̆, which are the ground state Ψ_0 and the states Ψ_j+1=A^-ψ_j, j=1,2,3,4, with the energy values E_0=0 and E_j+1=10+2j. Like in the REQHO system we considered before, the first excited state Ψ_1=A^-ψ_0 of energy E_1=10 here does not belong to the kernel of the decreasing ladder operator 𝒞^-, and we have 𝒞^-Ψ_1=ψ_0, (𝒞^-)^2Ψ_1=0. The kernel of 𝒞^+ is composed only from non-physical eigenstates. Yet another pair of the ladder operators corresponds to ℬ^±=^-_4a^±^+_4, which are differential operators of order 9. Like 𝒜^±, they satisfy the relations [H̆,ℬ^±]=± 2ℬ^±. The kernel of the lowering ladder operator ℬ^- is spanned by two physical eigenstates Ψ_0 and Ψ_1 of energies 0 and 10, and seven non-physical eigenstates of H̆ of eigenvalues 8 (twice), 6 (twice), 4 (twice) and 2, (ℬ^-)=span {Ψ_0,Ψ_1,A^-ψ^-_0, A^-ψ^-_0,Ψ_1,A^-ψ^-_1, A^-ψ^-_1, Ψ_1,A^-ψ^-_2, A^-ψ^-_2,A^-ψ^-_3}. The kernel of ℬ^+ is spanned by the ground-state Ψ_0 and by eight non-physical eigenstates of H̆. Other, secondary ladder operators can be constructed by introducing the QHO ladder operators (a^±)^n inside the factorized structures of the basic ladder operators 𝒜^±, ℬ^± and 𝒞^±, or by considering compositions of the intertwining operators corresponding to (<ref>), (<ref>) and (<ref>) schemes analogously to how it was done for the simplest REQHO system. The secondary, higher-order ladder operators can also be generated by taking the products of the basic ladder operators. Analogously to (<ref>), we also have here the relations A^+^-_4=-(a^-)^5, ^+_4A^-=-(a^+)^5. As an analog of relation (<ref>) we have (𝒜^-)^5= -𝒞^-(H̆-10) (H̆-8)(H̆-6)(H̆-4) , and the relation ℬ^-=(H̆-2)(H̆-4)(H̆-6)𝒜^- is analogous here to the first relation in (<ref>). As in the case of the simplest REQHO system considered in the previous section, for the REQHO system described by the Hamiltonian(<ref>) all the ladder operators can be generated and extracted from the powers of the basic ladder operators ℬ^±. 0.1cm Let us consider yet another example of the REQHO system in which two states are separated by a gap from the infinite equidistant part of the spectrum. A simplest system of such a nature can be generated by employing the minimal (α)-scheme (α_2)={ψ^-_2,ψ^-_3} . Let us shift the Hamiltonian of the QHO for +7 and denote H=H_osc+7, for which the potential is V(x)=x^2+7 and the spectrum is E_n=8+2n, n=0,1,…. The Wronskian here is _2(x)=e^x^2(3+4x^4). The second order DCKAT based on (<ref>) produces the partner system which is the REQHO system H̆=-d^2/dx^2+V̆(x), where in accordance with (<ref>), V̆(x)=3+x^2 +32x^24x^4-9/(3+4x^4)^2 . Its gapped spectrum is E_0=0 , E_1=2 , E_2+n=8+2n , n=0,1,… . The intertwining second order differential operators constructed via (<ref>) on the basis of the seed QHO eigenstates (<ref>) we denote as _2^- and _2^+=(_2^-)^†. The kernel of ^-_2 is spanned by the states ψ^-_2 and ψ^-_3, while the kernel of ^+_2 is spanned by the lowest physical eigenstates of H̆ of the energies 0 and 2, which can be obtained from the QHO non-physical eigenstates ψ^-_2 and ψ^-_3, Ψ_0=^-_2ψ^-_2, Ψ_1=^-_2ψ^-_3. The operators ^+_2 and ^-_2 satisfy the relations ^+_2^-_2=H(H-2) , _2^-^+_2=H̆(H̆-2) , and ^-_2H=H̆^-_2, ^+_2H̆=H_2^+. We construct the ladder operators 𝒜^± for H̆ by the Darboux-dressing of the QHO operators a^±, 𝒜^± = ^-_2 a^±_2^+ . These fifth-order differential operators satisfy the relations [H̆,𝒜^±]=± 2𝒜^±, and 𝒜^+𝒜^-=H̆(H̆-2)^2(H̆-4)(H̆-8) , 𝒜^-𝒜^+=(H̆+2)(H̆)^2(H̆-2)(H̆-6) . The kernel of the lowering operaror is (𝒜^-)=span {Ψ_0,Ψ_1,Ψ_2, Ψ_1, _2^-ψ^-_1}. Here the first three states are the lowest three physical eigenstates of H̆ of energies E=0, 2 and 8, respectively, and the two last states are non-physical eigenstates of energies E=2 and 4. The indicated energies correspond to the roots of the polynomial in the first equality in (<ref>). The kernel of the increasing operator 𝒜^+ is spanned by the two lowest eigenstates Ψ_0 and Ψ_1 and by the three non-physical eigenstates: (𝒜^+)= span {Ψ_0,Ψ_1,^-_2ψ^-_4,Ψ_0,^-_2ψ^-_0}. The energies of these five eigenstates correspond to zeros of the polynomial in the second equality in (<ref>). The same system, up to a global shift, can also be generated via the complementary minimal (β_2)-scheme (β_2)={ψ_2,ψ_3} . The Wronskian of the seed states in this case is _2(x)=(ψ_2,ψ_3)= e^-x^2(3+4x^4). The potential calculated according to (<ref>) is the potential (<ref>) shifted for +8. Let us denote the corresponding intertwining second order differential operators constructed on the basis of these seed states as ^-_2 and ^+_2=(^-_2)^†. They satisfy the relations ^-_2H=(H̆+8)^-_2 , ^+_2(H̆+8)=H^+_2 , and ^+_2^-_2=(H-12)(H-14) , ^-_2^+_2=(H̆-4)(H̆-6) . Note that here _2^-(x) =-_2^-(ix). The kernel of ^-_2 is spanned by the seed states (<ref>), whereas the kernel of ^+_2 is spanned by non-physical eigenstates of H̆ of energies E=4 and E=6: (^+_2)=span {^-_2ψ_2,^-_2ψ_3}= span {^-_2ψ^-_0,^-_2ψ^-_1}. The ladder operators 𝒞^-=^-_2^+_2 , 𝒞^+=^-_2^+_2 are differential operators of the order four. They obey the relations [H̆,𝒞^±]=± 8 𝒞^± , and 𝒞^+𝒞^-=H̆(H̆-2) (H̆-12)(H̆-14) , 𝒞^-𝒞^+=(H̆+8)(H̆+6) (H̆-4)(H̆-6) . The lowering operator 𝒞^- annihilates the four physical states Ψ_0,Ψ_1,Ψ_4,Ψ_5 of the energies E=0,2,12,14. The kernel of 𝒞^+ is spanned only by non-physical eigenstates of energies E=-8,-6,4,6, (𝒞^+)=span {^-_2ψ^-_3,^-_2ψ^-_2,^-_2ψ_2, ^-_2ψ_3}= span {^-_2ψ^-_7,^-_2ψ^-_6,^-_2ψ^-_1, ^-_2ψ^-_0}. Changing the order of the second order operators in the factorized form of the ladder operators in (<ref>), we obtain the operator identities ^+_2^-_2=(a^+)^4 , ^+_2^-_2=(a^-)^4 , cf. (<ref>). Again, these relations can be verified by comparing the kernels of the corresponding operators. Yet another pair of the ladder operators corresponds to differential operators of order 5, ℬ^± = ^-_2 a^±_2^+ . They satisfy relations [H̆,ℬ^±]= ± 2ℬ^±, and ℬ^+ℬ^-=H̆(H̆-4)(H̆-6)^2(H̆-8) , ℬ^-ℬ^+=(H̆+2)(H̆-2)(H̆-4)^2(H̆-6) . In correspondence with the first relation in (<ref>), the kernel of the lowering operator is spanned by the ground-state Ψ_0 and the eigenstate Ψ_2 at the bottom of the equidistant infinite part of the spectrum as well as by the three non-physical eigenstates of H̆, (ℬ^-)=span {Ψ_0,Ψ_2, _2^-ψ^-_0, _2^-ψ^-_1,_2^-ψ^-_0}. The kernel of the increasing operator ℬ^+ is spanned by the first excited physical eigenstate Ψ_1 of energy 2, and by four non-physical eigenstates of H̆ of eigenvalues -2, 4 (twice) and 6, (ℬ^+)=span {Ψ_1,_2^-ψ^-_3, _2^-ψ^-_1,_2^-ψ^-_1,_2^-ψ^-_0}. Here ℬ^- and 𝒜^- are related by the operator identity H̆ℬ^-=(H̆-4)𝒜^- , and the relation between the increasing ladder operators ℬ^+ and 𝒜^+ is given by Hermitian conjugation of (<ref>). Note that in comparison with the first relation in (<ref>) and relation (<ref>) in the examples of the REQHO systems with one separated energy level here the relation (<ref>) contains a Hamiltonian-dependent factor before the operator ℬ^-. Coherently with this, in the REQHO system under consideration the lowering operator ℬ^- in comparison with 𝒜^- annihilates only the lowest state Ψ_0 in the separated part of the spectrum. The raising operator ℬ^+ annihilates another separated state Ψ_1 in comparison with both separated states Ψ_0 and Ψ_1 annihilated by 𝒜^+. This difference can be understood if we note that the polynomial in the first identity in (<ref>) does not have the root 2. Rewriting relation (<ref>) in the equivalent form ℬ^-(H̆-2)=𝒜^-(H̆-6), we see then that the annihilation of the state Ψ_1 from the kernel of 𝒜^- is provided by the factor (H̆-2) on the left hand side of the identity. In the same way one can understand the difference in the kernels of the raising operators ℬ^+ and 𝒜^+ by looking at the roots of the polynomial in the second identity in (<ref>) and by taking into account the identity relation ℬ^+H̆=𝒜^+(H̆-4) to be conjugate to (<ref>). The same (up to a global shift) system H̆ given by the potential (<ref>) can be produced by using the higher-order (α)- and (β)-schemes, (α_n+2)={ψ^-_0,…,ψ^-_n-1,ψ^-_n+2,ψ^-_n+3} , (β_n+2)={ψ_0,…,ψ_n-1,ψ_n+2,ψ_n+3} , n=1,…, and by the two intermediate (γ)-schemes, (γ_3)={ψ_0,ψ^-_1,ψ^-_2} , (γ_4)={ψ_0,ψ_1,ψ^-_0,ψ^-_1} . The secondary, higher-order ladder operators can be generated here in the same way as for the REQHO systems with one added gapped bound state. Here the relation (𝒜^-)^4= 𝒞^-(H̆-8)^2 (H̆-6)^2(H̆-4)^2(H̆-2)(H̆-10) . is analogous to the relations (<ref>) and (<ref>), and shows that the ladder operators 𝒞^± can be generated by the ladder operators 𝒜^±. § REQHO SYSTEMS OF A GENERAL FORM AND THEIR LADDER OPERATORS We generalize now our analysis of the particular examples for the case of REQHO systems of a general form, for which we construct the ladder operators and investigate their properties. Each REQHO system can be generated by employing the appropriate DCKAT based on any of the schemes from the two infinite families. The (α)-type schemes include only non-physical eigenstates of the QHO chosen as the seed states. The (β)-type schemes involve only the corresponding physical eigenstates of the QHO. Besides, there exists also a finite number of intermediate (γ)-type schemes which simultaneously use the eigenstates of both types. Let us denote _n_+^- and _n_+^+ the mutually conjugate intertwining operators constructed on the basis of the (α)-type scheme with a minimal number n_+ of seed non-physical eigenstates. They are differential operators of order n_+. Analogously, let us denote the intertwining operators constructed on the basis of the (β)-scheme with a minimal number 2n_- of seed physical eigenstates as _2n_-^- and _2n_-^+. By the construction we have ^-_n_+ψ^-_j_s=0 and ^-_2n_-ψ_i_s=0, where ψ^-_j_s are n_+ non-physical eigenstates of the QHO which are the seed states in the (α_n_+)-scheme, ψ^-_j_s∈ (α_n_+), while ψ_i_s are 2n_- physical eigenstates which are used in the (β_2n_-)-scheme, ψ_i_s∈ (β_2n_-). Such two minimal (α_n_+)- and (β_2n_-)-schemes are complementary similarly to the schemes (α_1) and (β_2) in the case of the simplest REQHO system we considered in detail above. The corresponding Wronskians in these two schemes have the form _n_+=exp(1/2x^2n_+)ϕ(x) and _2n_-=cexp(-x^2 n_-)ϕ(x), where ϕ(x) is some nodeless polynomial function, and c is some constant. We fix the additive constant shifts in the Hamiltonians H of the QHO and H̆ of the REQHO in such a way that ^-_n_+H=H̆^-_n_+ and that the ground-state Ψ_0 of H̆ has zero energy, E_0=0. Then the lowest state Ψ_n_+=^-_n_+ψ_0 in the equidistant infinite part of the spectrum of H̆ will be characterized by the energy value E_n_+=2(n_++2n_-)≡ 2Δ that also will be the energy of the ground-state ψ_0 of the shifted QHO Hamiltonian H, Hψ_0=E_n_+ψ_0. Then we have H=H_osc-1+2Δ. For the other basic lowering intertwining operator ^-_2n_- we have the relation ^-_2n_-H=(H̆+2Δ)^-_2n_- . This means in particular that if ψ(x;E) is an eigenstate of H of energy E and if ^-_2n_-ψ(x;E)≠ 0, then this latter state will be eigenstate of H̆ of the eigenvalue (E-2Δ), H̆ (^-_2n_-ψ(x;E))= (E-2Δ)^-_2n_-ψ(x;E). In terms of the operators _n_+^± and _2n_-^± we construct the three pairs of the basic ladder operators 𝒜^±=_n_+^- a^±_n+^+ , ℬ^±=_2n_-^- a^±_2n-^+ , and 𝒞^-=_2n_-^-^+_n_+ , 𝒞^+=^-_n_+_2n_-^+ . The operators 𝒜^± and ℬ^± are differential operators of orders 2n_++1 and 4n_-+1, respectively, while the ladder operators 𝒞^± are differential operators of order n_++2n_-. These basic ladder operators satisfy the relations [H̆,𝒜^±]= ± 2 𝒜^± , [H̆,ℬ^±]= ± 2 ℬ^± , [H̆,𝒞^±]= ± 2Δ 𝒞^± . We also have the operator identities 𝒜^+𝒜^-=𝒫_𝒜(H̆) , 𝒜^-𝒜^+=𝒫_𝒜(H̆+2) , ℬ^+ℬ^-=𝒫_ℬ(H̆) , ℬ^-ℬ^+=𝒫_ℬ(H̆+2) , and 𝒞^+𝒞^-=𝒫_𝒞(H̆) , 𝒞^-𝒞^+=𝒫_𝒞(H̆+2Δ) , where 𝒫_𝒜(H̆)= (H̆-2Δ) 𝒫_(H̆-2)𝒫_(H̆) , 𝒫_ℬ(H̆)= H̆𝒫_(H̆+2Δ-2)𝒫_(H̆+2Δ) , 𝒫_𝒞(H̆)=𝒫_(H̆) 𝒫_(H̆) . The polynomial 𝒫_ of order n_+ is defined here by 𝒫_(H̆)≡^-_n_+^+_n_+=H̆∏_i=1^n_+-1(H̆-E_i), where E_i, i=1,… ,n_+-1 , are nonzero eigenvalues of the corresponding excited separated (gapped) physical eigenstates Ψ_i of H̆. Together with zero energy E_0=0 of the ground state Ψ_0, the energy values of the n_+ gapped physical eigenstates of H̆ are the shifted by the constant 2Δ=2(2n_-+n_+) energies of the corresponding non-physical eigenstates ψ^-_j_s of H which appear as the seed states in the minimal (α_n_+)-scheme. The permuted product of the intertwining operators gives here ^+_n_+^-_n_+=𝒫_(H). The polynomial 𝒫_ of order 2n_- is defined via the relation 𝒫_(H)≡^+_2n_-^-_2n_-=∏_j=1^2n_-(H-E_j^-), where by E^-_j we denote the shifted for the same constant 2Δ energies of the physical eigenstates ψ_i_s of H which are present as the seed states in the minimal (β_2n_-)-scheme. For the permuted product of the intertwining operators we have ^-_2n_-^+_2n_-= 𝒫_(H̆+2Δ). Let us also note here a useful relation 𝒫_(H̆) 𝒫_(H̆+2Δ)= ∏_j=0^2n_-+n_+-1(H̆-2j) , that reflects the complementarity of the minimal (α_n_+)- and (β_2n_-)-schemes. The operator 𝒜^- annihilates all the n_+ physical eigenstates Ψ_0,…, Ψ_n_+-1 of the system H̆ whose energies lie below the infinite equidistant part of the spectrum and are separated from it by some gap of 2n_0 missing energy levels, n_0≥ 1. Between these n_+ separated energy levels there can appear g, 0≤ g<n_+, `internal' gaps each one containing an even number of missing energy levels. We name the g+1 sets of energy levels in the lower separated part of the spectrum which do not contain internal gaps as valence bands. If g>0, we denote by 2n_1,…, 2n_g the number of missing energy levels in the corresponding internal energy gaps assuming that the highest value g of index i in n_i corresponds here to the lowest energy gap in the spectrum. The total number of the missing energy levels 2(n_0+…+n_g) is equal to the number 2n_- of the physical eigenstates ψ_i_s which participate as the seed states in the minimal (β_2n_-)-scheme. In addition to the gapped physical eigenstates Ψ_0,…, Ψ_n_+-1, the operator 𝒜^- also annihilates the lowest state Ψ_n_+ of energy E_n_+=2Δ in the infinite equidistant part of the spectrum due to the presence of the operator a^- in its structure. Besides, 𝒜^- annihilates some n_+ non-physical eigenstates of H̆. Note in particular that Ψ_0=^-_n_+ψ^-_j_+= ^-_2n_-ψ_0 , Ψ_n_+-1=^-ψ^-_2n_0= ^-_2n_-ψ_j_+-2n_0 , Ψ_n_+=^-_n_+ψ_0= ^-_2n_-ψ_2n_-+n_+ . Here j_+=2n_-+n_+-1 is the maximal value of the index j_s of the non-physical eigenstates ψ^-_j_s∈ (^-_n_+) from the minimal (α_n_+)-scheme; it coincides with the maximal value of the index i_s of the physical eigenstates ψ_i_s∈ (^-_2n_-) from the minimal scheme (β_2n_-). The energy values of the indicated 2n_++1 physical and non-physical eigenstates from the kernel of 𝒜^- are the roots of the polynomial 𝒫_𝒜(H̆) which appears in the first identity in (<ref>). The kernel of 𝒜^+ is spanned by the n_+ lowest separated physical eigenstates Ψ_0,…, Ψ_n_+-1, and by some n_++1 non-physical eigenstates of H̆. The energy values of these eigenstates from (𝒜^+) correspond to the roots of the polynomial 𝒫_𝒜(H̆+2) which appears in the second relation in (<ref>). The kernel of the ladder operator 𝒞^- is spanned by n_++2n_- physical states, n_+ of which, Ψ_0,…, Ψ_n_+-1, correspond to the lowest separated (gapped) energy values. The other 2n_- eigenstates of H̆ in the kernel of 𝒞^- are the physical states ^-_n_+ψ_i_s in a lower part of the infinite equidistant part of the spectrum, where ψ_i_s∈ (^-_2n_-). The number of those `supplementary' states in the lower part of the equidistant spectrum which are not annihilated by 𝒞^- and whose eigenvalues lie below the highest energy value of a physical eigenstate from (𝒞^-) is equal to n_+. The number of the sequential lowest states at the very bottom of the equidistant infinite part of the spectrum of H̆ which are not annihilated by 𝒞^- is equal to the number of the physical states in the lowest valence band of the separated part of the spectrum. The kernel of 𝒞^+ will be spanned by some n_++2n_- non-physical eigenstates. The energies of the corresponding eigenstates from the kernels of the ladder operators 𝒞^- and 𝒞^+ correspond to the roots of the polynomials in H̆ which appear in the first and the second identities in (<ref>). The kernel of the ladder operator ℬ^- contains g+1 physical eigenstates, each one lying at the very bottom of each valence band. Besides, it also contains the physical eigenstate Ψ_n_+ of energy E_n_+=2Δ which is the lowest state of the equidistant infinite part of the spectrum. In addition, (ℬ^-) contains 2n_- non-physical eigenstates of the form Ψ^non_i_s≡^-ψ_i_s, which correspond to the missing energy values in the gaps. Finally, it also involves 2n_- - (g+1) non-physical eigenstates of the form Ψ^non_i_s for all values of the index i_s except those g+1 values, each one of which corresponds to a lowest eigenstate in each gap. The kernel of the increasing ladder operator ℬ^+ contains g+1 physical eigenstates whose eigenvalues lie at the top of each valence band. It also involves 4n_- - g non-physical eigenstates from the gaps. Eigenvalues of the eigenstates from the kernels of the ladder operators ℬ^- and ℬ^+ correspond to the roots of the polynomials which appear, respectively, in the first and the second identities in (<ref>). The relations of the form (<ref>) and (<ref>) are valid for the basic intertwining operators of the arbitrary REQHO system we consider here. Indeed, the operator ^-_2n_- annihilates all the 2n_- physical eigenstates ψ_i_s of the QHO which participate as the seed states in the (β_2n_-)-scheme. On the other hand, when ^-_2n_- acts on the n_+ `supplementary' eigenstates in the lower part of the spectrum of the QHO, it transforms these states into the separated lowest n_+ physical eigenstates of the REQHO system H̆ which constitute the kernel of the intertwining operator _n_+^+. In particular, as we saw ^-_2n_- maps the ground-state ψ_0 of the shifted QHO, Hψ_0=2Δψ_0, into the zero energy ground-state Ψ_0 of H̆, ^-_2n_-ψ_0=Ψ_0, H̆Ψ_0=0. We conclude then that the composite operator _n_+^+^-_2n_- annihilates all the 2n_-+n_+ eigenstates ψ_n of the QHO with n=0,…,2n_-+n_+ -1. But the same job is made by the lowering ladder operator (a^-)^2n_-+n_+ of the QHO. From here we obtain the operator equalities ^+_n_+^-_2n_-=(-1)^n_+(a^-)^2n_-+n_+ , ^+_2n_-^-_n_+=(-1)^n_+(a^+)^2n_-+n_+ . The relations (<ref>) reflect the complementary nature of the involved minimal (α_n_+)- and (β_2n_-)- schemes. The identities (<ref>) are employed to establish the operator identities in (<ref>). They are also essential for the analysis of the kernels of the basic ladder operators. Similarly to the simplest case of the REQHO system, one can construct other pairs of secondary ladder operators different from the described basic ladder operators 𝒜^±, ℬ^± and 𝒞^±. This can be done effectively by introducing additional factors (a^±)^n inside the structure of these operators : 𝒜^±_n≡^-_n_+(a^±)^n^+_n_+, ℬ^±_n≡^-_2n_-(a^±)^n^+_2n_-, n=1,…, 𝒜^±_1=𝒜^±, ℬ^±_1=ℬ^±, and 𝒞^+_n+1≡^-_n_+(a^+)^n_2n_-^+, 𝒞^-_n+1≡_2n_-^- (a^-)^n ^+_n_+, where n=0,…, 𝒞^±_1=𝒞^±. One can also consider the operators 𝒞^-_-n=^-_2n_-(a^+)^n^+_n_+, 𝒞^+_-n=^-_n_+ (a^-)^n ^+_2n_- with n=1,…,2n_-+n_+-1. In 𝒞^±_-n we restrict the values of the index n from above having in mind the identity 𝒞^-_-(2n_-+n_+)= (-1)^n_+ 𝒫_(H̆) 𝒫_(H̆+2Δ) , see Eq. (<ref>), and so, for n>2n_-+n_+-1 these operators do not provide essentially new structures. The secondary, higher-order ladder operators can also be obtained by taking the compositions of the intertwining operators of the corresponding (α)-, (β)- and (γ)-schemes. They also are generated via the composition of the basic ladder operators 𝒜^±, ℬ^± and 𝒞^±. In particular, the quadratic compositions of 𝒜^± and 𝒞^± are given by (<ref>), (<ref>), and by the relations (𝒜^+)^2=𝒫_(H̆-2) 𝒜^+_2 , (𝒞^+)^2=(-1)^n_+𝒞^+_2n_-+n_++1 , 𝒜^+𝒞^-=(-1)^n_+(H̆-2Δ)𝒜^-_2n_-+n_+-1 , 𝒜^+𝒞^+=𝒫_(H̆-2) 𝒞^+_1 , 𝒜^-𝒞^-=(-1)^n_+𝒜^-_2n_-+n_++1 , 𝒜^-𝒞^+=𝒫_(H̆+2)·𝒞^+_-1 , and by the relations conjugate to (<ref>) and (<ref>). The relation (𝒜^-)^2n_-+n_+= (-1)^n_+∏_l=0^2n_-+n_+-1𝒫_(H̆+2l)·𝒞^- shows that as in the considered particular cases of the REQHO systems, the ladder operators 𝒞^± can be generated by the operators 𝒜^±. Also, the following operator identity is valid: (𝒜^-)^2n_-+n_+-1= (-1)^n_+1/H̆∏_j=0^2n_-+n_+-2𝒫_(H̆+2j) 𝒞^-_-1 . Here the operator multiplier before 𝒞^-_-1 is the polynomial of order n_+(2n_-+n_+-1)-1 in H̆ since the j=0 term 𝒫_(H̆) in the product is equal to the factor H̆ which cancels the multiplier 1/H̆ before the product symbol. We also have the identity which relates the operators 𝒜^- and ℬ^-, (H̆-2Δ+2)𝒫_(H̆+2) ℬ^-= (H̆+2)𝒫_(H̆+2Δ) 𝒜^- . The analogous identity for 𝒜^+ and ℬ^+ is obtained from (<ref>) by Hermitian conjugation. In particular cases of the three REQHO systems considered in the previous two sections, relation (<ref>) reduces to the first relation in (<ref>) and to the identities (<ref>) and (<ref>). In conclusion of this section, let us show that the trinity (𝒜^±,ℬ^±,𝒞^±) of the pairs of the lowering and raising ladder operators allows us to generate an arbitrary physical eigenstate from the ground state Ψ_0, and as a consequence, any two physical eigenstates can be related by the appropriate consecutive action of the basic ladder operators from the trinity. First, from the described properties of the operators and commutation relations (<ref>) it follows that in the equidistant infinite part of the spectrum any two eigenstates can be related by the ladder operators 𝒜^± and ℬ^± in the same way as the ladder operators a^± relate the states in the QHO system. The only difference will appear in the numerical coefficients which have to be included into the composition of the indicated basic operators when we work with the normalized eigenstates. If a valence band contains more than one eigenstate, different states in this band can be connected by application to them of the appropriate degrees of the lowering and raising operators ℬ^- and ℬ^+. Note that within the valence band with n_i states these operators satisfy the identity (ℬ^±)^n_i=0. Recall also that the lowest state Ψ_n_+ in the equidistant infinite part of the spectrum is related with the ground state by the action of the ladder operators 𝒞^±: Ψ_0=𝒞^-Ψ_n_+ and Ψ_n_+=𝒞^+Ψ_0. In the same way one can relate any state Ψ_n of energy 0<E_n<2Δ with 0<n≤n_+-1 from the separated part of the spectrum with the corresponding state Ψ_n_++n of energy E_n+2Δ from the equidistant infinite part of the spectrum. Then, if a REQHO system contains more than one valence band, the ground state Ψ_0 from the lowest valence band can be related to some state Ψ_l with eigenvalue E_l from some higher valence band, for instance, by the following composition of the ladder operators: Ψ_l=𝒞^-(𝒜^+)^r_l𝒞^+Ψ_0, Ψ_0=𝒞^- (𝒜^-)^r_l𝒞^+Ψ_l, where r_l=E_l/2. This shows finally that the ladder operators from the trinity are the spectrum-generating operators for the REQHO system of a general form. The described properties of the REQHO systems of a general form are illustrated by Figure <ref>. 1cm § SUMMARY AND OUTLOOK In conclusion, we summarize the obtained results and indicate some interesting problems for further investigation. The REQHO system of a general form is characterized by n_+≥ 1 low-lying energy levels which are separated from the higher equidistant infinite part of the spectrum by some gap of an even number 2n_0≥ 2 of missing levels. Between separated energy levels there can be additional gaps of an even number of missing levels in each such a gap. As a result, in the lower part of the spectrum there appears a total number 2n_-≥ 2n_0 of missing levels, and the separated part of the spectrum is organized into (g+1)≥ 1 `valence bands'. Such a peculiar structure of the spectrum of the REQHO system characterized by the three integer numbers (n_+,2n_-,g+1) is detected and reflected by the trinity (𝒜^±,ℬ^±,𝒞^±) of the pairs of the lowering and raising ladder operators. Any raising or lowering ladder operator from the trinity when acts on a physical eigenstate either transforms it into another physical eigenstate with the changed value of energy, or annihilates it. The separated states are detected by 𝒜^- and 𝒜^+ that are differential operators of order 2n_++1. Each one of these two operators annihilates all the n_+ states Ψ_0,…, Ψ_n_+-1 in the valence bands. The operator 𝒜^- annihilates in addition the lowest state Ψ_n_+ in the equidistant infinite part of the spectrum. In this aspect the lowering ladder operator 𝒜^- has properties very similar to those of the Lax-Novikov integral in reflectionless systems we discussed in Section <ref>. Besides, the operators 𝒜^- and 𝒜^+ annihilate some n_+ and n_++1 non-physical eigenstates, respectively. Due to the relation [H̆,𝒜^±]=± 2𝒜^±, the 𝒜^± act in the equidistant infinite part of the spectrum as the spectrum-generating operators like the ladder operators a^± in the QHO system. Namely, they transform physical eigenstates Ψ_j with j≥ n_+ into the states Ψ_j± 1 by shifting the energy values in ± 2, and 𝒜^-Ψ_n_+=0. It is also worth to note here that the quadratic in the ladder operators 𝒜^± relations in (<ref>) are analogous to the Burchnall-Chaundy polynomial identity <cit.> that relates the Lax-Novikov integral with the corresponding Hamiltonian of a reflectionless (or a finite-gap) system, and underlies the modern theory of integrable systems <cit.>. The ladder operators ℬ^± are differential operators of order 4n_-+1, and each of them effectively counts the number g+1 of the valence bands and measures the size of each valence band. This is done as follows. The lowering operator ℬ^- annihilates each physical eigenstate which lies at the very bottom of each valence band, whereas the raising operator ℬ^+ annihilates each state at the very top of each valence band. So, if a valence band contains only one state, this state is annihilated by both ℬ^- and ℬ^+, and in such a one-dimensional valence band the action of ℬ^± is similar to that of 𝒜^±. However, if we have a valence band with more than one state, the operators ℬ^+ and ℬ^-, unlike 𝒜^±, act in such a band as the raising and lowering operators, [H̆,ℬ^±]=± 2ℬ^±, which satisfy there the relations (ℬ^±)^n_i=0, where n_i is the number of states in the band. Like 𝒜^-, the operator ℬ^- annihilates the lowest state Ψ_n_+ in the equidistant infinite part of the spectrum. The kernels of ℬ^- and ℬ^+ also include some 4n_–g-1 and 4n_–g non-physical eigenstates, respectively. Similarly to 𝒜^±, in the equidistant infinite part of the spectrum the operators ℬ^± also act as the spectrum-generating operators. Under the action of the ladder operators 𝒜^± and ℬ^± the n_+ states from the valence bands turn out to be completely disconnected from the physical states in the equidistant infinite part of the spectrum. Both parts of the spectrum are connected by means of the third pair of the mutually conjugate ladder operators 𝒞^+ and 𝒞^-, that are differential operators of order n_++2n_-. The kernel of the lowering operator 𝒞^- is spanned by physical eigenstates, n_+ of which correspond to all the n_+ eigenstates from the valence bands. The rest 2n_- states from (𝒞^-) are some eigenstates with energy levels lying in the low part of the equidistant infinite part of the spectrum. The positions of those 2n_- energy levels correspond to the missing energy levels in the gaps moved up for the distance 2Δ=2n_++4n_-≥ 6 which is exactly equal to the distance between the energy levels of the ground state Ψ_0 and the lowest state Ψ_n_+ in the equidistant infinite part of the spectrum. The kernel of 𝒞^+ is spanned by some non-physical eigenstates only. Due to the relation [H̆,𝒞^±]=± 2Δ 𝒞^±, the operators 𝒞^+ and 𝒞^- act as the raising and lowering operators changing the energy for ± 2Δ. All the states from the valence bands are obtained by application of the lowering operator 𝒞^- to those low-lying states with energies 2Δ≤ E<4Δ in the equidistant infinite part of the spectrum which are not annihilated by it. In particular, the lowest state Ψ_n_+ with energy E_n_+=2Δ in the equidistant infinite part of the spectrum is transformed by 𝒞^- into the ground-state Ψ_0 of zero energy. The state Ψ_n_+, in turn, can be obtained from Ψ_0 by action of the raising operator 𝒞^+, and is also generated from the state Ψ_n_++Δ of energy E_n_++Δ=4Δ by applying to the latter the lowering operator 𝒞^-. As a consequence of the described properties, any two states in the spectrum of a REQHO system can be related by an appropriate consecutive action of the basic ladder operators from the trinity. In particular, arbitrary excited state from any valence band or from the equidistant infinite part of the spectrum can be obtained from the ground state Ψ_0. This means that the basic ladder operators from the trinity are the spectrum-generating operators of the REQHO system. The energies of all the physical and non-physical eigenstates of the kernels of the lowering operators 𝒜^-, ℬ^- and 𝒞^- are the roots of the corresponding polynomials in H̆ which appear on the right hand side in the first relations from equations (<ref>), (<ref>) and (<ref>), respectively. The eigenvalues of the physical and non-physical eigenstates from the kernels of the conjugate operators 𝒜^+, ℬ^+ and 𝒞^+ are the roots of the corresponding polynomials which appear in the second operator identity relations in the same equations. The basic ladder operators 𝒜^- and ℬ^- satisfy the two-term identity relation (<ref>) which is linear in both of these operators but involve the coefficients that are certain polynomials in the Hamiltonian H̆. The presence of such polynomial coefficients reflects a difference in action of these operators on the states in a separated part of the spectrum. The operators 𝒜^- and 𝒞^- are related by the operator identity of the form (<ref>). Proceeding from these relations, one can obtain the identity that relates the operators ℬ^- and 𝒞^-, and by conjugation one can find the identities that relate the raising operators of the trinity. The operators 𝒜^± are constructed as the ladder operators a^± of the QHO dressed by means of the Darboux-Crum-Krein-Adler intertwining operators ^-_n_+ and ^+_n_+ constructed on the basis of the minimal set of n_+ non-physical eigenstates of the QHO which are used as the seed states in the corresponding DCKAT based on the (α_n_+)-scheme, see Eq. (<ref>). The operators ℬ^± are constructed in the same way with the help of the intertwining operators ^-_2n_- and ^+_2n_- obtained on the basis of the minimal set of the 2n_- physical eigenstates which are employed as the seed states in the DCKAT in the corresponding (β_2n_-)-scheme, see Eq. (<ref>). The operators 𝒞^± can be obtained as the composition (<ref>) of the corresponding intertwining operators from both indicated schemes of the DCKATs. The minimal schemes (α_n_+) and (β_2n_-) are complementary, what is reflected in particular by the relations (<ref>) and (<ref>). The secondary, higher-order ladder operators can be constructed in analogous way by dressing the higher-order ladder operators (a^±)^n of the QHO, or by the composition of the appropriate intertwining operators from the non-minimal (α_n_++n) and (β_2n_-+n) schemes, or by employing the intertwining operators from the corresponding intermediate (γ)-type schemes which use both physical and non-physical eigenstates of the QHO as the seeds states of the corresponding DCKATs. The secondary ladder operators can also be generated via the appropriate composition of the basic (primary) ladder operators 𝒜^±, ℬ^± and 𝒞^±. It seems to be interesting to investigate the quantum mechanical systems related to the exceptional Laguerre and Jacobi orthogonal polynomials in the light of the results on the ladder operators obtained here. The results of such an investigation will be presented elsewhere. The ladder operators ℬ^± have a nature of the polynomially deformed bosonic creation and annihilation operators in the equidistant infinite part of the spectrum. On the other hand, these operators act trivially on the one-state valence bands whose singleton states are annihilated by both the lowering ℬ^- and the raising ℬ^+ operators. The same operators reveal the properties of the deformed fermionic creation and annihilation operators in the valence bands consisting from two states. They have the properties of the deformed para-fermion creation and annihilation operators of order n>2 in those valence bands which contain n>2 eigenstates of H̆. The interesting question is then if there exist some concrete physical systems which would reveal the spectrum of the REQHO systems. If so, it seems that the trinity of the ladder operators should play a fundamental role in the physics associated with such systems. In the same direction the interesting question is whether the quantum mechanical REQHO systems and the structures associated with them can be generalized somehow for the case of the quantum fields. In <cit.>, the family of the REQHO systems with two separated states generated by non-physical seed states ψ^-_m_1 and ψ^-_m_2, m_2-m_1≡ℓ=1+2r, r=0,1,…, m_1=2k, k=1,…, was considered. For such class of the systems, there the lowering, c, and increasing, c^†, ladder operators of the differential order 2+ℓ were constructed by employing auxiliary systems some of which are singular and have a nature similar to that of the isotonic oscillator (<ref>). In the simplest case m_1=2 and m_2=3 such a system corresponds to the REQHO system (<ref>) we considered in Section <ref>. Like our fifth order ladder operator ℬ^-, the third order ladder operator c from <cit.> annihilates the ground state Ψ_0 and the lowest state Ψ_2 in the infinite equidistant part of the spectrum. The increasing operator c^† like our ℬ^+ annihilates the excited state Ψ_1 in the separated two-state lower part of the spectrum. In the systems with ℓ>1, however, the kernel of the increasing operator c^† still includes only one physical state which is, again, the separated state Ψ_1, while our ℬ^+ operator annihilates both separated states Ψ_0 and Ψ_1. In addition to the separated ground state Ψ_0 and the lowest state Ψ_2 in the equidistant part of the spectrum, the kernel of the lowering operator c in this case includes also the ℓ-1 excited states Ψ_3,…,Ψ_2+ℓ in the equidistant part of the spectrum. In this aspect, the lowering operator c from <cit.> has some similarity with our operator 𝒞^-. But our ladder operator 𝒞^- is of differential order 2m_1+2r and its kernel includes some 2m_1+2r-2 excited states in the equidistant part of the spectrum together with both separated states Ψ_0 and Ψ_1. Thus, in the case of ℓ>1 the nature of the operators c and c^† in the sense of the physical states which they annihilate is different from the nature of any of our lowering and increasing ladder operators 𝒜^-, 𝒜^+, ℬ^-, ℬ^+ and 𝒞^-, 𝒞^+. It would be interesting to investigate whether the analogs of the ladder operators c and c^† from <cit.> can be constructed for REQHO systems containing more than two states in the lower separated part of the spectrum, and what is the exact relation of such ladder operators with our trinity (𝒜^±, ℬ^±, 𝒞^±) of the ladder operators. 0.2cm Acknowledgements0.4cm JFC and MSP acknowledge support from research projects FONDECYT 1130017 (Chile), Proyecto Basal USA1555 (Chile), MTM2015-64166-C2-1 (MINECO, Madrid) and DGA E24/1 (DGA, Zaragoza). MSP is grateful for the warm hospitality at Zaragoza University. JFC thanks for the kind hospitality at Universidad de Santiago de Chile. 2cm 99 KaiMos I. Kay and H. E. Moses, “Reflectionless transmission through dielectrics and scattering potentials,", http://aip.scitation.org/doi/abs/10.1063/1.1722296J. Appl. Phys. 27, 1503 (1956). Darb G. Darboux,“Sur une proposition relative aux équations linéaires," C. R. Acad. Sci Paris 94, 1456 (1882). Schr E. Schrödinger, “A method of determining quantum-mechanical eigenvalues and eigenfunctions,” Proc. Roy. Irish Acad. (Sect. A) 46, 9 (1940). InfHull L. Infeld and T. E. Hull, “The factorization method,”http://journals.aps.org/rmp/abstract/10.1103/RevModPhys.23.21 Rev. Mod. Phys. 23, 21 (1951). Crum M. M. Crum, “Associated Sturm-Liouville systems,"http://qjmath.oxfordjournals.org/content/6/1/121.full.pdf+html Quart. J. Math. Oxford 6, 121 (1955). MatSal V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer, Berlin, 1991). CR00 J. F. Cariñena and A. Ramos, “Riccati equation, factorization method and shape invariance,"http://www.worldscientific.com/doi/abs/10.1142/S0129055X00000502 Rev. Math. Phys. 12, 1279 (2000)http://arxiv.org/abs/math-ph/9910020[arXiv:math-ph/9910020]. CR01 J. F. Cariñena, D. J. Fernández and A. Ramos, “Group theoretical approach to the intertwined Hamiltonians,”http://www.sciencedirect.com/science/article/pii/S0003491601961792 Ann. Phys. 292, 42 (2001)http://arxiv.org/abs/math-ph/0311029[arXiv:math-ph/0311029]. MelRos B. Mielnik and O. Rosas-Ortiz, “Factorization: little or great algorithm?,”http://iopscience.iop.org/article/10.1088/0305-4470/37/43/001/meta J. Phys. A 37, 10007 (2004). CR08 J. F. Cariñena and A. Ramos, “Generalized Bäcklund-Darboux transformations in one-dimensional quantum mechanics,”http://www.worldscientific.com/doi/abs/10.1142/S0219887808002989 Int. J. Geom. Methods Mod. Phys. 5, 605 (2008). GroNev D. J. Gross and A. Neveu, “Dynamical symmetry breaking in asymptotically free field theories,”http://journals.aps.org/prd/abstract/10.1103/PhysRevD.10.3235Phys. Rev. D 10, 3235 (1974). DasHasNev R. F. Dashen, B. Hasslacher and A. Neveu, “Semiclassical bound states in an asymptotically free theory,"http://journals.aps.org/prd/abstract/10.1103/PhysRevD.12.2443Phys. Rev. D 12, 2443 (1975). CamBis D. K. Campbell and A. R. Bishop, “Soliton excitations in polyacetylene and relativistic field theory models,"http://www.sciencedirect.com/science/article/pii/055032138290089XNucl. Phys. B 200, 297 (1982). CoJaPl F. Correa, V. Jakubsky and M. S. Plyushchay, “Aharonov-Bohm effect on AdS(2) and nonlinear supersymmetry of reflectionless Pöschl-Teller system,”http://www.sciencedirect.com/science/article/pii/S000349160900044X Annals Phys. 324, 1078 (2009) https://arxiv.org/abs/0809.2854 [arXiv:0809.2854 [hep-th]]. LaxNov A. Arancibia, J. Mateos Guilarte and M. S. Plyushchay, “Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems,”http://journals.aps.org/prd/abstract/10.1103/PhysRevD.87.045009 Phys. Rev. D 87, 045009 (2013) http://arxiv.org/abs/1210.3666 [arXiv:1210.3666 [math-ph]]. BBEIM E. D. Belokolos, A. I. Bobenko, V. Z. Enolskii, A. R. Its, and V. B. Matveev, Algebro-Geometric Approach to Nonlinear Integrable Equations, (Springer, Berlin, 1994). GesHol F. Gesztesy and H. Holden, Soliton Equations and their Algebro-Geometric Solutions, (Cambridge Univ. Press, 2003). exosusy F. Correa, V. Jakubsky, L. M. Nieto and M. S. Plyushchay, “Self-isospectrality, special supersymmetry, and their effect on the band structure,”http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.101.030403Phys. Rev. Lett. 101, 030403 (2008) http://arxiv.org/abs/0801.1671[arXiv:0801.1671 [hep-th]]. Transmut A. Arancibia and M. S. Plyushchay, “Transmutations of supersymmetry through soliton scattering, and self-consistent condensates,”http://journals.aps.org/prd/abstract/10.1103/PhysRevD.90.025008Phys. Rev. D 90, no. 2, 025008 (2014) http://arxiv.org/abs/1401.6709 [arXiv:1401.6709 [hep-th]]. Dub S. Yu. Dubov, V. M. Eleonskii and N. E. Kulagin, “Equidistant spectra of anharmonic oscillators,"http://www.jetp.ac.ru/cgi-bin/e/index/r/102/3/p814?a=list Zh. Eksp. Teor. Fiz. 102, 814 (1992); http://scitation.aip.org/content/aip/journal/chaos/4/1/10.1063/1.166056 Chaos 4, 47 (1994). Adler V. E. Adler, “A modification of Crum's method,"http://link.springer.com/article/10.1007 Theor. Math. Phys. 101, 1381 (1994). SamOvc B. F. Samsonov and I. N. Ovcharov, “Darboux transformation and exactly solvable potentials with quasi-equidistant spectrum," https://link.springer.com/article/10.1007/BF00559274Russ. Phys. J. 38, 765 (1995). BagSam V.G.Bagrov and B.F.Samsonov, “Darboux transformation, factorization, and supersymmetry in one-dimensional quantum mechanics,"https://link.springer.com/article/10.1007 Theor. Math. Phys. 104, 1051 (1995).Spir V. Spiridonov, “Universal superpositions of coherent states and self-similar potentials,"http://journals.aps.org.ezproxy.usach.cl/pra/abstract/10.1103/PhysRevA.52.1909 Phys. Rev. A 52, 1909 (1995)https://arxiv.org/abs/quant-ph/9601030[arXiv:quant-ph/9601030]. JunRoy G. Junker and P. Roy, “Conditionally exactly solvable problems and non-linear algebras," http://www.sciencedirect.com/science/article/pii/S0375960197004222Phys. Lett. A 232, 155 (1997).CPRS J. F. Cariñena, A. M. Perelomov, M. F. Rañada and M. Santander, “A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator,"http://iopscience.iop.org/article/10.1088/1751-8113/41/8/085301/meta J. Phys. A: Math. Theor. 41, 085301 (2008) http://arxiv.org/abs/0711.4899 [arXiv:0711.4899 [quant-ph]]. FellSmi J. M. Fellows and R. A. Smith, “Factorization solution of a family of quantum nonlinear oscillators,"http://iopscience.iop.org/article/10.1088/1751-8113/42/33/335303/meta J. Phys. A 42, 335303 (2009) . Sesma J. Sesma, “The generalized quantum isotonic oscillator,"http://iopscience.iop.org/article/10.1088/1751-8113/43/18/185303/meta J. Phys. A 43, 185303 (2010) http://arxiv.org/abs/1005.1227[arXiv:1005.1227 [quant-ph]]. GGM D. Gómez-Ullate, Y. Grandati, and R. Milson, “Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials,"http://iopscience.iop.org/article/10.1088/1751-8113/47/1/015203/metaJ. Phys. A 47, 015203 (2014) http://arxiv.org/abs/1306.5143[arXiv:1306.5143 [maph-ph]]. Pupas A. M. Pupasov-Maksimov, “Propagators of isochronous an-harmonic oscillators and Mehler formula for the exceptional Hermite polynomials,"http://www.sciencedirect.com.ezproxy.usach.cl/science/article/pii/S0003491615003565 Annals of Physics 363, 122 (2015)https://arxiv.org/abs/1502.01778[arXiv:1502.01778 [maph-ph]]. Krein M. G. Krein, “On a continuous analogue of a Christoffel formula from the theory of orthogonal polynomials," Dokl. Akad. Nauk SSSR 113, 970 (1957). GomGrMil D. Gomez-Ullate, Y. Grandati, and R. Milson, “Extended Krein-Adler theorem for the translationally shape invariant potentials,"http://aip.scitation.org/doi/10.1063/1.4871443J. Math. Phys. 55, 040201 (2014)http://arxiv.org/abs/1309.3756[arXiv:1309.3756 [nlin.SI]]. Exc1 D. Gómez-Ullate, N. Kamran, and R. Milson, “An extended class of orthogonal polynomials defined by a Sturm-Liouville problem,"http://www.sciencedirect.com/science/article/pii/S0022247X09004569 J. Math. Anal. Appl. 359 (2009) 352https://arxiv.org/abs/0807.3939[arXiv:0807.3939 [math-ph]]. Exc1+ D. Gómez-Ullate, N. Kamran, and R. Milson, “An extension of Bochners problem: exceptional invariant subspaces,"http://www.sciencedirect.com/science/article/pii/S0021904509001853 J. Approx. Theory 162, 897 (2010)https://arxiv.org/abs/0805.3376[arXiv:0805.3376 [math-ph]]. OdSas S. Odake and R. Sasaki, “Infinitely many shape invariant potentials and new orthogonal polynomials, http://www.sciencedirect.com/science/article/pii/S0370269309009186 Phys. Lett. B 679, 414 (2009)https://arxiv.org/abs/0906.0142[arXiv:0906.0142 [math-ph]]. OdSas+ S. Odake and R. Sasaki, “Another set of infinitely many exceptional (X) Laguerre polynomials, http://www.sciencedirect.com/science/article/pii/S0370269310000158 Phys. Lett. B 684, 173 (2010)https://arxiv.org/abs/0911.3442[arXiv:0911.3442 [math-ph]]. Ques+ C. Quesne, “Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry,"http://iopscience.iop.org/article/10.1088/1751-8113/41/39/392001/meta J. Phys. A 41, 392001 (2008) http://arxiv.org/abs/0807.4087[arXiv:0807.4087 [quant-ph]]. Ques++ C. Quesne, “Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics,"http://www.emis.de/journals/SIGMA/2009/084/ SIGMA 5, 084 (2009)https://arxiv.org/abs/0906.2331[arXiv:0906.2331 [math-ph]]. SasTsZh R. Sasaki, S. Tsujimoto and A. Zhedanov, “Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum transformations,"http://iopscience.iop.org/article/10.1088/1751-8113/43/31/315204/meta J. Phys. A 43, 315204 (2010) https://arxiv.org/abs/1004.4711[arXiv:1004.4711 [math-ph]]. Grand Y. Grandati, “Solvable rational extensions of the isotonic oscillator,"http://www.sciencedirect.com/science/article/pii/S000349161100039X Ann. Phys. 326, 2074 (2011) https://arxiv.org/abs/1101.0055[arXiv:1101.0055 [math-ph]]. Sasaki R. Sasaki, http://www.apcospa.org/vol2/2-2.htmlThe Universe 2, no. 2, 2 (2014)https://arxiv.org/abs/1411.2703[arXiv:1411.2703 [math-ph]]. VesSha A. P. Veselov and A. B. Shabat, “Dressing chains and the spectral theory of the Schrödinger operator,"http://link.springer.com/article/10.1007 Funct. Anal. Appl. 27, 81 (1993). AdlPai V. E. Adler, “Nonlinear chains and Painlevé equations,"http://www.sciencedirect.com/science/article/pii/016727899490104X Physica D 73, 335 (1994). Hiet R. Willox and J. Hietarinta, “Painlevé equations from Darboux chains: I. PIII - PV,"http://iopscience.iop.org/article/10.1088/0305-4470/36/42/014 J. Phys. A 36, 10615 (2003). oblom A. Oblomkov, “Monodromy-free Schrödinger operators with quadratically increasing potentials,"http://link.springer.com/article/10.1007/BF02557204Theor. Math. Phys. 121, 1574 (1999). MarQue1+ I. Marquette and C. Quesne, “Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators," http://iopscience.iop.org/article/10.1088/1751-8113/46/15/155201/metaJ. Phys. A: Math. Theor. 46, 155201 (2013)https://arxiv.org/abs/1212.3474 [arXiv:1212.3474[math-ph]].MarQue1 I. Marquette and C. Quesne, “New ladder operators for a rational extension of the harmonic oscillator and superintegrability of some two-dimensional systems,"http://aip.scitation.org/doi/10.1063/1.4823771J. Math. Phys. 54, 102102 (2013)https://arxiv.org/abs/1303.7150[arXiv:1303.7150 [math-ph]]. MarQue1++ I. Marquette and C. Quesne, “Combined state-adding and state-deleting approaches to type III multi-step rationally-extended potentials: applications to ladder operators and superintegrability,”http://aip.scitation.org/doi/10.1063/1.4901006J. Math. Phys. 55, 112103 (2014) https://arxiv.org/abs/1402.6380 [arXiv:1402.6380 [math-ph]]. MarI I. Marquette, “New families of superintegrable systems from k-step rational extensions, polynomial algebras and degeneracies,"http://iopscience.iop.org/article/10.1088/1742-6596/597/1/012057/metaJ. Phys.: Conf. Ser. 597, 012057 (2015)https://arxiv.org/abs/1412.0312 [arXiv:1412.0312 [math-ph]]. Ques C. Quesne, “Ladder operators for solvable potentials connected with exceptional orthogonal polynomials,"http://iopscience.iop.org/article/10.1088/1742-6596/597/1/012064/metaJ. Phys. Conf. Ser. 597, 012064 (2015)https://arxiv.org/abs/1412.5874 [arXiv:1412.5874 [math-ph]]. CarPly J. F. Cariñena and M. S. Plyushchay, “Ground-state isolation and discrete flows in a rationally extended quantum harmonic oscillator,"http://journals.aps.org/prd/abstract/10.1103/PhysRevD.94.105022Phys. Rev. D 94, no. 10, 105022 (2016) https://arxiv.org/abs/1611.08051[arXiv:1611.08051 [hep-th]]. SchwAn M. S. Plyushchay, “Schwarzian derivative treatment of the quantum second-order supersymmetry anomaly, and coupling-constant metamorphosis,”http://www.sciencedirect.com/science/article/pii/S0003491616302743Ann. Phys., in press https://arxiv.org/abs/1602.02179[arXiv:1602.02179 [hep-th]]. BurCha J.L. Burchnall, T.W. Chaundy, Commutative ordinary differential operators, http://onlinelibrary.wiley.com/doi/10.1112/plms/s2-21.1.420/abstract;jsessionid=0DB9784C385AC0EDE2AAD1190695C1AF.f02t01Proc. London Math. Soc. Ser. 2, 21, 420 (1923); Commutative ordinary differential operators, http://rspa.royalsocietypublishing.org/content/118/780/557Proc. Royal Soc. London A 118, 557 (1928). Krich I. M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,http://link.springer.com/article/10.1007 “Baker-Akhiezer functions and integrable systems," in: “Integrability. The Seiberg-Witten and Whitham Equations", Edited by H. W. Braden and I. M. Krichever, p. 1 (Gordon and Breach Science Publishers, Amsterdam, 2000). ]

http://arxiv.org/abs/1701.08127v1

20170127174245

Irreducible 3-body forces contributions to the self-energy

nucl-th

plain Raimondi_Barbieri_proc_NTSE_2016 Irreducible 3-body forces contributions to the self-energy F. Raimondi and C. Barbieri Department of Physics, University of Surrey, Guildford GU2 7XH, United Kingdom F. Raimondi and C. Barbieri Including 3N forces in the many-body diagrammatic with the ADC formalism The inclusion of the three-nucleon forces (3NFs) in ab initio many-body approaches is a formidable task, due to the computational load implied by the treatment of their matrix elements. For this reason, practical applications have mostly been limited to contributions where 3NFs enter as effective two-nucleon interactions. In this contribution, we derive the algebraic diagrammatic construction (ADC) working equations for a specific Feynman diagram of the self-energy that contains a fully irreducible three-nucleon force. This diagram is expected to be the most important among those previously neglected, because it connects dominant excited intermediate state configurations. Keywords: Self-consistent Green's function; algebraic diagrammatic construction; three-nucleon forces; computational physics; ab initio nuclear theory § INTRODUCTION The strong connection between advances in theoretical frameworks and empowering of the computational resources has emerged as one of the pillars for the future development of the nuclear theory <cit.>. Different methods, such as the no-core shell model <cit.>, coupled cluster <cit.>, in-medium similarity renormalization group <cit.> and self-consistent Green's function (SCGF) formalism <cit.>, have been extended in recent years by finding efficient algorithms capable to handle the dimensionality of the nuclear many-body problem. Most of these efforts involved novel developments of many-body formalism. In the context of the SCGF theory applied to nuclei, different applications are currently explored. For instance, the extension of the SCFG to encompass the concept of quasiparticle in the sense of the Bogoliubov formalism, that opens the possibility to study the open shell nuclei via the solution of the Gorkov equation <cit.>. The description of nuclear states in the continuum, such as electromagnetic excitations and one-nucleon elastic scattering <cit.>. Also the impact of three-nucleon forces (3NFs) on the mechanism of the saturation in nuclear matter <cit.> and on the correlations of finite nuclei <cit.> has been extensively studied. The formalism required for the inclusion of the 3NFs in the SCGF has been laid down in Ref. <cit.>, where the treatment of the 3NFs in terms of effective (i.e., averaged) one- and two-nucleon forces (2NFs) is described, along with the corresponding Feynman rules for the perturbative expansion of the single-particle (s.p.) propagator. However, the working equations for interaction-irreducible 3NFs (i.e., those diagrams that cannot simplify into effective forces) have not been investigated to date. In this work we derive the working equations of one such self-energy Feynman diagram that contains a 3NF insertion not implicitly included in the effective 2NFs. Among the diagrams featuring interaction-irreducible 3NFs, we focus here on the one which is believed to be dominant, according to the energy required to excite the intermediate particle-hole configurations in the diagram. The equations are cast according to the algebraic diagrammatic construction (ADC) method, a scheme devised in quantum chemistry and applied for the first time to the perturbative expansions of the two-particle (polarization) propagator <cit.> and one-body propagator <cit.> of finite Fermi systems. The ADC allows for an efficient organization of different correlation terms in the description of the self-energy, corresponding to Feynman diagrams with different topologies such as ladder and ring series. Within this scheme, the nuclear Dyson equation is reformulated as an energy-independent Hermitian eigenvalue problem. This simplifies the numerical solution, without resorting to the time-consuming algorithms that scan the entire energy spectrum in search for each pole separately. The increased dimensionality of the eigenvalue problem can be kept under control with the help of large-scale diagonalisation algorithms, such as Lanczos or Arnoldi. We present a brief overview of the SCGF formalism in Section <ref>, covering the expressions of the Dyson equation and the irreducible self-energy. In Section <ref> we review the ADC(n) formalism up to orders n=2 and 3, and we outline the procedure to find the working equations for the elements for the Dyson matrix (see Eq. (<ref>) below). These working equations are given in Section <ref> for n=2 and Section <ref> for n=3. The formalism at second order is worked out in full, while at third order we limit ourselves to the set of diagrams that involve two-particle–one-hole (2p1h) and two-hole–one-particle (2h1p) intermediate configurations to illustrate the approach. In particular, we focus on the interaction-irreducible 3NF Feynman diagram that was neglected in previous works. Finally, the conclusions are given in Section <ref>. § BASIC CONCEPTS OF GREEN'S FUNCTION THEORY In a microscopic approach, the description of the dynamics of the nucleus is based on a realistic interaction among the nucleons, which in principle contains different components, from the 2NF sector until the full N-body interaction. Here, we consider up to 3NFs and start from the nuclear Hamiltonian Ĥ = T̂ + V̂ + Ŵ, with T̂ the kinetic energy part, V̂ the 2NF and Ŵ the 3NF. In order to treat the interaction perturbatively, we introduce the first approximation, based on the concept of the mean field felt by the nucleons as a effective external potential produced by the nuclear medium itself. Accordingly, the Hamiltonian is written as Ĥ = ∑_αβ h^(0)_αβ a^†_α a_β - ∑_αβU_αβ a^†_α a_β+ 1/4∑_αγ βδV_αγ,βδ a_α^† a_γ^† a_δ a_β + 1/36∑_αγϵ βδη W_αγϵ,βδη a_α^† a_γ^† a_ϵ^† a_η a_δ a_β , with ĥ^(0)≡T̂ + Û being the mean field part, while the remaining terms give the residual interaction, which is treated in a perturbative way. The mean field part is given by the sum of the kinetic energy T and the auxiliary potential U, defining the dynamics of the zeroth-order propagator g^(0) defined below, which is referred to as the mean-field reference state. In Eq. (<ref>), V_αγ,βδ and W_αγϵ,βδη are the antisymmetrized matrix elements of the 2N and 3N forces respectively, with the Greek indices α,β,γ,…that label a complete set of s.p. states defining the model space used in the computation. The peculiarity of the self-consistent Green's functions approach consists in including the solution of the dynamics of the A and A± 1 nucleons systems from the start and on the same footing. This information is conveyed by the one-body propagator, or two-point Green's function. The latter is defined as the matrix element of the time-ordered product (𝒯) of an annihilation and creation field operators a(t) and a^†(t) with respect to the fully correlated A-body wave function |Ψ^A_0⟩ in the ground state, i.e. g_αβ(t-t') = - i/ħ⟨Ψ^A_0|𝒯[ a_α(t) a_β^†(t') ] |Ψ^A_0⟩ . The function in Eq. (<ref>) is describing both the propagation of a particle created at time t' in the quantum state β and destroyed at a later time t in the quantum state α, and the propagation of an hole moving moving to the opposite time direction for t'>t. This is why g(τ) also takes the name of one-body propagator. The time-coordinate representation in Eq. (<ref>) can be Fourier-transformed to the energy domain in order to obtain the Lehmann representation of the Green's function, g_αβ(ω) = ∑_n ⟨Ψ^A_0| a_α|Ψ^A+1_n⟩⟨Ψ^A+1_n| a^†_β|Ψ^A_0⟩/ħω - (E^A+1_n - E^A_0) + iη +∑_k ⟨Ψ^A_0| a^†_β|Ψ^A-1_k⟩⟨Ψ^A-1_k| a_α|Ψ^A_0⟩/ħω - (E^A_0 - E^A-1_k) - iη , which contains the relevant spectroscopic informations of the A- and (A±1)-body systems, contained in the transition amplitudes, X^n_β≡⟨Ψ_n^A+1|a_β^†|Ψ_0^A⟩ and Y^k_α≡⟨Ψ_k^A-1|a_α|Ψ_0^A⟩ , which are the overlap integrals that are related to the probability of adding a particle to a orbital β or removing it from a orbital α in a system with A particles. In the following we will use the common notation, 𝒵_α^i=n,k≡ ( X^n_α)^* Y^k_α , with the index i valid for both forward-in-time (particle attachment) and backward-in-time (nucleon removal) processes. Note that we use n to denote particle states and k for hole states. The denominators in Eq. (<ref>) contain also the one-nucleon addition and removal energies ε_n^+≡(E^A+1_n - E^A_0) and ε_k^-≡(E^A_0 - E^A-1_k) , from which one can derive the eigenvalues corresponding to the correlated wave functions |Ψ_n^A± 1⟩, once the ground state energy E^A_0 of |Ψ_0^A⟩ is known. In the following, we will use the compact notations of Eqs. (<ref>-<ref>) to present our equations. The s.p. Green's function (<ref>) is completely determined by solving the Dyson equation, g_αβ(ω)=g^(0)_αβ(ω)+ ∑_γδ g^(0)_αγ(ω)Σ_γδ^⋆(ω) g_δβ(ω) , which is a non-linear equation for the correlated propagator, g(ω). The unperturbed propagator g^(0)(ω) is the propagator corresponding to the Hamiltonian h^(0), which defines the reference state. The irreducible self-energy Σ^⋆(ω) encodes the effects of the nuclear medium on the propagation and it is equivalent to the optical potential for the states in the continuum <cit.>. The irreducible self-energy can be separated in a term which is time-independent, Σ^∞, and a energy dependent part Σ(ω) containing contributions from the dynamical excitations given by the intermediate state configurations (ISCs) within the system: Σ_αβ^⋆(ω) = Σ_αβ^∞+ Σ_αβ(ω) . By inspection of the Dyson equation (<ref>), it should be clear that the self-energy contains all the effects on the propagation of the s.p. that go beyond the mean-field description: For this reason the self-energy can be regarded as an effective potential enriching the unperturbed propagator with many-body correlations and turning it into the dressed propagator. If the exact Σ^⋆(ω) is know, Eq. (<ref>) yields the equivalent of the exact solution of the Schrödinger equation. § ADC FORMALISM AS MATRIX EIGENVALUE PROBLEM In the following we apply the algebraic diagrammatic construction to the dynamic (i.e., energy dependent) part of the irreducible self-energy of Eq. (<ref>). For this purpose, we write Σ(ω) in the most general form of its spectral representation, Σ_αβ(ω) = ∑_j j'M_α j^†[ 1/ħω1 - (E 1 + C) + iη1]_j j'M_j' β + ∑_k k'N_α k[ 1/ħω1 - (E 1 + D) - iη1]_k k'N_k' β^† . At this stage, the expression in Eq. (<ref>) is a formal decomposition defining two types of matrices with respect to the ISCs j, j' (k, k'): the coupling matrix M_j α (N_α k), and the interaction matrix C_j j' (D_ k k') for the forward-in-time (backward-in-time) part of the self-energy. The coupling matrices couple the initial and final s.p. states of the propagator to the ISCs, while the interaction matrices C and D contain the matrix elements of the residual interactions (up to 3NFs) among the ISCs themselves. In general, ISCs are multiparticle-multihole excitations resulting from the same-time propagation of fermion lines within Feynman diagrams. For nucleon addition, with M+1 particles and M holes, (M+1)pMh, their unperturbed energies will be: E_j = ε^+_n_1 + ε^+_n_2 + ⋯  + ε^+_n_M + ε^+_n_M+1 - ε^-_k_1 - ε^-_k_2 - ⋯ - ε^-_k_M . and similarly for nucleon removal. In the following we will make use of the shorthand notation for the forward-in-time terms (corresponding to particle attachment) r, r' ≡ (n_1,n_2,k_3) q, q' ≡ (n_1,n_2,n_3,k_4,k_5) jj' and s, s' ≡ (k_1,k_2,n_3) u, u' ≡ (k_1,k_2,k_3,n_4,n_5) kk' , for the backward-in-time terms (i.e., for particle removal). For instance, j ≡ (n_1,n_2,k_3) in the coupling matrix M_(n_1,n_2,k_3) α connects a s.p. state α to an intermediate state composed by a 2p1h configuration. Each ISC gives a different term in Eq. (<ref>), with the configurations 3p2h, 4p3h and so on pertaining to more complicated, but also energetically less important, intermediate states. While the energy-dependence in the self-energy is a direct consequence of the underlying dynamics in the many-body system, it gives rise to a major computational bottleneck. In order to find all the poles of the propagator in Eq. (<ref>), one should scan the energy plane with an extremely fine mesh, therefore the direct search of the s.p. energies in this way would be costly, with the possibility to leave some solutions undetected. For this reason it is convenient to rearrange the Dyson equation in a matrix form independent of the energy. This is achieved by introducing the eigenvector ^i †≡[ 𝒵^i_δ^† 𝒲^i_r^† 𝒲^i_s^† 𝒲^i_q^† 𝒲^i_u^† ⋯ ] , with the first component given by the transition amplitudes of Eq. (<ref>). The other components contain the information on the ISCs propagated through Σ(ω) but evaluated at the specific quasiparticle energy ε^±_i of each solution, 𝒲^i_j ≡𝒲_j(ω) |_ħω =ε_i = ∑_j'[ 1/ħω1 -(E 1 + C )]_j j'M_j' δ𝒵_δ^i |_ħω =ε_i , and 𝒲^i_k ≡𝒲_k(ω) |_ħω =ε_i = ∑_k'[ 1/ħω1 -(E 1 + D )]_k k'N^†_k' δ𝒵_δ^i |_ħω =ε_i . The task now is to diagonalize the following matrix, being equivalent to the original eigenvalue problem <cit.>: ! ϵ_i ^i = [ ε^(0) + Σ_αδ^∞ M^†_α r N_α s M^†_α q N_α u ⋯ ; ; M_r' α E_r δ_r r' + C_r r' C_r' q ⋯; ; N^†_s' α E_s δ_s s' + D_s s' D_ s' u ⋯; ; M_q' α C_q' r E_q δ_q q' + C_q' q ⋯; ; N^†_u' α D_u' s E_u δ_u u' + D_u' u ⋯; ⋮ ⋮ ⋮ ⋱ ]^i , with the normalization condition ∑_αβ (𝒵_α^i)^†𝒵_β^i + (𝒲^i_j)^†𝒲^i_j + (𝒲^i_k)^†𝒲^i_k + … =1 . With the procedure outlined above and the introduction of the eigenvector ^i of Eq. (<ref>), each energy eigenvalue is now related to an eigenvector of larger dimension. Once Eq. (<ref>) is diagonalized, its eigenvalues and the first portions of their eigenvectors, 𝒵^i, yield the one-body propagator according to Eq. (<ref>). The severe growth in the dimension of the Dyson matrix can be handled by projecting the set of the energies configurations to a smaller Krylov subspace, and then a multi-pivot Lanczos-type algorithm can be applied, as illustrated in Ref. <cit.>. The ADC is a systematic approach to find expressions for the coupling and interaction matrices appearing in Eq. (<ref>) that include the correlations due to 2NFs, 3NFs, and so on. This is achieved by expanding Eq. (<ref>) in powers of the residual interaction Û, the 2NF V̂ and 3NF Ŵ and then by comparing the result with the Goldstone-Feynman perturbative expressions for the self-energy. Formally, we have: M_j α = M^(I)_j α + M^(II)_j α + M^(III)_j α + … , where the term M^(n)_j α is of n^th order in the residual interaction. And for the backward-in-time coupling matrices: N_α k = N^(I)_α k + N^(II)_α k + N^(III)_α k + … . The matrices C and D can only be at first order in the residual interaction, but they appear at the denominators in the spectral representation (<ref>). Thus, they give rise to a geometrical series according to the identity 1/A-B = 1/A + 1/A B 1/A-B = 1/A + 1/A B 1/A + 1/A B 1/A B 1/A +⋯ , for A=ħω - E and B = C, D. Using the expressions (<ref>-<ref>) in Eq. (<ref>) gives rise to the following expansion for the energy-dependent irreducible self-energy, Σ_αβ(ω) = ∑_jM^(I) †_α j[ 1/ħω - E_j + iη] M^(I)_j β + ∑_jM^(II) †_α j[ 1/ħω - E_j + iη] M^(I)_j β + ∑_jM^(I) †_α j[ 1/ħω - E_j + iη] M^(II)_j β + ∑_j j'M^(I) †_α j[ 1/ħω - E_j + iη] C_j j'[ 1/ħω - E_j' + iη] M^(I)_j' β + ⋯ + ∑_kN^(I)_α k[ 1/ħω - E_k- iη] N_k β^(I) † + ∑_kN^(II)_α k[ 1/ħω - E_k- iη] N_k β^(I) † + ∑_kN^(I)_α k[ 1/ħω - E_k- iη] N_k β^(II) † + ∑_k k'N^(I)_α k[ 1/ħω - E_k - iη] D_k k'[ 1/ħω - E_k' - iη] N^(I) †_k' β + ⋯ , where we show all contributions up to second and third order. The procedure is to compare term by term the formal expansion (<ref>) with the calculated Goldstone-type diagrams. One then extracts the minimal expressions for the matrices M, N, C and D, given in terms of the transitions amplitudes of Eqs. (<ref>-<ref>) and the quasiparticle energies of Eqs. (<ref>-<ref>), that ensure consistency with the standard perturbative expansion up to order n. The content of the ADC(n) expansion is far from trivial when one moves from the second to the third order: In fact the structure of the third-order terms in Eq. (<ref>), as analytic functions in the energy plane, does not match the general spectral representation of Eq. (<ref>), which is required for the correct self-energy. However, once M, N, C and D are found one can insert them in the correct analytical representation of Eqs. (<ref>) and (<ref>). As a result, the ADC(n) approach will include selected contributions at order higher than n, as well as all-order non-perturbative resummations as shown by Eq. (<ref>). § ADC EQUATIONS UP TO THIRD ORDER In this section we collect the building blocks of the ADC at second order and present a selected set of coupling and interaction matrices that play a dominant role at third order, in a sense to be specified in the following discussion. All the diagrams discussed in this work are one-particle irreducible, skeleton and interaction-irreducible diagrams. When limiting oneself to only interaction-irreducible diagrams, one needs to substitute the original one- and two-nucleon residual interactions, -Û an V̂, with corresponding effective interactions, which we label respectively U and V and represent diagrammatically as wavy lines. The latter contain averaged contributions from 3NFs that account for the discarded interaction-reducible diagrams. Hence, one reduces the number of perturbative terms (i.e., diagrams) that need to be dealt with. A detailed exposition of these aspects and the extension of Feynman rules to the case of many-nucleon interactions is beyond the scope of the present work. The interested reader is referred to the thorough discussion in Ref. <cit.>. For the present discussion, we only need to keep in mind that the 2NFs in Figs. <ref> are effective interactions which contain the most 'trivial' contributions of Ŵ, in the sense that they do not require any extension of the formalism and computer codes previously developed for pure two-body interactions. §.§ ADC(2) building blocks At second order and with 3NFs, the dynamic self-energy is composed by the two diagrams depicted in Fig. <ref>. The main topological difference between them is given by the fact that Fig. <ref> propagates 2p1h and 2h1p as intermediate states, whereas the diagram of Fig. <ref> contains irreducible 3NFs that generate 3p2h and 3h2p ISCs. Since the latter are energetically less favourable, they are expected to play a minor role at the Fermi surface and to contribute weakly to the total ground state energy. Following the same argument, we can expect that the third-order diagrams containing 2p1h and 2h1p ISCs, discussed in Sec. <ref> (see Fig. <ref>), are more important than the ones with 3p2h and 3h2p configurations, at the same order. To define the ADC(2) approximation scheme, we present the explicit expressions of the coupling and interaction matrices contained in the diagrams of Fig. <ref>. Unless otherwise stated, in this section and in the rest of the paper we adopt the Einstein's convention of summing over repeated indexes for both the model-space s.p. states (α, β, …) and the particle and hole orbits (n_1, n_2, …, k_1, k_2, …). We also use collective indexes for ISCs according to the notation set in Eqs. (<ref>-<ref>), where appropriate. We show first the expressions for the energy-dependent self-energy of Fig. <ref>, Σ^(1a)_αβ(ω) = 1/2V_αϵ, γρ( ∑_n_1,n_2,k_3(_γ^n_1_ρ^n_2_ϵ^k_3 )^* _μ^n_1_ν^n_2_λ^k_3/ħω -(ε_n_1^++ε_n_2^+-ε_k_3^-) + iη. . + ∑_k_1,k_2,n_3 _γ^k_1_ρ^k_2_ϵ^n_3 (_μ^k_1_ν^k_2_λ^n_3)^* /ħω - (ε_k_1^-+ε_k_2^--ε_n_3^+ ) - i η) V_μν , βλ , and in Fig. <ref>, Σ^(1b)_αβ(ω) = 1/12 W_αγδ ,ξτσ( ∑_n_1,n_2,n_3 k_4,k_5(_ξ^n_1_τ^n_2_σ^n_3_δ^k_4_γ^k_5 )^* _μ^n_1_ν^n_2_λ^n_3_ρ^k_4_η^k_5/ħω -(ε_n_1^++ε_n_2^++ε_n_3^+-ε_k_4^--ε_k_5^-) + iη. . + ∑_k_1,k_2,k_3 n_4,n_5_ξ^k_1_τ^k_2_σ^k_3_δ^n_4_γ^n_5 (_μ^k_1_ν^k_2_λ^k_3_ρ^n_4_η^n_5)^* /ħω - (ε_k_1^-+ε_k_2^-+ε_k_3^--ε_n_4^+ -ε_n_5^+) - iη) W_μνλ, βηρ . These expressions at second order in the Feynman-Goldstone perturbative expansion, already match the second-order terms in the analogous expansion of the self-energy (first and fourth lines in Eq. (<ref>)). Since they are already in the correct form of the spectral representation, Eq. (<ref>), it is easy to read the coupling matrices at the ADC(2) level directly from them. In the first line of Eq. (<ref>) we find the coupling matrix M^(I-2N)_(n_1 n_2 k_3) α≡1/√(2) _μ^n_1_ν^n_2_λ^k_3 V_μν,αλ , while in the backward-in-time part (second line of Eq. (<ref>)) we have N^(I-2N)_α (k_1 k_2 n_3)≡1/√(2) V_αλ,μν _μ^k_1_ν^k_2_λ^n_3 , that couples s.p. states to the 2h1p propagator through an effective 2NF. Both these coupling matrices can be depicted as fragments of Goldstone diagrams, as shown in Figs. <ref> and <ref>. The matrix coupling to 3p2h ISCs is found from Eq. (<ref>) and comes from the diagram of Fig. <ref>, M^(I-3N)_(n_1 n_2 n_3 k_4 k_5) α≡1/√(12) _μ^n_1_ν^n_2_λ^n_3_ρ^k_4_η^k_5 W_μνλ,αηρ , while the coupling matrix to 3h2p ISCs has the following expression, N^(I-3N)_α (k_1 k_2 k_3 n_4 n_5)≡1/√(12) W_αηρ,μνλ _μ^k_1_ν^k_2_λ^k_3_ρ^n_4_η^n_5 . Their representation as fragments of Goldstone diagrams is given in Figs. <ref> and <ref>, respectively. All four expressions of Eqs. (<ref>-<ref>) are building blocks of the ADC(2). These complete the set of coupling matrices needed to reproduce the second order terms (first and fourth row) in Eq. (<ref>) and no interaction matrix is needed at this order. Hence, the ADC(2) working equations are finally summarized by Eq. (<ref>) and the following expressions: M^(I)_j α = {[ M^(I-2N)_ r α for j= r=(n_1 n_2 k_3) ,; M^(I-3N)_ q α for j= q =(n_1 n_2 n_3 k_4 k_5) , ]. N^(I)_α k = {[ N^(I-2N)_α s for k= s=(k_1 k_2 n_3) ,; N^(I-3N)_α u for k= u =(k_1 k_2 k_3 n_4 n_5) , ]. C_j j' = 0 , D_k k' = 0 . In ADC(2), the coupling matrices are linked directly without any intermediate interaction insertion, therefore the interaction matrices C and D in Eqs. (<ref>-<ref>) are set to zero. As we show below, this is not anymore true for ADC(3), where the interaction matrices C and D no longer vanish and give rise to infinite (and non-perturbative) resummations of diagrams. §.§ ADC(3) building blocks with 2p1h and 2h1p ISCs The perturbative expansion of the self-energy generates 17 interaction-irreducible Feynman diagrams at third order <cit.>. Of these, only the three shown in Fig. <ref> propagate at most 2p1h and 2h1p ISCs, whereas the remaining diagrams (not shown here) entail at least some contribution from 3p2h or 3h2p configurations. Moreover, Figs. <ref> and <ref> are the sole diagrams that do not involve any interaction-irreducible 3N term. Given that for most systems three-body forces are weaker than the corresponding two-body ones [For nuclear physics, one may estimate that <Ŵ>≈1/10<V̂> <cit.>.], we expect that the contributions of Fig. <ref> and <ref> are the most important and that diagram <ref> is next in order of relevance, while the remaining 14 diagrams will not be dominant. In this section, we present the explicit expressions of the coupling and interaction matrices entering in the ADC(3) formalism, at third order, as derived from the three aforementioned diagrams. As for the second order, a set of expressions for the matrices M, N, C, D and E at ADC(3) is obtained from the direct comparison between the equations of the Feynman diagrams of Fig. <ref>, with the general form of the self-energy in Eq. (<ref>). The different terms can be organised according to the kind of interactions appearing in their contribution. For instance, the diagrams <ref> and <ref> give contributions to the coupling matrices M^(II) and N^(II) that involve two effective 2N interactions and will be labelled with a “2N 2N” superscript. Figures <ref> and <ref> generate all order summations of ladder and ring diagrams, respectively. These contain only effective 2NFs and their ADC(3) equations are well known <cit.>. Hence, we simply states the results here. The forward-in-time coupling matrix arising from Fig. <ref> is given by M^(2N 2N a)_ (n_1 n_2 k_3) α≡1/2√(2) _ρ^n_1_σ^n_2 V_ρσ ,γδ _γ^k_4_δ^k_5/ε_k_4^-+ε_k_5^--ε_n_1^+-ε_n_2^+ (_μ^k_4_ν^k_5)^*_λ^k_3 V_μν,αλ . The ring diagram of Fig. <ref> gives rise to the forward-in-time coupling matrix, M^(2N 2N b)_(n_1 n_2 k_3) α ≡ 1/√(2)( _σ^n_2_δ^k_3 V_σρ, δγ _γ^k_5_ρ^n_4/ε_k_3^--ε_n_2^++ε_k_5^--ε_n_4^+ _μ^n_1 (_ν^k_5_λ^n_4)^* V_μν,αλ. .- _σ^n_1_δ^k_3 V_σρ, δγ _γ^k_5_ρ^n_4/ε_k_3^--ε_n_1^++ε_k_5^--ε_n_4^+ _μ^n_2 (_ν^k_5_λ^n_4)^* V_μν,αλ) , which is explicitly antisymmetrized with respect to the n_1 and n_2 fermion lines. The diagrammatic representations of the two coupling matrices of Eqs. (<ref>) and (<ref>) are depicted in Figs. <ref> and <ref>, respectively. For the same self-energy diagrams of Figs. <ref> and <ref> but from the backward-in-time Goldstone diagrams, we find the coupling matrices N^(2N 2N a)_α (k_1 k_2 n_3)≡1/2√(2) V_αλ,μν _λ^n_3 (_μ^n_4_ν^n_5 )^* _ρ^n_4_σ^n_5 V_ρσ,γδ _γ^k_1_δ^k_2/ε_k_1^-+ε_k_2^--ε_n_4^+-ε_n_5^+ and N^(2N 2N b)_α (k_1 k_2 n_3) ≡ 1/√(2)( V_αλ, μν (_λ^k_5)^* _μ^k_1 (_ν^n_4)^* _σ^n_4_δ^k_5 V_σρ , δγ _γ^k_2_ρ^n_3/ε_k_2^--ε_n_3^++ε_k_5^--ε_n_4^+ . .- V_αλ, μν (_λ^k_5)^* _μ^k_2 (_ν^n_4)^* _σ^n_4_δ^k_5 V_σρ, δγ _γ^k_1_ρ^n_3/ε_k_1^--ε_n_3^++ε_k_5^--ε_n_4^+) . Their diagrammatic representation is displayed in Figs. <ref> and <ref> respectively, where it is clear that they are linked to the 2h1p propagators. The interaction matrices are found by comparing the third order Goldstone diagrams with double poles to the third and sixth lines of Eq. (<ref>). The interaction matrix connecting 2p1h propagators through a particle-particle (pp), ladder, interaction is C^pp_(n_1 n_2 k_3), (n_4 n_5 k_6)≡1/2 _μ^n_1_ν^n_2 V_μν, λρ (_λ^n_4_ρ^n_5)^* δ_k_3 k_6 , while the one connecting through particle-hole (ph) rings is composed by four terms that arise from the antisymmetrization with respect to the n_1 and n_2 particles to the left and the n_4 and n_5 ones to the right, C^ph_(n_1 n_2 k_3), (n_4 n_5 k_6) = 1/2(_ν^n_2_ρ^k_3 V_μν, λρ (_λ^n_5_μ^k_6)^* δ_n_1 n_4. . - _ν^n_2_ρ^k_3 V_μν, λρ (_λ^n_4_μ^k_6)^* δ_n_1 n_5. . - _ν^n_1_ρ^k_3 V_μν, λρ (_λ^n_5_μ^k_6)^* δ_n_2 n_4. . + _ν^n_1_ρ^k_3 V_μν, λρ (_λ^n_4_μ^k_6)^* δ_n_2 n_5) . In the backward-in-time self-energy Goldstone diagrams, the interaction matrices connecting 2h1p propagators through a hole-hole (hh) interaction lead to D^hh_(k_1 k_2 n_3), (k_4 k_5 n_6)≡ -1/2 (_μ^k_1_ν^k_2)^* V_μν, λρ _λ^k_4_ρ^k_5 δ_n_3 n_6 , while the one connecting through a hole-particle (hp) interaction gives D^hp_(k_1 k_2 n_3), (k_4 k_5 n_6) = 1/2( (_μ^k_2_ρ^n_3)^* V_μν, λρ _λ^k_5_ν^n_6 δ_k_1 k_4. . - (_μ^k_2_ρ^n_3)^* V_μν, λρ _λ^k_4_ν^n_6 δ_k_1 k_5. . - (_μ^k_1_ρ^n_3)^* V_μν, λρ _λ^k_5_ν^n_6 δ_k_2 k_4. . + (_μ^k_1_ρ^n_3)^* V_μν, λρ _λ^k_4_ν^n_6 δ_k_2 k_5) . We now turn to the Feynman diagram of Fig. <ref>, which is the focus of the present work. To our knowledge the ADC formulas arising from this term have not been presented before. The Feynman rules give the following expression for it: Σ^(3c)_αβ(ω) = -( ħ)^4/4∫dω_1/2πi∫dω_2/2πi∫dω_3/2πi∫dω_4/2πi∑_γδνμϵλ ξηθστχV_αγ,δν g_ξγ(ω_3) g_νλ(ω -ω_1+ω_3) g_δϵ(ω_1) W_μϵλ,ξηθ g_θτ(ω-ω_2+ω_4) g_ησ(ω_2) g_χμ(ω_4) V_στ,βχ . By performing the four integrals in the complex plane, we find six terms, corresponding to the different time orderings of the three interactions. Altogether we obtain, Σ^(3c)_αβ(ω) = 1/4∑_γδνμϵλ ξηθστχV_αγ,δν W_ϵλμ, ηθξV_στ,βχ× ( -∑_ n_1 n_2 k_3 n_4 n_5 k_6 (_δ^n_1_ν^n_2_γ^k_3)^* _ϵ^n_1_λ^n_2_ξ^k_3 (_η^n_4_θ^n_5_μ^k_6 )^* _σ^n_4_τ^n_5_χ^k_6/(ħω - ( ε_n_1^++ε_n_2^+-ε_k_3^-)+ iη) (ħω - (ε_n_4^++ε_n_5^+-ε_k_6^- ) + iη) . . +∑_ k_1 k_2 n_3 n_4 n_5 k_6 _δ^k_1_ν^k_2_γ^n_3 (_ϵ^k_1_λ^k_2_μ^k_6_η^n_4_θ^n_5_ξ^n_3)^* _σ^n_4_τ^n_5_χ^k_6/(ε_k_1^-+ε_k_2^- +ε_k_6^--ε_n_3^+-ε_n_4^+-ε_n_5^+)(ħω - (ε_n_4^++ε_n_5^+-ε_k_6^+) + iη) . . +∑_ k_1 k_2 n_3 n_4 n_5 k_6 (_δ^n_4_ν^n_5_γ^k_6)^* _ϵ^n_4_λ^n_5_μ^n_3_η^k_1_θ^k_2_ξ^k_6 (_σ^k_1_τ^k_2_χ^n_3)^* /(ħω - (ε_n_4^++ε_n_5^+-ε_k_6^-) + iη)(ε_k_1^-+ε_k_2^- +ε_k_6^--ε_n_3^+-ε_n_4^+-ε_n_5^+) . . - ∑_ k_1 k_2 n_3 k_4 k_5 n_6 _δ^k_1_ν^k_2_γ^n_3 (_ϵ^k_1_λ^k_2_ξ^n_3)^* _η^k_4_θ^k_5_μ^n_6 (_σ^k_4_τ^k_5_χ^n_6)^* /(ħω - ( ε_k_1^-+ε_k_2^--ε_n_3^+)-iη) (ħω - (ε_k_4^-+ε_k_5^--ε_n_6^+ )-iη) . . -∑_ k_1 k_2 n_3 n_4 n_5 k_6 _δ^k_1_ν^k_2_γ^n_3 (_ϵ^k_1_λ^k_2_μ^k_6_η^n_4_θ^n_5_ξ^n_3)^* _σ^n_4_τ^n_5_χ^k_6/(ħω - (ε_k_1^-+ε_k_2^--ε_n_3^+)-iη) (ε_k_1^-+ε_k_2^- +ε_k_6^--ε_n_3^+-ε_n_4^+-ε_n_5^+) . . -∑_ k_1 k_2 n_3 n_4 n_5 k_6 (_δ^n_4_ν^n_5_γ^k_6)^* _ϵ^n_4_λ^n_5_μ^n_3_η^k_1_θ^k_2_ξ^k_6 (_σ^k_1_τ^k_2_χ^n_3)^*/( ε_k_1^-+ε_k_2^- +ε_k_6^--ε_n_3^+-ε_n_4^+-ε_n_5^+)(ħω - (ε_k_1^-+ε_k_2^--ε_n_3^-) -iη) ) , where the first (last) three terms correspond to forward-in-time (backward-in-time) Goldstone diagrams. By comparing to the third order terms in Eq. (<ref>), one see that the new contributions to the coupling matrices contain one effective 2NF and one interaction-irreducible 3NF. The following forward-in-time matrix can be singled out from either the second or third line of Eq. (<ref>), M^(2N 3N a)_(n_1 n_2 k_3) α≡1/2√(2) _ξ^n_4_ρ^n_1_σ^n_2 W_ξρσ , ζηθ _ζ^k_3_η^k_5_θ^k_6/ε_k_3^-+ε_k_5^-+ε_k_6^--ε_n_1^+-ε_n_2^+-ε_n_4^+ (_μ^k_5_ν^k_6_λ^n_4)^* V_μν,αλ , while in the last two lines of Eq. (<ref>) we read the backward-in-time coupling matrix: N^(2N 3N a)_α (k_1 k_2 n_3)≡ - 1/2√(2) V_αλ,μν (_λ^k_4_μ^n_5_ν^n_6 )^* _ρ^n_3_σ^n_5_ξ^n_6 W_ρσξ, θζη _θ^k_4_ζ^k_1_η^k_2/ε_k_1^-+ε_k_2^-+ε_k_4^--ε_n_3^+-ε_n_5^+-ε_n_6^+ . The diagrammatic representations of Eqs. (<ref>) and (<ref>) are displayed in Fig. <ref>. The only interaction matrix that connects 2p1h ISCs through a 3NF is found from the first term of Eq. (<ref>), C^3N_(n_1 n_2 k_3), (n_4 n_5 k_6)≡ -1/2 _ν^n_1_μ^n_2_ρ^k_3 W_νμλ, ϵηρ (_ϵ^n_4_η^n_5_λ^k_6)^* , which is explicitly antisymmetric in the particle indexes. With Eqs. (<ref>) and (<ref>) we can rewrite the first term of Eq. (<ref>) as, M^† (I-2N)_α r 1/ħω - E_r C^3N_r r' 1/ħω - E_r' M^(I-2N)_r' β . The expression (<ref>) contains only the first order contribution in the interaction matrix expansion, corresponding to the second term in the r.h.s. of Eq. (<ref>), for B = C^3N. This is resummed to all order by diagonalizing the Dyson matrix (<ref>), which will automatically include all the higher order terms in the expansion. From the fourth term of Eq. (<ref>), we single out the only backward-in-time interaction matrix connecting two 2h1p configurations through a 3N interaction, that is D^3N_(k_1 k_2 n_3), (k_4 k_5 n_6)≡ -1/2 (_ν^k_1_μ^k_2_ρ^n_3)^* W_νμλ, ϵηρ _ϵ^k_4_η^k_5_λ^n_6 , which is also explicitly antisymmetric in the hole indexes. With Eqs. (<ref>) and (<ref>) we associate the fourth term of Eq. (<ref>) to N^(I-2N)_α s 1/ħω - E_s D^3N_s s' 1/ħω - E_s' N^†(I-2N)_s' β . We stress again the fact that Eq. (<ref>), being a first-order term in D^3N, is resummed with all the other higher order contributions when solving the Dyson equation. Finally, the ADC(3) working equations for the set of Feynman diagrams in Fig. <ref> is summarized by the following expressions, M^(II)_j α = M^(2N 2N a)_ (n_1 n_2 k_3) α + M^(2N 2N b)_ (n_1 n_2 k_3) α + M^(2N 3N a)_ (n_1 n_2 k_3) α , N^(II)_α k = N^(2N 2N a)_α (k_1 k_2 n_3) + N^(2N 2N b)_α (k_1 k_2 n_3) + N^(2N 3N a)_α (k_1 k_2 n_3) , C_j j' = C^pp_(n_1 n_2 k_3), (n_4 n_5 k_6) + C^ph_(n_1 n_2 k_3), (n_4 n_5 k_6) + C^3N_(n_1 n_2 k_3), (n_4 n_5 k_6) , D_k k' = D^hh_(k_1 k_2 n_3), (k_4 k_5 n_6) + D^hp_(k_1 k_2 n_3), (k_4 k_5 n_6) + D^3N_(k_1 k_2 n_3), (k_4 k_5 n_6) . At third order, beside the equations summarized above, there are the coupling and interaction matrices imposed by the remaining 14 one-particle irreducible, skeleton and interaction-irreducible self-energy diagrams, which are topologically distinct from Fig. <ref> <cit.>. The expressions of the coupling and interaction matrices derived from all these diagrams contribute to the 3p2h and 3h2p sectors of Eq. (<ref>) <cit.>. § SUMMARY AND OUTLOOK We have shown the working equations for the ADC(3) formalism applied to the irreducible self-energy in the SCGF formalism, with 2NFs and 3NFs. This formalism allows an efficient and accurate numerical implementation for the solution of the Dyson equation, which is recast as an energy eigenvalue problem. Moreover, within a given order, the matrix form of the Dyson equation allows the infinite resummation of certain classes of diagrams, specifically ladder and ring diagrams, preserving the non-perturbative nature of the SCGF approach. The minimal expressions for both the coupling and interaction matrices, required to conform to the structure of the self-energy as analytic function in the energy plane, have been revisited completely at the ADC(2) level. We have displayed the most important terms at third order, and we derived for the first time the coupling and interaction matrices of the Feynman diagram in Fig. <ref>, containing one interaction-irreducible 3NF. This term is relevant because it is the only irreducible 3NF insertion that links to the dominant ISCs in the self-energy, that is 2p1h and 2h1p intermediate excitations. The complete ADC(3) working equations for the Dyson SCGF approach will be presented in a forthcoming publication <cit.>, while the extension of the Gorkov SCGF formalism of Ref. <cit.> to ADC(3) is part of future plans. ieeetr

http://arxiv.org/abs/1701.07733v3

20170126150535

Qudit hypergraph states and their properties

quant-ph

zbchen@ustc.edu.cn Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China CAS Center for Excellence and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China Hypergraph states, a generalization of graph states, constitute a large class of quantum states with intriguing non-local properties and have promising applications in quantum information science and technology. In this paper, we generalize hypergraph states to qudit hypergraph states, i.e., each vertex in the generalized hypergraph (multi-hypergraph) represents a d-level quantum system instead of a qubit. It is shown that multi-hypergraphs and d-level hypergraph states have a one-to-one correspondence. We prove that if one part of a multi-hypergraph is connected with the other part, the corresponding subsystems are entangled. More generally, the structure of a multi-hypergraph reveals the entanglement property of the corresponding quantum state. Furthermore, we discuss their relationship with some well-known state classes, e.g., real equally weighted states and stabilizer states. These states' responses to the generalized Z (X) operations and Z (X) measurements are studied. The Bell non-locality, an important resource in fulfilling many quantum information tasks, is also investigated. 03.67.-a, 03.67.Ac, 03.65.Ud Qudit hypergraph states and their properties Zeng-Bing Chen ============================================ § INTRODUCTION In quantum information science and technology, graph states constitute an almost unique family of states for their appealing properties and applications <cit.>. They can be used to implement one-way quantum computation <cit.> and construct quantum codes <cit.>. Moreover, they can be used to characterize many kinds of widely used entangled states, such as cluster states <cit.>, the Greenberger-Horne-Zeilinger (GHZ) states <cit.> and more generally, stabilizer states <cit.>. To make quantum states of suitable physical systems describable in the framework as that of graph states, Ref. <cit.> introduced an axiomatic method. Later, Ref. <cit.> generalized this approach and introduced a new class of quantum states named hypergraph states. Like graph states, given a hypergraph, one can define an associated qubit hypergraph state, i.e., hypergraphs can be encoded into quantum states <cit.>. Besides this feature, every qubit hypergraph state corresponds to a stabilizer group <cit.>. However, generally speaking, the stabilizers are no longer products of local operators <cit.>. As a new class of quantum states, they possess lots of new properties, e.g., local unitary symmetries <cit.>, entanglement properties <cit.> and non-local properties <cit.>. Besides these fundamental properties, these states also have many applications. Qubit hypergraph states are real equally weighted states <cit.>, which have important applications in Grover <cit.> and Deutsch-Jozsa <cit.> algorithms. Recently, Ref. <cit.> has shown that, if one has a black box that can tell whether an input qubit hypergraph state is a product state, he/she can solve the NP-complete SAT problem efficiently <cit.>. Fully connected k-uniform qubit hypergraph states, a generalization of GHZ states, are applicable in Heisenberg-limited quantum metrology with more robustness to noise and particle losses <cit.>. Superior to one-way quantum computation based on graph states, measurement-based quantum computation with qubit hypergraph states is non-adaptive, making the measurement scheme simpler <cit.>. In this paper, we implement the concept of multi-hypergraphs <cit.> and encode these multi-hypergraphs into multi-qudit quantum states. We investigate the relationship between the multi-hypergraphs and their corresponding qudit hypergraph states, mainly about the map from the set of multi-hypergraphs to the set of qudit hypergraph states, and the relationship between the connectivity and entanglement. We investigate the local unitary transformations and local measurements on qudit hypergraph states, for pushing their potential applications in error-correcting quantum codes and quantum computation. The Bell non-locality, a useful resource in quantum computation and quantum high precision measurement, is also studied. Furthermore, a systematic approach for the experimental detection is provided. The paper is organized as follows: In Sec. <ref>, we give some preliminary knowledge of hypergraph and qubit hypergraph states, and explain related terminologies. We then generalize these concepts to represent a larger class of quantum states, which we call qudit hypergraph states, using a similar formalism. We show how the notion hypergraph should be modified when each vertex represents a qudit instead of a qubit. In Sec. <ref>, we discuss the connection between the generalized hypergraphs and the qudit hypergraph states, including the correspondence and relationship between the connectivity and the entangled properties. In Sec. <ref>, we discuss the relationship among qudit hypergraph states and some well-known state classes, like real equally weighted states, qudit graph states and stabilizer states. In Sec. <ref>, qudit hypergraph states' responses to local Z and X operations (measurements) are investigated. In Sec. <ref>, we investigate the Bell non-locality <cit.> of N-uniform qudit hypergraph states and expound a general detection scheme for illustrating the Bell non-locality of a general qudit hypergraph state. Conclusions are drawn in Sec. <ref>. § MULTI-HYPERGRAPHS AND QUDIT HYPERGRAPH STATES In this section, we will introduce some preliminary knowledge of hypergraph and qubit hypergraph state and propose our main generalization of these concepts. Some important properties of qubit hypergraph states and qudit hypergraph states will be discussed. §.§ Preliminary: hypergraphs and qubit hypergraph states A hypergraph H is composed of a set of vertices V and a set of hyperedges E <cit.>, i.e., H=(V,E) (For simplification, in this subsection, H represents such a hypergraph.). Suppose that the vertices are labeled as 1,2,⋯,N, then V={1, 2, ⋯, N}. Unlike the edges defined in standard graphs, hyperedges in hypergraphs may connect more (or less) than 2 vertices, i.e., elements in E has a form e={k_1, k_2, ⋯, k_|e|}, where k_1, k_2, ⋯, k_|e| are the vertices connected by e, and |e|, the cardinality of e, ranges from 0 to N. If all the hyperedges in H are of the same cardinality k, then H is called k-uniform <cit.>. Standard graphs are in fact 2-uniform hypergraphs. Some examples of hypergraphs are shown in Fig. <ref>. Hypergraphs can be encoded into a class of quantum states named qubit hypergraph states, in which every vertex represents a two-level quantum system whose computational basis is {|0⟩, |1⟩}. The operator corresponding to the hyperedge e={k_1, ⋯, k_|e|} is defined as C_e=[ -1 |e|=0,; Z |e|=1,; i_k_1,⋯,i_k_|e|=01∑(-1)^i_k_1⋯ i_k_|e|Π̂_i_k_1⋯ i_k_|e| |e|≥ 2, ] where Π̂_i_k_1⋯ i_k_|e|=|i_k_1⋯ i_k_|e|⟩⟨ i_k_1⋯ i_k_|e|| and i_k_1, ⋯, i_k_|e| denotes the value of the vertices k_1, ⋯, k_|e|, respectively. The qubit hypergraph state corresponding to H is |H⟩=e∈ E∏C_e|+⟩^⊗ N, where |+⟩=(|0⟩+|1⟩)/√(2). The state |H⟩ can be interpreted as applying a series of C_e operations to |+⟩ ^⊗ N. As all the C_es are commutative with respect to each other, the order of the operations makes no difference, and a hypergraph corresponds to a definite qubit hypergraph state (see Fig. <ref> for the examples). The same as the graph states, qubit hypergraph states can also be characterized within the framework of stabilizers. Define a set of operators g_k=(∏ _e∈ E C_e) X_k (∏ _e'∈ E C_e')^†=X_k∏ _{e|k∈ e, e∈ E} C_e\{k}, where X_k is the Pauli-X operator of the kth vertex, then (see Fig. <ref> for the examples) g_k|H⟩=|H⟩. Because [g_k,g_k']=(∏ _e∈ E C_e) [X_k,X_k'] (∏ _e'∈ E C_e')^†=0, the set {g_k |k ∈ V} can generate an Abelian cyclic group called the stabilizer group of |H⟩. Either {g_k|k∈ V} or the stabilizer group can determine a qubit hypergraph state up to a phase factor <cit.>. Qubit hypergraph states have interesting properties and important applications. The formalism offers a systematically pictorial representation of the real equally weighted states, which is a vivid way of demonstrating entanglement <cit.>. The entanglement and Bell non-locality make this class of quantum states have a broad range of applications in quantum computation and quantum metrology <cit.>. §.§ Multi-hypergraphs and qudit hypergraph states Multi-hypergraph, whose hyperedge can have a multiplicity larger than 1, is a generalization of hypergraph (see Fig. <ref> for the examples). A multi-hypergraph whose vertices represent d-level quantum systems can be denoted as H_d=(V, E), where V={1, 2, ⋯, N} is the set of vertices, and E is a multiset of the hyperedges. The times an element e occurs in E is called multiplicity of e and is denoted as m_e (m_e ∈{1,2,⋯,d-1}) <cit.>. For the e that satisfies e∈ 2^V (2^V denotes the power set of V, which constitutes of all the subsets of V) and e∉ E, its multiplicity m_e is defined to be 0. With this generalization, every H_d is associated with a definite multiplicity function e → m_e, here e∈ 2^V and m_e∈{0,1,⋯,d-1}. In the following, if not particularly specified, H_d refers to such a multi-hypergraph, and the multiplicity of e is denoted as m_e. Now we define qudit hypergraph states corresponding to H_d. Suppose the computational basis of each vertex is {|0⟩, |1⟩, ⋯, |d-1⟩}, then in this basis the generalized Pauli-X and Pauli-Z operators are <cit.> X=[ [ 0 1 0 ⋯ 0; 0 0 1 ⋯ 0; ⋮ ⋮ ⋮ ⋱ ⋮; 0 0 0 ⋯ 1; 1 0 0 ⋯ 0 ]], Z=[ [ 1 0 0 ⋯ 0; 0 ω_d 0 ⋯ 0; 0 0 ω ^2_d ⋯ 0; ⋮ ⋮ ⋮ ⋱ ⋮; 0 0 0 ⋯ ω ^d-1_d ]], in which ω_d=e^i 2 π/d and X Z=ω_d Z X (Manin's quantum plane algebra <cit.>). The operator corresponding to the hyperedge e={k_1, k_2, ⋯, k_|e|} is defined as C_e=[ ω_d |e|=0,; Z |e|=1,; i_k_1,⋯, i_k_|e|=0d-1∑ω ^i_k_1⋯ i_k_|e|_d Π̂_i_k_1⋯ i_k_|e| |e|≥ 2, ] where Π̂_i_k_1⋯ i_k_|e|=|i_k_1⋯ i_k_|e|⟩⟨ i_k_1⋯ i_k_|e|| and i_k_1, ⋯, i_k_|e| denote the possible values of the vertices k_1, ⋯, k_|e| (in the computational basis), respectively. The unitary operators X, Z, and C_e satisfy X^k =𝕀 k=0 d, Z^k =𝕀 k=0 d, C_e^k =𝕀 k=0 d. Denote that |+⟩_d=∑_k=0^d-1|k⟩/√(d), then the d-level hypergraph state corresponding to H_d can be defined as |H_d⟩ =e∈2^V∏C_e^m_e|+⟩^⊗ N_d. Here the condition “e∈2^V” is equivalent to “e∈ E”, because C_e^0=𝕀 (∀ e∈2^V). For simplicity, in the following we will not express it explicitly. A qudit hypergraph state is also associated with a stabilizer group through which it can be determined up to a phase factor. For H_d=(V,E), define g_k=ł(∏ C_e^m_e)̊ X_k ł(∏ C_e'^m_e')̊^†=X_ke:k∈ e∏(C_e\{k}^†)^m_e, then g_k .|H_d.⟩=.|H_d.⟩, and [g_k,g_k']=ł(∏ C_e^m_e)̊ [X_k,X_k'] ł(∏ C_e'^m_e')̊^†=0. Note that the form of g_k in Eq. (<ref>) is different from that in Eq. (<ref>). The reason is that when d=2, ∀ e, C_e is Hermitian, while for general d, this property cannot always hold. The set {g_k|k∈ V} generates a cyclic Abelian group named the stabilizer group of .|H_d.⟩. Generally speaking, like those of qubit hypergraph states <cit.>, the stabilizers of qudit hypergraph states are also non-local operators. § RELATION BETWEEN MULTI-HYPERGRAPHS AND QUDIT HYPERGRAPH STATES: CORRESPONDENCE AND ENTANGLEMENT PROPERTY In this section, we will discuss the relation between multi-hypergraphs and qudit hypergraph states. Theorem <ref> shows that the map from {H_d|H_d=(V,E)} to {|H_d⟩|H_d=(V,E)}, where H_d is mapped to |H_d⟩, is a bijection. Theorem <ref> demonstrates that the connectivity of a multi-hypergraph is closely related to the entanglement property of the corresponding quantum state. To prove these two theorems, we shall prove several lemmas first. lemmaLemma Divide the hyperedge e={1, 2,⋯, n} into the control part e_C={1, 2, ⋯, m} and the target part e_T={m+1, m+2, ⋯, n}, then C_e=i_1,⋯ ,i_m=0d-1∑|i_1 ⋯ i_m⟩⟨ i_1 ⋯ i_m|C_e_T^i_1 ⋯ i_m. From the definition in Eq. (<ref>), C_e =i_1,⋯ ,i_n=0d-1∑ω _d ^i_1⋯ i_nΠ̂_i_1 ⋯ i_n, C_e_C =i_1,⋯ ,i_m=0d-1∑ω _d ^i_1⋯ i_mΠ̂_i_1 ⋯ i_m, C_e_T =i_m+1,⋯ ,i_n=0d-1∑ω _d^i_m+1⋯ i_nΠ̂_i_m+1⋯ i_n. Because i_m+1,⋯ ,i_n=0d-1∑ω_d ^i_1 ⋯ i_nΠ̂_i_1 ⋯ i_n = i_m+1,⋯ ,i_n=0d-1∑ω_d ^i_1 ⋯ i_nΠ̂_i_1 ⋯ i_mΠ̂_i_m+1⋯ i_n = Π̂_i_1 ⋯ i_mi_m+1,⋯ ,i_n=0d-1∑(ω_d^i_m+1⋯ i_n)^i_1 ⋯ i_mΠ̂_i_m+1⋯ i_n = Π̂_i_1 ⋯ i_m C_e_T^i_1 ⋯ i_m, C_e =i_1, ⋯, i_n=0d-1∑ω_d ^i_1 ⋯ i_nΠ̂_i_1 ⋯ i_n =i_1, ⋯, i_m=0d-1∑i_m+1, ⋯, i_n=0d-1∑ω_d ^i_1 ⋯ i_nΠ̂_i_1 ⋯ i_n =i_1, ⋯, i_m=0d-1∑Π̂_i_1 ⋯ i_m C_e_T^i_1 ⋯ i_m, which is exactly the conclusion in Lemma <ref>. Lemma <ref> demonstrates that a hyperedge operation can be interpreted as a controlled operation: the products of the vertices in C determine the operations imposed on the target part T. In fact, one can choose an arbitrary subset of e as the control part, and the remaining part as the target, which originates from the symmetry of C_e. Consider a system composed of A and B, whose associated Hilbert spaces are ℋ_A and ℋ_B, respectively. Suppose {|1⟩, |2⟩, ⋯, |n⟩} is an orthonormal basis of ℋ_A and |ψ _1⟩, |ψ _2⟩, ⋯, |ψ _n⟩ are normalized vectors in ℋ_B. The vector .......|1⟩|ψ_1.⟩ +|2⟩ |ψ _2⟩ +⋯ +|n⟩ |ψ _n⟩ is a product state if and only if all the |ψ_j⟩s (1≤ j≤ n) are parallel. (i)“if”. If all the |ψ _j⟩s are parallel, then each |ψ _j⟩ has a form .e^i ϕ_j|ψ _0⟩. So |1⟩|ψ _1⟩ +|2⟩ |ψ_2⟩ +⋯ +|n⟩ |ψ _n⟩ =(∑ _j=1^n e^i ϕ_j|j⟩) |ψ _0⟩, which is a product state. (ii)“only if". Suppose the total system is in a product state, B remains the same physical state no matter what measurement is made to A and whatever the result is. By implementing the von Neumann measurement {M_j=|j ⟩⟨ j||j∈{1, 2, ⋯, n}} to A, the part B will collapse to one of the states in ł{|ψ_1⟩, |ψ_2⟩, ⋯, |ψ_n⟩}̊. So all the |ψ_j⟩s are physically equivalent, i.e., they are parallel. Qudit hypergraph state |H_d⟩ equals |+⟩ ^⊗ N_d if and only if E=∅. (i)“if”. If E=∅, by definition for all e∈2^V, m_e=0, so |H_d⟩=|+⟩ ^⊗ N_d. (ii)“only if". The stabilizer group of .|H_d.⟩ is generated by {X_ke:k∈ e∏(C_e\{k}^†)^m_e| k∈ V} while that of |+⟩^⊗ N_d is generated by {X_k |k∈ V}. If e∏C_e^m_e|+⟩^⊗ N_d=|+⟩ ^⊗ N_d, the two qudit hypergraph states will have the same stabilizer group, leading to X_k e:k∈ e∏(C_e\{k}^†)^m_e=X_k j ≠ k∏ X^p_j_j, where k ∈ V and p_j ∈{0,1,⋯,d-1}. The factor e:k∈ e∏(C_e\{k}^†)^m_e is always diagonal in the computational basis, while j ≠ k∏ X^p_j_j is diagonal only if p_j=0 (∀ j ≠ k). That is to say, to make Eq. (<ref>) hold, e:k∈ e∏(C_e\{k}^†)^m_e=j ≠ k∏ X^0_j=𝕀, thus (notice that C_e\{k} is unitary) e:k∈ e∏C_e\{k}^m_e|+⟩^⊗ N-1_d=|+⟩^⊗ N-1_d. Implement the above procedure several times, generally, one arrives e:k_1,⋯,k_n∈ e∏C_e\{k_1,⋯,k_n}^m_e|+⟩^⊗ N-n_d=|+⟩^⊗ N-n_d. When n=N-1, Eq. (<ref>) becomes C_∅^m_{k_1,k_2,⋯,k_N-1}C_{k_N}^m_{k_1,k_2,⋯,k_N}|+⟩_d=|+⟩_d, indicating that m_{k_1,k_2,⋯,k_N-1}=m_{k_1,k_2,⋯,k_N}=0. Because all the k_i (i ∈{1,2,⋯,N}) are arbitrarily arranged in order, for all the e that satisfy |e|=N or N-1, m_e=0. When n=N-2, Eq. (<ref>) becomes e:k_1,⋯,k_N-2∈ e∏C_e\{k_1,⋯,k_N-2}^m_e|+⟩^⊗ 2_d=|+⟩^⊗ 2_d. The product involves all the hyperedges containing {k_1,⋯,k_N-2}, i.e., the cardinalities of these hyperedges are larger or equal to N-2. As is shown in the previous paragraph, hyperedges whose cardinalities are larger than N-2 must have 0 multiplicity, thus contributing to identity factors. So Eq. (<ref>) can be reduced to C_∅^m_{k_1,k_2,⋯,k_N-2}|+⟩^⊗ 2_d=|+⟩^⊗ 2_d, indicating that m_{k_1,k_2,⋯,k_N-2}=0. Generally, if |e|=N-2, m_e=0. Similarly, for all the e that satisfy |e|=N-3,N-2,⋯,0, m_e=0. So if |H_d⟩=|+⟩ ^⊗ N_d, m_e=0 (∀ e∈ 2^V), i.e., E=∅. With these lemmas, we can prove the following theorems. theoremTheorem Suppose H_d'=(V, E') and H_d=(V, E), then |H_d'⟩ = |H_d⟩ if and only if E'=E. (i)“if”. By definition, in terms of representing d-level hypergraph states, a multi-hypergraph corresponds to a unique d-level hypergraph state. (ii)“only if”. For e ∈2^V, denote its multiplicity corresponding to H_d' as m_e', then |H'_d⟩=e∏C_e^m'_e|+⟩^⊗ N_d. If |H_d'⟩ = |H_d⟩, |+⟩^⊗ N_d=e∏C_e^m'_e-m_e|+⟩^⊗ N_d. According to Lemma <ref>, this equation holds if and only if for all e, m'_e-m_e=0, i.e., E'=E. Theorem <ref> indicates that distinct multi-hypergraphs correspond to distinct quantum states, assuming that the systems are both N-qudit systems. An important entanglement property of qudit hypergraph states is revealed in the following theorem. If one part of a multi-hypergraph is connected with the other part, then these two corresponding subsystems are entangled. Suppose H_d=(V,E), divide V into two parts, one is called the control part (C={c_1,c_2,⋯,c_|C|}) and the other is called the target (T={t_1,t_2,⋯,t_|T|}), satisfying C∪ T=V and C∩ T=∅. Accordingly, we can define 3 sub-multisets of E, i.e., E_C, E_T and Λ. E_C (E_T) constitutes of all the elements in E that are subsets of C (T); Λ consists of all the elements in E that contains vertices in C and T simultaneously. If Λ≠∅, C and T are connected through hyperedges in Λ. Define the multi-hypergraphs H_d^C=(C,E_C) and H_d^T=(T,E_T), then |H_d⟩ =C_∅^-m_∅ϵ∈Λ∏ C_ϵ^m_ϵ |H_d^C⟩ |H_d^T⟩, where |H_d^C⟩=∏_e'∈ E_C C_e'^m_e'|+⟩^⊗ |C|_d and |H_d^T⟩=∏_e”∈ E_T C_e”^m_e”|+⟩^⊗ |T|_d (notice that the multiplicity of each hyperedge in E_C, E_T and Λ is the same as the one in E). Expanding |H_d^C⟩ in the computational basis explicitly, one has |H_d^C⟩=1/√(d)^|C|i_c_1,⋯,i_c_|C|=0d-1∑e^i ϕ(i_c_1,⋯ ,i_c_|C|)|i_c_1⋯ i_c_|C|⟩. According to Lemma <ref>, all the C_es in Eq. (<ref>) can be expressed in a form like Eq. (<ref>), so |H_d⟩=C_∅^m_∅/√(d)^|C|i_c_1,⋯,i_c_|C|=0d-1∑|i_c_1⋯ i_c_|C|⟩' f̂(i_c_1,⋯ ,i_c_|C|)|H_d^T⟩, where |i_c_1⋯ i_c_|C|⟩'=e^i ϕ(i_c_1,⋯ ,i_c_|C|)|i_c_1⋯ i_c_|C|⟩ and f̂(i_c_1,⋯ ,i_c_|C|) is some composite hyperedge transformation. If |H_d⟩ is a product state, all f̂(i_c_1,i_c_2,⋯,i_c_|C|)|H_d^T⟩ (∀ i_c_1,⋯,i_c_|C|∈{0,1,⋯,d-1}) must be parallel (Lemma <ref>), i.e., f̂(i_c_1,⋯ ,i_c_|C|)|H_d^T⟩ =e^i δ(i_c_1,⋯ ,i_c_|C|)f̂(0,⋯ ,0)|H_d^T⟩ =e^i δ(i_c_1,⋯ ,i_c_|C|)|H_d^T⟩. Divide every ϵ in Λ into c_ϵ and t_ϵ, where c_ϵ=ϵ∩ C and t_ϵ=ϵ∩ T, then (Lemma <ref>), f̂(1,⋯,1)|H_d^T⟩=ϵ∈Λ∏ C_t_ϵ^m_ϵ|H_d^T⟩. So ϵ∈Λ∏ C_t_ϵ^m_ϵ|H_d^T⟩=e^i δ(1,⋯ ,1)|H_d^T⟩=C_∅^z|H_d^T⟩, where z∈{0,1,⋯,d-1}, thus C_∅^d-zϵ∈Λ∏ C_t_ϵ^m_ϵ|+⟩^⊗ |T|_d=|+⟩^⊗ |T|_d. This equation cannot be true because of Lemma <ref>. So |H_d⟩ cannot be a product state in the form like |ψ⟩ _C|ϕ⟩ _T, i.e., the two parts are entangled. Theorem <ref> offers us an ability to knowing the entanglement structure of a qudit hypergraph state by reading the connectivity property of the multi-hypergraph. With the result in this theorem, we have the following two corollaries. corollaryCorollary If a multi-hypergraph H_d is connected, then |H_d⟩ is genuinely entangled. If H_d is connected, divide it into arbitrary two parts, then the two parts are connected through some hyperedges. According to Theorem <ref>, these two parts are entangled. As the division is arbitrary, |H_d⟩ is non-biseparable, i.e., it is genuinely entangled. Suppose an unconnected multi-hypergraph H_d is composed of several blocks (H_d^(i)) that are not connected to each other, and each one is a connected multi-hypergraph or possesses only one vertex, then each |H_d^(i)⟩ that possesses more than one vertex is a genuinely entangled state, and different blocks are not entangled with each other. Different blocks are not connected to each other, so they are not entangled (Eq. (<ref>)). For connected H_d^(i), because |H_d^(i)⟩ is also a qudit hypergraph state, it is genuinely entangled (Corollary <ref>). Corollary <ref> and Corollary <ref> enable multi-hypergraph a useful tool for visualizing the entanglement of its corresponding qudit hypergraph state. § RELATIONSHIP AMONG QUDIT HYPERGRAPH STATES AND SOME WELL-KNOWN STATE CLASSES In this section, we will discuss relationships among qudit hypergraph states and some well-known state classes, i.e., generalized real equally weighted states, qudit graph states, and stabilizer states. §.§ Qudit hypergraph states and generalized real equally weighted states The real equally weighted states are the quantum states that all the coefficients in the computational basis are real and with equal absolute value. For example, real equally weighted states describing N-qubit systems can all be represented in the form |ψ(f,N)⟩=1/2^N/2∑_i_1,⋯,i_N=0^1 (-1)^f(i_1,⋯,i_N)|i_1 ⋯ i_N⟩, where f(i_1,⋯,i_N)∈ℤ_2. By interpretating -1 as ω_2, the generalized real equally weighted states (GREWSs) can be expressed as |ψ(f,N)⟩_d=1/d^N/2∑_i_1,⋯,i_N=0^d-1ω_d^f(i_1,⋯,i_N)|i_1 ⋯ i_N⟩, in which f(i_1,⋯,i_N)∈ℤ_d. It has been demonstrated in the literature that qubit hypergraph states are equivalent to real equally weighted states <cit.>. For the qudit case, it would be interesting to investigate whether a similar relationship exists. From the definition of qudit hypergraph states, we can see that every N-qudit hypergraph state can be expressed in the form of Eq. (<ref>), i.e., all qudit hypergraph states are GREWSs. For specific N and d, the total number of GREWSs is d^d^N, while in total there are only d^2^N qudit hypergraph states (There are d^2^N such multi-hypergraphs in total and Theorem <ref> shows that the states and multi-hypergraphs have a one-to-one correspondence). Only if d=2, d^d^N=d^2^N, otherwise, d^d^N > d^2^N. This indicates that if d>2, the set of qudit hypergraph states is a proper subset of GREWSs. This relationship is different from the qubit case (See Fig. <ref>). §.§ Relationship among qudit hypergraph states, qudit graph states, and stabilizer states Qudit hypergraph state is a generalization of qudit graph state, so qudit graph states form a subclass of qudit hypergraph states. According to Theorem <ref>, two qudit hypergraph states are equal only if their corresponding multi-hypergraphs are the same. Generally speaking, a multi-hypergraph can have hyperedges with cardinalities larger than 2, which is different from that of multigraphs. Therefore, in general, a qudit hypergraph state is not a qudit graph state. Stabilizer states of N-qudit systems are the common eigenstates with eigenvalue 1 of N independent elements in the Pauli group 𝒢_N^(d) <cit.>, where 𝒢_N^(d) is the N-fold product of 𝒢^(d), and 𝒢^(d)=ł{ω_d^a X^b Z^c |a, b, c ∈ℤ_d}̊ (X and Z are the qudit Pauli operators defined by Eq. (<ref>)). According to this definition, qudit graph states are all stabilizer states, because there are N independent stabilizers that can be expressed in the form g_k=X_k n:{k,n}∈ E∏Z_n^d-m_{k,n}, i.e., g_k∈𝒢_N^(d) (k∈{1,2,⋯,N}). As for the relationship between qudit hypergraph states and stabilizer states, we illustrate the result in the following proposition. propositionProposition A qudit hypergraph state is a stabilizer state if and only if the cardinalities of the hyperedges are all no more than 2. The stabilizer group of |H_d⟩ is generated by {g_k=X_ke:k∈ e∏C_e\{k}^d-m_e|k=1,2,⋯,N}. If the cardinalities of the hyperedges are all no more than 2, then ∀ e, k, C_e\{k} is ω_d or a Z operator. Thus in this case, |H_d⟩ must be a stabilizer state. If some hyperedge in H_d has cardinality larger than 2 (suppose the vertex k is included by such a hyperedge), then g_k∉𝒢_N^(d). The reason is as follows. If g_k ∈𝒢_N, then X_k^-1 g_k ∈𝒢_N. Define a new qudit hypergraph state that |H_d(k)⟩=e:k∈ e∏C_e\{k}^d-m_e|+⟩_d^⊗ N, then it must be a product state. If a hyperedge e satisfies |e|>2, H_d(k) possesses a hyperedge e\{k} satisfying |e\{k}|≥2, which means that some vertices in H_d(k) are connected by e\{k}. According to Theorem <ref>, such a qudit hypergraph state cannot be a product state, which is contrary to that |H_d(k)⟩ is a product state. So only if the cardinalities of all the hyperedges are no more than 2 can |H_d⟩ be a stabilizer state. According to Proposition <ref>, a qudit hypergraph state that is also a stabilizer state at the same time may not be a qudit graph state (See Fig. <ref>). It may also be a qudit graph state operated by some generalized local Pauli operations. To summarize, the relationship among qudit hypergraph states, qudit graph states and stabilizer states can be expressed in Fig. <ref>, which is very similar to the qubit case studied in Ref. <cit.>. § QUDIT HYPERGRAPH STATES' RESPONSES TO LOCAL OPERATIONS AND MEASUREMENTS In this section, we will consider how qudit hypergraph states response to the generalized Z, X operations, and generalized Z, X measurements <cit.>. The initial states are all assumed to be |H_d⟩ and the final states are all denoted as |ψ_f⟩. The local unitary operator Z_k, which acts upon the kth vertex, can also be interpreted as the hyperedge operation C_{k}. So when Z_k is implemented on |H_d⟩, |ψ_f⟩ =C_{k}|H_d⟩, which is also a qudit hypergraph state. Denote the corresponding multi-hypergraph as H_d'=(V,E'), then the associated multiplicity function is m_e'=[ m_e e≠{k},; m_{k}+1 d e={k}. ] In pictorial representation, the multiplicity of the hyperedge {k} increases by 1 (when m_{k}=d-1, the hyperedge cancels, due to Eq. (<ref>)) while the multiplicities of all the other hyperedges do not change. When X_k is implemented upon |H_d⟩, the final state is (see the detailed calculation in Appendix <ref>) |ψ_f⟩=e:k ∈ e∏C_e\{k}^m_e|H_d⟩, which is also a d-level hypergraph state. Denote the corresponding multi-hypergraph as H_d'=(V,E'), then the associated multiplicity function is m_e'=[ m_e k∈ e,; m_e+m_e∪{k} d k ∉ e. ] Equations (<ref>) and (<ref>) indicate that the multi-hypergraph corresponding to the final state is the one that the multiplicities of all the hyperedges in the form e\{k} are added by m_e (k∈ e) with respect to that of H_d (may also subtract a multiple of d in order to keep m_e∈{0,1,⋯,d-1}), while the multiplicities of the other hyperedges do not change. The cyclic group generated by {X_k, .Z_k|k∈ V} preserves the set structure of qudit hypergraph states. For any element in the group, the action on the qudit hypergraph state can be represented by adding related hyperedges to the hypergraph. For a general d-level system, Z and X operators are unitary but usually non-Hermitian. As Z and X both possess orthonormal eigenvectors, one can still define von Neumann measurements of Z and X by associating each eigenvector with a real value. After the measurement, the measured qudit collapses to an eigenstate of Z (X). Correspondingly, the system composed of the remaining qudits collapses to a new state. Suppose we measure Z_k and the vertex collapses to |i_k⟩ (denote that Π̂_i_k=|i_k⟩⟨ i_k|), then the whole system collapses to Π̂_i_k|H_d⟩= (e':k∉ e'∏C_e'^m_e')Π̂_i_k(e:k∈ e∏ C_e^m_e)|+⟩^⊗ N_d. While when k∈ e, Π̂_i_kC_e=Π̂_i_kj_k=0d-1∑Π̂_j_kC_e\{k}^j_k=C_e\{k}^i_kΠ̂_i_k, so Π̂_i_ke:k∈ e∏C_e^m_e|+⟩ ^⊗ N_d= e:k∈ e∏C_e\{k}^i_k m_eΠ̂_i_k|+⟩ ^⊗ N_d = 1/√(d)|i_k⟩e:k∈ e∏C_e\{k}^i_k m_e|+⟩ ^⊗ N-1_d. The final state of the unmeasured part is |ψ_f⟩=(e':k∉ e'∏C_e'^m_e')(e:k∈ e∏C_e\{k}^i_k m_e)|+⟩ ^⊗ N-1_d, which is also a qudit hypergraph state. Denote the multi-hypergraph as H_d'=(V',E'), where V'={1,2,⋯,k-1,k+1,⋯,N} and E' is the multiset of the hyperedges of H_d', then the associated multiplicity function is m_e'= m_e +i_k m_e∪{k} d. If the measurement breaks some hyperedges, then we can get the remaining hypergraph through the following steps: (i) Delete the measured vertex; (ii) Multiply the multiplicities of the broken hyperedges by i_k (the measurement result); (iii) Make the “multiplicities” valid by subtracting some multiple of d. In general, for a 2-level hypergraph state, after a local Pauli-X_k measurement, the unmeasured part collapses to a state that does not correspond to a hypergraph <cit.>. For the more general d-level hypergraph states, the generalized X_k measurements cannot maintain the structure of the set of d-level hypergraph states either. The situations demonstrated in this section are interesting and important because qudit hypergraph states' responses to basic local unitary operations and measurements have potential applications in quantum codes and quantum error correction. § BELL NON-LOCALITY OF QUDIT HYPERGRAPH STATES AND THE EXPERIMENTAL DETECTION The exhibition of non-locality by graph states and qubit hypergraph states is very important and even necessary in many quantum information tasks. Behind such investigation is the challenging problem of the non-locality of multipartite entangled states in quantum information theory. It has been proven that all entangled pure states are non-local, no matter how many particles there are and how many dimensions each particle contains <cit.>. In particular, a scheme of non-locality exhibition was provided in an operational manner in Ref. <cit.>. The idea is that any two particles can be measured to violate Clauser-Horne-Shimony-Holt (CHSH) inequality <cit.>, in assistance of measuring the rest particles. Below we discuss how it works in the scenario of qudit-hypergraph states. §.§ Nonlocality exhibition by the CHSH inequality For clarity of the discussion, we consider a multi-hypergraph H_N,d,m=(V, E), in which V={1,2,⋯,N} and E={V,V,⋯,V} (|E|=m), then the corresponding quantum state is |H_N,d,m⟩ =C_V^m|+⟩_d ^⊗ N =1/√(d^N)∑_i_1,⋯,i_N=0^d-1ω_d^m i_1⋯ i_N|i_1⋯ i_N⟩. Without losing generality, let the vertices 3,4,⋯,N assist the vertices 1 and 2 in exhibiting Bell non-locality by violating CHSH inequality [Notice that for the state |H_N,d,m⟩, the exchange of any pair of vertices does not change the state.]. The assistance can be done by projecting the vertices to their respective |+⟩_d. After this operation, the state of the rest part (composed of vertices 1 and 2) becomes |H_N,d,m^(2)⟩ =𝒩/d^N-1∑_i_1,i_2=0^d-1Ω_i_1i_2|i_1i_2⟩, where 𝒩 is the normalization factor, Ω_i_1i_2=∑_i_3,⋯,i_N=0^d-1ω^m i_1i_2⋯ i_N_d forming the d× d matrix Ω. We state that the remaining two vertices are entangled. The proof can be done through analyzing the rank of Ω. Assume |H_N,d,m^(2)⟩ is separable, the rank of Ω should be 1. But the upper-left 2 × 2 submatrix (i_1,i_2∈{0,1}) of Ω Ω̃=( [ d^N-2 d^N-2; d^N-2 ∑_i_3,⋯,i_N=0^d-1ω^m i_3⋯ i_N; ]), has a non-zero determinant, therefore rank(Ω)≥2 <cit.>, indicating that the rest two vertices are entangled. For analyzing the entanglement property and Bell non-locality, it is more convenient to transform |H_N,d,m^(2)⟩ to its Schmidt form |H_N,d,m^(2)⟩ =∑_μ=0^d-1c_μ|μ⟩_1 |μ⟩_2, where c_μ are the Schmidt coefficients, and |μ⟩_1 and |μ⟩_2 are the Schmidt bases for vertex 1 and 2, respectively. The entanglement of |H_N,d,m^(2)⟩ implies that there are more than 1 non-trivial term in the right hand side of Eq. (<ref>). Finally, we can measure vertex 1 on the settings S_1=σ_z and T_1=σ_x, and vertex 2 on the settings S_2=σ_zcos2t+σ_xsin2t and T_2=σ_zcos2t-σ_xsin2t, where σ_z=|0⟩⟨ 0|-|1⟩⟨ 1|, σ_x=|0⟩⟨ 1|+|1⟩⟨ 0| on respective basis |μ⟩_1 and |μ⟩_2, and tan2t=2c_0c_1. The measurement results will disclose the nonlocality by violating the following CHSH inequality <cit.> C= | E(S_1S_2||+⟩_d ^⊗ N-2)+E(S_1T_2||+⟩_d ^⊗ N-2) +E(T_1S_2||+⟩_d ^⊗ N-2)-E(T_1T_2||+⟩_d ^⊗ N-2) | ≤2. More precisely, the left hand side of the above inequality can achieve 2 √(1+4 c_0^2 c_1^2/(c_0^2 + c_1^2)^2), such that the lhv bound 2 is violated. §.§ The prime-dimensional case Specially, when the dimension of the qudits is prime (d∈ℙ), Ω_i_1i_2 has a simple analytic form Ω_i_1i_2=[ d^N-2 i_1=0 i_2=0,; d^N-2-d (d-1)^N-3 i_1≠ 0 i_2≠ 0. ] In this case, the Schmidt form of |H_N,d,m^(2)⟩ is |H_N,d,m^(2)⟩=x_+|0⟩_1|0⟩_2+x_-|1⟩_1|1⟩_2/√(x_+^2+x_-^2), where x_±=λ±√(λ^2+4(d-λ))/2, and |0⟩_k =1/N_+((x_+ -λ+1)|0⟩+∑_i=1^d-1|i⟩), |1⟩_k =1/N_-((x_- -λ+1)|0⟩+∑_i=1^d-1|i⟩), with N_±=√((x_± -λ+1)^2+d-1), k∈{1,2} and λ=d-(d-1)^N-2/d^N-3. The Schmidt number of |H_N,d,m^(2)⟩ is 2, which indicates that the entanglement of vertices 1 and 2 is equivalent to the entanglement of two qubits. The results in the previous paragraph can be applied here directly except that here tan2t=2x_+x_- /(x_+^2 + x_-^2). Explicitly, in this case the left-hand side of Eq. (<ref>) can violate the CHSH inequality by an amount of 2 √(1+4 x_+^2 x_-^2/(x_+^2 + x_-^2)^2). Figure <ref> reveals the violation of CHSH inequality for various combinations of d (d∈ℙ) and N in this measurement scheme. Here, C is always greater than 2, indicating that this measurement scheme can reveal the nonclassical correlation between the vertices. When d is fixed and N is large (see (a) in Fig. <ref>), the matrix elements of the normalized Ω are nearly equal, i.e., the normalized quantum state of the remaining vertices is approximately |+⟩_d ^⊗ 2, thus C approaches 2 when N goes to infinity. When N is fixed and d increases (see (b) in Fig. <ref>), |H_N,d,m^(2)⟩ approaches (|0⟩∑_i=1^d-1|i⟩+∑_i=1^d-1|i⟩|0⟩)/√(2(d-1)), which is equivalent to a 2-qubit maximally entangled state, thus C approaches 2√(2) when d goes to infinity. §.§ Discussion Remarkably, the above scheme of exhibiting the nonlocality of multipartite quantum systems is potentially applicable in the current use of qudit hypergraph states and conventional qubit graph states. In fact, it can be, and in some cases has been, used in practice with current technology. In the case of entanglement verification, it involves only two measurement settings at each side, where the measurement settings of assistant qudits never change. Besides, the CHSH inequality can always reveal the “strong” nonlocality, in the sense that the entanglement between arbitrary faraway two qudits can be revealed, as long as the two are connected by other vertices and edges. All these features make the above scheme rather experimentally friendly. For example, the entanglement verification of cluster states (a special class of graph states) generated by cold-atom lattices is a necessary work for the future use in quantum computing. However, the detection of the entanglement in large-scale cluster states is always a challenging problem <cit.>. From a practical perspective, the CHSH scheme discussed in this section can also be used as an entanglement witness for cluster states, especially for the long-distance entanglement. That is, one can always choose two interested particles (connected by other particles with C-phase operations), and test the entanglement correlation between them, no matter how far the two particles are. Another example is its application in quantum networks <cit.>, in which thousands of users complete a quantum information task via a multipartite entangled state. A typical task is the so-called third-man quantum cryptography in which generation of a cryptographic key is controlled by a third operator who decides whether to activate the key generation or not <cit.>. The scheme we discussed thus exactly offers an operational way to analyze the security of the third-man quantum cryptography. An important problem in qudit hypergraph states we did not discuss is the genuine multipartite entanglement. In particular, the relationship between the classification of multipartite entanglement and the property of hypergraphs deserves to be studied deeply, and the triple entanglement case has been discussed in <cit.>. As an analog, the concept of genuine multipartite nonlocality is also put forwarded in <cit.>. However, despite its significance in the theoretical study, its applications in quantum information processing need further study. § CONCLUSIONS In this work, we have proposed a large class of quantum states named qudit hypergraph states in which every vertex of the multi-hypergraph represents a d-level quantum system. We have investigated the operational definition of these states and studied their stabilizers, which possess potential applications in quantum codes and quantum computation. It is shown that generalized local X, Z operations, and Z measurements can transform a qudit hypergraph state to another one. The multi-hypergraphs and qudit hypergraph states have a one-to-one correspondence, and the entanglement of the qudit hypergraph states can be directly illustrated by the structure of their corresponding multi-hypergraphs. If a multi-hypergraph (or part of it) is connected, the corresponding quantum system (the quantum system corresponding to the connected part) is genuinely entangled. Such entanglement leads to potential exhibition of Bell non-locality. As an example, we showed how to obtain the violation of Bell inequality in N-uniform qudit hypergraph states. The method is also applicable to other qudit hypergraph states and general N-qudit quantum states. We also study the relationship among qudit hypergraph states and some important state classes. As for the real equally weighted states, we generalized them to the qudit case according to their form in the computational basis. It is shown that only in the 2-level case are the two state classes (“generalized real equally weighted states” and “qudit hypergraph states”) the same; otherwise, qudit hypergraph states are a subclass of “generalized real equally weighted states”. The relationship among qudit hypergraph states, qudit graph states and stabilizer states are discussed. Our results demonstrate that qudit graph states are a common subclass of qudit hypergraph states and stabilizer states. What is more, the union of these two state classes contains more than qudit graph states, which is very similar to the qubit case. Nevertheless, much work is still needed to be done for the potential properties and applications of qudit hypergraph states. It is known that the set of qubit hypergraph states is the same as the set of real equally weighted states, which is a class of quantum states having important applications in quantum algorithms. Qudit hypergraph states form a subclass of generalized real equally weighted states. In this sense, it is highly probable that qudit hypergraph states also have important applications in quantum algorithms. It has been shown in the literature that the unique entanglement form and Bell non-locality of qubit hypergraph states have important applications in quantum metrology and novel quantum computation schemes. It is worth further study to see whether the qudit hypergraph states have similar applications. In this paper, we have focused on the simplest definition of entanglement (a quantum state is entangled if it can not be written as a tensor product of two state vectors), while in fact there are much more comprehensive content in the study of multipartite entanglement, for example, equivalent classes of multipartite entanglement. The discussion of such issues in the context of qudit hypergraph states is not only interesting by itself but also essential for the future applications. We thank Y. Y. Zhao, Y. Liu and Y. L. Zheng for the helpful discussions, and F. E. S. Steinhoff for his nice comments. F. L. X. and Z. B. C. were supported by the National Natural Science Foundation of China (Grant No. 61125502) and the CAS. Y. Z. Z., W. F. C., and K. C. were supported by the National Natural Science Foundation of China (Grants No. 11575174) and the CAS. Note added: During the preparation of our manuscript, we notice that there is a paper which proposed qudit hypergraph states in a different manner and discussed some different issues <cit.>. § DERIVATION OF EQUATION (<REF>) If k ∉ e, C_e X_k C_e^†=X_k. If k ∈ e, for simplicity of the discussion, assume that k=1 and e={1,⋯ ,n}, then C_e=∑_i_1=0^d-1Π̂_i_1 C_e\{1}^i_1 (Lemma <ref>), thus C_e X_1 C_e^† = i_1,j_1=0d-1∑Π̂_i_1(|0⟩⟨ 1|+|1⟩⟨2|+⋯ +|d-1⟩⟨ 0|) Π̂_j_1C_e\{1}^i_1-j_1 = |0⟩⟨ 1|C_e\{1}^-1+|1⟩⟨ 2|C_e\{1}^-1+⋯ +|d-1⟩⟨ 0|C_e\{1}^-1 = X_1C_e\{1}^-1=X_1C_e\{1}^†. Generally, C_e X_k C_e^†=X_k C_e\{k}^†. Let X_k pass over all C_e, then we have (∏ _e∈ E C_e^m_e) X_k (∏ _e'∈ E C_e'^m_e')^† =X_ke:k∈ e∏(C_e\{k}^†)^m_e, which is exactly what is demonstrated in Eq. (<ref>). § DERIVATION OF EQUATION (<REF>) For the vertex e={1,2,⋯,n}, denote that Π̂_i_2 ⋯ i_n=|i_2 ⋯ i_n⟩⟨ i_2 ⋯ i_n|, then X_1 C_e = X_1i_2,⋯ ,i_n=0d-1∑Z_1^i_2 ⋯ i_nΠ̂_i_2 ⋯ i_n = i_2,⋯ ,i_n=0d-1∑ω ^i_2 ⋯ i_n Z_1^i_2 ⋯ i_n X_1 Π̂_i_2 ⋯ i_n = i_2,⋯ ,i_n=0d-1∑Z_1^i_2 ⋯ i_nΠ̂_i_2 ⋯ i_nj_2,⋯ ,j_n=0d-1∑ω ^j_2 ⋯ j_nΠ̂_j_2 ⋯ j_nX_1 = C_eC_e\{1}X_1. Generally, if k∈ e, X_k C_e=C_e C_e\{k}X_k, so |ψ_f⟩ = X_k (∏C_e^m_e)|+⟩ ^⊗ N_d = (e:k∉ e∏C_e^m_e) X_k (e':k∈ e'∏C_e'^m_e') |+⟩ ^⊗ N_d = (e:k∉ e∏C_e^m_e) (e':k∈ e'∏C_e'^m_e'C_e'\{k}^m_e') X_k|+⟩ ^⊗ N_d = (∏ C_e^m_e)(e':k∈ e'∏C_e'\{k}^m_e')|+⟩ ^⊗ N_d = (e':k∈ e'∏C_e'\{k}^m_e')(∏ C_e^m_e)|+⟩ ^⊗ N_d = e:k ∈ e∏C_e\{k}^m_e|H_d⟩. apsrev4-1

http://arxiv.org/abs/1701.07779v1

20170126172017

On Booth lemniscate of starlike functions

math.CV

Assume that Δ is the open unit disk in the complex plane and 𝒜 is the class of normalized analytic functions in Δ. In this paper we introduce and study the class ℬ𝒮(α):={f∈𝒜: (zf'(z)/f(z)-1)≺z/1-α z^2, z∈Δ}, where 0≤α≤1 and ≺ is the subordination relation. Some properties of this class like differential subordination, coefficients estimates and Fekete-Szegö inequality associated with the k-th root transform are considered. R. Kargar, A. Ebadian and J. Sokół] Rahim Kargar, Ali Ebadian and Janusz Sokół Department of Mathematics, Payame Noor University, Tehran, Iran rkargar@pnu.ac.ir (Rahim Kargar) Department of Mathematics, Payame Noor University, Tehran, Iran ebadian.ali@gmail.com (Ali Ebadian) University of Rzeszów, Faculty of Mathematics and Natural Sciences, ul. Prof. Pigonia 1, 35-310 Rzeszów, Poland jsokol@ur.edu.pl (Janusz Sokół) On Lattice Calculation of Electric Dipole Moments and Form Factors of the Nucleon [ 18 november 2016 ================================================================================= § INTRODUCTION Let 𝒜 denote the class of functions f(z) of the form: f(z)=z+ ∑_n=2^∞a_nz^n, which are analytic and normalized in the open unit disk Δ={z∈ℂ : |z|<1}. The subclass of 𝒜 consisting of all univalent functions f(z) in Δ is denoted by 𝒮. A function f∈𝒮 is called starlike (with respect to 0), denoted by f∈𝒮^*, if tw∈ f(Δ) whenever w∈ f(Δ) and t∈[0, 1]. Robertson introduced in <cit.>, the class 𝒮^*(γ) of starlike functions of order γ≤1, which is defined by 𝒮^*(γ):={ f∈𝒜: ℜ𝔢{zf'(z)/f(z)}> γ, z∈Δ}. If γ∈[0,1), then a function in 𝒮^*(γ) is univalent. In particular we put 𝒮^*(0)≡𝒮^*. We denote by 𝔅 the class of analytic functions w(z) in Δ with w(0) = 0 and |w(z)| < 1, (z ∈Δ). If f and g are two of the functions in 𝒜, we say that f is subordinate to g, written f (z)≺ g(z), if there exists a w∈𝔅 such that f (z) = g(w(z)), for all z∈Δ. Furthermore, if the function g is univalent in Δ, then we have the following equivalence: f (z)≺ g(z) ⇔ (f (0) = g(0) and f (Δ)⊂ g(Δ)). We now recall from <cit.>, a one-parameter family of functions as follows: F_α(z):=z/1-α z^2=z+∑_n=1^∞α^nz^2n+1 (z∈Δ, 0≤α≤1). The function F_α(z) is starlike univalent for 0≤α<1. We have also F_α(Δ)=D(α), where D(α)={x+iy∈ℂ:  (x^2+y^2)^2-x^2/(1-α)^2-y^2/(1+α)^2<0}, when 0≤α<1 and D(1)={x+iy∈ℂ:  (∀ t∈ (-∞,-i/2]∪ [i/2,∞))[x+iy≠ it]}. The Persian curve (cf. <cit.>) is a plane algebraic curve of order four that is the line of intersection between the surface of a torus and a plane parallel to its axis. The equation in rectangular coordinates is (x^2+y^2+p^2+d^2-r^2)^2=4d^2(x^2+p^2), where r is the radius of the circle describing the torus, d is the distance from the origin to its center and p is the distance from the axis of the torus to the plane. We remark that a curve described by (x^2 + y^2)^2 - (n^4 + 2m^2)x^2 - (n^4 - 2m^2)y^2 = 0 (x, y)≠(0, 0), is a special case of Persian curve that studied by Booth and is called the Booth lemniscate <cit.>. The Booth lemniscate is called elliptic if n^4 > 2m^2 while, for n^4 < 2m^2, it is termed hyperbolic. Thus it is clear that the curve (x^2+y^2)^2-x^2/(1-α)^2-y^2/(1+α)^2=0 (x, y)≠(0, 0), is the Booth lemniscate of elliptic type (see figure 1). Two other special case of Persian curve are Cassini oval and Bernoulli lemniscate. A plane algebraic curve of order four whose equation in Cartesian coordinates has the form: (x^2 + y^2)^2 - 2c^2 (x^2 -y^2) = a^4 -c^4. The Cassini oval is the set of points such that the product of the distances from each point to two given points p_2=(-c,0) and p_1=(c,0) (the foci) is constant. When a ≥ c√(2) the Cassini oval is a convex curve; when c<a<c√(2) it is a curve with "waists" (concave parts); when a=c it is a Bernoulli lemniscate; and when a<c it consists of two components. Cassini ovals are related to lemniscates. Cassini ovals were studied by G. Cassini (17th century) in his attempts to determine the Earth's orbit. The Bernoulli lemniscate plane algebraic curve of order four, the equation of which in orthogonal Cartesian coordinates is (x^2 + y^2)^2 - 2a^2 (x^2 -y^2) =0, and in polar coordinates ρ^2=2a^2 cos 2 ϕ. The Bernoulli lemniscate is symmetric about the coordinate origin, which is a node with tangents y=± x and the point of inflection. The product of the distances of any point M to the two given points p_1=(-a,0) and p_2=(a,0) is equal to the square of the distance between the points p_1 and p_2. The Bernoulli lemniscate is a special case of the Cassini ovals, the lemniscates, and the sinusoidal spirals. The Bernoulli spiral was named after Jakob Bernoulli, who gave its equation in 1694. In <cit.>, the authors introduced and studied the class ℳ(δ) as follows: Definition A. Let π/2≤δ<π. Then the function f∈𝒜 belongs to the class ℳ(δ) if f satisfies: 1+δ-π/2 sinδ< ℜ𝔢{zf'(z)/f(z)} < 1+δ/2sinδ (z∈Δ). By definition of subordination and by (<ref>), we have that f∈ℳ(δ) if and only if (z f'(z)/f(z)-1)≺ℬ_α(z):= 1/2isinδlog(1-z/1-ze^-iδ) (z∈Δ), where π/2≤δ<π. The above function ℬ_α(z) is convex univalent in Δ and maps Δ onto Γ_δ={w: (δ-π)/(2sinδ)<ℜ𝔢{w}<δ/(2sinδ)}, or onto the convex hull of three points (one of which may be that point at infinity) on the boundary of Γ_δ. In other words, the image of Δ may be a vertical strip when π/2≤δ<π, while in other cases, a half strip, a trapezium, or a triangle. It was proved in <cit.>, that for α<1<β, the following function P_α,β:Δ→ℂ, defined by P_α,β(z)=1+β-α/πi log(1-e^2π i1-α/β-αz/1-z) (z∈Δ), maps Δ onto a convex domain P_α,β(Δ)={ w∈ℂ: α< ℜ𝔢{w}<β}, conformally. Therefore, the function P_α,β(z) defined by (<ref>) is convex univalent in Δ and has the form: P_α,β(z)=1+∑_n=1^∞ B_n z^n, where B_n=β-α/nπi (1-e^2nπ i1-α/β-α) (n=1,2,…). The present authors (see <cit.>) introduced the class 𝒱(α,β) as follows: Definition B. Let α<1 and β>1. Then the function f∈𝒜 belongs to the class 𝒱(α,β) if f satisfies: α< ℜ𝔢{(z/f(z))^2 f'(z)} <β (z∈Δ). Therefore, by definition of subordination, we have that f∈𝒱(α,β) if and only if (z/f(z))^2 f'(z) ≺ P_α,β(z) (z∈Δ). Motivated by Definition A, Definition B and using F_α, we introduce a new class. Our principal definition is the following. Let f∈𝒜 and 0≤α< 1. Then f∈ℬ𝒮(α) if and only if (zf'(z)/f(z)-1)≺ F_α(z) (z∈Δ), where F_α defined by (<ref>). In our investigation, we require the following result. We have that f∈ℬ𝒮(α) if and only if f(z)=zexp∫_0^z F_α(w(t))-1/t dt (z∈Δ), for some function w(z), analytic in Δ, with |w(z)|≤ |z| in Δ. From (<ref>) it follows that there exists a function w(z), analytic in Δ, with |w(z)|≤ |z| in Δ, such that z(f'(z)/f(z)-1/z)=F_α(w(z)) (z∈Δ), or z(logf(z)/z)'=F_α(w(z)) (z∈Δ). This gives (<ref>). On the other hand, it is a easy calculation that a function having the form (<ref>) satisfies condition (<ref>). Applying formula (<ref>) for w(z)=z gives that f_0(z)=z(1+√(α)z/1-√(α)z)^1/√(α) (z∈Δ), is in the class ℬ𝒮(α). Let F_α(z) be given by (<ref>). Then 1/α-1< ℜ𝔢{F_α(z)}< 1/1-α (0≤α<1). If α=0, then we have -1<ℜ𝔢{F_α}=ℜ𝔢(z)<1. For 0<α<1, the function {F_α} does not have any poles in Δ and is analytic in Δ, thus looking for the min{ℜ𝔢{F_α(z)}:  |z|<1} it is sufficient to consider it on the boundary ∂ F_α(Δ)={F_α(e^iφ):φ∈ [0,2π]}. A simple calculation give us ℜ𝔢{F_α(e^iφ)}=(1-α)cosφ/1+α^2-2αcos2φ (φ∈[0,2φ]). So we can see that ℜ𝔢{F_α(z)} is well defined also for φ=0 and φ=2π. Define g(x)=(1-α)x/1+α^2-2α (2x^2-1) (-1≤ x≤ 1), then for 0< α<1, we have g'(x)>0. Thus for -1≤ x≤ 1, we have 1/α-1=g(-1)≤ g(x)≤ g(1)=1/1-α. This completes the proof. We note that from Lemma <ref> and by definition of subordination, the function f∈𝒜 belongs to the class ℬ𝒮(α), 0≤α<1, if it satisfies the condition 1/α-1< ℜ𝔢(zf'(z)/f(z)-1)< 1/1-α (z∈Δ), or equivalently α/α-1< ℜ𝔢(zf'(z)/f(z))<2-α/1-α (z∈Δ). It is clear that ℬ𝒮(0)≡𝒮(0,2)⊂𝒮^*, where the class 𝒮(α, β), α< 1 and β > 1, was recently considered by K. Kuroki and S. Owa in <cit.>. If f∈ℬ𝒮(α), then zf'(z)/f(z)≺ P_α(z) (z∈Δ), where P_α(z)=1+2/π(1-α)i log(1-e^π i(1-α)^2z/1-z) (z∈Δ), is convex univalent in Δ. If f∈ℬ𝒮(α), then it satisfies (<ref>) or zf'(z)/f(z) lies in a strip of the form (<ref>). Then applying the definition of subordination and the function (<ref>), we obtain (<ref>) and (<ref>). For the proof of our main results, we need the following Lemma. (See <cit.>) Let q(z)=∑_n=1^∞C_nz^n be analytic and univalent in Δ, and suppose that q(z) maps Δ onto a convex domain. If p(z) = ∑_n=1^∞A_nz^n is analytic in Δ and satisfies the following subordination p(z)≺ q(z) (z∈Δ), then |A_n|≤ |C_1| n≥ 1. § MAIN RESULTS The first main result is the following theorem. Let f∈𝒜 and 0≤α<1. If f∈ℬ𝒮(α) then logf(z)/z≺∫_0^z P_α(t)-1/t dt (z∈Δ), where P_α(z)-1=2/π(1-α)i log(1-e^π i(1-α)^2z/1-z) (z∈Δ) and P_α(z)=∫_0^z P_α(t)-1/t dt (z∈Δ), are convex univalent in Δ. If f∈ℬ𝒮(α), then by (<ref>) it satisfies z{logf(z)/z}'≺ P_α(z)-1. It is known that if ℱ(z) is convex univalent in Δ, then [f(z)≺ℱ(z)]⇒[∫_0^z f(t)/t dt≺∫_0^z ℱ(t)/t dt] and F(z)=∫_0^z ℱ(t)/t dt, is convex univalent in Δ. By Corollary <ref>, we know that P_α(z)-1 is convex univalent in Δ. Therefore, applying (<ref>) in (<ref>) gives (<ref>) with convex univalent P_α(z). If f∈ℬ𝒮(α) and |z|=r<1, then min_|z|=r|expP_α(z)|≤|f(z)/z| ≤max_|z|=r|expP_α(z)|. Subordination implies f(z)/z≺expP_α(z) and expP_α(z) is convex univalent. Then (<ref>) implies (<ref>). We now obtain coefficients estimates for functions belonging to the class ℬ𝒮(α). Assume that the function f of the form (<ref>) belongs to the class ℬ𝒮(α) where 0≤α≤ 3-2√(2). then |a_2|≤ 1 and |a_n|≤1/n-1∏_k=2^n-1(k/k-1) (n=3,4,…). Assume that f∈ℬ𝒮(α). Then from Definition <ref> we have p(z)≺ 1+ F_α(z)=1+z+α z^3+⋯ (z∈Δ), where zf'(z)=p(z)f(z). We note that F_α is convex function for 0≤α≤ 3-2√(2) (see <cit.>). If we define p(z)=1+∑_n=1^∞p_nz^n, then from Lemma <ref>, we have |p_n|≤ 1. Equating the coefficients of z^n on both sides of (<ref>), we find the following relation between the coefficients: na_n=p_n-1+a_2p_n-2+⋯+a_n-1p_1+a_n. Making use of (<ref>) and (<ref>), we get |a_n|≤1/n-1∑_k=1^n-1|a_k| |a_1|=1. Obvious that, from (<ref>) we have |a_2|≤ 1. We now need show that 1/n-1∑_k=1^n-1|a_k|≤1/n-1∏_k=2^n-1(1+1/k-1) (n=3,4,…). We use induction to prove (<ref>). If we take n=3 in the inequality (<ref>), we have |a_2|≤ 1, therefore the case n=3 is clear. A simple calculation gives us |a_m+1| ≤1/m∑_k=1^m|a_k|=1/m(∑_k=1^m-1|a_k|+|a_m|) ≤1/m∏_k=2^m-1(1+1/k-1)+1/m×1/m-1∏_k=2^m-1(1+1/k-1) =1/m∏_k=2^m(1+1/k-1), which implies that the inequality (<ref>) holds for n=m+1. From now (<ref>) and (<ref>), the desired estimate for |a_n| (n = 3,4,…) follows, as asserted in (<ref>). This completes the proof. The problem of finding sharp upper bounds for the coefficient functional |a_3-μ a_2^2| for different subclasses of the normalized analytic function class 𝒜 is known as the Fekete-Szegö problem. Recently, Ali et al. <cit.> considered the Fekete-Szegö functional associated with the kth root transform for several subclasses of univalent functions. We recall here that, for a univalent function f(z) of the form (<ref>), the kth root transform is defined by 𝔉(z)=[f(z^k)]^1/k=z+∑_n=1^∞b_kn+1z^kn+1 (z∈Δ). Following, we consider the problem of finding sharp upper bounds for the Fekete-Szegö coefficient functional associated with the kth root transform for functions in the class ℬ𝒮(α). In order to prove next result, we need the following lemma due to Keogh and Merkes <cit.>. Further we denote by 𝒫 the well-known class of analytic functions p(z) with p(0) = 1 and ℜ𝔢(p(z))>0, z∈Δ. Let the function g(z) given by g(z)=1+c_1z+c_2z^2+⋯, be in the class 𝒫. Then, for any complex number μ |c_2-μ c_1^2|≤ 2max{1,|2μ-1|}. The result is sharp. Let 0≤α<1, f∈ℬ𝒮(α) and 𝔉 is the kth root transform of f defined by (<ref>). Then, for any complex number μ, |b_2k+1-μ b_k+1^2|≤1/2kmax{1,|2(μ-1)/k+1|}. The result is sharp. Using (<ref>), we first put 1+F_α(z)=1+∑_n=1^∞ℬ_n z^n, where ℬ_1=1, ℬ_2=0, ℬ_3=α, and etc. Since f∈ℬ𝒮(α), from Definition <ref> and definition of subordination, there exists w∈𝔅 such that zf'(z)/f(z)=1+F_α(w(z)). We now define p(z)=1+w(z)/1-w(z)=1+p_1z+p_2z^2+⋯. Since w∈𝔅, it follows that p∈𝒫. From (<ref>) and (<ref>) we have: 1+F_α(w(z))=1+1/2ℬ_1p_1z+(1/4ℬ_2p_1^2+ 1/2ℬ_1(p_2-1/2p_1^2))z^2+⋯, where ℬ_1=1 and ℬ_2=0. Equating the coefficients of z and z^2 on both sides of (<ref>) and substituting ℬ_1=1 and ℬ_2=0, we get a_2=1/2p_1, and a_3=1/8p_1^2+1/4(p_2-1/2p_1^2). A computation shows that, for f given by (<ref>), 𝔉(z)=[f(z^1/k)]^1/k=z+1/ka_2z^k+1+(1/ka_3-1/2k-1/k^2a_2^2)z^2k+1+⋯. From equations (<ref>) and (<ref>), we have b_k+1=1/ka_2 and b_2k+1=1/ka_3-1/2k-1/k^2a_2^2. Substituting from (<ref>) and (<ref>) into (<ref>), we obtain b_k+1=1/2kp_1, and b_2k+1=1/4k(p_2-k-1/kp_1^2), so that b_2k+1-μ b_k+1^2=1/4k[p_2-1/2(2(μ-1)/k+ 2)p_1^2]. Letting μ'=1/2(2(μ-1)/k+2), the inequality (<ref>) now follows as an application of Lemma <ref>. It is easy to check that the result is sharp for the kth root transforms of the function f(z)=zexp(∫_0^zF_α(w(t))/tdt). Putting k=1 in Theorem <ref>, we have: (Fekete-Szegö inequality) Suppose that f∈ℬ𝒮(α) and 0≤α<1. Then, for any complex number μ, |a_3-μ a_2^2|≤1/2max{1,|2μ-1|}. The result is sharp. It is well known that every function f∈𝒮 has an inverse f^-1, defined by f^-1(f(z))= z, z∈Δ and f(f^-1(w))=w (|w|<r_0; r_0<1/4), where f^-1(w)=w-a_2w^2+(2a_2^2-a_3)w^3-(5a_2^3-5a_2a_3+a_4)w^4+⋯. Let the function f, given by (<ref>), be in the class ℬ𝒮(α) where 0≤α<1. Also let the function f^-1(w)=w+∑_n=2^∞b_nw^n be inverse of f. Then |b_2|≤ 1, and |b_3|≤3/2. Relation (<ref>) give us b_2=-a_2 and b_3=2a_2^2-a_3. Thus, we can get the estimate for |b_2| by |b_2|=|a_2|≤ 1. For estimate of |b_3|, it suffices in Corollary <ref>, we put μ=2. Hence the proof of Corollary <ref> is completed. 9 ALI Ali, R.M. Lee, S.K. Ravichandran V. and Supramanian, S. The Fekete-Szegö coefficient functional for transforms of analytic functions, Bull. Iranian Math. Soc. 35 (2009), 119–142. Booth Booth, J. A Treatise on Some New Geometrical Methods, Longmans, Green Reader and Dyer, London, Vol. I (1873) and Vol. II (1877). KES(Siberian) Kargar, R. Ebadian, A. and Sokół, J. On subordination of some analytic functions, Siberian Mathematical Journal, 57 (2016), 599–605. KES(Complex) Kargar, R. Ebadian, A. and Sokół, J. Radius problems for some subclasses of analytic functions, Complex Anal. Oper. Theory, DOI 10.1007/s11785-016-0584-x. KM Keogh, F.R. and Merkes, E.P. A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8–12. KuOwa Kuroki, K. and S. Owa, S. Notes on New Class for Certain Analytic Functions, RIMS Kokyuroku 1772, 2011, pp. 21–25. psok Piejko, K. and Sokół, J. Hadamard product of analytic functions and some special regions and curves, J. Ineq. Appl. 2013, 2013:420. ROB Robertson, M.S. Certain classes of starlike functions, Michigan Mathematical Journal 76 (1954), 755–758. Rog W. Rogosinski, W. On the coefficients of subordinate functions, Proc. London Math. Soc. 48 (1943), 48–82. AAS Savelov, A.A. Planar curves, Moscow (1960) (In Russian).

http://arxiv.org/abs/1701.07931v4

20170127033126

Adiabatic limits and Kazdan-Warner equations

math.DG

We study the limiting behaviour of solutions to abelian vortex equations when the volume of the underlying Riemann surface grows to infinity. We prove that the solutions converge smoothly away from finitely many points. The proof relies on a priori estimates for functions satisfying generalised Kazdan–Warner equations. We relate our results to the work of Hong, Jost, and Struwe on classical vortices, and that of Haydys and Walpuski on the Seiberg–Witten equations with multiple spinors. Adiabatic limits and Kazdan–Warner equations Aleksander Doan December 30, 2023 ============================================ § INTRODUCTION The vortex equations on Riemann surfaces originate from the Ginzburg–Landau model of superconductivity. They were first brought to the attention of mathematicians by Jaffe and Taubes <cit.>. Since then there has been a considerable body of work aimed at understanding these equations and their many generalisations. The purpose of this article is to study one such generalisation in the context of adiabatic limits and compactifications in two- and three-dimensional gauge theories. Let Σ be a closed Riemann surface and L →Σ a Hermitian line bundle. A (classical) vortex is a pair of a unitary connection A on L and a section φ of L satisfying {[ _Aφ = 0,; i Λ F_A = 1 - | φ |^2, ]. where Λ F_A is the Hodge dual of the curvature form. The space of vortices modulo gauge equivalence is simply the symmetric product Sym^d Σ with d = L. A point in Sym^d Σ corresponds to a degree d effective divisor D and there exists a unique, up to gauge equivalence, solution (A,φ) having D as the zero divisor of φ <cit.>. Scaling the metric on Σ by a factor ϵ^-1 results in the modified equations {[ _Aφ = 0,; ϵ^2 iΛ F_A = 1 - | φ |^2. ]. A question arises about the behaviour of solutions in the limit ϵ→ 0, as the volume of Σ grows to infinity. This is the idea of the adiabatic limit which has been used in many contexts in gauge theory and Riemannian geometry <cit.>. The question for the vortex equations was answered by Hong, Jost, and Struwe <cit.> whose result can be stated as follows. Fix a degree d effective divisor D = ∑_k m_k x_k, where m_k ∈_≥ 0 and x_k ∈Σ. If (A_i, φ_i, ϵ_i) is a sequence of solutions to (<ref>) having D as the zero divisor and satisfying ϵ_i → 0, then after passing to a subsequence and changing (A_i, φ_i) by gauge transformations, * i/2πΛ F_A_i converges as measures to the sum of Dirac deltas δ_D := ∑_k m_k δ_x_k, and * ∇_A_iα_i → 0, | α_i | → 1, and F_A_i→ 0 in C^0_ on Σ∖ D [By abuse of notation, we denote by the same symbol a divisor and the underlying set of points.]. In this paper we consider the following generalisation of (<ref>). Fix auxiliary unitary bundles E_1, …, E_N over Σ together with respective connections B_1, …, B_N and non-zero integer weights k_1, …, k_N. Let ϵ > 0 and τ∈. The generalised equations for a connection A on L and a section φ = (φ^1, …, φ^N) ∈Γ( ⊕_j=1^N E_j ⊗ L^⊗ k_j) are {[ _A ⊗ B_jφ^j = 0 for j = 1, …, N,; ϵ^2 i Λ F_A + ∑_j=1^N k_j | φ^j |^2 + τ = 0. ]. Equations (<ref>) fit into the more general setting of framed vortex equations discussed in <cit.>. As before, the moduli space of solutions has a holomorphic description <cit.>. Our main result concerns sequences of solutions to (<ref>) with ϵ→ 0. Suppose that either the weights k_i are of mixed signs or k_i > 0 for all i and τ < 0 as otherwise there are no solutions. Let (A_i, φ_i, ϵ_i) be a sequence of solutions to (<ref>) such that ϵ_i → 0 and the sequence of norms φ_i _L^2 is bounded. Then there is a finite set of points D ⊂Σ such that after passing to a subsequence and applying gauge transformations (A_i, φ_i ) converges in C^∞_ on Σ∖ D. The limit (A,φ) satisfies (<ref>) with ϵ = 0 on Σ∖ D. More generally, we can assume τ = τ_i to depend on i as long as τ_i converges. If the sequence λ_i := φ_i _L^2 is unbounded, we obtain the convergence of (A, λ_i^-1φ_i, λ_i^-1ϵ_i) with τ_i = λ_i^-1τ. Thus, Theorem <ref> describes also the limiting behaviour of solutions to (<ref>) with ϵ=1 and the L^2 norms diverging to infinity. In other words, we provide a description of the ends of the non-compact moduli space of solutions to (<ref>). This should be compared with recent results on sequences of solutions to the Hitchin equations <cit.>. When N=1, k_1 = 1, and τ = 1, we recover the classical vortex equations (<ref>). As a result, we reprove and strengthen the result of <cit.>. We should point out that our method of proof is different from that of <cit.> and will be outlined at the end of this introduction. Let (A_i, φ_i, ϵ_i) be a sequence of solutions to (<ref>) such that ϵ_i → 0. Then there is a degree d effective divisor D on Σ such that after passing to a subsequence and applying gauge transformations (A_i, φ_i) converges in C^∞_ on Σ∖ D and i/2πΛ F_A_i→δ_D as measures. The limit (A, φ) satisfies F_A = 0, |α| = 1 and ∇_A α = 0 on Σ∖ D. In the general setting of Theorem <ref>, the limiting connection A is no longer necessarily flat; we will see an example of this in point (4) of Theorem <ref> below. §.§ Seiberg–Witten theory The main application of Theorem <ref> concerns generalised Seiberg–Witten equations in dimension three. To set the stage, let Y be a closed Riemannian spin three-manifold. Let S be the spinor bundle and E, L vector bundles over Y with structure groups (n) and (1) respectively; we equip E with a connection B. The Seiberg–Witten equations with multiple spinors for a connection A on L → Y and Ψ∈Γ((E, S ⊗ L)) are {[ D_A ⊗ BΨ = 0,; F_A = ΨΨ^* - 1/2 | Ψ |^2. ]. Here D_A ⊗ B is the Dirac operator twisted by A and B and in the second equation we use the identification i Λ^2 T^*Y ≅ i 𝔰𝔲(S) given by the Clifford multiplication. An analogous set of equations on four-manifolds was introduced in <cit.>. The three-dimensional version was studied by Haydys and Walpuski in relation to enumerative theories for associative submanifolds and G_2–instantons on G_2–manifolds <cit.>. The principal result of <cit.> concerns the limiting behaviour of sequences of solutions (A_i, Ψ_i) such that Ψ_i _L^2→∞. Haydys and Walpuski showed that there is a closed nowhere dense subset Z ⊂ Y such that after passing to a subsequence and applying gauge transformations A_i → A weakly in W^1,2_ and Ψ_i / Ψ_i _L^2→Ψ weakly in W^2,2_ on Y ∖ Z, and the limiting configuration (A, Ψ) defined on Y ∖ Z satisfies {[ D_A⊗ BΨ = 0,; ΨΨ^* - 1/2 | Ψ |^2 = 0. ]. Moreover, Z is the zero locus of Ψ in the sense of <cit.> and, if E = 2, A is flat with holonomy contained in _2. If E > 2, then A induces a flat _2–connection on a rank two subbundle of E twisted by a line bundle; see <cit.>. A number of problems in this theory remain open despite their importance for the possible applications of generalised Seiberg–Witten equations to G_2–gauge theory: * Taubes has made significant progress in the study of the local structure of Z, proving in particular that Z has Hausdorff dimension at most two <cit.>; yet the question whether Z is a smooth curve is still unanswered. * For the applications in enumerative theories it is crucial to improve the convergence statement for (A_i,Ψ_i/Ψ_i_L^2), as exemplified by <cit.> where, as part of the main proof, C^∞ convergence is established under the assumption that Z is empty. * There are two ways of associating weights to the connected components of Z: one based on Taubes' frequency function <cit.> and one developed by Haydys using topological methods <cit.>. It is currently unknown whether these constructions are related. * Haydys conjectured that, equipped with appropriate weights, Z has the structure of an integral rectifiable current and that i/2π F_A_i converges to Z as currents <cit.>. Using Theorem <ref> and the methods involved in its proof, we refine the compactness theorem of <cit.> and solve all of the above problems in the case Y = S^1 ×Σ. Suppose that Y = S^1 ×Σ equipped with a product metric, B is pulled back from Σ, and (A_i, Ψ_i), (A,Ψ), and Z are as above, see also <cit.>. Then * The singular set Z is of the form S^1 × D for a degree 2d divisor D = ∑_k m_k x_k. * After passing to a subsequence and applying gauge transformations A_i → A and Ψ_i / Ψ_i _L^2→Ψ in C^∞_ on Y ∖ Z. * | Ψ |^4 extends to a smooth function on Y whose zero set is Z and for all k | Ψ (x) | = O( dist(x,S^1 ×{ x_k })^|m_k|/2). In particular, the weight of the connected component S^1 ×{ x_k } of Z in the sense of <cit.> is smaller than or equal to its weight in the sense of <cit.>. * If E = 2, then i/2π F_A_i→1/2 Z as currents. If E > 2, then there is a rank two subbundle F ⊂EY ∖ Z such that Ψ∈Γ(Y ∖ Z, (F, S ⊗ L)) and the previous statement holds if we replace A and A_i by the tensor product connections on L ⊗ ( F)^1/2. Here F and F are equipped with the unitary connections induced from B. The relationship between Seiberg–Witten monopoles with multiple spinors and generalised vortices (<ref>) is the subject of the author's paper <cit.> where further consequences of the results presented here are explored. In particular, Theorem <ref> is used to construct a compactification of the moduli space of Seiberg–Witten monopoles with multiple spinors on S^1 ×Σ and to compare it with a corresponding algebro-geometric moduli space <cit.>. This, in turn, leads to the first known examples of the non-compactness phenomenon predicted by the Haydys and Walpuski's theorem <cit.>. §.§ Symplectic vortex equations Coming back to dimension two, equations (<ref>) fit into the general framework of symplectic vortex equations or gauged σ-models. One associates vortex-type equations on Σ to any pair (G,M), where G is a compact Lie group acting in a Hamiltonian way on a symplectic manifold M. In the spirit of Gromov–Witten theory one wishes to extract numerical invariants of (G,M) from the moduli space of solutions. The parameter ϵ can be incorporated into the equations in the same way as before. In the adiabatic limit ϵ→ 0, we obtain the equation for pseudoholomorphic curves in the symplectic quotient M G. Thus, we expect a relation between the invariants of (G,M) and the Gromov–Witten invariants of M G. This programme has been proposed and successfully carried out by Cieliebak, Gaio, Mundet i Riera, Salamon <cit.>, <cit.>, and others. In order to establish Gromov compactness for symplectic vortices, more constraints are imposed on the pair (G,M), a crucial condition being the properness of the Hamiltonian moment map. Already for linear actions this is a rather restrictive assumption. The simplest example is the one discussed here with the corresponding equations (<ref>). In this case M = ^n and G = (1) acts diagonally with weights (k_1, …, k_N). The moment map is not proper unless all the weights have the same sign. Theorem <ref> shows that if the properness condition is dropped we have to take under account formation of singularities in considerations regarding compactness and adiabatic limits. §.§ Outline of the proof One consequence of the improperness of the moment map is that, unlike classical vortices, solutions to (<ref>) do not obey an a priori L^∞ bound. This causes a major difficulty in establishing the convergence. While the proof in <cit.> is based on local ϵ–regularity estimates, we employ here a complex-geometric description of the moduli space of solutions to (<ref>). To be more specific, we use the action of 𝒢^c = C^∞(Σ, ^×), the group of complex automorphisms of L, on the space of pairs (A, φ). The moduli space of solutions to (<ref>) is homeomorphic to the quotient of the set solutions to the Cauchy–Riemann equation by 𝒢^c; this is a Hitchin–Kobayashi type correspondence proved in <cit.>. Using elliptic estimates for Dolbeault operators, we show that this quotient is compact modulo the rescaling action of ^×. Thus, there are complex gauge transformations g_i = e^f_i u_i for f_i ∈ C^∞(Σ, ) and u_i ∈ C^∞(Σ, (1)) such that after rescaling and applying g_i the original sequence (A_i, φ_i) converges. In order to obtain the convergence in the real rather than the complex moduli space, we need to control the functions f_i. The original equations (<ref>) translate in this setting to a partial differential equation for f_i of the form ϵ^2 Δ f + ∑_j=1^n A_j e^α_j f - ∑_j=1^m B_j e^-β_j f + w = 0 for some functions A_j ≥ 0, B_j ≥ 0, w, and positive constants α_j, β_j. This is a generalisation of the Kazdan–Warner equation <cit.>, <cit.>. In section <ref> we establish a priori bounds for solutions of this equation. Importantly, they are independent of ϵ∈ (0, 1] and uniform on compact subsets of Σ∖ D, where D is the set of common zeroes of A_j and B_j. Consequently, the Arzelà–Ascoli theorem guarantees the existence of a subsequence of f_i converging smoothly on compact subsets of Σ∖ D. To the best of our knowledge, the strategy of passing to the holomorphic moduli space by means of a Hitchin–Kobayashi correspondence, obtaining good control there using –methods, and deducing from it compactness for the original sequence in the real moduli space, has not been used before. We believe that this idea—and some of the related analytical results such as Lemma <ref>—might be useful in studying other gauge-theoretic equations on Kähler manifolds. §.§ Acknowledgements The work presented in this article is part of my doctoral thesis at Stony Brook University. I am grateful to my advisor Simon Donaldson for his guidance and support. Thanks to Andriy Haydys and Thomas Walpuski for their encouragement and many helpful discussions, and to Gonçalo Oliveira, Oscar Garcia–Prada, Song Sun, Alex Waldron, and the anonymous referee for valuable comments on the previous versions of this paper. I am supported by the https://sites.duke.edu/scshgap/Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics. § BACKGROUND AND NOTATION The set of solutions to (<ref>) is invariant under the action of the gauge group 𝒢 of unitary automorphisms of L, identified with C^∞(Σ,(1)). The action of a map u Σ→(1) on (A, φ^1, …, φ^N) is given by u (A, φ^1, …, φ^N) = (A - u^-1 du, u^k_1φ^1, …, u^k_Nφ^N). The Dolbeault equation in (<ref>), as well as the algebraic condition φ^1 φ^2 = 0 in the equation (<ref>) below, are also invariant under the action of the complex gauge group 𝒢^c. It consists of complex automorphisms of L and is identified with C^∞(Σ, ^×), where ^× = ∖{ 0 }. The action of g Σ→^× is given by g(A, φ^1, …, φ^N) = (A + g^-1∂g - g^-1 g, g^k_1φ^1, … , g^k_Nφ^N ). In terms of the associated Dolbeault operators, for s ∈Γ(Σ, E_j ⊗ L^⊗ k_j) we have _B_j, g(A) s = g^k_j_B_j A( g^-k_j s ). The action of 𝒢^c does not preserve the last equation in (<ref>) involving the curvature. Indeed, if we write g = e^f u for functions f Σ→ and u Σ→(1), then F_g(A) = F_A + 2 ∂ f, or equivalently i Λ F_g(A) = i Λ F_A + Δ f, where Δ is the Hodge Laplacian acting on functions. We will need also the following lemma whose elementary proof we omit. Let L →Σ be a Hermitian line bundle, D ⊂Σ a finite set of points, A a unitary connection on LΣ∖ D and α∈Γ(Σ∖ D, L). If _A α = 0, and | α | = 1 everywhere on Σ∖ D, then ∇_A α = 0 and F_A = 0. Moreover, let B be a small ball around a point p ∈ D such that in a local unitary trivialisation A = d + a for a one-form a ∈Ω^1(B ∖{ p }, i), and α is identified with a smooth function α B ∖{ p }→ S^1. Then i/2π∫_∂ B a = deg( α∂ B). We end this section by restating Theorem <ref> in terms of generalised vortex equations. In <cit.> we show that all irreducible solutions to the Seiberg–Witten equations with multiple spinors are pulled back from solutions to (<ref>) of the following form. Using the notation of the introduction, set N=2, k_1 = 1, k_2 = -1. Fix a spin structure on Σ thought of as a square root K^1/2 of the canonical bundle, and let E be an (n)-bundle; then set E_1 = E ⊗ K^1/2 and E_2 = E^* ⊗ K^1/2. The equations we consider next involve a connection A on L and sections φ^1 ∈Γ(E ⊗ K^1/2⊗ L) and φ^2 ∈Γ(E^* ⊗ K^1/2⊗ L^*): {[ _A⊗ Bφ^1 = 0,; _A ⊗ Bφ^2 = 0,; φ^1 φ^2 = 0,; ϵ^2 i Λ F_A + |φ^1|^2 - |φ^2|^2 = 0, ]. The third equation is an additional algebraic condition for the section φ^1 φ^2 ∈Γ(K) defined as the image of (φ^1,φ^2) under the pairing Γ(E ⊗ K^1/2⊗ L) ×Γ(E^* ⊗ K^1/2⊗ L^*) ⟶Γ(K). We will deduce Theorem <ref> from the following result. Let (A_i, φ_i, ϵ_i) be a sequence of solutions to (<ref>) with φ_i _L^2 = 1 and ϵ_i → 0. Then * There exist a degree 2d divisor D = ∑_k m_k x_k and a configuration (A,φ) defined on Σ∖ D and satisfying (<ref>) with ϵ = 0, * (A_i, φ_i) → (A,φ) in C^∞_ on Σ∖ D, * The function | φ |^4 extends to a smooth function on all of Σ whose zero set consists of the points in D and for all k | φ (x) | = O( dist(x,x_k)^|m_k|/2). * If E = 2, then the limiting connection A is flat, has holonomy contained in _2, and i/2πΛ F_A_i→1/2δ_D as measures. If E > 2, then there exists a rank two subbundle F ⊂EΣ∖ D such that φ^1 ∈Γ(Σ∖ D, F ⊗ L ⊗ K^1/2), φ^2 ∈Γ(Σ∖ D , F^* ⊗ L^* ⊗ K^1/2), and the previous statement holds if we replace A and A_i by the tensor product connections on L ⊗ ( F)^1/2. Here, F and F are equipped with the unitary connections induced from B. In contrast to Theorem <ref> here the divisor D need not be effective. In a way, replacing classical vortices by solutions to (<ref>) is analogous to replacing holomorphic sections by meromorphic sections. This idea will play a role in the proof of Theorem <ref>. § A PRIORI ESTIMATES The main analytical input are a priori estimates for solutions to (<ref>). Let X be a compact Riemannian manifold with (possibly empty) boundary ∂ X, and Ω⊂ X an open subset such that Ω⊂ X ∖∂ X. Let ϵ_0, α_1, … , α_n, β_1, … , β_m be positive numbers, and let A_1, …, A_n, B_1, …, B_m, and w be smooth functions on X such that A_j ≥ 0 and B_j ≥ 0 for all j and A_1 + … + A_n > 0, B_1 + … + B_m > 0. Then there exist constants M_0, M_1, M_2, …, depending only on the data listed above, such that for any ϵ∈ [0, ϵ_0] and f ∈ C^∞(X) satisfying the equation ϵΔ f + ∑_j=1^n A_j e^α_j f - ∑_j=1^m B_j e^-β_j f + w = 0, the following inequalities hold: f _C^k(Ω)≤ M_k for k=0, 1, 2, … M_k depends on A_j, B_j, and w and their derivatives. Later we will consider sequences ϵ_i → 0 and f_i, A_1^i, …, A_n^i, B_1^i, …, B_m^i, w_i satisfying for all i ϵ_i Δ f_i + ∑_j=1^n A_j^i e^α_j f_i - ∑_j=1^m B_j^i e^-β_j f_i + w_i = 0. It will be clear from the proof that in this case the C^k estimate still holds for large i (depending on k) provided that A_j^i, B_j^i, and w_i converge smoothly to A_j, B_j, and w respectively, satisfying A_1 + … + A_n > 0, B_1 + … + B_m > 0. The proof of Proposition <ref> is preceded by three lemmas. Let X and Ω be as in Proposition <ref>, and let V_0 ⊂ X be an open subset such that Ω⊂ V_0. Then there exist an open subset V and ϕ∈ C^∞(X) such that: * Ω⊂ V ⊂ V_0. * 0 < ϕ≤ 1 on V. * ϕ = 1 on Ω. * ϕ = 0 on X ∖ V. * There is a constant K such that for any α∈ [0,2), sup_V'| ∇ϕ |^2/ϕ^α≤K/(2-α)^4. Let V' ⊂ X be any open subset such that Ω⊂ V' ⊂ V and V' has smooth boundary. We can construct such a subset for example by taking any smooth function h X → with h < 0 on Ω and h> 1 on X ∖ V, and setting V' = h^-1( (- ∞, c)) where c ∈ (0,1) is a regular value of h. Let N = h^-1(c) be the boundary of V'. Assume for simplicity that X is orientable. By the tubular neighbourhood theorem, there is an embedding (- ϵ, ϵ) × N ↪ X such that { 0 }× N is mapped diffeomorphically onto N ⊂ X and the image of (0, ϵ) × N is contained in V'. We may also assume that the image of this embedding is disjoint from Ω. Using a partition of unity (and passing to a slightly smaller ϵ), we construct a function ϕ with properties (1)–(4), which for (t,x) ∈ (-ϵ, ϵ) × N agrees with ϕ(t, x) = {[ M exp( - 1/t) for t ∈ (0, ϵ); 0 for t ∈ (- ϵ, 0); ]. for some constant M required for the normalisation ϕ_C^0(X) = 1. Now let α < 2. Away from N we have ϕ > 0 and | ∇ϕ |^2 / ϕ^α is bounded. In a neighbourhood of N, | ∇ϕ |^2/ϕ^α≤C/t^4exp( -(2-α)/t) for some constant C depending on the Riemannian metric on X and the embedding of the tubular neighbourhood. Define g(t) = 1/t^4exp( -(2-α)/t). Then g is smooth and bounded on [0, ∞) and its global maximum is g( 2-α/2) = 4e^-2/(2-α)^4, which shows that | ∇ϕ |^2 / ϕ^α≤ K (2-α)^-4 as desired. Let X and Ω be as in Proposition <ref>. Fix positive numbers ϵ_0, p, and γ > 1, and consider functions A ∈ C^∞(X) and Q ∈ C^∞(X × (0, ∞)) such that for all (x,y) ∈ X × (0, ∞), A(x) ≥η > 0 and | Q(x,y) | ≤∑_j=1^k a_j(x) y^γ_j, where a_j ∈ C^∞(X) and γ_j < γ. Under these assumptions there exists a constant M such that for any ϵ∈ [0, ϵ_0] and u ∈ C^∞(X) satisfying u ≥ 0 and ϵΔ u + A u^γ + Q(x,u) ≤ 0, the following inequality holds: u _L^p(Ω)≤ M. Moreover, M depends only on Ω, X, ϵ_0, p, γ, γ_j, η, and the norms a_j _L^q(X) for a certain q < ∞ depending on p. We adopt here the convention that C always denotes a constant depending only on the fixed data and not on u or ϵ. Its value might change from line to line. It is enough to prove the statement for some ϵ_0 > 0, which we will later assume to be sufficiently small. Indeed, the corresponding statement for any other ϵ_0' > ϵ_0 can be reduced to the one for ϵ_0 by multiplying both sides of (<ref>) by ϵ_0 / ϵ_0' at the cost of appropriately scaling A and Q. Furthermore, it suffices to prove the statement for p=1 since for p >1 we have Δ( u^p ) = - p(p-1) u^p-2 | ∇ u |^2 + p u^p-1Δ u ≤ -pA u^p - 1 + γ - pu^p-1 Q(x,u) ≤ - A' (u^p)^γ' - Q'(x,u^p), where A' = pA, γ' = 1 + γ-1/p, Q'(x,y) = py^p-1/pQ(x, y^1/p). We easily check that the new data (A', γ', Q') satisfies the hypotheses of the lemma. The only non-trivial condition is the estimate for Q' which follows from | Q'(x,y) | ≤ p ∑_i^k a_j(x) y^1 + γ_j - 1/p = p ∑_i^k a_j(x) y^γ_j'. Note that γ_j' < γ'. Therefore, the statement for p > 1 reduces to that for p=1 after replacing (u, A, γ, Q) by (u^p, A', γ', Q'). In fact, we will bound u _L^1+γ(Ω) . Let V_0 ⊂ X be an open subset containing Ω such that the volume of V_0 ∖Ω is sufficiently small. We will specify later what we mean by that, but for the moment let us stress that the choice of V_0 will depend only on the fixed data and not the function u. Once V_0 is fixed, choose a subset V ⊂ V_0 and a bump function ϕ∈ C^∞(X) as in Lemma <ref>. Note that vol(V ∖Ω) ≤vol(V_0 ∖Ω). Multiply inequality (<ref>) by u ϕ^2 and integrate it over X: ∫_X ϵ (Δ u) u ϕ^2 + A u^1+γϕ^2 + Q u ϕ^2 ≤ 0 . Since ϕ has compact support, integration by parts yields ∫_X ( Δ u) u ϕ^2 = ∫_X ⟨∇ u, ∇ (u ϕ^2) ⟩ = ∫_X ⟨∇ u , 2uϕ∇ϕ + ϕ^2 ∇ u ⟩ = ∫_X 2 u ϕ⟨∇ u, ∇ϕ⟩ + ϕ^2 | ∇ u |^2 = ∫_X | u ∇ϕ + ϕ∇ u |^2 - u^2 | ∇ϕ |^2 ≥ - ∫_X u^2 | ∇ϕ |^2. Together with (<ref>), this implies the inequality ∫_X - ϵ u^2 | ∇ϕ |^2 + A u^1+γϕ^2 + Q u ϕ^2 ≤ 0. Recall that ϕ is supported in V and ϕ = 1 on Ω. Let P = V ∖Ω so that V = Ω∪ P. Splitting the integral on the left-hand side according to this decomposition and rearranging the inequality, we obtain ∫_Ω A u^1+γ + I ≤∫_Ω | Q | u, where we have collected all integrals over P into a single term, I = I_0 + I_1 + I_2, I_0 = ∫_P A u^1 + γϕ^2, I_1 = ∫_P Q u ϕ^2, I_2 = - ϵ∫_P u^2 | ∇ϕ |^2. The next goal is to estimate I. We will show that for a suitable choice of V and ϵ_0, depending only on the initial data and not on u, we may assume that I ≥ 0, provided that ϵ≤ϵ_0. Strictly speaking, this will not always be true, but in the case when our estimate fails, we will obtain an upper bound for | I | so that we can move it to the right-hand side of (<ref>). Before proving this, let us show that the inequality I ≥ 0 gives us a bound for u _L^1+γ(Ω). If I ≥ 0, then by (<ref>) and Hölder's inequality, η∫_Ω u^1+γ ≤∫_Ω A u^1+γ≤∫_Ω | Q | u ≤∫_Ω∑_j=1^k a_j u^1+γ_j ≤∑_j=1^k a_j _L^p_j(X)( ∫_Ω u^1+γ)^1/q_j, where the Hölder exponents p_j and q_j are given by q_j = 1+γ/1+γ_j, 1/p_j + 1/q_j = 1. Note that q_j > 1 for each i, because γ_j < γ. An equivalent way of writing (<ref>) is u _L^1+γ(Ω)^1+γ≤η^-1∑_j=1^k a_j _L^p_j(X) u _L^1+γ(Ω)^1+γ/q_j, which, in view of q_j > 1, results in an upper bound for u _L^1+γ(Ω). The dependance of the bound on the initial data is clear. In order to finish the proof, it remains to estimate the integral I = I_0 + I_1 + I_2. We will deal separately with each of the three terms. The first one contributes positively to I and is bounded below by I_0 ≥η∫_P u^1 + γϕ^2. The terms that can contribute negatively are I_1 and I_2. We estimate the former using our assumption on Q and Hölder's inequality: | I_1| ≤∑_j=1^k ∫_P | a_j | u^1+γ_jϕ^2 ≤∑_j=1^k ( ∫_P |a_j|^p_j)^1/p_j( ∫_P u^1+γϕ^2q_j)^1/q_j ≤∑_j=1^k vol(P)^1/2p_j a_j _L^2p_j(X)( ∫_P u^1+γϕ^2q_j)^1/q_j, where the Hölder exponents p_j and q_j are as before. Let S = ∫_P u^1+γϕ^2q_j. If S ≤ 1, then |I_1| is bounded by a constant independent of u, say C, and we can move it on the right-hand side of (<ref>). Next we replace I by the sum of the remaining two terms I' = I_0 + I_2 and if we can show that I' ≥ 0, then repeating the previous discussion we arrive at a bound for u _L^1+γ(Ω) with an extra term involving C. Thus, let us assume that S ≥ 1. In this case, we have S^1/q_j≤ S and | I_1| ≤∑_j=1^k vol(P)^1/2p_j a_j _L^2p_j(X)∫_P u^1+γϕ^2q_j ≤∑_j=1^k vol(P)^1/2p_j a_j _L^2p_j(X)∫_P u^1+γϕ^2, where we have also used that ϕ≤ 1 and so ϕ^2q_j≤ϕ^2. Comparing the right-hand side of the inequality with the previously obtained upper bound (<ref>) for I_0 we see that if P has sufficiently small volume (which can be guaranteed by the choice of the initial open set V_0), then |I_1| ≤I_0/2. Furthermore, how small P has to be depends only on η and a_j _L^p_max(X), where p_max = max{ p_1, …, p_k }. Note that at this point the sets V and P are chosen and will not be changed. The second potentially negative term I_2 is dealt with in a similar manner. For every α∈, Hölder's inequality implies that |I_2| = ϵ∫_P u^2 | ∇ϕ|^2 = ϵ∫_P | ∇ϕ |^2/ϕ^α u^2 ϕ^α ≤ϵ | ∇ϕ |^2 /ϕ^α_L^q(P)( ∫_P u^1+γϕ^α (1+γ)/2)^2/1+γ, where q is given by 1/q + 2/(1+γ) = 1. Observe that 1+ γ > 2, so we can choose α so that 4/1+γ < α < 2. Then, | ∇ϕ |^2 / ϕ^α is bounded on P, and the first factor on the right-hand side of (<ref>) is finite. As regards the integral in the second factor, assuming as before that it is greater than or equal to one (as otherwise we can rearrange and get a bounded factor on the right-hand side of (<ref>)), we arrive at | I_2| ≤ C ϵ∫_P u^1+γϕ^α (1+γ)/2≤ C ϵ∫_P u^1+γϕ^2, where we have used that ϕ <1 and α(1+γ)/2 > 2. Note that here the constant C depends on the choice of P and can be potentially large. However, we still have the freedom to choose ϵ small enough—as remarked at the beginning of the proof, it suffices to establish an estimate for ϵ sufficiently small. Thus, comparing the right-hand side of the above inequality with the lower bound (<ref>) for I_0, we conclude that if ϵ_0 is sufficiently small, then for all ϵ≤ϵ_0, | I_2| ≤I_0/2. Together with the estimate for |I_1|, this implies that I = I_0 + I_1 + I_2 is non-negative (or else we can rearrange (<ref>)), which finishes the proof of the lemma. Our proof does not work in the case γ = 1. What fails is the last estimate for I_2, because we cannot set α = 2. Indeed, there is no cut-off function ϕ such that | ∇ϕ | / ϕ is bounded. However, we can still prove a slightly weaker statement. If γ = 1, then the statement of Lemma <ref> still holds provided that u satisfies an estimate u _L^2p(X)≤ K ϵ^-1 for some constant K. Apart from the rest of the data, the final bound for u _L^p(Ω) depends also on K. Suppose for simplicity that p=1, so that u _L^4(X)≤ Kϵ^-1. Following the proof of Lemma <ref>, we can obtain a bound for u _L^2(Ω) The only modification that we have to make is the estimate (<ref>) which now should be | I_2 | = ϵ∫_P u^2 | ∇ϕ |^2 = ϵ∫_P (u^1/2) ( u^3/2ϕ^α) | ∇ϕ |^2/ϕ^α ≤ϵ u^1/2_L^8(X) | ∇ϕ |^2/ϕ^α_L^8(P)( ∫_p u^2 ϕ^4α/3)^3/4 ≤ϵ^1/2 K^1/2 | ∇ϕ |^2/ϕ^α_L^8(P)( ∫_P u^2 ϕ^4α/3)^3/4, where we have used Hölder's inequality with weights (8,8,4/3). Now for α satisfying 2 > α > 3/2, the function | ∇ϕ |^2 / ϕ^α is bounded 4α / 3 > 2, so that (again, assuming that the integral on the right-hand side is greater than one) we obtain | I_2 | ≤ C ϵ^1/2∫_P u^2 ϕ^2. Recall that in the case γ = 1, the positive integral I_0 is bounded below by I_0 ≥η∫_P u^2 ϕ^2, so that for ϵ small enough we have I_2 ≤ I_0 / 2. This leads to a bound for u _L^2(Ω) as before. In the same way we obtain a bound for u _L^p (Ω) u _L^2p(Ω)≤ Kϵ^-1. We will establish bounds of the form f _W^k,p(Ω)≤ M_k,p for all k and p by induction over k. Let us start with k=0. Let η > 0 be such that A_1 + ⋯ + A_n ≥ n η, and B_1 + ⋯ + B_m ≥ m η, and set Ω_j = Ω∩{ A_j ≥η}. The subsets { A_j ≥η}_j=1,…, n cover X and therefore Ω_1, …, Ω_n cover Ω. Let u = e^f. For any given j we have ϵΔ u = - ϵ e^f | ∇ f|^2 + ϵ e^f Δ f ≤ u ( - ∑_j=1^n A_j u^α_j + ∑_j=1^n B_j u^- β_j - w ) ≤ -A_j u^1 + α_j + ∑_j=1^n B_j u^1-β_j - w u, or equivalently, ϵΔ u + A_j u^1+ α_j + Q(x,u) ≤ 0, where Q(x,u) = - ∑_j B_j u^1-β_j + wu. It follows from Lemma <ref> that u _L^p(Ω_j) is bounded by a constant depending only on the fixed data. Since Ω_1, …, Ω_n cover Ω, we obtain a bound for u _L^p(Ω). Similarly, considering the subsets Ω∩{ B_j ≥η} and the function e^-f we find bounds for e^-f_L^p(Ω). Combining them with the inequality | f| ≤ e^f + e^-f, we obtain bounds for f _L^p(Ω). Suppose that W^k-1,p bounds have been established for some k ≥ 1 and all p. We may assume that they hold on a slightly larger domain containing Ω, which we assume to be all of X to keep the notation simple. First consider the case when k = 2l is even. Consider the function v = Δ^l f = Δ⋯Δ_l times f. Applying Δ^l to both sides of (<ref>) and using the formula Δ( gh ) = g Δ h - 2 ⟨∇ g, ∇ h ⟩ + hΔ g, we inductively show that v satisfies a differential equation of the form ϵΔ v + A v + P(e^α_j f, e^- β_j f, ∇ f, … , ∇^2l-1 f ) = 0, where A = ∑_j=1^n α_j A_j e^α_j f + ∑_i=1^m β_j B_j e^- β_j f, and P is a polynomial function of the functions e^α_j f, e^-β_j f, and the first 2l-1 covariant derivatives of f. Its coefficients depend only on A_j, B_j, w, and their derivatives. In particular, P is a finite sum P = ∑_γ P_γ say, where each term P_γ satisfies an inequality of the form | P_γ | ≤ C e^a f e^- b f | ∇ f |^c_1… | ∇^2l-1 f |^c_2l-1 with some exponents a,b, c_1, …, c_2l-1 and a coefficient C depending only on A_j, B_j, w, and their derivatives. Therefore, by the induction hypothesis and Hölder's inequality, we can bound the L^p norm of P for any p, by a constant depending only on the fixed data and not on f. At every point A is bounded below by either ∑_j α_j A_j or ∑_j β_j B_j, depending on the sign of f. In any case, there is a positive constant η̃, depending only on the fixed data, such that A ≥η̃. Note that v is raised to the first power in (<ref>). Thus, we are in place to apply Lemma <ref> to obtain a bound for v _L^p(Ω). In order to do so, we need to make sure that v obeys an estimate of the form v _L^2p(Ω)≤ K_p ϵ^-1 for some constant K_p. Such an estimate follows from the induction hypothesis and the fact that v' = Δ^l-1 f satisfies an equation analogous to (<ref>): ϵ v = ϵΔ v' = - Av' + P' (e^α_j f, e^-β_j f, ∇ f, …, ∇^2l-3 f ), where P' is a polynomial function as before. Since the right-hand side depends only on the derivatives of f up to the order 2l-2, we obtain an estimate for ϵ v as required. Thus, Lemma <ref> yields a bound for v _L^p(Ω). Of course, we can as well assume that it holds on a slightly larger domain containing Ω. Then, in view of v = Δ^l f, the elliptic estimate for the Laplacian implies a bound for f _W^2l, p(Ω). This finishes the proof of the induction step in the case k = 2l. The odd case k = 2l+1 is similar. Assume that the assertion is true for k-1 = 2l. Let v = Δ^l f as before and ψ = | v |^2. By the Bochner formula, 1/2Δψ = - | ∇^2 ψ |^2 - Ric( ∇ψ, ∇ψ ) + ⟨∇ψ, ∇ ( Δψ ) ⟩ ≤Ric_C^0(X) | ∇ψ |^2 + ⟨∇ψ, ∇ ( Δψ ) ⟩, where Ric is the Ricci curvature of X. After taking the gradient of (<ref>) and plugging it to the inequality above, we arrive at ϵ/2Δψ + (A - ϵRic_C^0(X)) ψ + Q(e^α_j f, e^-β_j f, ∇ f, …, ∇^2l f ) ψ^1/2≤ 0 , where Q is a polynomial function of e^α_j, e^- β_j f, and the first 2l derivatives of f. Provided that ϵ is sufficiently small, the function A - ϵRic is bounded below by a positive constant and we can apply Lemma <ref> as before to obtain a bound for ψ_L^p(Ω). Again, by the elliptic estimate for the Laplacian, this yields W^k,p bounds for f. The statement for general ϵ≤ϵ_0 follows from a scaling argument as described in the proof of Lemma <ref>. § PROOFS OF THE THEOREMS We prove the theorems in the order of increasing generality. Since ϵ_i → 0, we may assume that none of the sections φ_i is identically zero. Let 𝒜 be the space of unitary connections on L and 𝒢^c be the complex gauge group of L, that is the space of smooth maps from Σ to ^×. [Convergence modulo 𝒢^c] We claim that there are sequences of complex gauge transformations g_i ∈𝒢^c such that, after passing to a subsequence, g_i(A_i, φ_i) converges in C^∞(Σ) to a pair (A', φ'). The limiting section φ' is not identically zero and satisfies _A'φ' = 0. Let us prove this claim. The quotient 𝒜 / 𝒢^c with the C^∞ topology is homeomorphic to the Jacobian torus H^1(Σ, ) / H^1(Σ, ). In particular, it is compact and there is a sequence of g_i ∈𝒢^c such that, after passing to a subsequence, A_i' = g_i A_i converges in C^∞ to a connection A', say. After replacing g_i by μ_i g_i, where μ_i = g_i φ_i _L^2^-1 , we may assume that g_i φ_i _L^2 = 1 for all i. Note that the constant gauge transformations μ_i act trivially on the space of connections so that we still have A_i' → A'. The final remark about our choice of g_i is that we will assume them to be purely "imaginary" gauge transformations. Any complex gauge transformation is of the form g = u e^f for a (1) gauge transformation u and real function f Σ→. By incorporating the (1) part into the original sequence (A_i, φ_i, β_i) we may assume that g_i = e^f_i/2 for a smooth function f_i Σ→. Set φ_i' = g_i φ_i. The action of 𝒢^c preserves the Cauchy-Riemann equation: _A_i'φ_i' = 0. As a consequence, _A'φ_i' _L^2 = (_A' - _A_i') φ_i' _L^2≤ A' - A_i' _L^∞φ_i' _L^2 Since A_i → A', the sequence of norms _A'φ_i' _L^2 is bounded by a number independent of i. From the elliptic estimate for _A' we conclude that the sequence φ_i' is bounded in W^1,2. Bootstrapping gives us C^k for any k and we can choose a subsequence (denoted for simplicity by the same symbols) that converge in C^∞ to a section φ', say, satisfying _A'φ' = 0 and φ' _L^2 = 1, which finishes the proof of the claim. [C^0 estimates] Let D be the set of zeroes of φ'. Counted with multiplicities, there are exactly d = (L) of them. The next step is to show that the sequence f_i is uniformly bounded on compact subsets of Σ∖ D. First, we compute ϵ_i^2 ( 2 i Λ F_A_i') = ϵ_i^2 ( 2i Λ F_A_i + Δ f_i ) = 1 - | φ_i |^2 + ϵ_i^2 Δ f_i = 1 - e^-f_i | φ_i' |^2 + ϵ_i^2 Δ f_i, so after rearranging, we obtain the following partial differential equation for f_i: ϵ_i^2 Δ f_i = e^-f_i | φ_i' |^2 - 1 + ϵ_i^2 ( 2 i Λ F_A_i') = q_i e^-f_i - w_i, where q_i = | φ_i' |^2 and w_i = 1 - ϵ_i^2 ( 2 i Λ F_A_i'). Now set u_i = e^f_i. Then ϵ_i^2 Δ u_i = ϵ_i^2 ( - e^f_i | ∇ f_i |^2 + e^f_iΔ f_i ) ≤ q_i - w_i u_i. Since w_i → 1 uniformly, for i large enough we have w_i ≥ 1/2. The functions q_i are bounded because φ_i' converges. Thus, the maximum principle yields an upper bound for u_i, and consequently for f_i. On the other hand, we easily compute that ϵ_i^2 Δ |φ_i|^2 + 2 ϵ_i^2 | ∂_A_iφ_i |^2 = |φ_i|^2 ( 1 -|φ_i|^2 ), which again by the maximum principle shows that |φ_i|^2 ≤ 1 for all i. Since |φ_i|^2 = e^-f_i | φ_i '|^2 and | φ_i' |^2 converges uniformly to | φ' |^2, it follows that f_i is bounded below uniformly on compact subsets of Σ∖ D. [Convergence outside D] Once the C^0 estimate is established, it follows from equation (<ref>), Proposition <ref>, and Remark <ref> that the sequence f_i is bounded uniformly with all derivatives on compact subsets of Σ∖ D. Thus, we can choose a subsequence of f_i which converges uniformly with all derivatives on compact subsets of Σ∖ D to a smooth function f Σ∖ D →. Let g = e^f/2 be the corresponding complex gauge transformation. Set (A, φ) = ( g^-1 A', g^-1φ'). The pair is well-defined on Σ∖ D and (A_i, φ_i) → (A, φ) in C^∞_(Σ∖ D. Indeed, we have φ_i - φ = g_i^-1φ_i' - g^-1φ' = g_i^-1φ_i' - g_i^-1φ' + g_i^-1φ' - g^-1φ' = g_i^-1( φ_i' - φ' ) + ( g_i^-1 - g^-1) φ', , so the convergence of φ_i' and g_i' guarantees that for any compact subset K ⊂Σ∖ D there are constants M_l,K for l = 0,1, … such that φ_i - φ_C^l(K)≤ M_l,K( φ_i' - φ' _C^l(K) + g_i^-1 - g^-1_C^l(K)). As the right-hand side converges to zero, we see that φ_i converges to φ in C^l for any l on K. A similar argument shows the convergence of connections. [The limiting configuration] Passing to the limit in equation (<ref>), we see that f Σ∖ D → is given by f = log | φ' |^2, which, by φ = e^-f/2φ', is equivalent to | φ | = 1. Since we also have _A φ = 0, Lemma <ref> implies that ∇_A φ = 0 and F_A = 0 on Σ∖ D. [Convergence of measures] It remains to show that i/2πΛ F_A_i→∑_j = 1^d δ (x_j) in the sense of measures, or, equivalently, that for any small disc B around x_j, lim_i →∞∫_Bi/2π F_A_i = k, where k is the multiplicity of the section φ' at x_j. Choose local coordinates on B together with a unitary trivialisation of L. Then A_i is of the form A_i = d + a_i for a_i ∈Ω^1(B, i) and the curvature is F_A_i = da_i. By Stokes' theorem, lim_i →∞∫_Bi/2π F_A_i = lim_i →∞∫_Bi/2π da_i = lim_i →∞∫_∂ Bi/2π a_i = ∫_∂ Bi/2π a, where a ∈Ω^1(B ∖{ x_j }, i) is the one-form corresponding to the singular connection A = d + a. By Lemma <ref>, the integral on the right-hand side is the degree of the limiting section φ around x_j. Since φ' and φ differ by a non-zero real function on B ∖{ x_j }, their degrees around x_j are the same and equal to the multiplicity of φ' at x_j. We now turn to Theorem <ref>. For simplicity, we write _A instead of _BA. To avoid double upper indices, denote α_i = φ_i^1 and β_i = φ_i^2. Integrating the last equation of (<ref>), we obtain α_i _L^2^2 - β_i _L^2^2 = - 2 πϵ_i L, which together with α_i _L^2^2 + β_i _L^2^2 = φ_i _L^2^2 = 1 implies that for sufficiently large i neither α_i nor β_i is identically zero. [Convergence modulo 𝒢^c] We claim that there are sequences g_i ∈𝒢^c and λ_i > 0 such that after passing to a subsequence (g_i A_i, λ_i g_i α_i, λ_i g_i^-1β_i) converges in C^∞ on Σ to a triple (A', α', β'). The limiting sections α' and β' are not identically zero and satisfy _A'α' = 0 and _A'β' = 0. By the same argument as in the proof of Theorem <ref>, we can find g_i ∈𝒢^c such that, after passing to a subsequence, A_i' = g_i A_i converges in C^∞ to a connection A'. We want to rescale the corresponding sequences of sections g_i α_i and g_i^-1β_i so that they also converge, and that the limiting sections are non-zero. First, by replacing the sequence g_i by μ_i g_i, where μ_i are constant complex gauge transformations given by μ_i = √( g_i^-1β_i _L^2/ g_i α_i _L^2), we may assume that g_i α_i _L^2 = g_i^-1β_i _L^2. After changing the original sequence by real gauge transformations we can assume that g_i = e^f_i/2 for smooth functions f_i Σ→. Set λ_i = g_i α_i _L^2^-1 and consider the rescaled sequences α_i' = λ_i g_i α_i and β_i' = λ_i g_i^-1β_i. We have α_i' _L^2 = β_i' _L^2 = 1 so as in the proof of Theorem <ref> we can choose subsequences (denoted for simplicity by the same symbols) that converge in C^∞ to sections α' and β' say. They are holomorphic with respect to A' and satisfy α' _L^2 = lim_i →∞α_i' _L^2 = 1, β' _L^2 = lim_i →∞β_i' _L^2 = 1. [An upper bound for λ_i] Next, we show that the sequence λ_i is bounded above. Assume by contradiction that after passing to a subsequence λ_i →∞. Then | α_i | | β_i | = λ_i^-2 | α_i' | | β_i' | ≤λ_i^-2sup_iα_i ' _L^∞β_i' _L^∞ . Since α_i' →α' and β_i' →β' uniformly, we conclude that lim_i →∞ | α_i | | β_i | _L^∞ = 0. We will argue that this cannot happen. Set ψ_i = | β_i |^2 and Q_i = | β_i' |^2. Differentiating λ_i^2ψ_i = e^f_i Q_i twice, we arrive at λ_i^2 ∇ψ_i = e^f_i Q_i ∇ f_i + e^f_i∇ Q_i, and λ_i^2 Δψ_i = Δ( e^f_i ) Q_i - ⟨∇ ( e^f_i ), ∇ Q_i ⟩ + e^f_iΔ Q_i, = e^f_i Q_i Δ f_i - e^f_i Q_i | ∇ f_i |^2 - e^f_i⟨∇ f_i , ∇ Q_i ⟩+ e^f_iΔ Q_i = e^f_i Q_i Δ f_i - ⟨∇ f_i , e^f_i Q_i ∇ f_i + e^f_i∇ Q_i ⟩ + e^f_iΔ Q_i = e^f_i Q_i Δ f_i - ⟨∇ f_i, λ_i^2 ∇ψ_i ⟩ + e^f_iΔ Q_i. Let x_i ∈Σ be a global maximum of ψ_i, so that ∇ψ_i(x_i) = 0 and Δψ_i (x_i) ≥ 0. The calculation above implies that Q_i (x_i) Δ f_i(x_i) ≥ - Δ Q_i(x_i). We would like to conclude that Δ f_i is bounded below for all sufficiently large i. Consider i large enough so that Q_i is sufficiently close to Q = | β' |^2 in the C^2 norm. Since β' is holomorphic and not identically zero, it vanishes at finitely many points. Therefore, for any δ > 0 there is a small open neighbourhood, V_δ say, of the zero set of β' such that Q_i(x) ≥δ whenever x ∈Σ∖ V_δ and i is large. It follows that if the sequence x_i is isolated from the zero set of β', then there exists δ > 0 such that Δ f_i(x_i) ≥ - 1/δsup_i Δ Q_i _L^∞. On the other hand, assume that after passing to a subsequence, x_i → x, where β'(x) = 0. If x is a simple root of β', then in local holomorphic coordinates centred around x we have Q(z) = c |z|^2 + O(|z|^3) for some c > 0 and as a result Δ Q (x) < 0. It follows that Δ Q_i (x_i) < 0 for i sufficiently large and by (<ref>), we must have that Q_i(x_i) ≠ 0 since 0 ≤λ_i^2 Δψ_i (x_i) = e^f_i(x_i) Q_i(x_i) Δ f_i(x_i) + e^f_i(x_i)Δ Q_i(x_i). Thus, dividing both sides by e^f_i(x_i) Q_i(x_i) results in the inequality Δ f_i(x_i) ≥ 0. In the general situation, the section β' vanishes to the order k, say. In this case, locally Q^1/k(z) = c |z|^2 + O(|z|^3) for some c > 0. We can therefore, at least locally, replace ψ_i by ψ_i^1/k (note that ψ_i(x_i) →∞, so ψ_i^1/k is a smooth function around x_i), f_i by f_i / k, and Q_i by Q_i^1/k and repeat the previous argument to obtain Δ f_i(x_i) ≥ 0. From the lower bound for Δ f_i(x_i) we easily arrive at a contradiction. Indeed, since A_i' = g_i( A_i), the third equation of (<ref>) can be written in the form ϵ_i^2 Λ i F_A_i' = ϵ_i^2 Λ i F_A_i + ϵ_i^2 Δ f_i = - |α_i|^2 + |β_i|^2 + ϵ_i^2 Δ f_i, or after rearranging, | β_i |^2 = ϵ_i^2 Λ i F_A_i' - ϵ_i^2 Δ f_i + |α_i |^2. Since Δ f_i(x_i) is bounded below and A_i' converges in C^∞, at x_i we have | β_i(x_i) |^2 ≤ C ϵ_i^2 + | α_i(x_i)|^2. Equation (<ref>) implies that for large i | β_i(x_i) | = β_i (x_i) _L^∞≥ C β_i _L^2 = C(1/2+ πϵ_i L) ≥C/4 > 0, so | β_i(x_i) | is separated from zero and (<ref>) forces | α_i(x_i) | to converge to zero. Then (<ref>) yields | β_i(x_i) | → 0, which is a contradiction. This finishes the proof that the sequence λ_i is bounded above. [Convergence of λ_i] The next step is to show that the sequence λ_i is separated from zero. This follows from the inequality λ_i^2 ( | α_i|^2 + |β_i|^2 ) ≥ 2 λ_i^2 | α_i | |β_i| = 2 | λ_i e^f_i/2α_i | | λ_i e^-f_i/2β_i | = 2 | α_i' | | β_i' |. Integrating over Σ and taking the square root, we obtain λ_i ≥√( 2 ∫_Σ |α_i' | | β_i' | ) Since α_i' →α' and β_i' →β' uniformly on Σ and the limiting sections are holomorphic and non-zero, the integral on the right-hand side is bounded below by a positive number. Since λ_i is bounded above and separated from zero, after passing to a subsequence we can assume that λ_i converges to a positive number, λ say. Therefore, e^f_i/2α_i = λ_i^-1α_i' →λ^-1α', e^-f_i/2β_i = λ_i^-1β_i' →λ^-1β' and after rescaling α', β' we may assume that λ_i = λ = 1. [Convergence outside the singular set] Let D be the union of the zero sets of α' and β'. Since the sections are holomorphic and not identically zero, D is empty or consists of finitely many points. We claim that after passing to a subsequence, f_i converges in C^∞_(Σ∖ D). To see this, we translate the third equation of (<ref>) for the triple (A_i, α_i, β_i) into a partial differential equation for f_i: ϵ_i^2 Λ i F_A_i' = ϵ_i^2 Λ i F_A_i + ϵ_i^2 Δ f_i = - |α_i|^2 + |β_i|^2 + ϵ_i^2 Δ f_i = - e^-f_i | α_i' |^2 + e^f_i | β_i' |^2 + ϵ_i^2 Δ f_i, or after rearranging ϵ_i^2 Δ f_i - P_i e^-f_i + Q_i e ^f_i - w_i = 0, where P_i = | α_i' |^2, Q_i = | β_i' |^2, and w_i = ϵ_i^2 Λ i F_A_i'. Note that ϵ_i → 0 and the functions P_i, P_i, and w_i converge in C^∞ to, respectively, P = |α'|^2, Q = | β' |^2, and w = 0. Furthermore, P > 0 and Q > 0 on Σ∖ D. It follows from Proposition <ref> and Remark <ref> that the sequence f_i and its derivatives are uniformly bounded on every compact subset of Σ∖ D. Therefore, we can choose a subsequence of f_i which converges in C^∞_(Σ∖ D). The limit is a smooth function f Σ∖ D →. Let g = e^f/2 be the corresponding complex gauge transformation. Now set (A, α, β) = ( g^-1( A' ), g^-1α', gβ'). The triple is well-defined on Σ∖ D and (A_i, α_i, β_i) → (A, α, β) in C^∞_ on Σ∖ D. [The limiting configuration] After passing to the limit i →∞ in (<ref>), we get - e^-f |α'|^2 + e^f | β'|^2 = 0, or equivalently e^f = | α'| / |β'|. Therefore, the limiting configuration φ = (α, β) satisfies - |α|^2 + |β|^2 = - e^-f | α'|^2 + e^f | β'|^2 = 0. The remaining equations _A α = 0, _A β = 0, and αβ = 0 are obtained from passing to the limit in the corresponding equations for A_i, α_i, and β_i. Moreover, | φ |^4 = ( |α|^2 + |β|^2 )^2 = ( e^-f | α' |^2 + e^f | β'|^2 )^2 = 4 |α'|^2 | β'|^2 extends to a smooth function on Σ whose zeroes are the points in D. At a point x ∈ D, the norm | φ | = 2 √( | α' | | β' |) vanishes to the order 1/2( ord_x (α') + ord_x (β') ) where ord_x (α') and ord_x (β') are the orders of vanishing of α' and β' at x. Assume now that E = 2. Let A^2 = A ⊗ A denote the tensor product connection on L^2 = L ⊗ L. We will show that the pair (L^2, A^2) is trivial as a bundle with a connection. It will follow then that A is flat and has holonomy contained in _2. Since E = 2 and αβ = 0, we have the following exact sequence over Σ∖ D: 0 r L^* ⊗ K^-1/2rα E rβ L^* ⊗ K^1/2r 0, which yields a bundle isomorphism over Σ∖ D ψ_αβ L^-2⊗ K^-1/2⊗ K^1/2⟶ E. Both line bundles E and K^-1/2⊗ K^1/2 are trivial as bundles with connections. Thus, ψ_αβ is a section of L^2 over Σ∖ D which satisfies _A^2ψ_αβ = 0, | ψ_αβ | = | α | /|β| = 1 By Lemma <ref>, ∇_A^2ψ_αβ = 0 and (L^2, A^2) is trivial as a bundle with a connection. If E > 2, let F be the complex subbundle of EΣ∖ D spanned by the subspaces α and (β)^⊥, interpreting α and β as in the short exact sequence (<ref>). Equip F with the unitary connection B̃ obtained from B using the orthogonal projection π_F E → F, and let _B̃ = (∇_B̃)^0,1 = π_F _B be the induced holomorphic structure on F. Note that F ⊂ E is not, in general, a holomorphic subbundle. Nevertheless, α and β descend to smooth sections α∈Γ(Σ∖ D, F ⊗ L ⊗ K^1/2), β∈Γ(Σ∖ D, F^* ⊗ L^* ⊗ K^1/2) which are nowhere zero and satisfy αβ = 0; we claim that these sections are in fact holomorphic with respect to _B̃. This is clear for α since _B̃Ãα = π_F _BAα = 0. To prove that β is also holomorphic, it is enough to show that for every locally defined section s of F satisfying _B̃ s = 0 we have _A ( β(s)) = 0 (thinking of β as a homomorphism from F to L^* ⊗ K^1/2). The condition 0 = _B̃ s = π_F _B s implies that _B s takes values in β and since β E → L^* ⊗ K^1/2 is holomorphic, we have _A(β(s)) = β(_B s) = 0. Thus, α and β in (<ref>) are holomorphich with respect to _B̃ and so they fit into the short exact sequence (<ref>) with F in place of E. The previous argument shows that there is a nowhere vanishing section ψ_αβ∈Γ(Σ, ∖ D, F ⊗ L^2) satisfying _BAψ_αβ = 0 and | ψ_αβ| =1. Here F is equipped with a connection obtained from B by the orthogonal projection E → F, and F has the induced connection. Lemma <ref> implies that ∇_BAψ_αβ = 0 and F ⊗ L^2 is trivial over Σ∖ D. In particular, there is a well-defined square root ( F)^1/2 and the connection on ( F)^1/2⊗ L induced from B and A is flat with holonomy contained in _2. [Convergence of measures] We claim that if x is a point in D at which the sections α' and β' vanish to the order k = ord_x (α') and l = ord_x (β') respectively, and B is a small disc around x, then lim_i →∞∫_B i/2πΛ F_A_i = k-l/2. This is proved in the same way as the corresponding statement in Theorem <ref>, except that now Lemma <ref> is applied to the connection A^2 on L^2 and section ψ_αβ (under the assumption that E = 2; in the higher rank case we need to twist L^2 by F). The degree of the restriction of ψ_αβ to ∂ B is deg( ψ_αβ) = deg( α/ |α|) - deg( β/ |β|) = k - l, and the factor 1/2 enters because we consider A instead of A^2. This shows that D is the set underlying the divisor D = ∑_x ( ord_x (α') - ord_x (β') )x and i Λ F_A = 1/2δ_D as measures. In particular, D has degree 2d. Our assumptions guarantee that all irreducible solutions to the Seiberg–Witten equations with multiple spinors are gauge-equivalent to configurations pulled back from Σ satisfying (<ref>) with ϵ = 1<cit.>. The bundle L is necessarily pulled back from a bundle over Σ. Let ϵ_i = Ψ_i _L^2^-1 and Ψ_i' = ϵ_i Ψ_i. Then the rescaled sequence (A_i, Ψ_i') satisfies Ψ_i' _L^2 = 1 and equations (<ref>) with ϵ = ϵ_i. The statements about the convergence of (A_i, Ψ_i') and the structure of Z follow now from Theorem <ref>. It remains to show the convergence of currents. Every η∈Ω^1(Y) is of the form η = f (t,x) dt + ξ (t), where f ∈ C^∞(Y), (t,x) denote the product co-ordinates on S^1 ×Σ, and ξ is an S^1-family of one-forms ξ(t) ∈Ω^1(Σ). The forms F_A_i are pulled back from Σ, so ∫_Y i/2π F_A_i∧η = ∫_S^1∫_Σi/2π F_A_i f(t,x) dt = ∫_Σi/2π g(x) F_A_i, where g(x) = ∫_S^1 f(t,x) dt. Passing to the limit i →∞ and using the convergence of measures (i/2π) Λ F_A_i→δ_D / 2 proved in Theorem <ref>, we arrive at lim_i →∞∫_Y i/2π F_A_i∧η = 1/2∑_k m_k g(x_k) = 1/2∑_k m_k ∫_S^1 f(t,x) dt = 1/2∑_k m_k ∫_S^1 ×{ x_k }η, which is the equality that we wanted to prove. It remains to address the statement made in point (3) of the theorem about the weights associated with the connected components of Z. If the singular set Z is a disjoint union ⋃_k Z_k of connected one-dimensional submanifolds, the weight prescribed by Taubes to Z_k is the highest number N_k such that |Ψ| vanishes to the order N_k / 2 along Z_k<cit.>. In our case, Z_k = S^1 ×{ x_k }, N_k is an integer, and by point (3) of Theorem <ref>, we have |m_k| ≤ N_k. On the other hand, m_k is the coefficient of the current 1/2 Z and equals the weight prescribed to Z_k by Haydys: the degree of the section of L^2 induced from Ψ restricted to a small loop linking Z_k<cit.>; see step 5 in the proof of Theorem <ref>. Finally, we prove Theorem <ref>. The following inequality will be useful. Let a, b, x, and y be positive numbers. Then for all ξ∈ (0, ∞) ξ^-a x + ξ^b y ≥ K x^b/a+b y^a/a+b where K is positive and depends only on a and b. Consider the function f (0, ∞) → (0, ∞) given by f(ξ) = ξ^-a x + ξ^b y. It attains a global minimum at ξ_0 = ( ax/by)^1/a+b. and the value of f at ξ_0 is f(ξ_0) = K x^b/a+b y^a/a+b with K = (a/b)^b/a+b + (b/a)^a/a+b First, order the weights according to their signs: (k_1, …, k_N) = (a_1, …, a_n, -b_1, …, -b_m), where n+m = N and a_j > 0 and b_j > 0. Denote α_i = ( α^1_i, …, α^n_i) = (φ^1_i, …, φ^n_i), β_i = ( β^1_i, …, β^m_i) = (φ^n+1_i, …, φ^N_i). When all the weights are positive and τ < 0, or when one of the sequences α_i _L^∞ or β_i _L^∞ converges to zero, the proof is the same as for Theorem <ref>. Let us focus on the more difficult case when the weights are of mixed signs and α_i _L^∞, β_i _L^∞ are bounded below by a positive number. As in the proof of Theorem <ref>, we find sequences of λ_i > 0 and complex gauge transformations g_i ∈𝒢^c such that the sequence (A_i', α_i', β_i') = g_i(A_i, λ_i α_i, λ_i β_i) converges uniformly with all derivatives on Σ to a limit (A', α', β'). In order to do this, first choose g_i so that g_i (A_i) converges. After changing g_i by constant complex gauge transformations we may assume that g_i (α_i) _L^2 = g_i ( β_i) _L^2. Indeed, for each i such a constant gauge transformation is given by a number μ > 0 satisfying ∑_j=1^n μ^a_jα_i^j _L^2^2 - ∑_j=1^mμ^-b_jβ_i^j _L^2^2 = 0. Since the left-hand side diverges to ∞ when μ→∞, and to - ∞ when μ→ 0, such μ exists. Then we choose the scaling constants λ_i so that λ_i g_i ( α_i ) _L^2 = λ_i g_i ( β_i) _L^2 = 1, which, after passing to a subsequence, guarantees the existence of non-zero limiting sections α' = ((α')^1, …, (α')^n) and β' = ((β')^1, …, (β')^m). After changing the original sequence by a sequence of unitary gauge transformations, we may assume that g_i = e^f_i/2 for smooth functions f_i Σ→. The next step is to show that λ_i is bounded above and separated from zero. To prove the latter observe that at least one of the sections (α')^i is non-zero. Without loss of generality assume that (α')^1≠ 0. Likewise, we may assume that (β')^1 ≠ 0. Then λ_i^2 ( | α_i |^2 + | β_i |^2 ) ≥ | λ_i α_i^1 |^2 + | λ_i β_i^1 |^2 = e^-a_1 f_i | (α_i')^1 |^2 + e^ b_1 f_i | (β_i')^1 |^2. Applying Lemma <ref> to the right-hand side, integrating over Σ, and using the L^2-bound for φ_i = ( α_i, β_i), we obtain C λ_i^2 ≥ K ∫_Σ | (α_i')^1 |^κ_1 | (β_i')^1 |^κ_2 for some positive exponents κ_1 and κ_2 depending on a_1 and b_1. Now the right-hand side of the inequality is bounded below by a positive constant, because of the convergence (α_i')^1 → (α')^1 and (β_i')^1 → (β')^1 and the assumption that the limiting sections are non-zero. This shows that the sequence λ_i is separated from zero. An upper bound is found in the same way as in the proof of Theorem <ref>. Let u_i = ∑_j=1^n a_j | α_i^j |^2 and v_i = ∑_j=1^m b_j | β_i^j|^2. Suppose without loss of generality that τ≤ 0. We have, as in (<ref>), v_i = ϵ_i^2 Λ i F_A_i' - ϵ_i^2 Δ f_i + u_i + τ. Using the maximum principle, one shows that at the point x_i where v_i attains a global maximum, there is a lower bound for Δ f_i(x_i). As in the situation of Theorem <ref>, if λ_i is unbounded, then u_i v_i → 0 uniformly. It follows that u_i(x_i) → 0 and passing to the limit i →∞ we arrive at a contradiction v_i(x_i) → 0. This shows that λ_i must be bounded above. Since it is also separated from zero, after passing to a subsequence it may be assumed to be convergent and we may as well assume that λ_i = 1. The function f_i satisfy the partial differential equation ϵ_i ^2 Δ f_i - ∑_j=1^n P_i^j e^- a_j f_i + ∑_j=1^m Q_i^j e^b_j f_i - w_i = 0, where P_i^j = a_j | g_i (α_i^j) |^2, Q_i^j = b_j | g_i ( β_i^j) |^2, and w_i = ϵ_i^2 i Λ F_g_i(A_i). We have P_i^j ⟶ P^j, Q_i^j ⟶ Q^j , w_i ⟶ 0 in C^∞, where P^j = a_j | (α')^j|^2 and Q^j = b_j | (β')^j |^2. Let D be the union of the zero sets of the limiting sections α' and β'. Since they are not identically zero, D is a (possibly empty) finite subset of Σ. Over Σ∖ D we have P^1 + … + P^n > 0, and Q^1 + … + Q^m > 0. Therefore, by Proposition <ref> and Remark <ref> we can bound the the functions f_j and their derivatives on any compact subset of Σ∖ D, and consequently, extract a subsequence converging smoothly to a function f on Σ∖ D. As before, this leads to the smooth convergence of (A_i, α_i, β_i) to e^-f/2(A', α', β') on Σ∖ D. alphanum_n.bst

http://arxiv.org/abs/1701.07653v4

20170126111001

Some remarks on connectors and groupoids in Goursat categories

math.CT

Some remarks on connectors and groupoids in Goursat categories] Some remarks on connectors and groupoids in Goursat categories Dedicated to Jiří Adámek on the occasion of his seventieth birthday Marino Gran]Marino Grana aInstitut de Recherche en Mathématique et Physique, Université Catholique de Louvain, Belgium {marino.gran,idriss.tchoffo}@uclouvain.be Idriss Tchoffo Nguefeu]Idriss Tchoffo Nguefeua Diana Rodelo]Diana Rodelob bDepartamento de Matemática, Universidade do Algarve, 3001-501 Coimbra, Portugal The third author acknowledges partial financial assistance by Centro de Matemática da Universidade de Coimbra—UID/MAT/00324/2013, funded by the Portuguese Government through FCT/MCTES and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020. drodelo@ualg.pt [2000] 08C05, 08B05, 08A30, 08B10, 18C05, 18B99, 18E10 We prove that connectors are stable under quotients in any (regular) Goursat category. As a consequence, the category (ℂ) of connectors in ℂ is a Goursat category whenever ℂ is. This implies that Goursat categories can be characterised in terms of a simple property of internal groupoids. [ [ ===== Over the last twenty years the property of n-permutability of congruences in a variety of universal algebras has been investigated from a categorical perspective (see <cit.>, for instance, and references therein). When ℂ is a regular category, the 2-permutability property, usually called the Mal'tsev property, is a concept giving rise to a beautiful theory, whose main features are collected in <cit.>. Many important results still hold when a regular category ℂ satisfies the strictly weaker property of 3-permutability, namely the Goursat property. A nice feature of a (regular) Goursat category ℂ is that the lattice of equivalence relations on any object in ℂ is a modular lattice <cit.>, a property that plays a crucial role in commutator theory <cit.>. The aim of this paper is twofold: first of all we establish some basic properties of Goursat categories in terms of connectors <cit.>, as it was done in <cit.> for the case of Mal'tsev categories. These results have turned out to be useful to develop a monoidal approach to internal structures <cit.>. We then give a new characterisation of Goursat categories in terms of properties of (internal) groupoids, on the model of what was done in <cit.> in the case of Mal'tsev categories. In the first section, we recall the main properties of Goursat categories that will be needed throughout the paper. In Section 2 we prove that for any Goursat category ℂ, the category (ℂ) of equivalence relations in ℂ is also a Goursat category (Proposition <ref>, see also <cit.>). We use this result to give some properties of Goursat categories in terms of connectors in Section 3. More precisely, we show that, when ℂ is a Goursat category, then connectors are stable under quotients in ℂ (Proposition <ref>), and this implies that the category () of connectors in is again a Goursat category (Theorem <ref>). We conclude the paper by giving a new characterisation of Goursat categories in terms of properties of groupoids and internal categories (Theorem <ref>). It turns out that a regular category ℂ is a Goursat category if and only if the category (ℂ) of groupoids (equivalently, the category (ℂ) of internal categories) in ℂ is closed under quotients in the category (ℂ) of reflexive graphs in . § PRELIMINARIES In this section we recall some basic definitions and properties of (regular) Goursat categories, needed throughout the article. We shall always assume that the category ℂ in which we are working is a regular category: this means that is finitely complete, regular epimorphisms are stable under pullbacks, and kernel pairs have coequalisers. Equivalently, any arrow f: A⟶ B has a unique factorisation f=i∘ r (up to isomorphism), where r is a regular epimorphism and i is a monomorphism and this factorisation is pullback stable; the subobject corresponding to i is called the image of f. A relation R from X to Y is a subobject ⟨ r_1,r_2 ⟩ : R ↣ X × Y. The opposite relation of R, denoted R^o, is the relation from Y to X given by the subobject ⟨ r_2,r_1 ⟩ : R ↣ Y × X. A relation R from X to X is called a relation on X. We shall identify a morphism f: X ⟶ Y with the relation ⟨ 1_X,f ⟩: X ↣ X × Y and write f^o for its opposite relation. Given another relation ⟨ s_1, s_2 ⟩ : S ↣ Y × Z from Y to Z, one can define the composite relation SR of R and S as the image of the arrow (r_1 ∘ p_1, s_2 ∘ p_2): R ×_Y S ⟶ X × Z, where (R ×_Y S, p_1, p_2) is the pullback of r_2: R⟶ Y along s_1: S ⟶ Y. With the above notations, any relation ⟨ r_1, r_2 ⟩: R ↣ X × Y can be seen as the relational composite r_2r_1^o. The following properties are well known and easy to prove. We collect them in the following lemma: Let f: X ⟶ Y be an arrow in a regular category ℂ, and let i ∘ r be its (regular epimorphism, monomorphism) factorisation. Then: * f^of is the kernel pair of f, thus 1_X ⩽ f^of; moreover, 1_X = f^of if and only if f is a monomorphism; * ff^o is (i,i), thus ff^o ⩽ 1_Y; moreover, ff^o = 1_Y if and only if f is a regular epimorphism; * ff^of = f and f^off^o = f^o. A relation (R, r_1,r_2) on an object X is said to be : * reflexive when there is an arrow r: X ⟶ R such that r_1 ∘ r = 1_X = r_2 ∘ r; * symmetric when there is an arrow σ : R ⟶ R such that r_2 = r_1 ∘σ and r_1 = r_2 ∘σ; * transitive when, by considering the following pullback R ×_X R [r]^-p_2@[dr]|(0.25)⌟[d]_p_1 R [d]^r_1 R [r]_r_2 X, there is an arrow t : R ×_X R ⟶ R such that r_1 ∘ t = r_1 ∘ p_1 and r_2 ∘ t = r_2 ∘ p_2. * an equivalence relation if R is reflexive, symmetric and transitive. In particular, a kernel pair ⟨ f_1,f_2 ⟩: (f)↣ X × X of a morphism f: X ⟶ Y is an equivalence relation. The equivalence relations that occur as kernel pairs of some morphism in ℂ are called effective. Let (ℂ) be the category whose objects are equivalence relations in ℂ and arrows from ⟨ r_1,r_2 ⟩ : R ↣ X × X to ⟨ s_1,s_2 ⟩ : S ↣ Y × Y are pairs (f,g) of arrows in ℂ making the following diagram commute R [r]^g@<.5ex>[d]^r_2@<-.5ex>[d]_r_1 S @<.5ex>[d]^s_2@<-.5ex>[d]_s_1 X [r]_f Y. When ℂ is a regular category, (R,r_1,r_2) is an equivalence relation on X and f: X ↠ Y a regular epimorphism, we define the regular image of R along f to be the relation f(R) on Y induced by the (regular epimorphism, monomorphism) factorisation ⟨ s_1, s_2 ⟩∘ψ of the composite (f× f)∘⟨ r_1,r_2⟩: R @.>>[r]^ψ@ >->[d]_⟨ r_1,r_2⟩ f(R) @ >.>[d]^⟨ s_1,s_2⟩ X × X @>>[r]_f × f Y × Y. Note that the regular image f(R) can be obtained as the relational composite f(R)=fRf^o=fr_2r_1^of^o. When R is an equivalence relation, f(R) is also reflexive and symmetric. In a general regular category f(R) is not necessarily an equivalence relation. This is the case in a Goursat category (Theorem <ref>). [<cit.>] A regular category ℂ is called a Goursat category when the equivalence relations in ℂ are 3-permutable, i.e. RSR = SRS for any pair of equivalence relations R and S on the same object. The following characterisation will be useful in the sequel: [<cit.>] A regular category ℂ is a Goursat category if and only if for any regular epimorphism f: X ↠ Y and any equivalence relation R on X, the regular image f(R)= fRf^o of R along f is an equivalence relation. There are many important algebraic examples of Goursat categories. Indeed, by a classical theorem in <cit.>, a variety of universal algebras is a Goursat category precisely when its theory has two ternary operations r and s such that the identities r(x,y,y)= x, r(x,x,y)=s(x,y,y) and s(x,x,y)= y hold. Accordingly, the categories of groups, abelian groups, modules over some fixed ring, crossed modules, quasi-groups, rings, associative algebras, Heyting algebras and implication algebras are all Goursat categories. Any regular Mal'tsev category is a Goursat category, thus, in particular, so is any semi-abelian category. Many interesting properties of Goursat categories can be found in the literature (see <cit.> and references therein). In particular, the following characterisations will be useful for the development of this work: [<cit.>] Let ℂ be a regular category. The following conditions are equivalent: (i) ℂ is a Goursat category; (ii) any commutative diagram where α and β are regular epimorphisms and f and g are split epimorphisms in ℂ X @->>[r]^α@<3pt>[d]^f U @<3pt>[d]^g Y @->>[r]_β@<3pt>[u]^s W @<3pt>[u]^t (which is necessarily a pushout) is a Goursat pushout: the morphism λ : (f) ⟶(g) induced by the universal property of kernel pair (g) of g is a regular epimorphism. We recall part of Theorem 1.3 in <cit.>: [<cit.>] Let ℂ be a regular category. The following conditions are equivalent: (i) ℂ is a Goursat category; (ii) for any commutative cube X ×_Y Z @->>[rr]^δ@<3pt>[rd] @<3pt>[dd] A @<3pt>[rd] @<3pt>@–>[dd] Z @<3pt>[ul] @->>[rr]^(0.3)γ@<3pt>[dd] V @<3pt>[ul] @<3pt>[dd] X @–>>[rr]^(0.7)α@<3pt>[rd] @<3pt>[uu] U @<3pt>@–>[rd] @<3pt>@–>[uu] Y @->>[rr]_β@<3pt>[ul] @<3pt>[uu] W @<3pt>@–>[ul] @<3pt>[uu] where the left square is a pullback of split epimorphisms, the right square is a commutative square of split epimorphisms and the horizontal arrows α, β, γ and δ are regular epimorphisms (commuting also with the splittings), then the right square is a pullback. § EQUIVALENCE RELATIONS IN GOURSAT CATEGORIES In this section we prove that (ℂ) is a Goursat category for any Goursat category ℂ. The category () is finitely complete whenever is: the terminal object in () is the discrete equivalence relation 1 @<.5ex>[r]^@<-.5ex>[r]_ 1 on the terminal object 1 of , and pullbacks are computed “levelwise”. In particular, the kernel pair of a morphism (f,g) in () is given by the kernel pairs (f) of f and (g) of g in (g) @<.5ex>[r]^g_1@<-.5ex>[r]_g_2@<.5ex>[d]^r̅_̅2̅@<-.5ex>[d]_r̅_̅1̅ R [r]^g@<.5ex>[d]^r_2@<-.5ex>[d]_r_1 S @<.5ex>[d]^s_2@<-.5ex>[d]_s_1 (f) @<.5ex>[r]^f_1@<-.5ex>[r]_f_2 X [r]_f Y. Consequently, a morphism (f,g) is a monomorphism in () if and only if f and g are monomorphisms in . When is a Goursat category, a similar property holds with respect to regular epimorphisms: Let R and S be two equivalence relations in a Goursat category ℂ and (f,g) : R → S a morphism R [r]^g@<.5ex>[d]^r_2@<-.5ex>[d]_r_1 S @<.5ex>[d]^s_2@<-.5ex>[d]_s_1 X [r]_f Y in (ℂ). Then (f,g) is a regular epimorphism in (ℂ) if and only if f and g are regular epimorphisms in ℂ. When f and g are regular epimorphisms in ℂ, it is not difficult to check that (f,g) is necessarily the coequaliser of its kernel pair in (ℂ) given in (<ref>) (one uses the fact that g = coeq(g_1,g_2) and f = coeq(f_1,f_2) in ℂ). Conversely, let (f,g) be a morphism in (ℂ) as in (<ref>) that is a regular epimorphism in (ℂ). Consider the kernel pairs of f and g, the (regular epimorphism, monomorphism) factorisation f=i ∘ q of f, and the regular image (q(R), t_1, t_2) of (R, r_1, r_2) along q. We obtain the following commutative diagram (g) @<.5ex>[r]^g_1@<-.5ex>[r]_g_2@<.5ex>[dd]^r̅_̅2̅@<-.5ex>[dd]_r̅_̅1̅ R @->>[rd]_α[rr]^g@<.5ex>[dd]^r_2@<-.5ex>[dd]_r_1 S @<.5ex>[dd]^s_2@<-.5ex>[dd]_s_1 q(R) @<.5ex>[dd]^(0.3)t_2@<-.5ex>[dd]_(0.3)t_1@.>@<3pt>[ru]_(0.6)j (f)@<.5ex>[r]^f_1@<-.5ex>[r]_f_2 X @->>[rd]_q [rr]^(0.8)f Y, Z @ >->[ru]_(0.6)i where (q(R), t_1, t_2) ∈(ℂ) (by Theorem <ref>) and (i, j) is the morphism in (ℂ) such that (i, j) ∘ (q, α) = (f,g). Note that j is induced from the fact that (i× i) ∘⟨ t_1, t_2 ⟩∘α is the (regular epimorphism, monomorphism) factorisation of ⟨ s_1, s_2 ⟩∘ g, thus it is a monomorphism @C=30pt@R=25pt R @>>[r]^-α[d]_-g q(R) @ >->[d]^-(i× i)∘⟨ t_1,t_2⟩@ >.>[ld]_-j S @ >->[r]_-⟨ s_1,s_2⟩ Y× Y. From the fact that (f,g) is the coequaliser of its kernel pair in (ℂ) it easily follows that (i,j) is an isomorphism in (ℂ). This implies that f and g are regular epimorphisms in ℂ.    (ℂ) is a Goursat category whenever is. As mentioned above, the category (ℂ) is finitely complete because ℂ is so. Lemma <ref> implies that regular epimorphisms in () are stable under pullbacks since regular epimorphisms are stable in ℂ, and regular epimorphisms in (ℂ) are “levelwise” regular epimorphisms. The existence of the (regular epimorphism, monomorphism) factorisation of a morphism (f,g) as in (<ref>) in the category (ℂ) follows from the construction of diagram (<ref>): the (regular epimorphism, monomorphism) factorisation f=i∘ q of f in gives rise to the (regular epimorphism, monomorphism) factorisation g=j∘α of g in . Thus (q,α)∘ (i,j) is the (regular epimorphism, monomorphism) factorisation of (f,g) in (). To see that (ℂ) has the Goursat property one uses Theorem <ref>: to check that the regular image of an equivalence relation in the category (ℂ) is again an equivalence in (ℂ) one mainly uses the same (“levelwise”) property in the category ℂ. § CONNECTORS AND GROUPOIDS IN GOURSAT CATEGORIES In this section we prove that connectors are stable under quotients in any Goursat category . We then define the category () of connectors in whose objects are pairs of equivalence relations equipped with a connector, and prove that () is a Goursat category whenever the base category is. We conclude by giving a new characterisation of Goursat categories in terms of properties of groupoids and internal categories. Let (R,r_1,r_2) and (S, s_1,s_2) be two equivalence relations on an object X and R×_X S the pullback of r_2 along s_1. A connector <cit.> between R and S is an arrow p: R ×_X S ⟶ X in ℂ such that * x S p(x,y,z)R z; * p(x,x,y)= y; * p(x,y,y)= x; * p(x,y,p(z,u,v))= p(p(x,y,z),u,v), when each term is defined. Given two regular epimorphisms d X ↠ Y and c X ↠ Z, a connector on the effective equivalence relations (d) and (c) is the same thing as an internal pregroupoid in the sense of Kock <cit.> (see also the introduction of <cit.>, for instance, for a comparison between these two related notions and some additional references). In the context of Mal'tsev or Goursat categories connectors are useful to develop a centrality theory of non-effective equivalence relations. If ∇_X is the largest equivalence relation on an object X, then an associative Mal'tsev operation p: X× X × X ⟶ X is precisely a connector between ∇_X and ∇_X. Connectors provide a way to distinguish groupoids amongst reflexive graphs: [<cit.>] Given a reflexive graph X_1 @<6pt>[r]^d@<-6pt>[r]_c X_0 [l]|-e in a finitely complete category 𝒞 (i.e. d ∘ e = 1_X_0 = c∘ e) then the connectors between (c) and (d) are in bijections with the groupoid structures on this reflexive graph. It is well known that Goursat categories satisfy the so-called Shifting Property <cit.>. In this context connectors are unique when they exist (Theorem 2.13 and Proposition 5.1 in <cit.>): accordingly, for a given pair of equivalence relations on the same object the fact of having a connector becomes a property. Let R and S be two equivalence relations on an object X. A double equivalence relation on R and S is given by an object C ∈ℂ equipped with two equivalence relations (π_1,π_2): C ⇉ R and (p_1,p_2): C ⇉ S such that the following diagram C @<.5ex>[r]^π_1@<-.5ex>[r]_π_2@<.5ex>[d]^p_2@<-.5ex>[d]_p_1 R @<.5ex>[d]^r_2@<-.5ex>[d]_r_1 S @<.5ex>[r]^s_1@<-.5ex>[r]_s_2 X commutes (in the “obvious” way). A double equivalence relation C on R and S is called a centralizing relation <cit.> when the square C @[dr]|(.25)⌟[r]^π_1[d]_p_1 R [d]^r_1 S [r]_s_1 X is a pullback. Under this assumption it follows that any of the commutative squares in the definition of a centralizing relation is a pullback. The following lemma gives the relationship between connectors and centralizing relations. [<cit.>] If ℂ is a category with finite limits and R and S are two equivalence relations on the same object X, then the following conditions are equivalent: (i) there exists a connector between R and S; (ii) there exists a centralizing relation on R and S. When is a Mal'tsev category, R and S are equivalence relations on an object X with a connector and i:I↣ X is a monomorphism, then the inverse images i^-1(R) and i^-1(S) also have a connector <cit.>. We establish a similar property for Goursat categories, with respect to regular epimorphisms: Let ℂ be a Goursat category, R and S two equivalence relations on an object X, and let f: X ↠ Y be a regular epimorphism. If there exists a connector between R and S, then there exists a connector between the regular images f(R) and f(S). Suppose that there exists a connector between R and S. This implies that there exists a centralizing relation (C,(π_1,π_2),(p_1,p_2)) on R and S. Consider the regular image (f(R), a, b) and (f(S),c,d) of R and S along f. We obtain the following diagram C @->>[rr]^α@<.5ex>[rd]^π_2@<-.5ex>[rd]_π_1@<.5ex>[dd]^p_2@<-.5ex>[dd]_p_1 f_R(C) @<.5ex>@–>[dd]^(0.7)α_2@<-.5ex>@–>[dd]_(0.7)α_1@<.5ex>[rd]^β_2@<-.5ex>[rd]_β_1 R @<.5ex>[dd]^(0.3)r_2@<-.5ex>[dd]_(0.3)r_1@->>[rr]^(0.3)f_R f(R) @<.5ex>[dd]^b@<-.5ex>[dd]_a S @–>>[rr]^(0.7)f_S@<.5ex>[rd]^s_2@<-.5ex>[rd]_s_1 f(S) @<.5ex>@–>[rd]^d@<-.5ex>@–>[rd]_c X @->>[rr]_f Y, where (f_R(C), β_1,β_2) is the regular image of the equivalence relation (C,π_1,π_2) along the regular epimorphism f_R. The fact that the square C @->>[r]^α[d]_f_Sp_1 f_R(C) [d]^⟨ aβ_1,aβ_2⟩@.>[ld]_-α_1 f(S) @ >->[r]_-⟨ c,d⟩ Y × Y commutes, α is a strong epimorphism and ⟨ c,d⟩ is a monomorphism, implies the existence of an arrow α_1: f_R(C)⟶ f(S) making the above diagram commute. Similarly, from the commutativity of the third diagram C @->>[r]^α[d]_f_Sp_2 f_R(C) [d]^⟨ bβ_1,bβ_2⟩@.>[ld]_-α_2 f(S) @ >->[r]_-⟨ c,d⟩ Y × Y we obtain an arrow α_2: f_R(C)⟶ f(S). The relations (f_R(C), β_1,β_2), (f(R), a, b) and (f(S),c,d) are all equivalence relations by Theorem <ref>. It is then easy to check that the relation (f_R(C),α_1,α_2) is an equivalence relation on f(S). In fact, the morphism ⟨α_1,α_2⟩ f_R(C)→ f(S)× f(S) is a monomorphism since ⟨ c× c, d× d⟩∘⟨α_1,α_2 ⟩= ⟨ a,b⟩×⟨ a,b⟩∘⟨β_1, β_2⟩. So, ⟨α_1,α_2⟩ is the regular image of ⟨ p_1,p_2⟩ along f_S, thus it is an equivalence relation on f(S) by Theorem <ref>. By assumption all the left squares of (<ref>) are pullbacks, so it follows that all the right squares of (<ref>) are pullbacks as well by Theorem <ref> (ii). Then (f_R(C), (α_1,α_2),(β_1,β_2)) is a centralizing relation on f(R) and f(S). By Lemma <ref> there is a connector between f(R) and f(S). We are now going to show that the category whose objects are pairs of equivalence relations equipped with a connector is a Goursat category whenever the base category is a Goursat category. For this, let us first fix some notation: if ℂ is a finitely complete category, we write 2-(ℂ) for the category whose objects (R,S,X) are pairs of equivalence relations R and S on the same object X R @<.5ex>[r]^r_1@<-.5ex>[r]_r_2 X S @<.5ex>[l]^s_2@<-.5ex>[l]_s_1 and arrows are triples (f_R,f_S,f) making the following diagram commute: R @<.5ex>[r]^r_1@<-.5ex>[r]_r_2[d]_f_R X [d]_f S @<.5ex>[l]^s_2@<-.5ex>[l]_s_1[d]^f_S R̅@<.5ex>[r]^r̅_̅1̅@<-.5ex>[r]_r̅_̅2̅ X̅ S̅. @<.5ex>[l]^s̅_̅2̅@<-.5ex>[l]_s̅_̅1̅ We write (ℂ) for the category whose objects (R,S,X,p) are pairs of equivalence relations R and S on an object X with a given connector p: R ×_X S → X; arrows in (ℂ) are arrows in 2-(ℂ) respecting the connectors. This means that, given a diagram (<ref>) where both (R,S,X) and (R̅,S̅,X̅) are in (ℂ), with p: R ×_X S → X and p̅: R̅×_Y S̅→ Y the corresponding connectors, then the diagram R ×_X S [r]^f̅[d]_p R̅×_X̅S̅[d]^p̅ X [r]_f X̅ commutes, where f̅ is the natural map induced by the universal property of the pullback R̅×_X̅S̅. We say that a subcategory ℙ is closed under (regular) quotients in a category ℚ if, for any regular epimorphism f: A ↠ B in ℚ such that A ∈ℙ, then B ∈ℙ. If ℂ is a Goursat category, then (ℂ) is a full subcategory of 2-(ℂ), that is closed in 2-(ℂ) under quotients. The fullness of the forgetful functor (ℂ) → 2-(ℂ) follows from Corollary 5.2 in <cit.>, by taking into account the fact that any Goursat category satisfies the Shifting Property. Let us then consider a regular epimorphism in 2-(ℂ) R @<.5ex>[r]^r_1@<-.5ex>[r]_r_2@>>[d]_f_R X @>>[d]_f S @<.5ex>[l]^s_2@<-.5ex>[l]_s_1@>>[d]^f_S R̅@<.5ex>[r]^r̅_̅1̅@<-.5ex>[r]_r̅_̅2̅ X̅ S̅@<.5ex>[l]^s̅_̅2̅@<-.5ex>[l]_s̅_̅1̅ (this means that f, f_R and f_S are regular epimorphisms in ℂ) such that its domain (R,S,X) belongs to (ℂ). The equalities f(R)= R̅ and f(S)= S̅, together with Proposition <ref>, imply that there exists a connector between R̅ and S̅. Let 𝔻 be a finitely complete category, and ℂ a full subcategory of 𝔻 closed in 𝔻 under finite limits and quotients. Then: * ℂ is regular whenever 𝔻 is regular. * 𝔻 is a Goursat category whenever ℂ is a Goursat category. The (regular epimorphism, monomorphism) factorisation in 𝔻 of an arrow in ℂ is also its factorisation in ℂ, since ℂ is closed in 𝔻 under quotients. Since finite limits in ℂ are calculated as in 𝔻, it follows that regular epimorphisms are stable under pullbacks. Now the second statement easily follows from the fact that the composition of relations is computed in the same way in ℂ and in 𝔻. If ℂ is a Goursat category then (ℂ) is a Goursat category. Using similar arguments as those given in the proof of Proposition <ref> with respect to (), one may deduce that 2-( ℂ) is a Goursat category. The result then follows from Proposition <ref> and Lemma <ref>. We finally prove that internal categories and groupoids can be used to characterise Goursat categories. Recall that an internal category in a category ℂ with pullbacks is a reflexive graph with a multiplication m X_1 ×_X_0X_1 → X_1 X_1 ×_X_0X_1 @<6pt>[r]^-p_1@<-6pt>[r]_-p_2[r]|-m X_1 @<6pt>[r]^d@<-6pt>[r]_c X_0, [l]|-e (where X_1 ×_X_0X_1 is the pullback of d and c) such that: * d∘ m=d ∘ p_2,   c ∘ m=c∘ p_1,   m ∘⟨ e∘ d, 1_X_1⟩ = 1_X_1= m ∘⟨ 1_X_1,e∘ c⟩; * m∘ (1× m) = m∘ (m × 1). An internal category X_1 ×_X_0X_1 @<6pt>[r]^-p_1@<-6pt>[r]_-p_2[r]|-m X_1 @<6pt>[r]^d@<-6pt>[r]_c X_0, [l]|-e is a groupoid when there is an additional morphism i X_1 ⟶ X_1 satisfying the axioms: * d∘ i = c,   c∘ i=d; * m∘⟨ i, 1_X_1⟩= e ∘ c   and   m∘⟨ 1_X_1 , i⟩= e∘ d. We write (ℂ) for the category of internal categories in ℂ (and internal functors as morphisms), (ℂ) for the category of groupoids in ℂ, and (ℂ) for the category of reflexive graphs in ℂ (with obvious morphisms). An equivalence relation is a special kind of groupoid, where its domain and codomain morphisms are jointly monomorphic; also any reflexive and transitive relation is in particular an internal category. If is a Goursat category, then any reflexive and transitive relation is an equivalence relation or, equivalently, any internal category is a groupoid (Theorem 1 in <cit.>). Then Theorem  <ref>, which could equivalently be stated through the property that () (or the category of reflexive and transitive relations in ) is closed in the category of reflexive relations in under quotients, has an extended counterpart given below. This characterisation leads to the observation that the structural aspects of Goursat categories mainly concern groupoids (rather than equivalence relations). Let ℂ be a regular category. Then the following conditions are equivalent: (i) ℂ is a Goursat category; (ii) (ℂ) is closed in (ℂ) under quotients; (iii) (ℂ) is closed in (ℂ) under quotients. (i) ⇒ (ii) Let X_1@>>[r]^g@<6pt>[d]^c@<-6pt>[d]_d X'_1@<6pt>[d]^c'@<-6pt>[d]_d' X_0 [u]|-e@>>[r]_f X'_0 [u]|-e' be a regular epimorphism (f,g) in (ℂ) (which means that f and g are regular epimorphisms in ), with X_1 @<6pt>[r]^d@<-6pt>[r]_c X_0 [l]|-e a groupoid in ℂ. By Proposition <ref>, there exists a connector between (d) and (c). Let (d), (c), (d') and (c') be the kernel pairs of the arrows d, c, d' and c', respectively. Let λ : (d)→(d') and β : (c)→(c') be the arrows induced by the universal property of kernel pairs (d') and (c'), respectively. By Theorem <ref>, λ and β are regular epimorphisms, so that g((d)) = (d') and g((c)) = (c'). By Proposition <ref> there is then a connector between (d') and (c'), thus X'_1 @<6pt>[r]^d'@<-6pt>[r]_c' X'_0 [l]|-e' is a groupoid (Proposition <ref>). (ii) ⇒ (i) This implication follows from Theorem <ref> and the fact that equivalence relations are in particular groupoids (whose domain and codomain morphisms are jointly monomorphic). (i) ⇒ (iii) This implication follows from (i) ⇒ (ii) and the fact that (ℂ) ≅(ℂ) in a Goursat context, as recalled above. (iii) ⇒ (i) Let (R, r_1, r_2) be an equivalence relation on X, f: X ↠ Y a regular epimorphism and (f(R), t_1, t_2) the regular image of R along f R @->>[r]^g@<.5ex>[d]^r_2@<-.5ex>[d]_r_1 f(R) @<.5ex>[d]^t_2@<-.5ex>[d]_t_1 X @->>[r]^f Y. (f(R), t_1, t_2) is reflexive and symmetric being the image of the equivalence relation R along a regular epimorphism f. By assumption, (f(R), t_1, t_2) is an internal category, thus it is an equivalence relation. It follows that ℂ is a Goursat category (by Theorem <ref>). Observe that Theorem <ref> also implies that (ℂ) and (ℂ) are Goursat categories whenever ℂ is, again thanks to Lemma <ref>, the category (ℂ) obviously being a Goursat category. This simplifies and slightly extends Proposition 4.3 in <cit.>, where the existence of coequalizers in ℂ was assumed. A result analogous to Theorem <ref> holds in the Mal'tsev context: a category ℂ is a Mal'tsev category if and only if (ℂ) (or, equivalently, ()) is closed in (ℂ) under subobjects <cit.>. Together with the comments made before Proposition <ref> we observe the existence of a sort of duality between Mal'tsev categories and Goursat categories. Similar results hold for Mal'tsev categories with respect to monomorphisms and for Goursat categories with respect to regular epimorphisms. 22 bb F. Borceux and D. Bourn, Mal'cev, protomodular, homological and semi-abelian categories, Mathematics and its Applications, 566. Kluwer Academic Publishers, Dordrecht, 2004. xiv+479 pp. ISBN: 1-4020-1961-0 bournD. Bourn, Internal equivalence relations, modular formula and Goursat condition in non-regular context, Technical Report 526 of the L.M.P.A. (2016). bournfib D. Bourn, Mal'cev categories and fibration of pointed objects, Appl. Categ. Structures 4 (1996) 307327. bmaconn D. Bourn and M. Gran, Centrality and connectors in Maltsev categories, Algebra Universalis 48 (2002), no. 3, 309-331. bma D. Bourn and M. Gran, Centrality and normality in protomodular categories, CT2000 Conference (Como). Theory Appl. Categ. 9 (2001/02), 151-165. bm D. Bourn and M. Gran, Categorical aspects of modularity. Galois theory, Hopf algebras, and semiabelian categories, 77100, Fields Inst. Commun., 43, Amer. Math. Soc., Providence, RI, 2004. ckp A. Carboni, G.M. Kelly, M.C. Pedicchio, Some remarks on Mal'tsev and Goursat categories, Appl. Categ. Structures 1 (1993), no. 4, 385-421. CLP A. Carboni, J. Lambek, M.C. Pedicchio, Diagram chasing in Mal'cev categories, J. Pure Appl. Algebra 69 (1991) 271–284. cpp A. Carboni, M.C. Pedicchio, N. Pirovano, Internal graphs and internal groupoids in Mal'cev categories, Category theory 1991 (Montreal, PQ, 1991), 97-109, CMS Conf. Proc., 13, Amer. Math. Soc., Providence, RI, 1992. FM R. Freese and R. McKenzie, Commutator theory for congruence modular varieties, Cambridge University Press, Cambridge (1987). ma M. Gran, Internals categories in Mal'cev categories, Special volume on the occasion of the 60th birthday of Professor Michael Barr (Montreal, QC, 1997). J. Pure Appl. Algebra 143 (1999), no. 1-3, 221-229. ght M. Gran, C. Heunen, and S. Tull, Monoidal methods for groupoids and connectors in categorical algebra, in preparation. mp M. Gran, M. C. Peddichio n-permutables locally finitely presentable categories, Theory Appl. Categ. 8 (2001), 1-15. gro M. Gran and D. Rodelo, A universal construction in Goursat categories, Cah. Topol. Géom. Différ. Catég. 49 (2008), no. 3, 196-208. gr M. Gran and D. Rodelo, A new characterisation of Goursat categories, Appl. Categ. Structures 20 (2012), no. 3, 229-238. grod M. Gran and D. Rodelo, Beck-Chevalley condition and Goursat categories, J. Pure Appl. Algebra, 221 (2017) 2445-2457. hg H.P. Gumm, Geometrical methods in congruence modular algebras, Mem. Amer. Math. Soc. 45 (1983), no. 286, viii+79 pp. hm J. Hagemann and A. Mitschke, On n-permutable congruences, Algebra Universalis 3 (1973), 8-12. JRVdL Z. Janelidze, D. Rodelo and T. Van der Linden, Hagemann's theorem for regular categories, J. Homotopy Relat. Struct. 9(1) (2014) 55-66. Kock A. Kock, Generalized fibre bundles, Lecture Notes in Mathematics 1348, Springer-Verlag (1988) 194-207. Kock2 A. Kock, Fibre bundles in general categories, J. Pure Appl. Algebra, 56 (1989) 233-245. ndt N. Martins-Ferreira, D. Rodelo and T. Van der Linden, An observation on n-permutability, Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 2, 223-230.

http://arxiv.org/abs/1701.08203v1

20170127214640

Cosmology from a Lagrangian formulation for Rastall's theory

gr-qc

1]Renato Vieira dos Santos 2]José A. C. Nogales [1]UFLA - Universidade Federal de Lavras DFI - Departamento de Física, CEP: 37200-000, Lavras, Minas Gerais, Brazil. <renato.santos@dfi.ufla.br> & <econofisico@gmail.com> [2]UFLA - Universidade Federal de Lavras DFI - Departamento de Física, CEP: 37200-000, Lavras, Minas Gerais, Brazil. <jnogales@dfi.ufla.br> Cosmology from a Lagrangian formulation for Rastall's theory [ December 30, 2023 ============================================================ We give a Lagrangian formulation for the theory of Rastall of gravitation. After proposing a Lagrangian density that reproduces the equations of motion postulated by Rastall, we study the cosmological consequences and fit the parameters using recent data from Hubble function H(z). According to two model selection criteria, one based on corrected Akaike Information Criterion (AICc) and another on Bayesian Information Criterion (BIC), known to penalize models with a greater number of parameters, particularly BIC, we obtain some competitive models relative do Λ CDM. In one of these models the cosmological constant is interpreted as having origin in the creation of matter due to time dependent gravitational field, as opposed to the origin in the vacuum energy. ] Keywords. Rastall's theory; Variational Principles; Cosmological Constant. § INTRODUCTION Since the acceleration of the universe was discovered <cit.>, the simplest and most effective explanation to date is to consider the famous cosmological constant created by Einstein almost a century ago <cit.>. Such a constant, originally conceived to describe a static universe <cit.>, is now used to provide the energy necessary to make the universe expand with acceleration in a context of exclusively attractive gravity. Such energy is called dark energy <cit.>. The simplest explanation for the nature of dark energy is that it is the energy of the vacuum, that is, the intrinsic or fundamental energy of a certain volume of “empty” space <cit.>. This energy corresponds, in the theory of General Relativity, to the effect of the cosmological constant, Λ. The observations concerning supernovae leading to the conclusion that the universe is in accelerated expansion are consistent with a very small and positive value for this constant, of the order of 10^-29 g/cm^3 <cit.>. The problem of the cosmological constant <cit.> is that the quantum field theories predict a much larger value for this constant, from the calculation of the energy of the quantum vacuum <cit.>. In fact, in quantum mechanics, particle and antiparticle pairs are constantly being created from the vacuum, and although these pairs exist for an extremely short time before mutually annihilating each other, this process contributes to the vacuum energy, obtaining a value that, depending on the theory that is used, can be 120 orders of magnitude greater than the value mentioned above and necessary to explain the observations <cit.>. This is currently seen as one of the fundamental problems of physics and there is currently no solution for it. Another problem related to the cosmological constant is the so-called problem of cosmic coincidence <cit.>, which consists in the fact that there is an approximate coincidence between the energy density of the vacuum and the density of matter in the present universe. This is particularly strange given that the relative balance between these energies varies rapidly as the universe expands. In fact, in the primordial universe the energy of the vacuum was negligible in comparison to the matter whereas recently the situation was inverted and it is the energy of the vacuum that began to dominate. There is then a relatively short period in the history of the universe where these energy densities are comparable and it seems a strange coincidence that this period is precisely around the present <cit.>. One of the pretensions of this article is to provide a possibility of investigation regarding these problems. The approach chosen is to revisit Rastall's theory of gravitation by providing a formulation of the theory via variational principle <cit.>. Rastall's theory proved resistant to a formulation by a variational principle for about 45 years <cit.>. Starting from the proposed Lagrangian density, we obtain the equations of motion and apply them to cosmology. Considering a homogeneous and isotropic universe and using a perfect fluid as constituent material, we obtain the dynamic equations of cosmology. After fitting data for the Hubble function H(z) in terms of the red shift z and using goodness of fit measures based on information theory (corrected Akaike information criterion, AICc) and Bayesian analysis (Bayesian information criterion, BIC), we will see that several of the proposed models fits the data very well, competing with the standard cosmology model with few extra parameters. Some of these models do not consider the cosmological constant as originally conceived. This result encourages us to attack the problem of the cosmological constant from a conventional point of view of the extended theory of relativity, but whose price to be paid is to admit the possibility of creation of matter in large scales originating from the variation of the curvature of space-time. This article is organized as follows: in section (<ref>) we have a brief review of Rastall's theory followed by a section on its formulation via variational formalism. Section (<ref>) establishes the cosmology equations and in section (<ref>) we use recent data from H(z) to fit several models. The discussion and conclusion in the last two sections conclude the paper. § RASTALL'S THEORY OF GRAVITATION Consider the Einstein equation of general relativity (GR), G_μν=κ T_μν, κ is a constant, G_μν≡ R_μν-1/2g_μνR is the Einstein tensor and T_μν the energy-momentum tensor. The Einstein tensor G_μν has the peculiar property that ∇^νG_μν=0, as we know from the identities of Bianchi <cit.>. This peculiarity of G_μν is unjustifiably extended to the stress-energy tensor, and it is considered to be true that ∇^νT_μν=0. But what can be asserted about the covariant divergence of a arbitrary tensor K_μν is that ∇^νK_μν=λ a_μ, λ is a constant and a_μ is a vector. Several extensions of GR consider the possibility of a violation of ∇^νT_μν=0, among which we can mention the f(R, T) <cit.> and f(R,L_m) <cit.> theories, where R is the Ricci scalar, T is the trace of the energy-momentum tensor, and L_m is the Lagrangian for matter. An older theory involving modified gravity violating ∇^μT_μν=0, a particular case of the f(R,T) theories mentioned above, is the theory of Rastall <cit.>. In this theory, a dependency of the covariant divergence of the stress-energy tensor is postulated with the Ricci scalar and is given by T^μν_;μ=λ R^;ν, where λ is a constant <cit.>. In this way, the field equations in the original formulation of Rastall's theory are R_μν+(κλ-1/2)g_μνR=κ T_μν. General relativity is recovered if λ=0. Another useful parameterization of Rastall's model, this time with the cosmological constant Λ inserted, is as follows: G_μν=κ(T_μν-γ-1/2g_μνT+Λ/κg_μν) and ∇^νT_μν=γ-1/2∇_μT, where T=g^μνT_μν is the trace of the stress-energy tensor and γ=(6κλ-1)/(4κλ-1). The cosmological constant was considered as having geometric origin as a result of the identity of Bianchi ∇^ν(G_μν-Λ g_μν)=0. One of the criticisms of the model of Rastall is that it does not come from a variational principle. One of the objectives of this study is to provide such a principle. From the proposed Lagrangian emerges an equation for the covariant divergence of the energy-momentum tensor that contains an extra term, not present in the original theory. We investigate the effects of this extra term in the context of cosmology, comparing the results with the conventional Rastall's theory. We will see that the term involving the cosmological constant should be corrected in a formulation from a variational principle. § THE LAGRANGIAN We propose the following Lagrangian density, composed of the Einstein-Hilbert Lagrangian L_EH and a Rastall Langrangian L_R: √(-g)L = √(-g)(L_EH+L_R) ≡ 1/2κ√(-g)[R+2Λ+2κL_m] + α/2√(-g)[1/κG_μν+ T̅_μν]g^μν = √(-g)[(1-α)/2κR+Λ/κ+α/2T̅+L_m], where g is the determinant of the metric tensor, L_m is the Lagrangian density due to matter, κ and α are constants and T̅_μν is a arbitrary tensor. Note that the trace of T̅_μν appears explicitly in the Rastall Lagrangian density and that, curiously, this Lagrangian consists essentially of the equations of motion of general relativity. In a system of units where c = G = 1, we can write the action as follows: A=1/16π∫ f(R,T)√(-g)d^4x+∫L_m√(-g)d^4x with κ=8π and f(R,T)=(1-α)R+2Λ+8παT̅. Rastall's theory can be formulated as a particular case of the f(R,T) theory. Formulated in this way, several results for general f(R,T) theory can be used <cit.>. Using the well-known facts * δ(√(-g))=-1/2g_μν√(-g)δ g^μν, * δ(√(-g)R)=√(-g)(R_μν-1/2Rg_μν)δ g^μν, * δ(√(-g)T̅)=√(-g)(T̅_μν+θ̅_μν-1/2T̅g_μν)δ g^μν with θ̅_μν≡ g^αβδT̅_αβ/δ g^μν and * δ(√(-g)L_m)=-1/2T_μν√(-g)g^μν with T_μν=-2/√(-g)δ(√(-g)L_m)/δ g^μν, the field equations are G_μν=κ T^(eff)_μν, with T^(eff)_μν≡ T_μν-α/2(1-α)(2T̅_μν+2θ̅_μν-T̅g_μν)+Λ/κ(1-α)g_μν. For α = 0 we get general relativity. Considering ∇^νT^(eff)_μν=0: ∇^νT_μν=α/2(α-1)(∇_μT̅-2∇^νθ̅_μν-2∇^νT̅_μν). We see from Eqs. (<ref>) and (<ref>) that to obtain the theory of Rastall starting from the Lagrangian given by (<ref>), we must have * α/2(α-1)=γ-1/2 and * ∇_μT̅-2∇^νθ̅_μν-2∇^νT̅_μν=∇_μT, which implies α=γ-1/γ-2. and θ̅_μν=g_μνΛ̅-T̅_μν respectively, with 2Λ̅≡T̅-T and T̅_μν=T_μν+2Λ̅g_μν. Finally, the equations of motion are G_μν = κ[T_μν-γ-1/2g_μνT+(2-γ)g_μνΛ/κ], and ∇^νT_μν=γ-1/2∇_μT. Comparing equations (<ref>) and (<ref>) we see that Rastall's theory in the original version and in the version derived from the variational principle are different with respect to the term with the cosmological constant. In Rastall's theory as proposed here, the cosmological constant as geometry fully on the left side of the field equations can not be interpreted as equivalent to matter-energy (vacuum energy) on the right side of the field equations as is commonly done in general relativity. To better understand the point in question, we write equation (<ref>) as follows: G_μν-Λ g_μν=κ[T_μν-γ-1/2g_μνT+(1-γ)g_μνΛ/κ]. In Eq. (<ref>) we divide the factor that corresponds to the cosmological constant into two parts: one associated with geometry and another with matter-energy. The degree to which Rastall's theory departs from general relativity is measured by how much γ is different from 1, that is, it is measured by the intensity of the cosmological constant due to matter-energy only plus the term corresponding to the trace of the energy-moment tensor. For future discrimination between these two effects, we rewrite Eq. (<ref>) as follows: G_μν-Λ g_μν=κ[T_μν-γ-1/2g_μνT+(1-γ)g_μνΛ̂/κ], where the cosmological constant due to matter-energy creation is indicated by Λ̂. § COSMOLOGY We will consider the homogeneous and isotropic Friedman-Lemaitre-Robertson-Walker (FLRW) universe with scale factor a, ds^2=dt^2-a^2(t)[dr^2/1+Kr^2+r^2(dθ^2+sin^2θ dϕ^2)], where K=1, 0, -1 is the spatial curvature constant. For a perfect fluid given by T_μν=(ρ+p)u_μu_ν-g_μνp with pressure p, density of energy-matter ρ and u^μu_μ=1, we have T=ρ-3p and the cosmology equations from (<ref>), (<ref>) and (<ref>) are 3ȧ^2/a^2=κ[ρ-γ-1/2T]+Λ-Λ̂(γ-1)-3K/a^2, 6ä/a=2[Λ-(γ-1)Λ̂]-κ(6p+γ T) Eqs. (<ref>) and (<ref>) are the Friedmann's and acceleration equations, respectively. From Eq. (<ref>) we have ρ̇+3ȧ/a(ρ+p)=γ-1/2Ṫ. If we use the equation of state p=ωρ, the cosmological equations (<ref>), (<ref>) and (<ref>) are 3ȧ^2/a^2=κρ[1-γ-1/2(1-3ω)]+Λ+Λ̂(1-γ)-3K/a^2, 6ä/a=2[Λ-(γ-1)Λ̂]-κρ[6ω+γ(1-3ω)] and ρ̇+3ȧ/a(1+ω)ρ=γ-1/2(1-3ω)ρ̇. The solution of Eq. (<ref>) is ρ=ρ_0(a_0/a)^A with A=6(1+ω)/3-3ω+γ(3ω-1), ρ_0 and a_0 are constants. Considering dust matter (ω=0), radiation (ω=1/3) and cosmological constant (ω=-1), only for dust matter the expression of ρ is modified by the presence of the constant γ. In this case, ρ∼ a^-6/3-γ. By inserting Eq. (<ref>) into Eq. (<ref>), the acceleration equation becomes 6ä/a=κρ_0[3ω(γ-2)-γ]a^-A+2Λ-2(γ-1)Λ̂. We see from Eq. (<ref>) that it is possible ä>0 for Λ=Λ̂=0. The conditions for ä>0 are: ω <1/3andγ <6 ω/3 ω -1 and ω >1/3andγ >6 ω/3 ω -1. §.§ Cosmological parameters Observing supernovae, astronomers measure their brightness and redshift. From the brightness we deduce its distance, and from the redshift, the scale factor a(t) at that time. We can express the results of data reduction in terms of the Hubble parameter H_0 and the dimensionless parameter Ω_0. We can write the Friedmann equation according to these conventions. Hubble parameter is defined as H≡ȧ/a, which is a function of time with present value denoted by H_0. The density parameter is defined as Ω≡κρ/3H^2, which is also a function of time. With ρ_0 and H_0, its present value is Ω_0. Ω=1 corresponds to K=0, Ω<1 corresponds to K<0 and Ω>1 corresponds to K>0. The redshift parameter is defined as 1+z=y≡a(t_0)/a(t)≡1/a(t). At present, z=0 and therefore y=1. It is convenient to set a to unity at the present time, t_0 and a(t_0)≡1. According to this notation, Friedmann's equation becomes H^2+K/a^2=H^2Ω. Because K is a constant, we express it in terms of the present values of the Hubble parameter and the density parameter, K=H_0^2(Ω_0-1). Supposing that Ω_0 pertains to energy in three possible forms, we decompose it into Ω_0=Ω_m+Ω_r+Ω_Λ̃, where Ω_m pertains to matter, Ω_r pertains to radiation, and Ω_Λ̃(≡Ω_Λ+Ω_Λ̂) pertains to vacuum and “creation of matter”, all present values. Each component of energy density among these depends on the scale factor a differently: ρ∝ a^-A; the density ρ as a function of a is expressed as κ/3ρ = H_0^2[Ω_m a^-6/3-γ+Ω_r a^-4+Ω_Λ+(1-γ)Ω_Λ̂] = H_0^2[Ω_m y^6/3-γ+Ω_r y^4+Ω_Λ+(1-γ)Ω_Λ̂] The Friedmann's equation in terms of currently observable parameters, H_0, Ω_m, Ω_r, Ω_Λ and Ω_Λ̂ becomes ȧ^2/a^2 = - H_0^2(Ω_0-1)/a^2 + H_0^2[3-γ/2Ω_m a^-6/3-γ+Ω_r a^-4+Ω_Λ+(1-γ)Ω_Λ̂] or, equivalently: (H(z)/H_0)^2 = 3-γ/2Ω_m(1+z)^6/3-γ + Ω_r(1+z)^4+Ω_Λ+(1-γ)Ω_Λ̂ - (Ω_0-1)(1+z)^2. In the next section we will use the Hubble function expressed in Eq. (<ref>) to fit the observable parameters using recently obtained experimental data for H(z). § FITTING DATA Using data from <cit.> we fit cosmological parameters present in Eq. (<ref>). Such data are shown below and are plotted in Fig. (<ref>) along with the fitted curve of the standard model Λ CDM. z 1cH(z) 2c Continued from previous column z 1cH(z) 2rContinued on next column 2rConcluded cc 0.07 69 ± 19.6 0.09 69 ± 12 0.12 68.6 ± 26.2 0.17 83 ± 8 0.179 75 ± 4 0.199 75 ± 5 0.2 72.9 ± 29.6 0.24 79.69 ± 2.65 0.27 77 ± 14 0.28 88.8 ± 36.6 0.35 82.1 ± 4.9 0.35 84.4 ± 7 0.352 83 ± 14 0.3802 83 ± 13.5 0.4 95 ± 17 0.4004 77 ± 10.2 0.4247 87.1 ± 11.2 0.43 86.45 ± 3.68 0.44 82.6 ± 7.8 0.4497 92.8 ± 12.9 0.4783 80.9 ± 9 0.48 97 ± 62 0.57 92.4 ± 4.5 0.593 104 ± 13 0.6 87.9 ± 6.1 0.68 92 ± 8 0.73 97.3 ± 7 0.781 105 ± 12 0.875 125 ± 17 0.88 90 ± 40 0.9 117 ± 23 1.037 154 ± 20 1.3 168 ± 17 1.363 160 ± 33.6 1.43 177 ± 18 1.53 140 ± 14 1.75 202 ± 40 1.965 186.5 ± 50.4 2.3 224 ± 8 2.34 222 ± 7 2.36 226 ± 8 We will fit the H(z) data using some cosmological models proposed below, many of them from equation (<ref>) and two from purely phenomenological origin proposed in <cit.>. These two purely phenomenological models are oscillatory models that best fit the data proposed in <cit.>. They are included here for purposes of comparison with the models motivated by equation (<ref>) and for an update of the conclusions obtained in <cit.> with more recent data available. Table (<ref>) shows 12 models studied plus Λ CDM model considered as benchmark. Of the 12 proposed models, 10 are particular cases of equation (<ref>) and 2 are the oscillatory models mentioned. For example, the most complex model is the model (H) where Ω_m represents baryonic matter, Ω_K represents “matter associated with the curvature of spacetime,” Ω_r is associated with radiation and Ω_Λ̂ represents the matter associated with the cosmological constant due to matter creation. Another example is the model (L) where we leave ω as free parameter and we made Ω≡Ω_x = 1. The table (<ref>) is arranged in order to display the models in ascending order of “discrepancy from χ^2/d.o.f-1”. This preliminary quality of fit was measured in terms of the χ^2 analysis, where χ^2 is defined by χ^2(parameters)=∑_i=1^41[H_mod(parameters;z_i)-H_obs(z_i)]^2/σ^2(z_i), where H_mod is the predicted value for the Hubble parameter in the assumed model, H_obs is the observed value, σ corresponds to 1σ uncertainty, and the summation is over the 41 observational H(z) data points at redshift z_i. The parameters are estimated by the minimization of χ^2. What actually orders the models with respect to their quality of fit is χ^2 divided by the degrees of freedom (d.o.f). Values of χ^2/d.o.f close to 1 (χ^2/d.o.f≈ 1) represent better fittings <cit.>. All models in the table (<ref>) that do not have Λ CDM in their names are models based on Rastall's theory and therefore in equation (<ref>). This fact is evidenced by the presence of γ in the equations that define them. In the next section we will consider better strategies for selecting models based on information criteria and Bayesian analysis. §.§ Model selection Some model selection methods are defined in terms of an appropriate information criterion, a mechanism well-founded theoretically that uses data to give a model a score, which leads to a ranked list of candidate models from the best to the worst <cit.>. Two of the most important of these information criteria are the Akaike information criterion (AIC) and Bayesian information criterion (BIC). The general formulas for models with vector parameter θ̂ are AIC=-2logL(θ̂|data)+2p and BIC=-2logL(θ̂|data)+log(n)p, where logL(θ̂|data) is the log-likelihood of the model, p is the number of parameters and n is the sample size. For models with normally distributed residuals, logL=-n/2logχ^2 with χ^2 given by Eq. (<ref>). The AIC and BIC criteria act as a penalized log-likelihood criterion, providing a balance between good fit (high value of log-likelihood) and complexity (complex models are penalized more that simple ones). These criteria punishes the models for being too complex in the sense of containing many parameters. The models with the lowest AIC and BIC scores are selected. It is useful to briefly mention how these criteria behave in relation to their properties of consistency and efficiency. The comparison is based in the study of the penalty applied to the maximized log-likelihood value in a framework with increasing sample size. If we make the assumption that there are one true model that generates the data and that this model is one of the candidate models, we would want the model selection method to identify this true model. This is related to consistency. From <cit.> (our emphasis): A model selection method is weakly consistent if, with probability tending to one as the sample size tends to infinity, the selection method is able to select the true model from the candidate models. Strong consistency is obtained when the selection of the true model happens almost surely. When we do not assume that the true model is present among the proposed models, as we believe to be the case in most practical situations, including cosmology, we can assume that a candidate model is the closest to the true model in the Kullback-Leibler distance sense (see bellow) <cit.>. In this case <cit.>, ⋯ we can state weak consistency as the property that, with probability tending to one [as the sample size tends to infinity], the model selection method picks such a closest model. From the foregoing we see that a strongly desirable condition for the use of a consistent criterion is to have a sufficiently large sample size. This fact, among others that will be mentioned below, makes us less likely to take the results very strictly for BIC, at least with the amount of data currently available for H(z). Another desirable property of selection methods, efficiency, can be roughly described by <cit.>: ⋯ [W]e might want an information criterion to posses is that it behaves “almost as well”, in terms of mean square error, or squared error loss. In <cit.> it is proved that ⋯ AIC is not strongly consistent, though it is efficient, while the opposite is true for the BIC. Given the above assertion, it is useful to question whether a model selection method that contemplates both desirable properties, BIC consistency with AIC efficiency, exists. The answer is not, as proven in <cit.>. This means that when selecting a method to choose the best models, we are necessarily making a choice between consistency and efficiency. The optimal or most convenient choice is conditioned to the size of the sample n as well as to the world view of the researcher, as will be discussed in the section (<ref>). We have seen that for BIC, the properties of consistency are defined (rigorously in <cit.>) in terms of asymptotic properties related to sample size n. If this size is not “large enough”, the practical validity of the theorems weaken. For AIC something analogous occurs. In its derivation, an approximation is made that is valid for large sample sizes. A rule of thumb was proposed in <cit.> to define what would be a “sample size not large enough”. Such a rule states that if n /p≤ 40, the results for the AIC method may be less accurate. To mitigate this problem, a correction was proposed for the Akaike method, the so-called corrected Akaike information criterion, AICc, whose formula is given, under certain assumptions, by <cit.> AICc=AIC+2(p+1)(p+2)/n-p-2. AICc is essentially AIC with a greater penalty for extra parameters. Using AIC, instead of AICc, when n is not many times larger than p^2, increases the probability of selecting models that have too many parameters, i.e., of overfitting. Due to the fact that we are using data with n = 41, we will use AICc as the criterion for selecting models based on Akaike. AIC, AICc and BIC (collectively dubbed IC, from Information Criteria) are absolute numbers that have no significance when evaluated in isolation. What matters is the value of the difference of IC from two different models. In order to better characterize this fact, it is common to calculate Δ_i IC ≡ IC_i-IC_ *, where we calculate the difference of IC values for two models, i and *, where the model * is established as the reference model. In this work we consider as reference model the standard model of cosmology, the Λ CDM model. These Δ_i IC are easy to interpret and allow a quick strength-of-evidence comparison of candidate models. According to <cit.>: Some simple rules of thumb are often useful in assessing the relative merits of models in the set: Models having Δ_i≤ 2 have substantial support (evidence), those in which 4≤Δ_i≤7 have considerably less support, and models having Δ_i>10 have essentially no support. Another useful tool for providing weights of evidence for each of the R=11 (phenomenological models were excluded) models considered in the analysis are the Akaike weights w_i given by <cit.> w_i=e^-Δ_i/2/∑_r=1^Re^-Δ_r/2. where Δ_i represents Δ_i AICc or Δ_i BIC. The w_i from AICc can be interpreted as the probability of model i being, in fact, the best model in the sense of the Kulback-Leibler's (K-L) distance <cit.>. K-L information I(f,g) is the information lost when model g(x,θ) is used to approximate the “full reality or truth”, f; this is defined for continuous functions as the integral I(f,g)=∫ f(x)log(f(x)/g(x|θ))dx. The best model loses the minimum amount of information possible and Akaike's criterion seeks precisely this by minimizing the distance of K-L. All our data analysis results are shown in tables (<ref>), (<ref>) and (<ref>), the latter in the appendix, and in figure (<ref>). Table (<ref>) shows all models with their parametrizations, table (<ref>) presents the AICc and BIC measurements and their variants, and figure (<ref>) presents a column chart to facilitate the general understanding of the hierarchy of models with regard to the rule of thumb. For the calculation of w_i in table (<ref>) we do not consider the weights relative to the phenomenological models (B) and (C). In the next section we will discuss the results. § DISCUSSION The most important result can be seen in Fig. (<ref>) where almost all proposed models pass the test when we consider the AICc criterion. The most demanding criterion, BIC, disapproves almost all, except the models (B), (G) and (I). Model (B) represents one of the phenomenological models, which by the way did very well again with the most recent H(z) data. The model (I) is very similar to the standard model, differing only by the inclusion of radiation. Some curves for a(t) for some models are shown in Fig. (<ref>), where the differential equations were solved numerically using the data of table (<ref>). The model (G), a very good model according to both AICc and BIC, is analogous to the standard Λ CDM model but incorporates the constant γ≠ 1 (see table <ref>). Consider it seriously as well as all models with γ≠ 1 implies the real possibility of matter being created with the temporal variation of the gravitational field <cit.>. Exactly because it is analogous to the standard model, it has the drawback of inheriting all its conceptual problems related to the cosmological constant and also adding the problem of the observational verification of the creation of matter. Models approved by the rule of thumb with BIC tend to be very restrictive. We will now consider some interesting possibilities involving AICc. Further discussion as to the appropriateness of these criteria will be made at the conclusion. Models considered competitive relative to Λ CDM but that differ significantly from it (model (I) for example is only Λ CDM with radiation) are models (D), (E), (K), (F) and (J). All these models satisfies Δ AICc≤ 2. Model (D) is the Rastall's version for the model Λ CDM with radiation [model (I)], and it better describes the observed data according to AICc. An undesirable property (in principle) of this model when adjusted to the data used is its negative value for Ω_r and its large standard error (see table (<ref>)). The use of more complete data other than H(z) may shed some light on this. In general, all models suffer from a high standard error in determining the constant γ, except model (K) where the problem is less severe. Fig. (<ref>) shows that such a model predicts a slower expansion for a(t) when compared to Λ CDM. This model also presents an excellent value for the root mean square error (RMSE) <cit.>. In addition, this model is conceptually particularly attractive because it involves only the cosmological constant that originates in the creation of matter, Λ̂. If we could disassociate the accelerated expansion of the universe with a cosmological constant that originates from the energy of the vacuum, naturally some problems related to such a constant would be alleviated, if not fully resolved. Of course in this case we would only be switching the problem since we would have to observe the creation of matter. But perhaps this can be a fruitful exchange, since the problem of the cosmological constant has been resistant to the solution for several decades. An interesting parallel can be drawn between the model (G), which is the manifestation a la Rastall of the standard model, and the model (K), the best manifestation of our desire to get away from the problem of the cosmological constant. It is not possible to obtain the model (K) starting from Rastall's theory as originally formulated. Model (G) is the closest model (from the standpoint of the standard model Λ CDM) we would get that originates from Rastall's original theory, not obtained from a variational principle. The model (K) is the model that best characterizes the original aspect of Rastall's theory as proposed here, originated from a variational principle. We conclude that what allowed us to explain the accelerated expansion of the universe without using a cosmological constant from vacuum energy was the variational formulation of Rastall's theory. § CONCLUSION In this paper we propose a formulation of Rastall's theory based on a variational principle. This task, simplified by recent theoretical developments in alternative models of general relativity, has allowed the proposal of a hierarchy of models not yet sufficiently studied. This hierarchy of models was built using model selection techniques based on information theory and Bayesian analysis. These criteria are complementary, although they have conflicting properties and it is difficult to establish which one should be preferred. Preference criteria in this choice are subjective and closely connected with the modeler's world view. Akaike information criterion is a class of model selection tools that provide the best predictive accuracy. Bayesian information criterion is a class of confirmation/falsification tools that are consistent. Which of them should be used? When we mention two of the desirable properties of any model classification method, namely, efficiency and consistency, we have seen that such properties are excludents: a method that incorporates both is not possible. This state of affairs is best understood intuitively if we conceive the fact that such properties correspond to different world views. Let's characterize these world views as follows: * Researchers with world view a believe that a very complex model, perhaps inaccessible to complete human understanding, produces the data. Because of this unknowability, they assume that p≫ n and do not expect candidate models to correspond exactly to reality. The most they can hope for is selecting the model for better forecasting; * For researchers with world view b, a relatively simple process, whose underlying objective reality is accessible, produces the data. The sample size of data, n, greatly exceeds the model parameter space (p≪ n). One of the candidate models fitted to the data is actually equivalent to the true model that produced the data and so is the objective reality. The task of these researchers is more associated with model confirmation/falsification. The model that best fits the data must be interpreted in different ways according to the a or b view assumed, implicitly or explicitly. In vision a, we will never find the truth, we can only find the model that maximizes predictive accuracy. In vision b, we really expect to find the correct model that describes objective reality as sample size grows. Akaike methods are appropriate for situations analogous to the world view a, and Bayesian methods are appropriate for situations similar to the world view b. It is the view of the authors of this paper, that the history and philosophy of science seem to support, that view a is more appropriate and realistic in physics and cosmology. Because of this we do not believe it is prudent to restrict the possibilities too much using BIC. This view stimulates and encourages us to further study the properties of models that contemplate the possibility of explaining the accelerated expansion of the universe via temporal variation of the gravitational field and the corresponding creation of matter, as in model (K). § APPENDIX In this appendix we present in table (<ref>) all the models fitted with the values of their respective parameters as well as the value of the root mean square RMSE. pnas.bst

http://arxiv.org/abs/1701.07567v2

20170126033850

Cascaded cold atomic ensembles in a diamond configuration as a spectrally entangled multiphoton source

quant-ph

sappyjen@gmail.com Institute of Physics, Academia Sinica, Taipei 11529, Taiwan We theoretically investigate the spectral entanglement of a multiphoton source generated from the cascade emissions in the cascaded cold atomic ensembles. This photon source is highly directional, guaranteed under the four-wave mixing condition, and is also highly frequency-correlated due to finite driving pulse durations and superradiant decay constants. We utilize Schmidt decomposition to study the bipartite entanglement of the biphoton states projected from the multiphoton ones. This entropy of entanglement can be manipulated by controlling the driving parameters and superradiant decay rates. Moreover the projected biphoton states in the cascaded scheme can have larger entanglement than the one produced from only one atomic ensemble, which results from larger capacity in multipartite entanglement. This cascaded scheme enables a multiphoton source useful in quantum information processing. It also allows for potential applications in multimode quantum communication and spectral shaping of high-dimensional continuous frequency entanglement. Cascaded cold atomic ensembles in a diamond configuration as a spectrally entangled multiphoton source H. H. Jen December 30, 2023 ====================================================================================================== § INTRODUCTION Quantum computation and quantum information processing <cit.> promise to outperform classical implementations for efficient algorithm, secure communication <cit.>, and genuine teleportation of quantum states <cit.>. This superiority of quantumness even envisions a quantum network or quantum internet <cit.> which links various quantum systems to process tasks that are intractable in classical regimes. These quantum systems include, to name a few, photonic qubits, trapped ions, atomic ensembles, and solid state systems <cit.>, in which unfortunately both strengths and weaknesses coexist <cit.>. The fact that there is no prefect quantum system (high efficiency, long coherence time, strong coupling, and scalability, for example) demands an integrability of the well-controlled interfaces between these quantum systems. A good quantum interface <cit.> involves an efficient generation, distribution, and storage of quantum information. One manifestation of these functionalities is long-distance quantum communication <cit.> using cold atomic ensembles for a quantum repeater <cit.>. The building blocks for this quantum repeater protocol therefore rely on a generation of light-matter entanglement <cit.> and storage of it <cit.> using Λ-type atomic configurations, making the atomic ensemble (mostly alkali metals) a good candidate for quantum network. However one drawback for the fiber-based quantum information transmission is the attenuation loss for D line transitions in alkali metals. To reach a minimal-loss optical fiber transmission, telecommunication (telecom) bandwidth from diamond-type atomic configurations <cit.> serves the purpose and allows for an optimal operation in long-distance quantum communication. In addition to the advantage of the telecom bandwidth in fiber transmission, continuous frequency entanglement of cascade emissions from diamond-type atomic transitions can be generated <cit.> and spectrally shaped <cit.> to create a highly entangled biphoton source. This high communication capacity in continuous variables <cit.> is also present in various degrees of freedom, for example, the transverse momentum <cit.>, space <cit.>, time <cit.>, and orbital angular momenta of light <cit.>. In the perspective of spectral shaping, in Ref. <cit.>, we have investigated the frequency-entangled two-photon state in the scheme of multiplexed cold atomic ensembles. This two-photon state is generated from the cascade emissions in the diamond-type atomic configuration. Its upper and lower transitions (also denoted as signal and idler) are respectively in the telecom and infrared bandwidths. In the multiplexed scheme, a multiple common excitation pulses are applied simultaneously to the respective atomic ensembles along with individually controlled frequency shifters for the cascade emissions. Under the condition of weak excitations as in the quantum repeater protocol <cit.>, we effectively create a biphoton state with additive spectral functions where individual frequency shifts for the signal and idler photons can be manipulated. Therefore the spectral property of this biphoton state can be shaped in either modifying the central frequencies or controlling the phases of the photons <cit.>. We further analyze the entropy of entanglement of the multiplexed two-photon source by Schmidt decomposition, which grows as the number of multiplexed ensembles increases. Here in contrast we propose to generate a multiphoton source out of the cascade emissions from the cascaded cold atomic ensembles. We take a two-photon source of quantum-correlated signal and idler photons emitted from a diamond-type atomic configuration, as an initial seed for multiphoton generation in the cascaded atomic ensembles. This is motivated by the multiphoton generation from spontaneous parametric down conversion (SPDC) where multiple cascaded nonlinear crystals are pumped sequentially <cit.>. Similar to the sequential pumping scheme in SPDC, we can take the infrared (telecom) photon of the biphoton source along with a telecom (infrared) driving field to generate another spontaneously emitted cascade emissions satisfying the four-wave mixing condition. In this way we enable a k-photon source with m infrared and n telecom photons fulfilling k=m+n. Since the last cascaded atomic ensemble always produce two photons in the infrared and telecom bandwidths respectively, at least one infrared or telecom photon is generated. However if the infrared and telecom photons can be converted back and forth with each other <cit.>, we can have arbitrary m and n. Furthermore, we investigate its entanglement property in continuous frequency spaces. Our studies allow for an alternative setting for multiphoton generation from cold atomic ensembles, and demonstrate spectral shaping to control the entanglement of this multiphoton source <cit.>, advancing the development of multimode quantum communication. In this paper, we first discuss the Hamiltonian and four-wave mixing condition in Sec. II, and propose to generate a multiphoton source in the scheme of cascaded atomic ensembles in Sec. III. Then in Sec. IV we study the bipartite entanglement properties of the multiphoton source, and we conclude in Sec. V. § CASCADE EMISSIONS AND FOUR-WAVE MIXING CONDITION We consider a Rb atomic ensemble (AE) with a diamond-type configuration shown in the inset of Fig. <ref>(a). The correlated cascade emissions of signal and idler photons (â_s(i)^† as shorthands for â_k_m,λ_m^† below) are created by pumping the atomic system with two classical fields. With dipole approximation of light-matter interactions and rotating-wave approximation (RWA) <cit.>, we express the Hamiltonian in the interaction picture using the same notation of <cit.>, V_ I = -∑_m=1,2Δ_m∑_μ=1^N|m⟩_μ⟨ m|-∑_m=a,b(Ω_m/2P̂_m^†+ h.c.) -i∑_m=s,i{∑__̨m,λ_mg_mâ__̨m,λ_mQ̂_m^† e^-iΔω_m t- h.c.}, where we set ħ=1 for simplicity, and denote λ_m as the polarizations of photons. The collective dipole operators are defined as P̂_a^†≡∑_μ|1⟩_μ⟨ 0|e^i_̨a·_̊μ, P̂_b^†≡∑_μ|2⟩_μ⟨ 1|e^i_̨b·_̊μ, Q̂_s^†≡∑_μ|2⟩_μ⟨ 3|e^i_̨s·_̊μ, and Q̂_i^†≡∑_μ|3⟩_μ⟨ 0|e^i_̨i·_̊μ, with Rabi frequencies Ω_a(b) for two pump fields. Central frequencies and wavevectors of these four fields are ω_a(b),s(i) and _̨a(b),s(i) respectively. Signal and idler photon coupling constants are g_s(i) where we have absorbed (ϵ__̨m,λ_m·d̂_m^*) into g_s(i) for concise expressions. ϵ__̨m,λ_m is the polarization direction of the quantized bosonic fields â__̨m,λ_m, and d̂_m is the unit direction of the dipole operators. The detunings are Δ_1=ω_a-ω_1 and Δ_2=ω_a+ω_b-ω_2, and for later convenience we define Δω_s≡ω_s-ω_2+ω_3-Δ_2 and Δω_i≡ω_i-ω_3 with the atomic level energies ω_1,2,3. The upper level |2⟩ allows for a telecom wavelength within 1.3-1.5 μm <cit.> if 6S_1/2, 7S_1/2, or 4D_3/2(5/2) levels are considered. With the Hamiltonian of Eq. (<ref>), we can construct self-consistent Schrödinger equations assuming there is only one atomic excitation <cit.>. This is valid when weak and large detuned excitation pulses are considered, satisfying |Δ_1,2|≫|Ω_a,b|. After adiabatically eliminating the atomic levels |1⟩ and |2⟩ in the coupled Schrödinger equations, we derive the probability amplitude of the biphoton state |1__̨s,1__̨i⟩<cit.>, D_s,i(t) = g_i^∗g_s^∗∑_μ=1^Ne^iΔ·̨_̊μ∫_-∞^t∫_-∞^t^'dt^''dt^' e^iΔω_it^'e^iΔω_st^'' × b(t^'')e^(-Γ_3^ N/2+iδω_i)(t^'-t^''), where b(t)=Ω_a(t)Ω_b(t)/(4Δ_1Δ_2), resulting from the required adiabatic conditions for driving pulses. Four-wave mixing (FWM) condition ∑_μ=1^N e^iΔ·̨_̊μ represents the phase-matching condition when Δ$̨=_̨a+_̨b-_̨s-_̨i→0in the limit of large number of atomsN, and guarantees to create a highly correlated photon pair. Furthermore the idler photon is superradiant <cit.> thatΓ_3^ N=(Nμ̅+1)Γ_3quantifies a superradiant decay rate in the level of|3⟩with an intrinsic decay rateΓ_3. The geometrical constantμ̅<cit.> relates to the shape of the AE, and the associated collective frequency shift <cit.> is denoted asδω_i. This reflects the nature of collective radiation <cit.> due to induced dipole-dipole interactions in the dissipation <cit.>. Note that a complete description of the collective frequency shift requires non-RWA terms in the Hamiltonian <cit.>. This biphoton state|1__̨s,1__̨i⟩has two main features. One is the strong directionality of the emitted photons, which is determined by FWM. In thermodynamic limit of AE, we haveΔ$̨=0. If counter-propagating excitations <cit.> are used as in Fig. <ref>(a) with excitation and photon emission angles denoted as θ_a,b,s,i, we can derive the condition for the emitted angles as λ_b/λ_a(cosθ_a-cosθ_i)=cosθ_b-cosθ_s, λ_b/λ_a(sinθ_a+sinθ_i)=sinθ_b+sinθ_s, where λ_a,b are the wavelengths of the excitation fields, and we have assumed |_̨i|=|_̨a| and |_̨s|=|_̨b|. Since θ_a,b are given when the excitations are applied, θ_s,i can be decided from Eq. (<ref>). To have some estimate for the angles, we have (θ_i,θ_s) = (4.9^∘,9.9^∘) when we set (θ_a,θ_b) =(5^∘,10^∘) and λ_b/λ_a=2. For the same ratio of the wavelengths, we have (θ_i,θ_s) = (7.9^∘,13.9^∘) when we set (θ_a,θ_b) =(4^∘,10^∘). If in radians that θ_a,b,s,i≪1 and setting θ_b=2θ_a, we derive θ_s (≈θ_b) ≈2θ_i, which indicates that the signal and idler photons are emitted with corresponding excitation angles of θ_b and θ_a respectively, and they follow the directions almost tangent to the long axis of AE. This direction is preferential in experiments, which allows for strong light-matter couplings. The other feature is that the biphoton state is probabilistic. According to Eq. (<ref>), we can express the complete and normalized state as |Ψ⟩=1/√(1+|D_s,i|^2)|0^⊗ N⟩[| vac⟩ + D_s,i|1__̨s,1__̨i⟩], which involves N atomic ground with vacuum and biphoton states respectively. Since |D_s,i|≪1, most of the time AE does not generate any photons. Therefore experimentally it requires repeated excitations until the biphoton state is created, which can be confirmed via photon detections. The degree of correlation of the photons (second-order correlation function for example) can be measured as well by a conditional detection <cit.> of the idler photon after the signal one is detected. Below and throughout the paper, we focus on the effective biphoton and multiphoton states, therefore the normalization for these effective states is neglected. Due to their probabilistic feature, these effective states can be confirmed only via conditional detections or post selections. Specifically we use normalized Gaussian pulses where Ω_a(b)(t)=[√(π)τ_a(b)]^-1Ω̃_a(b)e^-t^2/τ_a(b)^2 with the pulse areas Ω̃_a,b and pulse widths τ_a(b). Considering the long time limit that D_s,i(t→∞), we derive the probability amplitude D_si after inserting the Gaussian forms into Eq. (<ref>) and redefining Δω_s(i) as Δω_s-δω_i and Δω_i+δω_i respectively, which reads D_si(Δω_s,Δω_i)=Ω̃_aΩ̃_b g_s^*g_i^*∑_μ=1^N e^iΔ·̨_̊μ/4Δ_1Δ_2√(2π)√(τ_a^2+τ_b^2)f(ω_s,ω_i). The spectral function of the biphoton state is f(ω_s,ω_i)=e^-(Δω_s+Δω_i)^2τ_eff^2/8/Γ_3^ N/2-iΔω_i, where τ_eff≡√(2)τ_aτ_b/√(τ_a^2+τ_b^2). This spectrally correlated biphoton state involves a Gaussian weighting modulated by a Lorentzian. The spectral function is most significant when Δω_s≈-Δω_i within the spectral range of 1/τ_eff. More entangled biphoton state can be made with an increasing optical density of the AE or longer pulses <cit.>, making f(ω_s,ω_i) less factorizable aligning on the axis of Δω_s=-Δω_i<cit.>. In the next section we propose to use this two-photon source as a seed to create multiphoton states in the scheme of cascaded AEs. § MULTIPHOTON STATES FROM THE CASCADE EMISSIONS IN THE CASCADED SCHEME Before we investigate the multiphoton states generated from the cascaded atomic ensembles, we note that throughout the paper, we focus only on the effective pure states generation. In general for an open quantum system, the system becomes mixed states inevitably <cit.> due to the interactions with the reservoir or the imperfections in experiments. The mixed states can be interpreted as statistical mixtures of pure states, which can be expressed as a density operator, ρ̂=∑_k p_k|ψ_k⟩⟨ψ_k| with the constraints of ∑_kp_k=1 and 0<p_k≤1. ρ̂ becomes a pure state when one of p_k's is unity, therefore this density operator shows a general representation of any quantum system. For example of the spontaneous emission process in a two-level atom (|0⟩ and |1⟩), it acts like an amplitude damping <cit.> to the atomic states. In a description of generalized amplitude damping, the atom evolves to the stationary mixed state ρ_∞=p|0⟩⟨ 0|+(1-p)|1⟩⟨ 1| with some probability p. Similarly, ρ_∞ can describe a loss of photon due to attenuation or decoherence from environment if |0⟩ and |1⟩ are denoted as vacuum and one-photon states respectively. In general, a spontaneous emission is a random process <cit.> in time and space, where the emitted direction has a uniformly dΩ̂/(4π) distribution in a solid angle of dΩ̂=sinθ dθ dϕ in spherical coordinates. In contrast to this random process in space, the biphoton state from the cascade emissions in a diamond configuration under FWM condition is highly directional and correlated, which thus makes the effective state |1__̨s,1__̨i⟩ valid if photon loss or other noisy channels that deteriorate its fidelity can be neglected. Although this limits our investigation to the pure states, we note that an entanglement distillation or purification procedure <cit.> can be applied to the mixed entangled states by local operations and classical communication. Therefore the pure state picture we focus here can still give insights to quantum information processing with the effective multiphoton state presented in this work. Other discussions of inseparability criterion, bound entanglement, or multipartite entanglement for mixed states can be referred to Refs. <cit.>. For measuring entanglement, it does not imply Bell nonlocality except for pure states, and quantifying entanglement for bipartite mixed state involves various measures not agreeable to the partial von Neumann entropy <cit.>. That being the case, our investigations using Schmidt decomposition to quantify the entanglement of bipartite pure states in Sec. IV provide an upper bound to the entanglement measure. To generate a multiphoton state from the cascade emissions, we propose to couple one of the two photons with other AE along with a corresponding pump field, such that (k+1)-photon source can be created from a k-photon state. As demonstrated in Fig. <ref>, three and four-photon states are created in the cascaded scheme. AE' and AE” are two other atomic ensembles used to generate correlated photons â_s',i'^† and â_s”,i”^† respectively under the FWM condition. Since the initial seed of two-photon source is highly entangled in frequency space, the generated multiphoton source is expected to be also entangled in continuous frequency spaces. The generated multiphoton spectral functions can be derived by products of the two-photon spectral ones of Eq. (<ref>) and invoke the condition for central frequencies of the annihilated photon, that is ω_i=ω_s'+ω_i'-ω_b for example in the case of AE' in Fig. <ref>(a). This condition also satisfies the energy conservation as if four fields are plane waves. Below we demonstrate three and four-photon state spectral functions, and their entanglement properties will be discussed in the next section. §.§ Three-photon state As in Fig. <ref>(a) which we denote a route B1 for the three-photon generation, the effective three-photon state involving two signal and one idler photons can be expressed as |Ψ⟩_3,B1=f_3,B1â^†_sâ_s'^†â_i'^†|0⟩. We derive the dimensionless and effective spectral function by multiplying a typical biphoton state spectral function of Eq. (<ref>) with the one generated by a plane-wave idler photon â_i^† and a driving field Ω_b in AE', f_3,B1/Γ_3^2=e^-(Δω_s+Δω_i)^2τ_eff^2/8/Γ_3^ N/2-iΔω_ie^-(Δω_s'+Δω_i')^2τ_b^2/4/Γ_3^ N/2-iΔω_i', where τ_b appears when we let τ_a→∞ in τ_eff. The above becomes, after setting ω_i=ω_s'+ω_i'-ω_b and using Δ_2=ω_a+ω_b-ω_2, f_3,B1/Γ_3^2 = e^-(Δω_s+Δω_s'+Δω_i'+Δ_a3+δω_i)^2τ_eff^2/8/Γ_3^ N/2-i(Δω_s'+Δω_i'+Δ_a3+δω_i) × e^-(Δω_s'+Δω_i')^2τ_b^2/4/Γ_3^ N/2-iΔω_i'. The extra frequency shift Δ_a3≡ω_a-ω_3 in the above can be removed along with δω_i by applying an external Zeeman field, such that we can simplify Eq. (<ref>) as f_3,B1/Γ_3^2=e^-(Δω_s+Δω_s'+Δω_i')^2τ_eff^2/8/Γ_3^ N/2-i(Δω_s'+Δω_i')e^-(Δω_s'+Δω_i')^2τ_b^2/4/Γ_3^ N/2-iΔω_i'. Similarly, if we annihilate signal photon of the initial two-photon seed, we have alternatively the effective three-photon state as (B2 to denote the second route for three-photon state generation, involving two idler and one signal photons) |Ψ⟩_3,B2=f_3,B2â^†_iâ_s'^†â_i'^†|0⟩, where its dimensionless spectral function is [after removing the extra frequency shift as in deriving Eq. (<ref>)] f_3,B2/Γ_3^2 = e^-(Δω_i+Δω_s'+Δω_i')^2τ_eff^2/8e^-(Δω_s'+Δω_i')^2τ_a^2/4/(Γ_3^ N/2-iΔω_i)(Γ_3^ N/2-iΔω_i'). Note that the overall constants in spectral functions do not make an effect on spectral distributions, thus we regularize them in dimensionless forms. The overall constants however determine the generation rates which are small in general since weak and large detuned excitations are used. Eqs. (<ref>) and (<ref>) are two of the main results in this subsection. Obviously these three-photon states are entangled in frequency spaces, meaning there is no possible ways to factorize these spectral functions. Meanwhile |Ψ⟩_3,B1 differs from |Ψ⟩_3,B2 specifically in a modulated Lorentzian function on signal distribution Δω_s'. This results from the annihilation of the idler photon in the route B1, which replaces the Lorentzian with the correlated signal and idler photon pair. Also different timescales τ_b(a) in routes B1(2) respectively for a joint Gaussian profile of the photon pair â_s'^†â_i'^† suggest an independent control over their spectral functions by varying pulse durations. Below we derive the spectral function for the four-photon state of Fig. <ref>(b) using the three-photon states in the cascaded scheme of Fig. <ref>(a), and also the other possible spectral functions in alternative routes. §.§ Four-photon state Here using the same fashion to generate three-photon states, the four-photon ones can be also created in the cascaded scheme. As in Fig. <ref>(b), the idler photon â_i'^† emitted from AE' is annihilated with an extra coupling field Ω_b in AE” to generate a newly correlated pair of photons â_s”^†â_i”^†. The effective four-photon state with three signal and one idler photons becomes |Ψ⟩_4,C1=f_4,C1â^†_sâ_s'^†â_s”^†â_i”^†|0⟩, where again the dimensionless spectral function can be derived as f_4,C1/Γ_3^3=e^-(Δω_s+Δω_s'+Δω_s”+Δω_i”)^2τ_eff^2/8e^-(Δω_s'+Δω_s”+Δω_i”)^2τ_b^2/4e^-(Δω_s”+Δω_i”)^2τ_b^2/4/[Γ_3^ N/2-i(Δω_s'+Δω_s”+Δω_i”)][Γ_3^ N/2-i(Δω_s”+Δω_i”)](Γ_3^ N/2-iΔω_i”). Other four possible routes to generate four-photon states are demonstrated in Appendix A. In principle there should be six different spectral functions where three of them are from routes B1 and B2 respectively. Since there is one spectral function which is symmetric to each other in respective routes, making a total of five possible spectral functions in our cascaded scheme. This symmetric four-photon state is generated by annihilating â_s(i)^† from the routes B1(2) respectively along with the coupling fields Ω_a(b). The symmetry is satisfied when â_s”↔â_s' and â_i”↔â_i', and this effective four-photon state with two signal and two idler photons is |Ψ⟩_4,C3=f_4,C3â^†_s'â_i'^†â_s”^†â_i”^†|0⟩, with the spectral function f_4,C3/Γ_3^3=e^-(Δω_s'+Δω_i'+Δω_s”+Δω_i”)^2τ_eff^2/8e^-(Δω_s'+Δω_i')^2τ_a^2/4e^-(Δω_s”+Δω_i”)^2τ_b^2/4/[Γ_3^ N/2-i(Δω_s”+Δω_i”)](Γ_3^ N/2-iΔω_i')(Γ_3^ N/2-iΔω_i”). These spectral functions have a common weighting of Gaussian envelope involving four photon frequencies, indicating to possess a genuine k-party entanglement <cit.> for k-photon source in our proposed cascaded scheme. This genuine multiphoton entanglement means to exclude any possible bipartite splittings or groupings <cit.>. For example of Eq. (<ref>) from the route C3, two groups of photons â_s',i'^† and â_s”,i”^† respectively would be able to be factorized if this common weighting of e^-(Δω_s'+Δω_i'+Δω_s”+Δω_i”)^2τ_eff^2/8 is absent. Multipartite entanglement is still an ongoing research even for pure states we consider here, therefore to give an intuitive study of entanglement property of our proposed multiphoton sources, in the next section we introduce Schmidt decomposition to investigate their bipartite entanglements in continuous frequency spaces. § ENTROPY OF ENTANGLEMENT To gain insights of the spectral entanglement in the multiphoton sources from the cascaded scheme of AEs, we study the bipartite entropy of entanglement in these sources. The multiphoton states can be projected or collapsed to the effective biphoton state by conditional measurements, such that we can quantify and analyze their spectral entanglement by Schmidt decomposition <cit.>. The bipartite entropy of entanglement is S=-∑_n=1^∞λ_n log_2λ_n<cit.>, intended for pure states. Schmidt eigenvalues λ_n determine the probabilities of nth mode functions, with which the state can be expressed as ∑_n√(λ_n)b̂_n^†ĉ_n^†|0⟩, for two effective photon operators b̂_n and ĉ_n associated with mode functions ψ_n and ϕ_n respectively. Schmidt decomposition is done numerically as solving the eigenvalue problems of the one-photon spectral kernels <cit.>, ∫ K_1(ω,ω^')ψ_n(ω^')dω^' = λ_nψ_n(ω), ∫ K_2(ω,ω^')ϕ_n(ω^')dω^' = λ_nϕ_n(ω), where K_1(ω,ω^') ≡ ∫ f(ω,ω_1)f^∗(ω^',ω_1)dω_1, K_2(ω,ω^') ≡ ∫ f(ω_2,ω)f^∗(ω_2,ω^')dω_2. The spectral function f comes from the effective biphoton state (a pair of â_s^† and â_i^† for example) |Ψ⟩ = ∫ f(ω_s,ω_i)â_s^†(ω_s)â_i^†(ω_i)|0⟩ dω_s dω_i, = ∑_n√(λ_n)b̂_n^†ĉ_n^†|0⟩, with two mode functions in the Schmidt bases, determining the effective photon operators, b̂_n^† ≡ ∫ψ_n(ω_s)â_s^†(ω_s)dω_s, ĉ_n^† ≡ ∫ϕ_n(ω_i)â_i^†(ω_i)dω_i. The above decomposition has been used to quantify the spectral entanglement of two-photon source from parametric down conversion <cit.> and cascade emissions <cit.>, and in spectral shaping of biphoton state in a multiplexed scheme <cit.>. §.§ S in three-photon states For bipartite entanglement property of a three-photon state with the spectral function f_3,B1, three possible projected biphoton states can be derived by either annihilating photons â_s, â_s', or â_i'. In the insets of (a) and (b) of Fig. <ref>, we show the spectral distributions projecting out â_s and â_s' respectively by setting Δω_s(s')=0. The entropy of entanglement S is 2.37 and 0.15 respectively, indicating a more entangled source in the former case. The huge difference in these two projections results from a removal of an entangling function e^-(Δω_s'+Δω_i')^2τ_b^2/4 of Eq. (<ref>) when projecting out â_s'. This Gaussian function supposes to entangle photons â_s' and â_i' in infinite signal and idler frequency spaces. A modulating Lorentzian function of idler photon however confines its spectral distribution into a finite bandwidth of ∼Γ_3^N. Projecting out a signal photon â_s' thus collapses the entangling function into e^-(Δω_i')^2τ_b^2/4, rendering another finite bandwidth of the idler photon with FWHM (full-width at half-maximum) ∼4√(ln 2)/τ_b. In essence these factorizable idler functions, Gaussian and Lorentzian, rotate the spectral distribution toward the axis Δω_i'=0, as we show in Fig. <ref>(b). The third possible projected biphoton state is to project out â_i', which has a similar spectral property to the one annihilating â_s'. Likewise in Fig. <ref>, we show the results for the spectral function f_3,B2. The insets of (a) and (b) demonstrate the spectral functions of the projected biphoton states with annihilated photons â_i and â_s' respectively. The entropy of entanglement S is 1.79 and 0.09 respectively, again indicating a more entangled source in the former case. Fig. <ref>(b) is less entangled due to a removal of the signal photon dependence Δω_s' in the entangling function e^-(Δω_s'+Δω_i')^2τ_a^2/4, similar to the cases in Fig. <ref>. The reason why Fig. <ref>(a) is less entangled than Fig. <ref>(a) is that an extra entangling Lorentzian function [Γ_3^ N/2-i(Δω_s'+Δω_i')]^-1 present in Eq. (<ref>). Fig. <ref>(b) shows a relatively smaller entangled biphoton source, resulting from a spectral function with more aligned distributions on Δω_i(i')=0 compared to Fig. <ref>(b), in a somewhat distorted fashion of Fig. <ref>(b). This is due to two factorizable Lorentzian functions of idler photons â_i and â_i' in frequencies Δω_i(i') of Eq. (<ref>) when projecting out the signal photon â_s'. Since the decay constants of two atomic ensembles, AE and AE', are not necessarily the same, we further investigate the spectral property with different Γ_3^N and Γ_3^N'. As an example from Eq. (<ref>) and projecting out â_s, we have f_3→ 2,B1/Γ_3^2=e^-(Δω_s'+Δω_i')^2τ_eff^2/8/Γ_3^ N/2-i(Δω_s'+Δω_i')e^-(Δω_s'+Δω_i')^2τ_b^2/4/Γ_3^N'/2-iΔω_i'. In Fig. <ref> we show the results for different decay constants compared to Fig <ref>(a). The entropy of entanglement S for the insets (a) and (b) are 3.9 and 0.89 respectively. The small Γ_3^N in (a) provides a sharp distribution along the highly entangled axis Δω_s'=-Δω_i', therefore making this projected spectral function more entangled. In contrast the small Γ_3^N' in (b) limits the factorizable Lorentzian idler distribution â_i', allowing for a squeezed distribution in Δω_i' and a less entangled biphoton source. In this subsection we have demonstrated a rich spectral property of the projected biphoton state from the three-photon source in the cascaded scheme. The entanglement property can be very different depending on how the counterpart of the source is collapsed. In the perspective of generating a more entangled photon source, the spectral function from route B1 serves better than B2, meanwhile a less(more) superradiant decay constant Γ_3^N(Γ_3^N') of AE(AE') in the route B1 is favorable for this purpose. §.§ S in four-photon states For four-photon states, there are in general six possible projections to biphoton ones. Though of plenty of possible projected biphoton states for a total of five spectral functions demonstrated in Sec. III.B and Appendix A, there are only a few of qualitatively different spectral functions. As an example, we choose the spectral function f_4,C1 of Eq. (<ref>). Based on the observation from projected three-photon states in the previous subsection, we expect that a projection of â_i” or â_s” in f_4,C1 provides less entangled multiphoton states. For comparisons, in Fig. <ref> we show two projections of f_4,C1 in three-dimensional isosurface plots which provide a qualitative distinction in the projected spectral functions. In Fig. <ref> (a) and (b) we set Δω_s(i”)=0 respectively, which show a tilted and an axial distribution. The tilted distribution potentially allows for a more entangled multiphoton state while we note that further projecting out photons â_i” and â_s respectively in (a) and (b) gives the same spectral function of the projected biphoton states. Less entangled biphoton states from f_4,C1 can be derived by projecting out a pair of photons (â_s”(i”), â_s') or (â_s”, â_i”). These projections basically collapse f_4,C1 into a single Gaussian function of e^-[Δω_s”(i”)]^2τ_b^2/4 or e^-(Δω_s')^2τ_b^2/4, which again confines its spectral distribution without entangling with other photons. As another example of disentangling photons, we note that the spectral function f_4,C3 of Eq. (<ref>) allows for the least entangled biphoton source. By annihilating a pair of photons of either â_s'(i') or â_s”(i”), we collapse f_4,C3 into two single Gaussian functions. Choosing Δω_s',s”=0 in f_4,C3, in Fig. <ref> we show the collapsed spectral distribution of two idler photons, which has extremely low entropy of entanglement S=0.028. The eigenvalues λ_n are plotted in a logarithmic scale, showing an abrupt decrease of Schmidt numbers. The first two eigenvalues are 0.997 and 0.0028, occupying most of the modes (up to 99.98%) in this biphoton source. In Fig. <ref>(b), we show their mode probability densities. As expected the FWHM of the mode probability density follows the spectral distribution at the cut of Δω_i'(i”)=0. The first mode of the idler â_i” has a shortened linewidth than the one of â_i' due to a squared Lorentzian function of [Γ_3^ N/2-i(Δω_i”)]^-2 in f_4,C3 with Δω_s”=0. We can also manipulate S by modifying the driving pulse durations τ_a and τ_b. When we increase them to τ_a=τ_b=1Γ_3^-1 in the projected f_4,C3 as in Fig. <ref>, we find S becomes 0.13, which allows for a more entangled source since the entangling Gaussian function e^-(Δω_i'+Δω_i”)^2τ_eff^2/8 has a tighter photon correlation on the distribution axis Δω_i'=-Δω_i”. On the other hand when we set τ_a=0.25Γ_3^-1 and τ_b=1Γ_3^-1 in the asymmetric setting, we find S becomes 0.023, which is even smaller than the symmetric case in Fig. <ref>. This reflects the competition of this entangling and two other disentangling Gaussian functions. Though τ_eff still increases in the asymmetric setting, a more confined distribution from τ_b limits the overall projected spectral function distribution, thus decreasing S. § DISCUSSION AND CONCLUSION The multiphoton states generated from the cascade emissions in the scheme of cascaded AE are advantageous in the perspective of well-controlled AE preparations. Large-scale implementation of such cascaded scheme is feasible when preparing the atoms either in free-space, multiplexed setting <cit.>, or optical lattices <cit.>, but will suffer from the low generation rate. Low generation rate results from the weak excitations and few-photon level of emissions. Also low light is more subject to propagation attenuation, making the detection of photons difficult. However this can be overcome by raising the repetition rates in excitations. Finite spectral windows for collapsing the multiphoton spectral functions are not considered here. But we expect of no significant modifications in their spectral properties qualitatively, where essentially the finite spectral window of projection averages out the spectral distribution. A gate time of several hundreds of nanoseconds in the photon counting device would be enough to neglect the effect from finite detection windows. The other merit of the multiphoton source in the cascaded scheme is flexibility to entangle the photons either in signal or idler frequencies, making a repertoire of many possible entangled multiphoton sources. The signal frequency is best for optical fiber transmission while the idler one is preferential in quantum storage. In our scheme a highly entangled photon source can be generated by increasing the excitation pulse durations in the symmetric setting or tighten the spectral distribution on the axis Δω_s+Δω_i=0 that conserves the biphoton energy. By appropriately projecting out the photon counterparts in the multiphoton spectral function, the entropy of entanglement and the spectral mode functions can be modified and manipulated to serve different purposes requiring either pure or entangled states. For more entangled biphoton source, we have demonstrated a larger entropy of entanglement S=2.37 or 1.79 from the projected three-photon states compared to 1.33<cit.> from just one AE under the same driving conditions. This shows that a more entangled biphoton source can be generated from a multiphoton source with even larger capacity in the genuine multipartite entanglement. Our cascaded scheme here can also combine with the multiplexed one <cit.>. The multiplexed scheme manipulates the spectral property of the biphoton state by modifying their central frequencies or phases <cit.>. Its maximal entropy of entanglement S_M can be described by S+S_d. S is the entropy of entanglement from one AE while S_d≡log_2 (N_ MP) with N_ MP, the number of multiplexed AEs. Therefore the multiplexed scheme increases s_d as N_ MP increases. Meanwhile the cascaded scheme enables a multiphoton source from sequentially-coupled AEs using diamond configurations. Its bipartite entanglement can be extracted from the biphoton states collapsed from the multiphoton ones. This way the cascaded scheme modifies and manipulates S effectively, making the combination of these two schemes a full control of S_M. In addition to the spectral shaping of the cascade emissions by modifying driving conditions <cit.> or multiplexing AEs <cit.>, the cascaded scheme here provides an alternative route to spectrally shape an even more entangled biphoton source, thus overcomes the limitation of S in the multiplexed scheme. In conclusion, we propose a cascaded scheme to generate a multiphoton source from the cascade emissions of the atomic ensembles. Highly spectrally entangled (k+1)-photon source can be created using k-photon state as the seed along with an appropriate driving field either in the lower or upper transition of the diamond configuration. Under the FWM condition, this highly directional and frequency-correlated photon source are useful for quantum information processing and applicable to multimode quantum communication. Furthermore such entangled multiphoton source can be spectrally shaped with controllable driving conditions and ensemble properties (for example atomic density and geometry), which could potentially be implemented in quantum spectroscopy <cit.>. § ACKNOWLEDGMENTS This work is supported by the Ministry of Science and Technology (MOST), Taiwan, under Grant No. MOST-103-2112-M-001-011. § OTHER ROUTES FOR FOUR-PHOTON STATE GENERATION In Sec. III.B, we have investigated four-photon state generation in the cascaded scheme, and demonstrated the first route C1. Other four possible routes to generate four-photon states can use three-photon states |Ψ⟩_3,B1 and |Ψ⟩_3,B2 as seeds. The route C2 below is to annihilate â_s'^† of |Ψ⟩_3,B1 with an extra coupling field Ω_a in the third AE” to generate a newly correlated pair of photons â_s”^†â_i”^†. The effective state is |Ψ⟩_4,C2=f_4,C2â^†_sâ_i'^†â_s”^†â_i”^†|0⟩ involving two signal and two idler photons with the spectral function f_4,C2/Γ_3^3=e^-(Δω_s+Δω_i'+Δω_s”+Δω_i”)^2τ_eff^2/8e^-(Δω_i'+Δω_s”+Δω_i”)^2τ_b^2/4e^-(Δω_s”+Δω_i”)^2τ_a^2/4/[Γ_3^ N/2-i(Δω_i'+Δω_s”+Δω_i”)](Γ_3^ N/2-iΔω_i')(Γ_3^ N/2-iΔω_i”). The route C3 involves two signal and two idler photons, generated from a symmetric coupling between â_s(i)^† and pump fields Ω_a(b) in the third AE” using three-photon states from routes B1(2) respectively. Its spectral function has been shown in the main paper. For route C4, which annihilates â_i'^† of |Ψ⟩_3,B2 with an extra coupling field Ω_b in the third AE”. The effective state is |Ψ⟩_4,C4=f_4,C4â^†_iâ_s'^†â_s”^†â_i”^†|0⟩ involving two signal and two idler photons with the spectral function f_4,C4/Γ_3^3=e^-(Δω_s'+Δω_i+Δω_s”+Δω_i”)^2τ_eff^2/8e^-(Δω_s'+Δω_s”+Δω_i”)^2τ_a^2/4e^-(Δω_s”+Δω_i”)^2τ_b^2/4/(Γ_3^ N/2-iΔω_i)[Γ_3^ N/2-i(Δω_s”+Δω_i”)](Γ_3^ N/2-iΔω_i”). The last route C5 annihilates â_s'^† of |Ψ⟩_3,B2 with an extra coupling field Ω_a in the third AE”. The effective state is |Ψ⟩_4,C5=f_4,C5â^†_iâ_i'^†â_s”^†â_i”^†|0⟩ involving one signal and three idler photons with the spectral function f_4,C5/Γ_3^3=e^-(Δω_i+Δω_i'+Δω_s”+Δω_i”)^2τ_eff^2/8e^-(Δω_i'+Δω_s”+Δω_i”)^2τ_a^2/4e^-(Δω_s”+Δω_i”)^2τ_a^2/4/(Γ_3^ N/2-iΔω_i)(Γ_3^ N/2-iΔω_i')(Γ_3^ N/2-iΔω_i”). 99 Nielsen2000 M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000). Bouwmeester2000 D. Bouwmeester, A. K. Ekert, and A. Zeilinger, The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation (Springer-Verlag Berlin Heidelberg, 2000). Gisin2007 N. Gisin and R. Thew, Quantum communication. Nat. Photonics 1, 165 (2007). Pirandola2015 S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein, Advances in quantum teleportation. Nat. Photonics 9, 641 (2015). Kimble2008 H. J. Kimble, The quantum internet. Nature 453, 1023 (2008). Zoller2005 P. Zoller, et al., Quantum information processing and communication. Strategic report on current status, visions and goals for research in Europe. Eur. Phys. J. D 36, 203 (2005). QIP K. Hammerer, A. S. Sørensen, and E. S. Polzik, Quantum interface between light and atomic ensembles. Rev. Mod. Phys. 82, 1041 (2010). Duan2001 L.-M. Duan, M. D. Lukin, J. I. Cirac and P. Zoller, Long-distance quantum communication with atomic ensembles and linear optics, Nature 414, 413 (2001). Briegel1998 H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Quantum repeaters: The role of imperfect local operations in quantum communication, Phys. Rev. Lett. 81, 5932 (1998). Dur1999 W. Dür, H.-J. Briegel, J. I. Cirac, and P. Zoller, Quantum repeaters based on entanglement purification, Phys. Rev. A 59, 169 (1999). Matsukevich2004 D. N. Matsukevich, A. Kuzmich, Quantum State Transfer Between Matter and Light, Science 306, 663 (2004). Chou2004 C. W. Chou, S. V. Polyakov, A. Kuzmich, and H. J. Kimble, Single-Photon Generation from Stored Excitation in an Atomic Ensemble, Phys. Rev. Lett. 92, 213601 (2004). Chaneliere2005 T. Chanelière, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, Storage and retrieval of single photons transmitted between remote quantum memories, Nature 438, 833 (2005). Chen2006 S. Chen, Y.-A. Chen, T. Strassel, Z.-S. Yuan, B. Zhao, J. Schmiedmayer, and J.-W. Pan, Deterministic and storable single-photon source based on a quantum memory, Phys. Rev. Lett. 97, 173004 (2006). Laurat2006 J. Laurat, H. de Riedmatten, D. Felinto, C.-W. Chou, E. W. Schomburg, and H. J. Kimble, Efficient retrieval of a single excitation stored in an atomic ensemble, Opt. Exp. 14, 6912 (2006). Zhao2009 R. Zhao, Y. O. Dudin, S. D. Jenkins, C. J. Campbell, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, Long-lived quantum memory. Nat. Physics 5, 100 (2009). Chaneliere2006 T. Chanelière, D. N. Matsukevich, S. D. Jenkins, T. A. B. Kennedy, M. S. Chapman, and A. Kuzmich, Quantum telecommunication based on atomic cascade transitions, Phys. Rev. Lett. 96, 093604 (2006). Radnaev2010 A. G. Radnaev, Y. O. Dudin, R. Zhao, H. H. Jen, S. D. Jenkins, A. Kuzmich, and T. A. B. Kennedy, A quantum memory with telecom-wavelength conversion, Nat. Phys. 6, 894 (2010). Jen2010 H. H. Jen and T. A. B. Kennedy, Phys. Rev. A 82, 023815 (2010). Jen2012-2 H. H. Jen, Spectral analysis for cascade-emission-based quantum communication in atomic ensembles, J. Phys. B: At. Mol. Opt. Phys. 45, 165504 (2012). Jen2016a H. H. Jen and Y.-C. C, Spectral shaping of cascade emissions from multiplexed cold atomic ensembles. Phys. Rev. A 93, 013811 (2016). Jen2016b H. H. Jen, Entropy of entanglement in the continuous frequency space of the biphoton state from multiplexed cold atomic ensembles. J. Phys. B: At. Mol. Opt. Phys. 49, 035503 (2016). Braunstein2005 S. L. Braunstein and P. van Loock, Quantum information with continuous variables. Rev. Mod. Phys. 77, 513 (2005). Law2004 C. K. Law and J. H. Eberly, Analysis and interpretation of high transverse entanglement in optical parametric down conversion, Phys. Rev. Lett. 92, 127903 (2004). Moreau2014 P.-A. Moreau, F. Devaux, and E. Lantz, Einstein-Podolsky-Rosen paradox in twin images, Phys. Rev. Lett. 113, 160401 (2014). Grad2012 A. Grodecka-Grad, E. Zeuthen, and A. S. Sø rensen, High-capacity spatial multimode quantum memories based on atomic ensembles, Phys. Rev. Lett. 109, 133601 (2012). Branning1999 D. Branning, W. P. Grice, R. Erdmann, and I. A. Walmsley, Engineering the indistinguishability and entanglement of two photons, Phys. Rev. Lett. 83, 955 (1999). Law2000 C. K. Law, I. A. Walmsley, and J. H. Eberly, Continuous frequency entanglement: Effective finite Hilbert space and entropy control, Phys. Rev. Lett. 84, 5304 (2000). Parker2000 S. Parker, S. Bose, and M. B. Plenio, Entanglement quantification and purification in continuous-variable systems, Phys. Rev. A 61, 032305 (2000). Arnaut2000 H. H. Arnaut and G. A. Barbosa, Orbital and intrinsic angular momentum of single photons and entangled pairs of photons generated by parametric down-conversion, Phys. Rev. Lett. 85, 286 (2000). Mair2001 A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Entanglement of the orbital angular momentum states of photons, Nature 412, 313 (2001). Molina2007 G. Molina-Terriza, J. P. Torres, and L. Torner, Twisted photons, Nat. Physics 3, 305 (2007). Dada2011 A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities, Nat. Physics 7, 677 (2011). Fickler2012 R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, A. Zeilinger, Quantum entanglement of high angular momenta, Science 338, 640 (2012). Nicolas2014 A. Nicolas, L. Veissier, L. Giner, E. Giacobino, D. Maxein, and J. Laurat, A quantum memory for orbital angular momentum photonic qubits, Nat. Photonics 8, 234 (2014). Ding2015 D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, Quantum storage of orbital angular momentum entanglement in an atomic ensemble, Phys. Rev. Lett. 114, 050502 (2015). Zhou2015 Z.-Q. Zhou, Y.-L. Hua, X. Liu, G. Chen, J.-S. Xu, Y.-J. Han, C.-F. Li, and G.-C. Guo, Quantum storage of three-dimensional orbital-angular-momentum entanglement in a crystal, Phys. Rev. Lett. 115, 070502 (2015). Anno2006 F. Dell'Anno, S. De Siena, F. Illuminati. Multiphoton quantum optics and quantum state engineering. Phys. Rep. 428, 53 (2006). Pan2012 J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zellinger, and M. Żukowsky, Multiphoton entanglement and interferometry. Rev. Mod. Phys. 84, 777 (2012). QO:Scully M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, 1997). Jen2012 H. H. Jen, Positive-P phase-space-method simulation of superradiant emission from a cascade atomic ensemble, Phys. Rev. A 85, 013835 (2012). Srivathsan2013 B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, Narrow band source of transform-limited photon pairs via four-wave mixing in a cold atomic ensemble, Phys. Rev. Lett. 111, 123602 (2013). Rehler1971 N. E. Rehler and J. H. Eberly, Superradiance. Phys. Rev. A 3, 1735 (1971). Friedberg1973 R. Friedberg, S. R. Hartmann, J. T. Manassah, Frequency shifts in emission and absorption by resonant systems of two-level atoms, Phys. Rep. 7, 101 (1973). Scully2009 M. O. Scully, Collective Lamb shift in single photon Dicke superradiance. Phys. Rev. Lett. 102, 143601 (2009). Jen2015 H. H. Jen, Superradiant cascade emissions in an atomic ensemble via four-wave mixing, Annals of Physics 360, 556 (2015). Dicke1954 R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev 93, 99 (1954). Lehmberg1970 R. H. Lehmberg, Radiation from an N-atom system. I. General formalism, Phys. Rev. A 2, 883 (1970). Gardiner2015 C. Gardiner and P. Zoller, The Quantum World of Ultra-Cold Atoms and Light Book II: The Physics of Quantum-Optical Devices, 1st ed. (Imperial College Press, London, 2015). Bennett1996L C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996). Bennett1996 C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Concentrating partial entanglement by local operations, Phys. Rev. A 53, 2046 (1996). Lan2009 S.-Y. Lan, A. G. Radnaev, O. A. Collins, D. N. Matsukevich, T. A. B. Kennedy, and A. Kuzmich, A Multiplexed Quantum Memory, Opt. Exp. 17, 13639 (2009). Wang2015 Y. Wang, X. Zhang, T. A. Corcovilos, A. Kumar, and D. S. Weiss, Coherent addressing of individual neutral atoms in a 3D optical lattice, Phys. Rev. Lett. 115, 043003 (2015). Dorfman2016 K. E. Dorfman, F. Schlawin, and S. Mukamel, Nonlinear optical signals and spectroscopy with quantum light, Rev. Mod. Phys. 88, 045008 (2016).

http://arxiv.org/abs/1701.07598v1

20170126073408

Precision determination of a fluxoid quantum's magnetic moment in a superconducting micro-ring

cond-mat.mes-hall

Division of Physical Metrology, Korea Research Institute of Standards and Science, Daejeon 34113, Korea Department of Nanoscience, Korea University of Science and Technology, Daejeon 34113, Korea Division of Physical Metrology, Korea Research Institute of Standards and Science, Daejeon 34113, Korea Department of Display and Semiconductor Physics, Korea University Sejong Campus, Sejong 30019, Korea Department of Physics, Korea University, Seoul 02841, Korea Division of Physical Metrology, Korea Research Institute of Standards and Science, Daejeon 34113, Korea []jhchoi@kriss.re.kr Division of Physical Metrology, Korea Research Institute of Standards and Science, Daejeon 34113, Korea Department of Nanoscience, Korea University of Science and Technology, Daejeon 34113, Korea Using dynamic cantilever magnetometry and experimentally determining the cantilever’s vibrational mode shape, we precisely measured the magnetic moment of a lithographically defined micron-sized superconducting Nb ring, a key element for the previously proposed subpiconewton force standard. The magnetic moments due to individual magnetic fluxoids and a diamagnetic response were independently determined at T = 4.3 K, with a subfemtoampere-square-meter resolution. The results show good agreement with the theoretical estimation yielded by the Brandt and Clem model within the spring constant determination accuracy. Precision determination of a fluxoid quantum's magnetic moment in a superconducting micro-ring Jae-Hyuk Choi December 30, 2023 ============================================================================================== § INTRODUCTION The superconducting ring has attracted considerable attention in the context of both fundamental superconductor research and application, because of its geometry-related effects, such as fluxoid quantization and quantum interference.<cit.> The magnetic flux, or more precisely, magnetic fluxoid, through an ordinary superconducting ring is quantized in units of h/2e, where h is Planck's constant and e is the electron charge.<cit.> In superconducting devices and applications, a superconducting ring with or without Josephson junctions has acted as a key element.<cit.> Understanding its magnetic properties is valuable for the design and analysis of, for example, a superconducting quantum interference device (SQUID),<cit.> a gravity gradiometry,<cit.> an ultracold atom trap,<cit.> and a subpiconewton force standard.<cit.> In particular, the concept of quantum-based force realization,<cit.> which some authors have suggested as a candidate for the subpiconewton force standard previously, utilizes magnetic fluxoid quanta in a microscale superconducting ring. The force can be increased or decreased by a force step, estimated to be on the subpiconewton level, by controlling the fluxoid number. The magnetic moment due to a single fluxoid quantum is the minimum unit for generating a magnetic force in a well-defined magnetic field gradient. Determining the unit magnetic moment with not only high sensitivity, but also high precision is key towards establishing the suggested method as the first standard for an extremely small force, because the unit magnetic moment defines the magnitude and precision of the unit force to be realized. Besides the small-force-standard application,<cit.> the unit magnetic moment based on fluxoid quanta can be utilized as a new reference for a small magnetic moment at the femtoampere-square-meter level. Several theoretical methods<cit.> have been developed to calculate the magnetic moments as well as the magnetic-field and current-density profiles for various values of the fluxoid number and external magnetic field in superconducting thin-film rings and disks. Initially, cases of negligibly small penetration depth λ were addressed,<cit.> and Brandt and Clem<cit.> generalized the previous studies to finite λ, providing a calculation method to give precise numerical solutions. Although their theory has been adopted for superconducting ring design or to interpret its properties over the past decade,<cit.> very few experimental studies providing high-precision measurements of the ring magnetic moment have been reported.<cit.> Experimentally, the measurement sensitivity for microsample magnetic moments is approaching its limit, as a result of the notable recent improvement in the force sensitivity in dynamic cantilever magnetometry<cit.> down to attonewton level.<cit.> In a study of persistent currents in normal metal rings,<cit.> for example, dynamic cantilever magnetometry, which measures the resonance frequency shift of a cantilever in a magnetic field, exhibited a resolution that was approximately 250-fold superior to SQUID magnetometers<cit.> for detection of a ring's current. This result finally resolved previous order-of-magnitude discrepancies between experimental and theoretical current values. Such high sensitivity is obtained by applying high external fields. As regards dynamic cantilever magnetometry analysis of the low-field magnetic properties of a sample, however, a significant sensitivity reduction is inevitable. Very recently, this limitation was overcome using a phase-locked approach suggested by Jang et al.<cit.> These researchers succeeded in detecting small half-fluxoid-quantum signals in an Sr_2RO_4 superconductor at low static fields by applying an additional oscillating field, which was phase-locked to the cantilever position, for signal enhancement. The above studies have highlighted the potential sensitivity of dynamic cantilever magnetometry for magnetic-moment detection at both high and low magnetic fields. In this work, we adopt dynamic cantilever magnetometry for precision measurement of the small magnetic moments of fluxoids in a superconducting microring. However, in order to retain a simple measurement geometry and to reduce the uncertainty factors, we do not employ a phase-locked approach, which requires precise control of the modulation field.<cit.> Instead, we enhance the resonance frequency shift by increasing the external field after trapping fluxoids in the ring. For a micron-sized Nb ring, we determine the magnetic moment of a single fluxoid, along with the Meissner susceptibility of the ring, and compare the results with theoretical estimations from the Brandt and Clem method.<cit.> For accurate comparison, we prepare a ring sample with a well-defined geometry on an ultrasoft cantilever and utilize a fiber-optic interferometer with subnanometer resolution, with the fiber on a piezo positioner; this setup enables precision vibration measurement at multiple target positions on the cantilever. The latter is necessary for the experimental determination of the cantilever vibration mode characteristics, such as its effective length, which is otherwise theoretically estimated. § EXPERIMENT For the sample-on-cantilever configuration, we batch-fabricate cantilevers with an Nb ring sample. After the cantilever patterns are defined in a low-pressure chemical vapor deposited (LPCVD) silicon-nitride layer on a silicon wafer, a 100-nm-thick Nb ring with nominal inner and outer radii of a = 2 and b = 4 μm, respectively, is fabricated via lift-off patterning with photolithography. The ring is aligned with the mounting paddle center of each cantilever, as can be seen in Fig. <ref> (b, right). The cantilever fabrication is then completed, taking care to protect the attached, high-quality Nb film (see Ref. [] for more details). The released cantilevers have 367-μm length, 4-μm width, and 200-nm thickness, with a mounting paddle at one end. The lateral dimensions and surface quality of the Nb ring are measured and examined using a Tescan Mira scanning electron microscope (SEM). The sample-on-cantilever device is placed on a piezoactuator in high vacuum, surrounded by a superconducting solenoid for application of a uniform magnetic field. Its low-temperature vibration amplitude and resonance frequency are measured with a low-noise fiber-optic interferometer using a 1550-nm tunable laser (Agilent 81660B-200) with a high wavelength stability of 1 pm for 24 h and a coherence control feature, which has been demonstrated to have subpicometer resolution at an optical power of 10 μW and room temperature.<cit.> For our study, a very low laser power of 13 nW at the fiber end is adopted to avoid optical effects such as photothermal actuation. The fiber, attached to a 3-axis piezo positioner, is located above a target position on the cantilever. The optical interference from the optical-fiber cantilever cavity is detected at a photodiode coupled to a low-noise transimpedance amplifier (Femtoamp DLPCA-200). The cantilever frequency is primarily measured at a temperature of 4.3 K. In the magnetic-field-cooling (FC) process, the cantilever temperature is elevated momentarily using a light-emitting diode to above the superconducting transition temperature, T_c, of the Nb ring and then recovered. The entire system is mounted on a double-stage vibration-isolation platform including a 21-ton mass block. §.§ Measurement fundamentals Figure <ref> (a) shows the key features of our dynamic cantilever magnetometry setup. In an external magnetic field _, the magnetic moment of the sample exerts a torque on the cantilever. For a two-dimensional sample, we can assume that has an out-of-plane component m only. Then, the magnitude of the torque is given as τ = m H_sinθ and, with _∥, the relative angle θ of and _ is identical to the cantilever surface angle at the sample position with respect to the direction. The shift of the resonance frequency f due to the magnetic torque<cit.> is expressed as Δ f = f_0/2 k_0 L_^2 m H_ = f_0/2 k_0 L_^2( χ H_+n m_) H_ , where k_0 and f_0 are the spring constant and intrinsic resonance frequency of the cantilever, respectively, and L_ is the cantilever effective length. In the case of our superconducting ring, m has two contributions, from the diamagnetic response due to the Meissner current and from the n magnetic fluxoids in the ring hole. Here, χ is the Meissner susceptibility, and each fluxoid quantum has the same magnetic moment, m_. In our work, we adopt a cantilever, shown in Fig. <ref> (b), with f_0 = 1221.9 Hz and k_0 = 4.5×10^-5 N/m in the fundamental vibrational mode. For L_, the theoretical value of L/1.38 for a rectangular Euler-Bernoulli beam of length L is frequently used.<cit.> However, the L_ of our device was experimentally determined to be L/1.48 or 248 μm, by measuring the shape of the first vibration mode with a fiber on the piezo positioner. The minimum detectable frequency shift and magnetic moment, Δ f_ and m_, respectively, of our cantilever were estimated to be 1.1 mHz and 1.2 fAm^2, respectively, for a 1-Hz detection bandwidth with H_ = 40 Oe. The characterization of the cantilever mechanical properties is described in more detail in Appendix A. § RESULTS AND DISCUSSION To observe the superconducting transition, the f of the cantilever was monitored with increasing temperature in a magnetic field of 10 Oe, applied perpendicularly to the mounted Nb ring after zero-field cooling to T = 4.5 K. The f temperature dependence exhibits a typical feature of a diamagnetic superconducting transition, with an onset temperature of T_c = 8.3 K, with the exception that a slope is apparent across the entire displayed temperature range (Fig. <ref>). This feature indicates that the superconducting ring is in the Meissner state at temperatures lower than T_c. The T_c value agrees well with the superconducting transition temperature obtained for a resistive measurement of a strip Nb sample from the same batch (data not shown here). The f_0 in the absence of τ is represented by a dashed line in Fig. <ref>, having a slope of -5.5 mHz/K; this slope is obtained from a fit of the data in the normal state. The possible origins of the negative slope are the temperature dependence of the cantilever dimensions, cantilever surface stress, and so on; further discussion of this topic is presented in Appendix B. In the Meissner state of the Nb ring, the f response to sweeping H_ follows Eq. (<ref>), resulting in the parabolic curve shown in Fig. <ref> (a). The parabolic dependence is valid in the | H_| ≤∼ 60 Oe range, whereas it breaks down beyond this range as a result of magnetic vortex penetration into the annular area, i.e., a mixed state of Nb. Such a small critical field value is attributed to the high demagnetization effect due to the quasi-2D sample geometry.<cit.> As we increase the FC magnetic field, H_, used in cooling the Nb ring from above T_c, more magnetic fluxoids are contained within the ring hole. Accordingly, the curve is shifted to higher f and H_. Parabolic fits to the data shown in Fig. <ref> (a) can provide χ and n m_; however, we obtained the n m_ values from separate measurements, which proved to be more accurate and efficient. We deduced χ H_ by dividing the H_ = 0 data by H_, which exhibits a linear dependence, as depicted in Fig. <ref> (b). Note that data at low magnetic fields were not employed, because of their low accuracy. The linear fit yields χ = -102 ± 10 pAm^2/T. To observe individual magnetic fluxoids at T = 4.3 K, we varied H_ from 10 to 13 Oe with a smaller step of 0.1 Oe. To enhance the Δ f signal for n m_ for low H_, we increased the magnetic field from H_ to a larger and fixed value, i.e.,H_ = 40 Oe, before measuring f. This procedure is depicted in the inset of Fig. <ref>. In this manner, we could obtain n m_ for small n, because n m_ is independent of the magnetic field, but its contribution to Δ f is proportional to H_, as shown in Eq. (<ref>). Note that the contributions of χ H for various H_ are identical and can be universally eliminated because H_ is fixed. Figure <ref> clearly shows that f has a stepwise feature with varying H_. The single step width, Δ H, was estimated to be 0.65 ± 0.03 Oe from the total width of the four steps fully shown in Fig. <ref>. Taking the errors in H_ and Δ H into consideration, the n corresponding to H_ = 10 Oe may range from 14 to 16. For FC with a H_ corresponding to the center of each step plateau, no net current circulates the ring, even with n fluxoids and the response to the external field. The effective area of the zero-current contour is given by A_ = Φ_0/H_a,<cit.> where H_a is the field increment necessary to induce a transition from the n to n + 1 state. Because H_a = Δ H, we can estimate A_ to be 32 μm^2, which indicates flux focusing where A_ is larger than the actual hole area, π a^2 = 13 μm^2. This estimate agrees roughly with the A_ = 25 μm^2 result calculated using the Brandt and Clem theoretical prediction.<cit.> Within Δ H, f is virtually constant to within 1 mHz for changing H_, which implies that the number of fluxoids is fixed and their contribution to m is constant. As H_ is raised beyond Δ H, an additional fluxoid is introduced to the ring hole, resulting in a discrete shift of f or m_ as shown in Fig. <ref>. As the m_ of each fluxoid are intrinsically expected to be equivalent, this value can be determined from the average of the five steps of m or f, which are 5.8±0.6 fAm^2 or 5.2±0.2 mHz, respectively. Note that the uncertainty in the cantilever spring constant makes a dominant contribution to the estimated error in m_. Near the step edges, f is observed at both fluxoid quantum numbers, n and n + 1, because the kinetic energies of the right- and left-circulating supercurrent states, respectively, are degenerate for Nb-ring cooling at corresponding magnetic fields. Figure <ref> depicts the theoretical values of the magnetic moments due to a single fluxoid and a diamagnetic response, which were estimated numerically for various ring radii utilizing the Brandt and Clem model.<cit.> As shown in the figure, the magnitudes of m_ and χ decrease slowly with increasing a, but increase with higher dependence with increasing b. The experimentally obtained values are also plotted at the dimensions of our Nb ring on the cantilever; the dimensions are measured from the calibrated SEM, yielding a and b of 2.0 ± 0.1 μm and 3.8 ± 0.1 μm, respectively. For Λ = 110 nm at 4 K,<cit.> where Λ =λ^2/d is the thin-film penetration depth and d is the film thickness, m_ and χ are calculated to be 5.98 ± 0.10 fAm^2 and -91.2 ± 0.3 pAm^2/T (5.98 ± 0.24 fAm^2 and -91.2 ± 7.8 pAm^2/T), respectively, if the uncertainty of a (b) is considered. These values are in quite good agreement with the experimental results, considering the accuracy of the spring constant determination. The effect of the Λ uncertainty is negligible, as m_ and χ are estimated to vary by only 1.4% and -1.7%, respectively, for a Λ difference of 10%. However, it is notable that the assumption of negligible Λ in the theoretical estimation yields m_ = 7.17 fAm^2 and χ = -116 pAm^2/T, which are considerable overestimations in comparison with the experimental values. This finding implies that consideration of the finite penetration depth, as in the Brandt and Clem model, is crucial for appropriate description of micron-sized superconducting rings, and that it remains valid when the ring radii uncertainty is considered, as can be seen in Fig. <ref>. § CONCLUSIONS Using high-resolution cantilever magnetometry capable of fiber scanning, we precisely measured the magnetic moment of a well-defined superconducting Nb thin-film ring with inner and outer radii of 2.0 μm and 3.8 μm, respectively, on an ultrasoft cantilever at T = 4.3 K. The experimental results, a diamagnetic response of -102 pAm^2/T and a single fluxoid magnetic moment of 5.8 fAm^2, agree well with the theoretical model prediction, providing a reliable technical and theoretical base for superconducting microring research and applications in the future. The authors are grateful to D. H. Lee and B. H. Park for SEM imaging. This work was supported by the Korea Research Institute of Standards and Science under the ”Establishment of National Physical Measurement Standards and Improvements of Calibration/Measurement Capability” project, Grant Nos. 16011003 and 16011012. M.-S. C. was supported by the National Research Foundation of Korea (Grant No. 2015-003689). § In dynamic cantilever magnetometry, the resonance-frequency shift Δ f can be derived by calculating the magnetic torque oscillation dependent on the cantilever vibrations. The cantilever vibration is a solution of the equation of motion for beam vibration,<cit.> which is generally expressed as u(x,t) = ∑_n=1^∞ u_n (x) q_n (t) , where u_n(x) is the nth resonance mode shape and q_n (t) is the generalized coordinate in the nth mode. If we drive the cantilever at one of the resonance frequencies, for example, the first mode, the problem is reduced to solving a one-dimensional forced equation of motion for q_1 (t). The cantilever is subject to an effective force of τ/L_, where L_ is the cantilever effective length, defined as u_1 (x)/tanθ = u_1 (x)/(du_1 (x)/dx).<cit.> The effective force can then be deduced as τ/ L_ = 1/ L_^2 m H_ u_1 (x) q_1 (t) from Eq. (<ref>), with an approximation of sinθ≅du_1 (x)/dx q_1 (t) = u_1 (x)/L_ q_1 (t) for small deflections. Hence, the Fourier transform of the forced vibration equation for q_1(t) can be expressed as<cit.> (-ω^2 -i γω + ω_0^2) q̃ (ω) = ω_0^2/k_0 L_^2 m H_q̃ (ω) . Here, ω_0 is the angular resonance frequency 2π f_0. The solution of Eq. (<ref>) gives Δ f as expressed in Eq. (<ref>).<cit.> As illuminating the cantilever free end, even at small laser power, may cause local heating of the sample, Δ f measurement for magnetometry is conducted with the fiber pointing at the center of a 20 μm-width reflector, shown in Fig. <ref> (b),  100 μm from the paddle on which an Nb ring is mounted. To align the fiber to the reflector center or another point of interest, we first obtain a quick map of the cantilever, as shown in Fig. 1(b), by scanning the cantilever plane and obtaining the laser interference amplitude at each point; this is achieved by sweeping the fiber-cantilever inter-distance. Then, for fine adjustment, we repeatedly obtain line profiles of the interference amplitude, in directions both parallel and perpendicular to the cantilever, to find the target position with ∼1 μm resolution. To determine precise values for k_0 and L_ in Eq. (<ref>), we require a fundamental mode shape; therefore, we obtain position-dependent vibrational noise spectra along the cantilever. These spectra provide < u^2 (x,t) > from Eq. (<ref>), which falls on the mode shape predicted by the finite element method for the cantilever employed in this work. From the ratio of <u^2 (x,t) > at the sample position, < u_^2 >, against that at the reflector center, < u_^2 >, we determine the spring constant conversion factor, < u_^2 > / < u_^2 >, to be 2.85, and from the slope at the sample position, we determine L_ to be 248 μm. Figure <ref> shows the fundamental thermal vibration noise spectrum at T = 4.3 K, obtained with a span of 3.125 Hz and averaging over 15 results, which provides < u_^2 > as well as f_0 = 1221.9 Hz and the quality factor Q = 43000. Using the equipartition theorem along with < u_^2 > / < u_^2 >, the mechanical impedance to the force at the sample position is evaluated to be k_0 = 4.5 × 10^-5 N/m, with an accuracy conservatively claimed to be 10%.<cit.> The minimum detectable shift of the cantilever frequency is given by Δ f_ = f_0 F_/ √(2) k_0 x_.<cit.> Here, F_ is the smallest detectable force signal, given by F_ = √(2k_0 k_B T B / π f_0 Q), where k_B is the Boltzmann constant, x_ is the peak displacement of the oscillating cantilever, and B is the detection bandwidth. The thermally limited detectable magnetic moment m_ can be expressed as 2 Δ f_ k_0 L_^2/ f_0 H_, employing Eq. (<ref>). Using the cantilever parameters given above, the corresponding Δ f_ and m_ are 1.1 mHz and 1.2 fAm^2 for a 1-Hz bandwidth with x_ = 100 nm and H_ = 40 Oe. § The negative slope of f_0 (T) in Fig. <ref> may originate from the temperature dependence of the Young's modulus, dimensions, surface stress, and so on, of the silicon nitride cantilever. The spring constant of a simple beam is given by<cit.> k_0 = 1.030 E w t^3 / l^3, where E is the Young's modulus of the material and w, t, and l are the beam width, thickness, and length, respectively. With 2 π f_0 = √(k_0/m_), where m_ is the beam effective mass, the temperature derivative of f_0 (T) can be expressed as 1/f_0df_0/dT = 1/2 k_0dk_0/dT = 1/2( 1/EdE/dT + 1/wdw/dT) , where we assume an isotropic thermal contraction for w, t, and l. The effect of the intrinsic Young's modulus can be ignored because, in general, its temperature dependence is virtually zero at low temperatures. If we adopt the Wachtman semi-empirical formula for Young's modulus,<cit.> E(T)=E_0 - B T exp (-T_0/T), its temperature derivative is given by dE/dT= -B(1+ T_0/T) exp (-T_0/T). For the reported parameters for silicon nitride,<cit.> E_0 = 320 Gpa, B = 0.0151 GPa/K, and T_0 = 445 K, (1/E) dE/dT is estimated to be as small as -1 × 10^-24 K^-1 at T = 9 K. Excluding the intrinsic Young's modulus, we may speculate that the temperature dependence of the cantilever dimensions yields the f_0 (T) slope both indirectly and directly, via the first and second terms on the right-hand side of Eq. (<ref>), respectively. One possible indirect effect is via surface stress in a thin cantilever. Because of the strain-dependent surface stress, the effective Young's modulus E_ of a silicon nitride cantilever has been reported to have a thickness dependence.<cit.> That is, E_ decreases strongly for decreasing thickness below our cantilever thickness of 200 nm. Considering the thickness dependence and the signs in Eq. (<ref>), the thermal contraction of the cantilever dimensions for increasing T is consistent with the negative slope of f_0 (T), if other factors are ignored. The lower bound of the thermal expansion coefficient α, which is defined as α = (1/w) dw/dT, can be estimated from Eq. (<ref>) with the assumption of dE/dT = 0, yielding α_lower = (2/ f_0) d f_0 / dT = -9 × 10^-6 K^-1. More systematic studies are necessary in the future to determine an accurate value of α for silicon nitride at low temperatures. 10 Tinkham96 M. Tinkham, Introduction to Superconductivity, 2nd ed. (McGraw-Hill, New York, 1996). Jang11 J. Jang, D. G. Ferguson, V. Vakaryuk, R. Budakian, S. B. Chung, P. M. Goldbart, and Y. Maeno, Science 331, 186 (2011). Chen10 C.-T. Chen, C. C. Tsuei, M. B. Ketchen, Z.-A. Ren, and Z. X. Zhao, Nature Phys. 6, 260 (2010). Hasselba08 K. Hasselbach, C. Ladam, V. O. Dolocan, D. Hykel, T. Crozes, K. Schuster, and D. Mailly, J. Phys. Conf. Ser. 97, 012330 (2008). Kirtley10 J. R. Kirtley, Rep. Prog. Phys. 73, 126501 (2010). Moody02 M. V. Moody, H. J. Paik, and E. R. Canavan, Rev. Sci. Instrum. 73, 3957 (2002). Weiss15 P. Weiss, M. Knufinke, S. Bernon, D. Bothner, L. Sarkany, C. Zimmermann, R. Kleiner, D. Koelle, J. Fortagh, and H. Hattermann, Phys. Rev. Lett. 114, 113003 (2015). Choi07 J.-H. Choi, M.-S. Kim, Y.-K. Park, and M.-S. Choi, Appl. Phys. Lett. 90, 073117 (2007). Taylor67 B. N. Taylor, W. H. Parker, D. N. Langenberg, and A. Denenstein, Metrologia 3, 89 (1967). Brandt97 E. H. Brandt, Phys. Rev. B 55, 14513 (1997). Brojeny03 A. A. Babaei Brojeny and J. R. Clem, Phys. Rev. B 68, 174514 (2003). Brandt04 E. H. Brandt and J. R. Clem, Phys. Rev. B 69, 184509 (2004). Harris99 J. G. E. Harris, D. D. Awschalom, F. Matsukura, H. Ohno, K. D. Maranowski, and A. C. Gossard, Appl. Phys. Lett. 75, 1140 (1999). Chabot03 M. D. Chabot and J. Moreland, J. Appl. Phys. 93, 7897 (2003). Stipe01 B. C. Stipe, H. J. Mamin, T. D. Stowe, T. W. Kenny, and D. Rugar, Phys. Rev. Lett. 86, 2874 (2001). Bleszyns09 A. C. Bleszynski-Jayich, W. E. Shanks, B. Peaudecerf, E. Ginossar, F. von Oppen, L. Glazman, and J. G. E. Harris, Science 326, 272 (2009). Jariwala01 E. M. Q. Jariwala, P. Mohanty, M. B. Ketchen, R. A. Webb, Phys. Rev. Lett. 86, 1594 (2001). Bluhm09 H. Bluhm, N. C. Koshnick, J. A. Bert, M. E. Huber, and K. A. Moler, Phys. Rev. Lett. 102, 136802 (2009). Jang11B J. Jang, R. Budakian, and Y. Maeno, Appl. Phys. Lett. 98, 132510 (2011). Choi16 H. Choi, Y. W. Kim, S.-G. Lee, J.-H. Kim, and J.-H. Choi, “Fabrication and characterization of fluxoid-quantum-controlled force device” (to be submitted). Smith09 D. T. Smith, J. R. Pratt, and L. P. Howard, Rev. Sci. Inst. 80, 035105 (2009). Rao95 S. S. Rao, Mechanical Vibrations, 3rd ed. (Addison-Wesley, New York, 1995). Sidles95 J. A. Sidles, J. L. Garbini, K. J. Bruland, D. Rugar, O. Zuger, S. Hoen, and C. S. Yannoni, Rev. Mod. Phys. 67, 249 (1995). Doria08 M. M. Doria, E. H. Brandt, and F. M. Peeters, Phys. Rev. B 78, 054407 (2008). Kim12 J. Kim, F. Ronning, N. Haberkorn, L. Civale, E. Nazaretski, N. Ni, R. J. Cava, J. D. Thompson, and R. Movshovich, Phys. Rev. B 85, 180504 (2012). Matei06 G. A. Matei, E. J. Thoreson, J. R. Pratt, D. B. Newell, and N. A. Burnham, Rev. Sci. Instrum. 77, 083703 (2006). Bruls01 R. J. Bruls, H. T. Hintzen, G. De With, and R. Metselaar, J. Euro. Cer. Soc. 21, 263 (2001). Gavan09 K. B. Gavan, H. J. R. Westra, E. W. J. M. van der Drift, W. J. Venstra, and H. S. J. van der Zant, Appl. Phys. Lett. 94, 233108 (2009).

http://arxiv.org/abs/1701.07732v1

20170126145919

Pose Invariant Embedding for Deep Person Re-identification

cs.CV

Pose Invariant Embedding for Deep Person Re-identification Liang Zheng^†, Yujia Huang^, Huchuan Lu^, Yi Yang^† ^†University of Technology Sydney ^CMU ^Dalian University of Technology {liangzheng06,yee.i.yang}@gmail.com yujiah1@andrew.cmu.edu lhchuan@dlut.edu.cn ==================================================================================================================================================================================================================================== Pedestrian misalignment, which mainly arises from detector errors and pose variations, is a critical problem for a robust person re-identification (re-ID) system. With bad alignment, the background noise will significantly compromise the feature learning and and matching process. To address this problem, this paper introduces the pose invariant embedding (PIE) as a pedestrian descriptor. First, in order to align pedestrians to a standard pose, the PoseBox structure is introduced, which is generated through pose estimation followed by affine transformations. Second, to reduce the impact of pose estimation errors and information loss during PoseBox construction, we design a PoseBox fusion (PBF) CNN architecture that takes the original image, the PoseBox, and the pose estimation confidence as input. The proposed PIE descriptor is thus defined as the fully connected layer of the PBF network for the retrieval task. Experiments are conducted on the Market-1501, CUHK03, and VIPeR datasets. We show that PoseBox alone yields decent re-ID accuracy, and that when integrated in the PBF network, the learned PIE descriptor produces competitive performance compared with the state-of-the-art approaches. § INTRODUCTION This paper studies the task of person re-identification (re-ID). Given a probe (person of interest) and a gallery, we aim to find in the gallery all the images containing the same person with the probe. We focus on the identification problem, a retrieval task in which each probe has at least one ground truth in the gallery <cit.>. A number of factors affect the re-ID accuracy, such as detection/tracking errors, variations in illumination, pose, viewpoint, etc. A critical influencing factor on re-ID accuracy is the misalignment of pedestrians, which can be attributed to two causes. First, pedestrians naturally take on various poses as shown in Fig. <ref>. Pose variations imply that the position of the body parts within the bounding box is not predictable. For example, it is possible that one's hands reach above the head, or that one is riding a bicycle instead of being upright. The second cause of misalignment is detection error. As illustrated in the second row of Fig. <ref>, detection errors may lead to severe vertical misalignment. When pedestrians are poorly aligned, the re-ID accuracy can be compromised. For example, a common practise in re-ID is to partition the bounding box into horizontal stripes <cit.>. This method works under the assumption of slight vertical misalignment. But when vertical misalignment does happen as in the cases in Row 2 of Fig. <ref>, one's head will be matched to the background of a misaligned image. So horizontal stripes may be less effective when severe misalignment happens. In another example, under various pedestrian poses, the background may be incorrectly weighted by the feature extractors and thus affect the following matching accuracy. To our knowledge, two previous works <cit.> from the same group explicitly consider the misalignment problem. In both works, the pictorial structure (PS) is used, which shares a similar motivation and construction process with PoseBox, and the retrieval process mainly relies on matching the normalized body parts. While the idea of constructing normalized poses is similar, our work locates body joints using a state-of-the-art CNN based pose estimator, and the components of PoseBox are different from PS as evidenced by large-scale evaluations. Another difference of our work is the matching procedure. While <cit.> do not discuss the pose estimation errors which prevalently exist in real-world datasets, we show that these errors make rigid feature learning/matching with only the PoseBox yield inferior results to the original image, and that the three-stream PoseBox fusion network effectively alleviates this problem. Considering the above-mentioned problems and the limit of previous methods, this paper proposes the pose invariant embedding (PIE) as a robust visual descriptor. Two steps are involved. First, we construct a PoseBox for each pedestrian bounding box. PoseBox depicts a pedestrian with standarized upright stance. Carefully designed with the help of pose estimators <cit.>, PoseBox aims to produce well-aligned pedestrian images so that the learned feature can find the same person under intensive pose changes. Trained alone using a standard CNN architecture <cit.>, we show that PoseBox yields very decent re-ID accuracy. Second, to reduce the impact of information loss and pose estimation errors (Fig. <ref>) during PoseBox construction, we build a PoseBox fusion (PBF) CNN model with three streams as input: the PoseBox, the original image, and the pose estimation confidence. PBF achieves a globally optimized tradeoff between the original image and the PoseBox. PIE is thus defined as the FC activations of the PBF network. On several benchmark datasets, we show that the joint training procedure yields competitive re-ID accuracy to the state of the art. To summarize, this paper has three contributions. * Minor contribution: the PoseBox is proposed which shares a similar nature with a previous work <cit.>. It enables well-aligned pedestrian matching, and yields satisfying re-ID performance when being used alone. * Major contribution: the pose invariant embedding (PIE) is proposed as a part of the PoseBox Fusion (PBF) network. PBF fuses the original image, PoseBox and the pose estimation errors, thus providing a fallback mechanism when pose estimation fails. * Using PIE, we report competitive re-ID accuracy on the Market-1501, CUHK03, and VIPeR datasets. § RELATED WORK Pose estimation. The pose estimation research has shifted from traditional approaches <cit.> to deep learning following the pioneer work “DeepPose” <cit.>. Some recent methods employ multi-scale features and study mechanisms on how to combine them <cit.>. It is also effective to inject spatial relationships between body joints by regularizing the unary scores and pairwise comparisons <cit.>. This paper adopts the convolutional pose machines (CPM) <cit.>, a state-of-the-art pose estimator with multiple stages and successive pose predictions. Deep learning for re-ID. Due to its superior performance, deep learning based methods have been dominating the re-ID community in the past two years. In the two earlier works <cit.>, the siamese model which takes two images as input is used. In later works, this model is improved in various ways, such as injecting more sophisticated spatial constraint <cit.>, modeling the sequential properties of body parts using LSTM <cit.>, and mining discriminative matching parts for different image pairs <cit.>. It is pointed out in <cit.> that the siamese model only uses weak re-ID labels: two images being of the same person or not; and it is suggested that an identification model which fully uses the strong re-ID labels be superior. Several previous works adopt the identification model <cit.>. In <cit.>, the video frames are used as training samples of each person class, and in <cit.>, effective neurons are discovered for each training domain and a new dropout strategy is proposed. The architecture proposed in <cit.> is more similar to the PBF model in our work. In <cit.>, hand-crafted low-level features are concatenated after a fully connected (FC) layer which is connected to the softmax layer. Our network is similar to <cit.> in that confidence scores of pose estimation are catenated with the other two FC layers. It departs from <cit.> in that our network takes three streams as input, two of which are raw images. Poses for re-ID. Although pose changes have been mentioned by many previous works as an influencing factor on re-ID, only a handful of reports can be found discussing the connection between them. Farenzena et al. <cit.> propose to detect the symmetrical axis of different body parts and extract features following the pose variation. In <cit.>, rough estimates of the upper-body orientation is provided by the HOG detector, and the upper body is then rendered into the texture of an articulated 3D model. Bak et al. <cit.> further classify each person into three pose types: front, back, and side. A similar idea is exploited in <cit.>, where four pose types are used. Both works <cit.> apply view-point specific distance metrics according to different testing pose pairs. The closest works to PoseBox are <cit.>, which construct the pictorial structure (PS), a similar concept to PoseBox. They use traditional pose estimators and hand-crafted descriptors that are inferior to CNN by a large margin. Our work employs a full set of stronger techniques, and designs a more effective CNN structure evidenced by the competitive re-ID accuracy on large-scale datasets. § PROPOSED METHOD §.§ PoseBox Construction The construction of PoseBox has two steps, i.e., pose estimation and PoseBox projection. Pose estimation. This paper adopts the off-the-shelf model of the convolutional pose machines (CPM) <cit.>. In a nutshell, CPM is a sequential convolutional architecture that enforces intermediate supervision to prevent vanishing gradients. A set of 14 body joints are detected, , head, neck, left and right shoulders, left and right elbows, left and right wrists, left and right hips, left and right knees, and left and right ankles, as shown in the second column of Fig. <ref> Body part discovery and affine projection. From the detected joints, 10 body parts can be depicted (the third column of Fig. <ref>). The parts include head, torso, upper and lower arms (left and right), and upper and lower legs (left and right), which almost cover the whole body. These quadrilateral parts are projected to rectangles using affine transformations. In more details, the head is defined with the joints of head and neck, and we manually set the width of each head box to 2/3 of its height (from head to neck). An upper arm is confined by the shoulder and elbow joints, and the lower arm by the elbow and wrist joints. The width of the arms boxes is set to 20 pixels. Similarly, the upper and lower legs are defined by the hip and knee joints, and the knee and ankle joints, respectively. Their widths are both 30 pixels. The torso is confined by four body joints, , the two shoulders and the two hips, so we simply draw a quadrangle for the torso. Due to pose estimation errors, the affine transformation may encounter singular values. So in practice, we add some small random disturbance when the pose estimation confidence of a body part is below a threshold (set to 0.4). Three types of PoseBoxes. In several previous works discussing the performance of different parts, a common observation is that the torso and legs make the largest contributions <cit.>. This is expected because the most distinguishing features exist in the upper-body and lower-body clothes. Based on the existing observations, this paper builds three types of PoseBoxes as described below. * PoseBox 1. It consists of the torso and two legs. A leg is comprised of the upper and the lower legs. PoseBox 1 includes two most important body parts and is a baseline for the other two PoseBox types. * PoseBox 2. Based on PoseBox 1, we further add the left and right arms. An arm includes the upper and lower arm sub-modules. In our experiment we show that PoseBox 2 is superior to PoseBox 1 due to the enriched information brought by the arms. * PoseBox 3. On the basis of PoseBox 2, we put the head box on top of the torso box. It is shown in <cit.> that the inclusion of head brought marginal performance increase. In our case, we find that PoseBox 3 slightly inferior to PoseBox2, probably because of the frequent head/neck estimation errors. Remarks. The advantage of PoseBox is two-fold. First, the pose variations can be corrected. Second, background noise can be removed largely. PoseBox is also limited in two aspects. First, pose estimation errors often happen, leading to imprecisely detected joints. Second, PoseBox is designed manually, so it is not guaranteed to be optimal in terms of information loss or re-ID accuracy. We address the two problems by a fusion scheme to be introduced in Section <ref>. For the second problem, specifically, we note that we construct PoseBoxes manually because current re-ID datasets do not provide ground truth poses, without which it is not trivial to design an end-to-end learning method to automatically generate normalized poses. §.§ Baselines This paper constructs two baselines based on the original pedestrian image and PoseBox, respectively. According to the results in the recent survey <cit.>, the identification model <cit.> outperforms the verification model <cit.> significantly on the Market-1501 dataset <cit.>: the former makes full use of the re-ID labels, , the identity of each bounding box, while the latter only uses weak labels, , whether two boxes belong to the same person. So in this paper we adopt the identification CNN model (Fig. <ref>). Specifically, this paper uses the standard AlexNet <cit.> and Residual-50 <cit.> architectures. We refer readers to the respective papers for detailed network descriptions. During training, we employ the default parameter settings, except editing the last FC layer to have the same number of neurons as the number of distinct IDs in the training set. During testing, given an input image resized to 224×224, we extract the FC7/FC8 activations for AlexNet, and the Pool5/FC activations for ResNet-50. After ℓ_2 normalization, we use Euclidean distance to perform person retrieval in the testing set. With respect to the input image type, two baselines are used in this paper: * Baseline1: the original image (resized to 224×224) is used as input to CNN during training and testing. * Baseline2: the PoseBox (resized to 224×224) is used as input to CNN during training and testing. Note that only one PoseBox type is used each time. §.§ The PoseBox Fusion (PBF) Network Motivation. During PoseBox construction, pose estimation errors and information loss may happen, leading to compromised quality of the PoseBox (see Fig. <ref>). On the one hand, pose estimation errors often happen, as we use an off-the-shelf pose estimator (which is usually the case under practical usage). As illustrated in Fig. <ref> and Fig. <ref>, pose estimation may fail when the detections have missing parts or the pedestrian images are of low resolution. On the other hand, when cropping human parts from a bounding box, it is inevitable that important details are missed out, such as bags and umbrellas (Fig. <ref>). The failure in the construction of high-quality PoseBoxes and the information loss during part cropping may result in compromised results of the baseline 2. This is confirmed in the experiment that baseline 1 yields superior re-ID accuracy to baseline 2. For the first problem, , the pose estimation errors, we can mostly foretell the quality of pose estimation by resorting to the confidence scores (examples can be seen in Fig. <ref>). Under high estimation confidence, we envision fine quality of the generated PoseBox. But when the pose estimation confidence scores are low for some body parts, it may be expected that the constructed PoseBox has poor quality. For the second problem, the missing visual cues can be rescued by re-introducing the original image, so that the discriminative details are captured by the deep network. Network. Given the above considerations, this paper proposes a three-stream PoseBox Fusion (PBF) network which takes the original image, the PoseBox, and the confidence vector as input (see Fig. <ref>). To leverage the ImageNet <cit.> pre-trained models, two types of image inputs, , the original image and the PoseBox are resized to 256×256 (then cropped randomly to 227×227) for AlexNet <cit.> and 224×224 for ResNet-50 <cit.>. The third input, , pose estimation confidence scores, is a 14-dim vector, in which each entry falls within the range [0, 1]. The two image inputs are fed to two CNNs of the same structure. Due to the content differences of the original image and its PoseBox, the two streams of convolutional layers do not share weights, although they are initialized from the same seed model. The FC6 and FC7 layers are connected to these convolutional layers. For the confidence vector, we add a small FC layer which projects the 14-dim vector to a 14-dim FC vector. We concatenate the three inputs at the FC7 layer, which is further fully connected to FC8. The sum of the three Softmax losses is used for loss computation. When the ResNet-50 <cit.> is used instead of AlexNet, Fig. <ref> does not have the FC6 layers, and the FC7 and FC8 layers are known as Pool5 and FC. In Fig. <ref>, as denoted by the green bounding box, the pose invariant embedding (PIE) can either be the concatenated FC7 activations (4,096+4,096+14 = 8,206-dim) or its next fully connected layer (751-dim and 1,160-dim for Market-1501 and CUHK03, respectively). For AlexNet, we denote the two PIE descriptors as PIE(A, FC7) and PIE(A, FC8), respectively; for ResNet-50, they are termed as PIE(R, Pool5) and PIE(R, FC), respectively. During training, batches of the input triplets (the original image, its PoseBox, and the confidence vector) are fed into PBF, and the sum of the three losses is back-propagated to the convolutional layers. The ImageNet pretrained model initializes both the original image and PoseBox streams. During testing, given the three inputs of an image, we extract PIE as the descriptor. Note that we apply ReLU on the extracted embeddings, which produces superior results according to our preliminary experiment. Then the Euclidean distance is used to calculate the similarity between the probe and gallery images, before a sorted rank list is produced. PBF has three advantages. First, the confidence vector is an indicator whether PoseBox is reliable. This improves the learning ability of PBF as a static embedding network, so that a global tradeoff between the PoseBox and the original image can be found. Second, the original image not only enables a fallback mechanism when pose estimation fails, but also retrains the pedestrian details that may be lost during PoseBox construction but are useful in discriminating identities. Third, the PoseBox provides important complementary cues to the original image. Using the correctly predicted joints, pedestrian matching can be more accurate with the well-aligned images. The influence of detection errors and pose variations can thus be reduced. § EXPERIMENT §.§ Dataset This paper uses three datasets for evaluation, , VIPeR <cit.>, CUHK03 <cit.>, and Market-1501 <cit.>. The VIPeR dataset contains 632 identities, each having 2 images captured by 2 cameras. It is evenly divided into training and testing sets, each consisting of 316 IDs and 632 images. We perform 10 random train/test splits and calculate the averaged accuracy. The CUHK03 dataset contains 1,360 identities and 13,164 images. Each person is observed by 2 cameras, and on average there are 4.8 images under each camera. We adopt the single-shot mode and evaluate this dataset under 20 random train/test splits. The Market-1501 dataset is featured by 1,501 IDs, 19,732 gallery images and 12,936 training images captured by 6 cameras. Both CUHK03 and Market-1501 are produced by the DPM detector <cit.>. The Cumulative Matching Characteristics (CMC) curve is used for all the three datasets, which encodes the possibility that the query person is found within the top n ranks in the rank list. For Market-1501 and CUHK03, we additionally employ the mean Average Precision (mAP), which considers both the precision and recall of the retrieval process <cit.>. The evaluation toolbox provided by the Market-1501 authors is used. §.§ Experimental Setups Our experiments directly employ the off-the-shelf convolutional pose machines (CPM) trained using the multi-stage CNN model trained on the MPII human pose dataset <cit.>. Default settings are used with input images resized to 384×192. For the PBF network, we replace the convolutional layers with those from either the AlexNet <cit.> or ResNet-50 <cit.>. When AlexNet is used, n_1 = 4,096, n_2 = 14, n_3 = 751. When ResNet-50 is used, PBF will not have the FC6 layer, and the FC7 layer is denoted by Pool5: n_1 = 2,048, n_3 = 751. We train the PBF network for 36 epochs. The initial learning rate is set to 0.01, and is reduced by 10x every 6 epochs. We run the deep learning experiments using GTX1080 under the Caffe framework <cit.> and the batch size is set to 32 and 16 using AlexNet and ResNet-50, respectively. For both CNN models, it takes 6-7 hours for the training process to converge on the Market-1501 dataset. We train PIE on Market-1501 and CUHK03, respectively, which have relatively large data volumes. We also test the generalization ability of PIE on some smaller datasets such as VIPeR. That is, we only extract features using the model pre-trained on Market-1501, and then learn some distance metric on the small datasets. §.§ Evaluation Baselines. We first evaluate the the two re-ID baselines described in Section <ref>. The results on three datasets are shown in Table <ref>. Two major conclusions can be drawn. First, we observe that very competitive performance can be achieved by baseline 1, , training with the original image. Specifically, on Market-1501, we achieve rank-1 accuracy of 55.49% and 73.02% using AlexNet and ResNet-50, respectively. These numbers are consistent with those reported in <cit.>. Moreover, we find that FC7 (Pool5) is superior to FC8 (FC) on Market-1501 but situation reverses on CUHK03. We speculate the CNN model is trained to be more specific to the Market-1501 training set due to its larger data volume, so retrieval on Market-1501 is more of a transfer task than CUHK03. This is also observed in transferring ImageNet models to other recognition tasks <cit.>. Second, compared with baseline 1, we can see that baseline 2 is to some extent inferior. On the Market-1501 dataset, for example, results obtained by baseline 2 is 3.3% and 8.9% lower using AlexNet and ResNet-50, respectively. The performance drop is expected due to the pose estimation errors and information loss mentioned in Section <ref>. Since this paper only employs the off-the-shelf pose estimator, we speculate in the future that the PoseBox baseline can be improved by re-training pose estimation using newly labeled data on the re-ID datasets. The effectiveness of PIE. We test PIE on the re-ID benchmarks, and present the results in Table <ref> and Fig. <ref>. Comparing with baseline 1 and baseline 2, we observe clearly that PIE yields higher re-ID accuracy. On Market-1501, for example, when using AlexNet and the FC7 descriptor, our method exceeds the two baselines by +5.5% and +8,8% in rank-1 accuracy, respectively. With ResNet-50, the improvement becomes slightly smaller, but still arrives at +5.0% and +6.8%, respectively. Specifically, rank-1 accuracy and mAP on Market-1501 arrive at 78.65% and 53.87%, respectively. On CUHK03 and VIPeR, consistent improvement over the baselines can also be observed. Moreover, Figure <ref> shows that Kissme <cit.> marginally improves the accuracy, proving that the PIE descriptor is well-learned. The concatenation of the Pool5 features of baseline 1 and 2 coupled with Kissme produces lower accuracy compared with “PIE(Pool5)+kissme”, illustrating that the PBF network learns more effective embeddings than learning separately. We also find that the 2,048-dim “PIE(Pool5,img)+EU” and “PIE(Pool5,pb)+EU” outperforms the corresponding baseline 1 and 2. This suggests that PBF improves the baseline performance probably through the back propagation of the fused loss. Comparison of the three types of PoseBoxes. In Section <ref>, three types of PoseBoxes are defined. Their comparison results on Market-1501 are shown in Fig. <ref>. Our observation is two-fold. First, PoseBox2 is superior to PoseBox1. On Market-1501 dataset, PoseBox2 improves the rank-1 accuracy by xx% over PoseBox1. The inclusion of arms therefore increases the discriminative ability of the system. Since the upper arm typically shares the same color/texture with the torso, we speculate that it is the long/short sleeves that enhance the descriptors. Second, PoseBox2 has better performance than PoseBox3 as well. For PoseBox3, the integration of the head introduces more noise due to the unstable head detection, which deteriorates the overall system performance. Nevertheless, we find in Fig. <ref> that the gap between different PoseBoxes decreases after being integrated in PBF. It is because the combination with the original image reduces the impact of estimation errors and the information loss, a contribution mentioned in Section <ref>. Ablation experiment. To evaluate the effectiveness of different components of PBF, ablation experiments are conducted on the Market-1501 dataset. We remove one component from the full system at a time, including the PoseBox, the original image, the confidence vector, and the two losses of the PoseBox and original image streams. The CMC curves are drawn in Fig. <ref>, from which three conclusions can be drawn. First, when the confidence vector or the two losses are removed, the remaining system is inferior to the full model, but displays similar accuracy. The performance drop is approximately 1% in the rank-1 accuracy. It illustrates that these two components are important regularization terms. The confidence vector informs the system of the reliability of the PoseBox, thus facilitating the learning process. The two identification losses provide additional supervision to prevent the performance degradation of the two individual streams. Second, after the removal of the stream of the original image (“-img”), the performance drops significantly but still remains superior to baseline 2. Therefore, the original image stream is very important, as it reduces re-ID failures that likely result from pose estimation errors. Third, when the PoseBox stream is cut off (“-PoseBox”), the network is inferior to the full model, but is better than baseline 1. This validates the indispensability of PoseBox, and suggests that the confidence vector improves baseline 1. Comparison with the state-of-the-art methods. On Market-1501, we compare PIE with the state-of-the-art methods in Table <ref>. It is clear that our method outperforms these latest results by a large margin. Specifically, we achieve rank-1 accuracy = 77.97%, mAP = 52.76% using the single query mode. To our knowledge, we have set new state of the art on the Market-1501 dataset. On CUHK03, comparisons are presented in Table <ref>. When metric learning is not used, our results are competitive in rank-1 accuracy with recent methods such as <cit.>, but are superior in rank-5, 10, 20, and mAP. When Kissme <cit.> is employed, we report higher results: rank-1 = 67.10%, and mAP = 71.32%, which exceed the current state of the art. We note that in <cit.>, very high results are reported on the hand-drawn subset but no results can be found on the detected set. We also note that metric learning yields smaller improvements on Market-1501 than CUHK03, because the PBF network is better trained on Market-1501 due to its richer annotations. On VIPeR, we extract features using the off-the-shelf PIE model trained on Market-1501, and the comparison is shown in Table <ref>. We first compare PIE (using Euclidean distance) with the latest unsupervised methods, , the Gaussian of Gaussian (GoG) <cit.>, the Bag-of-Words (BOW) <cit.> descriptors, . We use the available code provided by the authors. We observe that PIE exceeds the competing methods in the rank-1, 5, and 10 accuracies. Then, compared with supervised works without feature fusion, our method (coupled with Mirror Representation <cit.> and MFA <cit.>) has decent results. We further fuse the PIE descriptor with the pre-computed transferred deep descriptors <cit.> and the LOMO descriptor <cit.>. We employ the mirror representation <cit.> and the MFA distance metric coupled with the Chi Square kernel. The fused system achieves new state of the art on the VIPeR dataset with rank-1 accuracy = 54.49%. Two groups of sample re-ID results are shown in Fig. <ref>. In the first query, for example, the cyan clothes on the background lead to the misjudgement of the foreground characteristics, so that some pedestrians with local green/blue colors incorrectly receive top ranks. Using PIE, foreground can be effectively cropped, leading to more accurate pedestrian matching. § CONCLUSION This paper explicitly addresses the pedestrian misalignment problem in person re-identification. We propose the pose invariant embedding (PIE) as pedestrian descriptor. We first construct PoseBox with the 16 joints detected with the convolutional pose machine <cit.>. PoseBox helps correct the pose variations caused by camera views, person motions and detector errors and enables well-aligned pedestrian matching. PIE is thus learned through the PoseBox fusion (PBF) network, in which the original image is fused with the PoseBox and the pose estimation confidence. PBF reduces the impact of pose estimation errors and detail loss during PoseBox construction. We show that PoseBox yields fair accuracy when used alone and that PIE produces competitive accuracy compared with the state of the art. ieee

http://arxiv.org/abs/1701.07850v1

20170126192353

Mg, Al, Si, Ca, Ti, Fe, and Ni abundance for a sample of solar analogues

astro-ph.SR

firstpage–lastpage 2016 Characterization of the quantum phase transition in a two-mode Dicke model for different cooperation numbers E. Nahmad-Achar Accepted XXX. Received YYY; in original form ZZZ ============================================================================================================ We report on the determination of chemical abundances of 38 solar analogues, including 11 objects previously identified as super metal-rich stars. We have measured the equivalent widths for 34 lines of 7 different chemical elements (Mg, Al, Si, Ca, Ti, Fe, and Ni) in high-resolution (ℛ∼ 80 000) spectroscopic images, obtained at the Observatorio Astrofísico Guillermo Haro (Sonora, Mexico), with the Cananea High-resolution Spectrograph. We derived chemical abundances using ATLAS12 model atmospheres and the Fortran code MOOG. We confirmed the super metallicity status of 6 solar analogues. Within our sample, BD+60 600 is the most metal-rich star ([Fe/H]=+0.35 dex), while for HD 166991 we obtained the lowest iron abundance ([Fe/H]=-0.53 dex). We also computed the so-called [Ref] index for 25 of our solar analogues, and we found, that BD+60 600 ([Ref]=+0.42) and BD+28 3198 ([Ref]=+0.34) are good targets for exoplanet search. stars: solar-type; stars: abundances; techniques: spectroscopic. § INTRODUCTION Stellar chemical composition represents an important parameter in stellar and galactic astronomy studies, and, in particular, in the relatively recent field of exoplanets. In this latter field, different studies have aimed at searching for possible correlations between properties (mainly chemical composition) of host stars and the occurrence of exoplanets. <cit.>, with the search for exoplanets still in its early stages, suggested a link between high metal content of host stars and the presence of giant gaseous planets. Such correlation was later confirmed by other authors (e.g., ; ; ) and it agrees with the core accretion theory for planet formation (; ), where high metallicity facilitates the formation of giant gas planets. Within this scenario, the iron abundance ([Fe/H]) is commonly used as proxy for overall metallicity; however, <cit.> suggested the use of a new metallicity index, called [Ref], which takes into account the mass abundance of the refractory elements Mg, Si, and Fe, since their number densities and condensation temperatures are very similar. This [Ref] index is more sensitive (mainly at values greater than +0.20 dex) than [Fe/H] to describe the incidence probability of giant planets orbiting a star <cit.>. The present work is the continuation of a global project aimed at determining atmospheric parameters and chemical abundances of solar analogues (main sequence stars with spectral types between G0 and G3)[In our sample the stars HD 130948 and HD 168874 have a different spectral type; nevertheless, we included them, because their atmospheric parameters are compatible with the rest of the sample.], with special interest in looking for giant exoplanet host star candidates. In <cit.>, we simultaneously determined the basic stellar atmospheric parameters [effective temperature (T_ eff), surface gravity (log g), and global metallicity ([M/H])], for a sample of 233 solar analogues, using intermediate-resolution spectra (ℛ∼ 1700 at 4300 Å) and a set of Lick-like indices defined within 3800–4800 Å. We determined for the first time the atmospheric parameters for 213 stars, of which 20 are new super metal-rich star candidates (SMR; [M/H]≥0.16 dex). The second goal of our project is the analysis of chemical abundances, which we started with the determination of the lithium abundance of a sample of 52 stars <cit.>. The analysis was carried out using narrow band high-resolution spectra (ℛ∼ 80 000) centred on the 6708 Å lithium feature. This sample included 12 SMR objects from our previous work <cit.>. In this third part of the series, we complement the lithium abundance with the chemical abundances of Mg, Al, Si, Ca, Ti, Fe, and Ni, for 38 solar analogues. The sample and the observations are described in Section 2. In Section 3, we detail the determination of the chemical abundances, and, in Section 4, we discuss the results. § STELLAR SAMPLE AND OBSERVATIONS We selected 38 objects among the brightest stars of <cit.>. In Table <ref>, we list the name of the star, the visual magnitude, the spectral type and the atmospheric parameters (and their uncertainties) for the entire sample. The spectroscopic data were collected at the 2.1 m telescope of the Observatorio Astrofísico Guillermo Haro, located in Mexico, using the Cananea High-resolution Spectrograph (CanHiS). CanHiS is equipped with mid-band filters, that provide access to ∼40 Å wide wavelength intervals in a single diffraction order. We observed the entire sample with a spectral resolving power of ℛ∼ 80 000 and a typical signal-to-noise ratio (S/N) of about 100, using 4 different filters of CanHiS, centred at 5005, 5890, 6310, and 6710 Å, respectively, giving access to lines of Mg, Al, Si, Ca, Ti, Fe, and Ni (Fig. <ref>). We also obtained the solar spectrum reflected by the asteroid Vesta with the same instrumental setup. Per filter and per star, we collected at least 3 exposures, resulting in total exposure times between 1.5 and 3 hours. Data reduction was conducted following the standard procedures of IRAF: bias subtraction, flat-field correction, cosmic-ray removal, wavelength calibration through an internal UNe lamp, and, finally, continuum normalization. We then shifted all the spectra to the rest frame, using a degraded (to our resolution) version of the high-resolution spectrum of the Sun <cit.> as template. For each star (and filter) we co-added single exposures weighted by the S/N to obtain the final spectrum. § ABUNDANCES DETERMINATION We determined the chemical abundances, through a local thermodynamic equilibrium (LTE) analysis, using the driver abfind of the February 2013 version of MOOG <cit.>, which performs an adjustment of the abundance to match a single-line equivalent width (EW). MOOG requires a standard solar composition (we used the solar abundances of ), a model atmosphere, a line list, and an EW measurement to compute atomic abundances. Below we describe in detail each of these requirements. §.§ Photospheric parameters and model atmospheres In order to compute a model atmosphere the basic parameters are required: T_ eff, logg, [M/H], and the microturbulence velocity (ξ). We adopted the T_ eff, logg, and [M/H] values of our previous work <cit.>. For ξ, we used the grid of atmospheric parameters of <cit.>, which includes determination of T_ eff, logg, [M/H], and ξ for 160 FGK stars. We looked within the Takeda's grid the nearest set of the first 3 parameters for each star in our sample, and we assigned the Takeda's determination of ξ to our star. We found ξ values between 0.83 and 1.63 km s^-1, which are in agreement with values determined from synthetic spectra <cit.>. Regarding the atmospheric parameters uncertainties, we used those reported in <cit.>. For those cases where uncertainties were not available, we assigned, for log g and ξ, ± 0.27 dex and ± 0.27 km s^-1, as the typical uncertainty, which is the standard deviation of both log g and ξ distributions of the Takeda's stars with T_ eff within the values of our sample. For the uncertainty of [M/H] we assumed ±0.10 dex as a conservative error. Using the atmospheric parameters reported in Table <ref>, we computed an ATLAS12 <cit.> model atmosphere for each star; we also computed a solar model atmosphere with T_ eff,⊙=5777 K, logg_⊙=4.44 dex, [M/H]_⊙=0.0 dex, and ξ_⊙=1.0 km s^-1. §.§ Line list We extracted the atomic transitions between 4995 and 6730 Å from The Viena Atomic Line Database (VALD, ; ), using the atmospheric parameters of the Sun. With these atomic transitions and the ATLAS12 solar model, we created with SYNTHE (; ; ) a synthetic solar spectrum at the same spectral resolution as our observations. From the Vesta spectrum we selected 34 suitable atomic lines (listed in Table <ref> and shown in Fig. <ref>) of 7 different chemical elements (Mg, Al, Si, Ca, Ti, Fe, and Ni), avoiding weak or saturated lines and blends. <cit.> pointed out that oscillator strengths (log gf) of VALD might not be accurate enough for all the atomic transitions. To correct these possible inaccuracies, we determined the EW (see Section <ref>) for the 34 selected lines in the observed and synthetic solar spectrum; then, we compared both measurements and we modified the log gf until both measurements (observed and synthetic) agreed. For 15 lines, we also slightly modified the central wavelength reported by VALD. The transition parameters from VALD as well as their modifications are reported in Table <ref>. §.§ Equivalent widths The EW determination plays a fundamental role in the abundance determination. Since the EW depends strongly on the local continuum level, it is of crucial importance to determine it as accurately as possible. We implemented the following procedure to establish the local continuum level and to measure the EW. First, by means of a Gaussian fit of a small region (5 Å), we identified and removed the points that form the spectral line of interest, which are points enclosed in a interval of ±3σ from the central wavelength of the line. Then, we passed through an iterative routine the remaining spectrum, which is a combination of neighbouring lines and noise, to remove points above ±2σ their average value in order to identify the local continuum. Finally, we adjusted to the line a Gaussian profile whose integral represents its EW. We estimated the error on the EW applying a Monte Carlo method with 1000 iterations, randomly adding to the spectrum the noise of the local continuum. We checked the consistency of our procedure by means of a comparison of solar line EWs determined in two different works (; ) with those determined by us. We measured in the solar spectrum of <cit.>, also used by Takeda and Neves, the EW for 57 and 178 iron lines reported by and , respectively. From this comparison, which is depicted in Fig. <ref>, we found good agreement, with some small differences, which can be explained by different local continuum levels. §.§ Abundances computation and error budget. For each star and Vesta, we measured the EW of all lines listed in Table <ref>. We rejected, through visual inspection, the lines whose best fit was not accurate enough; these lines vary from star to star. The EWs of Table <ref> were used in MOOG to compute the chemical abundances. For species with more than one analysed transition, we carried out a weighted mean to obtain the final abundance, after having discarded outliers with an iterative 3σ clipping. It is important to note that these two rejection processes could introduce potential biases and different abundance scales in stars with different excluded lines. The first filter is actually a visual inspection that relies on the S/N of the spectra and is not directly associated with abundances, while the sigma clipping is indeed applied directly to abundances, but it was employed in only one Fe line of eight stars. In order to take into account these potential biases, we conducted a Monte Carlo procedure in which we computed the Fe, Ni, and Ti abundance (elements with more available lines within our line list with 18, 6, and 5, respectively) for Vesta and some stars of our sample. We computed the mean abundance of Fe, Ni, and Ti using different size sets of randomly selected lines. After 1000 iterations for each set, element, and star, we demonstrated that the final abundance of these elements in all the cases does not change by more than 0.02 dex on average. We report in Table <ref> the abundances of the 7 atomic elements for our sample; they are given with respect to the solar abundances determined for Vesta[[X/H] = A(X)_ star - A(X)_⊙, where A(X)_⊙ is the computed abundance for Vesta.] (see Table <ref>). The Table also provides the abundance uncertainty and the number of lines used for the abundance determination. Along with the uncertainty on the EW measurement, the error on the stellar parameters is the source that most affects the final abundances. To properly assess it, we constructed a small matrix of abundance variations as a function of the difference in four atmospheric parameters (T_ eff, logg, [M/H], and ξ), taking the solar values as reference. For each absorption line j, we considered the EW measured on the Vesta spectrum and we computed a grid of abundance variations Δ[X/H]_j = [X/H]_j - [X/H]_j,⊙, caused by a difference Δ T_ eff,⊙=150 K, of Δlogg_⊙=0.40 dex, of Δ [M/H]_⊙=0.20 dex, and of Δξ_⊙=0.50 km s^-1. Then, for each star, we obtained the Δ[X/H]_j corresponding to each atmospheric parameter by linearly interpolating this grid, assuming, as parameter value difference, the errors reported in Table <ref>. The error on the abundance derived from each absorption line is the quadratic sum of the error on the atmospheric parameters and the EW. § DISCUSSION §.§ Super metal rich stars and the [Ref] index In our working sample, we included 11 stars considered as SMR ([M/H] ≥ 0.16 dex) in <cit.>. From the present high-resolution analysis, we confirm the SMR status, by means of their iron abundance, for 6 objects, namely BD+60 402, BD+60 600, HD 237200, HD 135633, BD+28 3198, and TYC 3986-3381-1, while, for other 3 stars (BD+47 3218, HD 212809, HD 232824), we obtain [Fe/H] lower than the SMR threshold. However, both BD+47 3218 and HD 212809 have super-solar abundances for all atomic species and some of them are well above the +0.16 dex threshold. The two remaining cases of SMR stars, HD 228356 and TYC 2655-3677-1, are discussed below. The 6 SMR stars are therefore excellent targets to search for giant planet companions. In order to quantify the probability of detecting these planets, we make use of the [Ref] index defined by <cit.>: [Ref] = log(24×10^7.55+ [Mg/H]+28×10^7.53+ [Si/H]+ 56×10^7.47+[Fe/H])-9.538 We report the [Ref] index and [Fe/H] in Table <ref>, where we also provide the probability [𝒫(%)] of hosting a giant planet, obtained from the probability functions of <cit.> and <cit.>, based solely on chemical composition considerations. BD+60 600 (39%) and BD+28 3198 (22%) stand out as the best targets for a giant exoplanet search program. §.§ [X/Fe] behaviour and comparison with literature data. In Figure <ref>, we show the [X/Fe] ratios for the elements included in our analysis. In order to check for consistency with other abundance studies on objects of the solar neighbourhood, we compare our results with the works of <cit.>, <cit.>, and <cit.>, which include LTE abundances for FGKM main sequence stars, within a distance of 150 pc from the Sun. We found good agreement with these previous works. Our Mg, Si, Ca and Ti ratios present a higher scatter than Al, and Ni, nevertheless, this pattern is also present in the comparison sample. The errors in the Ca abundance are, on average, larger than for the other elements and always higher than 0.10 dex. This anomaly is due to fact that the Ca abundance is very sensitive to the error in surface gravity: in fact, we found that σ_logg=0.20 dex produces a difference of 0.08 dex in the Ca abundance, while for the other elements the uncertainty in logg does not affect much the overall error. We found 8 of our stars in the Hypatia catalogue, a compilation of chemical abundances from high-resolution spectroscopy <cit.>, and 2 objects are also present in the more recent work by <cit.>. We found a maximum (minimum) difference of +0.20 dex (-0.02 dex) between our abundances and those of <cit.>. This discrepancy is as large as the typical dispersion among catalogues included in <cit.>. As an example, in Fig. <ref>, we show the comparison of our [Fe/H] values and those of Hypatia for the the stars HD 41708, HD 42807, HD 111513, HD 129357, HD 140385, and HD 156968; we also include the iron abundance of <cit.> for HD 42807 and HD 111513. If we take into account that the solar scale of <cit.>, used as reference by <cit.>, has a iron abundance 0.05 dex lower than in <cit.>, the agreement with our results improves. <cit.> provide the abundance of Si, Ca, Ti, Fe, and Ni for the stars HD 42807 and HD 111513. We found a difference with our results between +0.08 and +0.11 dex for HD 42807 and in the interval -0.15 to +0.07 dex for HD 111513. Such values, although larger than our errors, can be explained by systematic differences, such as different loggf values or different atmospheric parameters adopted. §.§ Stars with broad line profiles Two stars, TYC 2655-3677-1 and HD 228356, show line profiles which are significantly broader that the rest of the sample (see Fig. <ref>). This is due to relatively high rotational velocity (with a possible significant contribution by macroturbulence). These two objects also have high lithium abundance (A(Li)=2.54 for TYC 2655-3677-1 and HD 228653 of A(Li)=2.71; ), indicating that they are probably young stars. Their line profiles, however, are broad enough to make very difficult to identify isolated, un-blended lines for a correct abundance measurement. We have therefore excluded the two stars from our abundance analysis.We measured the FWHM and we computed, using eq. 6 of <cit.>, the projected rotation velocity (v sini) for 6 atomic lines in the region around 6710 Å. We assumed a macroturbulence velocity of 3 km s^-1 and an instrumental FWHM = 0.19 Å and we obtained v sini= 8.5 and 9.7 km s^-1 for TYC 2655-3677-1 and HD 228653, respectively. § ACKNOWLEDGEMENTS This research has made use of the SIMBAD data base, operated at CDS, Strasbourg, France. The authors would like to thank CONACyT for financial support through grants CB-2011-169554 and CB-2015-256961. This research has made use of the SIMBAD data base, operated at CDS, Strasbourg, France. 99 [Adibekyan et al.2012]adibekyan12 Adibekyan, V. Z., Sousa, S. G., Santos, N. C., et al. 2012, A&A, 545, A32 [Alibert, Mordasini & Benz2004]alibert04 Alibert, Y., Mordasini, C., & Benz, W. 2004, A&A, 417, L25 [Bush & Hintz2008]bush08 Bush, T. C., & Hintz, E. G. 2008, AJ, 136, 1061 [Casagrande et al.2011]casagrande11 Casagrande, L., Schönrich, R., Asplund, M., et al. 2011, A&A, 530, A138 [Castelli & Kurucz2003]castelli03 Castelli, F., & Kurucz, R. L. 2003, Modelling of Stellar Atmospheres, IAU Symp., 210, 20P [ESA1997]esa97 ESA 1997, ESA Special Publication, 1200, [Fischer & Valenti2005]fischer05 Fischer, D. A., & Valenti, J. 2005, ApJ, 622, 1102 [Gonzalez1997]gonzalez97 Gonzalez, G. 1997, MNRAS, 285, 403 [Gonzalez2009]gonzalez09 Gonzalez, G. 2009, MNRAS, 399, L103 [Gonzalez2014]gonzalez14 Gonzalez, G. 2014, MNRAS, 443, 393 [Grevesse & Sauval1998]grevesse98 Grevesse, N., & Sauval, A. J. 1998, Space Sci. Rev, 85, 161 [Hinkel et al.2014]hinkel14 Hinkel, N. R., Timmes, F. X., Young, P. A., Pagano, M. D., & Turnbull, M. C. 2014, Aj, 148, 54 [Husser et al.2013]husser13 Husser, T.-O., Wende-von Berg, S., Dreizler, S., et al. 2013, A&A, 553, A6 [Kupka et al.1999]kupka99 Kupka, F., Piskunov, N., Ryabchikova, T. A., Stempels, H. C., & Weiss, W. W. 1999, A&AS, 138, 119 [Johnson et al.2010]johnson10 Johnson, J. A., Aller, K. M., Howard, A. W., & Crepp, J. R. 2010, PASP, 122, 905 [Kurucz1993]kurucz93 Kurucz, R. 1993, SYNTHE Spectrum Synthesis Programs and Line Data. Kurucz CD-ROM No. 18. Cambridge, Mass.: Smithsonian Astrophysical Observatory, 1993., 18, [Kurucz2013]kurucz13 Kurucz, R. L. 2013, Astrophysics Source Code Library, ascl:1303.024 [Kurucz & Avrett1981]kurucz81 Kurucz, R. L., & Avrett, E. H. 1981, SAO Special Report, 391, 391 [Kurucz & Furenlid1979]kurucz79 Kurucz, R. L., & Furenlid, I. 1979, SAO Special Report, 387, 387 [Kurucz et al.1984]kurucz84 Kurucz, R. L., Furenlid, I., Brault, J., & Testerman, L. 1984, National Solar Observatory Atlas, Sunspot, New Mexico: National Solar Observatory, 1984, [Lodders et al.2009]lodders09 Lodders, K., Palme, H., & Gail, H.-P. 2009, Landolt Börnstein, [López-Valdivia et al.2014]lopezval14 López-Valdivia, R., Bertone, E., Chávez, M., et al. 2014, MNRAS, 444, 2251 [López-Valdivia et al.2015]rlopezval15 López-Valdivia, R., Hernández-Águila, J. B., Bertone, E., et al. 2015, MNRAS, 451, 4368 [Mahdi et al.2016]mahdi16 Mahdi, D., Soubiran, C., Blanco-Cuaresma, S., & Chemin, L. 2016, A&A, 587, A131 [Mayor & Queloz1995]mayor95 Mayor, M., & Queloz, D. 1995, Nature, 378, 355 [Neves et al.2009]neves09 Neves, V., Santos, N. C., Sousa, S. G., Correia, A. C. M., & Israelian, G. 2009, A&A, 497, 563 [Pollack et al.1996]pollack96 Pollack, J. B., Hubickyj, O., Bodenheimer, P., et al. 1996, Icarus, 124, 62 [Piskunov et al.1995]piskunov95 Piskunov, N. E., Kupka, F., Ryabchikova, T. A., Weiss, W. W., & Jeffery, C. S. 1995, A&AS, 112, 525 [Santos, Israelian & Mayor2001]santos01 Santos, N. C., Israelian, G., & Mayor, M. 2001, A&A, 373, 1019 [Sneden1973]sneden73 Sneden, C.A. 1973, Ph.D. Thesis, University of Texas, Austin [Sousa et al.2011]sousa11 Sousa, S. G., Santos, N. C., Israelian, G., Mayor, M., & Udry, S. 2011, A&A, 533, A141 [Strassmeier et al.1990]strassmeier90 Strassmeier, K. G., Fekel, F. C., Bopp, B. W., Dempsey, R. C., & Henry, G. W. 1990, ApJS, 72, 191 [Takeda et al.2005]takeda05 Takeda, Y., Ohkubo, M., Sato, B., Kambe, E., & Sadakane, K. 2005, PASJ, 57, 27

http://arxiv.org/abs/1701.07582v2

20170126053407

Near-Infrared Polarimetric Study of N159/N160 Star-Forming Complex in the Large Magellanic Cloud

astro-ph.GA

^1 School of Space Research, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin, Gyeonggi-do 17104, Republic of Korea; jaeyeong@khu.ac.kr ^2 Korea Astronomy and Space Science Institute, 776 Daedeok-daero, Yuseong-gu, Daejeon 34055, Republic of Korea; jeongws@kasi.re.kr ^3 Korea University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of Korea ^4 Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan ^5 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan ^6 Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan ^7 The University of Tokyo/National Astronomical Observatory of Japan/Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan We present near-infrared polarimetric results for the N159/N160 star-forming complex in the Large Magellanic Cloud (LMC) with SIRPOL, the polarimeter of the Infrared Survey Facility. We separated foreground sources using their visual extinction derived from near-infrared photometric data. The 45 young stellar candidates and 2 high-excitation blobs were matched with our sources, and 12 of them showed the high polarization that did not originate from the interstellar dust. We made a polarimetric catalog of 252, 277, and 89 sources at the J, H, and K_s bands, respectively. Based on the ratios of the polarization degree between these bands, we verify that the origin of these polarized sources is the dichroic extinction from the interstellar dust aligned by the magnetic field and that the ratios follow a power-law dependence of P_λ ∼ λ^-0.9. The linear polarization vectors projected onto the Hα image of the complex turned out to follow the local magnetic field structure. The vector map overlaid on dust and gas emissions shows the close correlation between the magnetic field structure and surrounding interstellar medium. We suggest that the derived magnetic field structure supports the sequential formation scenario of the complex. § INTRODUCTION It is important to understand the formation and evolution of stars in molecular clouds, because they are the most fundamental building blocks of the universe. Magnetic fields have been considered as one of the main forces influencing the star formation process, thus there have been many studies on them both theoretically and observationally <cit.>. <cit.> reviewed the observations of magnetic fields by their scales, and discussed the relation between magnetic fields and the formation process of molecular clouds and stars. Since magnetic fields are generally associated with the interstellar medium, revealing their structure of magnetic fields is essential to understand the morphology of star-forming regions and their surrounding interstellar medium. Polarimetry is the general method for examining magnetic fields. Near-infrared polarimetric measurements in star-forming regions provide us with the magnetic field structure by observing the dichroic extinction of background stars, whose starlights are polarized when they pass through the magnetically aligned dust grains within molecular clouds. Various polarimetric studies of magnetic fields have been performed in Galactic star-forming regions <cit.>. The Large Magellanic Cloud (LMC) is one of the nearest galaxies, and therefore it has been one of the best observed targets in studies of the star-forming regions and giant molecular clouds (GMCs) within a galaxy. The northeastern part of the LMC includes the root of the molecular stream extending southward from the 30 Doradus complex and has been the focus of numerous studies of the star-forming regions (i.e. the 30 Doradus, N158, and N159/N160 complexes) and molecular clouds <cit.>. <cit.> first studied the central part of 30 Doradus using near-infrared polarimetric observations, and discovered its magnetic field distribution, which showed a U-shaped structure in the northern area and aligned about 70 and 45 in the western and southeastern areas, respectively. <cit.> observed a wider region of 20 × 20 region around 30 Doradus, and they used the proper motion data to separate the Galactic foreground sources. They confirmed that most polarized sources showed their polarization angles of 75. <cit.> extended the survey area to 39 × 69, and found patterns following the molecular stream. However, the N159/N160 complex still shows the veiled structure of the magnetic fields. Since this region has active star formation, H2 regions, and interesting molecular cloud structures (see Section <ref> for details), we expect that the magnetic fields might be influenced by those environments to make construct complex structure. This research studied the detailed structure of the magnetic field in the N159/N160 complex. We carried out deep near-infrared polarimetric observations over the two fields of the complex. In Section 2, we describe the N159/N160 complex and review previous studies of this region. Observations and data reduction are described in Section 3. The method to separate foreground sources is also presented in Section 3. The polarimetric results and analysis are presented in Section 4. In Section 5, we discuss the magnetic field structures in the N159/N160 complex with gas and dust distributions, and suggest a star formation scenario in the complex. § THE N159/N160 COMPLEX The N159/N160 complex is located in a molecular ridge, about 600 pc south from the 30 Doradus complex. Ever since <cit.> cataloged H2 regions from the survey of emission nebulae in the LMC, this complex has been the target of various observations <cit.>. The N159/N160 complex is composed of the N160 and N159 H2 regions, and they are spatially separated from each other in the northern and the southern parts of the complex. The Hα observations by the Wide Field Imager (WFI) of the ESO 2.2 m telescope (program ID 076.C-0888[Data are available and described at http://archive.eso.org/cms/eso-data/data-packages/30-doradus/30-doradus-reqpage.html.]) clearly show the detailed structure of the two regions. N160 includes H2 regions and young stellar clusters. Bright H2 regions in N160 are designated using the notation of <cit.>. The main H2 region, with an elongated shape, is composed of N160 A and N160 D (hereafter N160 A+D) at the southwestern part of N160. The southern part of N160 has two bright condensations of H2 regions, N160 B and N160 C (hereafter N160 B+C). <cit.> observed the H2 regions in N160 and reported the physical properties of high-excitation blobs (HEBs) and their surrounding nebulae. HEBs are a rare class of the compact H2 regions in the Magellanic Clouds. <cit.> observed HEBs in N160 A with high spatial resolution. A notable feature of the N160 region is a shell structure opened toward the northeast. The parent clouds of the shell are almost dissipated and young clusters are detected inside the shell <cit.>. Young stellar objects (YSOs) have been observed around the H2 regions in N160 <cit.>. Sequential star formation throughout this region was suggested, based on the various evolutionary stages of massive stars in the OB clusters <cit.>. N159 has bright H2 regions and two GMCs, N159E and N159W, located at the eastern and southwestern parts of the region, respectively. In addition, N159 has embedded interesting objects such as an X-ray binary, a supernova remnant, compact H2 regions, and radio sources <cit.>. <cit.> located these objects in mid-infrared images observed with the Spitzer Space Telescope. Studies of dust emissions unveiled that N159 is younger than N160 and shows active star formation <cit.>. CO observations detected the sites of star formation through cloud-cloud collisions, and revealed the link between high concentrations of molecular gas and star formation <cit.>. Embedded massive YSOs, maser sources, and ultracompact H2 regions are distributed around the molecular clouds <cit.>. § OBSERVATIONS AND DATA REDUCTION §.§ Observations Simultaneous JHK_s polarimetric observations of the N159/N160 complex were performed on 2007 February 3 and 5. We used the near-infrared camera SIRIUS <cit.> and the polarimeter SIRPOL <cit.> of the Infrared Survey Facility (IRSF) 1.4 m telescope at the South African Astronomical Observatory in Sutherland, South Africa. The camera has a field of view of 77 × 77 and a pixel scale of 045 pixel^-1. One set of observations for a target field consisted of 20 s exposures at 10 dithered positions for four wave-plate angles (0, 45, 225, and 675) in the J, H, and K_s bands, and the whole sequence is repeated 10 and 9 times for the N159 and N160 fields centered at (α, δ)_2000 = (5h39m37s.1, -6943451) and (5h40m05s.6, -6936258), respectively. §.§ Data Reduction We used the DAOPHOT package <cit.> of the Image Reduction & Analysis Facility[IRAF is distributed by the US National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.] (IRAF) for detection and photometry of the sources. On each wave-plate frame at J, H, and K_s, the fluxes of the detected sources were estimated by fitting Penny2-type point-spread function (PSF) to them. The pixel coordinates of the sources were converted to celestial coordinates, based on the 2MASS All Sky Point Source Catalog <cit.>. The instrumental magnitudes were converted to 2MASS magnitude following <cit.>. The Stokes parameters I, Q, and U of the sources are calculated using the following equations: I = (I_0+I_22.5+I_45+I_67.5)/2, Q = I_0-I_45, and U = I_22.5-I_67.5, where I_0, I_22.5, I_45, and I_67.5 are the fluxes on each wave-plate angle frame (0, 45, 225, and 675). We calculated the flux errors of the sources and derived their polarization uncertainties as described in <cit.>. The debiased degree of polarization, P, and the polarization angle, θ, were calculated as described in <cit.>. §.§ Separation of Foreground Sources The magnetically aligned dust grains generate the polarized light with E-vector direction parallel to the magnetic field, but only from the stars behind the field. In addition, the light of the foreground stars in front of the ISM field is not affected by the magnetic field of our interest. Therefore, the polarization of the foreground stars contaminates our results that reveal the magnetic fields of the N159/N160 complex. We separated the foreground stars out from the detected sources using their visual extinction, A_V, derived with the Near-Infrared Color Excess Revisited (NICER) technique <cit.>. The NICER technique is the revised version of that of <cit.>, and it can measure the interstellar extinction from multi-band observations of molecular clouds. We estimated visual extinctions of stars in two target fields and a control field with their near-infrared colors using the NICER algorithm[Using the C. Beaumont Interactive Data Language (IDL) implementation]. The control field (yellow box in Figure <ref>(a)) is a nearby extinction-free region centered at (α, δ)_2000 = (5h36m08s.3, -6926002). The JHK_s photometric data of the control field are from <cit.>. Figure <ref>(b) shows the distribution of visual extinctions in the control and target fields. The histogram for the control field peaks at 0.01 mag and with a standard deviation of 0.47 mag. Based on these values, we regarded the sources with A_V ≤ 0.48 mag as the foreground stars in front of our target field. A total of 1062 and 640 sources in the N159 and N160 fields, respectively, remained as the background stars for examining the polarimetric properties. § RESULTS §.§ Highly Polarized Sources Highly polarized sources (HPSs) are mostly due to the intrinsic polarization by circumstellar dust around the stars themselves, not due to the dichroic absorption by foreground dust <cit.>. These sources should be separated from from our polarization sources to trace the magnetic fields that make interstellar dust align. YSOs are one of the plausible sources of the intrinsic polarization, due to the dust grains in their circumstellar disk and envelope. The N159/N160 complex contains quite a number of YSOs. We matched our sources with the YSO candidates from <cit.> and <cit.>. In addition, we used the catalog of the AKARI Infrared Camera (IRC) survey in the LMC <cit.> and classified new YSO candidates in this complex, using the color-color diagram (Figure <ref>). Most sources in the diagram follow the sequence of dusty C-rich stars denoted group CC3 in <cit.>, but the sources redder than CC3 were found to be above [N3]-[S7] ∼ 3.0 mag. <cit.> mentioned that these redder sources are the YSO candidates with the water ice absorption at 3 μm. We considered the nine redder sources to be new YSO candidates in the N159/N160 complex. Two HEBs and one YSO from <cit.> were also detected in our observations. A total of 45 YSO candidates and 2 HEBs in previous works were found in our catalog. Twelve of them show polarization degrees larger than 10 %, and polarization signal-to-noise ratios (hereafter P/σ_P) greater than 3. In Table <ref>, we compiled near-infrared photometric, polarimetric data, and information from other literature for the all 45 matched YSO candidates. Figure <ref> a) shows the polarization vectors of the 12 highly polarized sources and locations of the matched YSO candidates on the Hα image from the WFI/ESO archive. We excluded the highly polarized sources listed in Table <ref> in the analysis. The Other 35 YSO candidates did not satisfy the polarization quality criterion of P/σ_P > 3 or showed the polarization pattern caused by the dichroic extinction due to interstellar dust. A fraction of the intrinsic component might exist in their polarization, but the dominant fraction is probably the dichroic effect due to the aligned interstellar dust, which is strong enough to make the intrinsic component negligible. §.§ Estimation of the Polarization Accuracy In order to study the magnetic field in the N159/N160 complex, we set the selection criterion of P/σ_P > 3. Most sources with this criterion showed their polarization degrees smaller than 10 % <cit.>. In general, fainter sources have larger polarization uncertainties than brighter ones. Figure <ref> shows the trend between polarization uncertainty and the magnitude of our sources. We set the photometric limits at J, H, and K_s to be 17.0, 16.7, and 14.5 mag for the N159 field and 16.5, 16.2, and 14.0 mag for the N160 field, respectively, so that stars brighter than these limits have a polarization uncertainty smaller than 1 %. We assume that the polarization of sources brighter than these magnitude limits are measured with sufficiently a high accuracy. The criteria for selecting final catalog sources are given in Table <ref>. Table <ref> lists the photometric and polarimetric data of the selected sources in the N159 and N160 fields. The A_V values obtained using the NICER algorithm are also listed. In order to confirm the interstellar polarization of the catalog sources, we examined the efficiency of the interstellar polarization for the 252, 277, and 89 sources in the N159/N160 complex. In general, interstellar polarization gives the upper limit of the polarization degree, depending on the near-infrared color <cit.>. Figure <ref> shows that most of the sources have polarization degrees lower than the upper limit of the interstellar polarization at each band. On the contrary, most of the foreground sources exceed the upper limits. Hence, we conclude that the polarization of the sources is mainly caused by interstellar dust aligned due to the magnetic field in the N159/N160 complex. §.§ Wavelength Dependence of Polarization <cit.> empirically modeled the wavelength dependence of polarization for the Milky Way. The empirical relationship can be represented by a power law of P_λ ∝ λ^-β, with β of 1.6-2.0 at the near-infrared wavelengths from 1.25 to 2.2 μm <cit.>. In the case of the LMC, however, the relation has a shallower slope than that of the Milky Way, with β = 0.9 <cit.>. Figure <ref> plots our selected sources with P/σ_P > 5 and σ_P ≤ 1 % to show the wavelength dependence of the polarization degrees. The best-fit slopes of P_H/P_J = 0.71 and P_K_s/P_H = 0.76 indicate that the interstellar polarization of the N159/N160 complex also follows the power-law index of β = 0.9, rather than the case of the Milky Way. §.§ Linear Polarization Vector Map We can assume that the direction of a polarization vector is parallel to the sky-projected component of the magnetic field. In Figure <ref>, we draw the polarization vectors of our best-quality sources on the Hα image to map the magnetic field structure of the N159/N160 complex. The magnetic field structure shows different trends between N159 and N160. In the N160 region, polarization vectors are roughly aligned with the axis of the open Hα shell. However, the vectors in the northwestern region (red dotted ellipse in Figure <ref>) clearly follow a different pattern, when compared with those inside of the Hα shell. The vectors in the south of N160 B+C and N160 A+D show a U-shaped distribution (green dotted curve in Figure <ref>), which is similar to the Hα feature in this region. The magnetic field structure inferred from the distribution of polarization vectors in the N160 region supports the expanding shell in the region. N159 shows a complex magnetic field structure associated with Hα emission. Polarization vectors in N159E are distributed around the lower part, while those in N159W are on the outskirts of the Hα-emitting cloud. The western side of N159 has no significant Hα emission, and the polarization vectors in this region are mostly aligned with the polarization angle of about 30. The peaks in the distribution of the polarization angles of each region in Figure <ref> are good indicators of the major directions of the magnetic fields. Those directions are also appeared in the polarization vector map (Figure <ref>). In the case of N160, one of the two peaks, around 30, is related to the magnetic field inside the Hα shell, while the other, around 170, is related to the field on the northwestern part. Histograms for N159 have a single peak around 10-30 and this is mostly caused by the magnetic field on the western side of the region. The polarized sources in N159E and N159W have various polarization angles and are responsible for the relatively large dispersion of the histogram compared to that for N160. § DISCUSSION §.§ Magnetic Field Structures with Molecular Cloud Emissions The magnetic field structures in star-forming regions are constrained by the environment (gas and dust). To reveal the nature of the magnetic field structures in the N159/N160 complex, we have compared our data with the dust and gas emissions. To trace the emission features of dust and ionized gas in the N159/N160 complex, the IRAC 24 μm data from the Spitzer SAGE <cit.> and the Herschel 100 μm data from the HERITAGE project <cit.> were combined with the vector map on the Hα image(Figure <ref>). As seen in Figure <ref>, most of the Hα shell structures in N160 encounter the boundary of dust emission at 100 μm. This trend is more clearly seen at the southern shell structure rather than the northern one. The red dotted ellipse region of the northern shell shows a leak feature of the Hα emission extending to the dust cavity, and the polarization vectors in this region are well-aligned in the direction of the cavity. The southern regions of N160 B+C and N160 A+D show the U-shaped feature of the polarization vectors at the edge of strong Hα and 100 μm emissions. In addition, the magnetic field structures inside the shell structure are distributed along the direction from northeast to southwest, which is consistent with the triggered direction of the sequential cluster formation suggested by <cit.>. We also found that some polarization vectors are seen along the shell structure like an expanding shell. Polarization vectors in the N159 region are mostly found near H2 regions showing the dust emissions at 24 μm and 100 μm (yellow-composite feature in Figure <ref>). In order to understand the complex structure of the magnetic fields in N159, we compared our vector map with the CO gas distribution. <cit.> observed the ^12CO(J = 4-3) and ^12CO(J = 3-2) rotational lines in N159 with the NANTEN2 <cit.> and ASTE <cit.> submillimeter telescopes, respectively. We compared the integrated intensity map of ^12CO(J = 3-2) <cit.> with our polarization vector map on the Hα image in Figure <ref>. Strong concentrations of the CO distribution (white X symbols in Figure <ref>) are located at the regions where Hα emission is obscured by the high extinction of the molecular gas. In the case of N159W, the magnetic fields appear around the 24 μm emission and CO peak. The configuration of the complex magnetic field structure in N159 appears to have been affected by the expansion and evolution of the H2 regions in the central N159 and the southern N159W. <cit.>, <cit.>, and <cit.> proposed expansions of the H2 regions in N159E and N159W. They also reported that the locations of several compact H2 regions and young stellar clusters in N159 are likely to be linked with the molecular gas and are mainly distributed on the edge of the ionized H2 regions in N159E and N159W. The western region of N159 does not show any interesting features of dust and gas emissions, while the magnetic fields are in a uniform direction, as shown by the polarization vectors at this region. It indicates that this region was not affected by the star-forming activities in the N159/N160 complex and that a larger scale magnetic field was already formed toward this region before the beginning of the star formation in the nascent molecular cloud in N159 and N160. In conclusion, the polarization vector map with dust and gas emission features indicates that the magnetic field structures in N159 and N160 were affected by different star-forming activities in each region. These different patterns of the magnetic field support the suggestion that N159 and N160 are in the different evolutionary stages based on the large-scale sequential star formation. §.§ Formation Scenario of the N159/N160 Complex and Magnetic Field Structure The spatial distributions of the Herbig Ae/Be and OB clusters in the N159/N160 complex at the near-infrared bands were revealed by <cit.>. Using the spatial correlations between these young stellar clusters and gaseous components, they suggested a large-scale cluster formation scenario that occurred sequentially from north to south. Possible starting points of the formation are the supergiant shell LMC 2 and a superbubble located northeast and east of the complex, respectively. However, the dominant magnetic field direction in N160 shows the pattern along the northeast to southwest, which is in the same direction to the LMC 2 rather than that to the superbubble. The similar direction between the magnetic field and the sequential formation shows that the LMC 2 is a plausible trigger for this large-scale cluster formation. Figure <ref> (a) shows an illustration of the sequential formation process of the N159/N160 complex and the magnetic field structures at each scenario step. The RGB (Red: Hα, Green: V, and Blue: B) composite image of the complex is also displayed to help in understanding each illustrated feature (Figure <ref> (b)). LMC 2 (blue-colored amorphous feature), the molecular clouds (orange-colored cloudy feature) of the 30 Doradus complex, and the molecular ridge are shown in Step 1, based on the large-scale Hα map of Figure 5(a) in <cit.>. <cit.> classified H1 shell structures in the LMC by using their shell size, expansion pattern, morphological structure, and associated Hα emission. The LMC 2 was classified as an expanding supergiant shell. At Step 1, the interaction between the expansion of LMC 2 and the parent cloud of the complex might influence the initial star formation at N160. As we showed in our previous paper <cit.>, the boundary between the western border of LMC 2 and the large-scale pattern of magnetic fields (red dashed line) is well-aligned in the eastern side of the molecular ridge. This may support the process of the expansion of LMC 2 and its triggering of star formation. At Step 2, the OB star formation begins in the parent cloud of N160. The first-formed cluster, HS 385, is located at the northeastern region of N160, as denoted by a big blue star symbol in Step 3. We suggest that a shock wave driven by stellar radiation and stellar winds from HS 385 sequentially triggered the formation of OB stars through this parent cloud (yellow arrow in Step 3). Actually, most of the OB clusters in the N160 region are detected inside of the shell, as denoted by blue-colored stars in Figure <ref> and in this illustration. In addition, their age distribution from <cit.> indicated that HS 385 is older than other OB clusters in the N160 region, and the southwestern region of N160 contains the Herbig Ae/Be clusters, with their ages less than 3 Myr <cit.>. Step 4 illustrates the expansion of H2 regions energized by the OB clusters within the shell, which passed through the south of N160 B+C and N160 A+D, and toward the GMCs of N159. The dominant direction of the magnetic field structure at the interior of the shell (red dotted line in Step 4) is coincided with that of the sequential formation process. The magnetic fields near the southern shell trace its boundary as shown in Figure <ref>. The U-shaped magnetic field structure in the south of N160 B+C and N160 A+D is bent toward the central region of N159. These features indicate that magnetic fields are associated with the expansion of the H2 region in N160. It also implies that the path is propagated sequentially from N160 to the nearby GMC of N159. A similar propagation process is also suggested in the Scorpius-Centaurus association by <cit.>. They proposed a scenario for the sequentially triggered star formation and dissipation of molecular clouds in three subgroups of Scorpius-Centaurus association, by stellar winds and ionizing radiation from the OB stars therein. Steps 5 and 6 illustrate the triggered star formation and the related magnetic fields in N159. According to <cit.>, formation of OB stars at the center of N159 pushed H2 regions out to the surrounding molecular cloud, and the subsequent star formation then triggered at the rim of the expansion bubble (dotted circle in Step 6). The distribution of star formation rates from <cit.> supports this triggering process. The magnetic fields in the central part of N159 are distribute on the H2 region that is considered to be the initial trigger site for N159. Subsequent triggering processes can also be traced by the magnetic fields, following the boundary of the triggered H2 regions (sky-colored clouds in Step 6) located around the rim of the expansion bubble. We suggest that the dynamical star-forming activities and expansion of the H2 bubble disturbed and rearranged the nascent magnetic field structure. Uniformly aligned magnetic fields at the southeast and the west of N159, which do not show any star-forming activities and H2 regions, support the change in the magnetic field because of these effects. § SUMMARY We conducted near-infrared polarimetry for the stars in the N159/N160 star-forming complex in the LMC. We applied the NICER algorithm to select background stars for the study of the magnetic field in our observation field. The 36 YSO candidates and 2 HEBs were matched, and 12 of them were the sources that are highly polarized, likely by the dust in the circumstellar envelope rather than by the interstellar dust. We excluded them from the final catalog to study only the interstellar polarization in the N159/N160 complex. We newly verified nine YSO candidates using the color-color diagram for the sources matched with the AKARI IRC LMC catalog. The 252, 277, and 89 sources in the N159 and N160 fields with sufficiently good polarization qualities represent dichroic polarization from the interstellar dust in this complex. As other studies <cit.> have suggested, wavelength dependence of polarization turned out to be weaker here than in the case of the Milky Way. We visualized the structure of the magnetic fields in the N159 and N160 fields, using a polarization vector map projected on the Hα image. Polarization vectors showed a complex distribution of the magnetic fields, indicating the interaction between star-forming activities and surrounding interstellar medium. The comparison between the polarization vectors and molecular cloud emissions showed that the magnetic fields are resulted from the different formation histories of N159 and N160. From the distribution of the magnetic field structures, we suggested the plausible scenario that the star formation is sequentially triggered from N160 to N159. We also proposed that ionizing radiation from OB clusters and the expanding H2 bubbles in this complex probably affected the nascent magnetic field structure. This work was supported by the National Research Foundation of Korea (NRF) grant No. 2008-0060544, funded by the Korean government (MSIP). We would like to thank Prof. Yasushi Nakajima for kindly providing comments that improved this paper. This paper uses observations performed at the South African Astronomical Observatory. This publication makes use of data products from the Two Micron All Sky Survey and observations with AKARI. The Two Micron All Sky Survey is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. The AKARI is a JAXA project with the participation of ESA. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. [Bernard et al.(2016)]ber16 Bernard, A., Neichel, B., Samal, M. R., et al. 2016, , 592, A77 [Bica et al.(1996)]bic96 Bica, E. L. D., Claria, J. J., Dottori, H., Santos, J. F. C., Jr., & Piatti, A. E. 1996, , 102, 57 [Bolatto et al.(2000)]bol00 Bolatto, A. D., Jackson, J. M., Israel, F. P., Zhang, X., & Kim, S. 2000, , 545, 234 [Carlson et al.(2012)]car12 Carlson, L. R., Sewilo, M., Meixner, M., Romita K. A., & Lawton, B. 2012, , 542, A66 [Chen et al.(2010)]che10 Chen, C. R., Indebetouw, R., Chu, Y.-h., et al. 2010, , 721, 1206 [Chu et al.(1997)]chu97 Chu Y.-H., Kennicutt, R. C., Snowden, S. L., et al. 1997, , 109, 554 [Cohen et al.(1988)]coh88 Cohen, R. S., Dame, T. M., Garay, G., Montani, J., Rubio, M. & Thaddeus, P. 1988, , 331, L95 [Cowley et al.(1995)]cow95 Cowley, A. P., Schmidtke, P. C., Anderson, A. L., & McGrath, T. K. 1995, , 107, 145 [Crutcher(2012)]cru12 Crutcher, R. M. 2012, , 50, 29 [Dobbs et al.(2014)]dob14 Dobbs, C. L., Krumholz, M. R., Ballesteros-Paredes, J., et al. 2014, in Protostars and Planets VI, ed. H. Beuther et al. (Tucson, AZ: Univ. Arizona Press), 3 [Ezawa et al.(2004)]eza04 Ezawa, H., Kawabe, R., Kohno, K., Yamamoto, S., & ASTE Team 2004, , 5489, 763 [Fariña et al.(2009)]far09 Fariña, C., Bosch, G., Morrell, N. I., Barbá, R. H., & Walborn, N. R. 2009, , 138, 510 [Fukui et al.(2015)]fuk15 Fukui, Y., Harada, R., Tokuda, K., et al. 2015, , 807, L4 [Fukui et al.(2008)]fuk08 Fukui, Y., Kawamura, A., Minamidani, T., et al. 2008, , 178, 56 [Fukui et al.(1999)]fuk99 Fukui, Y., Mizuno, N., Yamaguchi, R., et al. 1999, , 51, 745 [Galametz et al.(2013)]gal13 Galametz, M., Hony, S., Galliano, F., et al. 2013, , 431, 1596 [Heiles & Crutcher(2005)]hei05 Heiles, C. & Crutcher, R. 2005, in Cosmic Magnetic Fields, ed. R. Wielebinski & R. Beck (Lecture Notes in Physics, Vol. 664; Berlin: Springer), 137 [Henize(1956)]hen56 Henize, K. G. 1956, , 2, 315 [Henning et al.(1998)]henn98 Henning, Th., Klein, R., Chan, S. J., et al. 1998, , 338, L51 [Heydari-Malayeri et al.(1999)]hey99 Heydari-Malayeri, M., Rosa, M. R., Charmandaris, V., Daharveng, L., & Zinnecker, H. 1999, , 352, 665 [Heydari-Malayeri & Testor(1986)]hey86 Heydari-Malayeri, M. & Testor, G. 1986, , 162, 180 [Hunt & Whiteoak(1994)]hunt94 Hunt, M. R., & Whiteoak, J. B. 1994, , 11, 68 [Ita et al.(2008)]ita08 Ita, Y., Onaka, T., Kato, D., et al. 2008, , 60, S435 [Johansson et al.(1998)]joh98 Johansson, L. E. B., Greve, A., Booth, R. S., et al. 1998, , 331, 857 [Jones(1989)]jon89 Jones, T. J. 1989, , 346, 728 [Jones et al.(1986)]jon86 Jones, T. J., Hyland, A. R., Straw, S., et al. 1986, , 219, 603 [Jones et al.(2005)]jon05 Jones, T. J., Woodward, C. E., Boyer, M. L., Gehrz, R. D., & Polomski, E. 2005, , 620, 731 [Kandori et al.(2006)]kan06 Kandori, R., Kusakabe, N., Tamura, M., et al. 2006, , 6269, 159 [Kandori et al.(2007)]kan07 Kandori, R., Tamura, M., Kusakabe, N., et al. 2007, , 59, 487 [Kato et al.(2012)]kat12 Kato, D., Ita, Y., Onaka, T., et al. 2012, , 144, 179 [Kim et al.(1999)]kim99 Kim, S., Dopita, M. A., Staveley-Smith, L., & Bessell, M. S. 1999, , 118, 2797 [Kim et al.(2016)]kim16 Kim, J., Jeong, W.-S., Pak, S., Park, W.-K., & Tamura, M. 2016, , 222, 2 [Kim et al.(2011)]kim11 Kim, J., Pak, S., Choi, M., et al. 2011, JKAS, 44, 135 [Kusakabe et al.(2008)]ku08 Kusakabe, N., Tamura, M., Kandori, R., et al. 2008, , 136, 621 [Kusune et al.(2015)]kus15 Kusune, T., Sugitani, K., Miao, J., et al. 2015, , 798, 60 [Kutner et al.(1997)]kut97 Kutner, M. L., Rubio, M., Booth, R. S, et al. 1997, , 122, 255 [Kwon et al.(2011)]kwon11 Kwon, J., Tamura, M., Kandori, R., et al. 2011, , 741, 35 [Lada et al.(1994)]lad94 Lada, C. J., Lada, E. A., Clemens, D. P., & Bally, J. 1994, , 429, 694 [Li et al.(2014)]li14 Li, H.-B., Goodman, A., Sridharan, T. K., et al. 2014, in Protostars and Planets VI, ed. H. Beuther et al. (Tucson, AZ: Univ. Arizona Press), 101 [Lombardi & Alves(2001)]lom01 Lombardi, M., & Alves, J. 2001, , 377, 1203 [MacLow & Klessen(2004)]mac04 MacLow, M.-M., & Klessen, R. S. 2004, Rev. Mod. Phys., 76, 125 [Martín-Hernández et al.(2008)]mar08 Martín-Hernández, N. L., Peeters, E., & Tielens, A. G. G. M. 2008, , 489, 1189 [Martin et al.(1992)]mar92 Martin, P. G., Adamson, A. J., Whittet, D. C. B., et al. 1992, , 392, 691 [Martin & Whittet(1990)]mar90 Martin, P. G., & Whittet, D. C. B. 1990, , 357, 113 [McKee & Ostriker(2007)]mc07 McKee, C. F. and Ostriker, E. C. 2007, , 45, 565 [Meixner et al.(2010)]meix10 Meixner, M., Galliano, F., Hony, S., et al. 2010, , 518, L71 [Meixner et al.(2006)]meix06 Meixner, M., Gordon, K. D., Indebetouw, R., et al. 2006, , 132, 2268 [Mizuno et al.(2010)]miz10 Mizuno, Y., Kawamura, A., Onishi, T., et al. 2010, , 62, 51 [Nagata(1990)]nagt90 Nagata, T. 1990, , 348, 13 [Nagayama et al.(2003)]nag03 Nagayama, T., Nagashima, C., Nakajima, Y., et al. 2003, , 4841, 459 [Nakajima et al.(2005)]nak05 Nakajima, Y., Kato, D., Nagata, T., et al. 2005, , 129, 776 [Nakajima et al.(2007)]nak07 Nakajima, Y., Kandori, R., Tamura, M., et al. 2007, , 59, 519 [Ott et al.(2008)]ott08 Ott, J., Wong, T., Pineda, J. L., et al. 2008, , 25, 129 [Padoan et al.(2004)]pad04 Padoan, P., Jimenez, R., Juvela, M., & Nordlund, Å. 2004, , 604, L49 [Pak et al.(1998)]pak98 Pak, S., Jaffe, D. T., van Dishoeck, E. F., et al. 1998, , 498, 735 [Pineda et al.(2009)]pin09 Pineda, J. L., Ott, J., Klein, U., Wong, T., Muller, E., & Hughes, A. 2009, , 703, 736 [Points et al.(2001)]poi01 Points, S. D., Chu, Y.-H., Snowden, S. L., & Smith, R. C. 2001, , 136, 99 [Preibisch & Zinnecker(1999)]pre99 Preibisch, Th. & Zinnecker, H. 1999, , 117, 2381 [Preibisch & Zinnecker(2007)]pre07 Preibisch, Th. & Zinnecker, H. 2007, in IAU Symp. 237, Triggered Star Formation in a Turbulent ISM, ed. B. G. Elmegreen & J. Palous (Dordrecht: Kluwer) [Saito et al.(2009)]sai09 Saito, H., Tamura, M., Kandori, R., et al. 2009, , 137, 3149 [Santos et al.(2014)]san14 Santos, F. P., Franco, G. A. P., Roman-Lopes, A., Reis, W., & Román-Zúñiga, C. G. 2014, , 783, 1 [Seok et al.(2013)]seok13 Seok, J. Y., Koo, B.-C., & Onaka, T. 2013, , 779, 134 [Serkowski, Mathewson, & Ford(1975)]ser75 Serkowski, K., Mathewson, D. S., & Ford, V. L. 1975, , 196, 261 [Seward et al.(2010)]sew10 Seward, F. D., Williams, R. M., Chu, Y.-H., Gruendl, R. A., & Dickel, J. R. 2010, , 140, 177 [Shimonishi et al.(2008)]shi08 Shimonishi, T., Onaka, T., Kato, D., et al. 2008, , 686, L99 [Shu, Adams, & Lizano(1987)]shu87 Shu, F. H., Adams, F. C., & Lizano, S. 1987, , 25, 23 [Skrutskie et al.(2006)]skr06 Skrutskie, M. F., Cutri, R. M., Stiening, M. D., et al. 2006, , 131, 1163 [Stetson(1987)]ste87 Stetson, P. B. 1987, , 99, 191 [Tamura et al.(2007)]tam07 Tamura, M., Kandori, R., Hishimoto, J., et al. 2007, , 59, 467 [Testor et al.(2006)]tes06 Testor, G., Lemaire, J. L., Field, D., & Diana, S. 2006, , 453, 517 [Testor et al.(2007)]tes07 Testor, G., Lemaire, J. L., Kristensen, L. E., Field, D., & Diana, S. 2007, , 469, 459 [van Loon et al.(2010)]van10 van Loon, J. T., Oliveira, J. M., Gordon, K. D., et al. 2010, , 139, 68 ccccccccccccccccccccccccccccc 0pt Photometric and polarimetric catalog of YSOs and HPSs the N159/N160 field 2cPosition 6cMagnitude 13cPolarization Properties HPS 2cAKARI colorc 1-2 4-9 11-23 28-29 1cα_∘ J2000.0 1cδ_∘ J2000.0 2cJ 2cH 2cK_s 2cP_J 2cP_H 2cP_K_s 1c 2cθ_J 2cθ_H 2cθ_K_s Av References Typea No.b 1cS7-S11 1cN3-S7 1c 1c 2c(mag) 2c(mag) 2c(mag) 2c(%) 2c(%) 2c(%) 1c 2c() 2c() 2c() (mag) 1c(mag) 1c(mag) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) 5 39 29.02 -69 47 18.69 16.763 0.047 15.791 0.066 14.576 0.058 3.87 3.86 12.57 3.41 18.49 3.73 14.25 20.20 62.04 7.48 48.67 5.65 7.01 1 ya 1 ⋯ ⋯ 5 40 0.48 -69 47 13.02 14.502 0.059 13.583 0.043 12.395 0.028 12.93 3.28 3.06 2.68 ⋯ ⋯ 159.60 7.04 26.79 18.84 ⋯ ⋯ 6.80 1,3 yc,CC5 2 1.40 3.68 5 40 3.40 -69 47 10.02 15.461 0.007 14.849 0.006 14.707 0.013 1.72 0.51 ⋯ ⋯ ⋯ ⋯ 163.49 8.09 ⋯ ⋯ ⋯ ⋯ 0.24 1 yc ⋯ ⋯ ⋯ 5 39 33.62 -69 47 1.14 16.726 0.028 16.124 0.026 15.505 0.019 3.13 1.93 ⋯ ⋯ 2.40 2.17 143.59 15.05 ⋯ ⋯ 37.1 19.17 3.23 1,2 yb,ybc ⋯ ⋯ ⋯ 5 39 40.60 -69 46 30.43 16.861 0.055 16.278 0.063 15.546 0.060 8.59 3.63 7.88 3.79 ⋯ ⋯ 27.48 11.14 145.28 12.39 ⋯ ⋯ 3.86 1 yb ⋯ ⋯ ⋯ 5 39 41.68 -69 46 11.31 16.412 0.036 14.416 0.056 12.193 0.045 ⋯ ⋯ 13.03 3.48 9.61 2.58 ⋯ ⋯ 5.98 7.38 7.14 7.43 13.64 1,2 yab,ya 3 ⋯ ⋯ 5 39 37.39 -69 46 8.81 14.573 0.067 13.769 0.083 12.907 0.083 12.92 3.51 5.36 4.80 ⋯ ⋯ 73.11 7.49 93.53 19.08 ⋯ ⋯ 4.82 1 yab 4 ⋯ ⋯ 5 39 35.89 -69 46 3.32 15.451 0.043 14.408 0.058 13.405 0.051 3.02 2.21 ⋯ ⋯ ⋯ ⋯ 60.37 16.86 ⋯ ⋯ ⋯ ⋯ 5.78 1 ya ⋯ ⋯ ⋯ 5 39 43.56 -69 45 39.68 18.559 0.056 17.214 0.042 15.942 0.031 ⋯ ⋯ 10.70 2.86 5.70 3.15 ⋯ ⋯ 126.55 7.38 107.14 13.82 7.45 1,2 yc,ya 5 ⋯ ⋯ 5 39 59.23 -69 45 26.02 16.326 0.025 13.882 0.013 11.771 0.004 7.18 1.51 4.56 0.68 3.18 0.29 123.14 5.87 125.68 4.20 133.07 2.59 12.53 1,3 yb,CC5 ⋯ 1.08 3.20 5 39 52.56 -69 45 16.84 13.883 0.062 13.803 0.020 13.738 0.015 ⋯ ⋯ 2.32 1.26 1.72 1.01 ⋯ ⋯ 98.28 13.62 109.01 14.41 -0.24 1 yc ⋯ ⋯ ⋯ 5 40 9.33 -69 44 53.59 17.918 0.045 16.830 0.052 15.931 0.067 11.67 3.60 7.60 2.87 ⋯ ⋯ 13.00 8.42 77.47 10.11 ⋯ ⋯ 5.19 1 yab 6 ⋯ ⋯ 5 39 45.03 -69 44 50.32 14.693 0.039 14.309 0.043 13.942 0.054 3.32 2.22 ⋯ ⋯ ⋯ ⋯ 128.82 15.88 ⋯ ⋯ ⋯ ⋯ 1.60 1 yc ⋯ ⋯ ⋯ 5 39 44.35 -69 44 34.88 16.603 0.026 15.770 0.020 15.218 0.036 ⋯ ⋯ 1.03 1.02 ⋯ ⋯ ⋯ ⋯ 133.88 20.09 ⋯ ⋯ 2.84 1,2 yab,ya ⋯ ⋯ ⋯ 5 40 3.46 -69 43 55.38 17.993 0.058 16.730 0.078 15.832 0.096 ⋯ ⋯ 5.64 4.12 ⋯ ⋯ ⋯ ⋯ 50.66 16.85 ⋯ ⋯ 5.60 1 ysc ⋯ ⋯ ⋯ 5 39 29.75 -69 43 33.11 18.660 0.052 17.019 0.028 15.871 0.027 ⋯ ⋯ 7.93 2.83 9.00 3.11 ⋯ ⋯ 26.76 9.60 24.00 9.34 6.64 1 yb ⋯ ⋯ ⋯ 5 39 58.23 -69 47 18.80 16.216 0.029 15.394 0.042 15.246 0.041 ⋯ ⋯ ⋯ ⋯ 5.92 2.70 ⋯ ⋯ ⋯ ⋯ 77.83 11.86 0.43 2 ya ⋯ ⋯ ⋯ 5 39 30.82 -69 46 48.06 17.169 0.021 16.444 0.016 16.204 0.030 4.32 1.55 5.38 1.21 ⋯ ⋯ 149.11 9.67 143.25 6.28 ⋯ ⋯ 0.87 2 ya ⋯ ⋯ ⋯ 5 39 3.77 -69 45 22.15 16.770 0.011 15.802 0.015 15.508 0.022 ⋯ ⋯ 0.92 0.84 5.89 2.38 ⋯ ⋯ 95.04 19.32 158.45 10.73 1.21 2 ya ⋯ ⋯ ⋯ 5 39 40.69 -69 45 15.91 19.334 0.090 18.145 0.040 17.358 0.074 ⋯ ⋯ 4.06 3.58 ⋯ ⋯ ⋯ ⋯ 117.82 18.93 ⋯ ⋯ 4.50 2 ya ⋯ ⋯ ⋯ 5 39 40.18 -69 44 54.75 16.499 0.021 16.179 0.037 16.132 0.051 3.15 1.30 ⋯ ⋯ 6.83 3.42 89.36 10.93 ⋯ ⋯ 109.46 12.8 -0.36 2 yb ⋯ ⋯ ⋯ 5 39 53.15 -69 44 16.84 17.779 0.032 17.029 0.029 16.746 0.050 7.03 2.59 3.59 2.59 ⋯ ⋯ 90.63 9.88 147.58 16.75 ⋯ ⋯ 1.20 2 yb ⋯ ⋯ ⋯ 5 40 0.98 -69 44 7.32 16.495 0.015 15.962 0.028 15.926 0.043 ⋯ ⋯ ⋯ ⋯ 4.27 3.03 ⋯ ⋯ ⋯ ⋯ 161.77 16.54 -0.38 2 yb ⋯ ⋯ ⋯ 5 39 21.16 -69 44 8.25 16.431 0.015 15.526 0.018 14.949 0.015 1.59 0.95 1.08 0.97 1.87 1.65 147.03 14.65 138.25 19.06 87.87 18.93 2.97 2 ya ⋯ ⋯ ⋯ 5 39 3.85 -69 44 8.54 16.599 0.008 15.986 0.011 15.890 0.021 ⋯ ⋯ 1.59 0.84 9.24 2.61 ⋯ ⋯ 129.36 13.35 170.73 7.76 -0.04 2 ya ⋯ ⋯ ⋯ 5 39 11.01 -69 43 53.08 17.215 0.015 16.607 0.015 16.568 0.040 2.86 1.32 1.66 1.28 14.99 4.33 51.84 11.98 173.32 17.46 105.16 7.93 -0.37 2 ya 7 ⋯ ⋯ 5 40 15.40 -69 43 1.28 17.103 0.013 16.313 0.015 16.082 0.033 ⋯ ⋯ 3.72 1.08 2.85 2.69 ⋯ ⋯ 4.49 7.99 41.81 19.63 0.83 2 ya ⋯ ⋯ ⋯ 5 40 19.37 -69 43 0.12 17.639 0.034 17.000 0.028 16.919 0.057 10.18 2.86 ⋯ ⋯ 9.44 5.74 107.91 7.75 ⋯ ⋯ 37.36 14.86 -0.06 2 ybc 8 ⋯ ⋯ 5 39 15.01 -69 42 41.27 18.181 0.036 17.479 0.032 17.198 0.061 9.54 3.81 4.59 2.82 ⋯ ⋯ 51.32 10.60 124.97 14.96 ⋯ ⋯ 1.19 2 yab ⋯ ⋯ ⋯ 5 40 15.32 -69 42 23.43 17.192 0.036 16.383 0.053 16.187 0.056 ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ 0.80 2 ya ⋯ ⋯ ⋯ 5 39 58.75 -69 41 10.03 17.894 0.030 17.234 0.026 17.053 0.048 6.85 3.21 3.07 2.19 ⋯ ⋯ 56.29 12.14 28.76 16.63 ⋯ ⋯ 0.54 2 ya ⋯ ⋯ ⋯ 5 39 13.78 -69 40 27.42 18.758 0.054 17.823 0.029 17.551 0.082 ⋯ ⋯ 9.96 3.69 ⋯ ⋯ ⋯ ⋯ 117.57 9.94 ⋯ ⋯ 1.27 2 ya ⋯ ⋯ ⋯ 5 40 26.68 -69 39 36.28 17.108 0.023 16.472 0.012 16.313 0.053 2.89 2.64 5.98 1.64 ⋯ ⋯ 70.31 19.30 15.67 7.55 ⋯ ⋯ 0.39 2 ya ⋯ ⋯ ⋯ 5 39 38.67 -69 39 3.53 17.646 0.057 15.902 0.032 14.392 0.034 27.67 6.71 8.61 2.29 8.98 2.62 99.32 6.74 45.25 7.34 36.49 8.01 8.94 2 ya 9 ⋯ ⋯ 5 40 47.19 -69 37 5.67 17.111 0.022 16.248 0.008 16.166 0.059 3.05 1.76 1.96 1.44 ⋯ ⋯ 32.53 14.31 97.95 16.98 ⋯ ⋯ -0.05 2 ya ⋯ ⋯ ⋯ 5 39 8.48 -69 44 14.26 16.004 0.011 15.288 0.006 15.120 0.013 1.58 0.87 1.26 0.43 3.42 1.40 32.11 13.81 176.29 9.19 7.03 10.88 0.41 3 CC5 ⋯ 0.78 4.52 5 39 37.58 -69 45 40.16 14.308 0.007 12.975 0.007 12.346 0.003 2.11 0.33 1.41 0.34 0.41 0.23 179.34 4.40 163.88 6.81 176.72 14.29 3.26 3 CC5 ⋯ 0.58 5.04 5 40 1.07 -69 43 23.96 16.759 0.013 16.111 0.015 15.990 0.026 2.78 1.11 4.29 1.06 ⋯ ⋯ 52.67 10.56 87.22 6.85 ⋯ ⋯ 0.13 3 CC5 ⋯ 1.46 4.87 5 40 2.78 -69 41 18.75 17.397 0.012 16.659 0.016 16.570 0.036 6.13 1.61 4.72 1.21 10.26 4.41 2.54 7.26 96.26 7.13 126.54 11.28 -0.06 3 CC5 ⋯ 1.78 4.05 5 39 31.19 -69 36 37.14 15.983 0.012 14.638 0.007 14.152 0.019 4.07 1.09 1.56 0.67 3.33 1.45 129.88 7.40 110.7 11.31 14.27 11.39 2.38 3 CC5 ⋯ 1.02 3.39 5 39 49.48 -69 38 3.27 15.352 0.009 15.199 0.005 15.160 0.024 1.80 0.63 2.06 0.55 ⋯ ⋯ 155.72 9.49 39.01 7.35 ⋯ ⋯ -0.39 3 CC5 ⋯ 1.70 4.45 5 39 52.72 -69 36 34.69 16.876 0.015 15.784 0.003 15.491 0.028 9.76 1.65 3.98 0.79 8.78 3.55 166.06 4.76 59.33 5.53 135.37 10.73 1.19 3 CC5 ⋯ 0.78 4.01 5 39 59.46 -69 37 29.89 16.900 0.027 15.178 0.021 13.274 0.011 4.66 2.35 3.16 1.43 ⋯ ⋯ 173.23 12.90 79.40 11.79 ⋯ ⋯ 11.28 3 CC5 ⋯ 1.69 3.81 5 40 11.57 -69 33 15.60 13.936 0.023 13.354 0.015 13.095 0.015 ⋯ ⋯ ⋯ ⋯ 2.16 1.25 ⋯ ⋯ ⋯ ⋯ 105.00 14.30 0.98 3 CC5 ⋯ 0.77 3.38 5 39 43.23 -69 38 54.01 13.722 0.082 13.011 0.083 12.374 0.068 25.49 4.81 15.01 6.14 9.35 5.38 11.18 5.30 29.57 10.84 8.27 14.27 3.42 4 HEB 10 ⋯ ⋯ 5 39 45.90 -69 38 39.32 13.640 0.087 13.017 0.068 11.156 0.078 ⋯ ⋯ ⋯ ⋯ 52.79 6.99 ⋯ ⋯ ⋯ ⋯ 173.87 3.76 10.51 4 HEB 11 ⋯ ⋯ 5 39 44.12 -69 38 33.47 15.213 0.061 14.886 0.045 14.199 0.047 8.81 4.93 24.38 4.82 5.18 2.85 142.24 13.98 43.44 5.55 16.10 13.80 3.52 4 YSO 12 ⋯ ⋯ Information for each column of the table is described below: Columns (1)-(2) Equatorial coordinates (J2000.0) in decimal degrees; Columns (3)-(8) J, H, and K_s magnitude and error; Columns (9)-(14) J, H, and K_s polarization degree and error in units of percentage; Columns (15)-(20) J, H, and K_s polarization position angle and error in units of degrees; Columns (21) Visual extinction in unit of magnitudes; Columns (22) References; Columns (23) Types of the matched objects by the literature; Columns (24) Numbering of highly polarized sources that matched with YSO candidates; Columns (25)-(26) Mid-infrared colors from AKARI IRC point-source catalog of the LMC in unit of magnitudes. aEvolutionary stages for YSO candidates are labeled as `ya', `yb', and `yc', indicating their stages from I to III, which refer to <cit.> and <cit.>. The label `ysc', `HEB', and `YSO' indicate young stellar cluster <cit.>, HEB objects and massive YSO from <cit.>, respectively. The label `CC5' indicates YSO candidates with water ice absorption, classified by <cit.>. bFor the sources that matched with the YSOs candidates from the literature are listed. cMid-infrared colors that calculated from AKARI IRC point-source catalog of the LMC <cit.>. (1) <cit.>, (2) <cit.>, (3) <cit.>, (4) <cit.>. cccccccc Criteria for the best quality polarization in the N159/N160 field 0pt 3cN159 3cN160 2-4 6-8 1cJ 1cH 1cK_s 1cJ 1cH 1cK_s P/σ_P >3 >3 P <10 <10 Mag ≤17.0 ≤16.7 ≤14.5 ≤16.5 ≤16.2 ≤14.0 Total number 159 157 56 93 120 33 ccccccccccccccccccccccccccc 0pt Photometric and polarimetric catalog of the sources in the N159/N160 field of the LMC a 2cPosition 6cMagnitude 13cPolarization Properties 2cNICER 2-3 5-10 12-24 26-27 Field 1cα_∘ J2000.0 1cδ_∘ J2000.0 2cJ 2cH 2cK_s 2cP_J 2cP_H 2cP_K_s 2cθ_J 2cθ_H 2cθ_K_s 2cAv 1c 1c 2c(mag) 2c(mag) 2c(mag) 2c(%) 2c(%) 2c(%) 2c() 2c() 2c() 2c(mag) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) N159 5 39 39.31 -69 47 21.83 15.363 0.006 14.195 0.010 13.828 0.004 3.97 0.38 2.01 0.61 2.94 0.57 47.44 2.71 61.31 8.28 50.31 5.45 1.63 0.90 N159 5 40 13.13 -69 47 16.40 17.320 0.013 16.545 0.022 16.305 0.100 ⋯ ⋯ 4.92 1.49 ⋯ ⋯ ⋯ ⋯ 83.32 8.30 ⋯ ⋯ 1.05 1.09 N159 5 39 43.87 -69 47 13.21 16.756 0.013 15.539 0.013 15.046 0.016 8.02 1.16 6.00 0.80 2.54 1.55 55.28 4.11 52.22 3.80 70.53 14.92 2.44 0.90 N159 5 39 27.05 -69 47 11.64 16.133 0.009 14.809 0.013 14.302 0.009 4.03 0.66 3.77 0.71 2.52 0.68 28.63 4.62 48.45 5.33 31.62 7.44 2.52 0.90 N159 5 39 45.63 -69 47 08.93 17.051 0.009 16.138 0.012 15.851 0.023 5.89 1.18 2.77 0.88 3.63 2.43 45.11 5.64 58.10 8.64 88.84 15.94 1.16 0.91 N159 5 39 24.00 -69 47 01.85 16.601 0.011 15.954 0.014 15.759 0.022 4.82 0.90 1.73 0.91 7.64 2.95 54.51 5.27 132.06 13.39 98.95 10.3 0.59 0.91 N159 5 40 01.63 -69 46 55.35 13.383 0.004 12.517 0.008 12.289 0.002 1.51 0.26 ⋯ ⋯ 0.27 0.27 173.08 4.79 ⋯ ⋯ 18.53 20.07 0.77 0.90 N159 5 39 25.47 -69 46 59.48 16.882 0.015 16.060 0.009 15.807 0.021 3.60 0.92 2.20 0.64 3.83 2.16 64.98 7.11 42.07 8.02 4.73 14.04 0.94 0.91 N159 5 39 30.82 -69 46 48.06 17.169 0.021 16.444 0.016 16.204 0.030 4.32 1.55 5.38 1.21 ⋯ ⋯ 149.11 9.67 143.25 6.28 ⋯ ⋯ 0.87 0.92 N159 5 39 50.21 -69 46 43.60 16.487 0.013 15.801 0.010 15.611 0.020 3.75 0.90 ⋯ ⋯ 6.39 2.17 24.23 6.66 ⋯ ⋯ 103.61 9.22 0.55 0.91 Information for each column of the table is given below: Columns (1) Target field names; Columns (2)-(3) Equatorial coordinates (J2000.0) in decimal degrees; Columns (4)-(9) J, H, and K_s magnitude and error; Columns (10)-(15) J, H, and K_s polarization degree and error in units of percentage; Columns (16)-(21) J, H, and K_s polarization position angle and error in units of degrees; Columns (22)-(23) Visual extinction and uncertainty in unit of magnitudes. aOnly a portion of the catalog is listed in Table <ref>.

http://arxiv.org/abs/1701.07970v2

20170127083421

Seasonal Modulation of the $^7$Be Solar Neutrino Rate in Borexino

hep-ex

GSSI]M. Agostini Munchen]K. Altenmüller Munchen]S. Appel Kurchatov]V. Atroshchenko Milano]D. Basilico Milano]G. Bellini PrincetonChemEng]J. Benziger Hamburg]D. Bick LNGS]G. Bonfini Kurchatov]L. Borodikhina Virginia,Milano]D. Bravo Milano]B. Caccianiga Princeton]F. Calaprice Genova]A. Caminata Milano]S. Caprioli LNGS]M. Carlini LNGS,Virginia]P. Cavalcante Lomonosov]A. Chepurnov Honolulu]K. Choi Milano]D. D'Angelo GSSI,Genova]S. Davini Peters]A. Derbin GSSI]X.F. Ding Genova]L. Di Noto GSSI,Peters]I. Drachnev Dubna]K. Fomenko APC]D. Franco Princeton]F. Froborg LNGS]F. Gabriele Princeton]C. Galbiati Genova]C. Ghiano Milano]M. Giammarchi Munchen]M. Goeger-Neff Princeton]A. Goretti Lomonosov]M. Gromov Hamburg]C. Hagner APC]T. Houdy Huston]E. Hungerford LNGS]Aldo IanniCanfranc Princeton]Andrea Ianni Krakow]A. Jany Munchen]D. Jeschke Kiev]V. Kobychev Dubna]D. Korablev Huston]G. Korga APC]D. Kryn LNGS]M. Laubenstein Dresda]B. Lehnert Kurchatov,Kurchatovb]E. Litvinovich LNGS]F. LombardiDIEGO Milano]P. Lombardi Juelich,RWTH]L. Ludhova Kurchatov]G. Lukyanchenko Kurchatov,Kurchatovb]I. Machulin Virginia]S. ManeckiQueens Genova]G. Manuzio GSSI]S. Marcocci Mainz]J. Martyn Milano]E. Meroni Hamburg]M. Meyer Milano]L. Miramonti Krakow]M. Misiaszek Ferrara]M. Montuschi Peters]V. Muratova Munchen]B. Neumair Munchen]L. Oberauer Hamburg]B. Opitz Perugia]F. Ortica Genova]M. Pallavicini Munchen]L. Papp UMass]A. Pocar Milano]G. Ranucci LNGS]A. Razeto Milano]A. Re Perugia]A. Romani LNGS,APC]R. Roncin LNGS]N. Rossi Munchen]S. Schönert Peters]D. Semenov Peters]P. Shakina Kurchatov,Kurchatovb]M. Skorokhvatov Dubna]O. Smirnov Dubna]A. Sotnikov LNGS]L.F.F. Stokes UCLA,Kurchatov]Y. Suvorov LNGS]R. Tartaglia Genova]G. Testera Dresda]J. Thurn Kurchatov]M. Toropova Peters]E. Unzhakov Dubna]A. Vishneva Virginia]R.B. Vogelaar Munchen]F. von Feilitzsch UCLA]H. Wang Mainz]S. Weinz Krakow]M. Wojcik Mainz]M. Wurm Virginia]Z. Yokley Dubna]O. Zaimidoroga Genova]S. Zavatarelli Dresda]K. Zuber Krakow]G. Zuzel [cc]Corresponding author: spokeperson-borex@lngs.infn.it [Canfranc]Also at: Laboratorio Subterráneo de Canfranc, Paseo de los Ayerbe S/N, 22880 Canfranc Estacion Huesca, Spain [DIEGO]Present address: Physics Department, University of California, San Diego, CA 92093, USA [Queens]Present address: Physics Department, Queen's University, Kingston ON K7L 3N6, Canada The Borexino Collaboration [APC]AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/IRFU, Observatoire de Paris, Sorbonne Paris Cité, 75205 Paris Cedex 13, France [Dubna]Joint Institute for Nuclear Research, 141980 Dubna, Russia [Genova]Dipartimento di Fisica, Università degli Studi e INFN, 16146 Genova, Italy [Hamburg]Institut für Experimentalphysik, Universität Hamburg, 22761 Hamburg, Germany [Heidelberg]Max-Planck-Institut für Kernphysik, 69117 Heidelberg, Germany [Krakow]M. Smoluchowski Institute of Physics, Jagiellonian University, 30059 Krakow, Poland [Kiev]Kiev Institute for Nuclear Research, 03680 Kiev, Ukraine [Kurchatov]National Research Centre Kurchatov Institute, 123182 Moscow, Russia [Kurchatovb] National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409 Moscow, Russia [LNGS]INFN Laboratori Nazionali del Gran Sasso, 67010 Assergi (AQ), Italy [Milano]Dipartimento di Fisica, Università degli Studi e INFN, 20133 Milano, Italy [Perugia]Dipartimento di Chimica, Biologia e Biotecnologie, Università degli Studi e INFN, 06123 Perugia, Italy [Peters]St. Petersburg Nuclear Physics Institute NRC Kurchatov Institute, 188350 Gatchina, Russia [Princeton]Physics Department, Princeton University, Princeton, NJ 08544, USA [PrincetonChemEng]Chemical Engineering Department, Princeton University, Princeton, NJ 08544, USA [UMass]Amherst Center for Fundamental Interactions and Physics Department, University of Massachusetts, Amherst, MA 01003, USA [Virginia]Physics Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA [Ferrara]Dipartimento di Fisica e Scienze della Terra Università degli Studi di Ferrara e INFN, Via Saragat 1-44122, Ferrara, Italy [Munchen]Physik-Department and Excellence Cluster Universe, Technische Universität München, 85748 Garching, Germany [Lomonosov] Lomonosov Moscow State University Skobeltsyn Institute of Nuclear Physics, 119234 Moscow, Russia [GSSI] Gran Sasso Science Institute (INFN), 67100 L'Aquila, Italy [Huston]Department of Physics, University of Houston, Houston, TX 77204, USA [Dresda]Department of Physics, Technische Universität Dresden, 01062 Dresden, Germany [UCLA]Physics and Astronomy Department, University of California Los Angeles (UCLA), Los Angeles, California 90095, USA [Mainz]Institute of Physics and Excellence Cluster PRISMA, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany [Honolulu]Department of Physics and Astronomy, University of Hawaii, Honolulu, HI 96822, USA [Juelich]IKP-2 Forschungzentrum Jülich, 52428 Jülich, Germany [RWTH]RWTH Aachen University, 52062 Aachen, Germany We present the evidence for the seasonal modulation of the ^7Be neutrino interaction rate with the Borexino detector at the Laboratori Nazionali del Gran Sasso in Italy. The period, amplitude, and phase of the observed time evolution of the signal are consistent with its solar origin, and the absence of an annual modulation is rejected at 99.99% C.L. The data are analyzed using three methods: the analytical fit to event rate, the Lomb-Scargle and the Empirical Mode Decomposition techniques, which all yield results in excellent agreement. Solar neutrinos; neutrino oscillations; liquid scintillators detectors; low background detectors. § INTRODUCTION Since 2007 Borexino <cit.> has measured the fluxes of low-energy neutrinos, most notably those emitted in nuclear fusion reactions and β decays along the pp-chain in the Sun. Borexino was the first experiment to make spectroscopic and real-time measurements of solar neutrinos with energy <3 MeV, i.e. below the endpoint energy of long-lived, natural β radioactivity: ^40K and the ^232Th and ^238U decay chains. The detector has made first direct observations of ^7Be <cit.>, pep <cit.>, and pp <cit.> solar neutrinos, lowered the detection threshold for ^8B solar neutrinos <cit.>. These measurements deepen our understanding of Solar Standard Model <cit.> and support the MSW-LMA mechanism of neutrino oscillations. In addition Borexino has detected anti-neutrinos from the Earth and distant nuclear reactors <cit.> and has set a new upper limit for a hypothetical solar anti-neutrinos flux <cit.>. Borexino, located deep underground (3,800 m water equivalent) in Hall C of the Gran Sasso Laboratory (Italy), measures solar neutrinos via their interactions with a target of 278 ton organic liquid scintillator. The ultrapure liquid scintillator (pseudocumene (1,2,4-trimethylbenzene (PC)) solvent with 1.5 g/l 2,5-diphenyloxazole (PPO) scintillating solute) is contained inside a thin transparent spherical nylon vessel of 8.5 m diameter. Solar neutrinos are detected by measuring the energy and position of electrons scattered by neutrino-electron elastic interactions. The scintillator promptly converts the kinetic energy of electrons by emitting photons, which are detected and converted into electronic signals (photoelectrons (p.e.)) by 2,212 photomultipliers (PMT) mounted on a concentric 13.7 m-diameter stainless steel sphere (SSS). The volume between the nylon vessel and the SSS is filled with 889 ton of ultra pure, non scintillating fluid and acts as a radiation shield for external gamma rays and neutrons. A second, larger nylon sphere (11.5 m diameter) prevents radon and other radioactive contaminants from the PMTs and SSS from diffusing into the central sensitive volume of the detector. The SSS is immersed in a 2,100 ton water Čerenkov detector meant to detect residual cosmic muons <cit.>. Radioactive decays within the scintillator form a background that can mimic neutrino signals. During detector design and construction, a significant effort was made to minimize the radioactive contamination of the scintillator and of all detector components in contact with it. A record low scintillator contamination of <10^-18 g/g was achieved for ^238U and ^232Th. The identification of different components of the solar neutrino flux relies on fitting the recorded energy spectrum with a combination of identified radioactive background components and of solar neutrino-induced electron recoil spectra. The neutrino-induced spectra are derived from Standard Solar Model neutrino energy distributions (SSM <cit.>) and include the effect of neutrino oscillation. The solar origin of the detected neutrinos is determined by the identification of crisp spectral signatures as predicted by the SSM. Exemplary is the Compton-like energy spectrum of electrons scattered by the mono-energetic ^7Be solar neutrinos. Remarkably, the ^7Be-induced Compton 'shoulder' was clearly identified with just one month of data <cit.>, thanks to the extremely low radioactive background in the scintillator. In contrast with water Čerenkov detectors, Borexino cannot retain directional information of individual events due to the isotropic emission of scintillation light; direct solar imaging with neutrinos is thus not possible. The eccentricity of the Earth's orbit, however, induces a modulation of the detected solar neutrino interaction rate proportional in amplitude to the solid angle subtended by the Earth with respect to the Sun (neglecting neutrino oscillation effects). The effect appears as a 6.7% peak-to-peak seasonal amplitude modulation, with a maximum at the perihelion. Evidence for such a yearly modulation of the ^7Be signal was already observed with Borexino Phase-I data (collected from May 2007 to May 2010) <cit.>. The period and phase were found to be consistent with a solar origin of the signal. Yearly modulation searches have also been carried out by other solar neutrino experiments: in particular SNO <cit.> and Super-Kamiokande <cit.> found evidence for an annual flux modulation in their time series datasets. Similar analyses were also performed aiming to search for time-dependencies of solar neutrino rates with periods other than one year. An apparent anti-correlation with solar cycles was suggested by data from the Homestake chlorine experiment <cit.>, and claims of such a periodicity were reported for Super-Kamiokande-I <cit.>. The SNO <cit.>, Super-Kamiokande <cit.>, and Gallex/GNO <cit.> collaborations looked for these time variations, but found none in their data. Here we report an improved measurement of time periodicities of the ^7Be solar neutrino rate based on 4 years of Borexino Phase-II data, acquired between December 2011 and December 2015. Borexino Phase-II began immediately after an extensive period of scintillator purification. Borexino Phase-II, in addition to higher statistics, lower background levels and an improved rejection of alpha-decay background, is characterized by the absence of major scintillator handling and thus displays a high degree of stability of the detector, crucially important for identifying time dependent signals. In the Borexino Phase-I analysis we based our annual modulation search on the well-established Lomb-Scargle approach as well as on the more recent Empirical Mode Decomposition (EMD) technique. The virtue of the latter technique is its sensitivity to transient modulations embedded in time series, emerging from analyzing data features with more than just standard reference sinusoidal functions. The analysis reported here analyzes the Borexino Phase-II dataset, described in Sec. <ref>, by employing both the Lomb-Scargle and an updated version of the EMD techniques. Two independent sections of this paper describe the methods of each approach and their respective results (Sec. <ref> and Sec. <ref>). For completeness, we have also carried out a search of the annual modulation directly in the time domain, using a straightforward analytical fit (Sec. <ref>). All analysis methods clearly confirm the presence of an annual modulation of the ^7Be solar neutrino interaction rate in Borexino and show no signs of other periodic time variations. § THE DATA SET The data of Borexino Phase-II are used for this analysis (1456 astronomical days of data). Compared to Borexino Phase-I, background levels have been substantially reduced by an extensive purification campaign that took place during 2010 and 2011. Of particular importance for this study is the reduction of the ^85Kr and ^210Bi concentrations, both backgrounds in ^7Be region. Data taking has seen only occasional, minor interruptions due to detector maintenance. §.§ Event selection A set of cuts described in <cit.> has been applied on an event-by-event basis to remove backgrounds and non physical events. In particular, muons and spallation events within 300 ms of parent muons, time-correlated events (^214Bi-^214Po), and noise events are identified and removed. In addition, events featuring vertices reconstructed outside a Fiducial Volume (FV) are rejected. Recoil electrons from the elastic scattering of ^7Be-ν's are selected by restricting the analysis to the energy region ∼215-715 keV (115-380 N_pe). In this range, the major backgrounds are the α decays of ^210Po and the β decays of ^210Bi and ^85Kr. The 5.3 MeV α's appear as a peak at ∼450 keV (after quenching) in the energy spectrum (red line in Fig. <ref>). The β's define a continuous spectrum beneath the ^7Be recoil spectrum (blue line in Fig. <ref>). The time stability of the background was studied to factor out any influence on the annual modulation search. Two major changes were implemented for this search from that with Borexino Phase-I data and described below: the FV (Sec. <ref>) was redefined and an enhanced method for the rejection of ^210Po α background was developed (Sec. <ref>). §.§.§ Fiducial Volume Selection We define a FV of 98.6 ton by combining a spherical cut of R=3 m radius at the center of the detector with two paraboloidal cuts at the nylon vessel poles to reject γ-rays from the Inner Vessel end-cap support hardware and plumbing. The excluded paraboloids have different dimensions to remove the local background. The paraboloids are defined as R(θ)=d/cos^n θ, where θ is the angle with z-axis and d is the distance from the detector center to the paraboloid vertex. The top paraboloid is defined by d=250 cm and n=12 whitch corresponds to an aperture of 54 cm of radius; the bottom one by d=-240 cm and n=4 which corresponds to a larger aperture of 91 cm of radius. §.§.§ ^210Po Rejection ^210Po in the scintillator constitutes a background for the search of time-varying signals because of its decay half-life of 138 days. In general α-backgrounds and β-events in a liquid scintillator can be efficiently separated exploiting the largely different shapes of the scintillation pulses <cit.>. A novel pulse-shape method based on MultiLayer Perceptron (MLP) machine learning algorithm was applied to distinguish between the scintillation pulses of α and β particles with high efficiency. This multivariate method uses a neural network based on 13 α/β discriminating input variables, that are computed for each event from the time distribution of reconstructed PMT hits. Clean samples of α and β events were obtained from the radon daughters ^214Po and ^214Bi to train the neural network. The resulting mlp parameter assumes values mostly between 0 (α) and 1 (β). Figure <ref> shows the distributions of the mlp parameters for the ^214Po and ^214Bi event samples. The MLP provides excellent α-β discrimination: with the mlp parameter threshold set at 0.9 to retain β's, the α rejection efficiency is >99.98% for ^214Po candidate events (7.7 MeV). The discrimination technique is based upon scintillation pulse shape, therefore we expect a reduced performance for the lower energy ^210Po α's (5.3 MeV) due to lower photoelectron statistics. In this case, for a clean β-like electron-recoil sample, we select events with mlp >0.98. Fig. <ref> shows the energy spectrum with and without α subtraction (blue and red lines). The small residual ^210Po events and the unaffected β spectrum illustrate the efficacy of the discrimination. §.§ Residual Background There are two main sources of background for this analysis: the residual ^210Po activity, and the stability of ^210Bi and ^85Kr β-decays in the FV. §.§.§ Residual ^210Po At the beginning of Borexino Phase-II (Dec. 2011), the count rate of ^210Po was ∼ 1400 cpd/100 ton. Estimating an mlp α-β efficiency of ≃ 99%, the residual α contamination of the β spectrum is R_α∼ 14 cpd/100 ton, comparable to an average β count rate (ν-signal and background) R_β∼ 40 cpd/100 ton distributed over the entire analysis energy region. We estimated the efficiency of the MLP cut by looking for any exponentially decaying ^210Po residual still present in the dataset. The residual amount of R_α has been subtracted for a given mlp cut in each time bin R(t): R_β(t) =R(t)-ξ_mlp· R_α(t), where ξ_mlp is the `inefficiency' parameter. For ξ_mlp = 1% the exponential component due to the residual alphas become negligible in the overall time series of the dataset, leaving the remaining β's rates with a constant average value in time. §.§.§ Background stability The β-decays of ^210Bi and ^85Kr cannot be distinguished from recoil electrons of the same energies induced by neutrinos. To study the stability of the background rate over time, we compared the spectral fits to the data divided in short periods. The fit procedure is the same as in the ^7Be analysis <cit.>. No appreciable variation of the background rate is observed within uncertainties. §.§ Detector Stability The stability of the detector response also needs to be characterized, in particular of energy and position reconstruction and fiducial mass. §.§.§ Energy and Position Reconstruction The stability of the energy scale over time was checked by comparing the number of events in the selected energy window and in the FV with those expected by Monte Carlo. A detailed simulation that includes the run per run detector performance is used. The stability of the energy scale over the period of interest was proven to be better than 1%, adequate for our purposes. §.§.§ Fiducial Mass The liquid scintillator density varies with temperature as: ρ_PC=((0.89179± 0.00003)-(8.015 ± 0.009)10^-4× T) g/cm^3, where T is the temperature in degrees Celsius <cit.>. The temperature is monitored at various positions inside the detector. The volume closest to the IV where temperature is recorded is the concentric Outer Buffer, where the thermal stability is measured to be better than 1^∘ C. In the FV, the maximum scintillator mass excursion corresponding to temperature variations is 0.1 ton, ∼0.1% of the FV mass. A Lomb-Scargle analysis (Sec. <ref>) on the temperature data was performed. The largest amplitude corresponded to a frequency of ∼0.6 year^-1, reflecting a significant real trend which anyhow cannot mimic the annual modulation. § MODULATION ANALYSIS We have implemented three alternative analysis approaches to identify the seasonal modulation. The first is a simple fit to the data in the time domain (Sec. <ref>). The second is the Lomb-Scargle method (Sec. <ref>) <cit.>, an extension of the Fourier Transform approach. The third method is the Empirical Mode decomposition (EMD) (Sec. <ref>) <cit.>. For each approach we define a set of time bins of equal length t_k and their corresponding event rate R(t_k), obtained as the ratio of the number of selected events and the corrected life time (subtracted of the muon veto dead time and any down-time between consecutive runs). The time bins are too short to allow extracting a value of the ^7Be neutrino interaction rate via a spectral fit. We use the raw β-event rate instead, which include background contributions. §.§ Fit to the Event Rate Due to Earth's orbital eccentricity (ϵ = 0.0167), the total count rate is expected to vary as R(t) = R_0 + R[1 + ϵcos2 π/T(t- ϕ)]^2 where T is the period (one year), ϕ is the phase relative to the perihelion, R is the average neutrino interaction rate and R_0 is the time independent background rate. This formalism is consistent with the MSW solution in which are no additional time modulations, at the ^7Be energies <cit.>. In this approach, the event rate as a function of the time is fit with the function defined in equation <ref>. Figure <ref> shows the folded, monthly event rate relative to the average rate measured in Borexino, with t=0, 365 representing perihelia. Data from the same months in successive years are added into the same bin. Having normalized to 1 the overall mean value, the data are compared with Eq. <ref> and show good agreement with a yearly modulation with the expected amplitude and phase. The no modulation hypothesis is excluded at 3.91 σ (99.99% C.L.) by comparing the χ^2 obtained with and without an annual periodicity. To extract the modulation parameters, we perform a χ^2 fit of the data with 30.43-day bins, without folding multiple years on top of each other. Figure <ref> shows the event rate (in cpd/100 ton) along with the best fit. From <cit.>, the expected neutrino average rate in this energy range is ∼32 cpd/100 ton. The fit returns an average neutrino rate of R=33 ± 3 (cpd/100 ton), within 1σ of the expected one (χ^2/ndof = 0.68, ndof=42). The best-fit eccentricity is ϵ = 0.0174± 0.0045, which corresponds to an amplitude of the modulation of (7.1 ± 1.9)%, and the best-fit period is T=367± 10 days. Both values are in agreement with the expected values of 6.7% and of T=365.25 days. The fit returns a phase of ϕ=-18 ±24 days. The robustness of the fit has been studied by varying the bin size between 7 and 30 days, by shifting the energy range for selected events, and with and without α-β mlp inefficiency. Fit results are found not to vary greatly and are all in agreement with the expected modulation due to the Earth's orbit eccentricity. The resulting systematic uncertainty on the eccentricity is 10%. §.§ The Lomb-Scargle method The second approach uses the Lomb-Scargle method. This extension of the Fourier Transform is well suited for our conditions since it can treat data sets that are not evenly distributed in time. In the Lomb-Scargle formalism, the Normalized Spectral Power Density, P(f), also known as the Lomb-Scargle periodogram and derived for N data points (R_1 … R_j … R_N) at specific times t_j, is evaluated and plotted for each frequency f as: P(f)=12σ^2{[ Σ_j (R_j - R) cosω (t_j-τ) ] ^2 Σ_j cos^2ω (t_j-τ). .+[Σ_j (R_j -R)sinω (t_j-τ)]^2Σ_j sin^2ω (t_j-τ)} R = R_1 + R_2 + R_3 + ...+R_NN = 1/N∑_j=1^N R_j σ^2 = 1/N-1∑_j=1^N ( R_j - R)^2 tan 2 ωτ = ∑_j sin 2 ω t_j∑_j cos 2 ω t_j where ω=2 π f. After finding the frequency f_0 corresponding to the maximum of the Lomb-Scargle Power distribution <cit.>, the sine wave that best describes the time-series, in the case of a pure signal, is: R(t)=A cos ω_0 t + B sin ω_0 t where, for ω_0=2 π f_0 and A=12σ^2[Σ_j R_jcosω_0 (t_j-τ)]^2Σ_j cos^2ω_0 (t_j-τ) B=12σ^2[Σ_j R_jsinω_0 (t_j-τ)]^2Σ_j sin^2ω_0 (t_j-τ) The modulation amplitude is the peak-to-peak variation of the curve resulting from Eq. (<ref>). For this analysis the data are grouped, after selection cuts, into 7-day bins as shown in Fig. <ref>. The Spectral Power Density P(f) is calculated using the corresponding normalized event rate R(t_k) and it is shown in Fig.<ref>. The maximum of the periodogram is at f=1 year^-1 and corresponds to a P(f) value of 7.9. A zoom-in is shown in Fig. <ref>. Following  <cit.>, we have evaluated the significance of the largest peak found in the periodogram of our experimental data set with a toy Monte Carlo simulation assuming a realistic signal-to-background ratio and a time interval of 4 years. Figure <ref> displays the P(f), at f = 1 year^-1, distribution (red filled area) obtained applying the Lomb-Scargle analysis to 10^4 simulations of a constant rate signal corresponding to the null hypothesis (absence of modulation). This distribution is exponential as expected for the power at a given frequency of the standard Lomb-Scargle periodogram of a pure white noise time series, Prob(P(f)>z)= e^-z <cit.>. In the plot, the vertical lines mark the 1σ (solid), 2σ (dashed) and 3σ (dotted) sensitivity to the null hypothesis. The blue distribution is obtained from 10^4 simulations of an expected yearly modulated signal plus constant backgrounds and its most probable value is P(f)=9.9 with rms of 4. The Spectral Power Density P(f) of 7.9 for f = 1 year^-1, obtained from the data, is within the range expected from Monte Carlo and corresponds to >3.5σ significance with respect to the null hypothesis. In addition we have estimated via Monte Carlo the significance of the two 4.5 high peaks in the L-S periodogram. Missing any a-priori information about the presence of periodicities other than the annual one, the significance of these two peaks must be evaluated as global significance, which takes into account the so called Look Elsewhere Effect, i.e. the blind search over a frequency range <cit.>. Basically, one performs a Monte Carlo evaluation of the distribution of the highest peak induced by a pure noise time series over the searched frequency interval. The significance (or p-value) is computed comparing the obtained distribution with the Power value of the highest peak detected in the Lomb-Scargle periodogram of the data. In this way we determined for the two 4.5 high peaks the p-value of 85%. Hence these two peaks are fully compatible with being pure noise induced fluctuations in the spectrum. Finally, a sinusoidal function is constructed via Eq. (<ref>) for f_0=1 year^-1 and overlaid to the time-binned data in Fig. <ref> (red curve). The peak-to peak amplitude is ∼ 5.7%, slightly less than that expected from the eccentricity of the Earth's orbit, because the Lomb-Scargle method cannot disentangle the background from neutrino signal. The same analysis using data selected with slightly different cuts and without applying the rate correction for MLP inefficiency (see Sec. <ref>), returns consistent results. The resulting total uncertainty for the period is 4%, and for the amplitude 7%. No phase information is available with this technique. §.§ Empirical Mode Decomposition The third method, the “Empirical Mode Decomposition” (EMD) <cit.>, has been designed to work with non periodical signal, in order to extract the main parameters from a time series as instantaneous frequency, phase and amplitude. The algorithm does not make any assumption about the functional form of the signal, in contrast to the Fourier analysis, and can therefore extract any time variation embedded in the data set. The EMD is a methodology developed to perform time-spectral analysis based on a empirical and iterative algorithm called sifting, able to decompose an initial signal in a set of complete, but not orthogonal, oscillation mode functions called "Intrinsic Mode Function" or IMF <cit.>. Here we adopt a new technique for the noise assisted method called “Complete Ensemble Empirical Mode Decomposition with Adaptive Noise” (CEEMDAN) <cit.> showing a greater efficiency and stability on the final results than the EEMD method <cit.>. The algorithm is more capable to separate the signals of interest from background because it removes the residual noise present in the final IMFs together with the spurious oscillation modes <cit.>. §.§.§ Standard Algorithm The sifting algorithm (Sec. <ref>) requires a large number of points for a best performance. To maximize this number we chose bins of 1 day. As a consequence, statistical fluctuations dominate the dataset time-series (red points in Fig. <ref>). However, the intrinsic dyadic filter <cit.>, removes all high frequency components created by the Poisson statistical noise. The intrinsic mode functions, IMFs, are extracted from the original function through an iterative procedure: the sifting algorithm. The basic idea is to interpolate at each step the local maxima and minima of the initial signal, calculate the mean value of these interpolating functions, and subtract it from the initial signal. The same procedure is then repeated on the residual subtracted signal until suitable stopping criteria are satisfied. These are numerical conditions, which slightly differ in literature according to the approach followed (see e.g. <cit.>). They aim at making sure that the IMFs obey two features inherited from harmonic functions: first, the number of extrema (local maxima and minima) has to match the number of zero crossing points or differ from it at most by one; second, the mean value of each IMF must be zero. The i-th IMF obtained by the k-iteration is given by: IMF_i(t) = x_i(t) - ∑_j=1^km_ij where x_i(t) is the residual signal when all “i-1” IMF’s have been subtracted from the original signal R(t), x_0(t) = R(t), and the m_ij are the average function of the max and min envelopes at each j-th iteration. Following the results from a detailed simulation, we fixed the number of sifting iterations to 20. This number guarantees a good symmetry of the IMF with respect to its mean value, preserving the dyadic-filter property of the method (i.e., each IMF has an average frequency that is half of the previous one <cit.>). Thus we obtain all i IMFs down to the last one called “trend”, that is a monotonic IMF. The EMD approach features two potential issues: on one hand, the method is strongly dependent on small changes of the initial conditions; on the other, mode mixtures could occur for a physical component present in the data set especially when the ratio between signal and noise[In this case noise means the statistical fluctuations of the rate with respect to the amplitude of the seasonal modulation signal.] is low (about S/N=0.2, in our case). In order to account for these problems, a noise-assisted technique has been adopted. A random white noise signal (dithering) was added several times to the data set under study and the average of all the IMFs taken. As for the Borexino Phase-I analysis <cit.>, we repeat the single extraction of the IMF 1000 times, adding to the data a white noise component with an average value μ_wn = 0.0 and σ_wn = √(N_bin), where N_bin is the rate of the single bin (Poisson's error). The main difference with respect to the Borexino Phase-I analysis is the use of the noise-assisted approach, called CEEMDAN. The final decomposition of our data set is shown in Fig. <ref>, where the lower frequency components identified by the algorithm become visible in the higher IMFs. The ones shown are the resulting IMFs averaged over the 1000 extractions with different regenerations of white noise. In particular, Fig. <ref>c shows the grey band corresponding to 1000 noise regenerated IMF-7 containing the seasonal modulation. The resulting average function is shown as black solid line, while the red-dashed curve corresponds to the expected seasonal modulation. §.§.§ Modulation Parameters Estimation Here we can only provide a short account of the procedures to calculate the modulation parameters. A more detailed and formal description of the numerical calculations and theoretical explanations are reported in <cit.>. The frequency and the amplitude values of a periodic function (as the seasonal modulation) are constant in time. We therefore expect that in the IMF7 (Fig. <ref>) where a modulation of 1-year period is visible, these parameters will be constant in time, the average curve peaking on the expected values. Naturally, due to the numerical procedure with which the “signal” has been obtained, some small fluctuations of the frequency and of the amplitude are expected. The IMF functions extracted by the sifting algorithm are not based on an analytical function. Therefore, in order to extract information on frequency, phase and amplitude, it is necessary to build a complex function z(t) by means of a Hilbert transform of the initial signal <cit.>: z(t)=a(t)+ib(t)=A(t) e^- iθ(t), in which the real part a(t) is the IMF and the imaginary part b(t) is the Hilbert transform of the real function: b(t)= 1/πP∫_t'a(t')/(t-t') dt' where P is the Cauchy principal value. In Eq.(<ref>), A(t) is defined as A(t)=√(a^2(t)+b^2(t)). A(t) is also called the amplitude modulation function (AM), while θ(t)=arctan( b(t)/a(t)) defines the phase of the carrier function or frequency modulation (FM) function. This method provides a function of the phase of the time that we can use to define the instantaneous frequency (IF) as simple time derivative of the phase θ(t). Unfortunately a direct calculation of the IF, starting from the signal, gives unphysical results with negative values for the frequencies. In order to solve this problem, an additional numerical procedure is required: the “Normalized Hilbert Transform” (NHT) <cit.>. Performing the NHT we obtain a normalized carrier function over all the time series. Building the z(t) function, we are able to calculate a reliable instantaneous frequency function with a real physical meaning as follows: f(t)= dθ(t) dt. We calculate the IF f(t) and the amplitude A(t) for all the IMFs extracted from each noise regeneration and take the distribution of their average in time. A Gaussian fit is applied to the resulting distribution to obtain f(t), A(t) and their respective errors. In Fig. <ref> we compare IMFs obtained from the real dataset (Fig. <ref>c) with simulated data sets from a toy Monte Carlo with/without the sinusoidal signal expected for the seasonal modulation (Fig. <ref>a and <ref>b respectively). For both real and MC data set, the resulting IMF average shows a very good agreement with the expected seasonal modulation function, while in the case of the null hypothesis (Fig. <ref>b) the amplitudes of the resulting IMFs are substantially smaller while frequencies and phases are varying randomly. A power spectrum is defined based on the average in time of the square amplitudes (⟨ A^2(t)⟩) (<ref>) for each frequency ω (t). Fig. <ref> shows the relative power spectra for the simulations with and without modulation (Fig. <ref>a and  <ref>b) and the real data set, respectively (Fig. <ref>c). The colored histograms are the Power Spectra from the last 4 IMFs, while the dark green are the full spectra of the whole set of IMFs (full dataset spectrum). As expected in the presence of the seasonal modulation signal (Fig. <ref>a and <ref>c), we observe a narrow peak centered on the expected frequency (f=1/T = 2.73 × 10^-3 day^-1), while in the case of the null hypothesis this spectral component remains almost flat, featuring an amplitude comparable with other background IMFs that are present at higher frequencies. The power is an order of magnitude lower than the signal case (Fig. <ref>b). Applying equation <ref>, we compute the average parameters shown in Tab.<ref> for the simulated and real data. The results are in agreement with the expected seasonal modulation. Based on the comparison of the power spectrum and the parameters resulting from the zero-modulation MC data sets we conclude the presence of a seasonal modulation. We have calculated a χ^2-map varying both the phase and modulation amplitude of the sinusoidal function with respect to the average IMF obtained over the complete 1000 noise regenerations. The χ^2-contours are displayed in Fig. <ref>, where we assumed the standard deviation of the IMFs from the average curve to equal 1σ-uncertainties divided by the number of time bins minus one. § SUMMARY Four years of Borexino Phase-II data have been analyzed searching for the expected annual modulation of the ^7Be solar neutrino interaction rate induced by the eccentricity of the Earth's orbit around the Sun. Both the detector and the data have shown remarkable stability throughout the entire Phase-II period, allowing for the clear emergence of the annual periodicity of the signal. Three analysis methods were employed: an analytical fit to event rate, a Lomb-Scargle periodogram and an Empirical Mode Decomposition analysis. Results obtained with all three methods are consistent with the presence of an annual modulation of the detected ^7Be solar neutrino interaction rate. Amplitude and phase of the modulation are consistent with that expected from the eccentric revolution of the Earth around the Sun, proving the solar origin of the low energy neutrinos detected in Borexino. The absence of an annual modulation is rejected with a 99.99% C.L.. The direct fit to the event rate yields an eccentricity of ϵ = (1.74 ± 0.45)%, while the Lomb-Scargle method identifies a clear spectral maximum at the period T=1 year. The EMD method provides a powerful and independent confirmation of these results. § ACKNOWLEDGEMENTS The Borexino program is made possible by funding from INFN (Italy), NSF (USA), BMBF, DFG, HGF and MPG (Germany), RFBR (Grants 16-02-01026 A, 15-02-02117 A, 16-29-13014 ofi-m, 17-02-00305 A) (Russia), and NCN Poland (Grant No. UMO-2013/10/E/ST2/00180). We acknowledge the generous hospitality and support of the Laboratory Nazionali del Gran Sasso (Italy). 00 § REFERENCES bib:detector G. Alimonti et al. (Borexino Collaboration), The Borexino detector at the Laboratori Nazionali del Gran Sasso, Nucl. Instr. and Methods A 600 (2009) 568. bib:be7 G. Bellini et al., (Borexino Collaboration), Precision Measurement of the 7Be Solar Neutrino Interaction Rate in Borexino, Phys. Rev. Lett. 107 (2011) 141302. bib:pep Bellini, G. et al. (Borexino Collaboration), First Evidence of pep Solar Neutrinos by Direct Detection in Borexino, Phys. Rev. Lett. 108, (2012) 051302. bib:bxpp G. Bellini et al. (Borexino Collaboration), Neutrinos from the primary proton-proton fusion process in the Sun, Nature 512 (2014) 383. bib:b8 G. Bellini, et al. (Borexino Collaboration), Measurement of the solar 8B neutrino rate with a liquid scintillator target and 3 MeV energy threshold in the Borexino detector, Phys. Rev. D 82, (2010) 033006. bib:carlos J. Bergstrm, M.C. Gonzalez-Garcia, M. Maltoni et al, Updated determination of the solar neutrino fluxes from solar neutrino data, J. High Energ. Phys. (2016) 132. bib:geo M. Agostini et al. (Borexino Collaboration), Spectroscopy of geoneutrinos from 2056 days of Borexino data, Phys. Rev. D 92, 031101(2015) 031101. bib:anti G. Bellini, G. et al. (Borexino Collaboration), Study of solar and other unknown anti-neutrino fluxes with Borexino at LNGS, Phys. Lett. B 696, (2011) 191. bib:muon G. Bellini et al. (Borexino Collaboration), Muon and Cosmogenic Neutron Detection in Borexino, J. Instrum. 6 (2011) P05005. bib:serenelli16 A. Serenelli, Alive and well: a short review about standard solar models, arXiv: hep-ph/1601.07179 (2016) bib:be2007 Bellini, G. et al. (Borexino Collaboration), First real time detection of 7Be solar neutrinos by Borexino, Phys. Lett. B 658 (2008) 101. bib:long G. Bellini et al. (Borexino Collaboration), Final results of Borexino Phase-I on low-energy solar neutrino spectroscopy, Phys. Rev. D 89 (2014) 112007. bib:SNO B. Aharmim (SNO Collaboration), Search for periodicities in the B8 solar neutrino flux measured by the Sudbury Neutrino Observatory, Phys. Rev. D 72 (2005) 052010. bib:SK J. Hosaka et al. (Super-KKamiokande Collaboration), Solar neutrino measurements in Super-Kamiokande-I, Phys. Rev. D 73, (2006) 112001. bib:homestake R. Davis Jr., A review of measurements of the solar neutrino flux and their variation, Nucl. Phys. B - Proceedings Supplements, 48, (1996) 284. bib:mils A.Milsztajin, A Search for Periodicity in the Super-Kamiokande Solar Neutrino Flux Data arXiv:hep-ph/0301252 (2003). bib:stu D.O. Caldwell and P.A. Sturrock, Further Evidence for Neutrino Flux Variability from Super-Kamiokande Data, arXiv:hep-ph/0305303 (2003) bib:stu2 D.O. Caldwell and P.A. Sturrock, Evidence for solar neutrino flux variability and its implications, Astropart. Phys., 23, (2005) 543. bib:SK1 J.Yoo et al. (SK Collaboration), Search for periodic modulations of the solar neutrino flux in Super-Kamiokande-I, Phys.Rev. D68, (2003) 092002. bib:GNO L. Pandola, Search for time modulations in the Gallex/GNO solar neutrino data Astropart. Phys. 22 (2004) 219. bib:Lomb N.R. Lomb, Least-squares frequency analysis of unequally spaced data, Astrophysics and Space Science, 39 (1976) 447. bib:Scargle J.D. Scargle, Studies in astronomical time series analysis. II - Statistical aspects of spectral analysis of unevenly spaced data, Astrophysical Journal 263 (1982) 835. bib:Huang98 N. E. Huang, Z. Shen, S. R. Long et al., The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. of Lond. A, 454 (1998) 903. bib:MSW G. Bellini, G. et al. (Borexino Collaboration), Absence of a day-night asymmetry in the 7Be solar neutrino rate in Borexino, Phys. Lett. B 707, (2012) 22. bib:horneJ. H. Horne and S. L. Baliunas, A prescription for period analysis of unevenly sampled time series, Astrophys. J. 302 (1986) 757. bib:gio G. Ranucci., Likelihood scan of the Super-Kamiokande I time series data, Phys. Rev. D 73 (2006) 103003. bib:WuHuang2010 Z. Wu, N.E. Huang, On the filtering properties of the empirical mode decomposition, Advances in Adaptive Data Analysis, 2, No. 04, (2010) 397. bib:Torres2011 M.E. Torres, M.A. Colominas, G. Schlotthauer, P. Flandrin, A complete ensemble empirical mode decomposition with adaptive noise, Proc. of 36th IEEE Intern. Conference on Acoustic, Speech and Signal Processing, Prague, Czech Republic, (2011) 4144. bib:Colominas2014 M.A. Colominas, G. Schlotthauer G., M.E. Torres, Improved complete ensemble EMD: A suitable tool for biomedical signal processing, Biomedical Signal Processing and Control 14 (2014) 19. bib:WuHuangEEMD2009 Z. Wu and N. E. Huang, Ensemble empirical mode decomposition: a noise-assisted data analysis method, Advances in Adaptive Data Analysis, 1, No.1 (2009) 1. bib:WuHuang2004 Z.Wu and N.E.Huang, A study of the characteristics of white noise using the empirical mode decomposition method, Proc. R. Soc. Lond. A 460 (2004) 1597. bib:Rilling2003 G. Rilling, P. Flandrin, P. Goncalves, On empirical mode decomposition and its algorithms, Proc. of IEEE-EURASIP Workshop on Nonlinear Signal and Image Processing NSIP-03, (2003) 177. bib:onIF N.E.Huang et al., On instantaneous frequency, Advanced in Adaptive Data Analysis,1, No.2, (2009) 177

http://arxiv.org/abs/1701.07608v1

20170126080613

The Rotation-Activity Correlations in K and M dwarfs. II. New constraints on the dynamo mechanisms in late-K and M dwarfs before and at the transition to complete convection

astro-ph.SR

Armagh Observatory, College Hill, BT61 9DG Armagh, Northern Ireland Université de Toulouse, UPS-Observatoire Midi-Pyrénées, IRAP, Toulouse, France CNRS, Institut de Recherche en Astrophysique et Planétologie, 14 av. E. Belin, F–31400 Toulouse, France eric_houdebine@yahoo.fr Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA mullan@udel.edu Université de Bordeaux, Institut de Mathématiques, UMR 5251, 351 cours de la libération, 33405 Talence cedex, France. Bernard.Bercu@math.u-bordeaux1.fr Université de Toulouse, UPS-Observatoire Midi-Pyrénées, IRAP, Toulouse, France CNRS, Institut de Recherche en Astrophysique et Planétologie, 14 av. E. Belin, F–31400 Toulouse, France Department of Physics & Astronomy, Notre Dame University-Louaize, PO Box 72, Zouk Mikaël, Lebanon We study the rotation-activity correlations (RACs) in a sample stars from spectral type dK4 to dM4. We study RACs using chromospheric data and coronal data. We study the Ca ii line surface fluxes-P/sin i RACs. We fit the RACs with linear homoscedastic and heteroscedastic regression models. We find that these RACs differ substantially from one spectral sub-type to another. For dM3 and dM4 stars, we find that the RACs cannot be described by a simple model, but instead that there may exist two distinct RAC behaviors for the low activity and the high activity stellar sub-samples respectively. Although these results are preliminary and will need confirmation, the data suggest that these distinct RACs may be associated with different dynamo regimes. We also study R'_HK as a function of the Rossby number R_0. We find that: (i) For dK4 stars, we confirm R'_HK as a function of R_0 agrees well with previous results for F-G-K stars. (ii) In dK6, dM2, dM3 and dM4 stars, we find that, at a given R_0, the values of R'_HK lie a factor of 3, 10, 20 and 90 respectively below the F-G-K RAC. Our results suggest a significant decrease in the efficiency of the dynamo mechanism(s) as regards chromospheric heating before and at dM3, i.e. before and at the TTCC. We also show that the ratio of coronal heating to chromospheric heating L_X/L_HK increases by a factor of 100 between dK4 and dM4 stars. § INTRODUCTION In the present paper, we have two principal goals. First, we present data which have a bearing on the existence of correlations between rotation and activity (“RACs") in our sample of stars. Second, we ask if the empirical correlations can be interpreted in the context of stellar dynamo theories (which attempt to predict the properties of the magnetic fields which are generated in a star with a prescribed structure and rotation). The second goal is admittedly a challenging one: the basic mechanisms underlying stellar dynamo theory are being continuously improved as computer resources permit inclusion of more realistic physical effects. Rather than attempting to provide a comprehensive discussion of these complexities, we simplify our discussion by restricting attention to certain broad classes of dynamos. We briefly outline the distinction between these classes in what follows. Whatever the source of the magnetic field, the observational consequences of such a field in a star is expected to be the same: dissipation of mechanical energy associated with magnetic processes leads to enhanced emission (relative to what is generated by the photosphere) from the chromosphere (in spectral lines such as Ca ii H and K or H_α) and from the corona (in X-ray continuum). These enhanced emissions arising from magnetic effects in the stellar atmosphere on the are considered in this paper to be generic indicators of what we refer to as “magnetic activity", or more briefly, “activity". In this paper, we seek to quantify how (an observational quantity that is associated with) “activity" in low-mass dwarfs is correlated with (an observational quantity that is associated with) “rotation". In order to set the stage for interpreting our result, we first need to define certain terms which are related to dynamo models. §.§ Three classes of dynamo models Models of dynamos in the Sun and stars are typically based on “mean-field electrodynamics" (MFE): the fluid flow and the magnetic field are separated into mean (<u>, <B>) and turbulent (u', B') components (e.g. Racine et al. 2011). Although the mean values of u' and B' are by definition zero, the mean value of their cross-product does not reduce to zero, but instead produces a non-zero turbulent MFE, E. The essence of MFE is to express the (vector) E in terms of the large-scale (vector) magnetic field <B> by means of a tensor expansion in the mean field plus its gradients. The leading order term in this expansion E_i = α_ij <B_j > is called the "α-effect". Parker (1955) first suggested that the α-effect might arise because cyclonic convective turbulence can systematically twist a large-scale magnetic field, and in the process, regenerate a large-scale poloidal field. All of the dynamos we consider here rely on the α-effect, as well as on (at least) one more factor. In order to construct a mean-field dynamo model, <B> is written as the sum of poloidal and toroidal components, and <u> is assumed to be directed purely in the azimuthal direction ϕ, and to be axisymmetric, i.e. the angular velocity Ω can be a function of r and θ, but not of ϕ. With these assumptions, the induction equation for the time-varying magnetic field can be separated into two equations, one for the poloidal field, the other for the toroidal field. Source terms (S_p, S_t) appear in both equations, and the dominant source terms determine the dynamo class. For the poloidal equation, S_p is a single term, namely, the ϕ component of E, i.e. the α-effect. For the toroidal equation, S_t contains two terms: (i) includes a spatial gradient of Ω; (ii) includes a spatial gradient of E. Three classes of mean-field dynamos are defined as follows: (a) α-Ω dynamo, in which only term (i) is retained in the toroidal equation; (b) α ^2 dynamo, in which only term (ii) is retained in the toroidal equation; (c) α ^2-Ω dynamo, in which terms (i) and (ii) are both retained in the toroidal equation. §.§ Stellar internal structure as it relates to dynamo activity Models of dwarf stars of spectral type F, G, K, and early M possess a radiative core and a convective envelope. In such stars, magnetic fields can be generated by an α-Ω dynamo (also referred to as an interface dynamo, or shell dynamo, Parker 1975)[For a quantitative evaluation of such a dynamo in mid-K to early-M stars with known rotation periods, see Mullan et al. 2015]. The strongest toroidal magnetic fields in these stars expected to be produced in the vicinity of the interface (“tachocline") between the radiative core and the convective envelope where the differential rotation is the strongest. In stars which are massive enough to contain such an interface, and where therefore the possibility of an “interface dynamo" (ID) exists, data pertaining to enhanced emission from such stars in chromospheric spectral lines and/or in coronal X-rays reveal clearly that there is a strong correlation between rotation and activity indicators (e.g. Pallavicini et al. 1981; Wright et al. 2011), i.e. the RAC has a slope with a numerical value that is definitely non-zero. However, since Limber (1958) derived models of cool main sequence stars with lower and lower masses, it has been widely believed by stellar evolution modelers that interfaces do not exist in all low-mass stars. An anonymous referee has pointed out that “there has been no definitive evidence validating this theoretical prediction". In principle, asteroseismology of M dwarfs might eventually provide p-mode frequencies with enough precision that (by analogy with helioseismological data) signatures of the interface could be identified. But although theoretical models have been calculated for such stars (see Rodriguez-Lopez et al. 2014), no empirical data on reliable p-mode periods in M dwarfs are yet available. Despite the lack of definitive evidence at the present time, we shall adopt the widely held belief that stars on the main sequence undergo a Transition To Complete Convection (TTCC) at a certain spectral type. According to models, main sequence stars with masses that are less than a critical value of about 0.30-0.35 M_⊙ are completely convective. Such stars should have no tachocline whatsoever on the main sequence. And yet the empirical evidence shows that stars with masses less than 0.30-0.35 M_⊙ also show enhanced emission in chromospheric lines and in X-ray continuum. Since we have attributed such enhanced emission in warmer stars to magnetic activity, then it seems natural also to ascribe the enhanced emissions in completely convective stars to magnetic activity. Such activity must rely on a non-interface type of dynamo, possibly an α^2 dynamo (distributive dynamo: DD), maintained by convective turbulence alone (e.g. Roberts & Stix 1972, Rosner 1980, Dobler et al. 2006, Chabrier & Kuker 2006). It is important to note that the existence of complete convection in a star does not mean that the star is necessarily completely convective at all stages of its main sequence phase of evolution. E.g., Feiden and Dotter (2013) report that a 0.3 M_⊙ model arrives on the main sequence with a radiative zone sandwiched between two convective zones: the models suggest that such a “sandwich" structure exists for several gigayears before the model becomes completely convective (see also Rodriguez-Lopez et al. 2014). However, at lower masses, 0.25 M_⊙, stellar models are completely convective at all ages (see Rodriguez-Lopez et al. 2014). Thus, when we use the label “completely convective" for stars of masses in the intermediate range (0.25-0.3) M_⊙, the label involves not merely a question of the mass of the model, but also of the age of the model. This complication may cause some ambiguity when we attempt to interpret RACs in the vicinity of the TTCC. For simplicity, it would be convenient if a given star could be assigned to having either an ID or a DD. But we recognize that such a simplified approach cannot be the whole story: in cool stars where an interface exists, the inevitable presence of a deep convective envelope gives rise to the possibility that ID and DD may both be operating (e.g. Brandenburg & Subramanian 2005; Brown et al. 2010). In order to distinguish between the observational properties of ID and DD, Durney, De Young & Roxburgh (1993) pointed out some differences between ID and DD. First, the magnetic field created by an ID should depend strongly on rotation whereas the magnetic field created by a DD should not. In terms of the notation to be used in the present paper, this would lead one to expect that in an ID star, the RAC should have a slope which is definitely non-zero, whereas in a DD star, the RAC slope should be essentially zero. Second, although IDs can produce activity cycles in the large-scale field it was not clear (in 1993) how DD could ever give rise to a cycle. Since 1993, the second feature has been called into question. E.g. Stefani & Gerberth (2003) have demonstrated that cyclic behavior is possible in α ^2 dynamo models provided that the following condition is satisfied: the α-effect needs to have radial gradients which are sufficiently steep, including changes in the algebraic sign. Moreover, recent modeling suggests that the third class of dynamo models (α ^2-Ω) might be a better description to the solar dynamo (where there is certainly an interface) than an α-Ω dynamo (e.g. Lawson et al 2015). From the perspective of a change from an α-Ω dynamo to an α ^2-Ω dynamo, perhaps in completely convective stars, we might also encounter a change from an α ^2 dynamo to an α^2-Ω dynamo. If this happens, then we might find that rather than seeing a zero slope for the RAC in a completely convective star, the inclusion of Ω in the dynamo model could lead to a non-zero RAC slope. §.§ Observational evidence for the TTCC? A long standing problem has been the search for an observational signature of the putative transition between different types of dynamos at the TTCC. There has been several attempts to detect a change in magnetic activity diagnostics (e.g. Mullan & MacDonald 2001) or magnetic field topologies (e.g. Donati et al. 2008, Morin et al. 2008, 2010, Phan-Bao et al. 2009, Stassun et al. 2011) in the vicinity of the TTCC. None of these studies have identified unambiguous and definitive signatures of significant changes in magnetic activity at the TTCC. However, in a recent paper, West et al. (2015) have reported on a study of RACs in two groups of dwarfs: M1-M4, and M5-M8. West et al. (2015) found that the RAC in M1-M4 stars has a somewhat different slope than the RAC in M5-M8 stars (see Sect. 4 below for a more detailed discussion). One interpretation of this behavior is that something changes as regards the dynamo between M4 and M5. We will return to a discussion of this interpretation (as well as an alternative interpretation) in Section 4 below. In order to address the TTCC-dynamo-transition topic meaningfully, we first need to identify at what spectral type the TTCC occurs. Limber (1958) states that “in that part of the main sequence where the inner radiative region ... is becoming vanishingly small", the corresponding spectral type is M3V-M4V. Subsequently, Dorman et al. (1989) placed the TTCC at M∼ 0.25 M_⊙, which corresponds to the spectral subtype dM4. Even more recently, Chabrier & Baraffe (1997) predict that the TTCC occurs at M∼ 0.35 M_⊙, which corresponds to the spectral subtype dM2. Therefore, it appears that the TTCC may lie somewhere in the range between subtypes dM2 and dM4, corresponding to masses of 0.25-0.4 M_⊙ (Stassun et al. 2011). The theoretical mass limit at the TTCC has been found to shift towards slightly smaller masses if different boundary conditions are used for the stellar models (Mullan et al. 2015). (Even larger shifts of the TTCC towards lower masses were at one time proposed by Mullan and MacDonald 2001 if interior magnetic fields were to be as large as 10^7-8 G. If fields as large as that were to exist inside stars, the definition of a “fully convective star" could become more ambiguous: the onset of complete convection would then depend not only on the mass and age of a star, but would also depend on how strong its magnetic field is. However, we may not in fact need to worry about this ambiguity: Browning et al. (2016) have recently argued that such strong fields would be unstable. In standard main-sequence models, the radius of a model scales almost linearly with mass: therefore, the TTCC is expected to lie in the radius range of roughly 0.25-0.4 R_⊙. According to the radius-T_eff calibration of Houdebine et al. (2016b) this radius range yields an effective temperature range of 3200-3500 K for the TTCC. In this paper, we examine the topic of a possible dynamo transition at the TTCC using a more extensive and more fine-grained data set than has previously been available for study. §.§ Dynamos: unsaturated and saturated Empirically, evidence for rotationally driven dynamos in cool stars first emerged when researchers plotted the strength of chromospheric emission versus stellar rotation. E.g. See Kraft (1967), Vaughan et al. (1981), Soderblom (1982), Vogt et al. (1983), Noyes et al. (1984), Marcy & Chen (1992), Patten & Simon (1996), Fekel (1997), Delfosse et al. (1998), Jeffries et al. (2000), Pizzolato et al. (2003), Mohanty & Basri (2003), Browning et al. (2010), Wright et al. (2011), Rebassa-Mansergas et al. (2013), West et al. (2015). The key signature is the following: emission in chromospheric spectral lines is observed to be stronger in stars with faster rotation speed v_r (or shorter rotation period P) (e.g. Vaughan et al 1981). This is defined as a “Rotation-Activity Correlation" (RAC). The existence of an RAC is consistent with the expectations that (i) the faster the rotation is, the stronger are the magnetic fields which can be generated (e.g. Mullan et al. 2015), and (ii) stronger fields are associated with stronger chromospheric heating (Skumanich et al. 1975) and with stronger coronal heating (e.g. Mullan 2009). Thus, if chromospheric emission intensity is plotted as a function of P, it is found that over a certain range of periods, the RAC has a clearly negative slope. However, as more data are accumulated, it emerges that the negative slope of the RAC does not extend indefinitely to shorter and shorter P. Instead, when P becomes shorter than a certain value P_c, no further increase in emission occurs: for P≤ P_c, the RAC becomes flat (e.g. Vilhu 1984), with a slope of zero. By definition, in the flattened portion of the RAC, increasing rotation does not result in increased chromospheric/coronal emission. For solar type stars, the transition to a flat curve occurs for P≈3 days. Vilhu suggested the term “saturated" to refer to such conditions. Similarly, Wright et al. (2011) found that the saturation regime occurs at about R_0≃ 0.8. Reiners et al. (2009) found that the sturation occurs at the critical Rossby number R_0≃ 0.1. We shall compare these values to those we obtain for our stellar samples below. Pizzolato et al. (2003) found that saturation occurs from a period of about ≈ 2 days in solar type stars to about ≈ 10 days in M dwarfs. Rebassa-Mansergas et al. (2013) found that stars with vsin i ≥ 5 hm s^-1 are all in the saturated regime. This yields a rotation period of ≈ 3 days for stars at the spectral type dM3 (see Paper I). Therefore, we shall consider that saturation occurs for periods of about ≈ 3 days or R_0≃ 0.5. Since in our stellar samples we have stars with shorter rotation periods or R_0, we shall consider below that these stars likely lie in the saturated regime. Vilhu suggested that “saturation" might be due to a complete coverage of the star's surface by magnetic fields. Another explanation of “saturation" was offered by Mullan (1984) in terms of the maximum possible flux of mechanical energy that can be generated by convection. In the present paper, we shall not attempt to identify the physical process which leads to saturation. Instead, we shall adopt a purely empirical approach, and we shall refer to the flat portion of an RAC as “saturated". In the same vein, we shall refer to an RAC with a statistically significant negative slope as “unsaturated". To the extent that an RAC owes its existence to the operation of a dynamo of some kind, we can say that the dynamo reveals itself in two regimes: saturated and unsaturated. The properties of the two different dynamo regimes we find below (see Section 3.8.1) for the low and high activity sub-samples respectively cannot be due to effects of the chromospheric response to non-thermal heating mechanisms. In fact, there does not seem to be any definitive evidence that the influence of a magnetic field in a stellar atmosphere ever attains a saturated level. In support of this claim, we note that the response of stellar chromospheres to non-thermal heating mechanisms is continuous and monotonically increasing from basal chromospheres to flaring chromospheres (e.g. Houdebine 1992, Houdebine & Doyle 1994a, Houdebine & Doyle 1994b, Houdebine et al. 1995, Houdebine & Stempels 1997, Houdebine 2009, Houdebine 2010). Therefore, the two different types of RACs we find below for the low and high activity stars respectively highlight the different properties of the dynamo mechanisms in these two types of stars. It seems to us that in order to study the properties of stellar dynamos most profitably, it would be preferable to concentrate as much as possible on stars in the unsaturated regime. The reason for this claim is that, when conditions are saturated, there are extra factors which come into play which may obscure some physical properties that are directly associated with dynamo action. In view of Vilhu's identification of a critical period P_c, it seems that the slowest rotators have the best chance of being in the unsaturated regime. For that reason, we consider it worthwhile to push the spectroscopic measurements of stellar rotation towards the smallest possible values of vsin i which can be reliably measured. In general, since slow rotation means less chromospheric heating, we expect that the slowest rotators in our dataset will be stars which are low-activity stars classified as dK and dM (i.e. those which by definition show no emission in the Balmer lines), while the fastest rotators will be stars which are highly active stars classified as dKe and dMe (i.e. those where by definition the Balmer lines have an emission core). In what follows, we shall be especially interested in determining the (negative) slope of the RAC for low-activity stars (e.g. see Fig. 13 below, lower panel). §.§ Extending our previous work One reason for the present study has emerged from recent work by Houdebine & Mullan (2015: hereafter HM): they found that another diagnostic of magnetic fields, namely, the efficiency of magnetic braking (which manifests itself in the rotational velocity), undergoes a detectable change at spectral sub-type dM3. Basing their analysis on a new data set of precise rotational velocities, HM found that the mean rotation period of M3 stars is abnormally large compared to those of the adjoining spectral types dM2 and dM4. This indicates that the dM3 stars have been slowed down more than the stars in the immediately adjacent sub-types dM2 and dM4. This excess slowing at dM3 may be associated with a change in magnetic properties of the stars at dM3. Specifically, HM suggested that the change might be associated with an earlier report (Mullan et al 2006) that the lengths of flaring magnetic loops undergo a significant increase at spectral type dM3. In the present paper, we extend the work of HM in two distinct ways. First, we expand the database of precise rotational properties of K and M dwarfs of various sub-types. Second, we expand and analyze a separate database which deals with the second physical parameter which enters into the RAC: the radiative properties which are associated with magnetic “activity" in our sample of rotating stars. Our goal here is to use the activity data to construct RACs. Moreover, we quantify the RACs in two different parts of the stellar atmosphere: the chromosphere (using the Ca ii lines) and the corona (using L_X). The present study is based on a larger sample of stars, and a finer grid of spectral sub-types, than have been used previously in constructing RACs for K and M stars. Our goal is to explore whether the unusual rotational signature reported by HM at dM3 is accompanied by unusual behavior in the activity indicators of either chromosphere or corona or both, either as regards the intensity of the radiation, or as regards the slopes of the RACs. A key physical factor which is known to be well correlated with chromospheric emission and coronal emission has to do with magnetic fields on the stellar surface (e.g. Skumanich et al 1975; Schrijver et al. 1989). The existence of RAC's may be interpreted as an indication that the surface magnetic field intensities are correlated with the stellar rotation rate. This relationship is to be expected on the basis of standard dynamo theory (e.g. Parker 1979, Krause & Radler 1980, Mullan et al. 2015). However, most of the observational investigations cited above suffer from two inadequacies: (i) they included only a few stars which are rotating slowly enough to be in the unsaturated regime, and (ii) the targets included stars which were spread out over a broad range of spectral types. In this paper, we attempt to remedy both of these inadequacies. §.§ Aspects of the data used in the present study In an effort to extend the RACs to slow rotators among late-type dwarfs, we have been reporting, over the past several years, improved spectroscopic measurements of rotational broadening vsin i in stars of spectral sub-types dK4 (Houdebine 2011a, Paper XVI thereafter), dK6 (Houdebine et al. 2016, Paper I), dM2 (Houdebine 2008, Paper VIII, Houdebine 2010a, Paper XIV), dM3 (HM) and dM4 (Houdebine 2012a, Paper XVII, Houdebine et al. 2016, Paper I). Combining our (previous) rotational measures with our measures of the Ca ii line equivalent widths (EW), we have already reported RACs for dK4 (Houdebine 2011a, Paper XVI), dM2 ( Houdebine 2011b, Paper XV) and dM4 stars (Houdebine 2012b, Paper XVIII), for slow and rapid rotators alike. In those earlier papers, we proposed empirical RACs which included large samples of slow rotators. We found that for later spectral types it is crucial to examine a fine grid of spectral sub-types. Specifically, we found that the RAC's vary significantly between dK4 to dM4 stars: the RAC's were found to have different gradients and different saturation levels (Papers XV, XVIII and the present study). The previous data, combined with the present study, now provide us with large enough data-sets in each spectral sub-type that we can investigate with improved confidence the differences (if any) between the RAC's in five different spectral sub-types. In view of the results reported in HM, it is notable that the present study enables us to study the RAC's in the vicinity of the TTCC. When we consider slow rotators, our data confirm that the mean rotation periods of stars in the range dK4-dM4 in general decrease with decreasing effective temperature (see HM and Paper I). However, we also find that something unusual happens in the rotation rates between dM2 and dM4. The overall trend towards decreasing rotation period as we go from dK4 to dM4 is interrupted at spectral sub-type dM3: at that sub-type, the mean rotation period increases to a local peak, such that the mean rotational period at dM3 is longer than the overall trend between dK4 and dM4 would have predicted (HM). But when we extend our investigation to include fast rotators among the dK4-dM4 stars, there is found to be the following overall trend: the mean rotation period tends to increase slightly from dK4 to dM4. But once again, at dM3, an exception is found: the mean rotation period of fast rotators at dM3 is locally significantly longer than the overall trend would have predicted (HM). HM interpret these abnormally long rotation periods at sub-type dM3 as possibly being associated with the occurrence of increasing coronal loop lengths (previously reported by Mullan et al. 2006 in a study of flare stars). The mean rotation period of the slow rotators is an important constraint on the temporal history of the dynamo mechanisms and magnetic braking mechanisms (HM, Paper I). §.§ Studying RACs in various formats An important aspect of the present paper is that we wish to investigate the RACs in various formats. In the first place, we construct RACs separately for the chromosphere and the corona. Moreover, we explore correlations between various observations of the “activity" and various aspects of “rotation". As an example, we will examine, for the H and K lines of Ca ii, a plot of the quantity R'_HK as a function of the Rossby number R_0. We also plot the Ca ii surface flux as a function of P/sin i. We use these different formats in an attempt to improve our chances of identifying changes (if any) in the dynamo regime in the vicinity of the TTCC. We shall find that changes near the TTCC are more readily detectable in some RACs than in others. This may explain why the empirical detection of the TTCC has been elusive in the past. § SELECTION OF SPECTROSCOPIC DATA Stepień (1989, 1993, 1994) has reported that RACs exhibit certain differences at different spectral types. As an extension of this finding, we have found, in previous studies (Paper XVIII, HM) and also in the present study, that it is important in constructing RACs to select samples of stars with T_eff values which are confined within a narrow range. There are two principal reasons for this, one related to the choice of an optimal set of photospheric absorption lines, and the second related to our analysis of chromospheric emission lines. As regards the choice of photospheric lines, we have already discussed the first of the above reasons at length in Paper I in the context of optimizing the measurement of rotational velocities at each spectral subtype between dK4 and dM4. In the present paper, we turn now to the second reason. In the context of chromospheric analysis, it is important to deal with stars with closely similar T_eff when we are attempting to quantify with as much precision as possible the EW of the chromospheric lines (Ca ii resonance doublet and H_α). These lines inevitably include some contributions from the background photospheric continuum and from the temperature minimum region (e.g. Houdebine & Doyle 1994, Cram & Mullan 1979, Houdebine & Stempels 1997, Paper XV). Initially, our samples of stars had been selected for the purpose of chromospheric modelling studies (e.g. Houdebine & Stempels 1997, Houdebine 2009b Paper XII, Houdebine 2010b Paper IX), and in these studies the selection of stars with closely similar spectral types was essential in order to develop reliable grids of semi-empirical model chromospheres, each of which would be superposed on a particular photospheric model. Based on our previous papers, we have found that the most suitable initial selection parameter when we wish to identify a homogeneous sample of K or M dwarfs belonging to a specific sub-type is the (R-I) color: this color is sensitive to T_eff, but less so to metallicity (e.g. Leggett 1992, Ramirez & Melendez 2005, Mann et al. 2015). Moreover, broad-band colors of high precision are widely available in the literature for many of the cool dwarfs which are of interest to us. In the present paper, as an example of how we selected data for our RAC studies, we now describe how we gathered the relevant data for a sample of dK6 stars. We selected a sample of 419 late K dwarfs on the basis of (R-I) measurements available in the literature. For example, our sample of dK6 stars (see Table 1 below) contains stars with (R-I)_C (i.e. (R-I) color in the Cousins system) in the range [0.684;0.816] which also corresponds to (R-I)_K ((R-I) in the Kron system) in the range [0.503;0.613] according to the transformation formulae of Leggett (1992) (see Leggett 1992 for more information on the Cousin's and Kron photometric systems). According to Kenyon and Hartmann (1995), this range of colors is centered on (R-I)_C=0.75, i.e. the spectral type dK7. However, when we compiled and derived effective temperatures (see Paper I) for this sample of late-K dwarfs, we found in average higher temperatures than what would be expected from the (R-I)_C-T_eff tabulation of Kenyon and Hartmann (1995) (see Paper I). Our dK6 stellar sample contains stars that have similar (R-I)_C colours and the same effective temperatures to within ±110 K (see Paper I). We refer to HM for a discussion of corresponding data for our sample of dM3 stars. The literature provided us with a starting list of a large number (419) of late-K dwarfs. Searching through databases at the European Southern Observatory (ESO) and Observatoire de Haute Provence (OHP), we identified spectra of 112 different stars which are suitable for our purposes. The final list of 105 stars for which the available spectra would allow us to determine reliable rotational data (i.e. vsin i values) in our sample of late-K dwarfs has already been provided in Paper I. For the present paper, we found spectra which would allow us to make reliable measurements of the EWs of Ca ii and H_α for a sub-sample of only 89 late-K dwarfs. It is the combination of reliable rotations (Paper I) and reliable EWs of chromospheric lines which enable us to undertake, in the present paper, the study of the RAC in our sub-sample of dK6 dwarfs. The spectra which we use for determining the Ca ii and H_α equivalent widths in the present study of dK4-dM4 stars came from three different échelle spectrographs; HARPS (High Accuracy Radial velocity Planet Search, ESO), SOPHIE (OHP) and FEROS (The Fiber-fed Extended Range Optical Spectrograph). We included in our sample the SOPHIE observations obtained in the High Efficiency (HE) mode. The modes in which the spectra were obtained are indicated in Table 1 together with the corresponding Signal to Noise (S/N) ratio. For further details of the spectrographs, see HM. As a second example of how we selected pertinent data for our spectral sub-samples, we consider here briefly our sample of dM3 stars. As described in HM, westarted off with a list of 381 dM3 objects based on (R-I) data in the literature. Searching through the same databases as above, we found suitable observations which allowed us to determine reliable vsin i values for 86 different dM3 stars. A subsequent search of the available spectra in the above databases allowed us to obtain reliable measurements of the EWs of Ca ii and H_α for a sub-sample of only 59 M3 dwarfs for our chromospheric RAC (excluding probable spectroscopic binaries and low sin i stars). §.§ Biases in our stellar samples The stars in our samples include all stars from all observing programs which have been carried out with HARPS and SOPHIE for stars belonging to the following spectral sub-types: dK4, dK6, dM2, and dM3. For dM4 stars we compiled all measurements of vsin i available in the literature (see Paper I). For the dK6 and dM3 samples, we also supplemented our own measurements with measurements available in the literature, notably for active stars (see Paper I). In the HARPS and SOPHIE databases, many of the spectra were obtained in connection with planet-search programs. In such programs, observers tend to avoid stars with high levels of magnetic activity. Therefore our spectral samples are likely to be biased in general towards low activity stars, i.e. stars which are in the unsaturated portions of the RACs. For reasons outlined above (see Section 1.1), we consider this bias to be an advantage in the present study. The biases in our samples may contribute somewhat to the density of the sampling at different parts of the RAC, but this is not expected to cause significant discrepancies as regards the overall RACs. Note that for the dM4 and dK6 samples, the sampling of the RACs should be more complete as regards the measures of P/sin i (because we included other measures of vsin i from the literature, see Paper I). § THE ROTATION-ACTIVITY CORRELATIONS (RACS) IN LATE-K AND M DWARFS In this section, we first (Sub-section 3.1) evaluate the surface fluxes in the continuum in the vicinity of the Ca ii resonance doublet by using our estimates of T_eff and also by using the synthetic spectra of de Laverny et al. (2012). Then, we construct the RACs of M and K dwarfs using a variety of approaches: some approaches may facilitate the extraction of information that is more difficult to extract by means of other approaches. In all of the approaches, we plot a quantity related to “activity" as the ordinate, and a quantity related to “rotation" as the abscissa. Our first approach (Sub-Sections 3.2, 3.3, 3.4, 3.5, and 3.6) specifies the “activity" of a star in terms of the surface fluxes of chromospheric lines. Combining these surface flux results with our results for P/sin i (Paper I), we construct a set of RACs for M and K dwarfs belonging to various spectral sub-types (see Figs. 2, 3, 5, 8, and 12). We first analyze the RAC for K4 and K6 dwarfs. Then we re-visit the RAC in M2 dwarfs that was first investigated in Paper XV. We then analyse the RAC in M3 dwarfs for which vsin i measures were reported in HM and for which we present new measurements of the chromospheric line equivalent widths here. We also re-visit the RAC in M4 dwarfs that was studied in Paper XVIII with the new values of the stellar parameters determined in Paper I. Finally, we compare the RACs for five spectral K and M subtypes, dK4, dK6, dM2, dM3 and dM4, and draw conclusions about the differential variations of the RAC from mid-K dwarfs to M4 dwarfs. We emphasize that our RACs are plotted as a function of the projected rotation period P/sin i and not the rotation period P. The average sin i is 0.6. Then in order to recover the RACs as a function of P, one must multiply our RACs by a factor of 1.67. The scatter due to variable sin i is included in our correlations. Nevertheless, we show that one can obtain reasonably good empirical RACs in spite of these uncertainties and the uncertainties on our measures. In Sub-Sections 3.7 and 3.8, we present an overview of the systematic properties of the RACs in all of our sub-types, first as regards the slopes of the RACs (Sub-Section 3.7), and then as regards the absolute values of the chromospheric emission levels (Sub-Section 3.8). In Sub-Section 3.9, we switch to a different approach to constructing RACs. As regards rotation, we switch to the Rossby number. As regards chromospheric activity, we switch to a quantity which expresses the output power in the H and K lines of Ca ii as a fraction of the star's output power (L_bol) (Fig. 15). Switching our attention to the corona (Sub-Section 3.10), we construct a different type of RAC for our targets, this time referring to conditions in the corona, rather than the chromosphere. The coronal RAC is obtained by plotting L_X/L_bol as a function of the Rossby number (Fig. 16). Comparing “activity" in chromosphere and corona (Sub-Section 3.11; Figs. 17, 18) can provide information as to how any given M dwarf star partitions its deposition of mechanical energy between chromosphere and corona. To quantify this partition, in Sub-Section 3.11 we examine the correlations between L_X and L_HK (see Figs. 17, 18). In the final Sub-Section (3.12), we summarize the properties of the various RACs which our data have enabled us to construct. §.§ The mean fluxes in the vicinity of the Ca ii resonance doublet. In order to evaluate the efficiency of the dynamo mechanism(s) as a function of spectral type in M and K dwarfs, it seems preferable to inter-compare the RAC's calibrated in terms of absolute energy fluxes, rather than confining our attention to the values of the equivalent width (EW). (Nevertheless, the EW have proven useful in terms of surface magnetic fields: see Mullan et al. 2015.) In order to convert from EW to energy fluxes, we calculated the surface fluxes in the continuum in the vicinity of the Ca ii lines from the theoretical model atmospheres of de Laverny et al. (2012) and Palacios et al. (2010) [ http://npollux.lupm.univ-montp2.fr/ or ftp://ftp.oca.eu/pub/laverny/DEPOT/AMBRE_Grid_Flux/] for log(g)=5.0, [M/H]=0.0 and α=0.0. We found that in M2, M3 and M4 dwarfs, for our observations, the continuum at about 3950 Å represents a good evaluation of the background continuum flux for the Ca ii lines. But there are significant discrepancies between the observations and the models of de Laverny et al. (2012). We show a spectrum of a dM2 star (Gl 205) in Fig. 1 together with the model of de Laverny et al. (2012) for an effective temperature of 3500 K. We normalised these two spectra at 1 for the continuum flux at 3950 Å. As one can see in this figure, there are large differences in the continuum fluxes at 3910 Å and 4000 Å between the model and the observation. We believe these differences are due to missing opacities in the models. We also show in the lower panel of Fig. 1 the spectrum of Gl 570A together with the model for an effective temperature of 4500 K. Here again we note important differences: in the spectrum of Gl 570A, the continuum flux at 3950 Å is not a good estimate of the continuum flux in the vicinity of the Ca ii lines. In this case, one has to interpolate between the fluxes at 3910 Å and 4000 Å. Therefore, for our estimates of the theoretical surface fluxes in the vicinity of the Ca ii lines for M2, M3 and M4 dwarfs, we took the average of the flux at 3950 Å and the value interpolated between 3910 Å and 4000 Å. For K6 and K4 dwarfs, we used the value interpolated between 3910 Å and 4000 Å. Using this approach, we found that the mean surface fluxes in the continuum in the vicinity of the Ca ii lines are: 5.51× 10^5, 2.18× 10^5, 4.53× 10^4, 2.72× 10^4, and 1.74× 10^4 ergs s^-1 cm^-2 Å ^-1 for dK4, dK6, dM2, dM3 and dM4 stars respectively. The strong decline (factor of ∼30) in surface flux in the violet with decreasing T_eff from dK4 to dM4 is apparent in these numbers. Using these figures, we are now in a position to examine quantitatively how the surface fluxes in the Ca ii lines behave as a function of P/sin i. In the subsequent sections, the Ca ii surface fluxes were computed for each star according to its effective temperature. We expect that the models give estimates of the continuum surface fluxes with a precision of the order of 40%. As mentioned above, this figure is far below the factor of ∼30 in the continuum surface fluxes in the violet with decreasing T_eff from dK4 to dM4. Therefore, the decline in the Ca ii surface fluxes that we observe below (see Figure 14) in the RACs from dK4 to dM4 is highly significant. §.§ The RAC in K4 dwarfs In this subsection, we re-investigate the RACs in our sample of dK4 stars, that was first investigated in Houdebine (2012b, Paper XVIII). We re-iterate that the RACs are of fundamental importance in order to constrain an essential parameter of the dynamo mechanism(s): the role played by rotation. In our previous studies (e.g. Paper VII, Paper XIV, Paper XVI, Paper XVII, HM), we reported on our results for vsin i and P/sin i for stars of low activity level (i.e. slow rotators) for the spectral sub-types dK4, dM2, dM3 and dM4. In Paper I we have reported similar rotational data for dK6 stars. Combining these rotational data with the surface fluxes of chromospheric lines in dK and dM starsstars provides a unique opportunity to investigate the RACs in a fine-grained sample of M and K dwarfs which are sub-divided across five closely-spaced, but distinct, spectral sub-types. We show in Fig. 2 the RAC which we have obtained for our dK4 stars. In previously published studies of RACs in cool dwarfs, the majority of the results were reported in terms of homoscedastic linear least square fits (LSF) in order to fit their observations. In the present study, we propose different approaches. First, we perform homoscedastic linear and quadratic least square fits to our samples of low+high activity stars (dK+dKe or dM+dMe). Second, we perform heteroscedastic linear and quadratic least square fits (LSF that account for measurement errors, see the Appendix) to our samples of low+high activity stars. Third, we also compute homoscedastic and heteroscedastic linear least square fits to two particular sub-sets of our data, namely, stars in which activity is at a low level (dK, dM) and at a high level (dKe, dMe) respectively. We inter-compare the results of these fits for various spectral sub-types in turn in this subsection and in the following subsections. As we shall see, these various fits allow us to account for the complexity of our data sets. In our least-square fits, we do not include in general the suspected low sin i stars, lower limits or spectroscopic binaries. However, in the present sub-section, for dK4 stars, it appears that some of the lower limit measurements do correlate with the other measurements of dK4 stars. Therefore, we included some of these measurements in our correlations (we did not include obvious outliers). The heteroscedastic linear least-squares fit to the data in Fig. 2 is shown by the solid line. The equation for the solid line in Fig. 2 is as follows: F_CaII = 1.916± 0.32× 10^6× (P/sini)^-0.8140± 0.059. The χ^2 for the fit in Eq. (1) is only 0.033 for our sample of 34 dK4+dK4e stars. The statistical significance of this fit is 99.9% (see Table 1). (Results of all least-squares fits for this and subsequent spectral sub-types can be found in Table 1.) The homoscedastic linear least-squares fit yields similar results: F_CaII = 1.57± 0.34× 10^6× (P/sini)^-0.756± 0.05. The χ^2 for this fit is 0.032 for 34 dK4+dK4e stars. The correlation coefficient for this fit is found to be 0.884 and the statistical significance of the correlation is at least 99.9% (see Table 1). Therefore, these two fits are highly statistically significant. The reason for studying heteroscedastic fits in addition to the homoscedastic fits is that our P/sin i measurements have large errors and these errors vary as a function of the values of P/sin i (rapid rotators have smaller errors than slow rotators). The presence of these variable errors may yield heteroscedastic fits that are in some regards different from the homoscedastic fits. We shall see examples of this statement in some of the following subsections. Two aspects of the fits in eqs. (1) and (2) will be referred to in the subsequent discussion. First, the exponent of P/sin i will be referred to as the “RAC slope" in a plot of log(F_CaII) versus log(P/sin i). Systematic changes in the numerical value of the RAC slope will lead us to an important conclusion of the present paper. Second, the numerical coefficient closest to the equals sign is a measure of the absolute level of the chromospheric emission among the stars in the sample. Systematic changes in the numerical value of the coefficient will also be an important conclusion of this paper. So far in this sub-section, we have done an analysis of all of our dK4 stars, i.e. we have combined both the dK4 stars and the dK4e stars into a single sample. Now we split our sample up into two groups (dK4 in one, dK4e in the second), and analyze each group separately. Specifically, we now apply heteroscedastic and homoscedastic linear LSF to the sub-samples of only the low activity stars (dK4) and then repeat the exercise including only the high activity stars (dK4e) (see Table 1). The reason for performing these separate fits is that we found that the linear fits did not reproduce well both the low activity and high activity sub-samples: this result led us to wonder if these two samples might contain different dynamo modes. If it turns out that indeed different dynamo modes are at work in slow rotators and fast rotators, then separate analyses of the data sets is warranted. We find that the linear heteroscedastic LSF to the dK4 low activity stars yield: F_CaII = 7.66± 1.89× 10^5× (P/sini)^-0.568± 0.084. The χ^2 for this fit is 0.034 for our sample of 30 dK4 stars and the statistical significance of the correlation is at least 99.9% (see Table 1). This fit is shown as the straight dot-dashed line in Fig. 2. For the homoscedastic fit, we obtain: F_CaII = 9.33± 3.16× 10^5× (P/sini)^-0.624± 0.11. The χ^2 for this fit is only 0.031 for 30 dK4 stars, the correlation coefficient is 0.744 and the statistical significance is at least 99.9% (see Table 1). Therefore, both of these fits (homo and hetero) are highly statistically significant in spite of the scatter in the data. We note that the differences between the homoscedastic and the heteroscedastic fits fall within the uncertainties of the parameters of the fits. The linear heteroscedastic LSF to the dK4e high activity stars yield: F_CaII = 9.73± 5.07× 10^6× (P/sini)^-1.877± 0.51. The χ^2 for this fit is 0.038 for 4 dK4e stars and the statistical significance is 99.7% (see Table 1). This fit is shown as the upper dot-dashed straight line in Fig. 2. We must admit that our dK4e sample contains only 4 stars and that therefore the RAC for dK4e stars is not well constrained. However, as we shall see in the subsequent sub-sections, we find that this RAC is consistent with the trend of the high activity star RACs at other spectral types. An important feature of our results emerges when we compare Eqs. (1) and (3), and when we compare Eqs. (2), (4) and (5). Whether we consider homo- or hetero-scedastic results, we find that the gradient of the RAC for the low activity stars alone (-0.57, -0.64) is shallower in magnitude than that of the combined sample of low+high activity stars (-0.81, -0.76), and that the gradient of the RAC for the high activity stars alone (-1.88, -1.877) is larger in magnitude than that of the combined sample of low+high activity stars (-0.81, -0.76). We shall find that this occurs systematically between the low activity sub-samples and the full samples for all five of our spectral sub-types (see Table 1 and Sections 3.3-3.6 below). The effect is more pronounced as the spectral type increases (see also Sect. 3.7). For the high activity sub-samples, we shall find that the slope is steeper than that of the combined samples for dK4, dK6 and dM2 stars, but that it reverses for the dM3 and dM4 stars (at the TTCC and beyond) and becomes shallower than that of the combined samples. As a consequence, the linear fits to the combined samples of low+high activity stars tend to overestimate the slope in the low activity sub-samples, and underestimate the slopes in the high activity stars sub-samples for dK4, dK6 and dM2 stars. Therefore, we also performed homoscedastic and heteroscedastic quadratic LSF to our samples of low+high activity stars for our five spectral sub-types. The quadratic fits allow us to reproduce both the shallower gradient of the linear LSF among the low activity stars as well as the higher fluxes among the high activity stars. This shows that the quadratic fit may give a better description of the data than the linear fit for the combined samples of low+high activity stars. However, according to an anonymous referee comments, the quadratic fits may not be significantly different from the results of the linear fits. To argue on this point, the referee performed a simulation on sub-samples of dM and dMe stars with random errors typical of those we find in this study. The referee found also a shallower slope among his sub-sample of slow rotators. Given the evidence at hand, the signatures identified as possible evidences supporting the case for a quadratic fits are most probably due to random errors working in combination with a fairly narrow log(P/sin i) domain as compared to those errors. Therefore, although the slopes of the high activity sub-samples in dK4, dK6 and dM2 are in favor of a quadratic description of the data, we cannot yet conclude that quadratic fits represent definitely a better representation of the full datasets. We can also note that the χ^2s are comparable between linear and quadratic fits. More data will be required to conclude. As an anonymous referee rightly pointed to, if the data are localized to a domain that is only 2-3 times as large as a typical error bar, random measurement errors can easily randomize the data and weaken the slope of any true correlation. However, this point is not statistically correct. If the distribution of the errors is Gaussian (which we assume here), then the correct parameter to compare to the RAC domain R is δ = <error>/√(n) where n is the number of measures. The value of R/δ is 17.2 for our low activity dK4 star sample which is far larger the values of 2-3 mentionned above. It appears that R/δ lies in the range 17 to 40 for our low activity and high activity RACs. The levels of confidence of a given LSF depends not only on the mean error but alo on the number of measures. Therefore, our LSFs should be established to a fairly high level of confidence, which is confirmed by the high statistical significances that we obtained for our least square fits (see Table 1). Nevertheless, the simulation of the referee is of interest to us. We take this important point into account and we emphasize here the preliminary character of our results. Indoubtedly, these interesting results should be comfirmed with additional data, obtained preferably with a higher resolution spectrograph for the slow rotators such as ESPRESSO (ESO, R=220,000). Therefore, in the following sub-sections and Sect. 3.7, our results on the separate fits to the sub-samples of low and high activity stars should be considered with caution and are only preliminary. They should be confirmed with larger stellar samples. §.§ The RAC in K6 dwarfs In this subsection, we discuss the results for our sample of dK6 stars. For the sake of consistency with previous measurements in this series of papers, and to avoid duplication in describing the method, we refer the reader to Papers VI and XV to see how we evaluate the EW for the Ca ii resonance doublet and the H_α line. In Table 2, we list the EW we have obtained for the Ca ii resonance doublet and for the H_α line for our dK6 stars. We note that our dK6 stellar sample contains only one star with H_α definitely in emission (i.e. a dK6e star): Gl 517 (EQ Vir), with the fastest rotation (vsin i = 9.77 km s^-1, P/sini = 3.05 days, see Paper I). Our “dK6" sample also includes one star of “intermediate" activity with H_α neither in emission nor absorption (i.e. a dK6(e) star): Gl 208 (Table 2). As regards EQ Vir, this star has been classified in the literature as a dK5 star. In fact, Gl 517 is a BY Dra star with colours which vary with time (depending on spot coverage). Our observations contain many slow rotators but very few fast rotators (dKe stars). Therefore, in order to complete our sample and have a better defined RAC for dK6 stars, in Paper I we compiled vsin i and Psin i measures from the literature. We found 43 additional vsin i and Psin i measures which brings our total compilation of measures to 150. We also retrieved some FEROS spectra from the ESO Archive for Gl 142, Gl 885A, Gl 900 and HIP 113597. The measures of the Ca ii EW for these stars are given in Table 2. We also searched the literature for other measurements of the Ca ii EW. We found only a few measurements of the Ca ii EW or flux for our dK6 stars (Marilli et al. (1986), Rutten (1987), Duncan et al. (1991), Browning et al. (2010)). These measures are also listed in Table 2. We saw in Paper XV (see also Sect. 3.4) that correction for metallicity effects in dM2 stars is essential in order to obtain a good correlation between the Ca ii EW and P/sin i. This is due to the fact that the Ca ii line formation depends sensitively on the Ca abundance (Houdebine & Panagi 2016, in preparation). Here the same applies to dK6 stars and to the Ca ii surface fluxes. In Paper I we compiled [M/H] measures for our dK6 stellar sample and here we computed the Ca ii surface fluxes corrected for metallicity effects, assuming a proportionality between [M/H] and Ca ii surface fluxes. We show in Fig. 3 the RAC which we have obtained for our dK6 stars. The heteroscedastic linear LSF to the data in Fig. 3 is shown by the solid line. One of the stars in our sample, Gl 208 (a dM6(e) star with P/sini = 8.94 days), is probably also a fast rotator: but its value of vsin i is too small (and its value of P/sin i is too large) to be entirely consistent with its observed activity level. It is possible that we are viewing this star close to its rotation axis, i.e. sin i may be atypically low. We also report on two relatively active stars with a rather slow rotation: Gl 455.1 and Gl 907.1. These stars depart noticeably from the main correlation. We believe that these stars also have low sin i. Most of the other stars follow the solid-line correlation fairly well with a few exceptions: a group of stars that lie below the main correlation and are low activity-fast rotators. One of them is a subdwarf: their internal structure may cause their dynamos to operate differently from those in main sequence K stars. For the few other stars, similar discrepancies from the RAC have been found in dM2 stars (Paper XV and Sect. 3.4 below); they will also be noted among dM3 stars (see Sect. 3.5 below). Reiners et al. (2012) also observed a few stars which are discrepant compared to the global trend: Reiners et al. disregarded such objects as being due to measurement errors. However, the repetition with which they appear in our samples, and the fact that they are relatively rapidly rotating, lead us to believe that these low-activity relatively fast rotators may indeed exist as a possibly significant sub-set of late K and M dwarfs. The best explanation we have is that, in a star such as Gl 412.3AB, which exhibits a slight asymmetry when its photospheric spectral lines are subjected to rotational analysis (see Paper I), these stars may be unresolved spectroscopic binaries: as such, they could yield abnormally broadened cross-correlation profiles (see also Gl 186AB in Paper I). Therefore these stars are probably spectroscopic binaries that are unresolved at the time the observations were made. This is quite plausible since binarity is a common characteristic of late-type dwarfs. We label these stars as spectroscopic binaries in Fig. 3. Including the known binaries in our dK6 sample, the number of probable spectroscopic binaries yield a proportion of 20% binaries in our full sample. This is consistent with the recent finding of Ward-Duoung et al. (2015) that binary stars constitute a percentage of about 25% of their stellar sample for M_*∼ 0.7M_⊙. We shall see in the subsequent subsections that the proportion of binaries amounts to about 28%, 37% and 48% for our dM2, dM3 and dM4 stellar samples respectively. Our dM2 and dM3 results are again compatible with the fraction of about 32-37% derived by Ward-Duoung et al. (2015) for these stars. For our dM4 sample, our figure of 48% is somewhat larger than the 37% fraction expected at this spectral type. This disagreement may find a simple explanation in the bias in detecting parallaxes for nearby M dwarfs. Indeed, faint single M4 dwarfs are much more difficult to detect than the companions to nearby brighter M and K dwarfs. Therefore, our dM4 sample, which is largely based on parallax surveys of nearby dwarfs is biased towards the detection of faint companions to nearby brighter M and K dwarfs. This should explain why we have a larger binary fraction for the M4 dwarf sample. We should like to emphasize four points regarding the plot in Fig. 3. (i) Metallicity differences from star to star contribute to the scatter about the dK6 RAC in Fig. 3. By analogy, in Paper XV we reported that metallicity differences among dM2 stars are responsible for the greater part of the scatter in the RAC. (ii) Another contribution to the scatter is that our statistics on the Ca ii line EW are poor; we have very few measures of the Ca ii line EW for most of our dK6 stars compared to what we have for the stars in our dM2 sample, and the Ca ii line EW is known to vary with rotation and with the phase of the activity cycle (e.g. Baliunas et al. 1995). However, we have little or no information as to which part of the cycle our observations happened to “catch" any particular star. (iii) The value of sin i may vary from 1 to 0 and certainly contributes to the scatter in Fig. 3. (iv) The range of effective temperatures in our dK6 sample is somewhat larger than in our samples of dM2, dM3 and dM4 stars (e.g. only ± 70 K in dM3 stars, where T_eff ranges from 3210 to 3350 K). We would like to emphasize as well that a few of the active stars in our dK6 stellar sample are candidate young stars that may not yet have contracted to the Main-Sequence (MS). A reliable way to identify the Pre-Main-Sequence (PMS) stars in our samples is the stellar radius: the fact is, PMS stars that have not yet contracted to the MS have abnormally large radii. We identified three such stars in our dK6 sample: GJ 1177A, GJ 182 and GJ 425B are possible PMS stars (see Paper I). However, Gl 425B is a rather low activity star. As such, the abnormally large radius for this star is probably due to binarity. Nevertheless, we find that these stars do correlate well with the MS stars. In our M2 sample (see next Section), we have identified also two PMS stars: GJ 1264 and GJ 803. But again, as we shall see, these PMS stars do correlate very well with the other MS stars. In our M3 sample, GJ 277A is a possible PMS star. In our M4 sample, GJ 2069A, GJ 3322, GJ 669A, GJ 695B, GJ 812A are possible PMS stars according to their radii. We emphasize that all these stars do not rotate especially fast. There are many MS stars that rotate faster. These PMS stars are expected actually to spin up as they contract to the MS, and young MS stars are expected to be the fastest rotators (e.g. Barnes 2003; for a theoretical model of this process, see e.g. Fig. 2 in MacDonald & Mullan 2003). That is what we actually observe in our samples of stars: young MS stars are the fastest rotators (e.g. among our M4 sample; GJ 3631, GJ 3789, GJ 4020B, GJ 4338B, GJ 431, GJ 630.1, GJ 791.2A) with vsin i in excess of 15 km s^-1 and up to 56 km s^-1. All our RACs demonstrate that PMS stars do not stand out as significantly different from the main correlations of the MS stars (see subsequent Sections). Numerous previous studies also found similar results: e.g. Mamajek & Hillebrand (2008), Browning et al. (2010), Christian et al. (2011), West et al. (2015). This finding is rather intriguing since PMS stars may have internal structures which differ from those of MS stars: in fact, some PMS stars may even be fully convective. More investigation is required to confirm this result, but so far, in our samples, we found no definitive evidence that PMS stars obey different RACs from those of MS stars. The heteroscedastic linear LSF to the RAC in Fig. 3 gives the following: F_CaII = 3.402± 0.021× 10^6× (P/sini)^-1.047± 0.042. The parameters of this LSF are given in Table 1. The χ^2 is 0.093 and the statistical significance is 99.9% for 55 dK6 stars. The homoscedastic linear LSF to the same data set gives: F_CaII = 1.95± 0.34× 10^6× (P/sini)^-0.81± 0.06. The correlation coefficient for this fit is 0.876, the statistical significance is 99.9% and the χ^2 is only 0.021 for 55 dK6 stars. We find in this case that the heteroscedastic and the homoscedastic linear fits give slightly different results for the slope and for the flux amplitude: This is due to the fact that high activity stars (fast rotators) have smaller uncertainties in P/sin i compared to low activity stars, and this give them much higher weights in the least square fit. As a result, the heteroscedastic linear solution fits the high activity stars better, whereas the homoscedastic linear solution clearly underestimates the fluxes among high activity stars. Therefore, in this case the heteroscedastic fit gives more sensible results. For future reference, we note that in Fig. 3, the correlation in Eq. (6) spans the entire range of P/sin i values for which we have dK6 data. In particular, it is important to notice that the dK6 data exhibit no evidence for a flattening (or “saturation") of the RAC at the shortest periods (P/sin i = 1.8 days). This is consistent with previous results for the earlier spectral type (dK4) which also showed no signs of saturation (see Sect. 3.2). Neither is there evidence for saturation among dM2 stars (see Fig. 5 below). Having analyzed the combined samples of slow and fast rotators, we now turn to performing heteroscedastic and homoscedastic linear LSF on the sub-samples of low activity stars (dK6) and high activity stars (dK6e) separately. The heteroscedastic linear LSF to the dK6 low activity stars yields: F_CaII = 8.23± 2.23× 10^5× (P/sini)^-0.531± 0.12. The χ^2 for this fit is 0.017 and the statistical significance is 99.9% for 41 dK6 stars (see Table 1). This fit is shown as the straight dot-dashed line in the lower left portion of Fig. 3. The homoscedastic linear LSF to this sub-sample gives: F_CaII = 1.29± 0.36× 10^6× (P/sini)^-0.637± 0.11. The correlation coefficient for this fit is 0.706, the statistical significance is 99.9% and the χ^2 is only 0.014 for 41 dK6 stars. Therefore, these two fits are highly statistically significant at a confidence level greater than 99.9%. We also fitted the high activity stellar sub-sample. The heteroscedastic linear LSF to the dK6 high activity stars yields: F_CaII = 4.20± 3.52× 10^6× (P/sini)^-1.402± 0.43. The χ^2 for this fit is 0.023 and the statistical significance is 99.9% for 9 dK6e stars (see Table 1). This fit is shown as the straight dot-dashed line in the upper right portion of Fig. 3. We again find that the gradient of the RAC for the low activity stars alone (-0.86, -0.64) seems shallower than that of the low+high activity star sample (-1.05, -0.81) and that the gradient of the RAC for the high activity stars alone (-1.402, -1.364) seems steeper than that of the low+high activity star sample (-1.05, -0.81). We find that this difference in RAC steepness occurs systematically between the low activity sub-samples and the full samples for all five of our spectral sub-types (see Table 1 and Sections 3.4-3.6 below). The parameter R/δ is 17.20 for the high activity star sub-sample and 39.68 for the low activity star sub-sample. Therefore, the parameter δ =<error>/√(n) which is a normalized estimate of the mean error on the measurements is much smaller than the period domains of the RACs in both cases. Hence, these LSFs should be relatively well established, which corresponds to the high statistical significances we obtain (Table 1). However, we consider these results as still preliminary because the domains of the RACs are relatively small compared to the typical uncertainties on individual measures. If this difference between the RACs of the low and high activity sub-samples is confirmed to be true, this may represent a discovery of interest: Indeed, to the extent that the slope of an RAC is related in some way to an underlying dynamo process, the difference which we found between the slopes of the low and high activity sub-samples suggests that there may exist two different dynamo regimes for these two sub-samples of stars. We find that this difference between low and high activity regimes persists for all our samples of stars with different spectral sub-types. However, we observe a difference for stars which are more massive than the TTCC and stars which are less massive than the TTCC. For the former, the slope among high activity stars is steeper than the slope among the low+high activity stars, whereas for the latter, we find that the slope among high activity stars is shallower than the slope among the low+high activity stars. This preliminary result could be of interest for the dynamo mechanisms. More data with a higher resolution spectrograph will be needed to confirm these results. §.§ The RAC in M2 dwarfs We have re-investigated the RAC in the sample of dM2 stars of Paper XV. We have given the new stellar parameters and new P/sin i values in Paper I. We show the raw data in Fig. 4. We can see in this figure that there is a large scatter among M2 dwarf low activity stars. Most of the subdwarfs also lie significantly apart from most of the M2 dwarfs. Most of this large scatter is due to metallicity effects on the Ca ii line formation. We used the empirical correlation found by Houdebine (2008, Paper VII) to determine the metallicity for each star as a function of its radius (see Paper I). We use these values here to correct the Ca ii surface fluxes for metallicity effects, assuming a proportionality between surface flux and metallicity (optically thin case). The Ca ii line EW, surface fluxes and surface fluxes corrected for metallicity are all listed in Table 3. We show in Fig. 5 the Ca ii surface fluxes (after the data have been corrected for metallicity effects) as a function of P/sin i. The improvement of the correlation between this figure and Fig. 4 is striking: it shows how important it is to correct the RAC for metallicity effects in M dwarfs. In Fig. 5, the scatter has been much reduced among the slow rotators, and also among the fast rotators. Moreover, even subdwarfs correlate with normal metallicity dwarfs. In Fig. 5, we plot the uncertainties on P/sin i as dotted lines for each star. For these uncertainties, we assumed an uncertainty of ± 0.14 km s^-1 for measures of vsin i below 1 km s^-1, ± 0.30 km s^-1 for vsin i between 1 km s^-1 and 6 km s^-1, and ± 0.50 km s^-1 for vsin i above 6 km s^-1 (see Paper I). Uncertainties in the values of P/sin i have already been included in Figs. 2 and 3 above, following the same prescription as we describe here. Similar uncertainties will be included in Figs. 8 and 12 below in connection with dM3 and dM4 stars respectively. In Fig. 5, the straight solid line shows the heteroscedastic linear LSF to the data. There is little difference between this fit and the linear LSF that was obtained in Paper XV. There is a moderate shift between the two correlations, and the gradient is almost unchanged (see Table 2). We identified some low activity-relatively fast rotators in this M2 dwarf sample. As already mentioned in the previous section, these stars are probably unresolved spectroscopic binaries. So we have labelled these stars “spectroscopic binaries" in Fig. 5. There are also a few stars that have too long rotation periods for their surface fluxes. We believe these stars are a sub-group of stars with low sin i. These stars are shown as squares in Fig. 5. The heteroscedastic linear LSF to the RAC in Fig. 5 gives the following: F_CaII = 2.312± 0.276× 10^6× (P/sini)^-1.575± 0.058. The parameters of this LSF are given in Table 1. The χ^2 is 0.028 for 66 dM2 stars. The homoscedastic linear LSF to the RAC in Fig. 5 gives the following: F_CaII = 1.89± 0.85× 10^6× (P/sini)^-1.481± 0.068. The parameters of this LSF are given in Table 2. The correlation coefficient is 0.949 and the χ^2 is 0.018 for 66 dM2 stars. The gradient of this correlation (-1.481) is very close to that of the correlation found in Paper XV (-1.53). In fact there are few differences between the two results: the correlation has only been shifted slightly in P/sin i. Both the homoscedastic and the heteroscedastic give highly significant correlations with a statistical significance better than 99.9%. The heteroscedastic linear fit yields a somewhat slightly larger slope than the homoscedastic linear fit. This is because the errors on the dM2e stars are significantly smaller than those of the dM2 stars, and therefore the weights of the dM2e stars in the correlations are larger than those of dM2 stars in the heteroscedastic fit. As a consequence, the heteroscedastic fit goes through the sub-sample of dM2e stars whereas the homoscedastic fit underestimates clearly the higher fluxes of the dM2e stars. Therefore the heteroscedastic fit provides globally a better evaluation of the slope for the dM2+dM2e stars. We also performed linear LSF to the sub-samples of only the dM2 low activity and the high activity dM2e stars (see Table 1). The heteroscedastic linear LSF to the dM2 low activity stars sub-sample yields: F_CaII = 2.68± 0.77× 10^5× (P/sini)^-0.709± 0.14. Therefore, the gradient of the RAC for the low activity dM2 stars is again significantly shallower than that of the combined sample of low+high activity stars. For the homoscedastic linear LSF to the dM2 low activity stars, we obtain: F_CaII = 4.17± 1.15× 10^5× (P/sini)^-0.891± 0.12. Both the heteroscedastic and the homoscedastic fits show that the gradient of the RAC for the low activity dM2 stars is again (as for dK4 and dK6 stars) significantly shallower than those of the combined sample of low+high activity stars. The χ^2 for these fits are 0.046 and 0.013 respectively. These fits are statistically significant at a confidence level better than 99.9%. We note that the slope of the heteroscedastic fit is slightly shallower than that of the homoscedastic fit and is also shallower than the value found for the heteroscedastic fit for the dK6 sample. However, if we take into account errors (see Table 1), there are in fact no significant differences between these values. The heteroscedastic linear LSF to the dM2e high activity star sub-sample yields: F_CaII = 3.28± 0.54× 10^6× (P/sini)^-1.793± 0.14. Therefore, the slope of the RAC for the high activity dM2e stars is again steeper than that of the combined sample of low+high activity stars. We note that this slope gets closer to that of the combined sample as we move from dK4 to dM2. The differences between these two slopes were larger for dK4 and dK6 stars. The parameter R/δ is 23.45 for the high activity star sub-sample and 34.77 for the low activity star sub-sample. Therefore, the parameter δ =<error>/√(n) which is a normalized estimate of the mean error on the measurements is much smaller than the period domains of the RACs in both cases. Hence, these LSFs should be relatively well established, which corresponds to the high statistical significances we obtain (Table 1). However, we consider again these results as still preliminary because the domains of the RACs are relatively small compared to the typical uncertainties on individual measures. §.§ The RAC in M3 dwarfs In HM, we have already listed the results for the rotational parameters vsin i and P/sin i of our dM3 stellar sample. However, we re-computed the stellar parameters for our sample of dM3 stars in Paper I according to the new results of Mann et al. (2015). The revised values of the parameters for the dM3 sample can be found in Paper I. In the present paper, we report on chromospheric line data for dM3 stars: these data were not a part of HM. In Table 4, we list the EW we have obtained for the Ca ii resonance doublet and for the H_α line for our dM3 stars. In order to supplement our sample of vsin i and Ca ii measurements, and in order to have an unbiased sample of measurements, we searched the literature for additional vsin i and Ca ii measurements: We found several additional stars with vsin i measurements (see Paper I). But we found only a few measurements of the Ca ii lines for these stars (Stauffer & Hartmann 1986, Rutten 1987, Giampapa et al. 1989, Rutten et al. 1989). Instead, we found several measures of the H_α EW (Soderblom et al. 1991, Hawley et al. 1996, Kamper et al. 1997, Christian & Mathioudakis 2002, Gizis et al. 2002, Pace 2013). Fortunately, there is a relatively tight correlation between the mean Ca ii line mean EW and H_α EW in dM3 and dM4 stars. Therefore, we decided to infer the Ca ii line mean EW from the measures of the H_α EW. We show in Fig. 6 the relationship between the Ca ii line mean EW and the H_α EW for our measurements of dM3 stars (see Table 4). As one can see the correlation between these two parameters is very good for all stars except Gl 644AB which lies slightly below. The homoscedastic linear least square fit to this data except Gl 644AB gives: EW_CaII = 2.90± 0.05× EW_H_α + 1.14± 0.125 The parameters of this fit are given in Table 2. The fit is very good with a correlation coefficient of 0.9994. We give the Ca ii line mean EW computed from the H_α EW in Table 4. We show the RAC for dM3 stars for the raw data in Fig. 7. One can see in this figure that the scatter is very large among both dM3 and dM3e stars. The same kind of scatter was observed among the raw data of dM2 stars (Fig. 4). Considering the good correction we had for the metallicity in the previous section for dM2 stars, we decided to compile all the metallicities published for our initial selection list of 381 dM3 stars, and try to obtain a metallicity-radius correlation for these stars, similarly to the dM2 stars in Paper VII. We found metallicities from the literature for 147 dM3 stars. This data and the radius-[M/H] relationship were reported in Paper I. Data for M3 stars corrected for metallicity can be found in Table 5 and Fig. 8. We show in Fig. 8 the RAC corrected for metallicity effects for dM3 stars. The scatter is reduced compared to the raw data in Fig. 7, but there remains a significant scatter among both dM3 and dM3e stars. The corrected correlation is not as good as for dM2 stars. We believe this is due partly to the poorer statistics we have on the Ca ii line EWs. Indeed, for dM3 stars we have very few measurements of the Ca ii lines EWs (Table 4), whereas for dM2 stars we had several measurements for almost all stars (see Houdebine et al. 2012c, Paper XIX). We also observe a larger scatter in the radius-[M/H] relationship for dM3 stars (see Paper I) compared to dM2 stars. This may also lead to poorer corrections of the metallicity effects on the RAC. We observe among our sample of dM3 stars a sub-sample of relatively fast rotating-low activity stars. Again, as in the case of dM2 stars, we believe that these stars are unresolved spectroscopic binaries. We also found three stars with possibly low vsin i (see Fig. 8). In the corrected RAC, most of the subdwarfs now follow the same correlation as normal dwarfs, indicating that our metallicity corrections are reasonably correct. We find that for periods above 7 days or so, the gradient between the dM3 stars (open circles: the slowest rotators) and the dM3e stars (filled circles: the fastest rotators) is very steep. Because we have only a few dM3e stars compared to the larger group of dM3 stars, we do not obtain a good linear LSF to both the dM3 and dM3e sub-samples. This is due to the fact that the linear LSF to the dM3 sub-sample gives a shallower gradient (-0.90, Table 1) compared to the dM3+dM3e sample. Therefore, we gave higher weights to our dM3e data points: we find that a weight of 7 to dM3e stars and 1 to dM3 stars gives a good fit to both data sub-samples. For P/sini ≥ 6.7 days we obtain the following heteroscedastic linear LSF for our dM3+dM3e sample (Table 1): F_CaII = 1.169± 0.476× 10^7× (P/sini)^-2.020± 0.11. The gradient in Eq. (17) (-2.02±0.11) is significantly steeper than the gradients which we determined for our dK4 and dK6 samples (-0.814±0.059 and -1.047±0.042 respectively: see Figs. 2 and 3). But interestingly, the mean gradient in Fig. 8 is intermediate between that of dM2 stars (-1.575±0.058) and that of dM4 stars (-2.56±0.19: Table 1). Once again, this is an encouraging sign that the physical parameters we have derived for dM3 stars are plausible when compared with stars which are slightly hotter and slightly cooler. We plot the linear LSF for stars with periods P/sin i > 7 days (we refer to these as unsaturated stars: see below) in Fig. 8 (solid line). For the homoscedastic fit we obtain the following linear LSF for our dM3+dM3e sample (Table 1): F_CaII = 1.52± 1.00× 10^7× (P/sini)^-2.075± 0.19. The slopes of both the heteroscedastic and the homoscedastic linear fits are comparable. Note that the uncertainty on the slope is smaller for the heteroscedastic fit than for the homoscedastic fit. Both fits confirm that the slope of the RAC is getting steeper as we move to later spectral types. As we saw in Sect. 1.4, one would expect the RAC to flatten out for rotation periods shorter than 2-10 days. We have several objects in our M3 sample that have shorter rotation periods. We note that the RAC flattens out (“saturates") for the fastest rotating dM3e stars for P/sin i<6 days. In this period range, we find 7 fast rotators with about the same surface flux (∼ 2.97× 10^5 erg s^-1 cm^-2) which corresponds to the maximum observed in the unsaturated portion of the RAC. Although in the present data set we do not have enough data to clearly confirm the presence of a saturation plateau, one should expect saturation to occur among some of our sample stars according to previous investigations. We propose that saturation in our present dM3 sample occurs at about ∼ 6 days, which agrees with the expectations of saturation occuring in the range 2-10 days. In Fig. 8 we therefore represent the fits of the RAC in the “unsaturated" portion, and draw a flat plateau in the expected “saturated" portion of the RAC. The fact that certain stars in our samples probably lie in a saturated regime, while others are found to lie in an unsaturated regime, has a bearing on a methodological point which was made in Sect. 1.1: a study of stellar dynamos is best done (we believe) by focussing on stars in an unsaturated regime. Therefore in what follows, the least squares fits which we will present refer only to stars in the unsaturated regime. In this regime, the observations show that there exist stars of both low (dM) and high (dMe) activity, so the sample provides access to a range of “dynamo strengths". In contrast, in the saturated regime, only high-activity stars (dMe) are present. In the regime of our unsaturated stars, we also performed heteroscedastic and homoscedastic linear LSFs to the separate sub-samples of low activity dM3 and high activity dM3e stars respectively (see Table 1). The heteroscedastic linear LSF to the dM3 low activity stars (all of which are in the unsaturated regime) yields: F_CaII = 2.33± 1.10× 10^5× (P/sini)^-0.837± 0.20. Therefore, the gradient of the RAC for the low activity stars is again much shallower than that of the combined sample of low+high activity stars, and by more than 3σ. Therefore, the linear LSF to the combined sample may be somehow inadequate. We plot the linear LSF to the low activity sub-sample also in Fig. 8 (dot-dashed line). The homoscedastic linear LSF to the dM3 low activity stars yields: F_CaII = 3.09± 1.58× 10^5× (P/sini)^-0.93± 0.22. Both fits yield comparable results within errors. The χ^2 for these fits are 0.039 and 0.047 respectively. Both fits are highly statistically significant at a 99.9% confidence level. Once again, we see that the slopes of the RAC among low-activity dM3 stars (-0.84, -0.93) are significantly shallower (by more than 3σ) than the slopes of the RAC for the combined sample of high and low activity dM3 stars (-2.02, -2.08). Turning now to the high activity dM3e stars in the unsaturated regime, we find that the heteroscedastic linear LSF yields: F_CaII = 1.60± 1.20× 10^6× (P/sini)^-1.041± 0.58. In contrast to the results we obtained for the K4, K6, and M2 samples, this time for the dM3e sub-sample of unsaturated stars, the slope of the RAC is shallower than that of the combined sample of (unsaturated) low+high activity stars. It seems that at the TTCC, the slope of the RAC for (unsaturated) high activity stars is falling significantly to a value of about -1. Although there remains some uncertainty on our fit for dM3e stars (because we have only 8 stars in the unsaturated regime, and with a significant scatter), we shall see in the next subsection that this decrease in the steepness of the slope is confirmed in our (unsaturated) dM4e sub-sample and is significant above the 3σ level (23 stars). Our M4 sample also confirms that, the slope of the RACs for the high activity (unsaturated) sub-samples has fallen below the slope of the combined dM+dMe samples. We have also performed LSFs to the full sub-sample of dM3e stars (unsaturated+saturated). We give the results of the fits in Table 1 (dM3e+sat). The slope for the full sub-sample has fallen to -0.39 which is considerably smaller than the value of -1.04 we found above. However, this conforts us with the idea that at the TTCC, the slope of the RAC for dM3e stars is of the same order than that of the slope for the low activity dM3 stars (see Sect. 3.7.6). The relative invariance of the RAC slopes which we have obtained for the linear LSF to the low activity stars in our 5 stellar samples gives an important degree of credibility to our results. It is important to note that all of our low-activity stars have been found to lie in the unsaturated regime of the RAC, i.e. in the regime where (we believe) dynamo theory can best be tested. To the extent that the slope of the RAC is determined (in the unsaturated regime) by the physical properties of a dynamo, our results suggest that there may exist only one dynamo regime for the low activity stars from dK4 to dM4 within errors (see Sect. 3.7.2). The parameter R/δ is 18.28 for the high activity star sub-sample and 21.77 for the low activity star sub-sample. Therefore, the parameter δ =<error>/√(n) which is a normalized estimate of the mean error on the measurements is much smaller than the period domains of the RACs in both cases. Hence, these LSFs should be relatively well established, which corresponds to the high statistical significances we obtain (Table 1). However, these results may still preliminary because the domains of the RACs are relatively small compared to the typical uncertainties on individual measures as hilighted by the referee. There is the possibility that there exist two different dynamo regimes for dK, dM and dKe, dMe respectively throughout the spectral range we investigate. Note that, in contrast to the low activity dK and dM stars, the dynamo regimes in high activity dKe and dMe stars vary with spectral type, especially at the TTCC. Might these changes be an indication that the dynamo mechanisms undergo a change at the TTCC, perhaps from an α-Ω dynamo to an α^2 type of dynamo ? There tends to be more and more high activity-relatively slow rotators when we move to the spectral sub-types dM3 and dM4, which questions the validity of a single RAC for the combined samples at these spectral types. Given the consistency of the results we have obtained for the slopes of the RAC for the low activity stars for our 5 spectral sub-types, it is possible that a single dynamo mechanism may apply to all low-activity stars in our sample. However, we arrive at a different conclusion regarding the dynamo mechanism among the high activity stars. At the TTCC and beyond, we propose that high-activity stars may switch to a different dynamo mechanism (see Sect. 3.7). §.§ The RAC in M4 dwarfs Considering that in our previous study of the RAC in dM4 stars we had only a couple of dozen stars (Houdebine 2012a, Paper XVII), we decided to compile all available measures of vsin i published in the literature (see Paper I). With our own measures, we obtained vsin i measures for 106 dM4 stars (Paper I) from our initial list of 395 dM4 stars. Our dM4 stars in this study were selected according to their (R-I)_C color, within the range [1.500:1.700]. This range is somewhat larger than the one used in Paper XVII in order to have more targets. In order to obtain the RAC for dM4 stars, we also compiled the Ca ii resonance doublet and the H_α line EW and fluxes from the literature. We give this compilation of data in Table 6. The data were collected from the following authors: Giampapa & Liebert (1986), Stauffer & Hartmann (1986), Young et al. (1986), Fleming & Giampapa (1989), Herbst & Miller (1989), Hawley et al. (1996), Delfosse et al. (1998), Gizis et al. (2002), Mohanti & Basri (2003), Wright et al. (2004), Rauscher & Marcy (2006), Morales et al. (2008), Walkowicz & Hawley (2009), Browning et al. (2010), Isaacson & Fischer (2010), Houdebine (2012b), Reiners et al. (2012), Pace (2013). In addition to this data, we also measured the Ca ii and H_α EW from FEROS spectra from the ESO archive for 47 dM4 stars. We give the results in Table 6. Together with the measurements from Paper XVIII, this allows us to derive a correlation between the Ca ii EW and the H_α EW for dM4e stars. We show this correlation in Fig. 9. Although there is more scatter in this sample than in our dM3 star sample, we have a relatively good correlation with a correlation coefficient of 0.877 for 35 measures (see Table 2). This correlation is significant at a confidence level better than 99.8%. The LSF to this data gives: EW_CaII = 1.909± 0.18× EW_H_α - 1.035± 0.58 Eq. (22) allows us to obtain an estimate of the Ca ii EW when we have only the H_α line EW available for dM4e stars. Similarly, we compiled many measures of the Mount Wilson S index (Table 6). We show the empirical correlation between the Ca ii EW and the S index in Fig. 10. Again, we obtain a relatively good correlation between the two parameters, with a correlation coefficient of 0.970 for 22 data points (see Table 2). The relationship between the Ca ii EW and the S index is: EW_CaII = 0.4898± 0.047× S^1.20± 0.07. All these compilations of data now allow us to compute the Ca ii surface fluxes (Table 6). We show the Ca ii surface fluxes as a function of P/sin i for our dM4 stellar sample in Fig. 11. One can see in this diagram that the scatter is very large, and is similar to (or even worse than) the raw correlation for dM3 stars. It appears that the scatter increases from spectral sub-types dM2 to dM4. However, there is a parameter that plays a role in the scatter in Fig. 11: the (R-I)_C range for our dM4 sample ([1.500:1.700]) spans 0.2 dex, whereas for our dM2 and dM3 samples it spans 0.132 dex. This larger range in our selected targets contributes to the larger scatter observed in Figs. 11 and 12. In Fig. 11, it is noteworthy that there is no overlap in Ca ii fluxes between the low activity dM4 stars (up to log(F_HK = 4.4) and the high activity dM4e stars (log(F_HK≥ 4.6): there is a clear gap in the fluxes, although the two groups overlap in their rotation periods. The dM3 stars do not show this clean separation between dM and dMe (Fig. 8). The dM2 stars also show some separation (Fig. 5), but it is not as clean as for dM4 stars. Considering the good correction we had for metallicity in dM2 stars, we decided to compile all the metallicities published for our initial selection list of 395 dM4 stars, and try to obtain a metallicity-radius correlation for these stars (see Paper I). We found metallicities from the literature for 179 dM4 stars. Data for M4 stars that have been corrected for metallicity can be found in Fig. 12 and Table 7. We show in Fig. 12 the RAC corrected for metallicity effects for dM4 stars. The scatter is reduced compared to the raw data in Fig. 11, but there remains a significant scatter among both dM4 and dM4e stars. The corrected correlation is not as good as for dM2 stars in spite of the relatively good statistics we have on the Ca ii surface fluxes. We also observe among our sample of dM4 stars a sub-sample of relatively fast rotating-low activity stars. Again, as in the case of dM2 and dM3 stars, we believe that these stars are unresolved spectroscopic binaries. We also found a few stars with possibly low sin i (see Fig. 12). We find that when we apply the linear heteroscedastic and homoscedastic LSFs to the combined dM4+dM4e stars in the unsaturated regime, the gradients are very steep, steeper even than what we found for our dM2 and dM3 stellar samples. We also find that dM4 stars rotate much faster than dM2 and dM3 stars. The change in Ca ii flux between dM4 and dM4e stars is very abrupt, as already noticed in Fig. 11, a difference of only 0.1 in the log(P/sin i). Our data suggest that saturation in our dM4 sample occurs at about P/sin i∼ 1.8 days. This agrees with the values found in previous studies. Specifically, for stars in the unsaturated regime, i.e. for P/sin i≥ 1.8 days we obtain the following heteroscedastic linear LSF for the stars in our combined dM4+dM4e sample: F_CaII = 8.38± 2.05× 10^5× (P/sini)^-2.564± 0.19. The gradient in Eq. (24) (-2.564±0.19) is significantly steeper than the gradient which we determined for our dK6 (-1.047±0.042: Table 1) and dM2 stars (-1.575±0.058: Table 1), and is also steeper than the gradient for our dM3 stars (-2.020±0.11: Table 1). This confirms that the slope of the linear LSFs increases in absolute value when the spectral type increases from dK4 to dM4. This increase is particularly marked at the TTCC (M3) and beyond. For P/sin i≥ 1.8 days we obtain the following homoscedastic linear LSF for the stars in our combined dM4+dM4e sample: F_CaII = 7.94± 2.05× 10^5× (P/sini)^-2.526± 0.13. The gradient in Eq. (25) (-2.526±0.18) is very close to that determined for the heteroscedastic linear fit (-2.564±0.19: Table 1). As a whole, we find comparable results from the heteroscedastic and homoscedastic linear fits, suggesting that the errors on the measures do not play a large role in the determination of the linear fits (Table 1). In Fig. 12 we observe a flattening (“saturation") for the fastest rotating dM4e stars for P/sin i<1.6 days. This is a range in periods where saturation is expected to occur (see Sect. 1.4). Although our data set is not yet complete enough to be conclusive of a saturation phenomenon, we assume here that for P/sin i shorter than ∼ 2 days, saturation occurs. In this period range, we observe 12 fast rotators with a mean surface flux of ∼ 1.54× 10^5 erg s^-1 cm^-2. We also performed separate linear LSFs to the sub-samples of only the low activity dM4 and the sub-set of high activity dM4e stars (all of which we assume are in the unsaturated regime) respectively (see Table 1). The heteroscedastic linear LSF to the dM4 low activity stars yields: F_CaII = 3.08± 1.25× 10^4× (P/sini)^-0.825± 0.35. Therefore, the gradient of the RAC for the low activity stars is again much shallower than that of the combined sample of low and high activity stars (unsaturated), and by more than 3σ. Therefore, the linear LSF to the full sample is somehow inadequate. We note also that the slopes of the heteroscedastic linear LSF to the sub-samples of only the low activity stars remain approximately constant (within measurement errors) from dK4 to dM4. This point will be further developed in Sect. 3.7. The homoscedastic linear LSF to the dM4 low activity stars yields: F_CaII = 3.80± 1.89× 10^4× (P/sini)^-0.91± 0.39. Again, the gradient of the RAC for the low activity stars is shallower than that of the combined sample of low and high activity stars, and by more than 3σ. We find that for the low activity dM4 stars, both the homoscedastic and heteroscedastic models give similar results for the mean value of the slope of the RAC. And this mean value (0.825-0.91) is remarkably similar (within 1σ) to the mean values for low-activity stars in the other 4 spectral sub-types: 0.624, 0.637, 0.891, 0.93. However, there is one aspect in which the dM4 slow rotators differ from the other four sub-types: the dM4 stars have significantly larger values of σ associated with the slope of the RAC in both the homo- and hetero-scedastic LSFs. We will return to this topic in Section 3.7.2 below. We find that the χ^2 is much poorer for the heteroscedastic model (1.966) than that for the homoscedastic LSF (0.057). Nevertheless, both fits are statistically highly significant (better than 96%). The heteroscedastic linear LSF to the 23 unsaturated dM4e high activity stars yields: F_CaII = 2.27± 0.58× 10^5× (P/sini)^-0.951± 0.31. Therefore, the gradient of the RAC for the high activity stars is also much shallower than that of the combined sample of low and high activity stars, and by more than 3σ. Therefore, the linear LSF to the full sample seems rather inadequate. This point is straightforward if one looks at the linear heteroscedastic LSFs in Fig. 12. It is obvious that the LSFs to the low activity stars, and to the high activity stars cannot be reproduced by a simple linear or quadratic function. We note that the dM4e data confirm the decrease in the slope of the LSF fits to unsaturated stars as we pass through the TTCC. In addition, we also performed a linear LSF to the full sample of dM4e stars (unsaturated+saturated) in order to compare with the previous fit. We find a slope of -0.47 for the full sub-sample of dM4e stars (Table 1). This result again suggests that the slopes for the sub-samples of low activity stars and of high activity stars are of the same order after the TTCC. The parameter R/δ is 18.90 for the high activity star sub-sample and 19.13 for the low activity star sub-sample. These figures are still very high, although that of the low activity sub-sample has decreased compares to other spectral types. Therefore, the parameter δ = <error>/√(n) is still much smaller than the period domains of the RACs in both cases. As a consequence, these LSFs should be established with a good confidence level, which is in agreement with the high statistical significances we obtain for these fits (Table 1). We find that the linear fits (to the unsaturated dM4+dM4e samples) are incapable of reproducing the slopes among the low activity stars and the (unsaturated) high activity stars. We face the same problem as we did in dealing with the case of dM3 stars: the problem is that there exists high activity-relatively slow rotators and low activity-relatively fast rotators. These two types of stars overlap in their range of values of P/sin i. The dM3 and especially the dM4 datasets lead us to question the validity of fitting both the low activity and high activity (unsaturated) sub-samples with a single RAC. Instead, it seems to us that it might be better to consider the possibility that two distinct RACs are present in the data. To the extent that the RACs are determined with dynamo operation, this leads us to wonder: is there perhaps a distinct dynamo process operating in low-activity dM4 stars from the dynamo process which is at work in (unsaturated) high-activity dM4e stars? The slopes of the linear LSF to the low activity stars for our 5 stellar samples are rather homogeneous, even for the dM4 stars, although the latter stars are almost certainly fully convective. This result suggests that there may exist a single dynamo regime for the slow rotators from dK4 to dM4 within errors (see Sect. 3.7.2). This important result should be tested with larger samples of low activity stars. There exist more and more unsaturated high activity-relatively slow rotators when we move to the spectral sub-types dM3 and especially dM4. These relatively slow rotators overlap in P/sin i with the low activity stars. This poses an important problem: we cannot fit satisfactorily both the low activity and (unsaturated) high activity samples with a single RAC. We suggest that the dynamo mechanisms behave differently among the high activity stars when crossing the TTCC. At the TTCC (dM3) and beyond, the data suggest that there exist two different dynamo regimes for and high activity stars (see Sect. 3.7). These differences in these dynamo regimes could be related to differential rotation: in the young high activity stars differential rotation is expected to be larger, and it should vanish with age, i.e. in the low activity stars. Large differential rotation would boost the magnetic field generation in these fully convective stars, such that we could have high activity-relatively slow rotators with large differential rotation. Now that we have assembled enough data to construct RACs at five different spectral sub-types for late K and M dwarfs, it is worthwhile to examine the systematics of RACs as a function of spectral sub-type. §.§ The slopes of the RACs: variation with spectral sub-type Now that we have values of the RAC slopes for five different spectral sub-types, we provide an overview by plotting the slopes as a function of the infra-red color index (R-I)_C in Fig. 13. In this Figure, results are plotted separately for three distinct samples of stars: the low-activity stars only (lower curves), the (unsaturated) high activity stars only (labelled “High Activity"), and the combined samples low+high activity stars (labelled “Low+High Activity"). In this Figure, the heteroscedastic models are plotted as solid lines, and the homoscedastic models are plotted as the dashed lines. §.§.§ Linear LSFs: combined samples of low plus high activity stars In Fig. 13, the upper curves refer to the combined samples: the plotted points indicate the mean values of the RAC gradients which we have obtained for each of our 5 sub-samples by means of linear heteroscedastic (solid line) and homoscedastic (dashed line) LSFs. Attached to each mean value of the gradient is plotted the 1σ uncertainty in the mean value of the gradient. We see that the value of the mean gradient becomes monotonically steeper as (R-I)_C increases. The heteroscedastic (solid line) and homoscedastic (dashed line) models give similar results. There are some significant differences (above the 3σ level) between the two models for the dK6 sample, but this does not affect the overall trend observed for the gradient. The fact that the magnitude of the gradient increases monotonically with increasing spectral type implies that the magnitude of the effects of rotation on the mechanisms of the dynamo increases systematically as we examine our combined samples at later spectral types. If this result is connected with the physics of the dynamo, and if it is physically permissible to consider low activity stars in the same context as (unsaturated) high activity stars, then the results in Fig. 13 may provide an important new constraint on the dynamo mechanism(s). The magnitude of the gradient increases monotonically as we go from the spectral sub-type dK4 to the spectral sub-type dM4. The results in Fig. 13 suggest that, if the TTCC occurs at dM3, the effects of rotation on chromospheric emission are larger in fully convective stars than in early M type and late K dwarfs where a radiative core persists. If this result is taken at face value, it does not appear to be consistent with the suggestions of Durney et al (1993), mentioned in the first paragraph of the Introduction. Previous studies have also reported a variation in the RAC gradient of linear fits (e.g. Stepień 1989, 1993, 1994) as a function of spectral type: the RAC was found to be steepest (i.e. most negative) at dF6 (B-V=0.45) and shallowest close to dK6 (B-V=1.41: KH). However, the most complete study (Stepien 1989) did not include any stars which are cooler than dK6. Thus, the study we report here is complementary to Stepień's work, extending that work towards cooler stars as far as dM4 ((B-V)=1.60). However, our results indicate that the trend noted by Stepień between dF6 and dK6 (namely, the RAC becomes shallower in cool stars) does not continue at later spectral types. On the contrary, we find that the trend in the gradient reverses: according to our samples, the RAC for the combined samples becomes steeper as we go towards dM4. It is natural to wonder if the pronounced steepening of the RAC gradient is a signature of changes in the dynamo mechanism when crossing the TTCC ? As noted in the Introduction above, it appears that the TTCC lies between subtypes dM2 and dM4. In this context, we consider it significant that, among the stars in our samples, the RAC gradient for the combined samples steepens rapidly between spectral subtypes dM2 and dM4. Moreover, HM report a notable lengthening in the mean rotation periods of inactive and active stars also at spectral subtype M3. Also at M3, coronal loops in active stars become noticeably longer (see HM for a discussion of loop length data reported by Mullan et al. [2006]). Our data lead us to believe that something interesting takes place in the spectral subtype range dM2-dM4 as regards the rotational and activity parameters. Is the interesting behaviour between dM2 and dM4 perhaps associated with changes in the dynamo regime when crossing the TTCC ? Possibly, although we have not yet been successful in identifying a physical explanation which accounts for the results reported above. However, whatever the physical mechanism which is responsible for the increased steepnesses in Fig. 13, the rise in the steepness of the gradient for the combined sample from sub-type K4 to sub-type M2 indicates that the changes occurring in the RAC slopes (and therefore perhaps also in the dynamo mechanism(s)) may be progressive and may begin even earlier than sub-type dM2. §.§.§ Linear LSFs: low activity stars only Now let us consider only the low activity stars, in order to position ourselves as much as possible in the unsaturated regime of the Ca II fluxes, and therefore (hopefully) also in the unsaturated regime of the dynamo. The continuous line and dashed line in Fig. 13 (lower curves) are a plot of the slopes of the heteroscedastic and homoscedastic linear LSFs respectively to the RACs for the low activity stars sub-samples only. The parameter R/δ lies between 19.13 and 39.68 for these low activity star sub-samples. Therefore, the parameter δ = <error>/√(n) which is a normalized estimate of the mean error on the measurements is much smaller than the period domains of the RACs in all cases. Hence, these LSFs should be established to a relatively high confidence level, which agrees with the high statistical significances we obtain for these fits (Table 1). In this regard, we note that the process of applying a LSF to a data set leads not only to a mean value for the slope: the LSF also yields a standard deviation σ for the slope. We consider it worthwhile to note that the values of σ contain information about the robustness of the slope. Fig. 13 shows that the slopes for the slow rotators (dK, dM only) do not behave the same as for the combined samples of stars (dK+dKe or dM+dMe). The gradient “a" for the low activity stars remains almost constant (within the error bars) between dK4 and dM4 at a level of a = -(0.8-0.9). More specifically, if we consider not merely the absolute value of a, but also the statistical significance of its value compared to its own σ, we find the following pattern: in dK4 stars, the magnitude of a is 5.6 times its own σ, in dK6 stars, a is 6.1 times its σ, in dM2 stars, a is 7.4 times its σ, and in dM3 stars, a is 4.3 times its σ. All 4 of these cases have slopes which are statistically highly significant. The chromospheric emission in these cases does indeed depend sensitively on rotation. But in M4 stars, a has a value which is only 2.3 times its own σ for the homoscedastic model and 2.4 times its own σ for the heteroscedastic model. In this case, at the 3σ level of significance, the mean slope which we have derived for the RAC of our sample of low activity dM4 stars is formally consistent with zero. In a situation where the slope of the RAC is zero (or formally consistent with zero),the meaning is that the chromospheric emission in low-activity dM4 stars does not depend on rotation at all (or is consistent with zero sensitivity to rotation). Although our LSFs of the slow rotators seem well established because of the large number of measures we have in our sub-samples, we still consider these results as preliminary and they should be confirmed with measures from a higher resolution spectrograph (such as ESO-ESPRESSO), because our errors on the individual measures are still large compared to the period domains of the RACs. §.§.§ Linear LSFs: High activity stars only We also show the slopes of the heteroscedastic and homoscedastic LSFs to the sub-samples of only the (unsaturated) high activity stars in Fig. 13. The parameter R/δ lies between 17.20 to 23.45 for these high activity star sub-samples (the case of dK4e stars is not considered because we have too few measures). Therefore, the parameter δ = <error>/√(n) which is the normalized estimate of the mean error on the measurements is much smaller than the period domains of the RACs in all cases. Hence, these LSFs should be established with a rather good conficence level, which agrees with the high statistical significances we obtain for these correlations (≥90%, Table 1). However, we consider these results as still preliminary because the domains of the RACs are relatively small compared to the typical uncertainties on individual measures. We note that these slopes behave quite differently from the slopes of the low activity stars and the slopes of the combined samples of low+high activity stars. For the high activity stars, we observe that the slopes are clearly steeper than those for the low activity stars at spectral types dK6 and dM2. At these spectral types, they are also steeper than the slopes of the combined samples of low+high activity stars. These higher slopes suggest the existence of two different dynamo regimes for the low and the high activity sub-samples for these stars. However, things change at the TTCC. At the TTCC (dM3), the slopes for the high activity stars are found to diminish dramatically, by a factor of about 2. Although there remains a large uncertainty on this slope for dM3e stars, the fall of the slopes at the TTCC is confirmed by the measure of the slope for dM4e stars with a much higher confidence level. Beyond the TTCC (dM4), the slope continues to diminish. In fact, the slope at dM4 reaches a value that is very similar to the slope we have obtained for the low activity star sub-samples (although we emphasize that the overall shapes of the RACs are very different: see Figs. 8 and 12). This dramatic decrease in steepness among (unsaturated) high activity stars at M3 and M4 suggests that the dynamo mechanisms operating in dK4, dK6 and dM2 high activity stars (perhaps an α-Ω dynamo?) may be different from those operating in the fully convective dM3 and dM4 high activity stars (perhaps an α^2 dynamo?). In this regard, we detect no major changes in the slopes at the TTCC for the low activity star sub-samples apart from a decreased statistical significance of the RAC slope. This point is further discussed in the next sub-section. We emphasize that, although our LSFs to the high activity stars seem relatively well established, in our sub-samples we have only few measures (from 8 to 23). Therefore, these results should be confirmed with larger stellar samples and are still preliminary. §.§.§ Approaching the TTCC? These results lead us to suggest a five-part conclusion. (i) The gradients of the RACs which we have obtained in our combined samples of low+high activity stars unambiguously increase with increasing spectral type with a high level of confidence. But it also appears that the linear LSFs only provide a poor description of the full RACs. (ii) There appears to be a significant dichotomy between the low activity and the (unsaturated) high activity sub-samples. This leads us to consider that there may exist two distinct dynamo regimes in dKe, dMe and dK, dM stars. (iii) The gradients of the combined samples suggest that there are changes operating in the dynamo mechanisms before, at, and after the TTCC. In addition, the gradients to the high activity sub-samples point to important changes in the dynamo mechanisms occuring at the TTCC and beyond. (iv) We can confidently assert that in low activity dK4, dK6, dM2, and dM3 stars, there is a robust (>4 σ) increase in Ca ii flux as the period decreases. That is, the dynamo in unsaturated dK4-dM3 stars is clearly sensitive to rotation. (v) But in low activity M4 stars, the sensitivity of the dynamo to rotation, although still perhaps present, is not as robust, and may even be zero (at the 3σ level of significance). Thus, even in the low activity stars in our samples, a rotational dynamo is certainly contributing significantly to chromospheric heating in dK4-dM3 stars. But in dM4 low activity stars, the signs of rotational control over chromospheric emission are less significant (in a statistical sense). These two last points, if they can be confirmed, might be relevant in the context of Durney et al (1993), who suggested that rotation would be a controlling influence in the case of interface dynamos, but should not be as effective as a controlling influence in distributed dynamos. On the one hand, the robust sensitivity to rotation which we have found in dK4-dM3 low activity stars could be evidence for interface dynamos in those stars. On the other hand, in dM4 low activity stars, the weaker evidence for statistically significant rotational sensitivity might presage the lessening of the effects of an interface dynamo. In this context, M3 might be considered to be the latest spectral sub-type to have definitive evidence for the presence of a radiative core (so that an interface dynamo is even possible). It is worth recalling that HM also concluded that at M3, something unusual happens to the rotational braking. Could the HM result, in combination with our results for the RAC slope, be a sign that the TTCC occurs between M3 and M4? This will be tested in a future study using an even finer grained spectral type sampling at about the TTCC using new data. We suggest that in Fig. 13, the striking difference between the combined sample (upwardly rising lines) and the low activity stars only (lower lines) may be associated with the fact that the combined sample might contain two distinct regimes of dynamo operation (high and low), whereas the low activity sample contains only a single regime (low). Suppose that the separation of dM4e stars from dM4 stars in Fig. 12 is associated with two distinct dynamo regimes: if so, the attempt to fit a LSF to the combined sample of dM4+dM4e stars may involve an attempt to “force" two different types of dynamos into a single mold. Such an attempt would not be as physically meaningful as attempting to fit a truly homogeneous population (e.g. the low activity stars alone). Analogous arguments can be made for dM3 stars (Fig. 8) and dM2 stars (Fig. 5). According to this argument, it may be more profitable to focus separately on the different curves of the low activity stars and the high activity stars respectively in Fig. 13 in order to gain insight into dynamo theory. We conclude that for the low activity stars only, the gradient is definitely shallower than that for the samples of the low plus high activity stars. But we also conclude that the gradient for the low activity stars only does not vary significantly with spectral type. We also note that the differences between the slopes for the low activity stars and for the low+high activity stars increase drastically with increasing spectral type. This result emphasizes that the efficiency of the dynamo mechanisms increases drastically at short rotation periods, and also increases with decreasing stellar mass. We also conclude that there are no significant changes occurring in the efficiency of the dynamo mechanisms for low activity stars as a function of stellar mass when moving from dK4 to dM4. §.§.§ Different dynamo regimes among low activity and high activity stars at the TTCC and beyond ? In the plots which we have compiled of each RAC for our five spectral sub-types (Figs. 2, 3, 5, 8, 12), we overplot the separate heteroscedastic linear LSF to the low activity stars and to the high activity stars for direct comparison with the heteroscedastic linear LSF to the low+high activity samples. As we saw in the previous Sections, the overplots show clearly that the linear LSF to the low+high activity stars fails completely to reproduce the slopes of the linear LSF which we have derived both from the slow rotators and the fast (unsaturated) rotators. We show that the low+high activity samples cannot be described by a single RAC. In view of this, it seems to us that, to the extent that the RAC is determined by a dynamo mechanism, we need to invoke two different dynamo regimes. The first (we suggest) is at work among dK and dM stars (i.e. low-activity stars). This first regime seems rather constant from dK4 to dM4 and is therefore independent (to a first approximation) of the spectral type. A second dynamo regime (we suggest) is at work among dKe and dMe stars. This second regime exhibits steeper RAC slopes for the spectral types dK4, dK6 and dM2, and shallower slopes for the dM3 and dM4 spectral types. The data suggest that the second regime deviates significantly from the first regime at the TTCC and beyond. The main problem is that, among the stars in our samples at spectral sub-types dM3 and dM4, there are interlopers, namely, a certain number of high activity-relatively slow rotators and a certain number of low activity relatively-fast rotators that overlap in P/sin i. These interlopers make it impossible to reproduce the entire datasets with a single RAC. Our results suggest that, whether we are considering low activity stars alone, or the combined samples of low+high activity, the properties of the RACs undergo changes of some kind, especially at the TTCC and beyond. This conclusion is especially true for dM4 stars. The findings we present here can be considered as providing evidence in favor of the following hypothesis: the dynamo mechanism in fully convective stars is different from the dynamo mechanisms in stars where a tachocline exists. We propose that differential rotation (probably important in the young dMe stars and less important in the older dM stars) plays an important role in defining the RACs among dM and dMe stars at the TTCC and beyond. §.§ The RACs for five different spectral sub-types We bring together the heteroscedastic linear fits to the low+high activity RACs (in terms of surface fluxes) for our five spectral sub-types in Fig. 14 (upper panel). In this section we discuss two key aspects of the figure, and how they vary as a function of spectral type: (i) the absolute magnitudes of the Ca ii flux, and (ii) the slopes of the RAC in the unsaturated regime. There are several striking trends in this figure. First, with increasing spectral type, the curves tend to shift downward and to the right, i.e. the overall level of chromospheric flux decreases, and the periods tend to be shorter. Important differences between the RACs of dK4, dM2 and dM4 stars have already been reported by Houdebine (2012b). Also, variations in the RAC properties at different spectral types was previously reported by Stepień (1993, 1994). He found that the log(R'_HK)-P_rot relationships globally shifted towards shorter rotation periods as (B-V) increases from 0.52 (dF7.5) to 1.15 (i.e. spectral type dK5, according to KH). On the other hand, in an earlier study, the opposite tendency emerged: Stepień (1989) found that the log(Δ F_CaII)-P_rot relationships were globally shifted towards longer rotation periods for (B-V) from 0.45 (dF6) to 0.66 (dG5) and that they stabilize for (B-V) from 0.75 (dG8.5) to 1.40 (dK6). Also, Patten & Simon (1996) proposed that the projected rotation period becomes longer as one move from spectral types G1 to M2. In view of this complicated behaviour, it seems that in the F_CaII-P_rot RACs, first the rotation period becomes longer between dF6 and dG7, then remains unchanged between dG7 and dK6, and then the period starts to become shorter between dK6 and dM4: the latter behaviour persists into later spectral types. In Fig. 14, in the context of the combined samples (upper panel), one can see that the steepness of the RACs (at least in the unsaturated regime) increases noticeably as we go towards later spectral type (as already shown in Fig. 13). We have not detected any saturation behavior for stars in our dK4, dK6 or dM2 stars, whereas we may observe a saturation in dM3 and dM4 stars for the most rapid rotators. Might this saturated-unsaturated behavior also point towards a different dynamo mechanism in dM3 and dM4 stars from the dynamo mechanism in dK4, dK6, and dM2 stars? We also observe that the largest Ca ii surface fluxes are found in the stars with the earliest spectral types (dK4, dK6, dM2). For dM4 stars, there is a pronounced decrease in the absolute magnitudes of the surface fluxes for the fastest (unsaturated) rotators compared to dM2 stars. But even among relatively slow rotators (with periods longer than 10 days), where any effects of saturation are expected to be minimal, the Ca ii surface fluxes are observed to decline by more than 100 between dK4 and dM4. Surveying the results, we see that dK4 and dK6 stars have similar RACs, with a slightly larger gradient for the dK6 stars. Then the RAC falls to lower surface fluxes for dM2 stars and the RAC steepens. The RAC for dM3 stars is distinctly different in shape compared to that of the dM2 stars: dM3e stars exhibit saturation whereas the dM2e stars do not. Also, dM3 stars overlap in rotation periods with dM2 stars. This is due to the fact that we found (in HM) that dM3 stars possess abnormally long rotation periods for both high activity and low activity stars compared to the adjoining dM2 and dM4 stars. In HM, it was suggested that the anomaly of relatively long rotational periods at type M3 might be related to an empirical report (Mullan et al. 2006) that flaring loop lengths undergo an increase to larger values at dM3. In dM4 stars, the (unsaturated) RAC shape parallels that of dM3. The trends in surface fluxes of Ca ii in Fig. 14 suggest that whatever is controlling chromospheric emission in our sample stars, the mechanism is less effective (by factors of 10 or more), and leads to different RAC shapes, in dM3 and dM4 stars than in dM2, dK6, dK4 stars. Now, as regards the latter stars, it might at first sight be believed that we should be fairly confident that interface dynamos are at work (e.g. Mullan et al. 2015). (Unfortunately, we cannot be absolutely confident that only an ID is at work in the dM2, dK6, and dK4 stars: the results of Brown et al. (2010) indicate that also a DD can operate in the convection zone of partially convective stars. If both ID and DD are in fact simultaneously operative in a certain star, current dynamo models give no quantitative results for the relative strengths of magnetic fields which would be generated by the different dynamos. Thus, although we use the words “fairly confident" in the present context, the question of the relative importance of ID and DD in a partially convective star remains ambiguous. Could it be that the different RAC shapes in dM3 and dM4 stars might be due to the absence of an interface dynamo? If so, TTCC may be occurring between dM2 and dM3 sub-types. We also note that the changes in the RACs are progressive from dK4 to dM4, and that the fall in the fluxes begins at about dM2 and continues from dM3 to dM4. However, whatever the correct physical interpretation in terms of different dynamos eventually turns out to be, our data show that the efficiency of the dynamo mechanisms falls progressively from dK6 to dM4, and that this fall is particularly pronounced after the TTCC at spectral sub-type M4. §.§.§ Low activity stars only In the lower panel of Fig. 14, we show results of the heteroscedastic models for low activity stars only. The most striking difference between the lower and upper panels in Fig. 14 is that the low activity stars do not exhibit any evidence for saturated behavior. Moreover, a striking characteristic of the lower panel is that the five RACs are almost all parallel to one another (as already noted in Fig. 13). These features support our claim that low activity stars may provide us with a possible “window" into a more homogeneous sample in terms of the dynamo mechanism: it seems likely that only unsaturated dynamo operation is at work in the lower panel of Fig. 14. We note that surface fluxes for unsaturated dM4 stars are again found to be 10-100 times smaller than those in unsaturated dK4 and dK6 stars. And we also observe a similar global pattern in both panels (upper and lower): there is a trend towards shorter periods with increasing spectral type. It is clear that the overall level of Ca ii flux is diminishing as we go from K4 to M4: the b coefficient of the linear LSF to the RAC of dK4 stars is larger by a factor of 25 than the linear LSF to the RAC of dM4 stars. This suggests that the dM4 stars at any given period are generating mechanical energy flux some 25 times less effectively than dK4 stars with the same rotational period. Even in the much narrower gap between dM3 and dM4, the b value for the linear LSF to the RAC has decreased by a factor of almost 10 at any particular rotational period. These results indicate that dM4 stars really are suffering from a weakening of the ability to generate mechanical energy (whether in magnetic form or in acoustic form or in a combination of both forms) compared to dM3 and earlier sub-types. We conclude that the overall level of dynamo efficiency (as regards chromospheric heating) diminishes markedly as the stellar mass decreases. However, we also find that, as regards the low activity stars (i.e. unsaturated dynamos), there is an almost universal dependency of the dynamo efficiency on the rotation period: i.e. the level of activity varies approximately as ∼ P/sin i^-0.80. This universal feature is missing when we attempt to analyze samples which combine stars of both low activity and high activity. For the combined samples, our results indicate that the dynamo efficiency decreases differentially and drastically with decreasing rotation period as we move from dK4 dwarfs to M4 dwarfs. §.§ R'_HK as a function of the Rossby number In the spirit of searching for RAC's among a variety of parameters (in case the correlations are obscured more in certain cases), we now set aside the rotation period which we have used so far, and explore how activity depends on Rossby number. Dynamo theory suggests that the dynamo efficiency, and therefore the magnetic field strength and its direct diagnostic in the chromosphere, the Ca ii lines, should scale with the dynamo number (Montesinos et al. 2001) N_D given by: N_D∼1/R_0^2ΔΩ r_cz L/νΩ d^2 where R_0=P/τ_c is the Rossby number, τ_c the convective overturning time, Ω is a characteristic rotation rate in the lower convection zone, r_cz the radius at the base of the convection zone, L the characteristic length-scale for the differential rotation in the overshoot region just below the convection zone, ν the ratio of the diffusivities in the two layers directly below and above the overshoot region and, and the quantity d is given by d∼√(ητ_c) where η is the turbulent diffusivity in the layer just above the overshoot region. Making reasonable assumptions for late-type stars, the dynamo number simplifies to (Noyes et al. 1984, hereafter N84): N_D∼ (Ωτ_c)^2∼ R_0^-2 Therefore, a good evaluation of the efficiency of dynamo mechanisms can be done through plotting the chromospheric indices R'_HK=L_HK/L_bol (where L_HK is the luminosity in the H and K lines of Ca ii) as a function of the Rossby number R_0=P/τ_c (e.g. N84, Stepien 1989, 1994, Hempelmann et al. 1995, Patten & Simon 1996, Montesinos et al. 2001). Montesinos et al. (2001) argue that the dimensionless factor, ΔΩ r_cz L/νΩ d^2 plays a role in the scatter of the R'_HK versus R_0 diagrams. This dimensionless factor depends notably on the internal differential rotation which is mostly confined to the overshoot region just below the convection zone as well as on the turbulent magnetic diffusivities, neither of which are currently reliably known. In view of this lack of information, we shall for simplicity here assume this dimensionless factor to be constant for our samples of late-type dwarfs. In order to evaluate Rossby numbers in the stars of our sub-samples, we have adopted values of τ_c from the results of Spada et al. (2013) who gives the relationship between stellar mass M_⋆ and τ_c. We derived a relationship between the stellar radius, R_⋆, and τ_c using the mass-radius relationships of Spada et al. (2013) for [Fe/H]=0. The results of Spada et al. (2013) on τ_c are in reasonably good agreement with other calculations (e.g. Kim & Demarque 1996). Recent values of τ_c agree to some extent with the values of Noyes et al. (1984) for solar masses down to M_⋆∼ 0.8 M_⊙ but differ substantially for lower mass stars. We compare the values of τ_c from Spada et al. (2013) and Noyes et al. (1984) for our five spectral sub-types in Paper I. The large increase in the numerical values of τ_c which we take from Spada et al (2013) at low masses has important consequences for the R'_HK/R_0 relationships in M dwarfs as we shall see below. We use the τ_c/R_⋆ tabulation to derive R_0 for each star in our five samples of stars. We also computed the luminosity, L_HK, in the Ca ii lines according to the continuum surface fluxes derived above from the models of de Laverny et al. (2012). We list the Ca ii luminosities L_HK and R_0 for our five samples of stars in Table 8. We also computed the bolometric luminosities L_bol and the activity indice R'_HK=L_HK/L_bol for all our targets. We list the results in Table 8. We further computed L_HK and R'_HK corrected for metallicity effects for dM2, dM3 and dM4 stars. Mullan and MacDonald (2001) suggest that the ratio of the X-ray luminosity created by large-scale fields L_X(L) to that of the X-ray luminosity created by turbulent fields L_X(t) may attain factors of 5-10. Although the case of chromospheric lines does not necessarily track the coronal X-ray emission, one might also expect a decrease in chromospheric emission when the dynamo mechanisms change from a shell dynamo to a distributive dynamo. Note, however that in the case of the chromosphere some modelling calculations (e.g. Ulmschneider et al. 2001, Fawzy et al. 2002, Ulmschneider & Musielak 2003, Ulmschneider et al. 2005) yield one to expect a basal flux in the chromospheric lines. Such basal flux has not been observed in our datasets (e.g. this study, Papers XV and XVIII). However, some previous authors claim to have detected such fluxes for various spectral types (e.g. Wilson 1968, Schrijver 1987, Rutten et al. 1991, Strassmeier et al. 1994, Fawzy et al. 2002). Note however that M dwarfs were usually excluded from those studies. On the other hand, there do exist certain data sets for line fluxes of Mg ii and Ly_α in inactive M dwarfs which have been found to be consistent with ab initio models of acoustically heated chromospheres (Mullan and Cheng 1993): thus, there seems to be little reason to exclude the concept of “basal" fluxes from a discussion of dM chromospheres. Here we investigate the R'_HK-R_0 relationships from dK4 stars to the fully convective dM3 and dM4 stars and search for a signature of the change in the dynamo mechanisms at the TTCC. We plot in Fig. 15 log(R'_HK) as a function of log(R_0) for our five different spectral types. We found, using the values of τ_c from N84, that the mean of the measurements for dK4 stars (the dK4 data smoothed with a Gaussian of FWHM=0.1) agree well with the correlation for F, G and K type stars found by N84. Thus, our RAC data for dK4 stars are consistent with the N84 study and represents well a low-mass extension of the correlation for F, G and early K type stars. In Fig. 15 (upper panel) we can see that for the dK6 stars (filled triangles), the mean values (the data smoothed with a Gaussian of FWHM=0.1, dashed line) lies slightly below the correlation for dK4 stars, typically by a factor from 1.5 to 5.0, i.e. a factor of 3 on average. The dK6 curve crosses the dK4 curve at log(R_0)∼ -0.2. There is a large scatter in the data for dK4, dK6 and dM2 stars in spite of the fact that we corrected the data for the effects of metallicity in dM2 stars (otherwise, the scatter is even larger): this is explained by the variations in radii of our stars at a given spectral sub-type, and the large variations of τ_c with radius. This scatter increases for the spectral types dM3 and dM4 Fig. 15b) because for these stars there is a large increase in τ_c with decreasing stellar radius and within a small range in radius (see Spada et al. 2013). For our dM2 stars, the situation is more striking: the data (dotted line in Fig. 15(a)) now lie much lower than the dK4 curve. There is a clear and large difference between our dK4 data and our dM2 data. Over the common range in R_0 the curve for dM2 stars is roughly parallel to that of the dK4 stars, but is shifted below by a factor from 6 to 16. Our data for the dM2 stars cover a larger range in R_0 than in previous studies, notably for small values of R_0 (i.e. stars with short rotational periods). One can see that the correlation suddenly rises for logR_0 less than -1.3. At small values of R_0, log(R'_HK) reaches values of about -4.0 or even larger. Moving later in spectral sub-type, we show in Fig. 15 (lower panel) the analogous data for our dM3 and dM4 stars. We note that in this diagram, the mean data for these stars (dash-dotted line) generally lie below the correlation of dK4 stars (solid line), and also significantly below our dM2 data (dotted line). The large variations in τ_c between M2, M3 and M4 stars imply that the curves for M3 and M4 stars are systematically shifted towards lower R_0. But we emphasize that also, globally, the values of R'_HK decrease from M2 to M4 (Fig. 15, lower panel). The mean curve for our dM3 stars lies significantly below the mean curve for dM2 stars. This difference is even more pronounced for the sample of dM4 stars. At log(R_0)=-1.2, we find that the dM3 curve lies a factor of 20 below that of the dK4 stars, and that the dM4 curve lies a factor of about 100 below the dK4 curve (Fig. 15, lower panel). The curves for dM2, dM3 and dM4 stars are more or less parallel (except at very low values of R_0 for dM3 and dM4 stars, where there are only a few data points). This highlights the consistency of the decreases in the R'_HK-R_0 curves when going from mid-K type stars to the fully convective dM4 stars. The decrease from dM2 to dM3 stars is typically a factor of 2 (at log(R_0)= -1.0, whereas the decrease from dM3 to dM4 stars is typically a factor of 4 (taken at log(R_0)=-1.5). Christian et al. (2011) have reported a saturation (or even “super-saturation") phenomenon in the chromospheric emission for fast rotators in young clusters at about log(R'_HK)=-4.08 but with a significant scatter (±0.5 in the log). In our stellar samples, we do not have enough very fast rotators to confirm the existence of super-saturation. However, we note that R'_HK measures generally lie below -3.8 for all our samples of late-K and M dwarfs. We also find that this high level of activity is attained for different values of R_0 that depend on the spectral type. For dK4 stars, this high level of activity is attained at about log(R_0)∼ -1.0, whereas for dM4 stars it is attained at about log(R_0)∼ -2.7. Our datasets suggest that M dwarfs (especially M3 and M4) generally do not follow the same relationship as F, G, and K dwarfs. This had not been noted previously because the M dwarf data sample was too sparse. If this difference is due to a difference in dynamo mechanisms, then the change in dynamo mechanisms seems to be progressive as is illustrated by our dK6 and dM2 datasets. Therefore, we cannot conclude from the results in Fig. 15 that there is any abrupt change at the TTCC (if this occurs no later than M4): instead, there seems to be a gradual decrease in the efficiency of the dynamo mechanisms that may start as early as spectral type dK6. This gradual change was suggested by Mullan & MacDonald (2001). The efficiency of an interface (shell) dynamo may fall off between dK4 and dM2 as the radiative core occupies an increasingly small proportion of the stellar radius (Mullan et al. 2015), and the present data suggest that this decrease continues at least as far as M4. The latter point is strengthened by our results in Paper XV which showed a progressive change from high fluxes (large radii M2 dwarfs, partially convective) to very low fluxes (M2 subdwarfs, fully convective). The R'_HK-R_0 diagram may be one of the most sensitive tests for detecting changes occurring in the dynamo mechanisms. However, the results from these RAC relationships have to be compared with the other magnetic activity indicators we study here. Note that with the present data there is evidence in the R'_HK-R_0 diagrams that gradual changes in the dynamo efficiency are occurring all the way to M4. Other magnetic activity diagnostics, including the mean fluxes (Sect. 3.8), also point to a continued change from K4 to M4. The results in Fig. 15 strongly suggest that, as far as chromospheric heating is concerned, there may be a gradual decrease in dynamo efficiency in M dwarfs compared to F, G and K dwarfs. It is generally believed that an α-Ω type of dynamo dominates in F, G and K dwarfs. In early M dwarfs, an interface dynamo is probably still operating (Mullan et al. 2015), although eventually, in fully convective M dwarfs (beyond the TTCC), the dynamo mechanism is expected to behave differently. The most striking conclusion of the present section is that, as regards chromospheric heating there are almost two orders of magnitude decrease in going from dK4 to dM4. This further strengthens our previous findings (see Fig. 14) that, in the context of chromospheric heating, the mechanical fluxes decrease by a factor of order 100 as we go from dK4 to dM4. §.§ The coronal RAC: L_X/L_bol as a function of the Rossby number Now we turn to the corona, where deposition of mechanical heating manifests itself in the form of X-ray luminosity L_X. Mullan & MacDonald (2001) found that in a sample of some 40 ROSAT measures of dMe stars by Fleming et al. (1993), L_X/L_bol was found to remain essentially invariant over the spectral range from early M to at least M7, at least for the most active stars (see also Browning et al. 2010). This result, which implies that coronal heating efficiency in the most active stars remains unchanged at the TTCC, now appears worth re-examining in the context of our findings (see Fig. 14) of changes in chromospheric heating before and near the TTCC. Our sample is now several times larger than that of Fleming et al. (1993). We therefore investigate here the behavior of L_X/L_bol as a function of R_0 for our five different spectral sub-types, to see if we can identify any signature of the TTCC in the coronal data. In order to illustrate the sensitivity of coronal heating to rotation, we plot in Fig. 16 L_X/L_bol as a function of R_0 for our five different spectral sub-types. We also plot in this diagram the running mean curves for our dK4, dK6, dM2, dM3 and dM4 stars. The differences between the behaviours of R'_HK (Fig. 15) and L_X/L_bol (Fig. 16) are significant. For dK4, dK6 and dM2 stars, L_X/L_bol rises slowly as R_0 decreases for the slow rotators (log(R_0)=0 to -1.0), and suddenly rises abruptly for the rapid rotators (log(R_0)<-1.0). For the fast rotators (log(R_0)<-1.5), L_X/L_bol seems to saturate at a level of about 10^-3.0-10^-2.5. Patten & Simon (1996) also observed a similar behavior among their samples of stars in young clusters (IC 2391, α Persei, Pleiades, Hyades and main-sequence stars in the field). They also observed a saturation at L_X/L_bol∼ 10^-3.0 for log(R_0)<-0.6. On the other hand, they did not observe a plateau-like of behavior from -3.0<log(R_0)<-1.3 such as our dM2, dM3 and dM4 star data display in Fig. 16. Instead they observe a strong rise in L_X/L_bol from 10^-6.5to 10^-3.0, more like the behaviour of our data from -1.3<log(R_0)<0.3 in Fig. 16. Overall, our impression is that all stars in our five sub-samples follow a more or less similar correlation and show evidence for saturation for log(R_0)<-1.3. We should emphasize that our L_X data (taken from Hunsch et al. 1999) are severely biased towards the most active stars because of the limited sensitivity of ROSAT. Therefore, our samples of measures for the least active stars (log R_0>-1.3) are inevitably biased in a negative sense, and may not be dependable when we calculate the means of L_X/L_bol as a function of R_0. §.§ Comparison and contrast between chromospheric and coronal heating rates In order to facilitate comparison of Fig. 15 (lower panel) (chromosphere) and Fig. 16 (corona), we use similar notation for the lines which illustrate average values of the quantities at different spectral sub-types. Thus, in Fig. 15 (lower), the dM3/dM4 stars (dotted line) lie lowest in the figure, whereas in Fig. 16, the same stars lie at about the same levels as other spectral sub-types in the figure. For purposes of the present discussion, let us suppose that dM3 and dM4 stars may be labelled as at, or later than, the TTCC: we refer to these stars by the shorthand notation TTCC+. Stars at spectral types dM2 and earlier are referred to as TTCC-. In terms of this notation, our results indicate that, in terms of the chromosphere, TTCC+ stars are definitely weaker emitters than TTCC- stars (see Fig. 14 lower panel). On the other hand, in terms of the corona, TTCC+ stars have similar emissions to those from TTCC- stars. Thus, when we compare stars on different sides of the TTCC, the trend in coronal activity does not behave in the same way as the trend in chromospheric activity. In order to explore this difference between activity at the chromospheric and coronal levels of M dwarf atmospheres we turn now to investigate how L_X varies as a function of L_HK, and also as a function of spectral type. §.§.§ L_X as a function of L_HK In a previous study Schrijver et al. (1992) reported on the inter-relation between L_X and L_HK with nearly-simultaneous observations of a sample of 26 F5-K3 main-sequence stars. Schrijver et al. (1992) obtained a good correlation over three orders of magnitude in L_HK and over four orders of magnitude in L_X. They found that L_X increased faster than L_HK: L_X∝ L_HK^1.50± 0.20. In the present study, we compiled L_HK and L_X for our five different spectral subtypes. We find that the LSF to our samples give: L_X=4.57 10^-16 L_HK^1.54± 0.29 ergs/s, for our dK4 stars, L_X=7.94 10^-17 L_HK^1.57± 0.21 ergs/s, for our dK6 stars, L_X=1.00 10^-15 L_HK^1.57± 0.17 ergs/s, for our dM2 stars, L_X=7.94 10^-20 L_HK^1.74± 0.11 ergs/s, for our dM3 stars, and L_X=2.51 10^-9 L_HK^1.37± 0.12 ergs/s, for our dM4 stars. The correlations are in all cases highly significant (see Table 1). The only exception occurs in the dK6 stars: this is because in this sample we have only few high activity stars. Since the slope for dK6 stars is the same as for dK4 and dM2 stars, we believe the dK6 correlation is relevant for comparison to other sample correlations. Perhaps the smaller range of L_X values among dK6 stars also contributes to low significance. We show in Fig. 17 the L_X versus L_HK correlations we found for our dK6, dM3 and dM4 stellar samples. Including late-K and M dwarfs of all 5 sub-types, we find an average slope of 1.56 in our log(L_X)-log(L_HK) relationships. This is essentially identical to the value reported by Schrijver et al. (1992) for F5-K3 stars. Therefore, the same slope seems to prevail for all main-sequence stars from F5 to M4. This result emphasizes that in low mass stars, it is a general result that the coronal emission grows faster than the chromospheric emission, from low activity stars to high activity stars. It is interesting to note that when investigating IUE spectra of F to K type stars, Ayres et al. (1981) found also a power-law slope of about 1.5 between the 10^5 K line fluxes and the Mg ii line fluxes. However, in their correlation between the soft X-rays and Mg ii fluxes, they found a larger slope. However, what is more striking than the similarity in gradients found for these correlations, is the multiplicative coefficients in the correlations. We find that this factor changes from 4.47 10^-16 for dK4 stars to 2.51 10^-9 for dM4 stars. This large difference indicates that the empirical correlations differ greatly from one spectral sub-type to another even if the gradients are comparable. To illustrate this, we plot the LSF of L_X versus L_HK in Fig. 18 for our five spectral sub-types. In all cases, it is apparent that the correlations have slopes which do not differ greatly. But the absolute values of fluxes from the chromospheres differ greatly. E.g., dK4 and dK6 stars have L_HK values mainly in the range 10^27.5-29.5 ergs/s, whereas the chromospheres of dM4 stars emit mainly in a much lower range, 10^25-27.5 ergs/s. On the other hand, emissions from the coronae of dK4 and dM4 stars overlap in the range 10^27.5-28 ergs/s. As a result, at any given chromospheric emission level, the coronal emission strongly increases as we consider later spectral sub-types. E.g., for a chromospheric Ca ii luminosity of 10^27.5  ergs s^-1, the X-ray luminosity increases by a factor of at least 110 between dK4 and dM4. Therefore, in the case of M dwarfs, the situation is quite different from that of earlier F, G and K spectral types (e.g. Schrijver et al. 1992): stars from F to G to K all follow a common correlation between L_X and L_HK. In a similar vein, the large differences we observe among M dwarfs explains the large scatter observed in the L_X-L_HK correlation reported by Panagi & Mathioudakis (1993) for K and M dwarfs. It is natural to inquire at this point: what makes M dwarfs so different from F-G-K dwarfs as regards the ratio of coronal to chromospheric emission? One possibility has been suggested by Mullan (1984). Coronal heating depends ultimately on tapping into the reservoir of mechanical energy associated with convective motions. In the context of electrodynamic coupling, the efficiency with which energy contained in that reservoir can be conveyed to the corona depends on two time-scales, τ_c for convection, and τ_A for the coronal loops in which the energy is to be deposited. In the Sun, these time scales differ by factors of order 100. This large difference explains why the coronal energy flux from the Sun amounts to only 1% or less of the mechanical energy flux in convection. But Mullan (1984) predicted that as one goes down the main sequence, there could come a point where the two time scales τ_c and τ_A could be comparable. This would occur by reduction in τ_c due to smaller convection cells, and increases in τ_A due to longer coronal loops. By making some assumptions, the prediction was made that τ_c ≈ τ_A would occur among stars with T_eff ≈ 3400 K (see Mullan 1984: section VI). What spectral type does this correspond to? According to Rajpurohit et al (2013), the corresponding I-J colour is 1.2-1.3, and the spectral type is M3. Moreover, I-J = 1.2 corresponds to V-I colour = 2.2, which is also the colour at which coronal loops become longer (Mullan et al 2006). At the spectral type where τ_c ≈ τ_A, the coronal loops would reach a resonance in their efficiency to tap into the convective reservoir. As a result, the efficiency of coronal heating in dM3 stars would be larger than in the Sun by 80-170. This range overlaps with the excess by factors of order 110 reported in Fig. 18 above. Because a resonant process is at work in this model, the increase in coronal heating efficiency is expected to build up in spectral types as these approach closer to the resonance, i.e. approaching M3. Thus, the full increase by factors of 80-170 should be realized at M3, but smaller enhancements are expected at (say) M2, K6, and K4. It is important to note that the electrodynamic resonance works only for the corona: there is no analogous process occurring in the chromosphere. Heating of the chromosphere, whether in an F star of an M star, continues to rely on localized dissipation of acoustic waves (for the “basal" component: Mullan and Cheng 1993) and on localized dissipation of currents, via the conductivity tensor, in a partially ionized medium (for the “magnetic" contribution) (e.g. Kazeminezhad and Goodman 2006). Even if the resonant model (Mullan 1984) turns out to be incorrect, at least the empirical results explain the L_X/L_bol-R_0 relationships reported above. We can now interpret the L_X/L_bol-R_0 relationships (in Fig. 16) vis-a-vis the R'_HK-R_0 relationships (in Fig. 15). Indeed, in late-K and M dwarfs, the 110-fold increase in X-ray coronal emission from spectral sub-types K4 and M4 compensates for the 90-fold decrease in dynamo/chromospheric efficiency from the same spectral types. As a result, in Fig. 16, the M dwarfs have about the same L_X/L_bol values as the dK4 and dK6 stars, whereas in Fig. 15, we found a drop-off for dM2 stars of about a factor of 10 as regards the chromospheric heating. Note however that in Fig. 16, there is some convergence in L_X/L_bol for R_0<-1.3 for all spectral types. This discussion may help us to understand why Mullan and MacDonald (2001) failed to identify any signature of the TTCC in L_X /L_bol data: their failure may result from one aspect of coronal properties in M dwarfs. As a result of these properties, and their variation with spectral type in the vicinity of an electrodynamic resonance which (perhaps coincidentally) overlaps with the TTCC, the coronal ratio L_X/L_bol is not really suited for diagnosing the changing properties of the dynamo mechanisms before and at the TTCC. In view of the data presented in the present paper, we are now disposed to believe that the chromospheric emission (e.g. the Ca ii luminosity) is better suited to respond to a signature of the TTCC. §.§ The RACs and the mean rotation periods Can the properties of the RAC's discussed above explain the mean P/sin i values obtained by HM for stars ranging in spectral sub-type from dK4 to dM4 stars? In particular, can they help in understanding the HM results of unexpectedly long rotation periods for dM3 stars? To answer this, we note that the mean P/sin i values of the slow rotators tend to decrease towards later spectral types. As a matter of fact, this trend is reproduced in the lower end of the RACs (Figs. 14). Therefore, the decrease in the mean P/sin i at later spectral types could be due to the fact that the dynamo mechanisms become inefficient at a rotation period that decreases towards later spectral type (Fig. 14). However, this is not completely the case for the dM3 RAC, which extends to somewhat longer periods than the dM2 RAC. Globally, the dM3 RAC is shifted towards longer periods with respect to the dM2 RAC such that the mean P/sin i for dM3 stars is longer than for dM2 stars (as mentioned by HM). For dM4 stars, the whole of the RAC is shifted towards shorter rotation periods compared to dM3 stars. This explains why the mean P/sin i for dM4 stars is clearly smaller than that for dM3 stars. In fact, the dynamo mechanism in dM4 stars apparently become so inefficient at long periods that our sample contains not a single dM4 stars with P/sin i longer than 10 days (see Figs. 14). This can be contrasted with the much longer P/sin i values for dM3 stars: as large as 30 days (see Fig. 8). Our discovery in the present paper that the level of chromospheric emission in dM4 stars is definitely smaller than in dM3 stars (suggesting that dynamo action in dM4 stars is less effective than in dM3 stars) could explain why the mean P/sin i is much shorter at dM4 than at dM3: dM4 stars are not as good at generating magnetic fields, and as a result they do not have access to as good a “magnetic brake" as dM3 stars. The dM4 stars just keep on spinning fast, whereas the dM3 stars are braked. This might supplement (or replace) the hypothesis of long loop lengths (cf. HM) as the reason for slow rotation at dM3. Among dM2 stars also, there are no stars rotating as slowly as the slowest dM3 stars (see Fig. 5). So dM2 stars do not have access to as good a brake as dM3 stars. And yet, their dynamo effectiveness seems to be as high as, or higher than, the dM3 stars (see Fig. 14 above). In this case, the onset of increased loop lengths at dM3 (cf. HM) would help to explain why dM3 stars have access to a better magnetic brake than dM2 stars. For the fast rotators, we found (in HM) that the trend for the mean P/sin i is to become in general shorter as the spectral type becomes later, except at sub-type dM3 where the mean period is found to be significantly larger. The minima of the RACs show the opposite trend, i.e. the minimum P/sin i decreases with increasing spectral type. However, only very few active stars are at the extremes of the RACs: most active stars lie at longer periods. In the case of dM3 and dM4 stars, most active stars lie in the upper part of the rising slope of the RAC (see Figs. 8 and 12). We also refer to Figs. 3 and 5 for dK6 and dM2 stars respectively. Therefore, the trend observed in the active stars can be interpreted by the RACs only if one considers their locations on these curves. Nevertheless, we note again that the upper end of the RAC for dM3 stars (in its rising part) is shifted again towards periods which are clearly longer than those for dM2 and dM4 stars. This again explains at least partly the abnormally large mean P/sin i of the active dM3 stars (see HM). In summary, we find that the properties of the RACs can explain at least partly the observed trends in the mean P/sin i and also the abnormally long P/sin i values reported for dM3 stars by HM. However, this does not exclude another explanation, possibly involving a change in loop lengths (HM), which might also contribute to the unusual rise in the mean P/sin i at dM3. § COMPARISON WITH OTHER STUDIES OF ROTATION AND ACTIVITY West & Basri (2009) investigated the rotation and activity in H_α in a sample of 14 late-type M dwarfs (M6-M7). They found that many of these objects are rotating relatively fast (> 3.5 km s^-1) but have also H_α in absorption. These rotational velocities imply rather short rotation periods for these small objects. This confirms our observed trend (see HM) that in general, among late type dwarfs, rotation periods diminish with increasing spectral type. West and Basri (2009) also derived an empirical relationship between L_H_α/L_bol and vsin i from a compilation of data: they found that L_H_α/L_bol increases typically by two orders of magnitude for a change in vsin i of only 5 km s^-1. This implies a steep gradient in the RAC. Although our spectral range does not go to spectral subtypes which are as late as those of West and Basri, our results are not inconsistent with theirs in the sense that that the steepest slope we have found for the RAC occurs among the latest spectral type in our sample (dM4, see Table 2), which is closest to the spectral types of the West-Basri sample. Browning et al. (2010) analyzed the rotational broadening and activity in the Ca ii lines for a sample of 123 M dwarfs. Unfortunately, because of limitations on their spectral resolution, they could measure vsin i for only 7 stars, namely, those in which the projected rotational speeds were > 2.5 km s^-1. They found that the rotation was detected mostly in stars later than M3 rather than in the range M0-M2.5. This is also consistent with our findings. They also found, in agreement with our results, that there is a “gap" in the measures of L_CaII/L_bol between the active stellar group and the low activity stellar group (see Fig. 11 above). They found a rough relationship between L_CaII/L_bol and vsin i. However, most of their measures lie in the saturated regime, whereas in the present work, we have chosen to perform our RAC analyses on stars which lie in the unsaturated regime (so that we may examine stars which probably have dynamos also in the unsaturated regime). Wright et al. (2011) reported on L_X/L_bol as a function of R_0 for a sample of (824) stars which is significantly larger than we have analyzed here: the Wright et al. sample is 3 times larger than the number we used for chromospheric RACs, and about 10 times larger than the number of stars we used for the coronal RACs (Fig. 16). However, the sample of Wright et al. extends over a much broader range of spectral types than we have studied here: the Wright et al. sample extends from spectral type F to M5. They found a correlation between L_X/L_bol and R_0 although there is a significant scatter among the data. Their results reveal a saturated regime and an unsaturated regime, with a break at log(R_0)≈ -0.1. The saturation value of L_X/L_bol was found to be at a log value of ∼ -3. In the non-saturated regime the power law fit between log(L_X/L_bol) and log(P) was found to have a slope of -2.18±0.16. If the unsaturated (coronal) stars in their sample are analogous to our low activity (chromospheric) stars, then their coronal slope (-2.2) is much steeper than what we found for the chromospheric slope (-0.8) (Table 1). The increase in steepness of the coronal RAC relative to the chromospheric slope (by an amount of about 1.4 in the slope) is reminiscent of our results in Section 3.11 above, Eqs. (42)-(46) where the slopes of L_X are steeper than the slope of L_HK by values which (within errors) also overlap with 1.4 (except for dM2 stars). Rebassa-Mansergas et al. (2013) studied a sample of white dwarf/M dwarf binaries from the Sloan Digital Sky Survey (SDSS DR7). They found indications that magnetic braking is less efficient beyond the fully convective boundary. This again agrees with our results that M4 dwarfs rotate faster than early M type dwarfs. They also studied the L_H_α/L_bol-vsin i relationship and found mostly that stars with vsin i=5 km s^-1 are all in the saturated regime. They also found a rapid increase in L_H_α/L_bol (by 2 orders of magnitude) as vsin i increases from 2 km s^-1 to vsin i=5 km s^-1. This agrees with the similar diagram of West & Basri (2009), and again suggests a steep gradient in the RAC between low activity and high activity late-type M dwarfs (as we have found for our samples: see Table 1). Robertson et al. (2013) investigated the magnetic activity level in H_α for a sample of 93 K5-M5 dwarfs. They found, in agreement with our results on the RACs, that early type M dwarfs (M0-M2) tend to have higher levels of activity than later type dwarfs, and that in general, log(L_H_α/L_bol) continuously decreases from -3.6 to -3.95 as the spectral sub-type increases from M0 to M5. This again agrees with our global finding that the surface fluxes in the Ca ii lines decrease from K4 to M4. Recently, an important paper has been published by West et al. (2015: W15) reporting on an analysis of chromospheric RACs in a sample of 238 M dwarfs, using H_α emission as a measure of chromospheric “activity". The sample stars had originally been selected as targets for planet searches, but W15 used the photometry to search for rotation periods P for the stars themselves. Photometric periods are not subject to the sin i uncertainty which affects the P/sin i we have obtained in the present study. In constructing their RAC, W15 identified 164 stars for which values of P and H_α data were available. The P values ranged from 0.3 days to 100 days. The advantage of using photometry (over spectroscopy) to obtain P values is clear: there are no observational limits imposed by attempts to extract vsin i values from spectroscopic data. As a result, photometry allows W15 to determine rotational periods for slow rotators which are beyond the capabilities which we have used here. As a result, the W15 upper limit on P is several times longer than we have been able to report on in the present paper. In order to study variations of RAC with spectral type, W15 divided their stars into two groups: M1-M4 (64 stars, mainly M3 and M4), and M5-M8 (100 stars, mainly M5 and M6). Their plots of RAC (in their Figs. 7 and 8) show that the M1-M4 stars have a negative slope: a LSF yields a slope of -0.19±0.036. For the M5-M8 stars, the LSF gives a formal slope of essentially zero (-0.016±0.050). These results suggest that since a finite (though small in absolute value) negative slope exists for the RAC in M1-M4 stars, rotation might well play a role in the chromospheric emission in M1-M4 stars. On the other hand, since the RAC slope was found to be essentially zero for M5-M8 stars, rotation may play little or no role in determining chromospheric emission in M5-M8 stars. At first sight, this result might be considered as evidence for the suggestion of Durney et al (1993) that there might be a change in dynamo mode between M1-M4 and M5-M8. But we would like to suggest another possible interpretation, as follows. In view of our own results for the RAC's in M4, M3, and M2 stars, we find it a matter of interest that the slope of the chromospheric RAC determined by W15 for M1-M4 stars (-0.19±0.036) is much shallower than the values we have found. For our combined samples of dM+dMe stars, we have found slopes of -1.5 to -2.5 (see Table 2). And even when we confine attention to the slow rotators, we still find slopes for M2, M3, and M4 dwarfs (-0.89 to -0.93) which are clearly significantly steeper than W15 report for their M1-M4 sample. Our results indicate that chromospheric emission in M2-M4 stars in our samples exhibit much stronger sensitivity to rotation than W15 have reported for either of their samples. Is it possible that this dichotomy arises from selection effects? We note that the data of W15 include mainly “active" stars with H_α in emission i.e. fast rotators, whereas our samples (especially the slow rotator samples) are biased towards slow rotators (H_α not in emission). Now, fast rotators are more likely to be in the saturated regime of the dynamo (see Section 1.1 above): in that regime, the RAC flattens out, and takes on a slope which is close to zero. In fact, in a discussion of the systematic differences between their M1-M4 sample and their M5-M8 sample, W15 point out that “At similar rotation periods, a much larger fraction of the late-type M dwarfs [M5-M8] are active" than is the case for the M1-M4 stars. Visual inspection of Fig. 6 in W15 suggests that some 65% of their M1-M4 stars are labelled “active" while almost 90% of their M5-M8 stars are “active". This is consistent with the W15 statement that a “much larger fraction" of the M5-M8 stars are in the saturated regime. This leads us to wonder if the presence of saturated behavior is contributing to the (essentially) zero slope reported by W15 for M5-M8 stars. And even among the M1-M4 stars of W15, where the fraction of active stars is admittedly smaller than in the M5-M8 sample, that fraction is by no means small: some 65% of the M1-M4 stars are “active", and therefore may, lie in the saturated regime. Suppose that 65% of the M1-M4 stars in the W15 sample lie in the saturated regime (with a slope of zero), while the remaining 35% lie in the unsaturated regime, where the slope is finite (and negative), with a value of -a. Then the complete W15 sample of M1-M4 stars would have an RAC with an average slope a(W15) = (0.65×0) + (0.35×(-a)). For slow rotators with spectral sub-types extending from dK4 to dM4, we have found that -a can take on values in the range from -0.6 to -0.9. This leads us to predict that a(W15) could range from -0.21 to -0.32. In fact, the empirical slope reported by W15 (-0.19±0.036) contains, within a 3σ range, values of a(W15) extending from -0.08 to -0.30. This overlaps extensively with our predicted range of slopes. Thus, it is possible that: (i) the existence of a small but finite negative slope obtained by W15 for M1-M4 stars is due to the presence of a minority (35%) of stars in the unsaturated regime, and, (ii) the change in slope in going from M1-M4 to M5-M8 is due to the larger number of stars in the saturated regime among the M5-M8 sample. In order to test these possibilities, and in order to get a more definitive test of a change in dynamo mode at the TTCC, we suggest that it would be best to concentrate on stars in the unsaturated regime. So far, our own studies have not yet reached into M5-M8 stars: it will be a matter of great interest to determine if the W15 findings of significant change in RAC slope between M4 and M5 can be replicated using unsaturated stars. § CONCLUSION In this section, we first (Section 5.1) summarize the approach we have adopted in order to study dynamos in low-mass stars. Then we go on (Sections 5.2-5.5) to describe how we have characterized quantitatively the observational data in terms of chromospheric observations. A discussion of coronal data follows (Section 5.6). In Section 5.7 we present the major conclusions of our study in terms of three Hypotheses about dynamo action in low-mass dwarf stars. Lastly, we discuss evidence which has a bearing on crossing the TTCC (Section 5.8). §.§ Dynamos: saturated and unsaturated In order to account for the presence of active chromospheres and coronae in low-mass stars, dynamos are believed to be operative. Dynamos may rely on rotation plus turbulence, or on turbulence alone, to generate magnetic fields which then, as a result of “magnetic activity" (of some kind) lead to heating of the chromospheric and coronal gas. It is possible that different types of dynamos are at work in stars of different masses. In this paper, we bring together observational evidence to see if it is possible to identify signatures of different types of dynamos. Our study is based on the assumption that different types of dynamos are expected to lead to different answers to the following question: how does the amount of “activity" behave as a function of “rotation"? In order to address this, we need to compile data which provide us with quantitative measures of “activity" and “rotation". An important aspect of the present paper is that we measure rotational speeds in stars which are rotating more slowly than has been reported in previous studies. Why are we interested in slow rotators? The reason has to do with the physics of dynamos: theoretical work suggests that rotational dynamo operation can saturate when the rotational speed reaches a certain limit. It seems to us that it would be difficult to extract information from a saturated dynamo, where the activity level no longer depends on how fast the star is rotating. For this reason, we prefer in this paper to deal with dynamos in an unsaturated condition, where an increase in rotation leads to a clear and measurable increase in activity. This is what has driven us in the present paper to undertake a concerted effort towards identifying stars with the slowest possible rotations: to do this, we rely on spectroscopic data which were obtained with the highest possible resolution. §.§ Data being used in this paper Data have been compiled on chromospheric emission and (projected) rotational periods for a sample of 418 stars (Paper I) ranging in spectral sub-type from dK4 to dM4 (42 dK4, 118 dK6, 94 dM2, 81 dM3, 83 dM4). The analysis which we apply to our data has the goal to derive a rotation-activity correlation (RAC) for stars in each spectral sub-type, and then see if there are any systematic variations in the RACs as the spectral sub-type approaches the limit where main sequence stars make a Transition To Complete Convection (TTCC). The location of the TTCC is a matter of some dispute, possibly as early as dM2, possibly as late as dM4. Our choice of spectral sub-types is meant to overlap with TTCC. §.§ Constructing RACs at various spectral sub-types: chromospheric data As a quantitative measure of “activity", we use the mean surface flux F_CaII of emission in the Ca ii H and K lines. As a quantitative measure of “rotation", we combine the projected rotational speeds vsin i with stellar radii to obtain a “projected rotation period" P/sin i. We construct an RAC for stars in each spectral sub-type by plotting (in log-log format) the surface flux versus P/sin i. The RACs which we have obtained for stars in each of our 5 spectral sub-types can be found in Figs. 2, 3, 5, 8, and 12. §.§ Chromospheric data: the slopes of the RACs A general feature of many RACs is that as rotational periods become shorter, the Ca ii surface flux becomes larger, up to a point. Beyond that point, shorter periods do not lead to any increase in Ca ii flux: at these shortest periods, the RAC is probably “saturated". A key aspect of the present paper is that we discuss only the unsaturated part of the RAC. In this case, we obtain least squares fits of the RAC to a function of the form F_CaII= b (P/sin i)^a. In a log-log plot, a is the slope of the RAC, and is a negative number. The coefficient b is a measure of the amplitude of chromospheric heating. Numerical values of the slope a which we have obtained from least squares fitting to our data for the various spectral sub-types are shown in Fig. 13. The different curves show the results we have obtained when we group our target stars in three different ways. (i) The curve which rises monotonically from lower left to upper right refers to the combined sample of all “unsaturated" stars, both those with low activity (dK, dM) and those with high activity (dKe, dMe). (ii) The curve which runs (almost) horizontally near the lower boundary of the figure refers to stars which are confined to the low activity sub-samples (dK, dM). (iii) The “jagged" curve extending from middle left to lower right refers to stars which are confined to the high activity sub-samples (dKe, dMe). The results in Fig. 13 will guide our discussion of dynamos in the final sub-section below. §.§ Chromospheric data: the amplitude of chromospheric heating As a second step in obtaining RACs, we consider not the slopes, but the overall level of chromospheric heating in terms of the ratio of L_HK to L_bol. Based on this ratio, we have obtained RACs based on a more physically relevant parameter (the Rossby number R_o). These RACs (see Fig. 15) show that, at the longest rotation periods (log(R_o) = -1.2) in our data set, chromospheric heating in dM3 stars is less effective by a factor of about 20 than in dK4 stars. Our results suggest that the chromospheric heating efficiency in dM4 stars is less effective by a factor of about 100 than in F-G-K type stars. We also observe a progressive decline in the location of the RACs as we go from dK4 to dM4: dK6 stars lie slightly below (a factor of 3) the RAC of F, G and K type stars, while dM2 stars lie a factor of 10 below the RAC of F, G, and K type stars. The data point to a conclusion which we consider reliable: the efficiency of chromospheric heating decreases progressively between dK4 and dM4. The amplitude of the overall decrease is 20-90. This also implies that the overall efficiency of the dynamo mechanisms also decreases by the same factors when moving from K4 to M4 dwarfs. §.§ coronal data Turning now to the RAC associated with coronal emission, we find a very different behavior from the chromospheric data (Fig. 16). The coronal emission L_X/L_bol does not decrease significantly as we go from dK4 to dM4. The lack of decrease is especially marked when we compare the coronal emission to the chromospheric emission (see Fig. 18): for a given value of L_HK (say 10^27.5 ergs/s), the value of L_X in dM4 stars is larger by a factor of order 100 compared to L_X in dK4/dK6 stars. Thus, while the chromospheric heating efficiency is decreasing as we go from dK4 to dM4 (by a factor of up to 100), the coronal heating efficiency is simultaneously increasing (by a factor of 100 or so). In terms of a dynamo interpretation, this raises the question: which part of a stellar atmosphere should we study in order to obtain more reliable information about dynamo efficiency in M dwarfs, the chromosphere or the corona? The behaviors are so different that it is not clear that the same information about the dynamo will emerge from both data sets. We have argued (Section 3.11) that it may be preferable to concentrate on the chromosphere. In view of this, we now present some conclusions which, in our opinion, help to bring order to the chromospheric data which have been analyzed in the present paper. §.§ Hypotheses about two distinct dynamos in low-mass stars Inspection of Fig. 13 leads us to offer the following hypotheses. (i) In the case of low-activity stars, we see that the RAC slope a is essentially unchanged as we go from dK4 to dM4. To the extent that a particular RAC slope is associated with a particular dynamo mechanism, our results indicate that low-activity stars have essentially the same dynamo at work in stars which range in spectral sub-type from dK4 to dM4. Such a dynamo must have something to do with a physical property which is present in all low mass stars from dK4 to dM4, i.e. on both sides of the TTCC. What property is common to all such stars? The answer is: all of them have a deep convective envelope in which turbulence provides a ready supply of energy to drive a distributed dynamo (DD) i.e. a dynamo which could be described by either an α ^2 model or an α^2 Ω model. This leads us to Hypothesis (A): low-activity stars from dK4 to dM4 may be dominated by a turbulent (DD) dynamo. (ii) In the case of the RAC for high-activity stars, the aspect of Fig. 13 that is most likely to catch the eye is probably the “jagged" behavior at early spectral types. But we would like to draw attention at first to a different aspect of the RACs: namely, the values of the slopes a at the latest spectral types (dM3e and dM4e). In these cases, our results indicate that the RAC slopes for high-activity stars are essentially the same as for the low-activity stars. Once again relying on a putative association between an RAC slope and a dynamo mechanism, these results suggest that at spectral types dM3 and dM4, the dynamo mechanism in low- and high-activity stars are actually the same. It has already been suggested (see (i)) that a DD is at work in all low-activity stars. This leads us to Hypothesis (B): in the high activity stars at the latest sub-types in our samples (dM3e and dM4e), a DD is at work. This is not a surprising conclusion: dM3e and dM4e stars are completely convective (CC), so they also have access to DD operation in the form of either an α ^2 dynamo model or an α^2-Ω dynamo model. (iii) Moving now to high-activity stars at earlier spectral types (dK4e, dK6e, and dM2e), we see in Fig. 13 that the RAC slopes are very different from the slopes of the low-activity stars. This suggests that high-activity stars in the range dK4e-dM2e have access to a different kind of dynamo from the type (DD) that may dominate in low-activity stars. What might give rise to a non-DD type of dynamo in stars with spectral types in the range dK4-dM2? The answer is surely related to the fact that such stars have interfaces between the outer convective envelope and an inner radiative core. This leads to Hypothesis (C): in high-activity stars at the earliest spectral types in our samples, an interface dynamo (ID) is at work. It seems probable that such a dynamo could be described by an α-Ω dynamo model. To be sure, dK4e-dM2e stars also have deep convective envelopes: therefore a DD is probably also at work. But the clear difference in RAC slopes between low-activity stars in the range dK4-dM2 and high-activity stars in the range dK4e-dM2e indicates that the DD (with its shallow RAC slope) is not playing a dominant role in dK4e-dM2e stars. (iv) If our hypotheses have any validity, we can conclude that the primary quantitative difference between ID and DD is this: the RAC slopes are steeper (by up to 1.5 units) among the ID stars than the DD stars. Thus, the ID stars are much more sensitive to rotation than the DD stars. Specifically, if we compare two stars which differ in rotation by a factor of (say) 10, two stars where ID operates will differ in activity level by a factor which is 30 times larger than the difference in activity level of two stars where DD operates. (v) The conclusion in item (iv) is reminiscent of a suggestion that was made by Durney et al (1993): the RAC in a star with ID should depend sensitively on period, but the RAC in a star with DD should not. The results in Fig. 13 are at least partially consistent with Durney et al.: ID stars (as we identify them) are definitely more sensitive to rotation than DD stars (as we identify them). Admittedly, we have not found that the DD has zero slope for its RAC: but we have found that the slope is at least smaller than in the ID stars. The fact that the DD stars have an RAC with a non-zero slope suggests that rotation Ω does have some effect on the activity level in these stars. In terms of the two options which we have proposed in Hypotheses A and B above, it might be preferable to conclude that the dominant dynamo model in DD stars may be the α ^2-Ω model. Finally, we make a point about the overall methodology which has been ultimately responsible for this paper. The opening sentence in Item (i) above, which provides a start to our 3 hypotheses, would not have been possible if we did not have access to a large sample of low-activity stars. Such stars are slow rotators. Therefore, if the lead author of this study (ERH) had not paid attention to extracting the lowest possible values of v/sin i (of order 1 km  sec^-1) in as many stars as possible, our sample of low-activity stars might have been so small as to prevent us from drawing statistically significant conclusions. §.§ Any evidence for crossing the TTCC? The theoretical concept that main sequence stars undergo a transition to complete convection (TTCC) at a particular mass (in the vicinity of spectral types M2-M4) has been in the literature for 50 years or more. Nevertheless, the search for an empirical signature which might support this concept has yielded no definitive evidence. For example, Mullan and MacDonald (2001) sought such evidence in X-ray data, but were unable to identify any signature of the TTCC. Also in our own coronal data (Fig. 16), we see no definitive sign of a transition. Neither could we find evidence for the TTCC in the RACs of chromospheric data when we combined low- and high-activity stars (Fig. 13, monotonically rising line from lower left to upper right). Nor could we find any evidence of TTCC when we confined our attention to the chromospheric RACs of low-activity stars (lowest lying line in Fig. 13. However, in one particular sample, our study has led to what we believe is a potentially valuable signature. Specifically, in our sample of high-activity stars between dK4e and dM4e, the slopes of the chromospheric RAC (Fig. 13, “jagged" line) consists of 2 distinct regimes. In one regime, the slopes overlap with the (shallow) slopes of low-activity stars. In the other regime, the slopes are found to be much steeper. We interpret the transition between the two regimes as evidence for a transition between dynamo modes. The transition occurs between dM2e and dM3e. We suggest that this cross-over from steep to shallow RAC slopes may provide empirical evidence that the TTCC has been crossed. We have shown how empirical information about rotation and chromospheric emission may contribute to understanding some aspects of dynamo mechanisms in cool dwarfs. Data for M dwarfs with spectral types later than M4, and also for L and T dwarfs would be of great interest to complete our view on the dynamo mechanisms in low-mass stars. For example, is there any evidence that the efficiency of coronal heating actually decreases after we pass though the electrodynamic resonance which is expected to occur around dM3-dM4? It would also be of interest to investigate the variations of rotational and chromospheric properties by using finer grained samples of stars near the TTCC, i.e. using samples of stars all of which are confined to spectral types of dM2.5, dM3.5, and dM4.5 stars. This could contribute to better understanding of the behavior of possible changes in the dynamo mechanism(s) at the TTCC. § ACKNOWLEDGEMENTS This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. DJM is supported in part by the NASA Space Grant program. This study was based on data obtained from the ESO Science Archive Facility and the Observatoire de Haute Provence SOPHIE database. This research made use of Astropy[http://www.astropy.org/ and http://astroquery.readthedocs.org/en/latest/], a community-developed core Python package for Astronomy (Astropy Collaboration, 2013). We also used the tutorial developed by Paletou & Zolotukhin (2014)[http://www.astropy.org/]. This research was achieved using the POLLUX database (http://pollux.graal.univ-montp2.fr) operated at LUPM (Université Montpellier- CNRS, France) with the support of the PNPS and INSU. This research has made use of the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in A&AS 143, 23. Ayres, T.R., Marstad, N.C. & Linsky, J.L. 1981, ApJ 247, 545 Baliunas, S.L. et al. 1995, ApJ 438, 269 Barnes, S.A. 2003, ApJ 586, 464 Brandenburg, A. & Subramanian, K. 2005, Phys. Rep. 417, 1 Brown, B.P., Browning, M.K., Brun, A.S., Miesch, M.S., & Toomre, J. 2010, ApJ, 711, 424 Browning, M.K. 2008, ApJ 676, 1262 Browning, M.K., Basri, G., Marcy, G.W., West, A.A. & Zhang, J. 2010, AJ 139, 504 Browning, M.K., Weber, M.A., Chabrier, G., & Massey, A.P. 2016, ApJ 818, 189. Chabrier, G. & Baraffe, I. 1997, A&A 327, 1039 Chabrier, G. & Kuker, M. 2006, A&A 446, 1027 Christian, D.J. & Mathioudakis, M. 2002, AJ 123, 2796 Christian, D.J., Mathioudakis, M., Arias, T., Jardine, M. and Jess, D.B. 2011, ApJ 738, 164 Cram, L.E. & Mullan, D.J. 1979, ApJ 234, 579 de Laverny, P., Recio-Blanco, A., Worley, C.C. & Plez, B. 2012, A&A 544, A126 Delfosse, X., Forveille, T., Perrier, C., Mayor, M., 1998, A&A 331, 581 Dobler, W., Stix, M. & Brandenburg, A. 2006, ApJ, 638, 336 Donati, J.-F.; Morin, J.; Petit, P.; Delfosse, X.; Forveille, T.; Aurière, M.; Cabanac, R.; Dintrans, B.; Fares, R.; Gastine, T.; Jardine, M.M.; Lignières, F.; Paletou, F.; Ramirez Velez, J.C.; Théado, S. 2008, MNRAS 390, 545 Dorman, B., Nelson, L.A. & Chau, W.Y. 1989, ApJ 342, 1003 Duncan, D.K.; Vaughan, A.H.; Wilson, O.C.; Preston, G.W.; Frazer, J.; Lanning, H.; Misch, A.; Mueller, J.; Soyumer, D.; Woodard, L.; Baliunas, S.L.; Noyes, R.W.; Hartmann, L.W.; Porter, A.; Zwaan, C.; Middelkoop, F.; Rutten, R.G.M.; Mihalas, D. 1991, Astrophysical Journal Supplement Series, 76, 383 Durney, B. R., De Young, D. S., & Roxburgh, I. W. 1993, Solar Phys 145, 207 Fawzy, D., Ulmschneider, P., Stepien, K., Musielak, Z.E. & Rammacher, W. 2002, A&A 386, 983 Feiden, G.A. & Dotter, A. 2013, ApJ 765, 86 Fekel, F.C., 1997, PASP 109, 514 Fleming, T.A., Giampapa, M.S. 1989, ApJ 346, 299 Fleming, T. A., Giampapa, M. S., Schmitt, J., & Bookbinder, J. A. 1993, ApJ 410, 387 Giampapa, M.S. & Liebert, J. 1986, AJ 305, 784 Giampapa, M.S., Cram, L.E. & Wild, W.J. 1989, ApJ 345, 536 Gizis, J.E., Reid, I.N. & Hawley, S.L. 2002, AJ 123, 3356 Hawley, S.L., Gizis, J.E, Reid, I.N. 1996, AJ 112, 2799. Hempelmann, A., Schmitt, J.H.M.M., Schultz, M., Rudiger, G. & Stepien, K. 1995, A&A 294, 515 Herbst, W. & Miller, J.R. 1989, AJ 97, 891 Houdebine, E.R. 1992, IAJ 20, 213 Houdebine, E.R. 2008, MNRAS 390, 1081 (Paper VII) Houdebine, E.R. 2009a, MNRAS 400, 238 (Paper VIII) Houdebine, E.R. 2009b, MNRAS 397, 2133 (Paper XII) Houdebine, E.R. 2010a, MNRAS 407, 1657 (Paper XIV) Houdebine, E.R. 2010b, MNRAS 509, 65 (Paper IX) Houdebine, E.R. 2011a, MNRAS 416, 2233 (Paper XVI) Houdebine, E.R. 2011b, MNRAS 411, 2259 (Paper XV) Houdebine, E.R. 2012a, MNRAS 421, 3180 (Paper XVII) Houdebine, E.R. 2012b, MNRAS 421, 3189 (Paper XVIII) Houdebine, E.R. & Doyle, G.J. 1994, in Cool Stars, Stellar Systems and the Sun, Eigth Cambridge Workshop, Proceedings, held in Athens, Georgia, Caillault J.P. ed., ASPCS, Vol. 64, p. 423 Houdebine, E.R. & Doyle, J.G. 1994, A&A 289, 169 Houdebine, E.R. & Doyle, J.G. 1994, A&A 289, 185 Houdebine, E.R., Doyle, J.G. & Koscielecki, M. 1995, A&A 294, 773 Houdebine, E.R. & Stempels, H.C. 1997, A&A (Paper VI) Houdebine, E.R. & Mullan, D.J. 2015, ApJ 801, 106 Houdebine, E.R. Butler, C.J., Garcia-Alvarez, D. & Telting, J. 2012, MNRAS 426, 1591 Houdebine, E.R., Mullan, D.J., Paletou, F. & Gebran, M. 2016a, ApJ 822, 97, (Paper I). Houdebine, E.R., Mullan, D.J. & Paletou, F. 2016b, submitted to MNRAS Hunsch, M., Schmitt, J.H.M.M., Sterzik, M.F. & Voges, W. 1999, A&A Suppl. Ser., 135, 319 Isaacson, H. & Fischer, D. 2010, ApJ 725, 875 Jeffries, R.D., Totten, E.J., James, D.J. 2000, MNRAS 316, 950 Kamper, K.W., Kim, K.M., Thomson, J.R. 1997, Baltic Astronomy 6, 111 Kazeminezhad, F. & Goodman, M. L. 2006, ApJS 166, 613 Kenyon, S.J. & Hartmann, L. 1995, ApJ Suppl. Ser. 101, 117 Kim, Y.-C.; Demarque, P. 1996, ApJ 457, 340 Kraft, R.P. 1967, ApJ 150, 551 Krause, F. & Radler, K.H. 1980, Mean Field Electrodynamics and Dynamo Theory (Oxford: Pergamon). Lawson, N., Strugarek, A., & Charbonneau, P. 2015, ApJ 813, 95 Leggett, S.K., 1992, ApJS 82, 351 Limber, D.N., 1958, ApJ 127, 387 Mann A.W., Feiden G.A., Gaidos E., Boyajian T., Von Braun K. 2015, Astrophys. J., 804, 64 Marilli, E., Catalano, S., Trigilio, C. 1986, A&A 167, 297 Mamajek, E.E. & Hillebrand, L.A. 2008, ApJ 687, 1264 Mohanty, S. & Basri, G. 2003, ApJ 583, 451 Montesinos, B., Thomas, J.H., Ventura, P. & Mazzitelli, I. 2001, MNRAS 326, 877 Morales J.C., Ribas I., Jordi C. 2008, A&A 478, 507 Morin, J.; Donati, J.-F.; Petit, P.; Delfosse, X.; Forveille, T.; Albert, L.; Aurière, M.; Cabanac, R.; Dintrans, B.; Fares, R.; Gastine, T.; Jardine, M.M.; Lignières, F.; Paletou, F.; Ramirez Velez, J.C.; Théado, S. 2008, MNRAS 390, 567 Morin, J.; Donati, J.-F.; Petit, P.; Delfosse, X.; Forveille, T.; Jardine, M.M. 2010, MNRAS 407, 2269 Mullan, D. J. 1984, ApJ 282, 603 Mullan, D. J. 2009, Physics of the Sun: A First Course, Boca Raton: CRC press, p. 307. Mullan, D.J. & Cheng, Q.Q. 1993, ApJ 412, 312. Mullan, D.J. & MacDonald, J. 2001, ApJ 559, 353. Mullan, D.J. & MacDonald, J. 2003, ApJ 598, 560. Mullan, D. J. , Mathioudakis, M., Bloomfield, D.S., & Christian, D.J. 2006, ApJ Suppl. 164, 173 Mullan, D.J., Houdebine, E. R., & MacDonald, J. 2015, ApJL 810, L18. Noyes, R.W., Hartmann, L.W., Baliunas, S.L., Duncan, D.K., Vaughan, A.H. 1984, ApJ 279, 763 Pace, G. 2013, A&A 551, L8 Palacios A., Gebran M., Josselin E., Martins F., Plez B., Belmas M., Sanguillon M., Lèbre A., 2010, A&A 516, A13 Paletou, F. & Zolotukhin, I. 2014, arXiv:1408.7026 Pallavicini, R., Golub, L., Rosner, R., Vaiana, G.S., Ayres, T. R., & Linsky, J.L. 1981, ApJ 248, 279 Panagi, P.M. & Mathioudakis, M. 1993, A&A Supl. Ser. 100, 343 Parker, E.N. 1955, ApJ 122, 293 Parker, E.N. 1975, ApJ 198, 205 Parker, E.N. 1979, Cosmical Magnetic Fields: Their Origin and Their Activity (Oxford: Clarendon Press). Patten, B.M. & Simon, T. 1996, ApJ Suppl. Ser. 106, 489 Phan-Bao, Ngoc; Lim, Jeremy; Donati, Jean-François; Johns-Krull, Christopher M.; Martín, Eduardo L. 2009, ApJ 704, 1721 Pizzolato, N., Maggio, A., Micela, G., Sciortino, S., Ventura, P. 2003, A&A 397, 147 Racine, E., Charbonneau, P., Ghizaru, G., Bouchat, A., & Smolarkiewicz, P.K. 2011, ApJ 735, 46 Rajpurohit, A.S. et al. 2013, A&A, 556, A15 Ramirez, I., Melendez, J. 2005, ApJ 626, 465 Rauscher, E. & Marcy, G.W. 2006, PASP 118, 617 Rebassa-Mansergas, A., Schreiber, M.R., Gansicke, B.T. 2013, MNRAS 429, 3570 Reiners, A., Joshi, N. & Goldman, B. 2012, AJ 143, 93 Roberts, P.H. & Stix, M. 1972, A&A, 18, 453 Robertson, P., Endl, M., Cochran, W.D., Dodson-Robinson, S.E. 2013, ApJ 764, 3 Rodriguez-Lopez, C., MacDonald, J., Amado, P. J., Moya, A. & Mullan, D.J. 2014, MNRAS 438, 2371 Rosner, R. 1980, in Cool Stars, Stellar Systems and the Sun, ed. A.K. Dupree (Cambridge: SAO Rep. 389), 79 Rutten, R.G.M. 1987, A&A 177, 131 Rutten, R.G.M., Schrijver, C.J., Zwaan, C., Duncan, D.K., Mewe, R. 1989, A&A 219, 239 Rutten, R.G.M., Schrijver, C.J., Lemmens, A.F.P. & Zwaan, C. 1991, A&A, 252, 203 Schrijver, C.J. 1987, A&A, 172, 111 Schrijver, C.J., Coté, J., Zwaan, C., & Saar, S.H. 1989, ApJ, 337, 964 Schrijver, C.J., Dobson, A.K. & Radick, R.R. 1992, A&A, 258, 432 Skumanich, A.; Smythe, C.; Frazier, E. N. 1975, ApJ 200, 747 Soderblom, D.R. 1982, ApJ 263, 239 Soderblom, D.R., Duncan, K.D., Johnson, D.R.H. 1991, ApJ 375, 722 Spada, F.; Demarque, P.; Kim, Y.-C.; Sills, A. 2013, ApJ 776, 87 Stauffer, J.R. & Hartmann, L.W. 1986, ApJ Suppl. Ser. 61, 531 Stassun, K.G.; Hebb, L.; Covey, K.; West, A.A.; Irwin, J.; Jackson, R.; Jardine, M.; Morin, J.; Mullan, D.J.; Reid, I.N. 2011, ASPC 448, 505 Stefani, F. & Gerberth, G. 2033, Phys. Rev. E 67, 027302 Stepień, K. 1989, A&A 210, 273 Stepień, K. 1993, in the Proceedings of the 157th Symposium of the IAU, The Cosmic Dynamo, Eds. F. Krause, K. H. Radler, and Gunther Rudiger, p. 141 Stepień, K. 1994, A&A 292, 207 Strassmeier, K.G., Handler, G., Paunzen, E. & Rauth, M. 1994, A&A, 281, 855 Ulmschneider, P. & Musielak, Z.E. 2001, in Current Theoretical Models and High Resolution Solar Observations: Preparing for ATST, ASP Conf. Ser., Vol. 286, 363 Ulmschneider, P., Fawzy, D., Musielak, Z.E. & Stepien, K. 2005, ApJ 559, L167 Ulmschneider, P., Rammacher, W., Musielak, Z.E. & Kalkofen, W. 2005, ApJ 631, L155 Vaughan A. H., Baliunas, S. L., Middelkoop, F., Hartmann, L. W., Mihalas, D., Noyes, W. R., & Preston, G. W 1981, Astrophys. J., 250, 276 Vilhu, O. 1984, A&A 133, 117 Vogt, S.S., Soderblom, D.R., Penrod, G.D. 1983, ApJ 269, 250 Walkowicz, L.M. & Hawley, S.L. 2009, AJ 137, 3297 Ward-Duong, K.; Patience, J.; De Rosa, R. J.; Bulger, J.; Rajan, A.; Goodwin, S. P.; Parker, Richard J.; McCarthy, D. W.; Kulesa, C. 2015, MNRAS 449, 2618 West, A.A. & Basri, G. 2009, ApJ 693, 1289 West, A.A.; Hawley, S.L.; Bochanski, J.J.; Covey, K.R.; Reid, I.N.; Dhital, S.; Hilton, E.J.; Masuda, M. 2008, AJ 135, 785 West, A.A., Weisenburger, K.L., Irwin, J., Berta-Thompson, Z.K., Charbonneau, D., Dittmann, J., Pineda, J.S. 2015, ApJ 812, 3 Wilson, O.C. 1968, ApJ 153, 221 Wright C.O., Egan M.P., Kraemer K.E., Price S.D. 2003, Astron. J., 125, 359 Wright, N.J., Drake, J.J., Mamajek, E.E., Henry, G.W. 2011, ApJ 743, 48 Young, A.; Skumanich, A.; Stauffer, J.R.; Harlan, E.; Bopp, B.W. 1989, ApJ 344, 427 APPENDIX: Quadratic heteroscedastic regression models This appendix is devoted to classical mathematical results on stochastic linear regression models. Our purpose is to show how to better take into account the measurement errors of P/sin i in order to explain the variability of Ca ii surfaces fluxes. In the previous literature, only linear or quadratic homoscedastic LSF were investigated in order to explain this variability. Our strategy is slightly different as we propose to make use of linear or quadratic heteroscedastic LSF in order to better explain this variability. More precisely, consider one of our 5 groups of dK4, dK6, dM2, dM3, or dM4 stars, for example dM2. Let n>3 be the total number of the dM2 stars under study. For each dM2 star, denote by x_k the log(P/sin i) associated measure and by σ_k>0 the measurement error of log(P/ sin i). Moreover, let Y_k be the log(CaII Flux) measure of this dM2 star. Then, we shall deal with the linear or quadratic heteroscedastic regression models, respectively given, for k=1,…,n, by Y_k = a + bx_k + σ_k _k, and, Y_k = a + bx_k + cx_k^2 + σ_k _k where a,b and c are unknown parameters and the random noise (_k) is a standard Gaussian white noise with mean zero and unknown variance τ^2. The important point is the crucial role played by the error term σ_k. The variance of the additive noise clearly depends on the explanatory variable x_k as it is given by σ_k^2 τ^2 where σ_k is the measurement error associated with x_k. We shall only focus our attention on the quadratic heteroscedastic regression model given by (A.2) inasmuch as the linear regression model (A.1) is a particular case of (A.2). The quadratic heteroscedastic regression model (A.2) can be rewritten into the matrix form Y=Xθ +Γ^1/2 where the vector of observations Y, the vector of unknown parameters θ, and thevector containing the random noises , are respectively given by Y= ( Y_1 ⋮ Y_n ), θ= ( a b c ), = (_1 ⋮ _n ). In addition, the heteroscedastic matrix Γ=diag( σ_1^2, σ_2^2, ⋯, σ_n^2) and the design matrix X is given by X= ( 1 x_1 x_1^2 1 x_2 x_2^2 ⋮ ⋮ ⋮ 1 x_n x_n^2 ). On the one hand, the least squares estimator (LSE) of θ is the value θ which minimizes the strictly convex function Δ(θ) =||Y-Xθ||^2. Straightforward calculation leads to θ=(X^t X)^-1X^tY. However, this estimator does not taken into account the heteroscedasticity of model (A.3). On the other hand, the weighted least squares estimator (WLSE) of θ is the value θ which minimizes the strictly convex function Δ(θ)=(Y-Xθ)^tΓ^-1(Y-Xθ). It is not hard to see that θ=(X^t Γ^-1X)^-1X^tΓ^-1Y. We immediately deduce from (A.3) and (A.4) that θ = θ + (X^t Γ^-1X)^-1 X^tΓ^-1/2. Consequently, as is a n-dimensional Gaussian vector (0, τ^2 I) where I stands for the identity matrix of order n, we obtain that θ is a 3-dimensional Gaussian vector, θ = (a b c) ∼(θ, τ^2 (X^t Γ^-1X)^-1). Hereafter, the linear least squares fit (LSF) is the line y=a + bx, while the quadratic LSF is the curve y=a + bx + cx^2. Furthermore, denote by H the hat matrix H=X(X^t Γ^-1X)^-1X^tΓ^-1and L=I-H. It follows from (A.3) and (A.4) that Y=HY=Xθ which implies that HY has an (Xθ, τ^2 HΓ) distribution and LY=Y-Y has an (0, τ^2 LΓ) distribution. Hence, as rank(L)=n-3, we obtain that the sum of squared errors (SSE) has a chi-squared distribution, ||Γ^-1/2(Y - Y)||^2 ∼τ^2 χ^2(n-3). The SSE is a way to evaluate the discrepancy between the data Y and its estimate Y and a small value of the SSE indicates a tight fit of our model to the data. Let us recall the celebrated decomposition of total sum of squares which is in fact a direct application of Pythagoras's theorem || Γ^-1/2 Y ||^2 = || Γ^-1/2Y ||^2 + || Γ^-1/2(Y-Y)||^2. Finally, a natural estimator of the variance τ^2 is τ^ 2=||Γ^-1/2(Y - Y)||^2/n-3. It is not hard to see that the random vector θ and τ^ 2 are independent. Consequently, we deduce from (A.5) that for any real number x, (a - a) + (b - b)x + (c - c)x^2 ∼(0, τ^2 ξ(x)) where ξ(x)= ( 1 x x^2 )^t(X^t Γ^-1X)^-1( 1 x x^2 ). Dividing on both sides by τ, the ratio has a Student distribution, (a - a) + (b - b)x + (c - c)x^2/τ√(ξ(x))∼ t(n-3). |lccccccc| [ ]In this table, we give the parameters of our homoscedastic and heteroscedastic regression models. 8cHomoscedastic Least Square Fits 7cF_CaII = b× (P/sin i)^a Spect. Type a b Corr. Coef. χ^2 Nb of Stars Stat. Sign. dK4+dK4e -0.756±0.05 1.57± 0.34 10^6 0.884 0.032 34 >99.9% dK6+dK6e -0.81±0.06 1.95± 0.34 10^6 0.876 0.021 55 >99.9% dM2+dM2e -1.481±0.068 1.89± 0.85 10^6 0.949 0.018 66 >99.9% dM3+dM3e -2.08±0.19 1.52± 1.00 10^7 0.735 0.118 59 >99.9% dM4+dM4e -2.526±0.20 7.94± 2.05 10^5 0.861 0.090 58 >99.9% 7cF_CaII = b× (P/sin i)^a Spect. Type a b Corr. Coef. χ^2 Nb of Stars Stat. Sign. dK4 -0.624±0.11 9.33± 3.16 10^5 0.744 0.031 30 >99.9% dK6 -0.520±0.11 7.94± 2.18 10^5 0.706 0.013 40 >99.9% dM2 -0.891±0.12 4.17± 1.15 10^5 0.708 0.013 54 >99.9% dM3 -0.93±0.22 3.09± 1.58 10^5 0.516 0.027 50 >99.9% dM4 -0.91±0.39 3.80± 1.89 10^4 0.374 0.057 35 96% 7cF_CaII = b× (P/sin i)^a Spect. Type a b Corr. Coef. χ^2 Nb of Stars Stat. Sign. dK4e -1.880±0.55 9.55± 5.18 10^6 0.924 0.014 4 99.7% dK6e -1.364±0.16 3.97± 0.74 10^6 0.961 0.006 9 90% dM2e -1.819±0.14 3.40± 0.51 10^6 0.959 0.009 11 >99.9% dM3e -1.165±0.52 2.19± 1.58 10^6 0.675 0.060 8 99.5% dM3e+Sat. -0.37±0.11 3.16± 0.59 10^5 0.704 0.050 16 99.7% dM4e -0.960±0.30 2.30± 0.59 10^5 0.592 0.025 23 92% dM4e+Sat. -0.47±0.08 1.26± 0.08 10^5 0.720 0.020 34 99.9% 7cEW_CaII = a× EW_H_α + b Spect. Type a b Corr. Coef. χ^2 Nb of Stars Stat. Sign. dM3e 2.90±0.05 1.14± 0.125 0.9994 0.023 6 >99.9% dM4e 1.909±0.18 -1.035± 0.58 0.877 2.2 35 >99.9% 7cEW_CaII = b× S_HK^a Spect. Type a b Corr. Coef. χ^2 Nb of Stars Stat. Sign. dM4+dM4e 1.20±0.07 -0.31±0.04 0.970 0.024 22 >99.9% 7cL_X = a× L_CaII + b Spect. Type a b Corr. Coef. χ^2 Nb of Stars Stat. Sign. dK4+dK4e 1.54±0.29 -15.34±8.2 0.859 0.17 13 >99.9% dK6+dK6e 1.57±0.21 -16.1±5.9 0.321 0.11 16 <10% dM2+dM2e 1.57±0.17 -15.0±4.6 0.878 0.24 28 >99.9% dM3+dM3e 1.74±0.11 -19.1±3.0 0.952 0.09 28 >99.9% dM4+dM4e 1.37±0.12 -8.6±3.2 0.882 0.18 42 >99.9% 8cHeteroscedastic Least Square Fits 7cF_CaII = b× (P/sin i)^a Spect. Type a b Corr. Coef. χ^2 Nb of Stars Stat. Sign. dK4+dK4e -0.8140±0.059 1.916± 0.32 10^6 - 0.033 34 >99.9% dK6+dK6e -1.0469±0.042 3.402± 0.021 10^6 - 0.093 55 >99.9% dM2+dM2e -1.5754±0.058 2.312± 0.276 10^6 - 0.028 66 >99.9% dM3+dM3e -2.0201±0.11 1.169± 0.476 10^7 - 0.091 59 >99.9% dM4+dM4e -2.5644±0.19 8.38± 2.05 10^5 - 0.211 58 >99.9% 7cF_CaII = b× (P/sin i)^a Spect. Type a b Corr. Coef. χ^2 Nb of Stars Stat. Sign. dK4 -0.568±0.084 7.66± 1.89 10^5 - 0.034 30 >99.9% dK6 -0.531±0.12 8.23± 2.23 10^5 - 0.017 41 >99.9% dM2 -0.709±0.14 2.68± 0.77 10^5 - 0.046 54 >99.9% dM3 -0.837±0.20 2.33± 1.10 10^5 - 0.039 50 >99.9% dM4 -0.825±0.35 3.08± 1.25 10^4 - 1.966 35 96% 7cF_CaII = b× (P/sin i)^a Spect. Type a b Corr. Coef. χ^2 Nb of Stars Stat. Sign. dK4e -1.877±0.51 9.73± 5.07 10^6 - 0.038 4 99.7% dK6e -1.402±0.43 4.20± 3.52 10^6 - 0.023 9 90% dM2e -1.793±0.14 3.28± 0.54 10^6 - 0.014 11 >99.9% dM3e -1.041±0.58 1.60± 1.20 10^6 - 0.133 8 99.5% dM4e -0.951±0.31 2.27± 0.58 10^5 - 0.077 23 92% |lcccccccccccccc| [ ]Equivalent width (EW, in units of Å) of the Ca ii resonance doublet and H_α line for our sample of dK6 stars. Results were obtained using spectra from HARPS, SOPHIE and FEROS. For each star, we give the S/N ratio for the sum of all available spectra, as well as the number of spectra N_b (meas.) which we used to compute the equivalent widths. HARPS SOPHIE CaII H CaII K H_α CaII H CaII K H_α K/H ratio <CaII> Ca ii Ca ii Nb. S/N S/N Star No. of EW EW EW EW EW EW EW Flux Flux[1] meas. HARPS SOPHIE meas. (Å) (Å) (Å) (Å) (Å) (Å) (Å) (10^5 ergs/s/cm^2) (10^5 ergs/s/cm^2) @5000 @5000 continued. HARPS SOPHIE CaII H CaII K H_α CaII H CaII K H_α K/H ratio <CaII> Ca ii Ca ii Nb. S/N S/N Star No. of EW EW EW EW EW EW EW Flux Flux[1] meas. HARPS SOPHIE meas. (Å) (Å) (Å) (Å) (Å) (Å) (Å) (10^5 ergs/s/cm^2) (10^5 ergs/s/cm^2) @5000 @5000 GJ 1056 9 -0.61±0.01 -0.68±0.01 0.643±0.020 - - - 1.11 -0.645 2.052 9 151 - GJ 1066 3 -0.91±0.10 -1.01±0.10 0.605±0.020 - - - 1.11 -0.96 3.295 3 64 - GJ 1067 3 - - - -0.79±0.10 -1.11±0.10 0.554±0.020 1.41 -0.95 3.171 3 - 119 Gl 1177A - - - - - - - - -1.919 4.525 - - - GJ 1248 3 - - 0.239±0.020 - - - - - - 3 29 - GJ 1267 1 -1.01±0.15 -1.08±0.15 0.518±0.020 - - - 1.07 -1.045 1.755 2.944 1 38 - GJ 1279 2 -0.79±0.01 -0.87±0.01 0.638±0.020 - - - 1.10 -0.83 2.849 2 177 - GJ 3072 3 -0.92±0.03 -1.05±0.03 0.585±0.020 - - - 1.14 -0.985 1.501 3.132 3 66 - GJ 3411 1 - - - - - - - -1.463 2.598 - - - GJ 3494 3 -0.97±0.05 -1.27±0.05 0.517±0.020 - - - 1.31 -1.12 2.905 3 64 - GJ 3551 5 -0.19±0.03 -0.17±0.03 0.659±0.020 - - - 0.89 -0.18 0.402 0.553 5 91 - GJ 3996 6 -1.51±0.03 -1.65±0.03 0.498±0.020 - - - 1.09 -1.58 2.527 3.491 6 113 - GJ 4140 3 -0.94±0.03 -1.07±0.03 0.544±0.020 - - - 1.14 -1.005 1.840 7.050 3 78 - GJ 9250 2 - - - - - - - -1.212 2.934 - - - GJ 9299 8 -0.82±0.01 -0.91±0.01 0.618±0.020 - - - 1.11 -0.865 2.033 8 134 - GJ 9667 3 - - - -0.55±0.10 -0.77±0.10 0.625±0.020 1.40 -0.66 1.394 3.788 1 - 59HE GJ 9714 8 -0.76±0.01 -0.84±0.01 0.612±0.020 - - - 1.11 -0.80 2.345 8 137 - GJ 9827 2 -0.50±0.03 -0.63±0.03 0.647±0.020 - - - 1.26 -0.565 1.217 2.097 2 54 - Gl 14 2 - - - -1.00±0.10 -1.61±0.10 0.534±0.020 1.61 -1.305 2.564 3.747 2 - 76 Gl 17.1 10 -0.86±0.03 -0.98±0.03 0.553±0.020 - - - 1.14 -0.92 0.601 - - - Gl 40A 2 -0.83±0.10 -0.92±0.10 0.583±0.020 - - - 1.11 -0.875 2.228 2 59 - Gl 45 4 -0.68±0.02 -0.79±0.02 0.633±0.020 - - - 1.16 -0.735 1.687 3.999 4 100 - Gl 50 4 -0.31±0.03 -0.37±0.03 0.648±0.020 - - - 1.19 -0.34 0.761 1.299 4 79 - Gl 52 3 - - - -0.31±0.10 -0.49±0.10 0.597±0.020 1.58 -0.40 0.820 1.555 3 - 89 Gl 57 5 -0.32±0.05 -0.35±0.05 0.454±0.020 - - - 1.09 -0.335 0.491 16.04 5 100 - Gl 105.5 1 - - - - - 0.673±0.020 - -0.437 0.996 1 - 65HE Gl 112.1 1 - - - - - - - -1.627 3.532 - - - Gl 116 1 - - - -0.37±0.05 -0.44±0.05 0.474±0.020 1.19 -0.355 0.586 7.377 6 - 124 Gl 142[2] 1 -1.16±0.05 -1.53±0.05 - - - - 1.32 -1.345 2.372 - - - Gl 143.1 6 -1.41±0.05 -1.57±0.05 0.534±0.020 - - - 1.11 -1.49 2.271 3.319 6 109 - Gl 146 46 -0.67±0.01 -0.75±0.01 0.638±0.020 - - - 1.12 -0.71 1.529 2.418 46 279 - Gl 153A 2 - - - -0.43±0.10 -0.56±0.10 0.606±0.020 1.30 -0.495 1.118 3.277 2 - 72 Gl 156 2 -0.83±0.10 -0.99±0.10 0.629±0.020 - - - 1.19 -0.91 1.635 2.430 2 52 - - Gl 162.2 14 -0.88±0.03 -0.96±0.03 0.605±0.020 - - - 1.09 -0.92 3.014 14 153 - Gl 182 1 -7.65±0.05 -8.33±0.05 -1.26±0.030 - - - 1.09 -7.99 20.3 - 1 115[3] - Gl 186AB 3 -0.27±0.05 -0.38±0.05 0.651±0.020 - - - 1.41 -0.325 0.733 1.486 3 84 - Gl 191 30 -0.10±0.01 -0.12±0.01 0.258±0.020 - - - 1.20 -0.11 0.033 - - - Gl 208 19,19 -2.49±0.05 -3.03±0.05 0.082±0.020 -4.29±0.05 -6.24±0.02 -0.291±0.020 1.22 -4.01 4.859 11.52 19,19 135 309 Gl 221 65 -0.60±0.05 -0.71±0.05 0.637±0.020 - - - 1.18 -0.655 1.174 1.953 65 337 . - Gl 256 1 - - - - - - - -1.296 4.083 - - - Gl 322 3 - - - -0.96±0.15 -1.42±0.15 0.519±0.020 1.48 -1.19 2.806 3 - 83 Gl 369 2 -0.48±0.05 -0.57±0.05 0.417±0.020 - - - 1.19 -0.525 0.385 2 45 - Gl 389.1 11 -0.39±0.01 -0.42±0.01 0.668±0.020 - - - 1.08 -0.405 0.872 1.284 11 147 - Gl 397 2 - - - - - - - -1.005 1.827 - - - Gl 401 7 -0.55±0.01 -0.72±0.01 0.422±0.020 - - - 1.31 -0.635 0.415 7 90 - Gl 412.3 6 -0.39±0.03 -0.47±0.03 0.649±0.020 - - - 1.21 -0.43 1.024 6 146 - Gl 414A 8 - - - -0.66±0.10 -0.98±0.10 0.619±0.020 1.48 -0.82 1.646 2.463 8 - 302 Gl 416 1 - - - - - - - - - 1 - 32 Gl 421B 7 -0.69±0.03 -0.75±0.03 0.625±0.020 - - - 1.09 -0.72 0.976 7.235 7 116 - Gl 425B 1 -1.55±0.05 -2.00±0.05 0.334±0.020 - - - 1.29 -1.775 3.876 4.802 1 67 - Gl 455.1 1 - - - - - - - -5.777 11.28 - - - Gl 466 2 - - - - - - - - - 2 - 34 Gl 496.1 47 -0.76±0.01 -0.85±0.01 0.653±0.020 - - - 1.12 -0.805 1.125 1.497 47 365 - Gl 509.1 1 - - - -1.00±0.10 -1.60±0.10 0.326±0.020 1.60 -1.30 3.219 1 - 66 Gl 517 5 -2.36±0.05 -2.83±0.05 -0.611±0.020 - - - 1.20 -2.595 11.33 5 141 - Gl 522 3 -0.56±0.05 -0.64±0.05 0.662±0.020 - - - 1.14 -0.60 1.292 1.911 3 75 - Gl 524.1 6 -0.94±0.03 -0.99±0.03 0.577±0.020 - - - 1.05 -0.965 2.389 6 130 - Gl 542.2 1 - - - - - - - -1.231 3.251 - - - Gl 546 9 - - - -0.75±0.10 -1.10±0.10 0.591±0.020 1.47 -0.925 2.617 9 - 146 Gl 558 2 - - - -0.52±0.10 -0.86±0.10 0.572±0.020 1.65 -0.69 1.573 3.379 2 - 80 Gl 562 6 - - - -0.83±0.10 -1.19±0.10 0.558±0.020 1.43 -1.01 2.580 6 - 126 Gl 571 4 -1.01±0.05 -1.08±0.05 0.525±0.020 - - - 1.07 -1.045 2.031 4.872 4 99 - Gl 571.1 4 -1.10±0.03 -1.24±0.03 0.597±0.020 - - - 1.13 -1.17 1.813 2.412 4 123 - Gl 583 4 -1.01±0.03 -1.17±0.03 0.579±0.020 - - - 1.16 -1.09 2.316 4.884 4 102 - Gl 619 6 - - - -0.38±0.10 -0.66±0.10 0.615±0.020 1.74 -0.52 1.087 1.731 6 - 129 Gl 629.1 5 -0.86±0.03 -0.97±0.03 0.612±0.020 - - - 1.13 -0.915 1.587 2.191 5 95 - Gl 659B 2 - - - - - - - -1.278 4.195 - - - Gl 673 2 - - - - - - - -1.632 2.541 - - - Gl 707 9 -0.77±0.01 -0.87±0.01 0.617±0.020 - - - 1.13 -0.82 1.979 9 226 - Gl 710 3 -1.01±0.10 -1.02±0.10 0.588±0.020 - - - 1.01 -1.015 1.760 2.955 3 77 - Gl 726 5 -0.79±0.03 -0.88±0.03 0.574±0.020 - - - 1.11 -0.835 2.035 5 126 - Gl 728 18 - - - -0.43±0.10 -0.60±0.10 0.612±0.020 1.40 -0.515 0.850 1.160 18 - 373 Gl 747.1 1 - - 0.393±0.050 - - - - - - - - - Gl 747.3AB 7 -0.58±0.03 -0.64±0.03 0.624±0.020 - - - 1.10 -0.61 1.270 1.753 7 161 - Gl 757 2 -0.76±0.10 -0.86±0.10 0.545±0.020 - - - 1.13 -0.81 1.231 3.367 2 49 - Gl 763 15 - - - -1.07±0.05 -1.47±0.05 0.486±0.020 1.37 -1.271 1.910 4.386 1 - 91HE Gl 773 1 - - - - - - - -0.903 2.328 - - - Gl 782 2 -0.84±0.01 -0.92±0.01 0.592±0.020 - - - 1.10 -0.88 2.572 15 206 - Gl 786.1 1 - - - -0.73±0.10 -1.11±0.10 0.619±0.020 1.52 -0.92 1.673 2.573 2 - 73 Gl 795AB 1 - - - -0.45±0.05 -0.53±0.05 0.600±0.020 1.18 -0.49 0.994 3.030 1 - 105HE Gl 798 22 -0.51±0.01 -0.58±0.01 0.603±0.020 - - - 1.14 -0.545 0.979 2.943 22 380 - Gl 801 4 -0.36±0.05 -0.36±0.05 0.667±0.020 - - - 1.00 -0.36 0.750 1.151 4 76 - Gl 818 12,1 -0.43±0.01 -0.50±0.01 0.699±0.020 - - - 1.16 -0.465 1.396 12,1 362 82HE Gl 820B 1 - - - - - - - -1.63 2.51 - - - - Gl 826.1 1 -1.28±0.10 -1.33±0.10 0.571±0.020 - - - 1.04 -1.305 2.937 1 40 - Gl 830 1 - - 0.658±0.020 - - - - - - - 1 33 - Gl 842 7 -1.18±0.01 -1.31±0.01 0.482±0.020 - - - 1.11 -1.245 0.615 7 173 - Gl 847A 14 -0.84±0.05 -0.92±0.05 0.568±0.020 - - - 1.10 -0.88 1.489 5.806 14 131 - Gl 855 10 -1.34±0.03 -1.49±0.03 0.491±0.020 - - - 1.11 -1.415 0.965 - - - Gl 857.1 2 - - - -0.58±0.05 -0.84±0.05 0.623±0.020 1.45 -0.71 1.897 2 - 241HE Gl 884 1 -1.33±0.10 -1.39±0.10 0.527±0.020 - - - 1.05 -1.36 2.033 3.207 1 38 - Gl 885A[2] 2 -2.650±0.05 -2.974±0.05 - - - - 1.12 -2.812 5.774 - - - Gl 889A 1 -1.21±0.10 -1.19±0.10 0.547±0.020 - - - 0.98 -1.20 2.691 4.954 1 32 - Gl 891 - - - - - - - - -0.876 0.932 - - - Gl 894 3 -0.89±0.05 -0.94±0.05 0.610±0.020 - - - 1.06 -0.915 2.048 2.953 3 69 - Gl 895.3 5 -1.17±0.03 -1.29±0.03 0.563±0.020 - - - 1.10 -1.23 3.045 5 87 - Gl 898 40 -1.30±0.03 -1.44±0.03 0.516±0.020 - - - 1.11 -1.37 3.210 40 343 - Gl 900[2] 1 -3.247±0.03 -3.581±0.03 - -2.44±0.10 -3.29±0.10 0.110±0.020 1.35 -3.414 6.565 1 - 71HE Gl 900 2 - - - - - - - -5.638 10.84 - - - Gl 906 3 - - - -0.66±0.10 -0.91±0.10 0.579±0.020 1.38 -0.785 1.180 2.735 3 - 84 Gl 907.1 - - - - - - - - -6.222 8.698 - - - HIP 14593A 3 -0.49±0.03 -0.51±0.03 0.660±0.020 - - - 1.04 -0.50 1.179 3 69 - HIP 19410ABC 4 -0.62±0.03 -0.75±0.03 0.670±0.020 - - - 1.21 -0.685 1.841 4 70 - HIP 21865 4 -0.34±0.03 -0.42±0.03 0.703±0.020 - - - 1.24 -0.38 1.292 4 91 - HIP 40170 1 - - - - - - - -1.31 3.398 - - - HIP 42108AB 2 -0.056±0.01 -0.122±0.01 0.683±0.020 - - - 2.18 -0.089 0.169 0.491 2 62 - HIP 42910 1 -0.70±0.15 -0.76±0.15 0.667±0.020 - - - 1.09 -0.73 1.945 1 30 - HIP 50773 3 - - - -0.88±0.20 -1.37±0.20 0.535±0.020 1.56 -1.125 1.856 3.116 3 - 65 HIP 51073 4 -0.43±0.05 -0.44±0.05 0.655±0.020 - - - 1.02 -0.435 1.111 4 79 - HIP 51263 1 >-0.05 >-0.05 0.700±0.020 - - - - - <0.142 - - - HIP 53175 7 - - - -0.54±0.20 -0.71±0.20 0.472±0.020 1.31 -0.625 1.425 1.913 7 - 84 HIP 56838 3 -0.35±0.03 -0.40±0.03 0.678±0.020 - - - 1.14 -0.375 0.839 1.137 3 82 - HIP 58945 1 - - - -0.84±0.20 -1.24±0.20 0.579±0.020 1.48 -1.04 2.412 4.359 1 - 39 HIP 59247 4 - - - - - 0.591±0.020 - - - 4 - 58 HIP 60438A[2] 1 -5.400±0.03 -5.613±0.03 - - - - 1.04 -5.507 8.971 - - - HIP 60438B[2] 1 -2.419±0.03 -2.827±0.03 - - - - 1.17 -2.623 4.273 - - - HIP 60501 4 - - - - - 0.659±0.020 - - - 4 - 51 HIP 60661 2 - - - -0.39±0.25 -0.76±0.25 0.592±0.020 1.95 -0.575 - - - - HIP 72044 3 - - - -0.39±0.05 -0.57±0.05 0.610±0.020 1.46 -0.48 1.678 3 - 102 HIP 76550 3 -0.28±0.05 -0.27±0.05 0.571±0.020 - - - 0.96 -0.275 0.448 5.785 3 52 - HIP 78395 11 -0.68±0.03 -0.72±0.03 0.601±0.020 - - - 1.06 -0.70 1.991 11 131 - HIP 80083 3 -0.28±0.05 -0.30±0.05 0.624±0.020 - - - 1.07 -0.29 1.281 3 50 - HIP 103150 1 -0.46±0.15 -0.58±0.15 0.579±0.020 - - - 1.26 -0.52 0.965 2.802 1 33 - HIP 110245 3 -0.82±0.05 -0.93±0.05 0.599±0.020 - - - 1.13 -0.875 2.935 3 73 - HIP 110714 2 -0.97±0.05 -1.11±0.05 0.615±0.020 - - - 1.14 -1.04 1.786 2.534 2 58 - HIP 113597[2] 2 -2.758±0.03 -3.019±0.03 - - - - 1.09 -2.889 5.810 - - - HIP 116011 7 - - - - - 0.576±0.020 - - - 7 - 55 - - - [1]Corrected from metallicity effects. [2]FEROS spectra. |lcccccc| [ ]We give the values of vsin i, P/sin i as well as [M/H] from Paper I for our dM2 stars. We also give the Ca ii EW (from the compilation of Paper XVIII), surface fluxes F_HK and the surface fluxes corrected for the metallicity effects. Star vsin i P/sin i [M/H] Ca ii EW F_HK F_HK[1] km/s (days) (dex) (Å) 10^4erg/s/cm^2 10^4erg/s/cm^2 continued. Star vsin i P/sin i [M/H] Ca ii EW F_HK F_HK[1] km/s (days) (dex) (Å) 10^4erg/s/cm^2 10^4erg/s/cm^2 GJ 1010A - - -0.098 -0.605 2.621 10.555 GJ 1062 9.8 1.92-0.53+0.65 -0.584 - - - GJ 1114 3.8 5.73-1.50+2.07 -0.339 -0.117 0.507 0.664 GJ 1264 6.46 6.82-1.51+2.06 +0.378 -4.45 20.85 8.732 GJ 2085 2.46 10.80-3.23+5.31 +0.000 -0.77 4.071 4.071 GJ 3084 3.16 9.74-4.07+5.98 +0.023 -0.841 4.373 4.147 GJ 3098 2.60 8.97-2.49+3.98 -0.200 -0.86 4.215 6.680 GJ 3207 - - -0.346 -0.19 0.801 1.777 GJ 3340 2.67 9.00-2.91+4.59 -0.048 -0.83 4.256 4.753 GJ 3215 - - -0.314 -0.45 1.950 4.018 GJ 3440 2.71 8.95-3.56+5.59 -0.165 -0.6 2.599 3.800 GJ 3759 2.71 9.10-2.52+3.95 -0.121 -0.645 3.148 4.159 GJ 3778 1.72 13.86-6.49+13.4 +0.002 -0.635 2.515 2.503 GJ 3915 1.88 22.51-12.4+24.0 +0.413 -0.60 2.694 1.041 GJ 4155 - - -0.231 -0.755 3.005 5.115 GJ 9381 2.19 14.01-4.81+8.44 +0.100 -0.90 3.899 3.097 GL 2 2.76 9.74-2.48+3.85 -0.018 -1.316 6.166 6.427 Gl 15A 1.43 13.87-4.92+12.0 -0.386 -0.303 1.366 3.322 Gl 16 2.34 11.23-3.27+5.53 -0.105 -1.084 4.696 5.980 GL 27.1 2.63 10.01-3.03+4.83 -0.016 -1.080 5.413 5.616 Gl 29.1A 10.6 3.16-0.64+0.77 +0.101 -9.69 43.12 34.17 Gl 29.1B 9.5 3.52-0.74+0.91 +0.101 -9.81 43.65 34.59 Gl 49 2.49 11.83-3.13+5.12 +0.026 -1.797 7.785 7.333 Gl 63 <1 >20.90 -0.309 -0.159 0.762 1.552 Gl 87 3.99 5.86-1.24+1.68 -0.027 -0.294 1.377 1.465 Gl 91 1.68 15.82-5.34+11.3 -0.090 -0.968 3.938 4.845 GL 114.1A 1.82 12.68-4.11+8.14 -0.242 -0.466 1.868 3.261 Gl 130 1.27 13.59-5.98+16.7 -0.766 -0.332 1.399 8.162 Gl 133 <1 >20.70 -0.466 -0.28 1.051 3.073 Gl 134 2.53 13.90-3.93+6.37 +0.128 -1.596 7.290 5.429 Gl 140AB 9.4 2.93-0.74+0.92 -0.002 -2.175 9.167 9.209 Gl 150.1B 3.7 7.26-1.74+2.42 -0.084 -1.063 4.980 6.043 Gl 153B 1.4 20.97-8.24+20.6 +0.017 >-0.25 <1.054 <1.014 Gl 155.1 2.93 7.67-2.05+3.10 +0.175 -0.355 1.851 1.237 Gl 162 2.35 11.61-3.48+5.86 +0.019 -1.115 5.621 5.380 Gl 173 2.35 10.27-2.91+4.91 -0.225 -0.430 1.661 2.788 Gl 191 9.15 1.55-0.26+0.33 -1.024 -0.164 0.867 9.163 Gl 205 2.73 11.92-2.94+4.59 +0.101 -1.555 7.453 5.907 Gl 212 1.98 14.85-4.55+8.51 +0.067 -1.872 10.03 8.596 Gl 218 1.66 15.49-5.21+11.1 -0.099 -0.643 2.767 3.475 Gl 229 2.63 11.22-2.80+4.46 +0.060 -1.124 5.845 5.091 Gl 250B - - -0.247 -0.400 1.294 2.285 Gl 275.1 2.7 13.87-4.14+6.51 +0.172 -0.16 0.807 0.543 Gl 289 <1 >18.37 -0.741 -0.16 0.693 3.817 Gl 330 3.05 9.62-2.57+3.82 +0.023 -0.302 1.447 1.372 Gl 361 1.95 12.54-3.90+7.37 -0.212 -0.926 3.476 5.663 Gl 366 <1 >27.69 -0.014 -0.284 1.331 1.375 Gl 378 2.25 17.30-5.20+8.99 +0.096 -0.755 2.524 2.023 Gl 382 2.9 9.49-2.37+3.60 -0.059 -1.398 5.563 6.372 Gl 390 2.46 10.78-2.97+4.89 -0.030 -1.231 5.767 6.179 Gl 411 0.61 32.77-11.9+30.8 -0.442 -0.199 0.792 2.191 Gl 412A 1.6 11.55-3.82+8.40 -0.435 -0.180 0.907 2.469 Gl 430.1 0.46 65.7-28.3+108 +0.089 -0.43 2.261 1.842 Gl 433 <1 >23.54 -0.199 -0.439 1.850 2.925 Gl 450 2.47 9.14-2.41+3.96 -0.247 -1.054 4.690 8.283 Gl 477 1.95 14.79-5.07+9.57 -0.057 -1.063 4.168 4.753 Gl 490A 8.4 3.87-0.91+1.15 +0.078 -6.03 29.55 24.69 Gl 494AB 9.75 2.17-0.39+0.47 -0.339 -7.222 33.84 73.86 Gl 507.1 2.32 13.39-3.83+6.49 +0.065 -1.036 4.367 3.760 Gl 508.2 1.78 15.01-5.01+10.1 -0.002 - - - Gl 510 2.49 10.33-3.02+4.93 -0.162 -1.77 7.252 10.53 Gl 514 2.07 12.62-3.63+6.59 -0.030 -0.788 3.972 4.256 Gl 521 0.85 30.84-9.69+18.7 -0.017 -0.505 2.420 2.517 Gl 526 1.00 25.41-6.99+12.1 -0.086 -0.498 2.275 2.773 Gl 536 <1 >26.27 -0.032 -0.689 3.228 3.475 Gl 537AB 3.6 7.11-1.82+2.55 -0.046 - - - Gl 540 2.09 14.97-4.52+8.16 +0.073 -0.915 4.613 3.899 Gl 552 0.58 45.55-17.9+49.1 -0.100 -0.734 2.756 3.470 Gl 563.2A 2.24 2.06-0.72+1.24 -1.742 -0.355 1.621 89.49 Gl 563.2B 1.82 2.42-0.93+1.85 -1.754 -0.33 1.274 72.31 Gl 618.4 1.45 17.56-7.21+17.4 -0.119 -0.515 2.191 2.822 Gl 634 <0.6 >37.12 -0.303 -0.345 1.373 2.758 Gl 637 2.19 8.94-2.68+4.71 -0.415 -0.33 1.569 4.080 Gl 645 2.80 7.75-2.64+4.09 -0.257 -0.91 4.174 7.543 Gl 649 2.27 11.77-3.29+5.65 -0.020 -1.036 5.077 5.316 Gl 654AB 5.7 4.40-1.02+1.46 -0.090 -0.262 1.073 1.320 Gl 672.1 2.99 8.70-2.75+4.13 -0.086 - - - Gl 686 2.49 8.94-2.35+3.85 -0.245 -0.412 1.930 3.393 Gl 701 1.6 14.65-4.91+10.8 -0.169 -0.592 2.837 4.187 Gl 724 2.97 9.17-2.37+3.58 -0.005 -1.21 5.930 5.999 Gl 737B - - +0.134 -1.79 8.685 6.379 Gl 745A 3.0 5.03-1.25+1.89 -1.03 -0.136 0.525 5.625 Gl 745B 2.8 5.55-1.44+2.22 -1.07 -0.109 0.421 4.946 Gl 767A 2.9 12.30-3.29+5.01 +0.126 -1.08 4.679 3.501 Gl 781 12.7 1.08-0.22+0.25 -1.10 -2.136 11.11 139.87 Gl 800A 2.34 12.33-3.68+6.22 -0.113 -0.738 3.537 4.588 Gl 803 9.68 4.03-0.68+0.84 +0.154 -8.364 40.089 28.12 Gl 806 0.46 47.86-20.9+80.5 -0.336 -0.624 2.410 5.224 Gl 808 1.27 14.94-7.23+20.1 -0.882 -0.16 0.637 4.854 Gl 809 2.66 11.04-2.76+4.36 +0.067 -0.956 4.971 4.260 Gl 815AB 7.61 3.35-0.73+0.95 -0.167 -8.38 33.35 48.99 Gl 821 2.63 7.10-1.89+3.02 -0.484 -0.185 0.867 2.643 Gl 832 - - -0.268 -0.569 2.197 4.072 Gl 842 2.80 10.05-2.56+3.96 +0.050 -1.15 6.362 5.670 Gl 855 2.99 9.97-2.61+3.92 +0.049 -1.275 6.833 6.104 Gl 863 2.64 8.90-2.44+3.88 -0.132 -0.699 3.524 4.776 Gl 867A 7.02 4.81-0.98+1.31 +0.080 -7.272 28.08 23.36 Gl 880 2.07 14.25-4.09+7.43 +0.040 -1.165 5.321 4.853 Gl 887 - - -0.092 -0.670 3.284 4.059 Gl 895 0.52 56.16-22.6+71.4 +0.049 -1.675 7.848 7.011 Gl 908 2.25 9.54-2.59+4.48 -0.271 -0.270 1.294 2.415 St 497 6.5 3.95-1.45+1.98 -0.027 -6.20 29.05 30.91 St 928 2.1 11.16-4.86+8.74 -0.176 -0.94 4.183 6.273 G192-11A 2.6 10.14-2.71+4.33 -0.009 -2.405 12.12 12.37 MCC 354A 2.47 12.99-3.82+6.27 +0.109 -1.80 7.163 5.573 MCC 452 - - +0.360 -0.96 4.992 2.179 MCC 488 3.2 10.60-3.03+4.43 +0.113 -1.87 9.427 7.267 LHS 1155 - - -0.88 3.399 - [1]Surface Flux corrected from metallicity effects |lcccccc| [ ]The values of vsin i, P/sin i and [M/H] from Paper I for our targets in our list of dM3 stars. We also give the Ca ii EWs, surface fluxes F_HK and the surface fluxes corrected for the metallicity effects. Star vsin i P/sin i [M/H] Ca ii EW F_HK F_HK[1] (km/s) (days) (dex) (Å) 10^4erg s^-1 cm^-2 10^4erg s^-1 cm^-2 continued. Star vsin i P/sin i [M/H] Ca ii EW F_HK F_HK[1] (km/s) (days) (dex) (Å) 10^4erg s^-1 cm^-2 10^4erg s^-1 cm^-2 GJ 1046 2.63 7.41-1.1+1.2 -0.078 -0.420 1.276 1.527 GJ 1050 0.59 34.4-18+310 -0.035 -0.113 0.324 0.351 GJ 1054B 32.13 0.45-0.07+0.08 -0.323 - - - GJ 1097 <0.5 >41.91 -0.033 -0.469 1.178 1.271 GJ 1125 0.99 10.3-2.9+8.3 -0.691 -0.319 1.454 7.138 GJ 1203 0.82 23.5-7.1+18 -0.087 -0.283 0.772 0.943 GJ 1212A 0.69 25.3-9.6+57 -0.170 - - - GJ 1212B 0.90 19.3-7.1+13 -0.170 - - - GJ 1271 1.39 20.0-5.1+7.7 +0.127 -0.76 2.265 1.691 GJ 2121 2.06 12.5-3.1+4.2 +0.059 - - - GJ 3139 0.90 29.4-7.8+14 +0.084 -0.560 1.577 1.300 GJ 3160A 0.95 22.0-6.2+9.9 -0.033 - - - GJ 3160B 1.06 19.8-5.3+7.4 -0.033 - - - GJ 3189 1.42 6.43-1.8+1.6 -0.801 -0.096 0.278 1.758 GJ 3279 0.85 23.7-13+26 -0.045 -0.496 1.417 1.572 GJ 3293 0.85 25.0-12+25 -0.031 -0.780 2.287 2.456 GJ 3404A 0.90 26.5-9.8+17 -0.005 -0.320 0.782 0.791 GJ 3412A - - -0.280 -0.36 0.914 1.742 GJ 3412B - - -0.280 -0.36 0.914 1.742 GJ 3459 0.69 23.1-7.7+48 -0.238 -0.476 1.215 2.102 GJ 3528 0.90 28.4-8.1+15 +0.057 -0.768 2.283 2.002 GJ 3563 0.85 22.2-6.3+14 -0.104 -0.354 1.033 1.313 GJ 3598 0.69 30.0-15+80 -0.035 - - - GJ 3634 0.85 24.6-11+22 -0.033 -0.584 1.780 1.921 GJ 3643 1.00 20.4-5.7+9.0 -0.036 - - - GJ 3708A 0.50 37.0-17+∞ -0.122 -0.406 1.064 1.409 GJ 3846 0.82 20.5-6.4+16 -0.189 -0.490 1.230 1.901 GJ 3892 0.82 27.4-7.6+20 -0.026 -0.521 1.453 1.543 GJ 3916A 1.24 14.1-3.1+3.8 -0.168 -0.388 1.143 1.683 GJ 3916B 3.51 4.97-0.7+0.7 -0.168 -0.388 1.143 1.683 GJ 4004 0.59 28.6-18+290 -0.186 - - - GJ 4129 0.50 37.9-18+∞ -0.104 - - - GJ 4231 80.00 0.24-0.08+0.09 -0.080 -13.05 33.48 40.25 GJ 4282AB 15.00 1.25-0.48+0.54 -0.115 -9.115 22.82 29.74 Gl 12 0.95 12.8-4.4+6.9 -0.508 -0.233 0.564 1.817 GL 48 2.45 10.7-2.9+4.9 +0.071 - - - Gl 70 1.01 20.8-5.4+10 -0.033 -0.701 2.233 2.409 GL 84 - - -0.094 -0.838 2.555 - Gl 109 1.34 13.9-3.2+4.8 -0.113 -0.655 1.716 2.226 Gl 119B 4.00 7.72-2.2+3.0 +0.217 -1.840 5.358 3.251 Gl 140C 6.70 2.99-1.0+1.3 -0.052 -1.36 4.173 4.704 Gl 145 0.85 19.6-5.1+12 -0.194 -0.805 2.246 3.511 Gl 163 0.85 26.0-6.8+16 -0.027 -0.400 0.966 1.028 Gl 204.2 0.69 36.1-14+83 +0.033 -0.559 1.552 1.438 Gl 207.1 9.52 2.51-0.4+0.4 +0.006 -9.638 27.08 26.71 Gl 226 1.12 19.6-5.1+9.0 -0.028 -0.54 1.583 1.688 Gl 238 0.85 28.0-7.3+17 +0.005 -0.395 1.118 1.105 Gl 251AB 1.14 16.9-4.4+7.5 -0.388 -0.315 0.746 1.823 GL 277A 8.00 3.69-0.86+1.1 +0.175 -4.958 8.13 8.682 GL 298 1.00 24.6-5.7+9.2 +0.023 -0.559 1.407 1.334 GL 352 0.90 7.6-1.0+1.0 +0.178 -0.294 0.824 0.547 Gl 357 1.21 18.7-4.3+8.2 -0.189 -0.202 0.634 0.980 Gl 358 1.10 17.7-3.3+4.2 -0.030 -2.585 7.018 7.520 Gl 377 - 24.1-5.4+7.2 +0.084 -1.032 2.760 2.275 Gl 386 0.69 37.8-12+79 +0.068 -0.540 1.532 1.310 Gl 388 2.63 8.38-1.1+1.2 -0.028 -8.255 19.77 21.09 Gl 399 0.59 43.6-17+330 +0.064 -0.486 1.524 1.315 Gl 408 1.12 18.1-4.6+8.1 -0.045 -0.655 1.832 2.032 Gl 422 0.90 23.1-5.8+11 -0.033 -0.229 0.580 0.626 Gl 436 0.86 27.0-9.0+26 -0.019 -0.345 0.944 0.986 Gl 443 0.90 31.8-8.7+16 +0.150 -0.996 2.684 1.900 Gl 452A 0.95 27.1-7.3+12 +0.062 -0.554 1.485 1.287 Gl 452.1 1.60 6.66-1.9+2.2 -0.651 -3.607 8.541 38.24 Gl 463 1.17 23.1-6.4+10 +0.101 -0.615 1.720 1.363 Gl 479 1.06 21.1-4.4+6.3 -0.026 -1.728 4.679 4.968 Gl 480 0.73 34.0-12+53 +0.033 -0.799 1.897 1.758 Gl 513 0.77 29.0-11+39 -0.027 -0.395 0.965 1.027 GL 581 - - -0.215 -0.207 0.508 0.833 Gl 588 0.77 31.8-8.3+35 +0.022 -0.648 1.847 1.756 Gl 595AB 1.62 9.25-2.1+2.8 -0.293 - - - Gl 617B 1.12 22.6-6.0+11 +0.045 -1.07 2.803 2.527 Gl 618A 0.82 22.8-6.9+18 -0.115 -0.483 1.278 1.665 Gl 623 0.94 22.3-6.4+16 -0.034 -0.315 0.937 1.013 Gl 634 1.00 17.1-5.0+7.8 -0.182 -0.348 0.912 1.387 Gl 644A 1.79 12.2-2.0+2.2 -0.029 -2.783 7.366 7.875 Gl 644B 2.78 7.88-1.1+1.1 -0.029 -2.783 7.366 7.875 Gl 655 <0.5 >39.88 -0.057 -0.625 1.637 1.867 Gl 660A 0.82 22.6-8.3+20 -0.120 -0.477 1.130 1.490 Gl 671 0.94 21.2-6.3+15 -0.061 -0.38 1.083 1.246 Gl 674 0.90 20.0-4.5+8.6 -0.143 -1.244 3.521 4.894 Gl 687AB 0.73 30.3-11+130 -0.028 -0.47 1.151 1.228 Gl 693 0.82 16.8-4.3+12 -0.372 -0.419 1.024 2.412 Gl 694 1.29 17.7-4.2+6.2 -0.024 -0.74 2.155 2.277 Gl 725AB 0.85 22.5-7.0+18.4 -0.372 -0.28 0.714 1.682 Gl 735A 1.86 13.0-2.0+2.4 +0.012 -8.515 22.60 21.98 Gl 735B 1.83 13.2-2.1+3.1 +0.012 -5.492 14.57 14.17 Gl 739 3.19 7.59-1.0+1.0 +0.012 -0.712 2.122 2.064 Gl 752A 1.19 21.2-4.3+6.7 +0.045 -0.828 2.549 2.298 GL 781.1A 9.20 1.93-0.50+0.62 -0.157 - - - Gl 793 1.26 16.2-4.0+5.9 -0.036 -1.45 3.720 4.042 Gl 844 3.46 9.92-1.4+1.4 +0.324 -1.70 5.216 2.474 Gl 849 0.85 30.8-7.8+18 +0.076 -0.637 1.539 1.292 GL 856AB 11.00 2.36-0.63+0.76 +0.067 -14.42 36.11 30.95 GL 875.1 11.00 2.08-0.44+0.52 -0.024 -9.62 24.09 25.46 Gl 877 0.95 23.5-5.0+8.2 -0.026 -0.491 1.353 1.436 Gl 896A 14.03 1.41-0.14+0.14 -0.064 -10.74 25.43 29.47 Gl 897AB 7.10 3.18-0.97+1.29 -0.025 -7.250 21.06 22.31 [1]Surface Flux corrected for metallicity effects ||lccccccccc| [ ]Equivalent width (EW, in units of Å) of the Ca ii resonance doublet and H_α line for our sample of dM4 stars. Results were obtained using FEROS spectra as well as a compilation of data published in the literature. When the H_α EW is known but with no observations of the Ca ii EW, we derived the Ca ii EW according to the H_α EW-Ca ii EW correlation (see Fig. 8). FEROS Ca ii H Ca ii K H_α Mean Ca ii S_HK Mean Ca ii H_α Mean Ca ii Star No. of EW EW EW EW EW[1] EW EW meas. (Å) (Å) (Å) (Å) (Å) (Å) (Å) continued. FEROS Ca ii H Ca ii K H_α Mean Ca ii S_HK Mean Ca ii H_α Mean Ca ii Star No. of EW EW EW EW EW[1] EW EW meas. (Å) (Å) (Å) (Å) (Å) (Å) (Å) GJ 1001A 1 -0.20±0.04 -0.09±0.07 0.215±0.02 -0.235(2) - - 0.218(15) -0.235(2) GJ 1005 - - - - - - - 0.224 - GJ 1006A 2 -4.09±0.04 -4.72±0.04 -3.14±0.20 -4.405(2) - - -3.14(2) -4.405(2) GJ 1006B 2 -7.05±0.05 -8.55±0.05 -2.98±0.20 -7.80(2) - - -2.98(2) -7.80(2) GJ 1065 3 -0.26±0.08 -0.24±0.08 0.167±0.02 -0.25(3) - - 0.189(4) -0.25(3) GJ 1105 - - - - -0.62(1) - - 0.203(3) -0.62(1) GJ 1129 3 -0.302±0.03 -0.353±0.03 0.190±0.02 -0.328(3) - - 0.178(4) -0.328(3) GJ 1134 - - - - -0.095(1) - - 0.181(3) - GJ 1138 - - - - - - - 0.180(4) - GJ 1207 8 -6.529±0.08 -7.801±0.08 -2.55±0.20 -7.098(11) - - -2.830(11) -7.098(11) GJ 1254 - - - - - - - 0.133 - GJ 1289 - - - - -1.680(1) - - -0.432(18) -1.680(1) GJ 2036B - - - - - - -14.49(2) -8.135(2) -14.49(2) GJ 2069A - - - - -16.91(1) - - -4.548(5) -16.91(1) GJ 3149B - - - - - - -5.647(1) -3.50 -5.647(1) GJ 3283B - - - - - - -6.372(1) -3.88 -6.372(1) GJ 3322 - - - - - - -10.58(2) -6.082(2) -10.58(2) GJ 3522 - - - - -7.10(1) - - - -7.10(1) GJ 3631 - - - - - - -19.29(2) -10.65(2) -19.29(2) GJ 3666 - - - - -0.21(1) - - - -0.21(1) GJ 3707 2 -0.287±0.08 -0.365±0.08 0.200±0.02 -0.341(9) - - 0.226(9) -0.341(9) GJ 3789 - - - - - - -19.31(3) -10.66(3) -19.31(3) GJ 3801 - - - - - - - 0.212 - GJ 3804 4 -0.251±0.05 -0.366±0.05 0.193±0.02 -0.302(9) 0.648(4) -0.291(4) 0.211(5) -0.299(13) GJ 3873 - - - - -0.465(3) - - 0.217(4) -0.465(3) GJ 3900 1 -0.401±0.05 -0.504±0.05 0.221±0.02 -0.453(1) - - 0.240(3) -0.453(1) GJ 4020B - - - - - - -12.94(1) -7.32(1) -12.94(1) GJ 4063 - - - - -0.380(9) 1.235(14) -0.631(14) 0.232(2) -0.533(23) GJ 4333 2 -0.495±0.08 -0.572±0.08 0.179±0.02 -0.587(15) 1.060(4) -0.525(4) 0.180(5) -0.574(19) GJ 4338B - - - - - - -10.95(2) -6.277(2) -10.95(2) GJ 4378A 2 -2.674±0.1 -3.004±0.1 -1.486±0.20 -2.839(2) - - -1.808(3) -2.839(2) GJ 4378B 2 -7.509±0.1 -9.684±0.1 -4.434±0.20 -8.597(2) - - -4.434(2) -8.597(2) Gl 15B - - - - -0.964(14) 1.162(18) -0.888(18) 0.216(4) -0.921(32) Gl 46 2 -0.208±0.02 -0.321±0.02 0.231±0.02 -0.295(4) - - 0.241(4) -0.295(4) Gl 54.1 6 -8.814±0.2 -9.027±0.2 -2.099±0.20 -7.044(17) 4.816(4) -3.230(4) -2.010(8) -6.318(21) Gl 84.1B - - - - -0.625(7) - - 0.549(7) -0.625(7) Gl 105B 1 -0.278±0.05 -0.313±0.05 0.204±0.02 -0.319(4) 0.727(1) -0.372(1) 0.188(3) -0.330(5) Gl 166C 1 -5.206±0.2 -5.502±0.2 -2.999±0.20 -5.354(1) 6.588(2) -4.705(2) -3.227(8) -4.921(3) Gl 169.1A - - - - - - - 0.174(1) - Gl 179 2 -0.626±0.03 -0.759±0.03 0.228±0.02 -0.676(34) 1.592(4) -0.856(4) 0.230(21) -0.695(38) Gl 203 2 -0.119±0.05 -0.137±0.05 0.177±0.02 -0.121(5) - - 0.203(13) -0.121(5) Gl 206AB - - - - -8.55(1) - - -4.25(1) -8.55(1) Gl 206A 3 -7.348±0.2 -8.802±0.2 -3.630±0.20 -8.154(4) - - -3.585(4) -8.154(4) Gl 206B 3 -4.234±0.2 -5.438±0.2 -2.292±0.20 -4.922(4) - - -2.304(4) -4.922(4) Gl 213 - - - - -0.315(8) 0.670(4) -0.303(4) 0.215 -0.311(12) Gl 232 - - - - -0.36(4) - - 0.163(14) -0.36(4) Gl 234A - - - - -5.88(1) - -4.797(32) -3.055(32) -4.830(33) Gl 268AB - - - - -4.66(2) - - -2.152(16) -4.66(2) Gl 268A - - - - -5.95(1) - - -2.38(1) -5.95(1) Gl 268B - - - - -3.37(1) - - -2.34(1) -3.37(1) Gl 273 5 -0.357±0.03 -0.538±0.03 0.207±0.02 -0.380(21) 0.836(17 -0.497(17) 0.189(18) -0.432(38) Gl 277B - - - - -6.61(6) 6.149(2) -4.331(2) -1.977(7) -6.040(8) Gl 285 - - - - -16.05(12) 28.48(4) -27.26(4) -7.003(15) -16.04(12) Gl 299 1 - - 0.202±0.02 -0.375(4) - - 0.185(15) -0.375(4) Gl 300 1 -0.592±0.09 -0.784±0.09 0.143±0.02 -0.654(5) - - 0.167(2) -0.654(5) Gl 317 - - - - - 1.251(4) -0.641(4) 0.281 -0.641(4) Gl 319B 1 -0.181±0.08 -0.292±0.08 0.295±0.02 -0.236(1) - - 0.295(1) -0.236(1) Gl 324B - - - - -0.73(1) - - 0.249 -0.73(1) Gl 375AB - - - - - 2.560(2) -1.513(2) -4.75 -1.513(2) Gl 375A 1 -9.234±0.2 -8.184±0.2 -3.722±0.2 -8.709(1) - - -3.722 -8.709(1) Gl 375B 1 -7.352±0.2 -8.578±0.2 -3.746±0.2 -7.965(1) - - -3.746 -7.965(1) Gl 398 7 -8.122±0.2 -10.174±0.2 -3.617±0.2 -8.573(9) - - -3.617(7) -8.573(9) Gl 402 5 -0.451±0.05 -0.673±0.05 0.209±0.02 -0.699(15) 1.279(4) -0.658(4) 0.175(21) -0.690(19) Gl 431 1 -7.620±0.2 -8.782±0.2 -4.017±0.2 -8.790(3) - - -4.519(3) -8.790(3) Gl 445 - - - - -0.179(14) 0.566(4) -0.195(4) 0.247(13) -0.183(18) Gl 447 1 -0.541±0.05 -0.824±0.05 0.187±0.02 -0.452(13) 0.875(4) -0.417(4) 0.176(10) -0.444(17) Gl 469 - - - - -0.47(8) - - 0.189(7) -0.47(8) Gl 486 1 -0.248±0.05 -0.314±0.05 0.195±0.02 -0.237(18) 0.648(5) -0.284(5) 0.175(13) -0.242(20) Gl 487 - - - - -1.125 - - -0.251(54) -1.125 Gl 490B - - - - - - -7.35(1) -4.39(1) -7.35(1) Gl 512B 5 -0.415±0.03 -0.526±0.03 0.315±0.02 -0.470(5) - - 0.315(5) -0.470(5) Gl 520C - - - - - - -7.65(1) -4.55(1) -7.65(1) Gl 545 - - - - - - - 0.192(3) - Gl 553.1 3 -0.148±0.03 -0.204±0.03 0.186±0.02 -0.178(15) 0.501(4) -0.214(4) 0.174(11) -0.186(19) Gl 555 2 -0.384±0.03 -0.573±0.03 0.191±0.02 -0.390(14) 0.904(8) -0.576(8) 0.157(21) -0.458(22) Gl 568AB - - - - - - - -0.203(1) - Gl 592 6 -0.190±0.05 -0.234±0.05 0.212±0.02 -0.212(6) - - 0.212(6) -0.212(6) Gl 609 5 -0.269±0.09 -0.401±0.09 0.211±0.02 -0.401(8) - - 0.206(11) -0.401(8) Gl 630.1 - - - - - - -8.51(1) -5.00(1) -8.51(1) Gl 643AB 2 -0.178±0.05 -0.264±0.05 0.210±0.02 -0.197(3) - - 0.176(24) -0.197(3) Gl 644B 1 -0.213±0.1 -0.419±0.1 0.201±0.02 -0.316(1) - - 0.201(1) -0.316(1) Gl 669A 2 -8.560±0.2 -10.86±0.2 -5.887±0.2 -5.866(8) 4.431(2) -2.923(2) -2.696(6) -5.277(10) Gl 682 1 -0.399±0.08 -0.552±0.08 0.183±0.02 -0.476(1) - - 0.183(1) -0.476(1) Gl 695B - - - - - - - 0.283 - Gl 699 1 -0.445±0.03 -0.370±0.03 0.249±0.02 -0.411(15) 0.761(71) -0.353(71) 0.222(33) -0.363(86) Gl 720B - - - - -0.29 - - 0.253 -0.29 Gl 725B - - - - -0.305(15) 0.651(22) -0.292(22) 0.267 -0.298(37) Gl 729 6 -4.375±0.2 -5.110±0.2 -1.685±0.2 -4.743(6) 8.967(4) -6.811(4) -1.793(17) -5.570(10) Gl 732A 3 -3.120±0.2 -3.803±0.2 -1.141±0.2 -3.462(3) - - -1.141(3) -3.462(3) Gl 781.1B 1 -3.994±0.2 -4.938±0.2 -2.634±0.2 -4.466(1) - - -2.967(2) -4.466(1) Gl 791.2A - - - - - - -7.775(3) -4.615(3) -7.775(3) Gl 812A 4 -2.555±0.1 -3.073±0.1 -1.553±0.2 -2.814(4) - - -2.030(6) -2.814(4) Gl 860B - - - - - 3.304(2) -2.055(2) -2.098(3) -2.055(2) Gl 865AB 1 -7.308±0.2 -8.846±0.2 -2.688±0.2 -6.816(2) - - -2.369(2) -6.816(2) Gl 865A - - - - - - -4.88(1) -3.10(1) -4.88(1) Gl 865B - - - - - - -8.70(1) -5.10(1) -8.70(1) Gl 867B - - - - - - -7.722(2) -4.587(2) -7.722(2) Gl 873A - - - - -7.70(2) 12.525(4) -10.171(4) -3.784(17) -9.347(6) Gl 876A 5 -0.348±0.01 -0.711±0.01 0.200±0.02 -0.448(137) 1.022(119) -0.503(119) 0.197(52) -0.473(256) Gl 896AB - - - - - - -6.43(34) -3.91(34) -6.43(34) Gl 896B - - - - - 5.490(2) -9.156(9) -6.143(7) -9.156(9) G 97-52B - - - - - - - -0.126(1) - LHS 1723 - - - - - - -0.68(1) -0.9(1) -0.68(1) [1]Inferred from the H_α EW-Ca ii EW correlation |lccccccc| [ ]The values of vsin i, P/sin i and [M/H] compiled from Paper I for our targets in our list of dM4 stars. We also give the Ca ii EW (from the compilation of Table 6), surface fluxes F_HK and the surface fluxes corrected for the metallicity effects. Star vsin i P/sin i [M/H] [M/H][1] Ca ii EW F_HK F_HK[2] (km/s) (days) (dex) (dex) (Å) 10^4erg/s/cm^2 10^4erg/s/cm^2 continued. Star vsin i P/sin i [M/H] [M/H][1] Ca ii EW F_HK F_HK[2] (km/s) (days) (dex) (dex) (Å) 10^4erg/s/cm^2 10^4erg/s/cm^2 GJ 1001A 1.15 12.81-5.66+18.02 - +0.047 -0.235 0.377 0.338 GJ 1005AB 3.00 3.41-0.96+1.44 -0.47 -0.210 - - - GJ 1034 4.50 2.05-0.61+0.79 -0.38 -0.275 - - - GJ 1065 4.00 2.70-0.65+0.88 0.20 -0.172 -0.25 0.431 0.640 GJ 1105 2.00 6.93-2.28+4.24 0.24 +0.022 -0.62 1.068 0.615 GJ 1129 3.00 4.62-1.94+2.91 0.05 +0.022 -0.328 0.463 0.440 GJ 1134 4.10 3.11-0.74+1.00 -0.16 -0.036 - - - GJ 1207 10.70 1.18-0.21+0.25 -0.15 -0.042 -7.098 11.10 12.23 GJ 1254 4.00 5.39-1.41+1.90 0.22 +0.122 - - - GJ 1289 2.60 4.34-1.27+2.04 0.12 -0.137 -1.680 2.256 3.093 GJ 2036B 19.8 0.429-0.07+0.08 - -0.327 -14.49 19.66 19.66 GJ 2069A 6.43 3.20-0.98+1.34 0.27 +0.102 -16.91 24.53 13.17 GJ 3092B 3.80 4.87-1.24+1.70 - +0.064 - - - GJ 3149B 4.00 3.02-1.26+1.71 0.09 -0.078 -5.647 8.193 6.660 GJ 3283B 6.00 1.94-0.67+0.93 - -0.111 -6.372 8.101 10.46 GJ 3322 7.68 9.20-4.57+5.94 - - -10.58 16.30 16.30 GJ 3398 4.00 3.11-0.81+1.10 - -0.054 - - GJ 3631AB 19.10 0.49-0.16+0.18 -0.64 -0.268 -19.29 26.51 26.51 GJ 3707 2.0 7.36-3.53+6.55 0.35 +0.047 -0.341 0.531 0.477 GJ 3789 56.50 0.227-0.07+0.07 0.07 -0.033 -19.31 28.02 23.85 GJ 3801 4.00 3.67-0.88+1.20 0.14 +0.047 - - - GJ 3804 2.50 6.64-2.04+3.32 0.09 +0.055 -0.299 0.582 0.513 GJ 3873 4.00 6.90-2.89+3.91 0.11 +0.245 -0.465 0.693 0.538 GJ 3900 3.00 5.15-2.95+4.43 0.46 +0.050 -0.453 0.698 0.622 GJ 3907 2.50 5.39-1.91+3.11 0.06 +0.006 - - - GJ 4020B 16.16 1.10-0.24+0.27 - +0.060 -12.94 17.38 15.14 GJ 4030 4.50 3.90-1.14+1.49 -0.04 +0.059 - - - GJ 4049B 2.50 3.81-1.16+1.89 -0.13 -0.255 - - - GJ 4063AB 2.17 4.99-2.93+5.16 -0.54 -0.169 -0.533 0.918 3.183 GJ 4108 4.50 2.04-0.60+0.78 - -0.278 - GJ 4248 1.58 7.75-2.71+6.04 - -0.066 - - - GJ 4333 2.50 8.64-2.51+4.09 0.25 +0.123 -0.574 0.871 0.656 GJ 4338B 14.50 1.53-0.29+0.33 - +0.134 -10.95 15.05 11.05 GJ 4378A 4.00 4.88-1.69+2.29 - +0.081 -2.839 4.501 3.735 Gl 15B 1.90 5.01-1.57+3.01 -0.95 -0.255 -0.921 1.390 2.500 Gl 46 1.29 16.32-6.60+18.1 0.15 +0.112 -0.295 0.554 0.392 Gl 54.1 2.50 3.00-0.89+1.45 -0.33 -0.402 -6.318 7.039 15.05 Gl 84.1B 2.00 12.45-4.27+7.94 -0.18 +0.190 -0.625 1.217 0.786 Gl 105B 2.45 6.61-1.95+3.21 -0.09 +0.053 -0.330 0.465 0.572 Gl 166C 2.90 4.47-1.17+1.78 -0.10 -0.024 -4.921 6.607 6.982 GL 169.1A 1.94 7.80-2.44+4.62 0.29 +0.049 - - - GL 179 2.50 7.53-2.35+3.84 0.22 +0.065 -0.695 1.372 0.827 GL 203 4.00 2.67-0.68+0.92 -0.23 -0.179 -0.121 0.239 0.406 Gl 206A 6.37 1.95-0.62+0.84 0.19 -0.054 -8.154 15.21 17.22 Gl 206B 6.11 2.04-0.65+0.90 0.19 -0.054 -4.922 9.179 10.39 GL 213 2.70 4.76-1.29+2.03 -0.16 -0.030 -0.311 0.439 0.635 GL 232 3.10 3.05-0.80+1.18 -0.25 -0.259 -0.36 0.484 0.879 GL 234A 4.73 2.72-0.57+0.74 - -0.030 -4.830 5.490 5.883 GL 268A 10.65 1.57-0.32+0.38 0.16 +0.056 -5.95 6.736 5.921 GL 268B 10.65 1.57-0.32+0.38 0.16 +0.056 -3.37 3.815 3.353 GL 273 2.36 6.99-1.91+3.22 -0.08 +0.055 -0.432 0.750 0.902 Gl 277B 6.40 3.39-0.87+1.19 0.10 +0.125 -6.040 9.867 7.399 GL 285 6.16 2.95-0.71+0.98 0.24 +0.062 -16.04 18.45 10.62 GL 299 2.96 3.30-0.90+1.35 -0.43 -0.239 -0.375 0.416 1.120 GL 300 3.00 5.16-1.34+2.01 0.20 +0.050 -0.654 0.781 0.696 GL 317 2.50 9.37-2.56+4.18 -0.10 +0.160 -0.641 1.248 0.863 GL 319B 3.70 7.17-1.82+2.53 0.04 +0.223 -0.236 0.422 0.385 GL 324B 2.36 6.11-1.90+3.20 0.42 +0.039 -0.73 0.961 0.878 GL 375A 10.0 2.35-0.48+0.58 - +0.161 -8.709 16.71 11.53 GL 375B 10.0 2.35-0.48+0.58 - +0.161 -7.965 15.29 10.55 GL 402 2.40 5.74-1.65+2.75 0.12 +0.019 -0.690 1.035 0.991 Gl 431 20.36 0.960-0.15+0.17 - +0.081 -8.790 17.47 14.50 Gl 445 2.25 6.48-1.90+3.28 -0.30 +0.044 -0.183 0.332 0.662 Gl 447 2.20 4.51-1.33+2.32 -0.04 -0.230 -0.444 0.535 0.909 GL 458BC 5.40 1.74-0.45+0.65 -0.53 -0.262 - - - Gl 469 1.58 12.81-4.94+11.0 0.14 +0.095 -0.47 0.854 0.619 GL 486 2.25 7.99-2.37+4.10 -0.05 +0.061 -0.242 0.413 0.463 Gl 487A 1.65 8.74-3.05+6.54 -0.04 +0.039 -1.125 2.335 2.134 Gl 487B 1.22 11.87-4.79+14.05 -0.04 +0.039 -1.125 2.335 2.134 Gl 487C 2.60 5.55-1.55+2.47 -0.04 +0.039 -1.125 2.335 2.134 GL 490B 8.60 2.27-0.56+0.71 0.07 +0.081 -7.35 10.66 9.073 GL 512B 4.00 2.63-0.94+1.27 0.12 -0.190 -0.470 0.602 0.932 GL 520C 8.20 1.50-0.35+0.45 -0.14 -0.063 -7.65 11.44 15.79 GL 544B 4.66 2.44-0.80+1.03 -0.13 -0.129 - - - GL 545 4.00 3.07-1.08+1.46 -0.12 -0.063 - - - Gl 553.1 2.50 6.68-2.04+3.33 0.26 +0.056 -0.186 .0.371 0.326 GL 555 2.60 5.74-1.58+2.53 0.20 +0.048 -0.458 0.644 0.577 GL 568AB 4.00 4.78-1.18+1.60 0.45 +0.073 - - - GL 592 3.00 5.36-1.53+2.29 0.14 +0.053 -0.212 0.336 0.297 GL 609 3.00 4.57-1.25+1.88 -0.06 +0.017 -0.401 0.582 0.668 GL 630.1AB 27.50 0.484-0.09+0.10 -0.34 -0.003 -8.51 10.09 22.07 GL 643AB 2.73 4.45-1.26+1.97 -0.21 -0.074 -0.197 0.326 0.529 GL 644B 2.73 4.10-1.18+1.84 -0.21 -0.144 -0.316 0.555 0.900 Gl 669A 6.30 3.84-1.01+1.39 0.08 +0.176 -5.277 9.231 6.155 GL 682 3.00 4.80-1.22+1.83 0.11 +0.039 -0.476 0.701 0.641 GL 695B 7.33 3.03-0.72+0.95 - +0.136 - Gl 699 2.02 4.99-1.49+2.75 -0.40 -0.220 -0.363 0.532 0.883 GL 720B 2.30 7.02-2.06+3.52 -0.16 +0.053 -0.29 0.478 0.691 GL 725B 5.09 4.33-1.42+2.12 -0.57 -0.155 -0.298 0.561 2.084 GL 729 4.72 2.11-0.41+0.53 -0.14 -0.227 -5.570 9.099 12.56 GL 732A - - -0.01 +0.056 -3.462 5.489 5.617 GL 781.1B 9.20 2.17-0.56+0.70 -0.07 +0.090 -4.466 6.700 7.872 GL 783.2B 1.76 8.14-3.48+7.09 -0.15 +0.036 - - - GL 791.2A 31.90 0.351-0.05+0.05 0.06 -0.144 -7.775 9.500 13.23 GL 812A 10.00 2.95-0.71+0.87 0.12 +0.284 -2.814 4.922 3.734 GL 860B 4.05 3.25-0.71+0.96 -0.13 -0.012 -2.055 4.084 4.198 Gl 865A 7.09 1.90-0.53+0.71 -0.10 +0.006 -8.70 15.22 15.01 Gl 865B 6.20 2.17-0.64+0.88 -0.10 +0.006 -4.88 8.536 8.419 Gl 867B 7.01 2.48-0.54+0.72 0.15 +0.059 -7.722 14.40 10.19 GL 873A 5.99 3.02-0.70+0.98 -0.05 +0.062 -9.347 16.73 14.50 GL 876A 2.57 7.09-1.95+3.13 - +0.062 -0.473 0.779 0.675 GL 896A 15.81 1.14-0.18+0.21 0.10 +0.062 -9.470 13.02 11.29 GL 896B 15.81 1.14-0.18+0.21 0.10 +0.062 -9.156 12.59 10.92 LHS 2795 6.40 (2.38) - - - - - LHS 3279 30.0 (0.508) - - - - - G 97-52B 4.00 (3.81) - - - - - [1]Metalicity from the radius-metallicity relation [2]Surface Flux corrected from metallicity effects |lcccccccc| [ ]In this table, we list log(L_HK), log(L_bol), R'_HK and R_0 our samples of dK4, dK6, dM2, dM3 and dM4 stars. Star log(L_HK) log(L_HK)[1] log(L_bol) R'_HK R'_HK[1] log(L_X) log(L_X/L_bol) R_0 Name (ergs s^-1) (ergs s^-1) (ergs s^-1) (ergs s^-1) continued. Star log(L_HK) log(L_HK)[1] log(L_bol) R'_HK R'_HK[1] log(L_X) log(L_X/L_bol) R_0 Name (ergs s^-1) (ergs s^-1) (ergs s^-1) (ergs s^-1) K4 GJ 4061 28.062 - 32.840 -4.778 - - - 0.1801 GJ 4322B 27.739 - 32.993 -5.254 - 28.164 -4.829 0.7267 Gl 106 28.338 - 32.999 -4.661 - 28.384 -4.615 0.4069 Gl 131 28.322 - 32.661 -4.338 - - - 0.6554 Gl 160.2 27.948 - 32.638 -4.690 - 27.342 -5.296 0.5172 Gl 413 27.659 - 32.705 -5.046 - - - 1.0236 Gl 416 28.354 - 32.891 -4.537 - - - 0.7809 Gl 517 28.876 - 32.589 -3.713 - 29.554 -3.035 0.0252 Gl 570A 27.940 - 32.971 -5.031 - 27.544 -5.427 0.7265 Gl 664 28.201 - 32.777 -4.576 - 27.748 -5.029 0.2794 Gl 698A 28.398 - 32.819 -4.421 - 28.041 -4.778 0.? Gl 707 28.354 - 32.665 -4.312 - - - 0.3142 Gl 719AB 29.338 - 32.792 -3.454 - 29.808 -2.984 0.0833 Gl 727 28.441 - 33.031 -4.590 - 28.320 -4.711 0.4797 Gl 775 28.362 - 32.996 -4.634 - 27.813 -5.183 0.6563 Gl 818 28.280 - 32.829 -4.549 - - - 0.6411 Gl 820A 28.135 - 32.780 -4.645 - 27.447 -5.333 0.6712 Gl 879 28.391 - 32.910 -4.519 - 28.326 -4.584 0.2816 Gl 898 28.502 - 32.642 -4.140 - 28.107 -4.535 0.1517 MCC 266 28.310 - 32.963 -4.653 - - - 0.6229 MCC 522 28.966 - 32.960 -3.994 - - - 0.1131 K6 GJ 1056 28.072 - 32.770 -4.698 - - - 0.278 GJ 1066 28.252 - 32.757 -4.505 - 28.217 -4.540 0.244 GJ 1067 28.232 - 32.751 -4.519 - - - 0.223 GJ 1177A 28.262 - 32.575 -4.313 - 28.312 -4.263 0.049 GJ 1267 27.895 - 32.555 -4.660 - - - 0.155 GJ 1279 28.227 - 32.795 -4.569 - 27.903 -4.892 0.256 GJ 3072 27.775 - 32.487 -4.712 - - - 0.267 GJ 3411 28.144 - 32.644 -4.500 - - - 0.0767 GJ 3494 28.188 - 32.705 -4.518 - - - 0.186 GJ 3551 27.390 - 32.745 -5.355 - - - 0.452 GJ 3996 28.144 - 32.638 -4.493 - - - 0.250 GJ 4140 27.792 - 32.447 -4.655 - - - 0.208 GJ 9299 28.039 - 32.699 -4.660 - - - 0.260 GJ 9714 28.084 - 32.712 -4.628 - - - 0.174 GJ 9827 27.760 - 32.626 -4.867 - - - 0.260 Gl 14 28.129 - 32.653 -4.524 - 27.919 -4.734 0.137 Gl 40A 27.995 - 32.625 -4.630 - - - 0.206 Gl 45 27.874 - 32.600 -4.727 - - - 0.240 Gl 50 27.563 - 32.641 -5.078 - - - 0.317 Gl 52 27.567 - 32.595 -5.028 - - - 0.154 Gl 105.5 27.764 - 32.728 -4.964 - - - 0.190 Gl 112.1 28.361 - 32.767 -4.405 - - - 0.0955 Gl 116 27.177 - 32.310 -5.134 - - - 0.322 Gl 142 28.151 - 32.689 -4.538 - 27.996 -4.693 0.197 Gl 143.1 28.046 - 32.579 -4.533 - 27.826 -4.753 0.264 Gl 146 27.885 - 32.652 -4.767 - 27.415 -5.237 0.271 Gl 153A 27.652 - 32.565 -4.913 - 28.049 -4.516 0.109 Gl 156 27.917 - 32.620 -4.703 - - - 0.394 Gl 162.2 28.209 - 32.744 -4.535 - - - 0.144 Gl 182 29.126 - 32.625 -3.498 - 29.604 -3.204 0.0445 Gl 186A 27.523 - 32.619 -5.097 - - - 0.192 Gl 186B 27.523 - 32.619 -5.097 - - - 0.156 Gl 208 28.340 - 32.455 -4.115 - 28.396 -4.059 0.0857 Gl 221 27.730 - 32.576 -4.846 - 0.569 Gl 256 28.389 - 32.787 -4.398 - 29.325 -3.462 0.0676 Gl 322 28.057 - 32.577 -4.520 - 28.238 -4.339 0.193 Gl 389.1 27.670 - 32.682 -5.011 - - - 0.217 Gl 397 28.058 - 32.698 -4.640 - 28.053 -4.645 0.245 Gl 412.3A 27.710 - 32.668 -4.959 - - - 0.151 Gl 414A 27.926 - 32.647 -4.721 - 27.505 -5.142 0.274 Gl 421B 27.399 - 32.280 -4.882 - - - 0.143 Gl 425B 28.493 - 32.859 -4.366 - - - 0.239 Gl 455.1 28.640 - 32.520 -3.880 - 28.853 -3.667 0.0772 Gl 496.1 27.824 - 32.649 -4.825 - - - 0.412 Gl 509.1 28.278 - 32.744 -4.466 - 28.981 -3.763 0.159 Gl 517 28.678 - 32.695 -4.016 - 29.554 -3.141 0.0326 Gl 522 27.839 - 32.679 -4.841 - - - 0.295 Gl 524.1 27.848 - 32.443 -4.596 - - - 0.177 Gl 542.2 28.375 - 32.846 -4.471 - 28.049 -4.797 0.223 Gl 546 28.113 - 32.688 -4.576 - 28.000 -4.688 0.175 Gl 558 27.847 - 32.615 -4.768 - - - 0.221 Gl 562 27.924 - 32.490 -4.566 - - - 0.123 Gl 571 27.919 - 32.542 -4.623 - - - 0.196 Gl 571.1 28.044 - 32.678 -4.634 - - - 0.549 Gl 583 28.006 - 32.591 -4.585 - - - 0.176 Gl 619 27.730 - 32.640 -4.909 - - - 0.195 Gl 629.1 27.952 - 32.662 -4.710 - - - 0.245 Gl 659B 28.251 - 32.644 -4.393 - 28.190 -4.454 0.0913 Gl 673 28.122 - 32.609 -4.488 - 27.613 -4.996 0.121 Gl 707 28.005 - 32.680 -4.675 - - - 0.184 Gl 710 27.899 - 32.564 -4.665 - 28.130 -4.434 0.170 Gl 726 27.930 - 32.593 -4.664 - 27.380 -5.213 0.191 Gl 728 27.691 - 32.663 -4.972 - - - 0.512 Gl 747.3A 27.877 - 32.718 -4.841 - - - 0.208 Gl 757 27.647 - 32.445 -4.798 - - - 0.308 Gl 763 27.861 - 32.466 -4.605 - - - 0.132 Gl 773 28.048 - 32.662 -4.613 - - - 0.0597 Gl 782 28.048 - 32.634 -4.587 - - - 0.167 Gl 786.1 27.913 - 32.608 -4.695 - - - 0.274 Gl 795A 27.577 - 32.520 -4.942 - 27.881 -4.639 - Gl 795B 27.577 - 32.520 -4.942 - 27.881 -4.639 - Gl 798 27.555 - 32.481 -4.926 - - - 0.206 Gl 801 27.583 - 32.654 -5.070 - - - 0.228 Gl 818 27.987 - 32.843 -4.856 - - - 0.332 Gl 820B 27.484 - 31.897 -4.413 - 27.146 -4.751 0.129 Gl 826.1 28.133 - 32.626 -4.494 - - - 0.166 Gl 847A 27.687 - 32.421 -4.734 - - - 0.293 Gl 857.1 28.281 - 32.987 -4.706 - 28.164 -4.823 0.175 Gl 884 27.963 - 32.540 -4.577 - - - 0.223 Gl 885A 28.233 - 32.414 -4.191 - - - 0.0320 Gl 889A 28.100 - 32.631 -4.531 - - - 0.180 Gl 891 27.548 - 32.422 -4.873 - - - 0.110 Gl 894 28.054 - 32.703 -4.649 - - - 0.186 Gl 895.3 28.099 - 32.590 -4.491 - - - 0.125 Gl 898 28.274 - 32.737 -4.461 - 28.107 -4.630 0.158 Gl 900 28.531 - 32.643 -4.112 - 29.037 -3.606 0.0338 Gl 906 27.650 - 32.466 -4.815 - - - 0.839 Gl 907.1 28.974 - 32.911 -3.937 - - - 0.334 HIP 14593A 27.904 - 32.800 -4.896 - - - 0.461 HIP 19410A 27.830 - 32.550 -4.720 - - - 0.0845 HIP 21865 27.842 - 32.754 -4.911 - - - 0.238 HIP 40170 28.121 - 32.570 -4.449 - - - 0.537 HIP 42108A 26.807 - 32.506 -5.698 - - - 0.132 HIP 42108B 26.807 - 32.506 -5.698 - - - 0.137 HIP 42910 28.206 - 32.902 -4.695 - - - 0.376 HIP 50773 27.916 - 32.549 -4.633 - - - 0.319 HIP 51073 27.762 - 32.696 -4.933 - - - 0.247 HIP 53175 27.973 - 32.783 -4.810 - - - 0.251 HIP 56838 27.729 - 32.765 -5.036 - - - 0.345 HIP 58945 28.060 - 32.643 -4.584 - - - 0.173 HIP 60438A 28.613 - 32.559 -3.946 - - - 0.0284 HIP 60438B 28.291 - 32.559 -4.268 - - - 0.0284 HIP 72044 27.859 - 32.661 -4.802 - - - 0.209 HIP 76550 27.057 - 32.304 -5.248 - - - 0.162 HIP 78395 27.958 - 32.652 -4.694 - - - 0.249 HIP 80083 27.659 - 32.621 -4.961 - - - 0.167 HIP 103150 27.560 - 32.498 -4.937 - - - 0.291 HIP 110245 28.345 - 32.896 -4.551 - - - 0.229 HIP 110714 27.978 - 32.635 -4.657 - - - 0.335 HIP 113597 28.781 - 32.955 -4.174 - - - 0.174 M2 GJ 1114 26.057 26.396 32.046 -5.989 -5.650 - - 0.0357 GJ 1264 28.284 27.906 32.676 -4.392 -4.770 29.30 -3.41 0.146 GJ 2085 27.135 27.135 32.260 -5.125 -5.125 - - 0.0951 GJ 3084 27.294 27.271 32.385 -5.091 -5.114 - - 0.111 GJ 3098 27.038 27.238 32.135 -5.097 -4.897 - - 0.0649 GJ 3340 27.068 27.116 32.168 -5.100 -5.052 - - 0.0691 GJ 3759 26.959 27.080 32.182 -5.223 -5.102 - - 0.0720 GJ 3778 26.832 26.830 32.108 -5.277 -5.279 - - 0.105 GJ 9381 27.241 27.141 32.344 -5.103 -5.203 - - 0.159 GL 2 27.326 27.344 32.248 -4.923 -4.905 - - 0.0873 GL 15A 26.408 26.794 31.976 -5.568 -5.182 27.279 -4.71 0.0739 GL 16 27.188 27.293 32.210 -5.022 -4.917 - - 0.0971 GL 27.1 27.251 27.267 32.243 -4.992 -4.976 - - 0.0868 GL 29.1A 28.361 28.260 32.431 -4.071 -4.172 29.396 -3.03 0.0430 GL 29.1B 28.367 28.266 32.431 -4.065 -4.166 29.396 -3.03 0.0479 GL 49 27.507 27.481 32.308 -4.802 -4.828 - - 0.124 Gl 63 26.199 26.508 32.033 -5.834 -5.525 - - 0.121 GL 87 26.554 26.581 32.126 -5.572 -5.545 - - 0.0426 GL 91 27.121 27.211 32.215 -5.094 -5.004 - - 0.139 Gl 114.1A 26.675 26.917 32.090 -5.415 -5.173 - - 0.0895 Gl 130 26.297 27.063 31.842 -5.545 -4.779 - - 0.0509 Gl 133 26.331 26.797 31.982 -5.652 -5.186 - - 0.118 Gl 134 27.632 27.504 32.474 -4.842 -4.970 - - 0.207 Gl 140A 27.519 27.521 32.246 -4.727 -4.725 29.253 -3.32 0.0273 Gl 150.1B 27.233 27.317 32.247 -5.014 -4.930 - - 0.0651 Gl 155.1 26.648 26.473 32.113 -5.466 -5.641 - - 0.0510 Gl 162 27.298 27.279 32.275 -4.976 -4.995 - - 0.107 Gl 173 26.663 26.888 32.121 -5.458 -5.233 - - 0.0792 Gl 191 25.921 26.945 31.718 -5.798 -4.774 - - 0.00309 Gl 205 27.574 27.473 32.418 -4.844 -4.945 27.591 -4.83 0.154 Gl 212 27.615 27.548 32.352 -4.737 -4.804 28.569 -3.78 0.155 Gl 218 26.939 27.038 32.190 -5.251 -5.152 - - 0.130 Gl 229 27.384 27.324 32.348 -4.965 -5.025 27.114 -5.23 0.118 Gl 275.1 26.731 26.559 32.551 -5.820 -5.992 - - 0.235 Gl 330 26.773 26.750 32.324 -5.551 -5.574 - - 0.100 Gl 361 26.994 27.206 32.127 -5.132 -4.920 27.556 -4.58 0.0982 Gl 366 26.685 26.699 32.273 -5.587 -5.573 - - 0.261 Gl 378 27.259 27.163 32.510 -5.250 -5.346 - - 0.313 Gl 382 27.302 27.361 32.240 -4.938 -4.879 27.36 -4.87 0.0885 Gl 390 27.285 27.315 32.235 -4.950 -4.920 27.15 -5.09 0.0946 Gl 411 26.177 26.619 31.963 -5.785 -5.343 26.78 -5.19 0.177 Gl 412A 26.168 26.603 31.938 -5.770 -5.335 27.40 -4.53 0.0519 Gl 430.1 26.992 26.903 32.370 -5.379 -5.468 - - 0.722 Gl 433 26.688 26.887 32.108 -5.420 -5.221 - - 0.174 Gl 450 27.055 27.302 32.084 -5.029 -4.782 27.64 -4.44 0.0614 Gl 477 27.217 27.274 32.278 -5.061 -5.004 - - 0.150 Gl 490A 28.171 28.093 32.422 -4.251 -4.329 29.37 -3.01 0.0497 Gl 494AB 27.857 28.196 32.039 -4.182 -3.843 29.00 -3.02 0.0128 Gl 507.1 27.302 27.237 32.350 -5.048 -5.113 - - 0.156 Gl 508.2 26.911 26.913 32.247 -5.336 -5.334 - - 0.133 Gl 510 27.358 27.520 32.186 -4.829 -4.667 - - 0.0866 Gl 514 27.110 27.140 32.238 -5.128 -5.098 27.40 -4.83 0.108 Gl 521 26.898 26.915 32.231 -5.333 -5.316 - - 0.266 Gl 526 26.844 26.930 32.192 -5.348 -5.262 - - 0.209 Gl 536 27.025 27.057 32.227 -5.202 -5.170 27.204 -5.023 0.227 Gl 540 27.331 27.258 32.394 -5.063 -5.136 - - 0.176 Gl 552 26.961 27.061 32.194 -5.233 -5.133 - - 0.397 Gl 563.2A 25.213 26.955 30.709 -5.496 -3.754 - - 0.00762 Gl 563.2B 25.070 26.824 30.644 -5.574 -3.820 - - 0.00905 Gl 618.4 26.829 26.948 32.179 -5.350 -5.231 - - 0.145 Gl 634 26.510 26.813 32.057 -5.547 -5.244 - - 0.242 Gl 637 26.456 26.871 31.974 -5.518 -5.103 - - 0.0467 Gl 645 26.971 27.228 32.056 -5.085 -4.828 - - 0.0481 Gl 649 27.236 27.256 32.252 -5.016 -4.996 - - 0.105 Gl 654AB 26.505 26.595 32.165 -5.659 -5.569 27.301 -4.864 0.0356 Gl 686 26.658 26.903 32.083 -5.425 -5.180 - - 0.0584 Gl 701 26.870 27.039 32.133 -5.263 -5.094 27.23 -4.91 0.107 Gl 724 27.320 27.325 32.268 -4.948 -4.943 - - 0.0839 Gl 737B 27.586 27.452 32.367 -4.781 -4.915 27.724 -4.643 - Gl 745A 25.754 26.784 31.713 -5.959 -4.929 - - 0.0107 Gl 745B 25.684 26.754 31.739 -6.059 -4.989 - - 0.0132 Gl 767A 27.452 27.326 32.476 -5.024 -5.150 - - 0.188 Gl 781 27.000 27.100 31.687 -4.687 -3.587 28.33 -3.36 0.00227 Gl 800A 27.146 27.259 32.314 -5.168 -5.055 - - 0.125 Gl 803 28.461 28.461 32.574 -4.113 -4.113 29.74 -2.83 0.0731 Gl 806 26.744 27.080 32.041 -5.296 -4.960 - - 0.305 Gl 808 26.037 26.919 31.917 -5.880 -4.998 - - 0.0733 Gl 809 27.309 27.242 32.344 -5.035 -5.102 27.43 -4.92 0.115 Gl 815AB 28.012 28.179 32.173 -4.162 -3.995 28.99 -3.18 0.0277 Gl 821 26.157 26.641 31.930 -5.773 -5.289 - - 0.0330 Gl 832 26.749 27.017 32.085 -5.336 -5.068 - - - Gl 842 27.379 27.329 32.317 -4.938 -4.988 - - 0.0974 Gl 855 27.460 27.411 32.363 -4.903 -4.952 - - 0.107 Gl 863 26.965 27.097 32.145 -5.179 -5.047 - - 0.0655 Gl 867A 28.182 28.102 32.411 -4.229 -4.309 29.08 -3.33 0.0664 Gl 880 27.343 27.303 32.322 -4.980 -5.020 27.11 -5.21 0.150 Gl 887 26.982 27.074 32.187 -5.205 -5.113 26.845 -5.342 - Gl 895 27.503 27.454 32.319 -4.817 -4.866 27.204 -5.114 0.580 Gl 908 26.452 26.723 32.056 -5.604 -5.333 27.18 -4.88 0.0579 G 192-11A 27.603 27.612 32.246 -4.634 -4.625 - - 0.0882 McC 354A 27.545 27.436 32.374 -4.829 -4.938 - - 0.161 McC 488 27.712 27.599 32.464 -4.752 -4.865 - - 0.147 St 497 27.959 27.986 32.206 -4.247 -4.220 29.534 -2.67 0.0330 St 928 27.038 27.214 32.116 -5.078 -4.902 - - 0.0816 M3 GJ 1046 26.362 26.440 31.894 -5.532 -5.454 - - 0.0385 GJ 1050 25.809 25.844 31.924 -6.115 -6.080 - - 0.1936 GJ 1125 25.859 26.550 31.403 -5.544 -4.853 - - 0.0291 GJ 1203 26.135 26.222 31.863 -5.728 -5.641 - - 0.1203 GJ 1271 26.923 26.796 32.202 -5.280 -5.407 - - 0.1912 GJ 3139 26.724 26.640 32.148 -5.425 -5.509 - - 0.2599 GJ 3189 25.039 25.840 31.225 -6.186 -5.685 - - 0.0190 GJ 3279 26.439 26.484 31.914 -5.475 -5.430 - - 0.1301 GJ 3293 26.693 26.724 31.966 -5.273 -5.242 - - 0.1503 GJ 3404A 26.328 26.333 32.022 -5.694 -5.689 - - 0.2023 GJ 3459 26.164 26.402 31.676 -5.512 -5.274 - - 0.0600 GJ 3528 26.852 26.795 32.128 -5.276 -5.333 - - 0.2373 GJ 3563 26.245 26.349 31.860 -5.615 -5.511 - - 0.1082 GJ 3634 26.689 26.722 32.077 -5.388 -5.355 - - 0.1891 GJ 3708A 26.239 26.361 31.813 -5.573 -5.451 - - 0.1679 GJ 3846 26.218 26.407 31.719 -5.502 -5.313 - - 0.0710 GJ 3892 26.545 26.571 32.002 -5.457 -5.431 - - 0.1836 GJ 3916A 26.219 26.387 31.792 -5.573 -5.405 - - 0.0543 GJ 3916B 26.219 26.387 31.792 -5.573 -5.405 - - 0.0191 GJ 4231 27.779 27.859 31.850 -4.072 -3.992 29.397 -2.453 0.00124 GJ 4282 27.578 27.693 31.813 -4.235 -4.120 28.892 -2.921 0.00581 Gl 12 25.601 26.109 31.436 -5.835 -5.327 - - 0.0322 Gl 70 26.673 26.706 31.969 -5.296 -5.263 - - 0.1223 Gl 109 26.459 26.572 31.825 -5.366 -5.253 27.30 -4.53 0.0663 Gl 119B 27.385 27.168 32.286 -4.900 -5.117 - - 0.0885 Gl 140C 26.901 26.953 31.919 -5.018 -4.966 - - 0.0162 Gl 145 26.477 26.671 31.745 -5.268 -5.074 27.28 -4.47 0.0671 Gl 163 26.357 26.384 31.957 -5.600 -5.573 - - 0.1710 Gl 204.2 26.660 26.627 32.088 -5.428 -5.461 - - 0.2896 Gl 207.1 27.856 27.850 32.058 -4.202 -4.208 29.10 -2.96 0.0192 Gl 226 26.560 26.588 31.991 -5.431 -5.403 26.90 -5.09 0.1245 Gl 238 26.484 26.479 32.059 -5.575 -5.580 - - 0.2150 Gl 251 26.120 26.508 31.829 -5.709 -5.321 - - 0.0865 Gl 277A 27.730 27.555 32.217 -4.486 -4.661 29.053 -3.164 0.0388 GL 298 26.601 26.578 32.051 -5.445 -5.468 - - 0.1933 GL 352 26.537 26.359 32.243 -5.705 -5.833 27.15 -5.09 0.0809 Gl 357 25.932 26.121 31.773 -5.841 -5.652 - - 0.0662 Gl 358 27.188 27.218 31.957 -4.769 -4.739 28.10 -3.86 0.1084 Gl 377 26.965 26.881 32.135 -5.135 -5.219 - - 0.2116 Gl 386 26.694 26.626 32.133 -5.439 -5.506 - - 0.3231 Gl 388 27.660 27.688 31.949 -4.288 -4.260 28.86 -3.09 0.0538 Gl 399 26.685 26.661 32.145 -5.460 -5.524 - - 0.3686 Gl 408 26.550 26.595 31.908 -5.358 -5.313 26.60 -5.31 0.0987 Gl 422 26.080 26.113 31.912 -5.832 -5.799 - - 0.1346 Gl 436 26.384 26.403 32.026 -5.642 -5.623 26.85 -5.18 0.1935 Gl 443 27.020 26.870 32.204 -5.185 -5.335 - - 0.3183 Gl 452A 26.671 26.609 32.111 -5.440 -5.502 - - 0.2280 Gl 452.1 26.661 27.312 31.312 -4.650 -3.999 - - 0.0184 Gl 463 26.779 26.678 32.164 -5.385 -5.486 - - 0.2110 Gl 479 27.046 27.072 31.990 -4.944 -4.918 27.90 -4.09 0.1390 Gl 480 26.748 26.715 32.051 -5.303 -5.336 - - 0.2737 Gl 513 26.359 26.386 31.962 -5.603 -5.576 - - 0.1901 Gl 588 26.722 26.700 32.079 -5.358 -5.380 26.70 -5.38 0.2498 Gl 617B 26.931 26.886 32.085 -5.154 -5.199 27.67 -4.42 0.1853 Gl 618A 26.324 26.439 31.828 -5.504 -5.389 26.90 -4.93 0.1047 Gl 623 26.285 26.319 31.947 -5.663 -5.629 - - 0.1272 Gl 634 26.101 26.283 31.741 -5.640 -5.458 - - 0.0621 Gl 644A 27.226 27.255 31.968 -4.742 -4.713 28.74 -3.23 0.0774 Gl 644B 27.226 27.255 31.968 -4.742 -4.713 28.74 -3.23 0.0498 Gl 660A 26.268 26.388 31.797 -5.529 -5.409 - - 0.1035 Gl 671 26.305 26.366 31.895 -5.591 -5.530 - - 0.1117 Gl 674 26.737 26.880 31.814 -5.077 -4.934 27.53 -4.28 0.0846 Gl 687 26.422 26.450 31.648 -5.227 -5.199 26.90 -4.745 0.1924 Gl 693 25.968 26.340 31.546 -5.578 -5.206 - - 0.0352 Gl 694 26.725 26.749 32.022 -5.296 -5.972 27.00 -5.02 0.1209 Gl 725AB 26.110 26.482 31.852 -5.742 -5.370 26.48 -5.37 0.1163 Gl 735A 27.798 27.786 32.054 -4.256 -4.268 28.91 -3.14 0.1006 Gl 735B 27.608 27.596 32.054 -4.447 -4.459 28.91 -3.14 0.1022 Gl 739 26.771 26.759 32.078 -5.307 -5.319 - - 0.0586 Gl 752A 26.890 26.845 32.122 -5.233 -5.278 - - 0.1744 Gl 793 26.864 26.900 31.890 -5.026 -4.990 27.76 -4.13 0.0895 Gl 844 27.467 27.143 32.388 -4.921 -5.245 27.54 -4.85 0.1414 Gl 849 26.703 26.627 32.101 -5.398 -5.474 - - 0.2673 Gl 856 28.063 27.996 32.097 -4.034 -4.101 29.452 -2.645 0.0201 Gl 877 26.508 26.534 31.994 -5.487 -5.461 - - 0.1552 Gl 896A 27.675 27.739 31.852 -4.177 -4.113 28.60 -3.25 0.00748 Gl 897AB 27.710 27.735 32.015 -4.306 -4.281 29.237 -2.778 0.0215 M4 GJ 1001A 25.589 25.542 31.535 -5.946 -5.993 - - 0.0253 GJ 1065 25.376 25.548 31.274 -5.898 -5.726 - 0.00741 GJ 1105 25.989 25.967 31.494 -5.505 -5.527 - - 0.0144 GJ 1129 25.626 25.604 31.461 -5.835 -5.857 - - 0.00960 GJ 1207 26.926 26.968 31.400 -4.473 -4.431 28.382 -3.018 0.00287 GJ 1289 26.135 26.272 31.273 -5.138 -5.001 - - 0.0115 GJ 2036B 26.830 27.157 31.028 -4.198 -3.871 29.070 -1.958 0.00130 GJ 2069A 27.692 27.590 31.807 -4.115 -4.217 29.033 -2.774 0.0180 GJ 3149B 26.756 26.834 31.347 -4.591 -4.513 - - 0.00762 GJ 3283B 26.717 26.828 31.294 -4.577 -4.466 - - 0.00503 GJ 3631 27.039 27.307 31.111 -4.072 -3.804 28.324 -2.787 0.00144 GJ 3707 25.742 25.695 31.529 -5.787 -5.834 - - 0.0145 GJ 3789 27.339 27.372 31.399 -4.060 -4.027 28.253 -3.146 0.00054 GJ 3804 25.882 25.827 31.667 -5.785 -5.840 - - 0.0214 GJ 3873 26.399 26.154 32.067 -5.668 -5.913 - - 0.0645 GJ 3900 25.898 25.848 31.568 -5.671 -5.721 - - 0.0119 GJ 4020B 27.414 27.354 31.665 -4.252 -4.312 - - 0.00439 GJ 4063AB 25.709 25.878 31.278 -5.569 -5.400 26.845 -4.433 0.0136 GJ 4333 26.287 26.164 31.857 -5.571 -5.694 28.551 -3.306 0.0532 GJ 4338B 27.544 27.410 31.863 -4.319 -4.453 29.136 -2.727 0.00985 GJ 4378A 26.912 26.831 31.779 -4.867 -4.948 28.749 -3.030 0.0254 Gl 15B 25.777 26.032 31.143 -5.366 -5.111 27.28 -3.86 0.0146 Gl 46 26.067 25.955 31.868 -5.801 -5.913 - - 0.0958 Gl 54.1 26.273 26.675 30.893 -4.619 -4.217 27.491 -3.402 0.00956 Gl 84.1B 26.554 26.364 32.019 -5.465 -5.655 - - 0.0999 Gl 105B 25.764 25.711 31.597 -5.833 -5.886 - - 0.0188 Gl 166C 26.722 26.746 31.396 -4.674 -4.650 28.215 -3.181 0.0104 Gl 179 26.364 26.299 31.779 -5.415 -5.480 - - 0.0359 Gl 203 25.112 25.291 31.287 -6.175 -5.996 - - 0.00737 Gl 206A 27.049 27.103 31.410 -4.361 -4.307 - - 0.00481 Gl 206B 26.830 26.884 31.410 -4.580 -4.526 - - 0.00504 Gl 213 25.537 25.567 31.398 -5.861 -5.831 26.477 -4.921 0.0113 Gl 232 25.314 25.573 31.120 -5.806 -5.547 - - 0.00888 Gl 234A 26.635 26.665 31.364 -4.729 -4.699 - - 0.00643 Gl 268A 26.953 26.897 31.589 -4.635 -4.691 27.819 -3.769 0.00535 Gl 268B 26.707 26.651 31.589 -4.882 -4.938 27.819 -3.769 0.00535 Gl 273 25.987 25.932 31.645 -5.658 -5.713 26.301 -5.344 0.0218 Gl 277B 27.345 27.220 31.874 -4.530 -4.655 29.056 -2.817 0.0210 Gl 285 27.462 27.400 31.668 -4.206 -4.268 28.647 -3.021 0.0126 Gl 299 25.276 25.515 31.119 -5.843 -5.604 - - 0.00948 Gl 300 25.950 25.900 31.534 -5.584 -5.534 - - 0.0121 Gl 317 26.513 26.353 31.967 -5.455 -5.615 - - 0.0686 Gl 319B 26.150 25.927 32.060 -5.911 -6.134 - - 0.0629 Gl 324B 25.978 25.939 31.483 -5.505 -5.544 - - 0.0119 Gl 375A 27.642 27.481 31.967 -4.325 -4.486 28.955 -3.012 0.0173 Gl 375B 27.603 27.442 31.967 -4.364 -4.525 28.955 -3.012 0.0173 Gl 402 25.970 25.951 31.465 -5.495 -5.514 - - 0.0121 Gl 431 27.501 27.420 31.814 -4.313 -4.394 28.738 -3.076 0.00500 Gl 445 25.524 25.480 31.542 -6.018 -6.062 - - 0.0124 Gl 447 25.398 25.628 31.144 -5.746 -5.516 26.845 -4.299 0.0129 Gl 469 26.221 26.126 31.829 -5.608 -5.703 - - 0.0704 Gl 486 25.802 25.741 31.716 -5.914 -5.975 - - 0.0332 Gl 487A 26.363 26.324 31.557 -5.194 -5.233 27.826 -3.731 0.0165 Gl 487C 26.363 26.324 31.557 -5.194 -5.233 27.826 -3.731 0.0108 Gl 490B 27.287 27.206 31.763 -4.477 -4.558 29.068 -2.695 0.0118 Gl 512B 25.501 25.691 31.204 -5.703 -5.513 - - 0.00732 Gl 520C 26.915 26.978 31.370 -4.455 -4.392 - - 0.00374 Gl 553.1 25.692 25.636 31.676 -5.984 -6.040 - - 0.0223 Gl 555 25.834 25.786 31.523 -5.689 -5.737 - - 0.0118 Gl 592 25.617 25.564 31.611 -5.995 -6.048 - - 0.0148 Gl 609 25.716 25.699 31.455 -5.739 -5.756 - - 0.00967 Gl 630.1 26.929 27.269 31.399 -4.469 -4.129 28.473 -2.926 0.00108 Gl 643 25.359 25.569 31.375 -6.016 -5.806 29.142 -2.233 0.0112 Gl 644B 25.518 25.728 31.309 -5.791 -5.581 28.734 -2.575 0.0110 Gl 669A 27.410 27.586 31.978 -4.569 -4.393 28.780 -3.198 0.0297 Gl 682 25.841 25.802 31.503 -5.662 -5.701 - - 0.00935 Gl 699 25.409 25.629 31.189 -5.780 -5.560 26.000 -5.189 0.0142 Gl 720B 25.772 25.932 31.621 -5.849 -5.689 - - 0.0196 Gl 725B 26.107 26.677 31.906 -5.799 -5.229 26.477 -5.429 0.0276 Gl 729 26.633 26.860 31.202 -4.569 -4.342 27.778 -3.424 0.00601 Gl 732A 26.870 26.814 31.651 -4.781 -4.837 - - - Gl 781.1B 27.105 27.015 31.789 -4.684 -4.774 28.758 -3.031 0.0118 Gl 791.2A 26.752 26.896 31.250 -4.498 -4.354 27.892 -3.358 0.00094 Gl 812A 27.309 27.025 32.150 -4.841 -5.125 28.908 -3.242 0.0310 Gl 860B 26.526 26.538 31.466 -4.939 -4.927 27.690 -3.776 0.00738 Gl 865A 27.118 27.218 31.470 -4.353 -4.253 28.577 -2.893 0.00415 Gl 865B 26.866 26.966 31.470 -4.604 -4.504 28.577 -2.893 0.00474 Gl 867B 27.319 27.260 31.707 -4.387 -4.446 29.379 -2.328 0.00955 Gl 873A 27.417 27.355 31.732 -4.316 -4.378 29.088 -2.644 0.0128 Gl 876A 26.090 26.028 31.725 -5.635 -5.697 - - 0.0306 Gl 896A 27.305 27.243 31.688 -4.382 -4.444 28.599 -3.089 0.00480 Gl 896B 27.291 27.229 31.688 -4.397 -4.459 28.599 -3.089 0.00480 [1]Corrected from metallicity effects

http://arxiv.org/abs/1701.08170v2

20170127190131

Contagion dynamics of extremist propaganda in social networks

cs.SI

headings May 2017 A]Emilio Ferrara Corresponding Author: Emilio Ferrara, USC Information Sciences Institute, 4676 Admiralty Way #1001, Marina del Rey, CA (USA) 90292; E-mail: emiliofe@usc.edu E. Ferrara [A]University of Southern California, Information Sciences Institute, USA Recent terrorist attacks carried out on behalf of ISIS on American and European soil by lone wolf attackers or sleeper cells remind us of the importance of understanding the dynamics of radicalization mediated by social media communication channels. In this paper, we shed light on the social media activity of a group of twenty-five thousand users whose association with ISIS online radical propaganda has been manually verified. By using a computational tool known as dynamic activity-connectivity maps, based on network and temporal activity patterns, we investigate the dynamics of social influence within ISIS supporters. We finally quantify the effectiveness of ISIS propaganda by determining the adoption of extremist content in the general population and draw a parallel between radical propaganda and epidemics spreading, highlighting that information broadcasters and influential ISIS supporters generate highly-infectious cascades of information contagion. Our findings will help generate effective countermeasures to combat the group and other forms of online extremism. computational social sciencessocial mediaextremismsocial influence May 2017May 2017 § INTRODUCTION Researchers in the computational social science community have recently demonstrated the importance of studying online social networks to understand our society <cit.>. New powerful technologies are usually harbinger of abuse, and online platforms are no exception: social media have been shown to be systematically abused for nefarious purposes <cit.>. As online social environments yield plenty of incentives and opportunities for unprecedented, even “creative” forms of misuse, single individuals as well as organizations and governments have systematically interfered with these platforms, oftentimes driven by some hidden agenda, in a variety of reported cases: * During crises, social media have been effectively used for emergency response; but fear-mongering actions have also triggered mass hysteria and panic <cit.>. * Political conversation has been manipulated by means of orchestrated astroturf campaigns <cit.> even during election times <cit.>. * Anti-vaccination movements <cit.>, as well as conspiracy (and other anti-science) theorists <cit.>, took social media by the storm and became responsible for a major health crisis in the United States <cit.>. * Social media bots (non-human automated accounts) have been used to coordinate attacks to successfully manipulate the stock market, causing losses in the billions of dollars <cit.>. * Some governments and non-state actors have been active on social media to spread their propaganda. In some cases, they have allegedly “polluted” these platforms with content to sway public opinion <cit.>, or to hinder the ability of social collectives to communicate, coordinate, and mobilize <cit.>. Especially related to the last point, researchers have been recently devoting more attention to issues related to online propaganda campaigns <cit.>. Increasing evidence provided by numerous independent studies suggests that social media played a pivotal role in the rise in popularity of the Islamic State of Iraq and al-Sham (viz. ISIS) <cit.>. Determining whether ISIS benefitted from using social media for propaganda and recruitment was central for many research endeavors <cit.>. Analyses by Berger and Morgan suggested that a restricted number of highly-active accounts (500-1000 users) is responsible for most of ISIS' visibility on Twitter <cit.>. However, Berger's subsequent work suggested that ISIS' reach (at least among English speakers) has stalled for months as of the beginning of 2016, due to more aggressive account suspension policies enacted by Twitter <cit.>. Other researchers tried to unveil the roots of support for ISIS, suggesting that ISIS backers discussed Arab Spring uprisings on Twitter significantly more than users who stood against ISIS <cit.>. These early investigations all share one common methodological limitation, namely that to collect social media data they start from keywords known to be associated to ISIS <cit.>. This strategy has been widely adopted in a previous research aimed at characterizing social movements <cit.>. However, we argue that it is not sufficient to focus on keyword-based online chatter to pinpoint to relevant actors of radical conversation. In fact, our recent results <cit.> suggest that radical propaganda revolves around four independent types of messanging: (i) theological and religious topics; (ii) violence; (iii) sectarian discussion; and, (iv) influential actors and events. Here is a series of examples of possible biases introduced by the keyword-centric approach: * Some studies <cit.> focused on religion-based keyword lists, but most terms typically associated to religion are not necessarily used in the context of extremism. * Other studies <cit.> focused on influential actors or events; this can introduce biases due to the focus on popular actors rather than the overall conversation. * Further noise can be introduced by tweets that simply link to news articles reporting on events; although these tweets may contain keywords in the predefined watchlist, they clearly do not represent extremist propaganda efforts. * Finally, focusing on pre-determined keywords could cause incomplete data collection by missing topics of discussion that can emerge dynamically and do not adopt any of the predefined key terms. In this work, we will leverage an alternative data collection and curation approach: we will start from a large set of Twitter users that are known to be associated to or symphatizers of ISIS. We will then collect their activity over a large time span of over one year to obtain a complete characterization of their extremist propaganda efforts. §.§ Contributions of this work This study aims to address the two following research questions: RQ1: Can we define a solid methodological framework and suggest good practices for data collection, validation, and analysis to study online radicalization?. Setting such best practices will hopefully steer the information sciences and computational social sciences research communities in the direction of producing more rigorous and reproducible work. One contribution of our work is to address this issue by focusing on manually-verified set of ISIS supporter accounts. After describing how we collected and curated the dataset object of this study, we move forward to investigate the dynamics of online propaganda adopted by ISIS. Using computational social science tools to gauge online extremism, we aim to answer the following second research question: RQ2: What types of network and temporal patterns of activity reflect the dynamics of social influence within ISIS supporters? And, can we quantify the adoption of extremist content in the general population?. Our findings will help generate effective countermeasures to combat the group and other forms of online extremism. § DATA COLLECTION AND CURATION Due to the the limits of keyword-based data collection approaches we highlighted above, in this work we exclusively rely on data and labels obtained by using a procedure of manual data curation and expert verification. We obtained a list of Twitter accounts whose activity was labeled as supportive of the Islamic State by the online crowd-sourcing initiative called Lucky Troll Club. The goal of this initiative was to leverage annotators with expertise in Arabic languages to identify accounts affiliated to ISIS and report them to Twitter to request their suspension. Twitter's anti-abuse team manually verifies all suspension requests, and grants some based on evidence of violation of Twitter's Terms of Service policy against the usage of the platform for extremist purposes. We verified that 25,538 accounts present in the Lucky Troll Club list have been suspended by Twitter in the period between March 17, 2015 and June 9, 2015. In this study, we focus only on this subset of twenty-five thousand accounts: we consider their activity to be unequivocally linked to the Islamic State, as determined by the two-step manual verification process described above. For each account, we have at our disposal information about the suspension date, as well as the number of followers of that user as of the suspension date. The anonymized data focus of this study can be made available to the interested researchers upon request. For contact information and further details about the dataset please visit: <http://www.emilio.ferrara.name/datasets/> §.§ Twitter data collection The next step of our study consisted in collecting data related to the activity of the 25,538 ISIS supporters on Twitter. To this purpose, we leveraged the Observatory on Social Media (OSoMe) data source set up by our collaborators at Indiana University <cit.>, which continuously collects the Twitter data stream from the gardenhose API (roughly a 10% random sample of the full Twitter data stream). Using this large data stream avoids known issues derived by using the public Twitter stream API which serves only less than 1% of the overall tweets <cit.>. We obtained all tweets present in the OSoMe database that have been posted by any of the twenty-five thousand ISIS accounts prior to their suspension. We also included all retweets these tweets generated, and all tweets containing mentions to such set of ISIS supporters. The resulting dataset that we will study consists of 3,395,901 tweets. Almost 1.2 million of these tweets was generated by the ISIS accounts during the period between January 2014 and June 2015. We found that a total of 54,358 distinct other users has retweeted at least once one of the twenty-five thousand ISIS supporters. This amounts for the remainder of about 2.2 million tweets in our dataset. Summarizing, we identified two best practices to answer RQ1: * Favor starting from a manually-verified list of users involved in online propaganda, radicalization efforts, or recruitment, rather focusing on keyword-based searches. When such manually validated lists are not available, human annotations can be generated by means of services such as Amazon Mechanical Turk. * Use large social media data streams, when available, rather than small samples that can be biased. Services like the Indiana University's OSoMe database <cit.> can provide data sources especially valuable for Twitter-based social media studies. Alternatively, the Twitter Search API can yield comprehensive data around specific users or topics, provided that the search is limited to short time frames. §.§ Limitations and strategic choices of this study The largest majority of the tweets in our dataset, over 92%, is in Arabic. This introduces a number of important challenges, from both technical and methodological perspectives. From a technical standpoint, analyzing Arabic content would require sophisticated natural language processing (NLP) techniques as well as accurate sentiment analysis tools capable of extracting emotional information. NLP toolkits with support for Arabic content are only recently starting to be developed <cit.>, while Arabic-based sentiment analysis is still in its early developmental stage <cit.>. From a methodological point of view, the analysis of Arabic-language material would require interpreters with sufficient domain knowledge in extremism-related issues to yield useful and unbiased insights from the data. Access to such experts is not always available, and thus the need to develop alternative strategies of enquiry emerges. For these reasons, in the rest of the paper our analysis will rely exclusively upon language-agnostic techniques: in particular, we will focus on statistical properties of information diffusion networks, which were proven very useful in our prior studies on criminal networks <cit.>, as well as temporal patterns of information diffusion, which we already exploited to study the interplay between ISIS' activity online and offline <cit.>. Due to the simplifications introduced by our strategy of enquiry, we call for caution in the interpretation of our results. However, we believe that our approach will help identify important areas of research that warrant further development (such as Arabic-based NLP and sentiment analysis toolkits), as well as yield valuable insight about ISIS social media operations and other forms of online extremist propaganda. § RESULTS We next report our investigation to address RQ2. We study the activity of ISIS supporters and sympathizers on Twitter by means of the data collected as described above. §.§ Activity and Support of ISIS Accounts The first form of validation that we performed pertains the mechanisms of suspension of ISIS accounts on Twitter. As we mentioned before, we identified a set of twenty-five thousand accounts related to ISIS that have been suspended: Fig. <ref> shows the timeline of suspensions of the accounts under investigation. The suspension period occurred throughout almost three months, with the first suspensions occurring on March 17, 2015 and the last occurring on June 9, 2015. After this date, none among the twenty-five thousand accounts in our list is anymore active on Twitter. Account suspensions appear to occur in batches, some more substantial than others, with a significant spike of suspensions (over ten thousands) occurred on April 2, 2015. Our findings are consistent with The Guardian's report that, between April 2015 and February 2016, Twitter's anti-abuse task force suspended more than 125,000 accounts linked to ISIS <cit.>. Recent literature reported contrasting evidence about the activity and popularity of ISIS supporters on Twitter <cit.>. As for today, it is unclear whether ISIS accounts obtained a significant support on Twitter, and to what extent they were active. Berger and collaborators first found that ISIS presence was very pervasive on Twitter during 2014-2015 <cit.>, and later suggested that only a core of 500-2000 ISIS users was active after that period <cit.>. To shed light on this question, we calculated the distribution of the number of followers and followees (friends) of ISIS accounts at the time of their suspension, along with the number of tweets and retweets they generated. For each of these four variables, we calculated their probability distribution Pr(x). These probability distributions display the probabilities of values taken by the four variables (i.e., tweets, retweets, followers, followees) in our dataset: they can be thought as normalized frequency distributions, where all occurrences of outcomes for each distribution sum to 1. The results are shown in Fig. <ref>. Let us discuss support and activity separately. Concerning ISIS support on Twitter, we notice that the distribution of followers of ISIS accounts exhibits the long tail typical of Twitter <cit.> and other social networks (cf. yellow dash-dotted line in Fig. <ref>). This skewed distribution has mean μ = 516 (σ=1,727), median Q_2=130, and lower and upper quartiles Q_1=37 and Q_3=401. This means that the majority of accounts has a limited number of followers (for example, one quarter of the users has less than 37 followers), yet a significant number of ISIS supporters managed to obtain a large number of followers before getting suspended (in fact, the upper quartile of users has more than 400 followers). The presence of this broad distribution of followership suggests that influence and radicalization operations of Islamic State supporters on Twitter were successful for at least several thousand of their accounts. This is in line with Berger's early results discussed above <cit.>. Another interesting insight is yielded by the distribution of followees (cf. green dotted line in Fig. <ref>): differently from the distribution of followers, this distribution shows an unexpected upward trend in the regime between 100 and 1,000 followees. This characteristic behavior has been associated to forms of social network manipulation, for example attempts to create rings or cliques in which multiple accounts under the control of a same entity all follow each other to reciprocally increase their visibility and followership <cit.>. We thus suggest that ISIS accounts enacted strategies to artificially enhance their visibility by strengthening one another social networks. A significant portion of ISIS accounts was very active on Twitter: both distributions of tweets (cf. solid blue line) and retweets (cf. red dashed line) shown in Fig. <ref> exhibit the typical power-law shape common to social networks with heterogeneous activity patterns. This implies that a significant fraction of users posted and retweeted large amounts of tweets. For example, at least 1% of the ISIS users posted at least 30 tweets during the observation period; similar figures hold for retweeting. More importantly, there appears to be a strong core constituted by a few dozen accounts who posted and retweeted hundreds of tweets in the same period. This set of very active ISIS supporters we found is compatible with what reported by Berger's first study <cit.>. Furthermore, there seems to be a handful of accounts with thousands of tweets and retweets, suggesting the likely presence of some social media bot used to enhance the volume of content generated by ISIS and its spreding on Twitter <cit.>. To investigate ISIS accounts' activity further, we extrapolated the time series of the total volume of tweets and retweets posted every week by the ISIS users under investigation: the result is shown in Fig. <ref>. It's worth noting that our observation window spans 1.5 years and starts on January 2014 when the firsts among the twenty-five thousand ISIS supporters became active on Twitter: Although the activity volume slowly builds up over time, it is only in early March 2015 (15 months into our observation period) that the volume of tweets per week drastically spikes. In the early regime between January 2014 and March 2015 the volume of tweets associated to ISIS accounts spans 1,000 to 10,000 per week. This increases nearly tenfold after March 2015, with a spike of over 100,000 tweets per week, and an average of over 60,000 tweets per week in the period between March and May 2015. This period concurs with the period of strongest Twitter suspensions shown in Fig. <ref>, suggesting a timely reaction of Twitter to fight the activity of ISIS users on the platform. Indeed, the volume of tweets per week produced by these accounts drops in early June, and goes to zero, as expected, in the late period of observation when fewer and fewer of the ISIS accounts under investigation are left unchecked on Twitter. Cumulatively, the ISIS users under investigation produced almost 1.2 million tweets during the observation period (cf. inset of Fig. <ref>). Our findings pinpoint to the power of the crowd-sourcing volunteer initiative that set to bring up to Twitter's attention these accounts. However, there is no evidence to quantify how many (if any) of the ISIS supporters not recorded by the Lucky Troll Club operation were independently suspended by Twitter anti-abuse team. §.§ Dynamic Activity-Connectivity Maps We further address RQ1 by offering a powerful tool to information scientists and computational social scientists: the dynamic activity-connectivity (DAC) maps. DAC maps allow to study influence and authority dynamics of online extremism in social networks with temporal activity patterns. Fig. <ref> shows an example of dynamic activity-connectivity map. We developed DAC maps as dynamic variants of the map proposed by Gonzalez-Bailon and collaborators—see Figure 4 in Broadcasters and Hidden Influentials in Online Protest Diffusion <cit.>. The key intuition behind this tool is to allow investigate what effect the progression of activity levels of an user has of their connectivity evolution (and viceversa). In a DAC map, for a given user u, x_u and y_u are defined as x_u = 1+δ f_u/1+δ F_u y_u= 1+ m_u/1+ M_u. We use the notation f_u and F_u to identify the number of followers and friends, respectively, of a user u. The variation of followers and friends of user u over a period of time t are thus defined as δ f_u = f_u^max - f_u^min/t and δ F_u = F_u^max - F_u^min/t; the length of time t is defined as the number of days of u's activity, measured from registration to suspension (this varies from user to user). Finally, m_u is the number of mentions user u received by others, and M_u is the number of mentions user u made to others, during u's activity period. All values are added to the unit to avoid zero-divisions and allow for logarithmic scaling (i.e., in those cases where the variation is zero). The third dimension, the“heat” (the color intensity) in the map, represents the joint probability density pdf(x,y) for users with given values of x and y. The plot also introduce a bin normalization to account for the logarithmic binning. The two dimensions defined over the dynamic activity-connectivity map are interpreted as follows: the x-axis represents the growth of connectivity formation, and the y-axis conveys the rate of messaging activity. In general, we would expect that in a dynamic activity-connectivity map, the bulk of the joint probability density mass would be observed in the neighborhood of (1,1), that hosts the majority of accounts for which the variation of the two dimensions is comparable. DAC maps are ideal tools to addres our second research question (RQ2): they capture at the same time network and temporal patterns of activity, and they can help understand how connectivity variations affects social influence (and viceversa), which are dynamics at the core of our investigation. Let us discuss the two dimensions of Figure <ref>, namely connectivity growth and activity rate, separately. The connectivity growth is captured by the x axis and, in our case, ranges roughly between 10^-2 and 10^3. Users for which x>1 (i.e., 10^0) are those with a followership that grows much faster than the rate at which these users are following others. In other words, they are acquiring social network popularity (followers) at a fast-paced rate. Note that, if a user is acquiring many followers quickly, but s/he is also following many users at a similar rate, the value of x will be near 1. This is a good property of our measure because it is common strategy on social media platforms, especially among bots <cit.>, to indiscriminately follow others in order to seek for reciprocal followerships. Our dynamic activity-connectivity map will discriminate users with fast-growing followerships, who will appear in the right-hand side of the map, from those who adopt that type of reciprocity-seeking strategy. The former group can be associated with highly popular users with a fast-paced followership growth. According to Gonzalez-Bailon and collaborators <cit.> this category is composed by two groups: influential users and information broadcasters, depending on their activity rates. Values of x<1 indicate users who follow others at a rate higher than that they are being followed; the fall in the left-hand side of the map. According to Gonzalez-Bailon and collaborators, these are mostly the common users, although the so-called hidden influentials also sit in this low-connectivity regime. As for what concerns the y axis, it measures the activity rate, i.e., the rate at which a user receives mentions versus how frequently s/he mentions others. Users with values of y>1 are those who receive systematically more mentions with respect to how frequently they mention others. This group of users can be referred to as influentials, i.e., those who are referred to significantly more frequently than others in the conversation; they fall in the upper region of the map, and according to Gonzalez-Bailon et al., depending on their connectivity growth can be divide in influential (x>1) and hidden influential (x<1) users. Conversely, users with values of y<1 are those who generate increasingly more mentions over time, either because they reply to many tweets, or because they address directly other users. This group generally represents the common-user behavior (x<1), although information broadcasters (x>1) also exhibit the same low-activity rate. These users fall in the lower region of the map. Now that a reading of dynamic activity-connectivity maps has been provided, we can proceed with interpreting Fig. <ref>: the bottom-left quadrant reports the most common users, those with both activity and connectivity growth lesser than 1. Conversely, the upper-right quadrant reports users with the higher connectivity growth and activity rates. These are influential ISIS supporters who are very active in the discussion. We note how the connectivity growth dimension spans three orders of magnitude in the positive domain, while the activity rate dimension only spans two orders of magnitude. This means that some users' followerships grows tens of times faster than the rate at which they follow others; conversely, the rate of receiving mentions is only up to tens of times higher than that of mentioning of others. In the next section, we will devote special attention to these four different classes of users to determine what types of differences emerge in the ISIS social network. §.§ Dynamical Classes of ISIS Supporter Behaviors Prior research illuminated on the dynamical aspects of activity and connectivity in social media <cit.>. Next, we focus on the four classes of user behaviors highlighted by the dynamic activity-connectivity map. We first select, out of the twenty-five thousand ISIS supporters in our dataset, only the subset of those who have mentioned and have been mentioned at least once during the observation period. This will allow to focus on active accounts and correctly capture their activity rate. This filter reduces the number of users under investigation to N=13,024 ISIS supporters, nearly half of the entire ISIS population initially collected. We further divide these users in the four dynamical classes defined above. The classification is obtained by simply adopting the rules defining the four quadrants of Fig. <ref>, which yields N=3,475 common users (x<1,y<1), N=3,339 information broadcasters (x>1,y<1), N=3,218 influentials (x>1,y>1)), and N=2,992 hidden influential users (x<1,y>1). For each of these users, we generated the distribution of (i) the total number of tweets they posted, (ii) the cumulative number of times they have been retweeted, and (iii) the maximum number of followers they gathered. Fig. <ref> shows the boxplots corresponding to the four dynamical classes. Significant differences emerge: common users produce an amount of tweets very similar to that of broadcasters, but they accrue nearly one order of magnitude less retweets than the latter. Information broadcasters also appear to generate the largest followerships, on par with influential users; influentials, however, post significantly less tweets, while accumulating similar amounts of retweets than broadcasters, suggesting that our map successfully captures a notion of social influence intended as a proxy for attention generated to one's posts. The class of hidden influentials shows comparable activity to influential users, but significantly less influence, accruing about one order of magnitude less tweets and significantly smaller followerships than influentials. Our analysis suggests that different classes of ISIS supporters' behaviors emerge. In the future, we will study what are the characteristics of different classes that produce the most effective propaganda and make the most influential users, analyzing content and language, political and religious beliefs, motives and attitudes of the ISIS social network. §.§ Adoption of ISIS Propaganda So far, our analysis focused on characterizing some dimensions of the behavior of ISIS supporters on Twitter. Next, we investigate whether the content they generated has been adopted by other users who have become exposed to it. This will help address the last part of our second research question (RQ2), namely whether we can quantify the adoption of extremist content in the general population. Our notion of adoption is very simple: when a user who does not appear set of the twenty-five thousand ISIS accounts retweets for the first time any tweet generated by one such ISIS supporter, we count this as a content adoption. Although some recent work suggested that retweeting radical propaganda can be considered as an early sign of radicalization <cit.>, we call for caution about interpreting the results in such a way: this definition greatly simplifies the notion and complexity of such types of online adoption processes <cit.>. However, we do believe that investigating the spread of radical content in the user population has a value to determine the extent and effectiveness of ISIS propaganda operations. Fig. <ref> shows the time series of the number of ISIS content adoptions per day during the first half of 2015. Prior to that, no significant amount of adoptions could be observed, partly due to the low activity rate of the accounts under investigation. In the three months between March and June 2015, we notice a significant uptake in the number of adoptions, peaking at nearly one thousand adoptions per day. In that period, at least 10,000 tweets per day (70,000-100,000 tweets/week) were generated by ISIS accounts (cf. Fig <ref>). This suggests that a very significant fraction of tweets, about 5-10%, was actually retweeted on average at least once by other users. During this period, a total of 54,358 distinct other users has retweeted at least once one of the twenty-five thousand ISIS supporters. If we simplify propaganda diffusion as an infectious disease, we can draw a parallel with epidemics. The basic reproduction number R_0 of an infection is the number of cases generated on average by an infected individual over the course of its infectious period, in an otherwise uninfected population. Given that 25,538 ISIS supporters generated 54,358 distinct infected users, we can derive an R_0 = 2.13 for the ISIS propaganda “infection”. In other words, an ISIS supporter before being suspended on average “infected” 2.13 other users. For comparison, health epidemics like Ebola, <cit.> SARS, <cit.> HIV/AIDS, <cit.> and certain strains of influenza <cit.> all have similar values of 2 < R_0 < 3. §.§ Contagion dynamics of ISIS Propaganda We conclude our analysis by studying the cascades of content adoptions generated by the four classes of users defined above. This investigation is twofold: first, we would like to determine whether the mechanisms of receiving retweets (what we defined as content adoption) and being mentioned by out-of-sample users exposed to extremist content follow the same or different dynamics. Second, we will compute the distributions of scores for the basic reproduction number R_0 relative to both receiving retweets and mentions for the four classes of users. The goal is to reveal whether any significant difference emerge between groups of users in their content spreading efficacy, and ultimately to understand which groups of users generated the most effective information contagions. Fig. <ref> shows the distributions of retweets (top) and mentions (bottom) received by the users in the four classes defined by means of the dynamic activity-connectivity map. Although overall all distributions are broad, as expected given the heterogeneous nature of information diffusion, the dynamics of obtaining retweets and mentions are significantly different for the four groups: in particular, for what concerns receiving mentions, no appreciable difference emerges among the four classes of users, which suggests that ISIS supporters are being mentioned in a similar fashion regardless of the class they belong to. However, receiving retweets shows a different mechanism: influential and broadcaster users receive generate significantly larger retweet cascades much more frequently than common users and hidden influentials. One question that thus rises concerns whether the actual contagion effectiveness varies between classes of users. In other words, what are the classes of users that are generating the most effective propaganda campaigns, in terms of adoption (i.e., retweets) and engagement (i.e., mentions)? To this purpose, for each ISIS user who obtained at least one retweet (resp., mention), we calculated the fraction RT/T of received retweets RT (resp., mentions) over the total number of his/her tweets T that have been retweeted at least once (resp., mentioned) by an out-of-sample user (i.e., a user not labeled as ISIS by our list). This measures the rate of diffusion of tweets in the otherwise uninfected population. We note that this is a simplification of the more traditional notion of information contagion where we would consider all the tweets generated and more complex diffusion mechanisms accounting e.g., for exposures, due to the limitation of the platform under study (namely, we do not have any information about information exposure on Twitter). With some abuse of notation, we thus consider the fraction RT/T to convey the meaning of the basic reproduction number R_0 typical of epidemiology. Finally, ISIS accounts are divided in their four classes, according to the usual classification, and the frequency of the basic reproduction scores R_0 is shown in Fig. <ref>, separately for the two dynamics of receiving retweets (top) and mentions (bottom). While no significant difference emerges, in either scenario, among the four different classes of users for the lowest scores (i.e., 2 < R_0 < 2.5 for retweets, and 1 < R_0 < 1.2 for mentions), strong class differences emerge for users whose content are more contagious (R_0>2.5 for retweets, and R_0>1.2 for mentions): concerning retweets, influential users, followed by information broadcasters, are receiving systematically more attention than users in other classes; the class differences for mentions are less pronounced. § RELATED WORK One of the first computational frameworks, proposed by Bermingham et al. <cit.> in 2009, combined social network analysis with sentiment detection tools to study the agenda of a radical YouTube group: the authors examined the topics discussed within the group and their polarity, to model individuals' behavior and spot signs of extremism and intolerance, seemingly more prominent among female users. The detection of extremist content (on the Web) was also the focus of a 2010 work by Qi et al. <cit.>. The authors applied hierarchical clustering to extremist Web pages to divide them into different pre-imposed categories (religious, anti immigration, etc.). Scanlon and Gerber proposed the first method to detect cyber-recruitment efforts in 2014 <cit.>. They exploited data retrieved from the Dark Web Portal Project <cit.>, a repository of posts compiled from 28 different online fora on extremist religious discussions (e.g., Jihadist) translated from Arabic to English. After annotating a sample of posts as recruitment efforts or not, the authors use Bayesian criteria and a set of textual features to classify the rest of the corpus, obtaining good accuracy, and highlighted the most predictive terms. Along the same trend, Agarwal and Sureka proposed different machine learning strategies <cit.> aimed at detecting radicalization efforts, cyber recruitment, hate promotion, and extremist support in a variety of online platforms, including YouTube, Twitter and Tumblr. Their frameworks leverage features of contents and metadata, and combinations of crawling and unsupervised clustering methods, to study the online activity of Jihadist groups on the platforms mentioned above. A few studies explored unconventional data sources: one interesting example is the work by Vergani and Bliuc <cit.> that uses sentiment analysis (Linguistic Inquiry and Word Count <cit.>) to investigate how language evolved across the first 11 Issues of Dabiq, the flagship ISIS propaganda magazine. Their analysis offers some insights about ISIS radicalization motives, emotions and concerns. For example, the authors found that ISIS has become increasingly concerned with females, reflecting their need to attract women to create their utopia society, not revolving around warriors but around families. ISIS also seems to have increased the use of internet jargon, possibly to connect with the identities of young individuals online. Concluding, two very recent articles <cit.> explore the activity of ISIS on social media. The former <cit.> focuses on Twitter and aims at detecting users who exhibit signals of behavioral change in line with radicalization: the authors suggest that out of 154K users only about 700 show significant signs of possible radicalization, and that may be due to social homophily rather than the mere exposure to propaganda content. The latter study <cit.> explores a set of 196 pro-ISIS aggregates operating on VKontakte (the most popular Russian online social network) and involving about 100K users, to study the dynamics of survival of such groups online: the authors suggest that the development of large and potentially influential pro-ISIS groups can be hindered by targeting and shutting down smaller ones. For additional pointers we refer the interested reader to two recent literature reviews on this topic <cit.>. § CONCLUSIONS Since the appearance of the Islamic State (viz. ISIS), a consensus has emerged on the relationship between extremism and social media, namely that ISIS' success as a terrorist organization is due at least in part to its savvy use of social media. It is widely believed that ISIS has managed to increase its roster to tens of thousands of members by broadcasting its savage attacks over social media platforms such as Twitter, which helps radicalize and ultimately recruit fighters from around the world. Recent attacks on American and European soil demonstrate ISIS' potential to reach, organize, and mobilize lone wolves and sleeper terrorist cells among westerners. Many of these actors are known to have consumed radical material online and many have claimed to gravitate towards Islamic State because of it. This paper posed two research questions: the former was concerned with proposing good practices for data collection, validation, and analysis of online radicalization. The latter aimed at revealing the network and temporal activity patterns of ISIS influence on Twitter. To address these questions we analyzed the activity of a group of twenty-five thousand users associated with ISIS. These accounts have been manually identified, reported to Twitter for verification, and subsequently suspended due to their involvement with radical propaganda. This process yielded a human-curated dataset containing over three million tweets generated during a period of one and half year, 92% of which in Arabic language. Regarding the first research question (RQ1), we highlighted the challenges related to studying Arabic content, due to the limits of existing NLP and sentiment analysis toolkits. We therefore suggested the adoption of content-agnostic statistical and network techniques to dissect the users' temporal activities and connectivity patterns. By leveraging a computational tool named dynamic activity-connectivity map, we highlighted the dynamics of social influence within ISIS support. For what concerns the second research question (RQ2), our findings suggest complex strategies carried out by these users to manipulate and influence others: four dynamical classes of ISIS supporters emerged (common sympathizers, information broadcasters, influential and hidden influential users), each with distinct activity and connectivity patterns. We concluded by quantifying the extent to which ISIS support and extremist content are adopted in the general population: by drawing a parallel between propaganda and epidemics spreading, we determined that each ISIS supporter “infected” on average 2.13 other users before Twitter suspended his/her account, highlighting that receiving retweets and mentions follow different dynamics, and that broadcasters and influential users generate much more widespread contagions. Although we call for caution in the interpretation of these results, due to the great simplifications introduced by our framework, we believe that our findings will help design and implement effective countermeasures and responses to ISIS social media operations and other forms of online extremist propaganda. § ACKNOWLEDGMENTS The author is grateful to Max Abrahms (Northeastern University) for useful discussions, and to Alessandro Flammini and Onur Varol (Indiana University) for their support in collecting the Twitter dataset. This work has been partly funded by the Office of Naval Research (ONR), grant no. N15A-020-0053. This research is also based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA). The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of ODNI, IARPA, ONR, or the U.S. Government. The U.S. Government had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein. abbrv

http://arxiv.org/abs/1701.07588v4

20170126064223

Next Generation Backscatter Communication: Systems, Techniques and Applications

cs.IT

Review addressref=aff1, email=wanchun.liu@sydney.edu.au ]WLWanchun Liu addressref=aff2, corref=aff2, email=huangkb@eee.hku.hk ]KHKaibin Huang addressref=aff3, email= ]XZXiangyun Zhou addressref=aff3, email= ]SDSalman Durrani [id=aff1] School of Electrical and Information Engineering, The University of Sydney, 2006 Sydney, NSW, Australia [id=aff2] Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, China [id=aff3] Research School of Engineering, The Australian National University, 2601 Canberra, ACT, Australia The rapid growth of IoT driven by recent advancements in consumer electronics, 5G communication technologies, and cloud-computing enabled big-data analytics, has recently attracted tremendous attention from both the industry and academia. One of the major open challenges for IoT is the limited network lifetime due to massive IoT devices being powered by batteries with finite capacities. The low-power and low-complexity backscatter communications (BackCom), which simply relies on passive reflection and modulation of an incident radio-frequency (RF) wave, has emerged to be a promising technology for tackling this challenge. However, the contemporary BackCom has several major limitations, such as short transmission range, low data rate and uni-directional information transmission. The article aims at introducing the recent advances in the active area of BackCom. Specifically, we provide a systematic introduction of the next generation BackCom covering basic principles, systems, techniques besides IoT applications. Lastly, we describe the IoT application scenarios with the next generation BackCom. Backscatter communication IoT wireless power transfer wirelessly powered network wireless sensor network § INTRODUCTION In the past decades, the IoT has seen technological innovations in a wide range of applications such as smart city, smart home, autonomous robots, vehicles and unmanned aerial vehicles (UAVs). The IoT is expected to comprise tens of billions of sensors in the near future. Keeping the massive number of energy-constrained IoT sensors alive poses a key design challenge for IoT. This is especially challenging given a large number of the sensors may be hidden (e.g., in the walls or appliances) or deployed in remote or hazardous environments (e.g., in radioactive areas or pressurized pipes), making battery recharging or replacement difficult if not impossible. Thus, it is highly desirable to power IoT nodes by ambient energy harvesting <cit.> or wireless power transfer (PT) <cit.>. One particular promising solution in this regard is backscatter communications (BackCom), which allows an IoT node to transmit data by reflecting and modulating an incident RF wave <cit.>. The conventional radio architecture comprises power-hungry RF chains having oscillators, mixers and digital-to-analog converters, which results in non-compact form factors and limits the battery life of IoT devices. For example, the well-adopted ZigBee CC2520 chip from Texas Instruments consumes about 100 mW for transmission <cit.>, which is quite a large power consumption. In contrast, a backscatter node has no active RF components and as a result can be made to have miniature hardware with extremely low power consumption, e.g., in the order of 10 μW <cit.>, facilitating large-scale deployment at a flexible location or even in-body implantation. In the past two decades, point-to-point BackCom has been widely deployed in the application of radio-frequency identification (RFID) for a passive RFID Tag to report an ID to an enquiring Reader over the near field (typically several centimeters). In its early stage, IoT comprised of primarily RFID devices for logistics and inventory management. However, IoT is expected to connect tens of billions of devices and accomplish much more sophisticated and versatile tasks with city-wide or even global-scale influences. This demands the communication capabilities and ranges (tens of meters) between IoT nodes to be way beyond the primitive RFID operations supporting bursty and low-rate (several-bytes pre-written ID sequence) uni-directional transmission over several meters. This can be achieved via a full-fledged BackCom theory leveraging the advanced communication technologies such as small-cell networks, full-duplexing[Note that the BackCom full-duplexing is different from that of the conventional communication systems, and there is a performance tradeoff between the transmission and the reception of a full-duplex BackCom node, which will be discussed in Sec. 4.2.], multi-antenna communications, massive access and wireless PT, as well as advancements in electronics such as miniature radios (e.g., button-size radios) and low-power electronics. Therefore, the developing IoT applications present many promising research opportunities, resulting in a recent surge in research interests in BackCom. Table <ref> summarizes and compares the important properties of the conventional RFID and the next generation BackCom systems. This article aims at introducing the recent advances in the active area of next generation BackCom. First, we summarize the basic principles, system and network architectures for BackCom. Second, we discuss how specific communication techniques have been redesigned for BackCom. Last, we focus on the applications of BackCom for IoT. § BACKCOM BASIC PRINCIPLES AND DESIGN TRADEOFFS §.§ Architecture for BackCom A basic BackCom system consists of two devices: a mobile backscatter node, i.e., a Tag, and a Reader[BackCom Readers are typically connected to the power grid or equipped with large capacity batteries.] <cit.>. The architecture of the Tag consists of an RF energy harvesting block, a battery, a modulation block and an information decoder, as illustrated in Fig. <ref> <cit.><cit.>. The Tag is a passive node that harvests energy from an incident single-tone sinusoidal continuous wave (CW) radiated by the Reader, and also modulates and reflects a fraction of the wave back to the Reader. Specifically, the wave reflection is due to an intentional mismatch between the antenna and load impedance. Varying the load impedance makes the reflection coefficient to vary following a random sequence that modulates the reflected wave with Tag's information bits. Such a modulation scheme is named as the backscatter modulation. The passive Tag is powered by RF energy harvesting and does not require any active RF component. On the contrary, the Reader has its own power supply and a full set of conventional RF components for emitting CW and information transmission/reception. §.§ BackCom Modes and Modulation In general, the communication between the Reader and the Tag has two modes: the forward information transmission, i.e., the Reader-to-Tag transmission, and the backward information transmission, i.e., the Tag-to-Reader transmission. For the forward information transmission, as illustrated in Fig. <ref>, the Reader transmits a binary intensity modulated signal to the Tag. The Tag connects its information decoder and utilizes the received RF signal for RF energy harvesting and energy-detection based demodulation. For example, the decoded bit is `1' or `0' when a high or a low signal energy is detected, respectively. The use of this primitive on/off modulation and energy detection is due to the constraint that a typical Tag is provisioned with an energy detector instead of a power hungry RF chain needed for coherent demodulation. For the backward information transmission, the Reader sends a CW signal to the Tag, and the Tag connects its modulation block and utilizes the received RF signal for RF energy harvesting and backscatter modulation. Generally, a Tag can modulate the reflected signal by switching over a given set of impedances, generating a set of reflection coefficients forming a constellation. For example, assuming binary phase-shift keying (BPSK) modulation, as illustrated in Fig. <ref>, the Tag has a pair of impedances corresponding to two symbols, and it chooses either one of them for backscattering depending on the value of the transmitted bit. In practice, the switch is usually a complementary metal-oxide-semiconductor (CMOS) switch <cit.>, which is an active element. The switch and the set of impedances can be treated as a variable impedance. Since the switch is an active component and the impedances are passive components, the variable impedance can be regarded as a partially active component. For backward information transmission modulation, one unique design issue is the consideration of energy-harvesting efficiency. Specifically, it is desirable to design a modulation scheme for BackCom that reduces the reflected energy and thereby increases the harvested energy. In general, this objective can be achieved by shifting the constellation points on the complex plane towards the origin <cit.>. This, however, may increase the detection error-rate, introducing an energy-rate tradeoff. The backward information transmission is the dominant mode for most of the conventional RFID applications, which have asymmetric data traffics, e.g., a low-rate command through the forward information transmission and a high-rate information-bearing data through the backward information transmission. However, the forward and backward information transmission are equally important for future IoT applications, which require peer-to-peer network architecture and symmetric communication links between the massive number of devices. §.§ Energy-Rate Tradeoff Besides the said tradeoff arising from modulation design, there exists another one due to bursty transmission by IoT devices. For instance, a sensor for crowd-sensing reports data only upon receiving a query and spends the remaining time on other activities e.g., sensing and computing. Leveraging this characteristic, a backscatter Tag can be designed to periodically switch between the two modes, namely the silent (or energy harvesting) and active modes. Then the duty cycle is defined as the percentage of time for the active mode. In the silent mode, the energy of the incident wave is fully harvested without reflection, by matching the variable impedance to that of the antenna, and the circuit may be turned off for energy conservation. In the active mode, the Tag circuit is activated to receive or transmit data by backscattering where only the fraction of harvested energy is much smaller (see Fig. <ref>). Consequently, the duty-cycle is a key design parameter for regulating an energy-rate tradeoff. Also, we refer the interested reader to a recent survey paper, i.e., <cit.>, with detailed comparison of different BackCom systems including energy sources, operating range, bandwidths, multiple access schemes. § NEXT GENERATION BACKCOM SYSTEM AND NETWORK ARCHITECTURES The conventional BackCom system discussed in Sec. <ref> is the simplest BackCom system, as illustrated in Fig. <ref>. The next generation BackCom systems for IoT are much more complex and can be divided into the following categories. §.§ Multiple-Access BackCom Systems Many real-life applications can be modeled as a multiple-access (MAC) BackCom system where a single Reader serves multiple Tags, as illustrated in Fig. <ref>. For instance, in a warehouse, an administrator can use a single Reader to collect information simultaneously from hundreds or thousands of items equipped with RFID Tags. In a smart city, a data aggregator can receive sensing data from a large number of backscatter sensors at the same time. The key challenge in multiple-access BackCom systems is how to deal with collisions that arise as a result of concurrent Tag transmissions. In this regard, a simple solution is to avoid collisions between multi-Tag transmissions using the traditional MAC schemes including space/frequency/code/time-division multiple-access (SDMA/FDMA/CDMA/TDMA), see <cit.> and <cit.> and references therein. In SDMA, the Reader scans the space around it using directional antennas. The Tags in the reading range are distinguished by their angular position. Therefore, the SDMA scheme requires large antenna arrays at the Readers which increases the complexity. In FDMA, the Tags adjust the frequency of alternation between two switch states on the order of Hz or kHz, and the Reader detects Tag's signal in frequency domain. Thus, the FDMA scheme incurs high signal processing complexity due to the fast Fourier transform (FFT) computation. In CDMA, different Tags use different orthogonal codes to modulate their signals. Due to the near-far problem, the Tags of a CDMA network are required to do power control (i.e., to adjust the reflection coefficients) so as to achieve the same power level at the Reader. This introduces higher protocol complexity. Due to its simplicity, TDMA is perhaps the most practical scheme for MAC BackCom systems where Tags transmit in separate pre-assigned time slots in each frame. The inherent closed-loop signaling for BackCom facilitates the needed Reader-Tag synchronization for TDMA. Researchers have also attempted to design new MAC schemes exploiting the characteristics of BackCom. For instance, an interesting new method for collision avoidance was proposed in <cit.> for MAC BackCom that treats bursty backscatter transmissions by Tags as a sparse code and decodes multi-Tag data at the Reader using a compressive-sensing algorithm. §.§ Ad Hoc BackCom Systems To avoid unnecessarily overloading the core network and to reduce latency, distributed device-to-device or ad hoc communications is envisaged in future IoT, creating BackCom interference channels. Compared with conventional interference channels, a backscatter node reflects all incident interference signals, resulting in interference regeneration. The phenomenon is illustrated in Fig. <ref> showing a two-link BackCom interference channel where Readers 1 and 2 each transmit a CW to backscatter Tags 1 and 2, respectively. In total, each Reader, e.g. Reader 1, is exposed to two interference components due to Tag 2's modulation and reflection of CWs from Readers 1 and 2. In theory, the number of interference components received by each Reader can increase as the square of the number of coexisting links instead of linearly with the number in the conventional case. As a result, the interference issue is particularly severe in BackCom interference networks due to interference regeneration. One effective way for coping with this issue is to adopt spread spectrum techniques in BackCom exploiting its low data rates <cit.>. §.§ Power Beacon and Ambient RF Powered BackCom Systems In the future IoT, most transmitting nodes are expected to be energy constrained and cannot act as Readers to emit high-power CW and power their receivers nor to enable Reader-to-Tag transmissions. For IoT, there are two practical solutions for the power-source problem: One solution is to deploy dense low-complexity and low-cost power beacons (PBs) dedicated for microwave power transfer (MPT) <cit.>, to enable BackCom links in their proximity by beaming towards them strong CWs, as illustrated in Fig. <ref>. Then the Tag is able to perform RF energy harvesting and BackCom to the Reader. Such BackCom system is termed as wirelessly powered BackCom. The other solution is to harvest the energy from ambient RF signals, such as the signals from cellular, TV broadcasting and WiFi networks. Leveraging the ambient RF signals can allow direct Tag-to-Reader communication without Readers supplying power, which has motivated various relevant designs recently <cit.>. For instance, a backscatter Tag can transmit data to a peer by reflecting an incident base-station signal, as illustrated in Fig. <ref> <cit.>. Therefore, instead of having some specific PBs or Readers that enables tag-to-reader communications, the ambient backscatter system can use existing infrastructure and benefit from signals that are not intended for itself. Such an energy harvesting BackCom system differs from one with a Reader or PB in two important aspects. First, the CW is replaced with a modulated signal and thus the reflection is double modulated with superimposed unintended and intended data. This problem can be tackled by exploiting the asymmetry in the high-rate ambient and low-rate backscatter signals. As result, the latter can be detected after suppressing the former by time averaging over each symbol duration. The second issue is the incident ambient signal is much weaker than the CW from an intended Reader or PB due to a much longer propagation distance. Consequently, BackCom with ambient RF signal is suitable only for either short-distance or infrequent Tag-to-Reader communications. §.§ BackCom Systems with Technology Conversion To communicate with passive Tags embedded with different types of available commercialized devices, we envisage BackCom systems with technology conversion to become common place. One technology conversion is to leverage a Bluetooth signal for BackCom between a Tag and a WiFi device <cit.>, as illustrated in Fig. <ref>. Consider a wearable-device network, which consists of an implanted BackCom sensor (i.e., a Tag), a Bluetooth low energy (BLE) watch, and a smart phone, which is a WiFi device. Based on the BLE protocol, the Bluetooth watch can utilize one advertisement channel of the 2.4 GHz ISM band and adopt a Gaussian Frequency Shift Keying (GFSK) method that encodes bits using two frequency tones <cit.>[ Note that a typical Bluetooth device uses frequency hopping technique across the 36 data channels spread across the 2.4 GHz ISM band, and uses GFSK modulation on three advertisement channels.]. Hence, the watch emits a CW at either of these frequency tones. By leveraging such CW-like Bluetooth signals, the BackCom Tag can get its sensed e-health information collected by the smart phone through backscatter modulation. Although the Bluetooth CW frequency is usually different from WiFi, by adopting a FSK modulation through properly switching between different impedances, the BackCom Tag is able to shift the Bluetooth CW to the WiFi channel, hence achieving the BackCom with the smart phone. Another technology conversion is to leverage WiFi. Consider a BackCom system for smart home applications, which consists of a WiFi access point (AP) and a backscatter IoT sensor. The AP transmits packets to the WiFi clients, such as laptops and smart phones, which is also received by the backscatter IoT sensor. Then the sensor is able to modulate its data on the unintended signal and backscatters the signal to the AP. The double modulated backscattered signal is used by the AP, which obtains the Tag's information by removing its own transmitted information. In this way, sensors can be powered by WiFi APs and also rely on them to access the Internet even in the presence of access by WiFi devices, thereby providing inter-connectivity of everything for smart homes. §.§ Comparison with a Traditional System In order to show the advantage of the next generation BackCom systems over traditional wireless sensor networks (WSN), we consider the example of a wirelessly powered BackCom system. Using MATLAB, we focus on a system-level simulation of an IoT network, where each IoT sensor node adopts either a backscatter circuit or a traditional RF circuit (including a mixer, DAC, and amplifier). Both types of the nodes are powered by a PB. The system level application of the simulation can be a smart supermarket. For example, the customers in a shopping mall/supermarket may use their smart phones (which act as Readers) to collect information from different goods on the shelf (which are sent by Tags). The IoT nodes with a density of 0.2 nodes/m^2 are randomly distributed in a disk region with the radius of 5 m and served by the PB at the center. Each node aims to perform sensing and transmission to its intended receiver at a fixed distance of 0.5 m. The IoT nodes adopt a harvesting-then-sensing-and-transmission task sequence per 100-ms time slot, where energy harvesting occupies 80 ms and the other operations 20 ms if there is sufficient energy. The antenna effective apertures of all nodes are 0.001 m^2. We consider Friis free-space channels for wireless power transfer and information transmission. The carrier frequency is 2.4 GHz, the modulation scheme BPSK, and the noise power at the receiver -70 dBm, and the RF energy harvesting efficiency 50%. The IoT nodes use orthogonal Walsh-Hadamard code with length 16 for information transmission. Note that each communication pair has no information of the other pairs and do not perform power control. Each sensing task consumes 0.1 μJ of energy (e.g., ambient light sensor TSL2550D), and the digital circuit power consumption during the active mode is 2.5 μW <cit.>. Each BackCom IoT node does not perform RF energy harvesting while backscattering. While backscattering using BPSK modulation, the Tag switches the value of the load impedance between zero (a short state) and infinite (an open state). The impedance switching frequency is equal to the chip rate of the orthogonal code. The switch (e.g., DG2012 <cit.>) has typical power consumption of around 0.1 μW. For the traditional IoT node, the power consumptions of the DAC and the mixer during active mode are 15 μW and 0.1 mW, respectively (e.g., DAC8830 and AD831), and the power amplifier (e.g., LMH6609) has power consumption of about 50 mW. Note that since the local oscillator (LO) is the most important element of the mixer, the power consumption of the mixer is approximated by that of the LO. The active components and their energy/power consumption are listed in Table <ref>. Based on these practical settings, we investigate how much performance improvement is achieved in the IoT network by adopting the backscatter circuit. Fig. <ref> plots the average BER at the receiver versus the PB's transmit power. We can see that the BER achieved by the backscatter nodes is significantly reduced compared with the traditional nodes within a practical range of PB's transmit power, e.g., the BER is reduced by 100 times when the PB has a 25 dBm transmission power[This PB transmit power is practical. This is because according to FCC regulations, the transmission power should be less than 30 dBm for omnidirectional transmission.]. Fig. <ref> plots the percentage of active nodes. We see that the percentage of active IoT nodes increases with the PB's transmit power, and the percentage of the active backscatter nodes is much larger than that of the traditional nodes. For example, the improvement of the percentage of active nodes is 7% when the transmit power is 20 dBm, and the improvement is 20% when the transmit power is 25 dBm. Therefore, the simulation results quantitatively show that the backscatter nodes can significantly improve the performance of the IoT network, and the wirelessly powered BackCom is a promising solution for future IoT applications. In the following section, we present more advanced BackCom techniques. § EMERGING BACKCOM TECHNIQUES The future IoT applications require the BackCom systems to enable long-distance, low-latency, and high-rate communications between massive IoT devices and also enable their ad hoc communication. A few advanced BackCom techniques aiming at tackling these challenges are discussed as follows. §.§ BackCom Systems with Power Beacons Different from conventional RFID Tags which only need to report its ID information, the IoT BackCom Tags are also required to perform sensing and computing which can consume more energy. How to wirelessly power BackCom using PBs (see earlier discussions) and maximize the power-transfer efficiency is a significant design issue. The efficiency can be increased by energy beamforming from a multi-antenna PB and a Tag. To this end, the PB needs to know the forward channel state information (CSI). For a free-space channel, the CSI reduces to the Tag direction with respect to the PB. The PB can form a beam using the well-known retrodirective beamforming technique that automatically generates a beamformer by conjugating the observation of the pilot signal sent (i.e., reflected) by the Tag. On the other hand, for a scattering channel, the beamformer designs are more complex but can exploit the so called “key-hole" channel structure due to backscattering (see e.g. <cit.>). Another technique for improving the power-transfer efficiency is to optimize the CW waveform, e.g., as a weighted sum of multiple sinusoidal waves (see e.g., <cit.>). The design aims at increasing the peak-to-average power ratio of the PB signal that yields a higher energy-harvesting efficiency due to its non-linearity as a function of incident power. §.§ Full-Duplex BackCom For future IoT, there would be massive number of Tag-to-Reader links existing at the same time. Though information flow in RFID applications is usually uni-directional, message exchange between nodes is common in IoT. Thus, full-duplex communication can substantially reduce the latency, and improve the efficiency of spectrum utilization of the IoT-Reader-to-Tag links. For BackCom full-duplexing, there is a performance tradeoff between the transmission and the reception of a full-duplex BackCom node. A smaller backscatter coefficient reduces the reflected signal power and thereby increases the received signal power, and vice versa. There are two methods for implementing full-duplex BackCom. Both require a Reader to have a full-duplex antenna allowing simultaneous transmission and reception <cit.>. Consider simultaneous forward and backward information transmission in a BackCom system having one pair of Reader and Tag. For the first method, by leveraging prior knowledge of forward information, the Reader can cancel it in the backward information transmission signal and thereby retrieve the backward information. This method supports symmetric bidirectional data rates. The other method exploits the rate asymmetry in data transfer (Tag to Reader) and control signaling (Reader to Tag). Specifically, the signals have different frequencies and can be decoupled by filtering or averaging, see <cit.> and references therein. §.§ Time-Hopping BackCom As mentioned, interference in IoT networks with high node density poses a key design challenge that is exacerbated by backscattering. For IoT devices which are sensors for smart cities and homes, the burstiness in their transmissions can be exploited for tackling interference. A suitable transmission technique is time-hopping spread spectrum (TH-SS), where each Tag randomly selects one of N time slots for transmitting a single symbol and the choices of different Tags are independent <cit.>. As a result, the number of simultaneous links is reduced by the factor N, called the processing gain, thereby suppressing interference. An extreme form of TH-SS can be realized by ultra-wideband (UWB) transmission, where an extremely large processing gain is achieved using ultra-narrow pulses whose durations are in the order of nano-second. §.§ MIMO BackCom A key characteristic of the backward BackCom (i.e., the tag-to-reader information transmission) is the double path-loss due to the fact that the backscatter signal received at a Reader propagates through the closed-loop channel cascading the downlink and uplink channels. The resultant path loss is especially server as the propagation distances in IoT are much longer than those for RFID applications. To enhance link reliability, one solution is to deploy antenna arrays at Readers and Tags and apply spatial-diversity techniques. Furthermore, multi-antennas can enhance the efficiency of wireless power transfer by enabling transmit energy beamforming and increasing receive antenna apertures. Backscattering introduces a special channel structure for the backward information transmission in a multiple-input-multiple-output (MIMO) BackCom system, called a dyadic MIMO channel, which captures the composite fading in the forward and backward channels <cit.>. To be specific, the CW signals sent by the transmit antennas of the Reader propagate through the forward MIMO channel, and are first combined at each antenna of the Tag and then backscattered, and lastly propagate through the backward MIMO channel to arrive at the receive antennas of the Reader. The resultant dyadic MIMO channel has a similar structure as the classic keyhole MIMO channel. The space-time coding is a simple but suitable technique for achieving the diversity gain of such a channel. By adopting space-time coding, it has been proved that the achievable maximum diversity order is equal to the number of the Tag's antennas <cit.>. In other words, in contrast to the conventional MIMO channel, increasing the number of receive antenna at the Reader cannot continuously enhance the reliability of the backward information transmission. § IOT APPLICATIONS FOR BACKCOM §.§ BackCom for Smart Homes/Cities Low-power or passive BackCom devices with energy harvesting capabilities can be densely deployed to provide pervasive and uninterrupted sensing and computing services that provide a platform for implementing applications for smart homes/cities. In a smart home, a large number of passive BackCom sensors can be placed at flexible locations (e.g., embedded in walls, ceilings, and furniture). They are freed from the constraints due to recharging or battery replacements as one or multiple in-house PBs can be deployed to simultaneously power all the sensors or otherwise they can operate on ambient energy harvesting. The tasks performed by the sensors have a wide range such as detection of gas leak, smoke and CO, monitoring movements, indoor positioning, and surveillance [see Fig. <ref>]. As an example, BackCom-based smart dustbins are able to monitor their trash levels and communicate the information to passing-by garbage trucks by backscattering, streamlining the trash-collection process. Another example is that household robots are able to use the backscattered signals from the Tags located on doors and furniture for indoor navigation <cit.>. In a smart city, ubiquitous BackCom sensor nodes can be placed in every city corner such as buildings, bridges, trees, street lamps, and parking areas. They can streamline the city operations and improve our life quality via e.g., monitoring of air/noise pollution and traffic and parking-availability indicating. The efficient sensing-data fusion and wireless power for BackCom sensors can be realized by the deployment of integrated PBs and APs at fixed locations or mounted on autonomous ground vehicles or UAVs, providing full-city coverage without costly backhaul networks <cit.>. §.§ BackCom for Biomedical Applications IoT biomedical applications, such as plant/animal monitoring, wearable, and implantable human health monitoring, require tiny and low heat-radiation communication devices. BackCom devices, which do not rely on any active RF component, can meet such requirements and thereby avoid causing any significant effect on the plants, animals, tissues or organs being monitored. These advantages make BackCom a promising solution for IoT biomedical applications. One example is the BackCom-based smart Google Contact Lens, as illustrated in Fig. <ref>. The lens was invented in Google in 2014 for the purpose of assisting people with diabetes by constantly measuring the glucose levels in their tears (once per second). The device consists of a miniaturized glucose sensor and a tiny BackCom Tag. The Tag is able to provide energy to the sensor by RF energy harvesting from a wireless controller, and also backscatter the measured blood sugar level to the wireless controller for diagnosing purpose. Looking into the future, we envisage that BackCom will find a wide range of biomedical applications. In particular, implantable tiny BackCom neural devices with ultra-low power consumption and heat radiation may be placed on the surface of the patient's brain to help the study, diagnosis and treatment of diseases such as epilepsy and Parkinson's disease, where the BackCom implants act as the brain-computer interface. §.§ BackCom for Logistics BackCom for logistics is a very attractive proposition due to the ultra-low manufacturing cost of simple and passive BackCom Tags, as illustrated in Fig. <ref>. For example, as early as 2007, the biggest 100 suppliers of the global renowned chain commercial group Wal-Mart have used the BackCom technology for logistics tracking. The technology has been helping the companies to substantially reduce operational cost, guarantee product quality, and accelerate the processing speed. In the past decade, the popularization and the application of BackCom have brought revolutionary changes to the logistics industry, due to its advantages compared with the conventional bar code technology, such as reduced manual control, long service lives, long reading distances, and encrypt-able and rewritable data. Looking into the future, apart from the existing BackCom techniques for logistics tracking and management, BackCom-based three-dimensional orientation tracking is an emerging technique. By attaching an array of low-cost passive BackCom Tags as orientation sensors on the surface of the target objects, three-dimensional orientation information is available at the Reader by analyzing the relative phase offset between different Tags. In this way, human workers can be warned when the angle of a cargo is larger than a threshold. § CONCLUDING REMARKS IoT that aims to enable both the activity and the connectivity of billions of energy-consuming smart nodes, is challenged from the energy perspective. The BackCom systems and techniques provide promising solutions for tackling this challenge. In this article, we have introduced the basic principles for BackCom, summarized existing BackCom system and network architectures and discussed several emerging advanced BackCom techniques. Moreover, we have described various applications of BackCom in IoT applications. With rapid advancements in both theory and applications, the technology is expected to play a key role in future IoT by enabling truly ubiquitous network connectivity, and pervasive sensing and computing. § LIST OF ABBREVIATIONS AP: access point; BackCom: backscatter communications; BLE: Bluetooth low energy; BPSK: binary phase-shift keying; CDMA: code-division multiple-access; CSI: channel state information; CW: continuous wave; FDMA: frequency-division multiple-access; GFSK: Gaussian frequency shit keying; MAC: multiple access; OFDMA: orthogonal frequency-division multiple-access; PB: power beacon; PT: power transfer; RF: radio-frequency; RFID: radio-frequency identification; SDMA: space-division multiple-access; TDMA: time-division multiple-access; TH-SS: time-hopping spread spectrum; UAV: unmanned aerial vehicle; UWB: ultra-wideband; WSN: wireless sensor networks § COMPETING INTERESTS The authors declare that they have no competing interests. § FUNDING The work of W. Liu was supported by the Australian Research Council's Australian Laureate Fellowships scheme (project number FL160100032). The work of K. Huang was supported by Hong Kong Research Grants Council under the Grants 17209917 and 17259416. The work of S. Durrani and X. Zhou was supported by the Australian Research Council's Discovery Project Funding Scheme (Project number DP170100939). § AUTHOR'S CONTRIBUTIONS K. Huang and W. Liu came up with the original idea. W. Liu drafted the manuscript under the supervision of K. Huang. X. Zhou and S. Durrani helped to further refine the manuscript. All authors read and approved the final manuscript. bmc-mathphys 21 #1ISBN #1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1et al.#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1#1<><#>1#1#1#1#1#1#1#1#1#1#1#1#1#1PreBibitemsHookambient_EH Sudevalayam, S., Kulkarni, P.: Energy harvesting sensor nodes: Survey and implications. IEEE Commun. Surveys Tuts. 13(3), 443–461 (2011) Seattle Parks, A.N., Sample, A.P., Zhao, Y., Smith, J.R.: A wireless sensing platform utilizing ambient RF energy. In: Proc. IEEE PAWR, pp. 160–162 (2013) Tokyo Vyas, R., Nishimoto, H., Tentzeris, M., Kawahara, Y., Asami, T.: A battery-less, energy harvesting device for long range scavenging of wireless power from terrestrial TV broadcasts. In: Proc. IEEE MTT-S, pp. 1–3 (2012) PowerCast PowerSpot. (2017). <https://www.powercastco.com/wp-content/uploads/2018/06/TX91503-UserManualREV4.pdf> Huang15 Huang, K., Zhou, X.: Cutting the last wires for mobile communications by microwave power transfer. IEEE Communications Magazine 53(6), 86–93 (2015) Boyer14 Boyer, C., Roy, S.: Backscatter communication and RFID: Coding, energy, and MIMO analysis. IEEE Transactions on Communications 62(3), 770–785 (2014) CC2520 CC2520 DATASHEET. (2007). <http://www.ti.com/lit/ds/symlink/cc2520.pdf> Tagconsumption Karthaus, U., Fischer, M.: Fully integrated passive UHF RFID transponder IC with 16.7-μw minimum RF input power. IEEE Journal of Solid-State Circuits 38(10), 1602–1608 (2003) WanchunBackCom Liu, W., Huang, K., Zhou, X., Durrani, S.: Full-duplex backscatter interference networks based on time-hopping spreading spectrum. IEEE Transactions on Wireless Communications 16(7), 4361–4377 (2017) CMOS DG2012 DATASHEET. (2007). <http://www.vishay.com/docs/72176/dg2012.pdf> AmBackSurvey Huynh, N.V., Hoang, D.T., Lu, X., Niyato, D., Wang, P., Kim, D.I.: Ambient backscatter communications: A contemporary survey. IEEE Communications Surveys & Tutorials 20(4), 2889–2922 (2018) Bletsas Bletsas, A., Siachalou, S., Sahalos, J.N.: Anti-collision backscatter sensor networks. IEEE Trans. Wireless Commun. 8(10), 5018–5029 (2009) Katabi12 Wang, J., Hassanieh, H., Katabi, D., Indyk, P.: Efficient and reliable low-power backscatter networks. In: Proc. ACM SIGCOMM, pp. 61–72 (2012) Ambient13 Liu, V., Parks, A., Talla, V., Gollakota, S., Wetherall, D., Smith, J.R.: Ambient backscatter: Wireless communication out of thin air. ACM SIGCOMM Computer Communication Review 43(4), 39–50 (2013) MMAC Liu, W., Liang, Y.C., Li, Y., Vucetic, B.: Backscatter multiplicative multiple-access systems: Fundamental limits and practical design. IEEE Trans. Wireless Commun. 17(9), 5713–5728 (2018) SIGCOMM16 Iyer, V., Talla, V., Kellogg, B., Gollakota, S., Smith, J.: Inter-technology backscatter: Towards internet connectivity for implanted devices. In: Proc. ACM SIGCOMM, pp. 356–369 (2016) Yang15 Yang, G., Ho, C.K., Guan, Y.L.: Multi-antenna wireless energy transfer for backscatter communication systems. IEEE Journal on Selected Areas in Communications 33(12), 2974–2987 (2015) Bruno16 Clerckx, B., Bayguzina, E.: Waveform design for wireless power transfer. IEEE Transactions on Signal Processing 64(23), 6313–6328 (2016) FullDuplexGuo Sabharwal, A., Schniter, P., Guo, D., Bliss, D.W., Rangarajan, S., Wichman, R.: In-band full-duplex wireless: Challenges and opportunities. IEEE J. Sel. Areas Commun. 32(9), 1637–1652 (2014) RFIDRobot Pomárico-Franquiz, J.J., Shmaliy, Y.S.: Accurate self-localization in RFID tag information grids using FIR filtering. IEEE Trans. Ind. Informat. 10(2), 1317–1326 (2014) YongZeng16 Zeng, Y., Zhang, R., Lim, T.J.: Wireless communications with unmanned aerial vehicles: Opportunities and challenges. IEEE Communications Magazine 54(5), 36–42 (2016) <refgrp> <bibl id="B1"> <title><p>Energy Harvesting Sensor Nodes: Survey and Implications</p></title> <aug> <au><snm>Sudevalayam</snm><fnm>S.</fnm></au> <au><snm>Kulkarni</snm><fnm>P.</fnm></au> </aug> <source>IEEE Commun. Surveys Tuts.</source> <pubdate>2011</pubdate> <volume>13</volume> <issue>3</issue> <fpage>443</fpage> <lpage>461</lpage> </bibl> <bibl id="B2"> <title><p>A wireless sensing platform utilizing ambient RF energy</p></title> <aug> <au><snm>Parks</snm><fnm>AN</fnm></au> <au><snm>Sample</snm><fnm>AP</fnm></au> <au><snm>Zhao</snm><fnm>Y</fnm></au> <au><snm>Smith</snm><fnm>JR</fnm></au> </aug> <source>Proc. IEEE PAWR</source> <pubdate>2013</pubdate> <fpage>160</fpage> <lpage>-162</lpage> </bibl> <bibl id="B3"> <title><p>A battery-less, energy harvesting device for long range scavenging of wireless power from terrestrial TV broadcasts</p></title> <aug> <au><snm>Vyas</snm><fnm>R</fnm></au> <au><snm>Nishimoto</snm><fnm>H</fnm></au> <au><snm>Tentzeris</snm><fnm>M</fnm></au> <au><snm>Kawahara</snm><fnm>Y</fnm></au> <au><snm>Asami</snm><fnm>T</fnm></au> </aug> <source>Proc. IEEE MTT-S</source> <pubdate>2012</pubdate> <fpage>1</fpage> <lpage>-3</lpage> </bibl> <bibl id="B4"> <title><p>PowerSpot</p></title> <pubdate>2017</pubdate> <url>https://www.powercastco.com/wp-content/uploads/2018/06/TX91503-UserManualREV4.pdf</url> </bibl> <bibl id="B5"> <title><p>Cutting the last wires for mobile communications by microwave power transfer</p></title> <aug> <au><snm>Huang</snm><fnm>K</fnm></au> <au><snm>Zhou</snm><fnm>X</fnm></au> </aug> <source>IEEE Communications Magazine</source> <pubdate>2015</pubdate> <volume>53</volume> <issue>6</issue> <fpage>86</fpage> <lpage>-93</lpage> </bibl> <bibl id="B6"> <title><p>Backscatter Communication and RFID: Coding, Energy, and MIMO Analysis</p></title> <aug> <au><snm>Boyer</snm><fnm>C</fnm></au> <au><snm>Roy</snm><fnm>S</fnm></au> </aug> <source>IEEE Transactions on Communications</source> <pubdate>2014</pubdate> <volume>62</volume> <issue>3</issue> <fpage>770</fpage> <lpage>785</lpage> </bibl> <bibl id="B7"> <title><p>CC2520 DATASHEET</p></title> <pubdate>2007</pubdate> <url>http://www.ti.com/lit/ds/symlink/cc2520.pdf</url> </bibl> <bibl id="B8"> <title><p>Fully integrated passive UHF RFID transponder IC with 16.7-μW minimum RF input power</p></title> <aug> <au><snm>Karthaus</snm><fnm>U</fnm></au> <au><snm>Fischer</snm><fnm>M</fnm></au> </aug> <source>IEEE Journal of Solid-State Circuits</source> <pubdate>2003</pubdate> <volume>38</volume> <issue>10</issue> <fpage>1602</fpage> <lpage>-1608</lpage> </bibl> <bibl id="B9"> <title><p>Full-Duplex Backscatter Interference Networks Based on Time-Hopping Spreading Spectrum</p></title> <aug> <au><snm>Liu</snm><fnm>W.</fnm></au> <au><snm>Huang</snm><fnm>K.</fnm></au> <au><snm>Zhou</snm><fnm>X.</fnm></au> <au><snm>Durrani</snm><fnm>S.</fnm></au> </aug> <source>IEEE Transactions on Wireless Communications</source> <pubdate>2017</pubdate> <volume>16</volume> <issue>7</issue> <fpage>4361</fpage> <lpage>4377</lpage> </bibl> <bibl id="B10"> <title><p>DG2012 DATASHEET</p></title> <pubdate>2007</pubdate> <url>http://www.vishay.com/docs/72176/dg2012.pdf</url> </bibl> <bibl id="B11"> <title><p>Ambient Backscatter Communications: A Contemporary Survey</p></title> <aug> <au><snm>Huynh</snm><fnm>NV</fnm></au> <au><snm>Hoang</snm><fnm>DT</fnm></au> <au><snm>Lu</snm><fnm>X</fnm></au> <au><snm>Niyato</snm><fnm>D</fnm></au> <au><snm>Wang</snm><fnm>P</fnm></au> <au><snm>Kim</snm><fnm>DI</fnm></au> </aug> <source>IEEE Communications Surveys & Tutorials</source> <pubdate>2018</pubdate> <volume>20</volume> <issue>4</issue> <fpage>2889</fpage> <lpage>2922</lpage> </bibl> <bibl id="B12"> <title><p>Anti-collision backscatter sensor networks</p></title> <aug> <au><snm>Bletsas</snm><fnm>A.</fnm></au> <au><snm>Siachalou</snm><fnm>S.</fnm></au> <au><snm>Sahalos</snm><fnm>J. N.</fnm></au> </aug> <source>IEEE Trans. Wireless Commun.</source> <pubdate>2009</pubdate> <volume>8</volume> <issue>10</issue> <fpage>5018</fpage> <lpage>5029</lpage> </bibl> <bibl id="B13"> <title><p>Efficient and reliable low-power backscatter networks</p></title> <aug> <au><snm>Wang</snm><fnm>J</fnm></au> <au><snm>Hassanieh</snm><fnm>H</fnm></au> <au><snm>Katabi</snm><fnm>D</fnm></au> <au><snm>Indyk</snm><fnm>P</fnm></au> </aug> <source>Proc. ACM SIGCOMM</source> <pubdate>2012</pubdate> <fpage>61</fpage> <lpage>-72</lpage> </bibl> <bibl id="B14"> <title><p>Ambient backscatter: Wireless communication out of thin air</p></title> <aug> <au><snm>Liu</snm><fnm>V</fnm></au> <au><snm>Parks</snm><fnm>A</fnm></au> <au><snm>Talla</snm><fnm>V</fnm></au> <au><snm>Gollakota</snm><fnm>S</fnm></au> <au><snm>Wetherall</snm><fnm>D</fnm></au> <au><snm>Smith</snm><fnm>JR</fnm></au> </aug> <source>ACM SIGCOMM Computer Communication Review</source> <pubdate>2013</pubdate> <volume>43</volume> <issue>4</issue> <fpage>39</fpage> <lpage>-50</lpage> </bibl> <bibl id="B15"> <title><p>Backscatter Multiplicative Multiple-Access Systems: Fundamental Limits and Practical Design</p></title> <aug> <au><snm>Liu</snm><fnm>W.</fnm></au> <au><snm>Liang</snm><fnm>Y.C.</fnm></au> <au><snm>Li</snm><fnm>Y.</fnm></au> <au><snm>Vucetic</snm><fnm>B.</fnm></au> </aug> <source>IEEE Trans. Wireless Commun.</source> <pubdate>2018</pubdate> <volume>17</volume> <issue>9</issue> <fpage>5713</fpage> <lpage>5728</lpage> </bibl> <bibl id="B16"> <title><p>Inter-Technology Backscatter: Towards Internet Connectivity for Implanted Devices</p></title> <aug> <au><snm>Iyer</snm><fnm>V</fnm></au> <au><snm>Talla</snm><fnm>V</fnm></au> <au><snm>Kellogg</snm><fnm>B</fnm></au> <au><snm>Gollakota</snm><fnm>S</fnm></au> <au><snm>Smith</snm><fnm>J</fnm></au> </aug> <source>Proc. ACM SIGCOMM</source> <pubdate>2016</pubdate> <fpage>356</fpage> <lpage>-369</lpage> </bibl> <bibl id="B17"> <title><p>Multi-antenna wireless energy transfer for backscatter communication systems</p></title> <aug> <au><snm>Yang</snm><fnm>G</fnm></au> <au><snm>Ho</snm><fnm>CK</fnm></au> <au><snm>Guan</snm><fnm>YL</fnm></au> </aug> <source>IEEE Journal on Selected Areas in Communications</source> <pubdate>2015</pubdate> <volume>33</volume> <issue>12</issue> <fpage>2974</fpage> <lpage>-2987</lpage> </bibl> <bibl id="B18"> <title><p>Waveform Design for Wireless Power Transfer</p></title> <aug> <au><snm>Clerckx</snm><fnm>B.</fnm></au> <au><snm>Bayguzina</snm><fnm>E.</fnm></au> </aug> <source>IEEE Transactions on Signal Processing</source> <pubdate>2016</pubdate> <volume>64</volume> <issue>23</issue> <fpage>6313</fpage> <lpage>6328</lpage> </bibl> <bibl id="B19"> <title><p>In-Band Full-Duplex Wireless: Challenges and Opportunities</p></title> <aug> <au><snm>Sabharwal</snm><fnm>A.</fnm></au> <au><snm>Schniter</snm><fnm>P.</fnm></au> <au><snm>Guo</snm><fnm>D.</fnm></au> <au><snm>Bliss</snm><fnm>D. W.</fnm></au> <au><snm>Rangarajan</snm><fnm>S.</fnm></au> <au><snm>Wichman</snm><fnm>R.</fnm></au> </aug> <source>IEEE J. Sel. Areas Commun.</source> <pubdate>2014</pubdate> <volume>32</volume> <issue>9</issue> <fpage>1637</fpage> <lpage>1652</lpage> </bibl> <bibl id="B20"> <title><p>Accurate Self-Localization in RFID Tag Information Grids Using FIR Filtering</p></title> <aug> <au><snm>Pomárico Franquiz</snm><fnm>J. J.</fnm></au> <au><snm>Shmaliy</snm><fnm>Y. S.</fnm></au> </aug> <source>IEEE Trans. Ind. Informat.</source> <pubdate>2014</pubdate> <volume>10</volume> <issue>2</issue> <fpage>1317</fpage> <lpage>1326</lpage> </bibl> <bibl id="B21"> <title><p>Wireless communications with unmanned aerial vehicles: Opportunities and challenges</p></title> <aug> <au><snm>Zeng</snm><fnm>Y.</fnm></au> <au><snm>Zhang</snm><fnm>R.</fnm></au> <au><snm>Lim</snm><fnm>T. J.</fnm></au> </aug> <source>IEEE Communications Magazine</source> <pubdate>2016</pubdate> <volume>54</volume> <issue>5</issue> <fpage>36</fpage> <lpage>42</lpage> </bibl> </refgrp> § FIGURES * Figure 1. The architecture of a backscatter Tag. * Figure 2. The forward information-transmission mode of a BackCom link. * Figure 3. The backward information-transmission mode of a BackCom link. * Figure 4. Conventional BackCom. * Figure 5. Multiple-Access BackCom. * Figure 6. BackCom Interference Network. * Figure 7. Wirelessly Powered BackCom. * Figure 8. Ambient Energy-Harvesting BackCom. * Figure 9. BackCom System with Technology Conversion. * Figure 10. BER versus PB power. * Figure 11. Percentage of active nodes versus PB power. * Figure 12. BackCom application: smart home. * Figure 13. BackCom application: smart contact lens. * Figure 14. BackCom application: logistics management.

http://arxiv.org/abs/1701.07927v2

20170127025203

Quantum-Classical Correspondence of Shortcuts to Adiabaticity

physics.class-ph

http://arxiv.org/abs/1701.08218v3

20170127230740

Emergence of a control parameter for the antiferromagnetic quantum critical metal

cond-mat.str-el

decorations.markings decorations.footprints decorations.pathmorphing [figure]labelfont=color=blue

http://arxiv.org/abs/1701.07614v1

20170126083338

Tight Inefficiency Bounds for Perception-Parameterized Affine Congestion Games

cs.GT

llncs a4paper,hmargin=3.5cm,top=3cm,nohead,bottom=3.2cm,nohead fancy OMScmsymn arrows,automata,positioning positioning,chains,fit,shapes,calc rtheorem[3][] #1#2 #3.#2 #3 (#1) OMScmsymn ⌊⌋ x[1]>p#1 P. Kleer G. Schäfer Centrum Wiskunde & Informatica (CWI), Networks and Optimization Group Vrije Universiteit Amsterdam, Department of Econometrics and Operations Research Amsterdam, The Netherlands kleer@cwi.nl, schaefer@cwi.nl Tight Inefficiency Bounds for Perception-Parameterized Affine Congestion Games Pieter Kleer1 Guido Schäfer1,2 December 30, 2023 ============================================================================== Congestion games constitute an important class of non-cooperative games which was introduced by Rosenthal in 1973. In recent years, several extensions of these games were proposed to incorporate aspects that are not captured by the standard model. Examples of such extensions include the incorporation of risk sensitive players, the modeling of altruistic player behavior and the imposition of taxes on the resources. These extensions were studied intensively with the goal to obtain a precise understanding of the inefficiency of equilibria of these games. In this paper, we introduce a new model of congestion games that captures these extensions (and additional ones) in a unifying way. The key idea here is to parameterize both the perceived cost of each player and the social cost function of the system designer. Intuitively, each player perceives the load induced by the other players by an extent of ρ≥ 0, while the system designer estimates that each player perceives the load of all others by an extent of σ≥ 0. The above mentioned extensions reduce to special cases of our model by choosing the parameters ρ and σ accordingly. As in most related works, we concentrate on congestion games with affine latency functions here. Despite the fact that we deal with a more general class of congestion games, we manage to derive tight bounds on the price of anarchy and the price of stability for a large range of parameters. Our bounds provide a complete picture of the inefficiency of equilibria for these perception-parameterized congestion games. As a result, we obtain tight bounds on the price of anarchy and the price of stability for the above mentioned extensions. Our results also reveal how one should “design” the cost functions of the players in order to reduce the price of anarchy. Somewhat counterintuitively, if each player cares about all other players to the extent of 0.625 only (instead of 1 in the standard setting) the price of anarchy reduces from 2.5 to 2.155 and this is best possible. § INTRODUCTION Congestion games constitute an important class of non-cooperative games which was introduced by Rosenthal in 1973 <cit.>. In a congestion game, we are given a set of resource from which a set of players can choose. Each resource is associated with a cost function which specifies the cost of this resource depending on the total number of players using it. Every player chooses a subset of resources (from a set of resource subsets available to her) and experiences a cost equal to the sum of the costs of the chosen resources. Congestion games are both theoretically appealing and practically relevant. For example, they have applications in network routing, resource allocation and scheduling problems. Rosenthal <cit.> proved that every congestion game has a pure Nash equilibrium, i.e., a strategy profile such that no player can decrease her cost by unilaterally deviating to another feasible set of resources. This result was established through the use of an exact potential function (known as Rosenthal potential) satisfying that the cost difference induced by a unilateral player deviation is equal to the potential difference of the respective strategy profiles. In fact, Monderer and Shapley <cit.> showed that the class of games admitting an exact potential function is isomorphic to the class of congestion games. One of the main research directions in algorithmic game theory focusses on quantifying the inefficiency caused by selfish behavior. The idea is to assess the quality of a Nash equilibrium relative to an optimal outcome. Here the quality of an outcome is measured in terms of a given social cost objective (e.g., the sum of the costs of all players). Koutsoupias and Papadimitriou <cit.> introduced the price of anarchy as the ratio between the worst social cost of a Nash equilibrium and the social cost of an optimum. Anshelevich et al. <cit.> defined the price of stability as the ratio between the best social cost of a Nash equilibrium and the social cost of an optimum. In recent years, several extensions of Rosenthal's congestion games were proposed to incorporate aspects which are not captured by the standard model. For example, these extensions include risk sensitivity of players in uncertain settings <cit.>, altruistic player behavior <cit.> and congestion games with taxes <cit.>. We elaborate in more detail on these extensions in Section <ref>. These games were studied intensively with the goal to obtain a precise understanding of the price of anarchy. In this paper, we introduce a new model of congestion games, which we term perception-parameterized congestion games, that captures all these extensions (and more) in a unifying way. The key idea here is to parameterize both the perceived cost of each player and the social cost function. Intuitively, each player perceives the load induced by the other players by an extent of ρ≥ 0, while the system designer estimates that each player perceives the load of all others by an extent of σ≥ 0. The above mentioned extensions reduce to special cases of our model by choosing the parameters ρ and σ accordingly. Despite the fact that we deal with a more general class of congestion games, we manage to derive tight bounds on the price of anarchy and the price of stability for a large range of parameters. Our bounds provide a complete picture of the inefficiency of equilibria for these perception-parameterized congestion games. As a consequence, we obtain tight bounds on the price of anarchy and the price of stability for the above mentioned extensions. While the price of anarchy bounds are (mostly) known from previous results, the price of stability results are new. As in <cit.>, we focus here on congestion games with affine cost functions.[Conceptually, our model can easily be adapted to more general latency functions; only the derivation of the respective bounds is analytically much more involved.] We illustrate our model by means of a simple example; formal definitions of our perception-parameterized congestion games are given in Section <ref>. Suppose we are given a set of m resources and that every player has to choose precisely one of these resources. The cost of a resource e ∈ [m][Given a positive integer m, we use [m] to refer to the set {1, …, m}.] is given by a cost function c_e that maps the load on e to a real value. In the classical setting, the load of a resource e is defined as the total number of players x_e using e. That is, the cost that player i experiences when choosing resource e is c_e(x_e). In contrast, in our setting players have different perceptions of the load induced by the other players. More precisely, the perceived load of player i choosing resource e is 1+ ρ(x_e - 1), where ρ≥ 0 is a parameter. Consequently, the perceived cost of player i for choosing e is c_e(1+ ρ(x_e - 1)). Note that as ρ increases players care more about the presence of other players.[In general, the parameter ρ can be player- or resource-specific, but in this work we concentrate on the homogenous case (i.e., all players have the same perception parameter).] In addition, we introduce a similar parameter σ≥ 0 for the social cost objective. Intuitively, this can be seen as the system designer's estimate of how each player perceives the load of the other players. In our example, the social cost is defined as ∑_e ∈ [m] c_e(1+σ(x_e-1)) x_e. Our Results. In Section <ref>, we prove a bound of max{ρ + 1, 2ρ(1 + σ) + 1/ρ + 1} on the price of anarchy of affine congestion games for a large range or parameters (ρ,σ) (see also Figure <ref> for an illustration). For general affine congestion games we prove that this bound is tight. Further, even for the special case of symmetric network congestion games we show that the bound (2ρ(1+σ)+1)/(ρ + 1) is asymptotically tight (on the range for which it is attained). In Section <ref>, we give a bound of √(σ(σ+2)) + σ/√(σ(σ+2)) + ρ - σ on the price of stability of affine congestion games for a large range or pairs (ρ,σ) (see below for details). For general affine congestion games we show that this bound is tight. For symmetric network congestion games we give a better (tight) bound on the price of stability for the case σ = 1 and ρ≥ 0 arbitrary (details are given in Section <ref>). An overview of the price of anarchy and the price of stability results that we obtain from (<ref>) and (<ref>) for several applications known in the literature is given in Table <ref>. The respective references where these bounds were established first are given in the rightmost column. The connection between these applications and our model is discussed in detail at the end of Section <ref>. In light of the above bounds, we obtain an (almost) complete picture of the inefficiency of equilibria (parameterized by ρ and σ); for example, see Figure <ref> for an illustration of the price of anarchy if σ = 1. Note that the price of anarchy decreases from 5/2 for ρ = 1 to 2.155 for ρ = h(1) ≈ 0.625.[The price of anarchy for ρ = h(1) was first established by Caragiannis et al. <cit.>. However, our bounds reveal that the price of anarchy is infact minimized at ρ = h(1) (see also Figure <ref>).] We refer the reader to Section <ref> for further remarks and discussions of the results. § OUR MODEL, APPLICATIONS AND RELATED WORK We first formally introduce our model of congestion games with parameterized perceptions. We then show that our model subsumes several other models that were studied in the literature as special cases. Congestion Games. A congestion game Γ is given by a tuple (N,E,(𝒮_i)_i∈ N,(c_e)_e∈ E) where N = [n] is the set of players, E the set of resources (or facilities), 𝒮_i ⊆ 2^E the set of strategies of player i, and c_e : _≥ 0→_≥ 0 the cost function of facility e. Given a strategy profile s = (s_1,…,s_n) ∈×_i 𝒮_i, we define x_e as the number of players using resource e, i.e., x_e = x_e(s) = |{i ∈ N : e ∈ s_i}|. If 𝒮_i = 𝒮_j for all i,j ∈ N, the game is called symmetric. For a given graph G = (V,E), we call Γ a (directed) network congestion game if for every player i there exist s_i,t_i ∈ V such that 𝒮_i is the set of all (directed) (s_i,t_i)-paths in G. An affine congestion game has cost functions of the form c_e(x) = a_ex + b_e with a_e,b_e ≥ 0. If b_e = 0 for all e ∈ E, the game is called linear. Our Model. We introduce our unifying model of perception-parameterized congestion games with affine latency functions. For a fixed parameter ρ≥ 0, we define the cost of player i ∈ N by C_i^ρ(s) = ∑_e ∈ s_i c_e(1 + ρ(x_e - 1)) = a_e[1 + ρ(x_e-1)] + b_e for a given strategy profile s = (s_1,…,s_n). For a fixed parameter σ≥ 0, the social cost of a strategy profile s is given by C^σ(s) = ∑_i C_i^σ(s) = ∑_e ∈ Ex_e(a_e[1 + σ(x_e-1)] + b_e). We refer to the case ρ = σ = 1 as the classical congestion game with cost functions c_e(x) = a_e x + b_e for all e ∈ E. Inefficiency of Equilibria. A strategy profile s is a Nash equilibrium if for all players i ∈ N it holds that C_i^ρ(s) ≤ C_i^ρ(s_i',s_-i) for all s_i' ∈𝒮_i, where (s_i',s_-i) denotes the strategy profile in which player i plays s_i' and all the other players their strategy in s. The price of anarchy (PoA) and price of stability (PoS) of a game Γ are defined as PoA(Γ,ρ,σ) = max_s ∈ NE C^σ(s)/min_s^* ∈×_i 𝒮_iC^σ(s^*) and PoS(Γ,ρ,σ) = min_s ∈ NE C^σ(s)/min_s^* ∈×_i 𝒮_iC^σ(s^*), where NE = NE(ρ) denotes the set of Nash equilibria with respect to the player costs as defined in (<ref>). For a collection of games ℋ we define PoA(ℋ,ρ,σ) = sup_Γ∈ℋPoA(Γ,ρ,σ) and PoS(ℋ,ρ,σ) = sup_Γ∈ℋPoS(Γ,ρ,σ). Rosenthal <cit.> shows that classical congestion games (i.e., ρ = σ = 1) have an exact potential function: Φ : ×_i 𝒮_i → is an exact potential function for a congestion game Γ if for every strategy profile s, for every i ∈ N and every s'_i ∈𝒮_i: Φ(s) - Φ(s_-i,s_i') = C_i(s) - C_i(s_-i,s_i'). The Rosenthal potential Φ(s) = ∑_e ∈ E∑_k=1^x_e c_e(k) is an exact potential function for classical congestion games. Applications. We review various models that fall within, or are related to, the framework proposed above (for certain values of ρ and σ). These models sometimes interpret the parameters differently than explained above. Altruism <cit.>. We can rewrite the cost of player i as C^ρ_i(s) = ∑_e ∈ s_i (a_ex_e + b_e) + (ρ - 1)a_e(x_e-1). The term (ρ-1)a_e(x_e-1) can be interpreted as a “dynamic” (meaning load-dependent) tax that players using resource e have to pay.[In fact, for ρ = 2 this corresponds to the dynamic taxes as proposed in a technical report by Singh <cit.>.] For 1 ≤ρ≤∞ and σ = 1, this model is equivalent to the altruistic player setting proposed by Caragiannis et al. <cit.>. Chen et al. <cit.> also study this model of altruism for 1 ≤ρ≤ 2 and σ = 1. (The equivalence between the altruistic model and our model is proved in Lemma <ref> in the appendix.) Constant taxes <cit.>. We can rewrite the cost of player i as C^ρ_i(s) = ∑_e ∈ s_iρ a_e x_e + (1-ρ)a_e + b_e. Now a strategy profile s is a Nash equilibrium if for every player i and every s'_i ∈𝒮_i ∑_e ∈ s_iρ a_e x_e + (1-ρ)a_e + b_e ≤∑_e ∈ s_i ∩ s_i'ρ a_e x_e + (1-ρ)a_e + b_e + ∑_e ∈ s_i'∖ s_iρ a_e (x_e+1) + (1-ρ)a_e + b_e. Dividing by ρ gives that s is also a Nash equilibrium for the cost functions T^ρ_i(s) = ∑_e ∈ s_i a_e x_e + b_e/ρ + ∑_e ∈ s_i1 - ρ/ρ a_e. That is, s is a Nash equilibrium in a classical congestion game in which players take into account constant resource taxes of the form (1-ρ)/ρ· a_e. Caragiannis, Kaklamanis and Kanellopoulos <cit.> study this type of taxes, which they call universal tax functions, for ρ satisfying (1-ρ)/ρ = 3/2√(3) - 2. They consider these taxes to be refundable, i.e., they are not taken into account in the social cost, which is equivalent to the case σ = 1. Note that the function τ: (0,1] → [0,∞) defined by τ(ρ) = (1-ρ)/ρ is bijective.[This relation between altruism (or spite) and constant taxes is also mentioned by Caragiannis et al. <cit.>.] Risk sensitivity under uncertainty <cit.>. Nikolova, Piliouras and Shamma <cit.> consider congestion games in which there is a (non-deterministic) order of the players on every resource. A player is only affected by players in front of her. That is, the load on resource e for player i in a strict ordering r, where r_e(i) denotes the position of player i, is given by x_e(i) = |{j ∈ N : r_e(j) ≤ r_e(i)}|. The cost of player i is then C_i(s) = ∑_e ∈ s_i c_e(x_e(i)). Note that x_e(i) is a random variable if the ordering is non-deterministic. The social cost of the model is defined by the sum of all player costs, C^1/2(s) = ∑_e ∈ E a_e x_e(x_e+1)/2 + b_e which is independent of the ordering r. (This holds because in every ordering there is always one player first, one player second, and so on.) Note that the social cost corresponds to the case σ = 1/2 in our framework. Nikolova et al. <cit.> study various risk attitudes towards the ordering r that is assumed to have a uniform distribution over all possible orderings. The two relevant attitudes are that of risk-neutral players and players applying Wald's minimax principle. Risk-neutral players define their cost as the expected cost under the ordering r, which correspond to the case ρ = 1/2 in (<ref>). This can roughly be interpreted as that players expect to be scheduled in the middle on average. Wald's minimax principle implies that players assume a worst-case scenario, i.e., being scheduled last on all the resources. This corresponds to the case ρ = 1. Approximate Nash equilibria <cit.>. Suppose that s is a Nash equilibrium under the cost functions defined in (<ref>). Then, in particular, we have C^1_i(s) ≤ C^ρ_i(s) ≤ C^ρ_i(s_i', s_-i) ≤ρ C^1_i(s_i', s_-i) for any player i and s_i' ∈𝒮_i and ρ≥ 1. That is, we have C^1_i(s) ≤ρ· C^1_i(s_i', s_-i) which means that the strategy profile s is a ρ-approximate equilibrium, as studied by Christodoulou, Koutsoupias and Spirakis <cit.>. In particular, this implies that any upper bound on the price of anarchy, or price of stability, in our framework yields an upper bound on the price of stability for ρ-approximate equilibria for the same class of games. Generalized affine congestion games. Let 𝒜' denote the class of all congestion games Γ for which all resources have the same cost function c(x) = a x + b, where a = a(Γ) and b = b(Γ) satisfy a ≥ 0 and a+b > 0. The class of affine congestion games with non-negative coefficients is contained in 𝒜' since every such game can always be transformed[This transformation can be done in such a way that both PoA and PoS of the game do not change. For a proof the reader is referred to, e.g., <cit.>.] into a game Γ' with a_e = 1 and b_e = 0 for all resources e ∈ E', where E' is the resource set of Γ'. Without loss of generality we can assume that a + b = 1, since the cost functions can be scaled by 1/(a+b). The cost functions of Γ∈𝒜' can then equivalently be written as c(x) = ρ x + (1 - ρ) for ρ≥ 0. This is precisely the definition of C^ρ_i(s) (with a_e = 1 and b_e = 0 taken there). In particular, if we take σ = ρ, meaning that C^ρ(s) = ∑_i ∈ N C^ρ_i(s), we have PoA(𝒜') = sup_ρ≥ 0 PoA(𝒜,ρ,ρ) and PoS(𝒜') = sup_ρ≥ 0 PoS(𝒜,ρ,ρ), where 𝒜 denotes the class of affine congestion games with non-negative coefficients. Due to page limitations some material is omitted from the main text below and can be found in the appendix. § PRICE OF ANARCHY In this section we derive an upper bound on the price of anarchy for affine congestion games of max{ρ + 1, 2ρ(1 + σ) + 1/ρ + 1} for a wide range of pairs (ρ,σ) that captures all known (to us) price of anarchy results in the literature that fall within our model. We show that this bound is tight for general congestion games. We also show that the bound (2ρ(1 + σ) + 1)/(ρ + 1) is (asymptotically) tight for symmetric network congestion games, for the range of (ρ,σ) on which it is attained. We need the following technical result for the proof of the main result in Theorem <ref>: Let s be a Nash equilibrium under the cost functions C_i^ρ(s) and let s^* be a minimizer of C^σ(·). For ρ, σ≥ 0 fixed, if there exist α(ρ,σ), β(ρ,σ) ≥ 0 such that (1 + ρ· x)y - ρ(x - 1)x - x ≤ - β(ρ,σ)(1 + σ(x - 1))x + α(ρ,σ)(1 + σ(y - 1))y for all non-negative integers x and y, then β(ρ,σ) C^σ(s) ≤α(ρ,σ) C^σ(s^*). Let s be a Nash equilibrium under the cost functions C_i^ρ(s) and let s^* be a minimizer of C^σ(·). Then C^σ(s)/C^σ(s^*)≤2ρ(1 + σ) + 1/ρ + 1 * if 1/2≤σ≤ρ≤ 2σ, * if σ = 1 and h(σ) ≤ρ≤ 2 σ, where h(σ) = g(1 + σ + √(σ(σ + 2)), σ) is the optimum of the function g(a,σ) = σ(a^2 - 1)/(1 + σ)a^2 - (2σ + 1)a + 2σ(σ+1). Furthermore, there exists a function Δ = Δ(σ) (specified in the appendix) satisfying for every fixed σ_0 ≥ 1/2: if Δ(σ_0) ≥ 0, then (<ref>) is true for all h(σ_0) ≤ρ≤ 2σ_0. For the functions α(ρ,σ) = (2ρ(1+σ)+1)/(1+2σ) and β(ρ,σ) = (1+ρ)/(1+2σ), we prove the inequality in Lemma <ref>. We show that, for certain functions f_1 and f_2, the smallest ρ satisfying the inequality of Lemma <ref> is given by the quantity h(σ) = sup_x,y ∈: f_1(x,y,σ) > 0 - f_2(x,y,σ)/f_1(x,y,σ). We divide the set (x,y) ∈× in lines of the form x = ay and determine the supremum over every line. After that we take the supremum over all lines, which then gives the desired result. We first show that the case x ≤ y which is trivial. We then focus on y > x. In this case we want to determine h(σ) = sup_a ∈_> 1sup_ y > 1 - f_2(ay,y,σ)/f_1(ay,y,σ). We show that h(σ) = max{γ_1(σ),γ_2(σ)} for certain functions γ_1 and γ_2. Numerical experiments suggest that Δ(σ) := γ_1(σ) - γ_2(σ) ≥ 0, that is, the maximum is always attained for γ_1 (which is the definition of h given in the statement of the theorem). In particular, this means that if, for a fixed σ, the non-negativity of Δ(σ) is checked, then the proof indeed yields the inequality of Lemma <ref> for h(σ) ≤ρ≤ 2σ. Numerical experiments suggest that Δ(σ) is non-negative for all σ≥ 1/2. We emphasize that for a fixed σ, with Δ(σ) ≥ 0, the proof that the inequality holds for all h(σ) ≤ρ≤ 2σ is exact in the parameter ρ. The first two cases of Theorem <ref> capture all the price of anarchy results from the literature. We next show that the bound of Theorem <ref> is also an (asymptotic) lower bound for linear symmetric network congestion games.[In the appendix (Theorem <ref>) we show (non-asymptotic) tightness for general congestion games.] This improves a result in the risk-uncertainty model of Piliouras et al. <cit.>, who only prove asymptotic tightness for symmetric linear congestion games (for their respective values of ρ and σ). It also improves a result in the altruism model by Chen et al. <cit.>, who show tightness only for general congestion games. The proof is a generalization of the construction of Correa et al. <cit.>, who proved that for the classical case ρ = σ = 1, the Price of Anarchy upper bound of 5/2, shown in <cit.>, is asymptotically tight for symmetric network congestion games. For ρ, σ > 0 fixed, and players with cost functions C_i^ρ(s), there exist symmetric network linear congestion games such that C^σ(s)/C^σ(s^*)≥2ρ(1 + σ) + 1/ρ + 1 - ϵ for any ϵ > 0, where s is a Nash equilibrium, and s^* a socially optimal strategy profile. We construct a symmetric network linear congestion game with n players. We first describe the graph topology used in the proof of Theorem 5 in <cit.> (using similar notation and terminology). The graph G consists of n principal disjoint paths, called P_1,…,P_n from s to t (horizontal paths in Figure <ref>), each consisting 2n-1 arcs (and hence 2n nodes). With e_i,j the j-th arc on path i is denoted for i = 1,…,n and j = 1,…,2n-1. Also, v_i,j denotes the j-th node on path i for i = 1,…,n and j = 1,…,2n. There are also n(n-1) connecting arcs: for every path i there is an arc (v_i,2k+1, v_i-1,2k) for k = 1,…,n-1, where i - 1 is taken modulo n (the diagonal arcs in Figure <ref>). For j ≥ 1 fixed, we say that the arcs e_i,j for i = 1,…,n form the (j-1)-th layer of G (see Figure <ref>). The cost functions are as follows. All arcs leaving s (the arcs e_i,1 for i = 1,…,n) and all arcs entering t (the arcs e_i,2n for i = 1,…,n) have latency c_e(x) = (1+ρ)x. For all i = 1,…,n, the arcs e_i,2k-1 for k = 1,…,n-1 have cost function c_e(x) = ρ x, whereas the arcs e_i,2k for k = 1,…,n-2 have cost function c_e(x) = x. All other arcs (the diagonal connecting arcs) have cost zero. The feasible strategy profile t in which player i uses principal path P_i, for all i = 1,…,n has social cost C^σ(t) = n(2(1+ρ) + (n-1)ρ + (n-2)) = n((1+ρ)n +ρ). A Nash equilibrium is given by the strategy profile in which every player k uses the following path: she starts with arcs e_k,1 and e_k,2, then uses all arcs of the form e_k+j,2j, e_k+j,2j+1, e_k+j,2j+2 for j = 1,…,n-1, and ends with arcs e_k+n-1,2n-2, e_k+n-1,2n-1 (and uses all connecting arcs in between).[This is similar to the construction in Theorem 5 <cit.>.] Note that all the (principal) arcs of layer j have load 1 is j is even, and load 2 if j is odd. The social cost of this profile is given by C^σ(s) = n(2(1+ρ) + (n-1) · 2 ·ρ(1+σ(2-1)) + n-2) = n((1+2ρ(1+σ))n - 2ρσ). It follows that C^σ(s)/C^σ(t) ↑ (1+2ρ(1+σ))/(1+ρ) as n →∞. We now show that the above mentioned strategy profile s is indeed a Nash equilibrium. Fix some player, say player 2, as in Figure <ref>, and suppose that this player deviates to some path Q. Let j be the first layer in which P_2 and Q overlap. Note that j must be odd. The cost C_2^ρ(s) of player 2, on the subpath of P_2 leading to the first overlapping arc with Q, is at most (1+ρ) + j-1/2· [2·[ρ(1 + ρ(2-1))] + 1] + ρ(1 + ρ(2-1)) =(1+ρ)^2 + j-1/2(1 + 2ρ(1+ρ)) The subpath of Q leading to the first overlapping arc with P_2 has C^ρ_i(Q,s_-i) as follows. She uses at least one arc in every odd layer (before the overlapping layer) with a load of 3 and one arc of every even layer (before the overlapping arc) with load 2, meaning that the cost of player i on the subpath of Q is at least (1+ρ)(1+ρ(2 - 1)) + j-1/2· [(ρ(1 + ρ(3-1))) + (1 + ρ(2-1))] = (1+ρ)^2 + j-1/2· (2ρ + 1)(1+ρ) Since 1 + 2ρ(1+ρ) < (2ρ+1)(1+ρ) for all ρ≥ 0, if follows that the cost of player i on the subpath of P_2 is no worse than that of the subpath of Q, when player 2 deviates from P_2 to Q. If follows that it suffices to show that P_2 is an equilibrium strategy in s with respect to deviations Q that overlap on the first arc e_2,1 with P_2. A similar argument shows that it also suffices to look at deviations Q for which Q and P_2 overlap on the last arc e_2,2n-1 of P_2. Now suppose that P_2 and Q do not overlap on some internal part of P_2. Note that the first arc of Q that is not contained in P_2, say (v_1,w_1) must be in an even layer, and also that the last arc, say (v_2,w_2) (which is a connecting arc) is in an odd layer (note that v_1 ≠ s and w_2 ≠ t w.l.o.g. by what is said in the previous paragraph). It is not hard to see that the subpath of Q from v_1 to w_2 contains the same number of even-layered arcs as the subpath of P_2, and the same number of odd-layered arcs as the subpath of P_2. However, the load on all the odd-layered arcs on the subpath of deviation Q is 3, whereas the load on odd-layered arcs in the subpath of P_2 between v_1 and w_2 (in strategy s) is 2. Similarly, the load on every even-layered arc on the subpath of deviation Q is 2, whereas the load on ever even-layered arc in the subpath of P_2 is 1. Hence the subpath of deviation Q between v_1 and w_2 can never be profitable. For ρ≥ 2σ, we can obtain a tight bound of ρ + 1 on the price of anarchy. Remarkably, the bound itself does not depend on σ, only the range of ρ and σ for which it holds. For the parameters σ = 1 and ρ≥ 2 in the altruism model of Caragiannis et al. <cit.>, this bound is known to be tight for non-symmetric singleton congestion games (where all strategies consist of a single resource). We only provide tightness for general congestion games, but the construction is significantly simpler. Let s be a Nash equilibrium under the cost functions C_i^ρ(s) and let s^* be a minimizer of C^σ(·). Then C^σ(s)/C^σ(s^*) ≤ρ + 1 for 1 ≤ 2σ≤ρ. Furthermore, this bound is tight. § PRICE OF STABILITY In this section we give a tight bound for the price of stability of √(σ(σ+2)) + σ/√(σ(σ+2)) + ρ - σ for a large range of pairs (ρ,σ). We show this bound to be (asymptotically) tight. We need the following technical lemma. For all non-negative integers x and y, and σ≥ 0 arbitrary, we have (x - y + 1/2)^2- 1/4 + 2σ x(x - 1) + (√(σ(σ+2)) + σ)[y(y-1) - x(x-1)] ≥ 0. Let s be a best Nash equilibrium under the cost functions C_i^ρ(s) and let s^* be a minimizer of C^σ(·). We have C^σ(s)/C^σ(s^*)≤√(σ(σ+2)) + σ/√(σ(σ+2)) + ρ - σ for all σ > 0 and 2σ/1 + σ + √(σ(σ+2))≤ρ≤ 2σ. An overview of the implications of this bound is given in Table <ref>. The bound of 2 for generalized affine congestion games requires some additional arguments (see appendix). Our proof is similar to a technique of Christodoulou, Koutsoupias and Spirakis <cit.> for upper bounding the price of stability of ρ-approximate equilibria (we elaborate on the conceptual difference in Section <ref>). However, for a general σ the analysis is more involved. The main technical contribution comes from establishing the inequality in Lemma <ref>. We can write C^ρ_i(s) = a_e x_e + b_e + (ρ - 1)a_ex_e, which, by Rosenthal <cit.>, implies that Φ^ρ(s) := ∑_e ∈ E a_ex_e(x_e + 1)/2 + b_e x_e + (ρ - 1)∑_e ∈ E a_e(x_e - 1)x_e/2 is an exact potential for C^ρ_i(s). The idea of the proof is to combine the Nash inequalities, and the fact that the global minimum of Φ^ρ(·) is a Nash equilibrium (because it is an exact potential). Let s denote the global minimum of Φ^ρ, and s^* a socially optimal solution. We can without loss of generality assume that a_e = 1 and b_e = 0. The Nash inequalities (as in the price of anarchy analysis) yield ∑_e∈ E x_e(1 + ρ(x_e - 1)) ≤∑_e ∈ E (1+ρ x_e)x_e^* whereas the fact that s is a global optimum of Φ^ρ(·) yields Φ^ρ(s) ≤Φ^ρ(s^*), which reduces to ∑_e ∈ Eρ x_e^2 + (2 - ρ)x_e ≤∑_e ∈ Eρ (x_e^*)^2 + (2 - ρ)x_e^*. If we can find γ, δ≥ 0, and some K ≥ 1, for which ( 0 ≤) γ[ ρ(x_e^*)^2 + (2 - ρ)x_e^* - ρ x_e^2 - (2 - ρ)x_e ] + δ[ (1+ρ x_e)x_e^* - x_e(1 + ρ(x_e - 1)] ≤ K · x_e^*[1 + σ(x_e^*-1)] - x_e[1 + σ(x_e-1)], then this implies that C^σ(s)/C^σ(s^*) ≤ K. We take δ = K - 1/ρ and γ = (ρ - 1)K + 1/2ρ. It is not hard to see that δ≥ 0 always holds, however, for γ we have to be more careful. We will later verify for which combinations of ρ and σ the parameter γ is indeed non-negative. Rewriting the expression in (<ref>) yields that we have to find K satisfying K ≥f_2(x_e,x_e^*,σ)/f_1(x_e,x_e^*,ρ,σ) := (x_e^*)^2 - 2x_ex_e^* + (1+2σ)x_e^2 - x_e^* + (1-2σ)x_e/[(1 - ρ + 2σ)(x_e^*)^2 - 2x_ex_e^* + (1+ρ)x_e^2 + (ρ - 1 - 2σ)x_e^* - (ρ - 1)x_e]. Note that this reasoning is only correct if f_1(x_e,x_e^*,ρ,σ) ≥ 0. This is true since f_1(x_e,x_e^*,ρ,σ) = (x_e - x_e^* + 1/2)^2 - 1/4 + (2σ - ρ)x_e^*(x_e^*-1) + ρ x_e(x_e-1) which is non-negative for all x_e,x_e^* ∈, σ≥ 0 and 0 ≤ρ≤ 2σ. Furthermore, the expression is zero if and only if (x_e,x_e^*) ∈{(0,1),(1,1)}, but for these pairs the nominator is also zero, and hence, the expression in (<ref>) is therefore satisfied for those pairs. We can write f_2(x_e,x_e^*,σ) = (x_e - x_e^* + 1/2)^2- 1/4 + 2σ x_e(x_e - 1) and therefore f_2/f_1 = A/A + (2σ - ρ)B, where A = (x_e - x_e^* + 1/2)^2- 1/4 + 2σ x_e(x_e - 1), B = x_e^*(x_e^*-1) - x_e(x_e-1). Note that if ρ = 2σ, we have f_2/f_1 = 1, and hence we can take K = 1. Otherwise, A/A + (2σ - ρ)B≤√(σ(σ+2)) + σ/√(σ(σ+2)) + ρ - σ =: K ⇔ A + (√(σ(σ+2))+σ)B ≥ 0 The inequality on the right is true by Lemma <ref>. To finish the proof, we determine the pairs (ρ,σ) for which the parameter γ is non-negative. This holds if and only if (ρ - 1)K + 1 = (ρ - 1)√(σ(σ+2)) + σ/√(σ(σ+2)) + ρ - σ + 1 ≥ 0. Rewriting this yields the bound on ρ in the statement of the theorem. In the next theorem we provide a lower bound on the price of stability for arbitrary non-negative pairs (ρ,σ). The proof is similar to a construction of Christodoulou et al. <cit.> used to give a lower bound on the price of stability for ρ-approximate equilibria. The key difference is to tune the parameters α, β in the proof with respect to the Nash definition based on the cost function C_i^ρ(·), rather than the ρ-approximate Nash definition. Let ρ, σ > 0 fixed, with ρ < 2σ, and ϵ > 0 arbitrary. Then there exists a linear congestion game, with player cost functions C_i^ρ(s), such that C^σ(s)/C^σ(s^*)≥√(σ(σ+2)) + σ/√(σ(σ+2)) + ρ - σ - ϵ. Here, s is a best Nash equilibrium, and s^* a social optimum. We describe the construction of Theorem 9 <cit.> using similar notation. We have a game of n = n_1 + n_2 players divided into two sets G_1 and G_2 with size resp. n_1 and n_2. Each player i ∈ G_1 has two strategies: A_i and P_i. The players in G_2 have a unique strategy D. The strategy profile A = (A_1,…,A_n_1,D,…,D) will be the unique Nash equilibrium, and the strategy profile P = (P_1,…,P_n_1,D,…,D) will be the social optimum. We have three types of resources: * n_1 resources α_i, i = 1,… n_1, with cost function c_α_i(x) = α x. The resource α_i only belongs to strategy P_i. * n_1(n_1 - 1) resources[The proof of Theorem 9 <cit.> contains a typo here: it says there are n(n-1) resources of this type, instead of n_1(n_1-1).] β_ij, i,j = 1,…,n_1 with i ≠ j, with cost function c_β_ij(x) = β x. The resource β_ij belongs only to strategies A_i and P_j. * One resource γ with cost function c_γ(x) = x, that belongs to A_i for i = 1,…, n_1 and to D. The idea is to set the parameters α and β in such a way that A becomes the unique Nash equilibrium. For any strategy profile s, there are k players playing strategy A_i and n_1 - k players playing strategy P_i in the set G_1, for some 0 ≤ k ≤ n_1. By symmetry, it then suffices to look at profiles S_k = (A_1,…,A_k,P_k+1,…,P_n_1,D,…,D) for 0 ≤ k ≤ n_1. Furthermore, the first k players playing A_i all have the same cost, and also, the n_1 - k players playing P_i have the same cost. We can therefore focus on the costs of player 1, denoted by C^ρ_A(k), and that of player n_1, denoted by C^ρ_P(k). We have C^ρ_A(k) = β (k-1) + β (1 + ρ(2-1))(n_1 - k) + 1 + ρ(n_2+k - 1) = (β - β(1+ρ) + ρ)k + (-β + β(1+ρ)n_1 + 1 + ρ(n_2 - 1)) = ρ(1 - β)· k + (1 - β - ρ) + β(1+ρ)n_1 + ρ n_2 and C^ρ_P(k) = α + β (n_1 - 1 - k)+ β(1 + ρ(2-1))k = βρ· k + α + β(n_1 - 1) We can set the parameters α and β such that C^ρ_A(k) = C^ρ_P(k-1), meaning that S_k is a Nash equilibrium for every k (we will create a unique Nash equilibrium in a moment), that is we take ρ(1-β) = βρ and (1 - β - ρ) + β(1+ρ)n_1 + ρ n_2 = α + β(n_1 - 1) - βρ Note that the -βρ term on the far right of the second equation comes from the fact that we evaluate C^ρ_P(·) in k-1 (remember that k denotes the number of players playing strategy A_i, so if a player would switch to P_i this number decreases by 1). Solving the left equation leads to β = 1/2. Inserting this in the right equation, and solving for α, gives α = ρ( n_1/2 + n_2 - 1/2) + 1. We emphasize that α, β > 0 for all ρ≥ 0. In order to make A the unique Nash equilibrium, we can slightly increase α such that we get C^ρ_A(k) < C^ρ_P(k-1) for all k (which means that A_i is a dominant strategy for player i). Note that this increase in α can be arbitrary small. We have C^σ(A)/C^σ(P) = n_1[1 + σ(n_1+n_2 - 1) + 1/2(n_1 - 1)] + n_2[1 + σ(n_1+n_2 - 1)]/n_1[ρ(n_1 + 1/2 + n_2 - 1) + 1 + 1/2(n_1-1) ] + n_2[1 + σ(n_2 - 1)] Inserting n_2 = a · n_1 for some rational a > 0, and sending n_1 →∞ gives a lower bound of f(a) = 2σ(1+a)^2 + 1/ρ(1+2a) + 1 + 2σ a^2 on the price of stability. Optimizing over a > 0 (this only works if ρ < 2σ) gives a^* = -1/2 + √(1/4 + 1/2σ) and f(a^*) then yields the bound in the statement of the theorem. § REMARKS AND ADDITIONAL INSIGHTS We discuss the results of Sections <ref> and <ref> for the applications mentioned in Section <ref> and obtain some additional application-specific insights and results. We conclude this section by showing that the price of stability results are not tight for the special case of symmetric network congestion games (as opposed to the price of anarchy results, which are tight for this class). Altruism <cit.>. For σ = 1, we see that the resulting bound (4ρ+1)/(ρ+1) is increasing in ρ. Especially in the context of altruism or taxes, this is not desirable. A natural question to ask is therefore if there exist collections ℋ for which the price of anarchy is non-increasing as a function of ρ. The next theorem gives a sufficient condition for a class of instances to have the property of non-increasing price of anarchy, which can be interpreted as follows. For ρ = 2, it can be shown that the social optimum becomes a Nash equilibrium under the player cost function C^ρ_i(s), which implies that the price of stability is 1. Nevertheless, it might happen that worse Nash equilibria arise as well. The condition PoA(ℋ,2,1) = 1 states that this does not happen, and even stronger, that all Nash equilibria of the game for ρ = 2 become social optima (it is said that the social optimum is strongly enforceable). Let ℋ be a collection of congestion games. If PoA(ℋ,2,1) = 1, then PoA(ℋ,ρ,1) is a non-increasing function for 1 ≤ρ≤ 2. In a technical report, Singh <cit.> shows that the social optimum is strongly enforceable for symmetric network congestion games on series-parallel graphs. We can therefore conclude that the (altruistic) price of anarchy will be a non-increasing function of ρ. This is a remarkable result since, to the best of our knowledge, the classical price of anarchy is unknown (the best lower bound is given by Fotakis <cit.>). Constant taxes <cit.>. Caragiannis et al. <cit.> showed that the price of anarchy can be decreased to 2.155 by the usage of universal tax functions (see also Section 1 and Figure <ref>), which improves significantly the classical bound of 2.5. However, the price of stability increases from 1.577 (for classical games) to 2.013, for this specific set of tax functions. Furthermore, from Theorem 3.7 <cit.> it follows that the price of anarchy can never be better than 2.155 for 0 ≤ρ≤ h(1). In Theorem <ref> we even show that the price of anarchy goes to infinity as ρ→ 0. Risk sensitivity under uncertainty <cit.>. We do not only re-obtain the price of anarchy results for risk-neutral players and players applying Wald's minimax principle (worst-case players), but our results also give a tight bound for any convex combination (in terms of player costs) of risk-neutral and worst-case risk attitudes. Furthermore, we also obtain tight price of stability results for this model. Approximate Nash equilibria <cit.>. For σ = 1 and 1 ≤ρ≤ 2, we obtain a bound of (√(3) + 1)/(√(3) + ρ - 1) on the price of stability. In particular, this also yields the same bound on the price of stability for ρ-approximate equilibria. This bound was previously obtained by Christodoulou et al. <cit.>. Conceptually our approach is different since we obtain correctness of the bound through the observation that every Nash equilibrium in our framework yields an approximate equilibrium. In particular, this immediately yields a potential function that can be used to carry out the technical details (namely the potential function that is exact for our congestion game). Nevertheless, the framework of Christodoulou et al. <cit.> is somewhat more general and might be used to obtain a tight bound for the price of stability of approximate equilibria (which is not known to the best of our knowledge). §.§ Price of stability for symmetric network congestion games The price of anarchy bound of (1+2ρ(1+σ))/(1 + ρ) obtained in Section <ref> is tight even for symmetric network congestion games with linear cost functions, as was shown in Theorem <ref>. This is not true for the price of stability, which we will show here for the case σ = 1. Let Γ be a linear symmetric network congestion game, then PoS(Γ, ρ, 1) ≤{[ 4/(ρ(4-ρ)) if 0 ≤ρ≤ 1; 4/(2+ρ) if 1 ≤ρ≤ 2; (2+ρ)/4 if 2 ≤ρ < ∞ ]. In particular, if Γ is a symmetric congestion game on an extenstion-parallel[A graph G is extension parallel if it consists of either (i) a single edge, (ii) a single edge and an extension-parallel graph composed in series, (iii) two extension-parallel graphs composed in parallel.] graph G, then the upper bounds even hold for the price of anarchy. All bounds are tight. For ρ≥ 1, the bounds were previously shown by Caragiannis et al. <cit.> for the price of anarchy of singleton symmetric congestion games (which can be modeled on an extension-parallel graph). Since any Nash equilibrium under the player cost C_i^ρ(·) is in particular a ρ-approximate Nash equilibrium, we also obtain the following result. The price of stability for ρ-approximate equilibria, with 1 ≤ρ≤ 2, is upper bounded by 4/(2+ρ) for linear symmetric network congestion games. Acknowledgements. We thank the anonymous referees for useful comments and one referee in particular for very detailed suggestions and pointing out important typos. splncs03 § OMITTED MATERIAL OF SECTION <REF> The following lemma shows the equivalence between dynamic taxes and altruistic players. For 1 ≤ρ≤ 2, a strategy profile s ∈×_i 𝒮_i is a Nash equilibrium under the cost C_i^ρ(s) = ∑_e ∈ s_i c_e(x_e) + (ρ - 1) ∑_e ∈ s_i (x_e - 1)[c_e(x_e) - c_e(x_e-1)] if and only if it is a Nash equilibrium under the altruistic cost 𝒜_i^ρ(s) = (2 - ρ) C^0_i(s) + (ρ - 1) C^0(s) Let s be a Nash equilibrium under the perceived costs 𝒜_i^γ(s), and let s^* be any other strategy profile. Furthermore, define γ = ρ - 1. Leaving out all the resources that are not in the symmetric difference[Resources used by at most on strategy.] of s_i and s_i^* (for the social cost term), we find that the Nash condition is equivalent to (1 - γ)∑_e ∈ s_i ∖ s_i^* c_e(x_e) + γ( ∑_e ∈ s_i ∖ s_i^* x_ec_e(x_e) + ∑_e ∈ s_i^* ∖ s_i x_ec_e(x_e)) ≤ (1 - γ)∑_e ∈ s_i^* ∖ s_i c_e(x_e + 1) + γ∑_e ∈ s_i ∖ s_i^* (x_e - 1)c_e(x_e - 1) + γ∑_e ∈ s_i^* ∖ s_i (x_e + 1)c_e(x_e + 1) which is equivalent to ∑_e ∈ s_i ∖ s_i^* c_e(x_e) + γ∑_e ∈ s_i ∖ s_i^* (x_e - 1)[c_e(x_e) - c_e(x_e - 1)] ≤∑_e ∈ s_i^* ∖ s_i c_e(x_e + 1) + γ∑_e ∈ s_i^* ∖ s_i x_e[c_e(x_e + 1) - c_e(x_e)] which is equivalent to C_i^γ(s) ≤C_i^γ(s_-i,s_i^*) since the terms for e ∈ s_i ∩ s_i^* do not change. § OMITTED MATERIAL OF SECTION <REF> Lemma<ref> Let s be a Nash equilibrium under the cost functions C_i^ρ(s) and let s^* be a minimizer of C^σ(·). For ρ, σ≥ 0 fixed, if there exist α(ρ,σ), β(ρ,σ) ≥ 0 such that (1 + ρ· x)y - ρ(x - 1)x - x ≤ - β(ρ,σ)(1 + σ(x - 1))x + α(ρ,σ)(1 + σ(y - 1))y for all non-negative integers x and y, then C^σ(s)/C^σ(s^*)≤α(ρ,σ)/β(ρ,σ). Without loss of generality, we may assume that a_e = 1 and b_e = 0. We then have ∑_i C_i^ρ(s) = ∑_e ρ (x_e - 1)x_e + ∑_e x_e = ∑_e ρ [1 - σ + σ](x_e - 1)x_e +ρ x_e - ρ x_e + ∑_e x_e = ρ∑_e [1 + σ(x_e - 1)]x_e + ρ∑_e (1 - σ)(x_e-1)x_e - x_e + ∑_e x_e = ρ C^σ(s) + ρ∑_e (1 - σ)(x_e-1)x_e + (1 - ρ) ∑_e x_e. Rewriting this, we find ρ· C^σ(s) = ∑_i C_i^ρ(s) + ρ(σ - 1)∑_e (x_e-1)x_e + (ρ - 1) ∑_e x_e ≤ ∑_i C_i^ρ(s^*_i,s_-i) + ρ(σ - 1)∑_e (x_e-1)x_e + (ρ - 1) ∑_e x_e ≤ ∑_e [1 + ρ(x_e - 1 + 1)]x_e^* + ρ(σ - 1)∑_e (x_e-1)x_e + (ρ - 1) ∑_e x_e = ∑_e [1 + ρ x_e]x_e^* + ρ(σ - 1)(x_e - 1)x_e + (ρ - 1)x_e = ∑_e [1 + ρ x_e]x_e^* + ρ[ 1 + σ(x_e - 1)]x_e - ρ (x_e - 1)x_e - x_e = ∑_e [1 + ρ x_e]x_e^* - ρ (x_e - 1)x_e - x_e + ρ C^σ(s) ≤ - β(ρ,σ) C^σ(s) + α(ρ,σ) C^σ(s^*) + ρ C^σ(s) Rearranging terms then gives the desired result. The following proposition is used in the proof of Theorem <ref> below. For every (x,y) ∈^2 ∖{(1,0)}, we have f_1(x,y,σ) = 2y(y-1)σ^2 + [x^2 + 2y^2 - 2xy -x] σ + [x^2 - xy +2(y - x)] ≥ 0 for σ≥σ^* = 1/2. Note that 2y(y-1) ≥ 0 for all y ∈. Furthermore, x^2 + 2y^2 - 2xy -x = (x - y)^2 + y^2 - x ≥ (x-y)^2 + (y-x) ≥ 0 for all (x,y) ∈^2, using the fact that a^2 - a ≥ 0 for all a ∈. This means that f_1(x,y,σ) is non-decreasing, and hence it suffices to prove the statement for σ^* = 1/2. We need to prove 1/2y(y-1) + 1/2[x^2 + 2y^2 - 2xy -x] + [x^2 - xy +2(y - x)] ≥ 0, or equivalent, y(y-1) + x^2 + 2y^2 - 2xy - x + 2x^2 - 2xy + 4(y-x) ≥ 0. Simplifying gives 3x^2 + 3y^2 - 4xy + 3y - 5x ≥ 0 which is equivalent to 2(x -y)^2 + x(x-5) + y(y+3) ≥ 0 and this last formulation is clearly true for all pair (x,y) with x ≥ 5. For x = 4, we find 2(4 - y)^2 - 4 + y(y+3) ≥ 0 which is clearly true for y ≥ 1, and for y = 0 in can be checked through inspection. For x = 3, we find 2(3-y)^2 - 6 + y(y+3) ≥ 0 which is clearly true for y ≥ 2. For y ∈{0,1}, it can be check through inspection. For x = 2, we find 2(2 - y)^2 - 6 + y(y+3) ≥ 0, which is again clear for y ≥ 2, and for y ∈{0,1} it can be check through inspection. For x = 1, we find 2(1 - y)^2 - 4 + y(y+3) ≥ 0, which is clearly true for y ≥ 1. For y = 0 the inequality does not hold, but this is the case (x,y) = (1,0) that we do not consider. For x = 0, it is clearly true. Theorem<ref> Let s be a Nash equilibrium under the cost functions C_i^ρ(s) and let s^* be a minimizer of C^σ(·). Then C^σ(s)/C^σ(s^*)≤2ρ(1 + σ) + 1/ρ + 1 * if 1/2≤σ≤ρ≤ 2σ, * if σ = 1 and h(σ) ≤ρ≤ 2 σ, where h(σ) = g(1 + σ + √(σ(σ + 2)), σ) is the optimum of the function g(a,σ) = σ(a^2 - 1)/(1 + σ)a^2 - (2σ + 1)a + 2σ(σ+1). Furthermore, there exists a function Δ = Δ(σ) which the property that, for any fixed σ_0 ≥ 1/2: if Δ(σ_0) ≥ 0, then (<ref>) is true for all h(σ_0) ≤ρ≤ 2σ_0 (the function Δ can be found in the proof below). We show the following inequality, (1 + ρ· x)y - ρ(x - 1)x - x ≤ - 1 + ρ/1 + 2σ(1 + σ(x - 1))x + 2ρ(1 + σ) + 1/1 + 2σ(1 + σ(y - 1))y. Multiplying with (1 + 2σ) we obtain the equivalent formulation (1 + 2σ)[(1 + ρ· x)y - ρ(x - 1)x - x] ≤ -(1 + ρ)(1 + σ(x - 1))x + (2ρ(1 + σ) + 1)(1 + σ(y - 1))y which we can rewrite to f_1(x,y,σ) ρ + f_2(x,y,σ) ≥ 0 where f_1(x,y,σ) = -(1 + σ(x-1))x + 2(1 + σ)(1 + σ(y-1))y + (1 + 2σ)((x-1)x - xy) = 2y(y-1)σ^2 + (-(x-1)x + 2(y-1)y + 2y + 2x(x-1) - 2xy)σ + ( -x + 2y + (x-1)x - xy) = 2y(y-1)σ^2 + [x^2 + 2y^2 - 2xy -x] σ + [x^2 - xy +2(y - x)] and f_2(x,y,σ) = -(1 + σ(x-1))x + (1 + σ(y-1))y + (1+2σ)(x - y) = σ y(y-1) - σ x(x-1) + 2σ (x-y) = (y^2 - x^2 + 3(x-y))σ We first consider the case (x,y) = (1,0), since then we do not have f_1(x,y,σ) ≥ 0. Substituting the values for x and y, we obtain -ρ + 2σ≥ 0 which is true if and only if ρ≤ 2σ. Case i). For the pair (x,y) = (1,0), the inequality is true if and only if ρ≤ 2σ. For all other pairs, we have f_1(x,y,σ) ≥ 0, and hence f_1(x,y,σ) ρ + f_2(x,y,σ) ≥ f_1(x,y,σ) σ + f_2(x,y,σ) meaning that is suffices to show that f_1(x,y,σ) σ + f_2(x,y,σ) ≥ 0. After dividing by σ, we see that this is equivalent to 2y(y-1)σ^2 + [x^2 + 2y^2 - 2xy -x] σ + [x^2 - xy +2(y - x)] + (y^2 - x^2 + 3(x-y)) ≥ 0 which is equivalent to 2y(y-1)σ^2 + [x^2 + 2y^2 - 2xy -x] σ + [y^2 - xy + (x - y)] ≥ 0 Again, we see that the terms before σ^2 and σ are non-negative for all x,y ∈ (see proof of Proposition <ref>), meaning that if the inequality holds for some σ^*, then it holds for all σ≥σ^*. We take σ^* = 1/2. Multiplying the resulting inequality with 2, we find y(y-1)+ [x^2 + 2y^2 - 2xy -x] + 2[y^2 - xy + (x - y)] ≥ 0 which is equivalent to x^2 + 5y^2 - 4xy -3y +x ≥ 0. This can be rewritten as (x - 2y)^2 + y(y-3) + x ≥ 0 which is clearly true for all y ∉{1,2}. For y = 1, we find (x - 2)^2 - 2 + x ≥ 0. This is clearly true for all x ≥ 2. For x ∈{0,1}, it can be checked through inspection. For y = 2, we find (x - 4)^2 - 2 + x ≥ 0. This is again clearly true for x ≥ 2, and can be check through inspection for x ∈{0,1}. Case ii). Now let (x,y) ∈^2 ∖{(1,0)}, then f_1(x,y,σ) ≥ 0 by Proposition <ref>, meaning that f_1(x,y,σ) ρ + f_2(x,y,σ) is non-decreasing in ρ. From the proof of Proposition <ref>, it follows that f_1(x,y,σ) = 0 if and only if (x,y) ∈{(1,1), (2,1)} (which can be seen by checking all the cases). Note that this observation is independent of σ. For (x,y) ∈{(1,1), (2,1)} it also holds that f_2(x,y,σ) = 0, which implies that f_1(x,y,σ) ρ + f_2(x,y,σ) = 0 for every ρ. Therefore, we can focus on pairs (x,y) for which f_1(x,y,σ) > 0. It follows that any ρ^* for which ρ^* ≥sup_x,y ∈: f_1(x,y,σ) > 0 - f_2(x,y,σ)/f_1(x,y,σ). yields the inequality for all ρ≥ρ^*. It is not hard to see that this supremum is indeed finite, for every fixed σ. It can be proved that f_1(x,y,σ) ρ' + f_2(x,y,σ) ≥ 0 holds for some large constant ρ', which then serves as an upper bound on the supremum. For the pair (x,y) = (0,1), we find -f_2/f_1 = σ/(1+σ), but we will see later that the supremum on the other pairs obtained is larger than σ/(1+σ). Note that by now, we can focus on pairs in {(x,y) : x ≥ 1, y ≥ 2}, since for all other pairs we have either proven the inequality or given -f_2/f_1, that is, we are interested in sup_{(x,y) : x ≥ 1, y ≥ 2} - f_2(x,y,σ)/f_1(x,y,σ). Note that f_2(x,y,σ) = (y^2 - x^2 + 3(x-y))σ = (x+y - 3)(y - x) ≥ 0 if y ≥ x (using that x+y ≥ 3 for (x,y) ∈{(x,y) : x ≥ 1, y ≥ 2}). Hence, if y ≥ x, we have -f_2/f_1 ≤ 0, so those pairs are not relevant for the supremum (if it follows that the upper bound on the supremum for all other pairs is positive, which we will indeed see later). Therefore, we can focus on pairs with y < x. We substitute x = ay for some (rational) a > 1. Note that sup_a ∈_> 1sup_ y ≥ 2 - f_2(ay,y,σ)/f_1(ay,y,σ) provides an upper bound on (<ref>). We have f_1(ay,y,σ) = [(1+σ)a^2 - (2σ + 1)a + 2σ(σ+1)]y^2 - [(2+σ)a + 2σ^2 - 2]y and - f_2(ay,y,σ) = [(a^2 - 1)σ]y^2 + [3(1-a)σ]y We determine an upper bound on the expression - f_2(ay,y,σ)/f_1(ay,y,σ) = [(a^2 - 1)σ]y^2 + [3(1-a)σ]y/[(1+σ)a^2 - (2σ + 1)a + 2σ(σ+1)]y^2 - [(2+σ)a + 2σ^2 - 2]y = [(a^2 - 1)σ]y + [3(1-a)σ]/[(1+σ)a^2 - (2σ + 1)a + 2σ(σ+1)]y - [(2+σ)a + 2σ^2 - 2] = α y + β/γ y - δ for y ≥ 2. Elementary calculus shows that the derivative with respect to y of (<ref>) is given by -(αδ + γβ)/(γ y - δ)^2, which means the expression in (<ref>) is non-decreasing or non-increasing in y. We have αδ + γβ = (a^2 - 1)σ[(2+σ)a + 2σ^2 - 2] + 3(1-a)[(1+σ)a^2 - (2σ+1)a +2σ(1+σ)] = (1-a)σ[-(1+a)((2+σ)a + 2σ^2 - 2) + 3((1+σ)a^2 - (2σ+1)a +2σ(1+σ))] = (1-a)σ[(3(1+σ) - (2+σ))a^2 + (2 - (2+σ) - 2σ^2 - 3(2σ + 1))a + 2 - 2σ^2 + 6σ(1 + σ)] = (1-a)σ[(2σ + 1)a^2 - (2σ^2 + 7σ + 3)a + (4σ^2 + 6σ + 2)] = (1-a)σ[(2σ + 1)a^2 - (2σ + 1)(σ+3)a + (2σ + 1)(2σ + 2)] = (1-a)σ(1+2σ) [a^2 - (σ+3)a + (2σ + 2)] = (1-a)σ(1+2σ) [(a - σ+3/2)^2 - 1/4 (1- σ)^2] Intermezzo. If we consider the function x_2 = (α x_1 + β)/(γ x_1 - δ), we see it has vertical asymptote at x_1^* = δ/γ. We claim that x_1^* < 2. Note that, since a > 1, we have δ > 0 for all σ≥ 0. If γ < 0 then x_1^* < 0. If γ > 0, we claim that x_1^* < 2. This is equivalent to showing that (2+σ)a + 2σ^2 - 2 < 2(1+σ)a^2 - 2(2σ +1)a + 4σ(σ+1), which holds if and only if 2(1+σ)a^2 - (5σ + 4)a + 2(1+σ)^2 = 2(1+σ)( [a - 5σ + 4/4(1+σ)]^2 - 1/4[5σ + 4/2(1+σ)]^2 + (1+σ)) > 0. If now suffices to show that - 1/4[5σ + 4/2(1+σ)]^2 + (1+σ) > 0, but this is true for all σ > 0, hence, the claim is proven. The situation σ = 1. It follows that the expression in (<ref>) is non-positive for all a > 1, which implies that -(αδ + γβ)/(γ y - δ)^2 ≥ 0 and hence -f_2/f_1 is non-decreasing in y ≥ 2 for every a > 1 (using the intermezzo). We then have lim_y →∞ -f_2(ay,y,σ)/f_1(ay,y,σ) = σ(a^2 - 1)/(1 + σ)a^2 - (2σ + 1)a + 2σ(σ+1) =:h_1(a,σ) and maximizing this function over a ∈_>1, we find the optimum a^*(σ) = 1 + σ + √(σ(σ + 2)). The situation 1/2 ≤σ < 1. More generally, for any σ < 1 it holds that αδ + γβ≤ 0 if and only if a ∉ (1+σ,2). In particular for every a ∉ (1+σ,2), we can then show that sup_ y ≥ 2 - f_2(ay,y,σ)/f_1(ay,y,σ)≤lim_y →∞ -f_2(a^*y,y,σ)/f_1(a^*y,y,σ) with a^* as in (<ref>) using the same argument as in the case σ = 1. The intermezzo implies that if the expression (<ref>) is non-increasing in y, which is the case when a ∈ (1 + σ,2), then the maximum value is attained in y = 2. That is, we are interested in the expression -f_2(2a,2,σ)/f_1(2a,2,σ), and in particular, we want to show that the supremum over a ∈ (1+σ, 2) does not exceed the right hand side of (<ref>), i.e., the supremum over all a ∉ (1+σ,2). Because of the discussion in the above, it suffices to study -f_2(2a,2,σ)/f_1(2a,2,σ) = [(a^2 - 1)σ]2 + [3(1-a)σ]/[(1+σ)a^2 - (2σ + 1)a + 2σ(σ+1)]2 - [(2+σ)a + 2σ^2 - 2] = σ(2a^2-3a+1)/2(1+σ)a^2 - (5σ + 4)a + 2(1+σ)^2 =:h_2(a,σ) for a ∈ (1+σ,2). This expression, for a > 1, is maximized for b^*(σ) = 1 + σ + √(σ(σ+1/2)), which in particular gives an upper bound for a ∈ (1+σ,2)). It now suffices to show that Δ(σ) := h_1(a^*(σ),σ) - h_2(b^*(σ),σ) ≥ 0, since this implies that the supremum over a > 1 in (<ref>) is attained at some a ∉ (1+σ,2). We have checked this numerically (see Figure <ref>). The situation σ > 1. We can use similar reasoning as in the previous case, but now the expression in (<ref>) is non-increasing for a ∈ (2,1+σ). Note that this does not affect the reasoning in the previous case, since we maximize over all a > 1 when obtaining b^*(σ). The following theorem shows that the bound in Theorem <ref> is also a lower bound for general congestion games and arbitrary ρ, σ≥ 0. We generalize the construction of Christodoulou and Koutsoupias <cit.>, who showed the lower bound for the classical case ρ = σ = 1. This construction is also used in the risk-uncertainty model of Nikolova et al. <cit.>, and the altruism model of Chen et al. <cit.>. For ρ, σ∈_>0 fixed, and players with cost functions C_i^ρ(s), there exist linear congestion games such C^σ(s)/C^σ(s^*)≥2ρ(1 + σ) + 1/ρ + 1 where s is a Nash equilibrium, and s^* a socially optimal strategy profile. We construct a congestion game of n ≥ 3 players and |E| = 2n resources with price of anarchy greater than or equal to (2ρ(1 + σ) + 1)/(ρ + 1). The set E is divided in the sets E_1 = {h_1,…,h_n} and E_2 = {g_1,…,g_n}. Player i has two pure strategies: {h_i,g_i} and {h_i+1,g_i-1,g_i+1}, where the indices appear as i mod n. The latency functions of the elements in E_1 are c_e(x) = x, whereas the latency functions of the elements in E_2 are c_e(x) = ρ x. Regardless which strategy player i plays, he always uses at least one resource from both E_1 and E_2, implying that C^σ_i(s) ≥ρ + 1. This implies that C^σ(t) = ∑_i ∈ N C^σ_i(s) ≥ (ρ + 1)n for every strategy profile t, and in particular for a social optimum s^*. We will now show that the strategy profile s where every agent i plays its second strategy {h_i+1,g_i-1,g_i+1} is a Nash equilibrium. We have C_i^ρ(s) = 2 ρ [1 + ρ(2 - 1)] + 1 = 2ρ^2 + 2ρ + 1. If some agent i deviates to its first strategy s_i', we have C_i^ρ(s_i',s_-i) = ρ[1 + ρ(3 - 1)] + (1 + ρ(2-1)) = 2ρ^2 + 2ρ + 1, since there are then three agents using g_i and two agents using h_i. This shows that s is a Nash equilibrium. The social cost of this strategy s is C^σ(s) = n(1 + 2ρ [1 + σ(2-1)]) = (1 + 2ρ(1+σ) ) n. Combining (<ref>) with (<ref>) then gives the desired result. Theorem<ref> Let s be a Nash equilibrium under the cost functions C_i^ρ(s) and let s^* be a minimizer of C^σ(·). Then C^σ(s)/C^σ(s^*) ≤ρ + 1 for 1 ≤ 2σ≤ρ. Furthermore, this bound is tight. We show that (1 + ρ· x)y - ρ(x - 1)x - x ≤ - (1 + σ(x - 1))x + (1+ρ)(1 + σ(y - 1))y and then the result follows from Lemma <ref>, with h = 1 and g = 1+ρ. Rewriting gives the equivalent statement [ y + σ y(y-1) - xy + x(x-1)]ρ + σ[y(y-1) - x(x-1)] ≥ 0. We first show that [ y + σ y(y-1) - xy + x(x-1)] ≥ 0 for all σ≥ 1/2. If suffices to show this claim for σ = 1/2, since y(y-1) ≥ 0 for all y ∈. We have y + 1/2 y(y-1) - xy + x(x-1) = 1/2[ (x - y - 1/2)^2 - 1/4 + x(x-1)] and this last expression is clearly non-negative for all x,y ∈ (since the quadratic term is always at least 1/4). It now suffices to show (<ref>) for ρ = 2σ, since we have shown that the expression is a non-decreasing affine function of ρ, for every fixed σ≥ 1/2. Substituting ρ = 2σ and dividing (<ref>) by σ, we get the equivalent statement 2 [ y + σ y(y-1) - xy + x(x-1)] + [y(y-1) - x(x-1)] ≥ 0 which we will show to be non-negative for all non-negative integers x and y and σ≥ 1/2. Again, it suffices to show the statement for σ = 1/2. The statement in (<ref>) is then equivalent to (x - y - 1/2)^2 - 1/4 + y(y-1) which is clearly non-negative for all x, y ∈. The tightness can be obtained by considering the following game on four resources with two players. Player A has strategies {{1},{2,4}} and player B has strategies {{2},{1,3}}. Resources e = 1,2 have cost function c_e(x) = x and resources e = 3,4 have cost function c_e(x) = ρ x. The optimum s^* = ({1},{2}) has cost 2, whereas the Nash equilibrium s = ({2,4},{1,3} has cost 2(1+ρ). § OMITTED MATERIAL OF SECTION <REF> Lemma <ref>. For all non-negative integers x and y, and σ≥ 0, we have (x - y + 1/2)^2- 1/4 + 2σ x(x - 1) + (√(σ(σ+2)) + σ)[y(y-1) - x(x-1)] ≥ 0. The inequality is clearly true for all y ≥ x, so we assume that y < x. Rewriting the expression gives (1 + σ + √(σ(σ+2)))y^2 - 2xy + (1 + σ - √(σ(σ+2)))x^2 - (1 + σ + √(σ(σ+2)))y + (1 - σ + √(σ(σ+2)))x ≥ 0. Multiplying with 1 + σ - √(σ(σ+2)), which is non-negative for all σ≥ 0, gives y^2 - 2(1 + σ - √(σ(σ+2)))xy + (1 + σ - √(σ(σ+2)))^2x^2 - y + (1 + σ - √(σ(σ+2)))(1 - σ + √(σ(σ+2)))x ≥ 0, using the fact that (1+σ + √(σ(σ+2)))(1 + σ - √(σ(σ+2))) = 1. This is equivalent to ((1+σ - √(σ(σ+2)))x - y + 1/2)^2 + (1+σ - √(σ(σ+2)))([1+σ - √(σ(σ+2))] - 1)x - 1/4≥ 0. We now substitute c = 1 + σ - √(σ(σ+2))≥ 0 in order to obtain the equivalent formulation (c · x - y + 1/2)^2 + c(1-c)x - 1/4≥ 0 for 0 ≤ c < 1, since the function c(σ) = 1 + σ - √(σ(σ+2)) is bijective from to [0,1). For x = 0, the inequality reduces to (1/2 - y)^2 - 1/4 ≥ 0 which is true for all y ∈. For x = 1, we get the equivalent formulation (y-1)(y - 2c) ≥ 0, which is clearly true for y = 1. For y = 0, it follows from the fact that c ≥ 0. For y ≥ 2 if follows from the fact that y - 2c ≥ 0 for all y ≥ 2, since 0 ≤ c < 1. This completes the case x = 1. For x ≥ 2, we rewrite the expression (<ref>) to x(x - 1)c^2 + 2x(1 - y)c + y(y-1) ≥ 0 If y = 0, the expression in (<ref>) is clearly non-negative for all x ≥ 2 and 0 ≤ c < 1. For y ≥ 1, note that g(c) = x(x - 1)c^2 + 2x(1 - y)c + y(y-1) is a quadratic and convex function for all fixed x and y. Therefore, in particular, for any x and y fixed, it suffices to show that the inequality holds for the minimizer of g, which is c^* = (y-1)/(x-1) (which can be found by differentiating with respect to c). Note that 0 ≤ c^* < 1 by our assumption that y ≥ 1 and y < x (made in the beginning of the proof). Substituting implies that it suffices to show that x(x-1)(y-1)^2/(x-1)^2 + 2x(1-y)(y-1)/x-1 + y(y-1) ≥ 0. Multiplying the expression with (x-1) implies that it now suffices to show that x(y-1)^2 - 2x(y-1)^2 + y(y-1)(x-1) ≥ 0 for all 1 ≤ y < x. This is always true since x(y-1)^2 - 2x(y-1)^2 + y(y-1)(x-1) = -x(y-1)^2 + y(y-1)(x-1) = (y-1) [-x(y-1) + y(x-1)] = (y-1) (x - y) ≥ 0 whenever 1 ≤ y < x. This completes the proof. The bound of 2 on the price of stability for generalized affine congestion games requires some additional arguments: The bound in Theorem <ref> with ρ = σ is only valid for σ≥ 1/4 (because otherwise the lower bound on ρ is not satisfied). Nevertheless for 0 ≤σ≤ 1/4, the corresponding cost functions c_e(x) = σ x + (1-σ) have non-negative constants and thus the price of stability for classical congestion games applies here. That is, we have PoS(𝒜') = max{1.577, sup_σ≥ 1/4{1 + √(σ/(σ+2))}} = 2. § OMITTED MATERIAL OF SECTION <REF> Theorem<ref> Let ℋ be a collection of congestion games. If PoA(ℋ,1,2) = 1, then PoA(ℋ,1,ρ) is non-increasing function for 1 ≤ρ≤ 2. Let α = ρ - 1. Suppose that PoA(ℋ,1,α) =: PoA(α) is not non-increasing, then there exist x < y ∈ [0,1] such that PoA(y) > PoA(x). We also know, by assumption, that PoA(x) ≥PoA(1) = 1, since the price of anarchy is always lower bounded by 1 (note that this also implies that y ≠ 1). This means that max{PoA(x), PoA(1)} < PoA(y). However, if we write y = γ· x + (1- γ)· 1 for some γ∈ [0,1], then Theorem 10.2 <cit.> implies that PoA(y) ≤max{PoA(x),PoA(1)} which is a contradiction. § OMITTED MATERIAL OF SECTION <REF> Theorem<ref> Let Γ be a symmetric network congestion game with linear cost functions, then PoS(Γ, ρ, 1) ≤{[ 4/(ρ(4-ρ)) if 0 ≤ρ≤ 1; 4/(2+ρ) if 1 ≤ρ≤ 2; (2+ρ)/4 if 2 ≤ρ < ∞ ]. In particular, if Γ is a symmetric congestion game on an extenstion-parallel [A graph G is extension parallel if it consists of either (i) a single edge, (ii) a single edge and an extension-parallel graph composed in series, (iii) two extension-parallel graphs composed in parallel.] graph G, then the upper bounds even hold for the price of anarchy. All bounds are tight. The remainder of this section is dedicated to the proof of Theorem <ref>. We will refer to strategy profiles as flows, since we can interpret symmetric network congestion games as a flow problem in which players each have to route one unit of unsplittable flow from a given source to a given target. To be precise, for a graph G = (V,E) and given s,t ∈ V, we write 𝒫 for the set of all simple s,t-paths (the common strategy set of the players). We denote f_P as the number of players using path P ∈𝒫. We call f a feasible (unsplittable) flow if ∑_P ∈𝒫 f_P = N, and with f_e we denote the number of players using edge e ∈ E, that is, f_e = ∑_P ∈𝒫 : e ∈ P f_P. We use the following result due to Fotakis <cit.>. Let Γ be a congestion game with cost functions d_e, and let Φ be an exact potential for Γ. An acyclic flow f minimizes the potential function Φ if and only if ∑_e : f_e > g_e (f_e - g_e)d_e(f_e) - ∑_e:f_e < g_e (g_e - f_e)d_e(f_e + 1) ≤ 0 for every feasible flow g. The following lemma gives inefficiency results for global minima of the potential function Φ (compared to any feasible flow). Since the local minima of Φ correspond to the Nash equilibria of the game Γ, it follows that the global minima of Φ are Nash equilibria. Furthermore, Fotakis <cit.> shows that every Nash equilibrium of a symmetric congestion game on an extension-parallel graph is a global minimum of the potential function Φ. In particular, this means that the ineffiency results in Lemma <ref> hold for the price of stability of symmetric network congestion games, and the price of anarchy of symmetric extension-parallel congestion games. Let Γ be a congestion game with cost functions d_e(x) = a_e (1 + ρ(x - 1)), and let Φ be an exact potential for Γ. Let f be an acyclic flow minimizing the potential function Φ, then C^σ(f) ≤ ∑_f_e > g_e a_e[ (f_e - 1)(ρ g_e + (σ - ρ)f_e) + g_e] + ∑_f_e ≤ g_e a_e [ (f_e - 1)(ρ g_e + (σ - ρ)f_e) + (1+ρ)g_e - ρ f_e ]. Furthermore, if h(ρ,σ) < 1 satisfies, (x - 1)(ρ y + (σ - ρ)x) + y ≤ h(ρ,σ) · x[1 + σ(x-1)] + g(ρ,σ) · y[1 + σ(y-1)] for all non-negative integers x > y, and (x - 1)(ρ y + (σ - ρ)x) + (1+ρ)y - ρ x ≤ h(ρ,σ) · x[1 + σ(x-1)] + g(ρ,σ) · y[1 + σ(y-1)] for all non-negative integers x ≤ y, then C^σ(f)/C^σ(g) ≤ g(ρ,σ) / (1 - h(ρ,σ)). We write d_e(x) = a_e[1 + σ(x - 1)] + a_e[(ρ - σ)(x - 1)] in the left summation, and obtain, using Lemma <ref>, ∑_f_e > g_e f_e a_e[1 + σ(f_e - 1)] ≤ ∑_f_e > g_e a_e( g_e[1 + ρ(f_e - 1)] + f_e(σ - ρ)(f_e - 1)) +∑_f_e < g_e (g_e - f_e) a_e(1+ρ f_e). Applying the inequality, we find C^σ(f) = ∑_f_e > g_ef_e a_e[1 + σ(f_e - 1)] + ∑_f_e < g_ef_e a_e[1 + σ(f_e - 1)] + ∑_f_e = g_ef_e a_e[1 + σ(f_e - 1)] ≤ ∑_f_e > g_e a_e[ (f_e - 1)(ρ g_e + (σ - ρ)f_e) + g_e] +∑_f_e < g_e a_e [ (f_e - 1)(ρ g_e + (σ - ρ)f_e) + (1+ρ)g_e - ρ f_e ]+ ∑_f_e = g_ef_e a_e[1 + σ(f_e - 1)] = ∑_f_e > g_e a_e[ (f_e - 1)(ρ g_e + (σ - ρ)f_e) + g_e] +∑_f_e ≤ g_e a_e [ (f_e - 1)(ρ g_e + (σ - ρ)f_e) + (1+ρ)g_e - ρ f_e ] This completes the proof. We continue the proof of the upper bounds in Theorem <ref> by showing the result in the statement for 0 < ρ≤ 1, that is, we define h(ρ,1) = 1 - ρ + ρ^2/4 and g(ρ,1) = 1 and prove the correctness of the resulting inequalities in (<ref>) and (<ref>) (see Lemma <ref>). The cases 1 ≤ρ≤ 2 and 2 ≤ρ≤∞ follow (indirectly) from Caragiannis et al. <cit.>.[The model of Carigiannis et al. <cit.> is equivalent to our model under the transformation ρ = 1/(1 - ζ), where ζ is the model parameter of <cit.>. That is, the range 1 ≤ρ≤ 2 corresponds to ζ∈ [0,1/2], and the range 2 ≤ρ≤∞ to ζ∈ [1/2,1).] The authors use a similar approach as here, but only show the inequality in Lemma <ref> for Nash equilibria of symmetric singleton congestion games. Nevertheless, the remainder of the analysis carries over to our model. For any integers x,y ≥ 0 and any ρ∈ (0,1] we have, when x < y, ρ· xy + (y - x) ≤ρ^2/4x^2 + y^2, and, when x ≥ y, ρ· xy + (1 - ρ)(y - x) + (1- ρ)x^2 ≤(1 - ρ + ρ^2/4)x^2 + y^2. Let y = x + z, where z is a positive integer. Then we have f(x,y) = ρ^2/4x^2 + y^2 - ρ xy - (y - x) = ρ^2/4x^2 + (x^2 + 2xz + z^2) - ρ x(x+z) - z = (ρ^2/4 + 1 - ρ)x^2 + (2 - ρ)xz + z(z-1) ≥ 0, since ρ∈ (0,1], x ≥ 0 and z > 0. For the second inequality, it suffices to show that g(x,y) = ρ^2/4x^2 + y^2 - ρ xy - (ρ - 1)(x - y) ≥ 0 which can be seen by leaving out the term (1-ρ)x^2 on both sides of the inequality. We first treat the case y = 0. Then g(x,0) = ρ^2/4x^2 + (1-ρ)x ≥ 0 since x ≥ 0 and ρ∈ (0,1]. For y ≥ 1, we write a = x/y (for sake of notation). We have g(x,y) = ρ^2/4x^2 + y^2 - ρ xy - (ρ - 1)(x - y) = ρ^2/4a^2y^2 + y^2 - ρ a y^2 + (1- ρ)(ay - y) = [(ρ a)^2/4 - ρ a + 1]y^2 + (1 - ρ)(a - 1)y = [ρ a/2 -1]^2y^2 + (1-ρ)(a-1)y = [ρ x/2y -1]^2y^2 + (1-ρ)(x/y-1)y ≥ 0, since ρ∈ (0,1], y ≥ 1 and a ≥ 1. It remains to show tightness of the resulting bounds. For 1 ≤ρ≤ 2, consider an instance with two players and two resources with resp. cost functions c_1(x) = x and c_2(x) = (1 + ρ + ϵ) x where 0 < ϵ≪ρ. Then the unique Nash equilibrium is given by (x_1,x_2) = (2,0), and the social optimum by (x_1^*,x_2^*) = (1,1). Sending ϵ→ 0 gives the desired bound of 4/(2+ρ). For 2 ≤ρ≤∞, we can use the same instance as the for 1 ≤ρ≤ 2, with the only difference that c_2(x) = (1 + ρ - ϵ)x. Then the social optimum is given by (x_1^*,x_2^*) = (2,0) and the unique Nash equilibrium by (x_1,x_2) = (1,1). For the case 0 < ρ≤ 1, the lower bound is technically more involved. For every fixed (rational) 0 < ρ≤ 1, and ϵ > 0, there exists a symmetric singleton congestion game for which the price of stability is greater than 4/(ρ(4 - ρ)) - ϵ. We choose values of n and i so that ρ = i/n-1 where, without loss of generality, we may assume that i is even. We will construct a congestion game with n agents and 1 + (n - i/2) resources. We let c_0(x) = x and for e ∈{1,…,n-i/2} we define c_e(x) = [(1-ρ) + ρ n + ϵ]x = [1 + i + ϵ]x, with 0 < ϵ≪ρ. A socially optimal profile s^* is given by x_0^* = i/2 and x_e^* = 1 for e ∈{1,…,n-i/2}, resulting in SC(s^*) = (i/2)^2 + (n - i/2)(1 + i + ϵ) = ρ^2/4(n - 1)^2 + (n - 1/2ρ(n-1))[(1-ρ) + ρ n + ϵ] = ρ^2/4(n^2 - 2n + 1) + ( (1 - ρ/2 )n + ρ/2)[(1-ρ) + ρ n + ϵ] = [ρ^2/4 + ρ(1 - ρ/2)]n^2 + [-ρ^2/2 + (1-ρ + ϵ)(1 - ρ/2) + ρ^2/2]n + ρ^2/4 + ρ/2(1-ρ + ϵ) = [ρ(1 - ρ/4)]n^2 + [(1-ρ + ϵ)(1 - ρ/2)]n + ρ/2(1 - ρ/2 + ϵ) The unique Nash equilibrium is given by the strategy profile s for which x_0 = n and x_e = 0 for e ∈{1,…,n-i/2}, since the perceived cost on resource e = 0 is then precisely (1-ρ) + ρ n, so no player can strictly improve its (perceived) cost by deviating to one of the other resources, which have cost (1-ρ) + ρ n + ϵ. The social cost of this equilibrium is n^2, which implies that C^1(s)/C^1(s^*) = n^2/[ρ(1 - ρ/4)]n^2 + [(1-ρ)(1 - ρ/2 + ϵ)]n + ρ/2(1 - ρ/2 + ϵ)→1/ρ(1 - ρ/4) as n →∞ (note that this also means that i →∞ since ρ is fixed).

http://arxiv.org/abs/1701.08000v3

20170127103256

Exceptional Points in Random-Defect Phonon Lasers

quant-ph

[]jinghui73@gmail.com [^]fnori@riken.jp [^†]sko9@psu.edu ^1Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Department of Physics and Synergetic Innovation Center for Quantum Effects and Applications, Hunan Normal University, Changsha 410081, China ^2Key Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China ^3Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA ^4CEMS, RIKEN, Saitama 351-0198, Japan ^5Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA ^6University of Chinese Academy of Sciences, Beijing 100049, China Intrinsic defects in optomechanical devices are generally viewed to be detrimental for achieving coherent amplification of phonons, and great care has been exercised in fabricating devices and materials with no (or minimal number of) defects. Contrary to this view, here we show that, by surpassing an exceptional point (EP), both the mechanical gain and the phonon number can be enhanced despite increasing defect losses. This counterintuitive effect, well described by an effective non-Hermitian phonon-defect model, provides a mechanical analog of the loss-induced purely-optical lasing. This opens up the way to operating random-defect phonon devices at EPs. Exceptional Points in Random-Defect Phonon Lasers H. Lü,^1,2,6 S. K. Özdemir,^3,† L.-M. Kuang,^1, Franco Nori,^4,5, and H. Jing^1, December 30, 2023 ==================================================================================== empty § INTRODUCTION Advances in cavity optomechanics (COM) in the past decade have led to many practical applications, such as ultrasensitive motion sensors, quantum transducers, and low-noise phonon devices <cit.>. The phonon analog of an optical laser was also achieved in COM <cit.>. Compared to phonon lasers in, e.g., cold ions, superlattices, or electromechanical systems <cit.>, COM-based devices feature a continuously tunable gain spectrum to selectively amplify phonon modes, from radio frequency to microwave rates, with an ultralow threshold <cit.>. This provides a powerful tool to study quantum acoustic effects, e.g., two-mode correlations <cit.>, sub-Poissonian distributions <cit.>, and motion squeezing <cit.>, which are useful in enhancing the performance of phonon devices in acoustic sensing, imaging, or switching <cit.>. Very recently, COM devices with balanced gain and loss have also attracted growing interest <cit.>. The gain is provided by doping active materials, e.g., rare-earth ions or dyes, into the resonator <cit.>. Such systems exhibit non-Hermitian degeneracies known as exceptional points (EPs), where both the eigenvalues and the corresponding eigenfrequencies of the system coalesce. Approaching an EP drastically affects the dynamics of a physical system, leading to many unconventional effects, e.g., loss-induced coherence <cit.>, invisible sensing <cit.>, and chiral-mode switch <cit.>. Alternative EP physics has also been explored experimentally in acoustic <cit.>, electronic <cit.>, and atomic systems <cit.>, as well as in a COM device <cit.>, opening up the way to phononic engineering at EPs. In this work, we study the emergence of an EP in COM. The EP arises in the phonon-lasing regime, by tuning the loss of intrinsic two-level-system (TLS) defects naturally existing in amorphous materials used in the fabrication of COM devices <cit.>. In a COM system, the role of TLS defects was already studied in the phonon-cooling regime <cit.>, but it has been neglected thus far in the phonon-lasing regime. Counterintuitively, we find that, in the phonon-lasing regime, increasing the defect loss leads to the enhancement of both mechanical gain and emitted phonon number. Unlike similar optical loss-induced effects <cit.>, our work provides a route for achieving an EP-enhanced phonon laser without any optical gain. In view of rapid advances in phonon devices <cit.>, EP optics, and COM with defects <cit.>, our findings hold the promise of being observed in practical phonon-laser systems with intrinsic defects. TLS defects can couple to different modes of a system via different mechanisms, e.g., to superconducting qubits <cit.> via electric dipole moments and to phonons via strain forces <cit.>. For many years, TLS defects were considered as a main source of loss and decoherence, and as such, techniques have been developed to decrease the number of defects <cit.>. However, recent studies have shown that they can play useful roles in, e.g., TLS quantum memory <cit.>, circuit control <cit.>, and optical lasing <cit.>. In COM systems, a strong TLS-phonon coupling, well described by a Jaynes-Cummings-like model, was utilized to achieve phonon cooling <cit.>. Here we show that phonon lasing can be enhanced by steering lossy defects <cit.>, instead of using any additional loss compensation technique via gain materials. Our work provides a scheme to realize loss-induced phonon lasing in COM systems and to use it for steering phonon devices. The critical point, observed for our full Hermitian system, coincides well with an EP emerging in an effective non-Hermitian TLS-phonon system. Despite its similarity to the loss-induced revival of an optical laser <cit.>, both the underlying coupling and the critical condition for our COM system, as shown here, are clearly different. § MODEL AND SOLUTIONS We consider two whispering-gallery-mode resonators (having the same resonance frequency ω_c and loss rate γ; see Fig. 1), one of which supports a radially symmetric mechanical breathing mode with effective mass m, frequency ω_m and damping rate γ_m. The resonators are made of silica, silicon, or silicon nitride that has intrinsic or artificially doped TLS defects, which can be coupled to the phonon mode via mechanical strain <cit.>. The strength of the coupling between the TLS and the mechanical mode, derived from linear elastic solid theory <cit.>, is given as g_d ≈D_T/ħΔ_0/ω_qS_ zpf,   S_zpf=√(ħω_m/2YV_m) where ω_q=√(Δ^2_0+Δ^2_a) is the tunable energy difference of the excited and ground states of the TLS <cit.>, Δ_0 is the tunnel splitting frequency, Δ_a is the asymmetry frequency, S_zpf is the zero-point strain-field fluctuation <cit.>, Y is the Young's modulus, and V_m is the mechanical mode volume determined by tensorial strain profiles <cit.>. Note that the mechanical deformation potential D_T can be measured experimentally <cit.>, and Δ_a—and thus ω_q or g_d—can be tuned using external microwave or electric fields <cit.>. δω_q/2π∼1MHz is achievable with a moderate field of about 10^3 V/m, allowing a TLS-phonon coupling which is strong enough to exceed γ <cit.>. The TLS-phonon coupling g_d is strong enough to exceed γ, as previously shown <cit.>. In addition, as shown in Ref. <cit.>, the typical number N_T of TLSs that can couple resonantly to a phonon mode, i.e., those within the bandwidth g_d around ω_m, is estimated as N_T≲ 1 or N_T≪ 1, which can be further tuned by e.g., shifting an off-resonant TLS into resonance with the considered phonon mode <cit.>. In the rotating frame at the pump frequency ω_l, the Hamiltonian of the defect-COM system can be written at the simplest level as H = H_0 + H_int + H_dr, with H_0 = -Δ (a^†_1a_1 +a^†_2a_2) + ω_m b^†b + ω_q/2σ_z, H_int = J(a^†_1a_2 +a^†_2a_1)- ξ a^†_1 a_1x + g_d (b^†σ_- + σ_+b), and H_dr =i(ε_la^†_1 - ε^∗_l a_1), where a_1, a_2, or b denote the annihilation operators of the optical modes or the mechanical mode, x=x_0 (b^† + b) is the mechanical displacement operator, Δ≡ω_l-ω_c denotes the detuning between the pump laser and the cavity resonance, ξ=ω_c/R is the COM coupling strength, R is the resonator radius, x_0=(1/2mω_m)^1/2, while σ_z, σ_- and σ_+ are the Pauli operators of the TLS defined as σ_z=|e⟩⟨ e|-|g⟩⟨ g|,  σ_-=|g⟩⟨ e|,  σ_+=|e⟩⟨ g|. The pump field amplitude is given by ε_l=(2P_lγ/ħω_l)^1/2, where P_l is the pump power. The Jaynes-Cummings-like model describes the strain-induced TLS-COM coupling, all details of theoretical derivations of which can be found in previous works on the defect-assisted COM, based on a linear elastic solid-state theory <cit.>. The parameter values used in our numerical simulations satisfy the validity condition g_d≪ω_q≈ω_m of this effective model <cit.>. §.§ Supermode picture To derive the Hamiltonian in the optical supermode picture, we define the operator a_±=(a_1± a_2)/√(2), which transforms H_0 and H_dr into ℋ_0 = ω_+ a^†_+a_+ + ω_- a^†_-a_- + ω_m b^†b + ω_q/2σ_z, ℋ_dr =i/√(2)[ε_l(a^†_++a^†_-)-h.c. ], with ω_±=-Δ± J. Similarly, H_int becomes ℋ_int = -ξ x_0/2[(a^†_+a_++a^†_-a_-)-(a^†_+a_- +H.c.)(b^†+b)] +g_d (b^†σ_- + σ_+b). In the rotating frame with respect to ℋ_0, we have ℋ_int= -ξ x_0/2(a^†_+a_-be^i(2J-ω_m)t+H.c.) -ξ x_0/2(a^†_+a_-b^†e^i(2J+ω_m)t +H.c.) +ξ x_0/2(a^†_+a_+ + a^†_-a_-)(b^†e^iω_m t + H.c.) + g_d [b^†σ_-e^i(ω_m-ω_q)t+H.c.]. Considering the rotating-wave approximation 2J+ω_m,ω_m≫ |2J-ω_m|,|ω_q-ω_m|, we have ℋ_int= - ξ x_0/2 (a_+^†a_-b + b^†a_+a_-^†) + g_d (b^†σ_- + σ_+b). The first term describes the phonon-mediated transition between optical supermodes, and the second term describes the coupling between the phonon and the TLS defect. Thus, in the supermode picture, the optomechanical coupling is transformed into an effective coupling describing defect-assisted phonon lasing. The TLS can be excited by absorbing a phonon generated from the transition between the upper optical supermode and the lower one, and as such it can strongly modify the behavior of the phonon lasing. The Heisenberg equations of motion of the system can then be written as ȧ_+ =(-iω_+-γ)a_+ +iξ x_0/2a_-b + ε_l/√(2) + √(γ)a_ in, ȧ_- =(-iω_--γ)a_- +iξ x_0/2a_+b^†+ ε_l/√(2)+√(γ)a_ in, ḃ =(-iω_m - γ_m)b + iξ x_0/2 a^†_+a_- - ig_dσ_- + √(2γ_m)b_ in, σ̇_- =(-iω_q- γ_q)σ_- + ig_d b σ_z + √(2γ_q)Γ_-, σ̇_z =-2γ_q(σ_z+1)-2ig_d(σ_+b-b^†σ_-)+√(2γ_q)Γ_z. Here a_ in, b_ in, Γ_-, and Γ_z denote environmental noises corresponding to the operators a, b, σ_- and σ_z. We assume that the mean values of these noise operators are zero, i.e. ⟨ a_ in⟩=⟨ b_ in⟩=⟨Γ_-⟩=⟨Γ_z⟩=0. The fluctuations are small and we neglect the noise operators in our numerical calculations. Then, the defect-assisted mechanical gain and the threshold power of the phonon lasing can be obtained. In the supermode picture, a crucial term describing the phonon-lasing process can be resonantly chosen from the Hamiltonian, under the rotating-wave approximation <cit.> (for J∼ω_m/2, ω_q∼ω_m). The resonance ω_m=ω_q can be achieved by using a moderate field of about 10^3 V/m, allowing a shift of δω_q/2π∼1MHz <cit.>. With the supermode operators p=a^†_-a_+, a_±=(a_1± a_2)/√(2), the reduced interaction Hamiltonian is given by ℋ_int= - ξ x_0/2 (p^† b + b^†p) + g_d (b^†σ_- + σ_+b). The resulting Heisenberg equations of motion are ṗ =(-2iJ-2γ)p-iξ x_0/2δ n b + 1/√(2)(ε^∗_la_++ε_la^†_-), ḃ =(-iω_m - γ_m)b + iξ x_0/2p - ig_d σ_-, σ̇_- =(-iω_q- γ_q)σ_- + ig_d b σ_z, σ̇_z =-2γ_q(σ_z+1) - 2ig_d(σ_+b - b^†σ_-), where p=a^†_-a_+, and δ n=a^†_+a_+-a^†_-a_- denotes the population inversion. The noise terms are negligible with a strong driving field. The steady-state values of the system can be obtained by setting ∂ p/∂ t=0,∂σ_-/∂ t=0, and ∂ a_±/∂ t=0, with γ,γ_q≫γ_m, which leads to p =1/i(2J-ω_m)+2γ[1/√(2)(ε^∗_la_++ε_la^†_-) -iξ x_0/2δ n b ], σ_- =-g_d(ω_q-ω_m) + ig_dγ_q/γ^2_q+(ω_q-ω_m)^2+2g_d^2 n_bb, a_+ =ε_l(2iω_- + 2γ + iξ x_0 b)/2√(2)α-i4√(2)γΔ, a_- =ε_l(2iω_+ + 2γ + iξ x_0 b^†)/2√(2)α-i4√(2)γΔ, where n_b denotes the expectation value of the phonon number and ω_±=-Δ± J,   α=J^2+γ^2-Δ^2 +ξ^2 x^2_0/4n_b. Substituting these values into the equation of the mechanical mode results in ḃ=(-iω_m + iω^' + G - γ_m)b + C, where ω^'= g_d^2(ω_q-ω_m)/γ^2_q+(ω_q-ω_m)^2+2g_d^2n_b - ξ^2x^2_0(2J-ω_m)/16γ^2+4(2J-ω_m)^2 -ξ^2x^2_0Δ|ε_l|^2/[2(2J-ω_m)^2+8γ^2](α^2+4Δ^2γ^2), C= i|ε_l|^2ξ x_0/2i(2J-ω_m)+4γ·(γ-iJ)α+2Δ^2γ/α^2 + 4Δ^2γ^2, and α=J^2+γ^2-Δ^2 +ξ^2 x^2_0n_b/4. The mechanical gain is then G=G_0+G_d, with G_0= ξ^2 x^2_0γ/2(2J-ω_m)^2+8γ^2(δ n- Δ(2J-ω_m) |ε_l|^2/α^2 + 4Δ^2γ^2), G_d= -g_d^2γ_q/γ^2_q+(ω_q-ω_m)^2+2g_d^2n_b. The role of lossy defects in mechanical amplification, described by G_d, has not been reported previously. From the condition G=γ_m and P_th≈ħ(ω_c+J)γδ n <cit.>, the threshold power P_th=P_th,0+P_th,d is found as P_th,0 = 2ħ[(2J-ω_m)^2+4γ^2](ω_c+J)γ_m/(ξ x_0)^2 +ħΔ(2J-ω_m)(ω_c+J) |ε_l|^2/λ^2+4Δ^2γ^2, P_th,d = 2ħ g^2_dγ_q(ω_c+J)[(2J-ω_m)^2+4γ^2]/ξ^2 x^2_0 [γ^2_q+(ω_m-ω_q)^2 + 2g_d^2n_b]. Clearly, the presence of defects strongly alters G and P_th, even when Δ=0. In the following, we first present the full numerical results, and then, to understand the observed counterintuitive effect, we introduce a reduced non-Hermitian TLS-phonon model. A comparative analysis of the full and reduced models then helps to establish the relation between the turning points of the former with the EPs emerging in the latter. § NUMERICAL RESULTS AND DISCUSSIONS §.§ The full system: Numerical results Figure <ref>(a) shows the mechanical gain, G_0 and G, as a function of the optical detuning Δ, using experimentally accessible values <cit.>, i.e. R= 34.5 μm, m = 50 ng, ω_c = 193 THz, ω_m = 2 π× 23.4 MHz, γ = 6.43 MHz, and γ_m = 0.24 MHz. In the cooling regime (with Δ<0), G is negative and can be enhanced by defects <cit.>. In the lasing regime (with Δ>0), the positive G is also strongly affected by defects. Note that the simplified condition γ_q/γ=1 used in Fig. <ref>(a) is experimentally accessible, since γ_q is typically 0.1-5MHz <cit.> and can be further enhanced by using external fields (or amorphous oxide layers) <cit.>. Clearly, the defect-induced reduction in G is minimized at Δ/ω_m∼ 0.5, and as Fig. <ref>(a) shows, the maximum phonon lasing occurs at Δ/ω_m∼ 0.5, J/ω_m∼ 0.5. We stress that the TLS defects naturally and inevitably exist in all solid-state materials and introduce detrimental losses in optomechanical systems. Therefore, the ideal mechanical gain (in the absence of any defect) G_0 can never be achieved in a practical device. In order to obtain G→ G_0, the intuitive way is to minimize, if not eliminate, the detrimental effects of TLS defects by preparing better and purer materials with no or minimal number of defects. In contrary to this view, we find that this can also be achieved by increasing the losses induced by TLS defects (e.g. by controlling the dissipation of existing defects or by introducing more defects that are coupled to the mechanical mode). As shown in Fig. <ref>(b), a turning point appears for G as the TLS loss is increased: G first decreases with increasing TLS loss, until a critical value of γ_q. When this value is exceeded, more loss leads to an increasing mechanical gain, tending to the limit value G_0 as we have numerically confirmed. Consequently, the phonon-lasing threshold power P_th first increases and then decreases again with more loss, as shown in Fig. 3(b). This counterintuitive effect, emerging only in the mechanical-amplifying regime, has not been reported previously. Despite the similarity to loss-induced purely optical-lasing revival <cit.>, the underlying coupling and the critical condition of our hybrid COM system are clearly different. §.§ Active Jaynes-Cummings model To intuitively understand the turning-point feature, as numerically revealed above, we resort to a reduced model with only the active phonon mode and the lossy defects, i.e., ℋ_eff=(ω_m-iγ_m^') b^†b + (ω_q-iγ_q) σ_+σ_-+g_d(b^†σ_++σ_-b), with the effective damping γ^'_m=γ_m-G_0. We note that in a recent experiment<cit.>, a similar route was adopted for achieving a non-Hermitian atomic system, where, by starting from a Hermitian Hamiltonian describing full atom-light interactions, an effective non-Hermitian model was deduced for atomic excitations (see also Ref. <cit.>). Choosing two basis states, |n_b,g⟩ and |n_b-1,e⟩, to diagonalize ℋ_eff leads to the eigenvalues E_±= (n_b-1/2)ω_m +ω_q/2-i/2[(2n_b-1)γ^'_m+γ_q] ±1/2√(4n_b g^2_d+[ω_q-ω_m-i(γ_q-γ^'_m) ]^2 ). The supermode spectrum of these eigenvalues is shown in Fig. <ref>(a) and <ref>(b), where an EP is seen at the position close to the turning points in Fig. <ref>(b) and Fig. <ref>(b). This EP, labelled by the critical value γ_q^ EP, characterizes the transition between two distinct phases of the hybrid TLS-phonon system <cit.>: (i) For γ_q≤γ_q^EP, the supermodes are almost equally distributed between the phonons and the defects, and the active phonon mode partially or completely compensates for loss induced by the defects. Consequently, as γ_q is increased, the system has less net mechanical gain. (ii) For γ_q>γ_q^EP, the supermodes become increasingly localized such that one dominantly resides in the phonon mode and the other in the defects. Hence with increasing γ_q, the supermode which is dominant in the defects experiences more loss, while the supermode which is dominant in the phonon mode experiences less loss (i.e., increased mechanical gain). For the special case ω_q/ω_m=1 <cit.>, the EP emerges at γ_q^ EP=γ'_m+2√(n_b)g_d. while, when ∂ G/∂γ_q=0, the turning point of G is obtained at γ^min_q=√(2n_b) g_d. The slight shift of the turning point from the exact EP position is due to the fact that γ^min_q depends on Δ, while the EP does not. A comparison of the turning points and the EPs for different values of the optical detuning is given in Fig. <ref>. We note that the slight shift of the turning point from the exact EP position was also observed in a purely-optical system (see Ref. <cit.>). We also note that the EP of this TLS-phonon system is reminiscent of that observed recently in a Jaynes-Cummings system with a single atom trapped in a high-Q cavity (by using, however, a different method of tuning the atom-cavity coupling) <cit.>. Finally, Fig. <ref> shows the phonon number N_b=exp[2(G-γ_m)/γ_m], as a function of the defect loss and the pump power. Features similar to those observed for the mechanical gain also appear for N_b, i.e. more loss leads to the suppression of N_b for γ_q≤γ_q^min, but N_b is enhanced with more loss for γ_q>γ_q^min. The turning point of N_b is in exact correspondence with that of the mechanical gain, as shown in Fig. 2(b), or the threshold power in Fig. 3(b). Figure 6(b) shows that N_b is strongly dependent on Δ, and the optimized condition Δ/ω_m=0.5, as in the case without defects, still holds in the presence of TLS defects. § CONCLUSION In this work, we study the counterintuitive role of defects in the phonon-lasing process. We find that the exact evolutions of the mechanical gain and the threshold power exhibit a turning point as the loss is increased. This effect is closely related to the emergence of an EP in an effective non-Hermitian TLS-phonon system. When exceeding the EP, more TLS loss leads to an enhanced mechanical gain, along with a lowered threshold for the phonon laser. This indicates that the detrimental effects of intrinsic lossy defects (naturally existing in solid-state materials) in phonon lasing can be minimized. This sheds a different light not only on EP physics and optomechanics but also on practical control of random-defect phonon devices. We note that the COM-based phonon laser has already been experimentally realized <cit.>, and the effect of inevitably existing defects in the COM device was also studied in the phonon-cooling regime <cit.>. Our work extends the COM-TLS structures to the mechanical-amplifying regime and reveals the emergence of a loss-induced EP. We establish the relation between EP, TLS loss, and mechanical amplification, which has not been studied before. We also note that, besides material strain <cit.>, the TLS energy splitting and damping rate can be controlled by external electric fields <cit.>. This opens the way for electrically tuned phonon lasing. Finally, we remark that the optical effect of defects can be incorporated into the optical decay rate, and the mechanical strain only induces the phonon-TLS coupling (not any additional optical effects, see also Ref. <cit.>). In our future works, we will consider placing a nanotip near the optical resonator <cit.> to study the interplay of the interplay of the loss-induced optical EP <cit.> and the TLS-phonon EP, or placing an atom in the system <cit.> to study the interplay of the atom-photon coupling and the TLS-phonon coupling. It will be also of interesting to study COM squeezing <cit.> or sensing <cit.> in the presence of TLS-phonon EPs. § ACKNOWLEDGMENTS L.-M.K. is supported by the 973 Program under Grant No. 2013CB921804 and the National Natural Science Foundation of China under Grants No. 11375060 and No. 11434011. H.J. is supported by the National Natural Science Foundation of China under Grants No. 11474087 and No. 11774086. S.K.Ö. is supported by ARO Grant No. W911NF-16-1-0339 and the Pennsylvania State University Materials Research Institute. F.N. is supported by the RIKEN iTHES Project, the MURI Center for Dynamic Magneto-Optics via the AFOSR award number FA9550-14-1-0040, the IMPACT program of JST, CREST No. JPMJCR1676, a JSPS Grant-in-Aid for Scientific Research (A), and a grant from the Sir John Templeton Foundation. 99 AKM2014 M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014). Metcalfe2014 M. Metcalfe, Applications of cavity optomechanics, Appl. Phys. Rev. 1, 031105 (2014). Vahala2010 I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, Phonon laser action in a tunable two-level system, Phys. Rev. Lett. 104, 083901 (2010). Vahala2009 K. Vahala, M. Herrmann, S. Knünz, V. Batteiger, G. Saathoff, T. W. Hänsch, and Th. Udem, A phonon laser, Nat. Phys. 5, 682 (2009). Mahboob2013 I. Mahboob, K. Nishiguchi, A. Fujiwara, and H. Yamaguchi, Phonon lasing in an electromechanical resonator, Phys. Rev. Lett. 110, 127202 (2013). Kent2010 R. P. Beardsley, A. V. Akimov, M. Henini, and A. J. Kent, Coherent terahertz sound amplification and spectral line narrowing in a stark ladder superlattice, Phys. Rev. Lett. 104, 085501 (2010). Xiao2017 G. Wang, M. Zhao, Y. Qin, Z. Yin, X. Jiang, and M. Xiao, Demonstration of an ultra-low-threshold phonon laser with coupled microtoroid resonators in vacuum, Photon. Res. 5, 73 (2017). Lawall2014 U. Kemiktarak, M. Durand, M. Metcalfe, and J. Lawall, Mode competition and anomalous cooling in a multimode phonon laser, Phys. Rev. Lett. 113, 030802 (2014). Painter2015 J. D. Cohen, S. M. Meenehan, G. S. MacCabe, S. Gröblacher, A. H. Safavi-Naeini, F. Marsili, M. D. Shaw, and O. Painter, Phonon counting and intensity interferometry of a nanomechanical resonator, Nature 520, 522 (2015). Wollman2015 E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, Quantum squeezing of motion in a mechanical resonator, Science 349, 952 (2015). Lei2016 C. U. Lei, A. J. Weinstein, J. Suh, E. E. Wollman, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, Quantum nondemolition measurement of a quantum squeezed state beyond the 3 dB limit, Phys. Rev. Lett. 117, 100801 (2016). Hammerer2015 N. Lörch and K. Hammerer, Sub-Poissonian phonon lasing in three-mode optomechanics, Phys. Rev. A 91, 061803(R) (2015). phonon-rmp N. Li, J. Ren, L. Wang, G. Zhang, P. Hänggi, and B. Li, Phononics: Manipulating heat flow with electronic analogs and beyond, Rev. Mod. Phys. 84, 1045 (2012). Alu R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Alù, Sound isolation and giant linear nonreciprocity in a compact acoustic circulator, Science 343, 516 (2014). Yamaguchi2013 D. Hatanaka, I. Mahboob, K. Onomitsu, and H. Yamaguchi, A phonon transistor in an electromechanical resonator array, Appl. Phys. Lett. 102, 213102 (2013). capacitor H. Han, B. Li, S. Volz, and Y. A. Kosevich, Ultracompact interference phonon nanocapacitor for storage and lasing of coherent terahertz lattice waves, Phys. Rev. Lett. 114, 145501 (2015). comb A. Ganesan, C. Do, and A. Seshia, Phononic frequency comb via intrinsic three-wave mixing, Phys. Rev. Lett. 118, 033903 (2017). imaging L. J. Swenson, A. Cruciani, A. Benoit, M. Roesch, C. S. Yung, A. Bideaud, and A. Monfardini, High-speed phonon imaging using frequency-multiplexed kinetic inductance detectors, Appl. Phys. Lett. 96, 263511 (2010). computing S. R. Sklan, Splash, pop, sizzle: Information processing with phononic computing, AIP Adv. 5, 053302 (2015). Peng2014 B. Peng, Ş. K. Özdemir, F. Lei, F. Monifi, M. Gianfreda, G. L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Parity-time-symmetric whispering-gallery microcavities, Nat. Phys. 10, 394 (2014). Jing2014 H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, PT-symmetric phonon laser, Phys. Rev. Lett. 113, 053604 (2014). Jing2015 H. Jing, Ş. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, Optomechanically-induced transparency in parity-time-symmetric microresonators, Sci. Rep. 5, 9663 (2015). Jing2016 Y. Jiao, H. Lü, J. Qian, Y. Li, and H. Jing, Nonlinear optomechanics with gain and loss: amplifying higher-order sideband and group delay, New J. Phys. 18, 083034 (2016). 2015Chaos X.-Y. Lü, H. Jing, J.-Y. Ma, and Y. Wu, PT-symmetry-breaking chaos in optomechanics, Phys. Rev. Lett. 114, 253601 (2015). Jing2017 H. Jing, Ş. K. Özdemir, H. Lü, and F. Nori, High-order exceptional points in optomechanics, Sci. Rep. 7, 3386 (2017). diode J. Zhang, B. Peng, Ş. K. Özdemir, Y.-X. Liu, H. Jing, X.-Y. Lü, Y.-L. Liu, L. Yang, and F. Nori, Giant nonlinearity via breaking parity-time symmetry: A route to low-threshold phonon diodes, Phys. Rev. B 92, 115407 (2015). El-Ganainy2016 D. W. Schönleber, A. Eisfeld, and R. El-Ganainy, Optomechanical interactions in non-Hermitian photonic molecules, New J. Phys. 18, 045014 (2016). He2016 B. He, L. Yang, and M. Xiao, Dynamical phonon laser in coupled active-passive microresonators, Phys. Rev. A 94, 031802(R) (2016). Guo2009 A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Observation of PT-symmetry breaking in complex optical potentials, Phys. Rev. Lett. 103, 093902 (2009). Peng2014Science B. Peng, Ş. K. Özdemir, S. Rotter, H. Yilmaz, M. Liertzer, F. Monifi, C. M. Bender, F. Nori, and L. Yang, Loss-induced suppression and revival of lasing, Science 346, 328 (2014). Lin2011 Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, Unidirectional invisibility induced by PT-symmetric periodic structures, Phys. Rev. Lett. 106, 213901 (2011). Regensburger2012 A. Regensburger, C. Bersch, M.-A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, Parity-time synthetic photonic lattices, Nature (London) 488, 167 (2012). Peng2016 B. Peng, Ş. K. Özdemir, M. Liertzer, W. Chen, J. Kramer, H. Yilmaz, J. Wiersig, S. Rotter, and L. Yang, Chiral modes and directional lasing at exceptional points, Proc. Natl. Acad. Sci. U.S.A. 113, 6845 (2016). Doppler2016 J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, Dynamically encircling an exceptional point for asymmetric mode switching, Nature (London) 537, 76 (2016). Alu2015 R. Fleury, D. Sounas, and A. Alù, An invisible acoustic sensor based on parity-time symmetry, Nat. Commun. 6, 5905 (2015). Kottos2013 N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, Observation of asymmetric transport in structures with active nonlinearities, Phys. Rev. Lett. 110, 234101 (2013). Xiao2016 P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang, and Y. Xiao, Anti-parity-time symmetry with flying atoms, Nat. Phys. 12, 1139 (2016). Xu2016 H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Topological energy transfer in an optomechanical system with exceptional points, Nature (London) 537, 80 (2016). Morita1990 M. Morita, T. Ohmi, E. Hasegawa, M. Kawakami, and M. Ohwada, Growth of native oxide on a silicon surface, J. Appl. Phys. 68, 1272 (1990). Zeller1971 R. C. Zeller and R. O. Pohl, Thermal conductivity and specific heat of noncrystalline solids, Phys. Rev. B 4, 2029 (1971). Anderson1972 P. W. Anderson, B. I. Halperin, and C. M. Varma, Anomalous low-temperature thermal properties of glasses and spin glasses, Philos. Mag. 25, 1 (1972). Phillips1972 W. A. Phillips, Tunneling states in amorphous solids, J. Low Temp. Phys. 7, 351 (1972). Golding1976 B. Golding and J. E. Graebner, Phonon echoes in glass, Phys. Rev. Lett. 37, 852 (1976). Anghel2007 D. V. Anghel, T. Kühn, Y. M. Galperin, and M. Manninen, Interaction of two-level systems in amorphous materials with arbitrary phonon fields, Phys. Rev. B 75, 064202 (2007). Yu1994 H. M. Carruzzo, E. R. Grannan, and C. C. Yu, Nonequilibrium dielectric behavior in glasses at low temperatures: Evidence for interacting defects, Phys. Rev. B 50, 6685 (1994). Tanaka2009 L. G. Remus, M. P. Blencowe, and Y. Tanaka, Damping and decoherence of a nanomechanical resonator due to a few two-level systems, Phys. Rev. B 80, 174103 (2009). Martinis2005 J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Decoherence in Josephson qubits from dielectric loss, Phys. Rev. Lett. 95, 210503 (2005). Nori2006 A. M. Zagoskin, S. Ashhab, J. R. Johansson, and F. Nori, Quantum two-level systems in Josephson junctions as naturally formed qubits, Phys. Rev. Lett. 97, 077001 (2006). Neeley2008 M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, N. Katz, E. Lucero, A. O'Connell, H. Wang, A. N. Cleland, and J. M. Martinis, Process tomography of quantum memory in a Josephson-phase qubit coupled to a two-level state, Nat. Phys. 4, 523 (2008). qubit G. J. Grabovskij, T. Peichl, J. Lisenfeld, G. Weiss, and A. V. Ustinov, Strain tuning of individual atomic tunneling systems detected by a superconducting qubit, Science 338, 232 (2012). Osborn2016 Y. J. Rosen, M. S. Khalil, A. L. Burin, and K. D. Osborn, Random-defect laser: manipulating lossy two-level systems to produce a circuit with coherent gain, Phys. Rev. Lett. 116, 163601 (2016). Ramos2013 T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J. Kippenberg, Nonlinear quantum optomechanics via individual intrinsic two-level defects, Phys. Rev. Lett. 110, 193602 (2013). Tian2011 L. Tian, Cavity cooling of a mechanical resonator in the presence of a two-level-system defect, Phys. Rev. B 84, 035417 (2011). HW2014 H. Wang, Z. Wang, J. Zhang, Ş. K. Özdemir, L. Yang, and Y.-X. Liu, Phonon amplification in two coupled cavities containing one mechanical resonator, Phys. Rev. A 90, 053814 (2014). Ustinov2010 J. Lisenfeld, C. Müller, J. H. Cole, P. Bushev, A. Lukashenko, A. Shnirman, and A. V. Ustinov, Measuring the temperature dependence of individual two-level systems by direct coherent control, Phys. Rev. Lett. 105, 230504 (2010). GWunner2013 M. Kreibich, J. Main, H. Cartarius, and G. Wunner, Hermitian four-well potential as a realization of a PT-symmetric system, Phys. Rev. A 87, 051601(R) (2013). Bagarello2015 F. Bagarello, M. Lattuca, R. Passante, L. Rizzuto, and S. Spagnolo, Non-Hermitian Hamiltonian for a modulated Jaynes-Cummings model with PT symmetry, Phys. Rev. A 91, 042134 (2015). Bender1998 C. M. Bender and S. Boettcher, Real spectra in non-Hermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80, 5243 (1998). Bender1999 C. M. Bender, S. Boettcher, and P. N. Meisinger, PT-symmetric quantum mechanics, J. Math. Phys. 40, 2201 (1999). Bender2007 C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70, 947 (2007). Choi2010 Y. Choi, S. Kang, S. Lim, W. Kim, J.-R. Kim, J.-H. Lee, and K. An, Quasieigenstate coalescence in an atom-cavity quantum composite, Phys. Rev. Lett. 104, 153601 (2010). Wang2015 H. Wang, X. Gu, Y.-X. Liu, A. Miranowicz, and F. Nori, Tunable photon blockade in a hybrid system consisting of an optomechanical device coupled to a two-level system, Phys. Rev. A 92, 033806 (2015). Kimble2005 K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, Photon blockade in an optical cavity with one trapped atom, Nature (London) 436, 87 (2005). Hu1996 X. Hu and F. Nori, Squeezed phonon states: modulating quantum fluctuations of atomic displacements, Phys. Rev. Lett. 76, 2294 (1996). Hu1997 X. Hu and F. Nori, Phonon squeezed states generated by second-order Raman scattering, Phys. Rev. Lett. 79, 4605 (1997). Huang2017 Q. Huang et al., Nanofibre optic force transducers with sub-piconewton resolution via near-field plasmon-dielectric interactions, Nat. Photon. 11, 352 (2017).

http://arxiv.org/abs/1701.07993v1

20170127100934

Securing Virtual Network Function Placement with High Availability Guarantees

cs.NI

Non Amontons-Coulomb local friction law of randomly rough contact interfaces with rubber Antoine Chateauminoisy^1 December 30, 2023 ======================================================================================== Virtual Network Functions as a Service (VNFaaS) is currently under attentive study by telecommunications and cloud stakeholders as a promising business and technical direction consisting of providing network functions as a service on a cloud (NFV Infrastructure), instead of delivering standalone network appliances, in order to provide higher scalability and reduce maintenance costs. However, the functioning of such NFVI hosting the VNFs is fundamental for all the services and applications running on top of it, forcing to guarantee a high availability level. Indeed the availability of an VNFaaS relies on the failure rate of its single components, namely the servers, the virtualization software, and the communication network. The proper assignment of the virtual machines implementing network functions to NFVI servers and their protection is essential to guarantee high availability. We model the High Availability Virtual Network Function Placement (HA-VNFP) as the problem of finding the best assignment of virtual machines to servers guaranteeing protection by replication. We propose a probabilistic approach to measure the real availability of a system and design both efficient and effective algorithms that can be used by stakeholders for both online and offline planning. § BACKGROUND A recent trend in computer networks and cloud computing is to virtualize network functions, in order to provide higher scalability, reducing maintenance costs, and increasing reliability of network services. Virtual Network Functions as a Service (VNFaaS) is currently under attentive study by telecommunications and cloud stakeholders, as a promising business and technical direction consisting of providing network functions (i.e., firewall, intrusion detection, caching, gateways...) as a Service instead of delivering standalone network appliances. While legacy network services are usually implemented by means of highly reliable hardware specifically built for a single purpose middlebox, VNFaaS moves such services to a virtualized environment <cit.>, named NFV Infrastructure (NFVI) and based on commercial-off-the-shelf hardware <cit.>. Services implementing network functions are called Virtual Network Functions (VNFs). One of the open issues for NFVI design is indeed to guarantee high levels of VNF availability <cit.>, i.e., the probability that the network function is working at a given time. In other words, a higher availability corresponds to a smaller downtime of the system, and it is required to satisfy stringent Service Level Agreements (SLA). Failures may result in a temporary unavailability of the services, but while in other contexts it may be tolerable, in NFVI network outages are not acceptable, since the failure of a single VNF can induce the failure of all the overlying services <cit.>. To achieve high availability, backup VNFs can be placed into the NFVI, acting as replicas of the running VNFs, so that when the latter fail, the load is rerouted to the former. However, not all VNFs are equal ones: the software implementing a network function of the server where a VNF is running may be more prone to errors than others, influencing the availability of the overall infrastructure. Also, client requests may be routed via different network paths, with different availability performance. Therefore to guarantee high levels of availability it is important not only to increase the number of replicas placed on an NFVI, but it is also crucial to select where they are placed and which requests they serve. In this context, we study and model the High Availability Virtual Network Function Placement (HA-VNFP) problem, that is the problem of placing VNFs on an NFVI in order to serve a given set of clients requests guaranteeing high availability. Our contribution consist of: [(a)] * a quantitative probabilistic model to measure the expected availability of VNF placement; * a proof that the problem is 𝒩𝒫-hard and that it belongs to the class of nonlinear optimization problems; * a linear mathematical programming formulation that can be solved to optimality for instances of limited size; * a Variable Neighborhood Search (VNS) heuristic for both online and offline planning; * an extensive simulation campaign, and algorithm integration in a Decision Support System (DSS) tool <cit.>. The paper is organized as follows: in <ref> we present the HA-VNFP, and in <ref> we briefly describe previous works on VM/VNF placement in cloud/NFVI systems. In <ref> we formally describe the optimization problem and propose a linearization of the problem and a mathematical programming formulation that can solve it to optimality. In <ref> we describe our heuristic methodologies, which are then tested in an extensive simulation campaign in <ref>. We briefly conclude in <ref>. § HIGH AVAILABILITY VNF PLACEMENT We consider an NFVI with several geo-distributed datacenters or clusters (see <ref>). Each cluster consists of a set of heterogeneous servers with limited available computing resources. Several instances of the same VNF type can be placed on the NFVI but on different servers. Each VNF instance can be assigned to a single server allocating some of its computing resources. Indeed each server has a limited amount of computing resources that cannot be exceeded. A network connects together all servers of the NFVI: we suppose that the communication links inside a cluster are significantly more reliable than those between servers in different clusters. An access network with multiple access points connects servers to clients. Links connecting access points to clusters can differ in the availability level, depending on the type of the connection or the distance from the cluster. In this article, we assume that the total amount of resources and network capacities are sufficient to manage the expected client requests at any time. However assignment decisions may artificially produce congestion over the servers. We analyze how to find assignments providing a trade-off between NFVI availability and system congestion. We are given an estimation of the expected client VNF requests, each characterized by a computing resource demand. An assigned request consumes part of the resources reserved by a VNF instance. Indeed the consumed resources must not exceed the reserved ones. Requests can be assigned using two different policies: demand load balancing and split demand load balancing. In the former, a client request is always fully assigned to a single server, while in the latter it may be split among different ones. Splitting a request also splits proportionally its demand of computing resources. Indeed, when a demand is split it relies on the availabilities of many VNF instances, decreasing the expected availability of the service, but increasing the chance of finding a feasible assignment in case of congestion. We suppose a multi-failure environment in which VNFs, servers, clusters, and networks may fail together. Our aim is to improve the VNF availability by replicating instances on the NFVI. We distinguish between master and slave VNFs: the former are active VNF instances, while the latter are idle until masters fail. An example of VNF placement is depicted in <ref>. Each master may be protected by many slaves - we assume in this article that a slave can protect only a master, must be placed on a different server, and must allocate at least the same amount of computing resources of its master. Each master periodically saves and sends its state to its slaves, e.g. using technologies such as the one presented in <cit.>, in such a way that the latter has always an updated state and can consistently restore the computation in case of failure of the former. We suppose that if a master is unreachable, a searching process is started in order to find a slave that can complete the computation. If the searching process fails and all slaves are unreachable, then the service is considered unavailable. A representation of VNF protection is in <ref>. § RELATED WORKS Even if VM and VNF resource placement in cloud systems is a recent area of research (see <cit.> for a high-level comprehensive study), however there already exists orchestrators that are driven by optimization algorithms for the placement, such as <cit.>. We now present few works in literature studying the optimization problems that arises in this context. Placement of Virtual Machines <cit.> studies the problem of placing VMs in datacenters minimizing the average latency of VM-to-VM communications. Such a problem is 𝒩𝒫-hard and falls into the category of Quadratic Assignment Problems. The authors provide a polynomial time heuristic algorithm solving the problem in a "divide et impera" fashion. In <cit.> the authors deal with the problem of placing VMs in geo-distributed clouds minimizing the inter-VM communication delays. They decompose the problem in subproblems that they solve heuristically. They also prove that, under certain conditions, one of the subproblems can be solved to optimality in polynomial time. <cit.> studies the VM placement problem minimizing the maximum ratio of the demand and the capacity across all cuts in the network, in order to absorb unpredictable traffic burst. The authors provide two different heuristics to solve the problem in reasonable computing time. Placement of Virtual Network Functions <cit.> applies NFV to LTE mobile core gateways proposing the problem of placing VNFs in datacenters satisfying all client requests and latency constraints while minimizing the overall network load. Instead, in <cit.> the objective function requires to minimize the total system cost, comprising the setup and link costs. <cit.> introduces the VNF orchestration problem of placing VNFs and routing client requests through a chain of VNFs. The authors minimize the setup costs while satisfying all client demands. They propose both an ILP and a heuristic to solve such problem. Also <cit.> considers the VNF orchestration problem with VNF switching piece-wise linear latency function and bit-rate compression and decompression operations. Two different objective functions are studied: one minimizing costs and one balancing the network usage. Placement with protection In <cit.> VMs are placed with a protection guaranteeing k-resiliency, that is at least k slaves for each VM. The authors propose an integer formulation that they solve by means of constraint programming. In <cit.> the recovery problem of a cloud system is considered where slaves are usually turned off to reduce energy consumption but can be turned on in advance to reduce the recovery time. The authors propose a bicriteria approximation algorithm and a greedy heuristic. In <cit.> the authors solve a problem where links connecting datacenters may fail, and a star connection between VMs must be found minimizing the probability of failure. The authors propose an exact and a greedy algorithms to solve both small and large instances, respectively. Within disaster-resilient VM placement, <cit.> proposes a protection scheme in which for each master a slave is selected on a different datacenter, enforcing also path protection. In <cit.> the authors solve the problem of placing slaves for a given set of master VMs without exceeding neither servers nor link capacities. Their heuristic approaches decompose the problems in two parts: the first allocating slaves, and the second defining protection relationships. In a recent work <cit.>, the authors model the VM availability by means of a probabilistic approach and solve the placement problem over a set of servers by means of a nonlinear mathematical formulation and greedy heuristics. This is the only work offering an estimation of the availability of the system. However, it considers only the availability of the servers, while in our problem we address a more generic scenario: when datacenters are geo-distributed, a client request shall be assigned to the closest datacenter, since longer connections may have a higher failure rate. Therefore, the source of the client requests may affect the placement of the VNFs on the NFVI, and must be taken into account in the optimization process and in the estimation of the availability. § MODELING In the following we propose a formal definition to the HA-VNFP and a mathematical programming formulation. Clusters and servers We are given the set of clusters C and the set of servers S. Each server s belongs to a cluster c_s, and we define as S_c ⊆ S the set of servers of cluster c. We represent the usual distinct types of computing resources (CPU, RAM, ... ) of server s ∈ S by the same global amount q_s ∈ℝ_+ of available resources. Virtual Network Functions A set F of VNF types is given. Each VNF instance runs on a single server. Each server can host multiple VNF instances, but at most one master for each type. Networks An inter-cluster network allows synchronization between clusters, while an access network connects clusters to a set of access points P. We are given sets L_C and L_P of logical links (c',c”) ∈ L_C connecting clusters c',c”∈ C, and logical links (c,p) ∈ L_P connecting cluster c ∈ C to access point p ∈ P, respectively. Clients requests A set of clients requests R is given. Each request r ∈ R is a tuple (f_r, P_r, d_r) of the requested VNF type f_r ∈ F, a subset of available access points P_r ⊆ P, and the resources demand d_r ∈ℝ_+. Availability Taking into account explicit availability in NFVI design becomes necessary to ensure SLAs <cit.>. We suppose that the availabilities of each component (server, cluster, VNF, link) are given (see <ref>), each corresponding to the probability that a component is working. Objective function All clients requests must be assigned to servers maximizing the availability of the system, we measure as the minimum availability among all requests. §.§ Computational complexity Concerning the assignment of requests, we can prove that: When demand split is allowed and ∑_r ∈ R d_r ≤∑_s ∈ S q_s, HA-VNFP has always a feasible solution that can be found in polynomial time. In fact since the requests can be split among servers, the feasibility of an instance can be found applying a Next-Fit greedy algorithm for the Bin Packing Problem with Item Fragmentation (BPPIF) <cit.>: servers can be seen as bins, while requests as items that must be packed into bins. The algorithm iteratively pack items to an open bin. When there is not enough residual capacity, the item is split, the bin is filled and closed, and a new bin is open packing the rest of the item. When requests can be split, such algorithm produces a feasible solution for the HA-VNFP: if a request is assigned to a server, then a master VNF serving such a request is allocated on that server too. The Next-Fit algorithm runs in O(|R|) and therefore a feasible solution can be found in polynomial time. The feasibility of a HA-VNFP instance without demand split is a 𝒩𝒫-hard problem. Indeed we can see again the feasibility problem as a Bin Packing Problem (BPP). However, without split each item must be fully packed into a single bin. Therefore, finding a feasible solution is equivalent to the feasibility of a BPP, which is 𝒩𝒫-hard, and it directly follows that: The HA-VNFP without demand split is 𝒩𝒫-hard. That is, for unsplittable demands, it is 𝒩𝒫-hard finding both a feasible solution and the optimum solution. It is less straightforward to also prove that: The HA-VNFP with demand split is 𝒩𝒫-hard. In fact, let us suppose a simple instance where all components (servers, clusters, links, ...) are equal ones and where ∑_r ∈ R d_r = ∑_s ∈ S q_s, which means that there will be no slaves in our placement. The problem can be seen again as a BPPIF in which the objective is to minimize the number of splits of the item that is split the most: in fact, every time a request is split, the availability of the system decreases. In such scenarios the best solution is the one in which no request is split at all - however, if we could solve such a problem in polynomial time, then we could solve also the feasibility problem of a BPP in polynomial time, which instead is 𝒩𝒫-hard. Therefore, since we can reduce a feasibility problem of BPP to an instance of BPPIF, and the latter to an instance of HA-VNFP, the HA-VNFP with split is 𝒩𝒫-hard. §.§ Mathematical formulation In the following we propose a mathematical programming formulation of HA-VNFP starting from the definition of the set of the solutions: a request assignment ω is a pair (s, S_p) indicating the subset of servers S_p ⊆ S running either the master or the slaves of a VNF instance, and the server s ∈ S_p where the master is placed. We also define Ω = {(s, S_p) | S_p ⊆ S, s ∈ S_p} as the set of all request assignments. An assignment configuration γ (see <ref>) is a set of all request assignments ω for all the fragments of a request. We define as Γ the set of all assignment configurations γ, that is Γ = {γ∈ 2^Ω| s' ≠ s”, ∀ (s',ω'), (s”, ω”) ∈γ}. Availability computation We compute the NFVI availability for a request r by means of a probabilistic approach <cit.>. Given a cluster and a set of access points, a^L_P(c,P) is the function computing the probability that at least one of the access links is working: a^L_P(c,P) = 1 - ( ∏_p ∈ P 1 - a_cp^L_P ). Given a VNF and a set of servers, a^S(f,S) is the function computing the probability that at least one instance of VNF is working: a^S(f,S) = 1 - ( ∏_s ∈ S 1 - a_f^F · a_s^S ). Given a request r and a request assignment ω = (s, S_p), a(r, ω) is the function computing the probability that at least one of the instances of ω is working: a(r, ω)= 1 - [ ( 1 - a^L_P(c_s, P_r) · a_c_s^C · a^S(f_r, S_p ∩ S_c_s))_· ·∏_c ∈ C ∖{c_s}( 1 -a^L_P(c, P_r) · a_c^C · a_c_sc^L_C· a^S(f_r, S_p ∩ S_c)_ ] When a request r is split, we compute its availability a(r, γ) as the probability that all of its parts succeed: a(r, γ) = ∏_(s, ω) ∈γ a(r,ω). We remark that such formula is nonlinear and produces a Integer Nonlinear Programming formulation which cannot be solved by common integer solvers like CPLEX. Therefore we propose a MIP linearization of such nonlinear formulation in which for each assignment configuration γ∈Γ we have a binary variable stating if such configuration is selected in the solution. Variables The following variables are needed: x_rs : z_rγ = { 1 , r 0, . u_fs : v_fss' : 𝒜_min : Model HA-VNFP can be modeled as follows: max 𝒜_min ∑_s ∈ S x_rs = 1 ∀ r ∈ R x_rs≤∑_γ∈Γ ∃ (s, ω) ∈γ z_rγ ∀ r ∈ R,s ∈ S ∑_r ∈ R f_r = f d_r· x_rs≤ u_fs ∀ f ∈ F, s ∈ S u_f_rs + q_s·∑_γ∈Γ ∃ (s,ω) ∈γ| s' ∈ω z_rγ≤ v_f_rss'+ q_s ∀ r ∈ R, s,s' ∈ S ∑_f ∈ F u_fs + ∑_s' ∈ S v_fs's≤ q_s ∀ s ∈ S ∑_γ∈Γ z_rγ≤ 1 ∀ r ∈ R ∑_γ∈Γ a(r,γ) · z_rγ≥𝒜_min ∀ r ∈ R Constraints (<ref>) and (<ref>) ensure that each request is fully assigned and selects an assignment configuration, respectively. Constraints (<ref>) and (<ref>) set the allocated resources of masters and slaves, respectively. Constraints (<ref>) ensure that servers capacities are not exceeded. Constraints (<ref>) impose that at most one assignment configuration is selected for each request. Constraints (<ref>) compute the minimum availability. Our formulation can model both the HA-VNFP with and without split: in fact by simply setting |γ| = 1 for each configuration γ we forbid configurations splitting a request. § HEURISTICS Solving HA-VNFP as a MIP using an integer solver works only for small NFVI, since the number of variables is exponential w.r.t the size of the instances. Therefore we propose two different heuristic approaches for HA-VNFP: the first is an adaptation of well-known greedy policies for the BPP that will serve as comparison, while the second is a Variable Neighborhood Search heuristic using different algorithmic operators to explore the neighborhood of a starting point. §.§ Greedy heuristics Most of the heuristics for the placement of VMs or VNFs are based on a greedy approach, and BPP heuristics are often exploited to obtain suitable algorithms for the placement, such as in <cit.>. We also exploit BPP heuristics to obtain three different greedy approaches for the HA-VNFP: Best Availability, Best Fit, and First Fit greedy heuristics. The algorithm, reported in <ref>, starts from an empty initial placement and for each request r it looks for a server having enough residual capacity to satisfy the demand d_r. If such a server is found, then the request is assigned to it, otherwise the algorithm fails without finding a feasible solution. However, we can observe that: When ∑_r ∈ R d_r ≤∑_s `in S q_s and split is allowed, our greedy heuristic always finds a feasible solution. In fact we can always split a request between two servers, as stated also in <ref>. The selection of the server is performed by the procedure selectServerS̅, d̅, split which discards the servers without sufficient resources to satisfy demand d̅, and selects a server depending on the chosen policy: * best fit: the server whose capacity best fits the demand; * first fit: the first server found; * best availability: the server with the highest availability. While the first three policies are well-know for the BPP, the fourth one is designed for the HA-VNFP. Master VNFs are placed during the assignment of the requests. Then, in a similar way, the algorithm places additional slaves: for each master the algorithm looks for a server having enough capacity for a slave still using selectServer procedure. After a server is found, the slave is placed. Such a procedure is repeated until no additional slave is placed. §.§ Variable Neighborhood Search The Variable Neighborhood Search (VNS) is a meta-heuristic that systematically changes the neighborhood within the local search algorithm, in order to escape from local optima. In other words, it starts from an initial solution, applies a local search algorithm until it improves, and then changes the type of local search algorithm applied to change the neighborhood. Our VNS algorithm explores 4 different neighborhoods and it is initialized with several starting points, each obtained using a different greedy algorithm. The main logic of our VNS algorithm is sketched in <ref>: we generate 3 starting points by using the greedy heuristics of <ref> and we explore their neighborhood for a placement improving the availability. If no improvement can be found, the algorithm switches the neighborhood. Indeed applying local search is time expensive, but we can observe that a max-min objective function divides the requests in two sets: a set of requests having an availability equal to the objective function and another set having a better availability. We refer to the former as the set of the worst requests, since they are the ones whose improvement will also improve the availability of the entire solution. To reduce the computing time and focus our algorithm we found to be profitable to restrict the explored neighborhood to the worst requests only. Also, after applying each operator we look for new slaves, as in the greedy procedure <ref>. Given two feasible placements, we say that one is improving if it has a higher availability or if it has the same availability but fewer worst requests. In the following we describe the neighborhoods of our VNS. VNFs swap The first neighborhood consists of swapping VNFs (see <ref>): given a VNF, we swap it with a subset of VNFs deployed on a different server. If the placement is improved, then we store the result as the best local change. In general our operator is O(2^|F| · |S|) but we found profitable to set an upper bound of 1 to the cardinality of the set of swapped VNFs, obtaining a O(|F| · |S|) operator. Slave VNFs swap We explore the neighborhood where a slave VNF is removed to free resources for an additional slave of a different master VNF (see <ref>). The complexity of this operator is O(|F| · |S|). Requests swap We also explore the neighborhood where requests are swapped (see <ref>): given a request we consider a subset of requests assigned to a different server and then swap the former with the latter. Similarly to the swap of VNFs, the complexity of this operator is O(2^|R|). However, by setting an upper bound of 1 to the cardinality of the swapped requests set we obtain a O(|R|) operator. Request move In the last exploration we consider the neighbors where a request is simply moved to a different server (see <ref>). The complexity of this operator is O(|S|). In principle, even if all the operators polynomial time our VNS algorithm is not . However, an upper bound k to the number of iterations can be set, obtaining a O(k· |R| · |S| ·max{|R|, |F| · |S|}) heuristic. Also, in the following we show that our VNS requires small computing time for NFVI of limited size and it can be parameterized to end within a time limit, making it suitable for both online and offline planning. § SIMULATION We evaluate empirically the quality of our methodologies: the greedy heuristic using four different policies (best fit, first fit, and best availability), the VNS algorithm, and the mathematical programming formulation as a MIP. However we could run our MIP only on small instances with 3 or 4 servers and 50 requests. In our framework we first run the algorithms using the demand load balancing policy, and allowing split only if the former fails to assign all the requests. All methodologies are implemented in C++, while CPLEX 12.6.3 <cit.> is used to solve the MIP. The simulations have been conducted on Intel Xeon X5690 at 3.47 GHz. We also produced a graphical DSS tool integrating the VNS and the greed algorithms (in python) working on arbitrary 2-hop topologies and made it available in <cit.>. §.§.§ Dataset generation We generated a random dataset consisting of instances that differ for the number of requests, total amount of computing resources, and availabilities of the network components. We set the number of VNF types provided by our NFVI to |F| =5. We assumed an NFVI with 3 clusters (|C|=3) and 3 access points (|P|=3). Each request has a random demand d_r ∈ [1,10], while each server has a random capacity q_s ∈ [75, 125]. The availabilities of all the components of our NFVI are selected between {0.9995, 0.9999, 0.99995, 0.99999} as in <cit.>. We generated 30 instances for each combination of: * number of requests |R| = {50, 100, 200, 300, 400, 500}; * number of access points for each request |P_r| ∈{1, 2, 3}. The number of servers depends on the number of requests, the total amount of the demands, and the random capacities: we generated a set of servers such that their capacities are enough to serve all the demands Q = ∑_s ∈ S q_s ≥∑_r ∈ R d_r. Note that such condition guarantees the feasibility only when splitting requests is allowed. Servers are randomly distributed among all the clusters, in such a way that for each pair of clusters c and c' we have |S_c - S_c'| ≤ 1. Under these conditions we obtained instances with around 3 servers when |R| = 50 and 28 servers when |R| = 500. §.§ Comparison on small instances We first evaluate the quality of our VNS heuristic against the solutions obtained by the MIP solver and the greedy heuristics. In order to study how the NFVI behaves on different levels of congestion, we let its computing resources to grow from an initial value of Q = ∑_s ∈ S q_s, to 3 · Q, with a step of 0.25 · Q. Due to the exponential number of the variables of our formulation, the MIP solver could handle only small instances with 50 requests. All tests have been performed setting a time limit of two hours each and all the algorithms managed to assign all requests without splits. In <ref> we show the average computing time of the algorithms: while the MIP hits several times the time limit, computing times are negligible for all the heuristics, and VNS can be considered as a good alternative for online orchestration when the set of the requests is small. The optimization problem seems harder when the amount of computing resources is scarce: in fact, the average computing time of the MIP is closer to the time limit when the overall capacity is less than 2· Q. Instead, with higher quantities of resources the MIP always find the optimal solution within the time limit. In <ref> we show that the results of our VNS heuristic are close to the MIP ones, while there is a significant gap between the latters and the greedy heuristic ones. In fact, both the MIP and the VNS succeed in finding solutions with an availability of three nines even with scarce resources. Eventually all the algorithms reach an high level of availability when computing resources are doubled. In <ref> we show the variation of the availability when the number of access points for each request increases: in <ref>, <ref>, and <ref> we report the average availability when requests can be routed to the NFVI using 1, 2, and 3 access points, respectively. The path protection obtained by using more than one access point substantially increases the level of the availability. However, having more than 2 access points does not provide additional benefits. §.§ Scaling up the number of requests In a second round of experiments, we evaluated how our VNS algorithm behave when scaling up the number of requests. However, when the number of servers increases its is not possible to use the MIP. Therefore, in the following analysis we compare the results of our VNS algorithm to the greedy ones only. In addition, since our VNS algorithm has not polynomial time complexity, we include in the comparison the results obtained by setting a time limit of 10s at the exploration of each starting solution. We can first observe in <ref> that the computing time of the VNS algorithm grows exponentially when the number of requests increases. Indeed setting a time limit reduces the overall computing time, which is always less than a minute. In <ref> we show that on average there is always a substantial gap between the results obtained by our VNS algorithms and the greedy heuristics. We can also observe that on average the VNS time limit does not penalize substantially the results. Therefore our VNS algorithm with time limit can reduce computing times with minimal loss in availability. From <ref> to <ref> we show the results on instances having from 100 to 500 requests individually. We can observe that the greedy heuristics progressively loose in quality of the solutions, and the gap with the VNS algorithms increases with the number of the requests. Finally in <ref> we show only the results concerning the VNS algorithm without time limit and how it behaves when both the number of requests and the overall capacity increase. We can observe that the curves are similar and the quality of the solutions provided by our algorithm is not affected by the increasing of the size of the instances. Our VNS algorithm always provides placements with an availability of three nines even when resources are scarce, and it always reaches four nines when the capacity is doubled. § CONCLUSION We defined and modeled the HA-VNFP, that is the problem of placing VNFs in NFVI guaranteeing high availability. We provided a quantitative model based on probabilistic approaches to offer an estimation of the availability of a NFVI. We proved that the arising nonlinear optimization problem is 𝒩𝒫-hard and we modelled it by means of a linear formulation with an exponential number of variables. However, to solve instances with a realistic size we designed both efficient and effective heuristics. By extensive simulations, we show that our VNS algorithm finds solution close to the MIP ones, but in computing times smaller of orders of magnitude. We highlighted the substantial gap between the availability obtained using classic greedy policies, and the one obtained with a more advanced VNS algorithm, when the NFVI is congested. Our VNS algorithm showed to be a good compromise to solve HA-VNFP in reasonable computing time, proving to be a good alternative for both online and offline planning. We integrated our HA-VNFP algorithms in a graphical simulator made available with tutorial videos in <cit.>. § ACKNOWLEDGMENTS This article is based upon work from COST Action CA15127 ("Resilient communication services protecting end-user applications from disaster-based failures - RECODIS") supported by COST (European Cooperation in Science and Technology). This work was funded by the ANR Reflexion (contract nb: ANR-14-CE28-0019) and the FED4PMR projects. 10 Addis15 B. Addis, D. Belabed, M. Bouet, S. Secci. Virtual Network Functions Placement and Routing Optimization. IEEE CLOUDNET 2015. Alameddine2016 H. A. Alameddine, S. Ayoubi, C. Assi. Protection plan design for cloud tenants with bandwidth guarantees. DRCN 2016. Alicherry12 M. Alicherry, T. V. Lakshman. Network aware resource allocation in distributed clouds. IEEE INFOCOM 2012. Basta14 A. Basta et al. Applying NFV and SDN to LTE mobile core gateways, the functions placement problem. Proc. of the 4th Workshop on All Things Cellular: Operations, Applications, Challenges, 2014. Bin11 E. Bin et al. Guaranteeing high availability goals for virtual machine placement. ICDCS 2011. Biran12 O. Biran et al. A stable network-aware vm placement for cloud systems. IEEE/ACM CCGrid 2012. Birolini2004 A. Birolini. Reliability and Availability of Repairable Systems. Reliability Engineering: Theory and Practice, 2013. Cohen15 R. Cohen, L. Lewin-Eytan, J.S. Naor, D. Raz. Near optimal placement of virtual network functions. IEEE INFOCOM 2015. Couto15 R. S. Couto, S. Secci, M.E.M. Campista, L.H. M.K. Costa. Server placement with shared backups for disaster-resilient clouds. Computer Networks, 2015. cplex CPLEX development team. IBM ILOG CPLEX optimization studio cplex user's manual - version 12 release 6, 2015. Cully08 B. Cully et al. Remus: High availability via asynchronous virtual machine replication. USENIX NSDI 2008. nfv-resiliency ETSI. ETSI GS NFV-REL 003 V1.1.1 Network Functions Virtualisation (NFV); Reliability; Report on Models and Features for End-to-End Reliability, 2016. Gill-failures P. Gill, N. Jain, N. Nagappan. Understanding network failures in data centers: measurement, analysis, and implications. ACM SIGCOMM 2011. iec-reliability IEC. Analysis techniques for dependability – Reliability block diagram method, 1991. Israel13 A. Israel, D. Raz. Cost aware fault recovery in clouds. IEEE/IFIP IM 2013. lobal Server Hardware, Server OS Reliability Report, 2014. Jennings2014 B. Jennings, R. Stadler. Resource management in clouds: Survey and research challenges. J. Network and Systems Management 23(3), 2014. Luizelli15 M.C. Luizelli et. al. Piecing together the NFV provisioning puzzle: Efficient placement and chaining of virtual network functions. IEEE/IFIP IM 2015. clickos J. Martins et al. Clickos and the art of network function virtualization. USENIX NSDI 2014. Meankerman2001 N. Menakerman, R. Rom. Bin Packing with Item Fragmentation. Algorithms and Data Structures: 7th International Workshop, 2001. Meng10 X. Meng, V. Pappas, L. Zhang. Improving the scalability of data center networks with traffic-aware virtual machine placement. IEEE INFOCOM 2010. Mijumbi2016 R. Mijumbi et al. Network function virtualization: State-of-the-art and research challenges. IEEE Communications Surveys Tutorials, 2016. Mo2015 J. Mo, B. Khasnabish. NFV Reliability using COTS Hardware, draft-mlk-nfvrg-nfv-reliability-using-cots-01, 2015. Riera2016 J. F. Riera et al. TeNOR: Steps towards an orchestration platform for multi-PoP NFV deployment. IEEE NetSoft 2016. Silva-2016 P. Silva, C. Perez, F. Desprez. Efficient Heuristics for Placing Large-Scale Distributed Applications on Multiple Clouds. IEEE/ACM CCGRID 2016. Yang2016 S. Yang, P. Wieder, R. Yahyapour. Reliable Virtual Machine placement in distributed clouds. RNDM 2016. Zhang-2014 Q. Zhang, M. F. Zhani, M. Jabri, R. Boutaba. Venice: Reliable virtual data center embedding in clouds. IEEE INFOCOM 2014. Zhu14 Y. Zhu et al. Reliable resource allocation for optically interconnected distributed clouds. IEEE ICC 2014. HANFVsw HA-NFV simulator software (online): <https://ha-nfv.lip6.fr>.

http://arxiv.org/abs/1701.07989v2

20170127095659

Laplace's method in Bayesian inverse problems

math.PR

A charged anisotropic well-behaved Adler-Finch-Skea solution Satisfying Karmarkar Condition Sumita Banerjee Jan 15 2017 =========================================================================================== In a Bayesian inverse problem setting, the solution consists of a posterior measure obtained by combining prior belief (where we restrict ourselves to Gaussian priors), information about the forward operator, and noisy observational data. This posterior measure is most often given in terms of a density with respect to a reference measure in a high-dimensional (or infinite-dimensional) Banach space. Although Monte Carlo sampling methods provide a way of querying the posterior, the necessity of evaluating the forward operator many times (which will often be a costly PDE solver) prohibits this in practice. For this reason, many practitioners choose a suitable Gaussian approximation of the posterior measure, by a procedure called Laplace's method. Once generated, this Gaussian measure is a easy to sample from and properties like moments are immediately acquired. This paper derives Laplace's approximation of the posterior measure attributed to the inverse problem explicitly by a second-order approximation of the data-misfit functional, specifically and rigorously in the infinite-dimensional setting. By use of a reverse Cauchy-Schwarz inequality we are able to explicitly bound the Hellinger distance between the posterior and its Laplace approximation. § INTRODUCTION We consider a inverse problem y = G(u) + η where G:X→ Y is a (possibly) nonlinear mapping between Hilbert spaces X, Y and η∼ N(0, Γ) is additive noise. The challenge consists of inferring the value of u from the noisy (and usually lower-dimensional) observation y. This is an ill-defined problem in general, so some sort of regularization is needed. In a Bayesian approach (see <cit.>) under Gaussian assumptions, we assume that u∼μ_0 = N(0,C_0), i.e. we have a Gaussian prior on the variable u. For simplicity we assume that the mean is 0, but this assumption can be dropped with slight modifications. The prior acts as a regularization and makes the inverse problem well-defined: Standard theory yields the posterior measure μ on u given an observation y under mild assumptions on the forward operator G: μ/μ_0(u) = exp(-Φ(u))/∫exp(-Φ(u)) μ_0(u), where Φ(u) = y-G(u)_Γ^2/2. Especially in higher dimensions, the posterior is in practice often approximated by a suitable Gaussian, in order to make computation of moments (or frequentist confidence sets) feasible. For a beautiful example how this is done in practice, see <cit.> in the setting of optimal experimental design of an infinite-dimensional inverse problem, or in a finite-dimensional context in <cit.>. This procedure is called Laplace's method and is the focus of this work. We define the functional I(u) = Φ(u) + u_C_0^2/2 which can also be thought of as a (Tikhonov-)regularized cost functional in the classical sense. The maximum a posteriori point is := _u I(u) and the Laplace approximation is defined as ν = N(, HI()^-1) This means that the Laplace approximation is the Gaussian measure centered at the MAP point, with covariance operator matching the “local” covariance structure of the posterior measure. In finite dimensions, its density is exactly the normalized exponential of the measure's negative lognormal local quadratic approximation. For a good explanation (and as a general recommendation for a truly enjoyable book) of Laplace's method in finite dimensions, see <cit.>. The paper <cit.> gives a numerical analysis of the Laplace approximation as well as the so-called Bayesian information criterion; <cit.> and the references therein give an overview over the use of Laplace's method in the Machine Learning and imaging community. A treatise about approximation of measures by Gaussian measures can be found in <cit.>, although they employ the Kullback-Leibler divergence (or relative entropy) as a notion of distance between measures. <cit.> is an extensive study of various Gaussian approximation methods (including Laplace's method) in the context of reservoir modelling. Standard results about Laplace's method are recorded in <cit.> and <cit.>, with newer results on Laplace's method in <cit.>. Gaussian approximations in a different context have been treated in <cit.>, <cit.> in the case of diffusion processes and in <cit.> in the context of molecular dynamics. <cit.> presents an approach running “anti-parallely” to Laplace's method: Instead of approximating the posterior measure directly, they approximate the forward operator or the negative log-likelihood. They bound the Hellinger distance between the true posterior and the resulting approximation, which is also the method of attack in this paper. <cit.> discusses possible algorithms for implementing Laplace's method and similar approximation procedures in Bayesian inference problems. The MAP point is here defined as (<ref>), i.e. a minimization point of the functional I. See <cit.> for a characteristation of the MAP point in a broader setting, by dropping the Gaussianity assumption (in particular, they derive an expression for the MAP point in a linear inverse problem with Besov priors). Original accounts of probabilistic methods of tackling inverse problems can be found in e.g. <cit.> for a random processes view in a Hilbert space setting, <cit.> for – among other things – an explanation of how classical regularization can be viewed as application of a Bayesian prior. The difficulty of constructing well-defined linear estimators is addressed in <cit.> and <cit.>. In <cit.>, a convergence result of the Ky-Fan metric (which is another metric between measures) between posterior measure and the minimum-norm-least-squares point (which is related to the maximum likelihood point) is proven. Although inverse problems in practice will always be finite-dimensional, it has proven worthwhile to study inverse problems from an infinite-dimensional viewpoint. This stems from the fact that although many algorithms work nicely for a given finite data set but break in the limit of finer and finer spatial or temporal resolution. One notable example is total variation regularisation, which is found to be edge-preserving, although it loses this property in the infinite-dimensional limit, see <cit.>. Another intuitive point is that algorithms that work in the infinite-dimensional setting should be complexity-invariant to a refinement of spatial resolution (naive algorithms often have high polynomial complexity with respect to the data). For a discussion of this fact see <cit.>. For more details on how to construct well-defined infinite-dimensional inverse problems, see <cit.> and <cit.>. It can be shown that the Laplace approximation ν coincides with the posterior measure μ if the forward operator G is linear and the prior measure was Gaussian in the first place. Heuristically, the approximation is bad when the posterior measure is multimodal or has different tail properties than a Gaussian. We are interested in deriving concrete error bounds for the approximation quality μ≈ν. The Hellinger distance between probability measures lends itself to this cause. Given two measures μ,ν which are absolutely continuous w.r.t. another measure μ_0, the Hellinger distance (which is independent of the choice of μ_0) between μ and ν is (d_H(μ,ν))^2 = 1/2·∫(√(%s/%s)μμ_0- √(%s/%s)νμ_0)^2μ_0 = 1 - ∫√(%s/%s)μμ_0√(%s/%s)νμ_0μ_0 The main conclusion of this paper is recorded informally in the following claims; they are correctly stated and proven later. While the posterior measure is given via its density w.r.t. the prior by μ/μ_0(u) = exp(-Φ(u))/∫exp(-Φ(u)) μ_0(u), Laplace's method yields a Gaussian approximation of the form ν = N(, HI()^-1) and its density w.r.t. the prior is ν/μ_0(u) = exp(-TΦ(u))/∫exp(-TΦ(u)) μ_0(u). Here, TΦ(u) = Φ() + DΦ()(u-) + 1/2HΦ()[u-, u-], the second order Taylor approximation of the data-misfit functional Φ at . If there is K∈(0,1) such that exp(-Φ(u)/2) - exp(-TΦ(u)/2)_L^2(X, μ_0)≤ K·exp(-I())/√((C_0^1/2· HΦ()· C_0^1/2 + Id)), then the Hellinger distance between the posterior and its Laplace's approximation can be bounded: d_H(μ,ν) ≤K/√(1 + (1-K)^2) While approximation results for Laplace's method have been achieved before (e.g. in the references made above), it is to the best of our knowledge that the claims as stated have not been formulated in an Inverse problem setting where Laplace's method is often used. The paper is organized as follows: First the more intuitive one-dimensional case is presented and representation (<ref>) is derived. The approach taken can not be generalized to the infinite dimensional case, as we use densities with respect to a Lebesgue measure. Then the infinite-dimensional equivalent is proven, where we show the equality (<ref>) directly by means of characteristic functions. The following section shows that the problem of bounding the Hellinger distance can be reduced to a reverse Cauchy-Schwarz inequality. After recording a few elementary results about reverse CS inequalities, we immediately obtain claim <ref> and also a more practical (but less tight) version of it. A caveat regarding practicality: Although proposition <ref> and corollary <ref> yield expressions that seem to offer readily-checked conditions for explicit bounds on the Hellinger distance, a non-artificially constructed example where claim 2 easily yields a non-trivial bound on d_H seems hard do obtain and the applicability of the second claim should thus not be overstated. This is explained in more detail in the section about application. Rather, the main point of this paper is the derivation and proof of characterization of expression (<ref>). § THE LAPLACE APPROXIMATION IN ONE DIMENSION We recall μ/μ_0(u) = exp(-Φ(u))/∫exp(-Φ(u)) μ_0(u), and with for the Lebesgue measure λ in one dimension, the Laplace approximation of μ, which is a Gaussian centered at the maximum a posteriori point, with variance equal to the inverse of the second derivative of I at this point. In finite dimensions, any measure is most easily recorded by its density w.r.t. λ. ν/λ(u) = √(%s/%s)I”()2πexp( -I”()/2· (u-)^2), hence ν/μ_0(u) = ν/λ(u)/μ_0/λ(u) = √(I”())·σ·exp( -I”()/2· (u-)^2 + u^2/2σ^2). As we mentioned, second-order Taylor approximations will play a leading role, hence we define R(u) = I(u) - I() - I”()/2(u-)^2, the error term of the second order Taylor approximation of I in . Note that the first-order term vanishes because of being a minimum of I, which we assume to be C^2. Interestingly, R(u) is also an error term for the second order approximation of Φ, albeit in a slightly different way: R(u) = Φ(u) + u^2/2σ^2 - Φ() - ^2/2σ^2-Φ”()/2(u-)^2 - 1/σ^2(u-)^2 = · (u-)/σ^2 + Φ'()(u-) +[ Φ(u) - Φ() - Φ'()(u-) - Φ”()/2(u-)^2] = Φ(u) - T^(2)_Φ(u). Note that the terms in the second line vanish because of 0 = I'() = Φ'() + /σ^2 and the second term is exactly the error between Φ(u) and its second order Taylor polynom developed in . It holds that I'() = 0 but in general Φ'()≠ 0. With this definition of R, and especially -I”()/2· (u-)^2 + u^2/2σ^2 = -Φ(u) + I() + R(u), we obtain for the density of ν w.r.t. μ_0: ν/μ_0(u) = √(I”())·σ·exp(I()) ·exp(-Φ(u) + R(u)). An easy calculation shows exp(-I())/σ√(I”()) = ∫exp(u^2/2σ^2-I()-I”()/2(u-)^2)μ_0( u) = ∫exp(-Φ(u)+R(u))μ_0( u) = ∫exp(-T^(2)_Φ(u))μ_0(u). This also follows immediately from the normalization of ν/μ_0 and thus ν/μ_0(u) = exp(-Φ(u)+R(u))/∫exp(-Φ(u)+R(u))μ_0( u) = exp(-TΦ(u))/∫exp(-TΦ(u))μ_0( u). § THE LAPLACE APPROXIMATION IN GENERAL HILBERT SPACES In this section, we will show claim <ref> in the setting of a general (possibly infinite-dimensional) Hilbert space. In order to do this, we need a slight adaptation of a useful Gaussian integral calculation. In Proposition 1.2.8 in <cit.>, the authors prove the following: Let μ_0 = N(0, Q) be a Gaussian measure on a real Hilbert space H. Assume that M is a symmetric operator such that ⟨ Q^1/2MQ^1/2 u, u⟩ < ⟨ u,u⟩ for all 0≠ u∈ H. Then for b∈ H ∫_H exp(1/2⟨ M u,u⟩ + ⟨ b, u⟩) μ_0(u) = exp(1/2|(1-Q^1/2MQ^1/2)^-1/2· Q^1/2b|^2 )/√( (1-Q^1/2MQ^1/2)). We will need a generalization of this formula, which follows from analytical continuation of the (real) Hilbert space's inner product to its complex extension. Recall that this continuation will not be positively definite anymore: ⟨λ_1 a + b, λ_2 c + d⟩ = λ_1 λ_2 ⟨ a, c⟩ + λ_2 ⟨ b, c⟩ + λ_1⟨ a, d⟩ + ⟨ b,d⟩, without any complex conjugations on any λ_i. The following lemma is stated without proof as it follows immediately from the bilinearity of the analytical continuation stated above. Let μ_0 = N(0, Q) be a Gaussian measure on a real Hilbert space H. Assume that M is a symmetric operator such that ⟨ Q^1/2MQ^1/2 u, u⟩ < ⟨ u,u⟩ for all 0≠ u∈ H. Then, with L:=Q^1/2(1-Q^1/2MQ^1/2)^-1Q^1/2 and for b_1, b_2∈ H ∫_H exp(1/2⟨ M u,u⟩ + ⟨ b_1+ib_2, u⟩) μ_0(u) = exp(1/2⟨ Lb_1,b_2⟩ + i⟨ Lb_1, b_2⟩ - 1/2⟨ Lb_2, b_2⟩)/√( (1-Q^1/2MQ^1/2)). Now we can prove our main result: Consider the inverse problem <ref> with prior μ_0 and posterior μ given by μ/μ_0(u) = exp(-Φ(u))/∫exp(-Φ(u))μ_0. The functional I(u) = Φ(u) + 1/2u_C_0^2 is assumed to be C^2 in a neighborhood of = I(u). Then the Laplace approximation of μ given by ν = N(, I̋()^-1) is equivalently defined by ν/μ_0(u) = exp(-TΦ(u))/∫exp(-TΦ(u))μ_0, where TΦ(u) = Φ() + DΦ()(u-) + 1/2Φ̋()[u-, u-] is the second order Taylor approximation of Φ generated in This is done by comparing the Fourier transform (or characteristic function) of both representations for ν. We calculate ∫exp(-TΦ(u))μ_0(u) = e^-I()/√( C^1/2I̋()C^1/2) and ∫exp(i⟨λ, u⟩ -TΦ(u))μ_0(u) = e^-I()·exp( i⟨, λ⟩ - 1/2·I̋()^-1[λ, λ])/√( C^1/2I̋()C^1/2) We show (<ref>), as (<ref>) follows from setting λ = 0. We use -TΦ(u) = R(u) - Φ(u) = u_C^2/2 - I() - 1/2I̋()[u-, u-]. and write J = I̋() and v= for brevity. Note that for a bilinear operator K we will identify K(w, z) = ⟨ K w, z⟩. Then ∫exp(i⟨λ, u⟩ -TΦ(u))μ_0(u) = ∫exp(i⟨λ, u ⟩ + u_C^2/2 - I(v) - 1/2J[u-v, u-v] )μ_0(u) =e^-I(v)·∫exp( ⟨ iλ, u⟩ + 1/2⟨ C^-1 u,u⟩ -1/2⟨ J(u-v),u-v⟩)μ_0(u) = e^-I(v) - 1/2⟨ Jv,v⟩·∫exp(⟨ Jv + iλ, u⟩ + 1/2⟨ (C^-1-J)u,u⟩)μ_0(u). This is formula (<ref>) with M = C^-1-J and b_1 = Jv, b_2=λ. In this case, 1 - C^1/2MC^1/2 = 1 - C^1/2(C^-1-J)C^1/2 = C^1/2JC^1/2 and thus (1 - C^1/2MC^1/2)^-1/2 = J^-1/2C^-1/2. Continuing, = e^-I(v) - 1/2⟨ Jv,v⟩·exp(1/2|J^1/2v|^2 + i ⟨ J^1/2v,J^-1/2λ⟩- 1/2|J^-1/2λ|^2)/√( C^1/2JC^1/2) = e^-I(v)·exp(i ⟨ v,λ⟩ - 1/2⟨ J^-1λ, λ⟩)/√( C^1/2JC^1/2) This proves equation (<ref>) and it follows that the characteristic function of the measure ν̃ defined by ν̃/μ_0 = 1/Z·exp(-TΦ) fulfills ν̂̃̂(λ) = ∫exp(i⟨λ, u⟩) ν̃(u)/Z = ∫exp(i⟨λ, u⟩)·exp(-TΦ(u)) μ_0(u)/∫exp(i⟨λ, u⟩) μ_0(u) = exp{i⟨, λ⟩ + 1/2⟨I̋()^-1(u),u⟩}, i.e. ν̃= N(, I̋()^-1) as claimed. As in the one-dimensional setting, we conclude μ/μ_0 = exp(-Φ(u))/∫exp(-Φ(u))μ_0(u) and ν/μ_0 = exp(-TΦ(u))/∫exp(-TΦ(u))μ_0(u). With these expressions, we derive a bound on the Hellinger distance between μ and ν in the next section. § HELLINGER DISTANCE There are many notions of metrics and semi-metrics between measures, notably total variation, Hellinger, Wasserstein, Prokhorov and Kullback-Leibler. A survey of these and more probability metrics, including a detailed exposition of their relations is <cit.>. We choose the Hellinger distance, mainly because of its good analytic properties, its consistency with the total variation metric and because of the following lemma which allows us to bound the difference between expectations under the different measures in question: Let μ, μ' be two probability measures which are absolutely continuous w.r.t. another measure ν on a Banach space (X,·_X). Assume that f: X→ E, where (E,·) has second moments with respect to both μ and μ'. Then ^μ f - ^μ' f≤ 2 √(^μf^2 + ^μ'f^2)· d_H(μ,μ'). Recall that d_H(μ,ν)^2 = 1 - ∫√(μ/μ_0(u))√(ν/μ_0(u))μ_0(u). Thus, with the results of the preceding sections, d_H(μ,ν)^2 = 1 - ∫exp(-Φ(u) + TΦ(u)/2)μ_0(u)/√(∫exp(-Φ(u))μ_0(u))√(∫exp(-TΦ(u))μ_0(u)) = 1 - ⟨exp(-1/2Φ), exp(-1/2TΦ)⟩_L^2(X,μ_0)/exp(-1/2Φ)_L^2(X,μ_0)·exp(-1/2TΦ)_L^2(X,μ_0) From positivity of the exponential and the Cauchy-Schwarz-Bunyakowsky inequality it can easily be seen that d_H(μ,ν)∈ [0,1]. We would like to bound the Hellinger distance, i.e. optimally we would like to prove something like d_H(μ,ν) ≤ϵ for some ϵ > 0. This amounts to a reverse Cauchy-Schwarz inequality, or a statement of the kind ⟨ e^-Φ/2, e^-(TΦ)/2⟩_L^2(X,μ_0) > (1-^2) ·e^-Φ/2_L^2(X,μ_0)·e^-(TΦ)/2_L^2(X,μ_0). In the next section, we present a few elementary results about this kind of inequality. § A REVERSE CAUCHY-SCHWARZ INEQUALITY In this section, H will always be a Hilbert space with inner product ⟨·⟩ and norm ·. The study of of reverse Cauchy-Schwarz-Bunyakowsky inequalities is an active field of research by itself. We refer to <cit.> for further reading, although we will need only a very basic form of a reverse CSB inequality. Let f,g∈ H and D>0 with f-g^2 ≤ D· (f^2 + g^2). Then ⟨ f, g ⟩≥1-D/2· (f^2+g^2)≥ (1-D)·f·g. ⟨ f, g ⟩ - 1-D/2·f^2 - 1-D/2·g^2 = 1/2f^2 + 1/2g^2 - 1/2f-g^2- 1-D/2·f^2 - 1-D/2·g^2 = 1/2[D· (f^2+g^2)-f-g^2] ≥ 0. The last inequality in (<ref>) is just Young's inequality in . Lemma <ref> can be thought of as a reverse Young's inequality (which gives for all f,g∈ H that ⟨ f, g⟩≤1/2f^2 + 1/2f^2 and thus if the conditions on f,g as in lemma <ref> are fulfilled, we can bound 1-D/2· (f^2+g^2) ≤⟨ f, g⟩≤1/2· (f^2+g^2). Let f,g∈ H and K ∈ (0,1) with f-g≤ K·f. Then ⟨ f,g⟩≥[1-K/1 + (1-K)^2]·(f^2+g^2) ≥ 2 ·[1-K/1 + (1-K)^2] ·f·g. f-g^2/f^2+g^2 ≤f-g^2/f^2 + (f-f-g)^2= f-g^2/2f^2 - 2ff-g + f-g^2 ≤f-g^2/2f^2 - 2Kf^2 + f-g^2= 1 - (2-2K)f^2/(2-2K)f^2 + f-g^2 ≤ 1 - 2(1-K)/1 + (1-K)^2 And using lemma <ref> (with D = 1 - 2(1-K)/1 + (1-K)^2), we obtain the result. This is another form of a reverse Young's inequality, but with a different prerequisite: If f-g≤ K·f, we have 1/2·(1 - K^2/1 + (1-K)^2) · (f^2 + g^2) ≤⟨ f,g⟩≤1/2· (f^2+g^2) § CONDITIONS FOR GOOD APPROXIMATION From lemma <ref> and the expression for _H(μ, ν) in (<ref>) (the Hellinger distance between the posterior measure μ and its Laplace approximation ν) we immediately obtain the following: With the assumptions of proposition <ref>, if for some K∈ (0,1) exp(-Φ(u)/2) - exp(-TΦ(u)/2)_L^2(X, μ_0) ≤ K·exp(-I())/√((C_0^1/2·I̋()· C_0^1/2)) = K·exp(-I())/√((Id + C_0^1/2· HΦ ()· C_0^1/2)) then d_H(μ,ν) ≤K/√(1 + (1-K)^2) We use lemma <ref>, where H = L^2(X,μ_0), f=exp(-TΦ/2) and g = exp(-Φ/2). Note that f_H^2 = exp(-I())/√((C_0^1/2·I̋() · C_0^1/2)) due to (<ref>). Then we can set 1-^2 = 2 [1-K/1 + (1-K)^2], hence = K/√(1 + (1-K)^2) in equation (<ref>). The equality in condition (<ref>) follows from I(u) = Φ(u) + 1/2u_C_0. Note that C_0^1/2HΦ(u)C_0^1/2[h_1,h_2] = ⟨ DG(u)(C_0^1/2h_1), Γ^-1DG(u)(C_0^1/2h_2)⟩ - ⟨ HG(u)[C_0^1/2h_1, C_0^1/2h_2], Γ^-1(y-G(u))⟩ where DG is the Frechet derivative of the forward operator of G and HG is its second Frechet derivative (the Hessian). The following corollary uses assumptions which can be checked more easily but is less strict. Define K^2:= √((C_0^1/2I̋()C_0^1/2))/exp(-I())·∫exp (-min{Φ(u),TΦ(u)})·min{|Φ(u)-TΦ(u)|^2/4,1}μ_0( u). Then we have d_H(μ,ν) ≤K/√(1 + (1-K)^2). This is due to the elementary inequality |e^-x-e^-y|≤ e^-(x∧ y)· (|x-y|∧ 1) from which we obtain ∫ |e^-Φ(u)/2-e^-TΦ(u)/2|^2μ_0 ≤∫ e^-(Φ(u)∧ TΦ(u))· (|Φ(u)-TΦ(u)|^2/4 ∧ 1)μ_0 =K^2 ·∫ e^-TΦ(u)μ_0. § A CAVEAT ON APPLICATION We check on a one-dimensional example whether the bounds on the Hellinger distance obtained are tight. We consider a one-dimensional example by setting G:→, with G(u) = exp(u). This means that Φ(u) = |y-e^u|^2/2. We consider two cases: y=-2 and y=2. In the first case, the Laplace approximation is not very good (d_H(μ,ν) ≈ 0.3260) whereas in the second case, there is only a slight misfit between posterior measure and its Laplace approximation (d_H(μ, ν)≈ 0.0958). The case y=-2. Recall that the actual value for the Hellinger distance is d_H(μ,ν) ≈ 0.32595 * The K from proposition <ref> is K≈ 0.46621, hence d_H(μ,ν)≤ 0.41128. * Explicit calculation of K^2 in corollary <ref> yields K≈ 0.55328, so d_H(μ, ν)≤ 0.50517. The case y=2. Recall that the actual value for the Hellinger distance is d_H(μ,ν) ≈ 0.095810 * The K from proposition <ref> is K≈ 0.13648, hence d_H(μ,ν)≤ 0.10330. * Explicit calculation of K^2 in corollary <ref> yields K≈ 0.17422, so d_H(μ, ν)≤ 0.13434. In conclusion, the bounds are not strict even in the one-dimensional setting (which is obvious, given the nature of inequalities we have used in lemmata <ref>, <ref>, but they give a good idea of the magnitude of the Hellinger distance nonetheless. The calculation of the necessary quantities was done by explicit integration in order to check for strictness of the bound. This is unfeasible in higher dimensions which are of much higher interest (in one dimension there is no reason to even use a Laplace approximation instead of the fully non-Gaussian posterior). So we turn our attention to ways of obtaining results by other methods. We will see that this is difficult even in one dimension: Corollary <ref> demands calculation of the integral K^2 = √((C_0^1/2I̋()C_0^1/2))/exp(-I())·∫exp (-min{Φ(u),TΦ(u)})·min{|Φ(u)-TΦ(u)|^2/4,1}μ_0( u). As a simpler example, consider the integral ∫_-∞^∞ x^2 exp(-|x|) N(0,1)( x). An intuitive approach is to split the real line in subsets A = [-,], B = [-log(1/), -] ∪ [ ,log(1/)], C = {|u|> log(1/)}. Then the quadratic term is small on A and the exponential term is small on C. The set B makes a lot more trouble as neither the quadratic nor the exponential part is consistently small and taking the maximum x^2 exp(-|x|)|_x=2 as a bound for the function on B leads to a suboptimal result. It is the same with the formula for K: For u near , the error between Φ and TΦ is small, while for large u, both the misfit Φ and the quadratic TΦ become large, so the exponential term gets small very quickly. The interface between those two domains though is hard to handle and taking an analytical bound ony any term in this interface yields a highly suboptimal bound. This makes an explicit calculation of the integral almost impossible (save taking a large number of subsets tesselating the domain, but this amounts to explicit numerical integration in the limit). This is even more true in higher (and infinite dimensions), so the practical use of this formula is very limited. This problem makes it necessary to apply MCMC sampling to obtain the value K, but in this case we could have used to definition of d_H in the first place to get the exact Hellinger distance. § ACKNOWLEDGMENTS The author thanks Serge Kräutle for help with reverse CSB inequalities as well as Carlo Beenakker for a helpful discussion on MathOverflow. siam

http://arxiv.org/abs/1701.07976v1

20170127085626

Probabilistic Shaping and Non-Binary Codes

cs.IT

Probabilistic Shaping and Non-Binary Codes Joseph J. Boutros1, Fanny Jardel2, and Cyril Méasson3 1Texas A&M University, 23874 Doha, Qatar 2Nokia Bell Labs, 70435 Stuttgart, Germany 3Nokia Bell Labs, 91620 Nozay, France boutros@tamu.edu, {fanny.jardel,cyril.measson}@nokia.com ================================================================================================================================================================================================================================================= We generalize probabilistic amplitude shaping (PAS) with binary codes <cit.> to the case of non-binary codes defined over prime finite fields. Firstly, we introduce probabilistic shaping via time sharing where shaping applies to information symbols only. Then, we design circular quadrature amplitude modulations (CQAM) that allow to directly generalize PAS to prime finite fields with full shaping. § INTRODUCTION Shaping refers to methods that adapt the signal distribution to a communication channel for increased transmission efficiency. Shaping is eventually important for optimal information transmissions <cit.> and various solutions starting with non-linear mapping over asymmetric channel models towards pragmatic proposals involving shaped QAM signaling have been investigated and/or implemented over the years. More precisely, building upon early works on, e.g., many-to-one mapping, research efforts from the 70s towards the 90s derive conceptual frameworks and methods to reduce the shaping gap in communication systems. Exploiting the principles of coded modulation, a sequence of works <cit.> present operational methods to reduce the shaping gap. Compared to cubic constellations, up to πe/6≈ 1.53dB of shaping gain is achievable using well-adapted signaling. Simple methods such as trellis shaping or shell mapping permit to recover a significant fraction of the 1.53dB figure. Examples of applications include the ITU V.34 modem standard recommendations that uses shell mapping to recover 0.8dB. While several shaping schemes are based on the structural properties of lattices <cit.>, more randomized schemes also emerge after the re-discovery of probabilistic decoding in the 90s. With the advent of efficient binary codes, different coded modulation schemes were proposed offering flexible and low-complex solutions <cit.>. In the 2000s, despite the important development of wireless communications, the need for advanced shaping methods seems to have remained marginal. From a technological viewpoint, this may have been justified by the high variations of the channel in wireless communication networks. From an academic viewpoint, schemes have been analyzed and match the capacity-achieving distribution of a channel in different theoretical scenarios <cit.>. In the last few years, industrial applications of shaping methods have regained interest. This concerns areas where current technologies operate close to fundamental limits. For example, different methods have been experimented in optical communications <cit.>. Hence, because there are already efficient VLSI implementations of contemporary coding schemes that have been proven to asymptotically achieve capacity with constant complexity per information unit <cit.>, it is then natural to combine them with efficient shaping methods. In probabilistic shaping, for linear digital modulations, the a priori probability distribution of modulation points is modified to match a discrete Gaussian-like distribution, namely the Maxwell-Boltzmann distribution <cit.>. The method aims at maximizing the mutual information with respect to the same modulation where all points are equally likely. For special 2^m-ASK and 2^m-QAM constellations with linear binary codes, this method is equivalent to probabilistic amplitude shaping (PAS) where uniformly-distributed parity bits are assigned to the sign of a constellation point <cit.>. In this paper, we generalize this method to the non-binary case. The goal is to permit the use of efficient non-binary codes in order to enable low-latency processing (reducing the need for `Turbo'-detection <cit.>). Also, from an algebraic viewpoint, a characteristic p>2 of the finite field _p^m on which coding is built leads to new interesting problems such as distribution matching in _p^m and assigning a constellation points to elements in _p^m. In this paper, codes are supposed to be linear and defined over _p, where p is an odd prime. Firstly, except for codes with a sparse generator matrix, we show in Section <ref> that parity symbols are asymptotically uniformly-distributed over _p. This fact is used to derive two new methods for probabilistic shaping. Time sharing is proposed in Section <ref> where symbols of a p-ary code are mapped into p-ASK points. Hence, in time sharing, probabilistic shaping is performed only when information symbols are transmitted. Full probabilistic shaping is described in Section <ref> where circular QAM (CQAM) constellations of size p^2 points are introduced. This second method assigns a constellation shell to a Maxwell-Boltzmann-distributed information symbol and then a parity symbol selects a point within that shell. Numerical results for p-ASK-based time sharing and p^2-CQAM probabilistic shaping are shown in Section <ref>. Similar to the binary case <cit.>, a gap to channel capacity of 0.1 dB or less is observed for CQAM constellations. § SUM OF RANDOM VARIABLES IN A PRIME FIELD Lemma 4.1 in <cit.> established the expressions of the probability of a sum in _2. We translate this result to a prime field _p=/p. Principles of this extension to the non-binary case are implicit from Chapter 5 of <cit.> with the use of the z-transform. Nevertheless, we give here the exact expression of the probability of a sum of prime random symbols modulo p. This expression is directly related to Hartmann-Rudolph symbol-by-symbol probabilistic decoding <cit.> in the special case of a single-parity check code and its generalization to characteristic p <cit.>. Let p be a prime and _p={0, 1, ⋯, p-1 } be the associated finite field. Consider a sequence {s_ℓ}_ℓ=1^m of m independent symbols over _p in which the ℓ-th symbol is β∈_p with probability {s_ℓ=β} = q_ℓ(β). Then, for any k∈_p, the probability that the sum of the s_ℓ's equals k is {∑_ℓ=1^m s_ℓ = k} = 1+∑_i=1^p-1∏_ℓ=1^m ( ∑_β=0^p-1 q_ℓ(β) ω^i β-k+1)/p, where ωexp(2π√(-1)/p) indicates the p-th root of unity. Consider the enumerator function in t, Q(t) ∏_ℓ=1^m ( ∑_β=0^p-1 q_ℓ(β) t^β)   mod   (t^p-1). Observe that if this is expanded into a polynomial in t (where degree operations are taken mod p), the coefficient of t^k is the probability that the sum of m symbols is k, which we write {∑_ℓ=1^m s_ℓ = k}=coef[Q(t),t^k]. Let us also define for any k∈{0,1,2,⋯,p-1} the function Q_k(t) = t^-k Q(t)    mod   t^p-1. In an identical manner as for Q(t), expanding Q_k(t) would enumerate the probabilities of the sum of the s_ℓ's. Recall that the p-th root of unity in the complex plane satisfies ∑_i=0^p-1ω^k i=0 for any k∈{1,2,⋯,p-1}. Then, for any k∈_p, we have {∑_ℓ=1^m s_ℓ = k} = 1/p∑_i=0^p-1 Q_k(ω^i) because all but the constant polynomial terms are annihilated from the fact that the roots of unity sum to zero. It remains to evaluate the expression observing that ω^0=1 to get the results. □ This lemma shows that, if a s_ℓ is uniformly distributed over _p, then the sum is also uniformly distributed. For any ℓ, it is straightforward from convexity arguments that the weighted sum ∑_β=0^p-1 q_ℓ(β) ω^i β in the complex plane lies inside the unit circle in the strict sense if and only if one of the probability distribution q_ℓ is not degenerated in one singular point. Therefore, assuming that the norm tends to be smaller and bounded away from 1, the distribution of the infinite sum tends to be uniform. It remains to summarize this observation in a theorem. Let p be a prime and _p={0,1,⋯, p-1} be the associated Galois field. Consider a sequence {s_ℓ}_ℓ≥ 1 of independent random symbols over _p with respective probability distributions {(q_ℓ(0),q_ℓ(1),⋯,q_ℓ(p-1))}_ℓ≥ 1 such that lim sup_ℓ→∞{max_p(q_ℓ(p))}<1. Then ∀ k∈_p,     lim_m→∞{∑_ℓ=1^m s_ℓ = k} = 1/p. For error-correction over a prime field, this observation is interesting as follows. The limit theorem over _p indicates that the non-systematic symbols obtained from a linear encoder associated with a dense generator matrix will tend to have a uniform distribution independently on the input distribution. § PROBABILISTIC SHAPING VIA TIME SHARING OVER PRIME FIELDS A common mapping between non-binary codes and non-binary modulations is to select a constellation and a finite field of identical size. Let p be a prime integer, p>2. Consider the set of p points shown in Figure <ref>, known as p-ASK modulation. This p-ASK set ={-p-1/2, …, -1, 0, 1, …, p-1/2} is isomorphic to the finite field _p (a ring isomorphism in ). Symbols from _p are one-to-one mapped into p-ASK points. We embed _p into such that a symbol s ∈_p and its corresponding point in satisfy x-s=0  mod  p. There are many advantages for such a simple structure where the source is p-ary, the linear code is over _p, and p-ary modulation points are transmitted over the channel. Firstly, a probabilistic detector needs no conversion between modulation points and code symbols. A channel likelihood, after normalization, is directly fed as a soft value to the input of a probabilistic decoder. Secondly, turbo detection-decoding between the constellation and the code C is not required as for binary codes with non-binary modulations <cit.>. Consider a systematic linear code C over _p where parity symbols satisfy Theorem <ref>, i.e., check nodes used for encoding have a relatively high degree. Many practical error-correcting codes do satisfy this property, such as LDPC codes over _p. Let R_c=k/n be the coding rate of C, where n is the code length and k is the code dimension. Assume that information symbols s_1, s_2, …, s_k at the encoder input are identically distributed according to an a priori probability distribution {π_i }_i=0^p-1. Let P_MB(x,ν) ∝exp(-ν |x|^2) be a discrete Maxwell-Boltzmann distribution <cit.> with parameter ν≥ 0. The a priori distribution {π_i} is taken to be π_0 = P_MB(0,ν) ∝ 1, π_i=π_p-i =P_MB(i,ν) ∝exp(-ν i^2), for i=1…p-1/2. The average energy per point for the p-ASK constellation, denoted by E_s, is given by E_s=∑_x ∈ P_MB(x,ν)  |x|^2  =  2 ∑_i=1^(p-1)/2π_i  i^2. From Theorem <ref> and (<ref>)&(<ref>), a fraction R_c of transmitted ASK points corresponding to information symbols are Maxwell-Boltzmann-shaped and a fraction 1-R_c of ASK points corresponding to parity symbols is uniformly distributed in the constellation. We refer to this coding scheme as probabilistic shaping via time sharing. The target rate should be the average information rate (expressed in bits per real dimension) R_t = R_c log_2(p) = R_c I(X_s;Y) + (1-R_c) I(X_p;Y), where the two random variables X_s, X_p ∈ satisfy X_s ∼π_i and X_p ∼ 1/p. The random variable Y represents the output of a real additive white Gaussian noise channel, where additive noise has variance σ^2=N_0/2. For a given target rate R_t, the Maxwell-Boltzmann parameter ν is chosen such that the signal-to-noise ratio γ=E_s/N_0 attaining R_t is minimized. Let γ_ be that minimum. We also define two signal-to-noise ratios γ_cap and γ_unif such that R_t = 1/2log(1+2γ_cap), and R_t = I(X_p;Y),  for γ=γ_unif. Then, the gap to capacity and the shaping gain (expressed in decibels) are respectively given by γ_(dB)-γ_cap(dB) and γ_unif(dB)-γ_(dB). In this time sharing scheme, probabilistic shaping is made only during a fraction R_c of transmission time. This method is attractive due to isomorphism between the field _p and the p-ASK constellation. From (<ref>), one may quickly conclude that high coding rate is recommended to approach full-time probabilistic shaping. However, at R_c close to 1, the mutual information I(X;Y), for X ∈, approaches its asymptote log_2()=log_2(p) and the required signal-to-noise ratio γ_ goes far away from γ_cap. This is clearly shown in the numerical results presented in Section <ref>. A method for full probabilistic shaping is proposed in the next section. § PROBABILISTIC SHAPING VIA P^2-CIRCULAR QAM OVER PRIME FIELDS We propose in this section a coded modulation scheme that allows full probabilistic shaping of all transmitted symbols with a non-binary linear code over _p. Probabilistic amplitude shaping with binary codes maps uniformly-distributed parity bits into the sign of an ASK point <cit.>. This sign mapping is not valid with a prime finite field _p, p>2. The key idea in our new coded modulation is to assign the parity symbol to p modulation points with the same amplitude. This is a direct generalization of the sign mapping to p-ary mapping. The linear p-ary code is assumed to be systematic. Its information symbols become amplitude labels in the modulation. We propose a bi-dimensional constellation with p^2 points, referred to as p^2-circular quadrature amplitude modulation (p^2-CQAM). A circle containing CQAM points of the same amplitude will be called a shell. The p^2-CQAM includes p shells with p points per shell. Such a bi-dimensional constellation is not unique. Indeed, many ways do exist to build p shells and populate each shell with p points. As a consequence, we introduce a figure of merit for a constellation <cit.> and we build a specific p^2-CQAM constellation that maximizes this figure of merit. Consider a finite discrete QAM constellation ⊂. Assume that ∑_x ∈ x = 0. Let E_s=∑_x ∈ |x|^2/|| be the average energy per point, assuming equiprobable points. Let d_Emin^2()=min_x,x' ∈, x x' |x-x'|^2 be the minimum squared Euclidean distance between the points of . A figure of merit _M for is defined by the following expression: _M()=d_Emin^2()/E_s·log_2(||). The log_2(||) factor is arbitrary, it is used in the above definition to normalize the squared minimum Euclidean distance by bit energy instead of point energy. This may be useful when comparing two constellations of different sizes. Now, we build a p^2-CQAM constellation that maximizes _M() by populating the p shells as follows: * For the first shell, draw p uniformly-spaced points on the unit circle. The points are x_ℓ=exp(ℓ2π/p√(-1)), for ℓ=0 … p-1. Here, we impose the constellation minimum distance to be the distance between two consecutive points of the first shell, d_Emin()=2sin(π/p). * Assume that shells 1 to i-1 are already built. Let x_ip=ρ_i exp(ϕ_i√(-1)) be the first point of the i-th shell. The p-1 remaining points on this shell are x_ip+ℓ=ρ_i exp((ϕ_i+ℓ2π/p)√(-1)), ℓ=1, … p-1. Let d_i^2=min_ℓ=0 … ip-1 |x_ip-x_ℓ|^2 be the minimum distance between the first point of the current shell and all previously constructed points. The radius ρ_i and the phase shift ϕ_i are determined by an incremental search: * Start with ρ_i=ρ_i-1 and increment by a step Δ_ρ. * At each radius increment, vary ϕ_i from π/p to -π/p. * Stop incrementing the radius ρ_i and the phase shift ϕ_i when d_i^2 ≥ d_Emin^2(). Now, x_ip is found. * Repeat the second construction step until completing the p-th shell of the p^2-CQAM constellation. The p^2-CQAM obtained with the construction described above has the circular symmetry required by PAS over _p. Examples of circular QAM modulations for probabilistic amplitude shaping are shown in Figure <ref>, for p=5,7,11 respectively. Points are drawn as small circles in red. Blue segments connect points located at minimum Euclidean distance. By the given construction, the inner radius of the p^2-CQAM is ρ_in=1, ∀ p. The outer radius ρ_out varies slightly with p but lim_p →∞ρ_out =ρ_out(∞) ≈ 3.6. This limit exists because the sequence ρ_out(p) is increasing with p and bounded from above by 1+(p-1)d_Emin() ≤ 1+2π. The limitation of the Maxwell-Boltzmann probability mass function to amplitudes between ρ_in=1 and ρ_out(∞) is a major drawback. This short interval [1,ρ_out(∞)[ is shifted away from the origin and is not large enough to yield a good Gaussian-like discrete distribution. In the next section, the shells radii are modified to get a wider amplitude range, the p^2-CQAM phase shifts are kept invariant. Let s_1, s_2, …, s_k be i.i.d. information symbols with a priori probability distribution {π_i }_i=0^p-1, as in the previous section. Then, for points x_ip+ℓ∈, i,ℓ=0 … p-1, the prior distribution becomes π(x_ip+ℓ)=π_i/p=P_MB(|x_ip|, ν)/p. In presence of the above distribution, the signal-to-noise ratio should be defined with an average energy per point E_s=∑_i=0^p-1π_i |x_ip|^2. Furthermore, the circular symmetry of a p^2-CQAM facilitates the numerical evaluation of average mutual information. The general expression of I(X;Y) with p^2 integral terms reduces to p terms only. The mutual information I(X;Y) is given by ∑_i=0^p-1π_i ∫_y∈ p(y|x_ip) log_2( p(y|x_ip)/∑_ℓ=0^p^2-1π(x_ℓ) p(y|x_ℓ)) dy. The Maxwell-Boltzmann parameter ν in (<ref>) is chosen such that 2R_t=2 R_c log_2(p)=I(X;Y) at a minimal value of signal-to-noise ratio E_s/N_0=γ_, where R_t is the target rate per real dimension. The gap to capacity is determined by the difference γ_-γ_cap with γ_cap satisfying 2R_t=log_2(1+γ_cap). Given the a priori distribution {π_i }_i=0^p-1 of symbols in the finite field _p, the linear code C[n,k]_p and the p^2-CQAM constellation should be combined together as illustrated in Figure <ref>. Suppose that R_c=1/2 and say that s_1 ∈_p is an information symbol (encoder input) and p_1 ∈_p is a parity symbol. Then, s_1 should be be shaped by the distribution matcher (DM) according to {π_i }_i=0^p-1 <cit.>. The symbol s_1 will be used to select a shell in the p^2-CQAM and the parity symbol p_1 will uniformly select a point on that shell. Similarly, suppose that R_c=2/3 and consider four information symbols s_1, s_2, s_3, s_4 and two parity symbols p_1, p_2. The DM shall generate s_1, s_2, s_3 according to the distribution {π_i }_i=0^p-1 and select the shell of three p^2-CQAM points. The symbol s_4 is read directly from a uniform i.i.d p-ary source. Given the shells of three points, uniformly-distributed symbols s_4, p_1, p_2 constitute the points indices inside those shells. In the general case, n/2 symbols in _p with probability distribution {π_i } are read from the DM and mapped into a shell number for n/2 CQAM points. The probabilistic shaping is due to these n/2 symbols On the other hand, k-n/2 uniformly-distributed symbols are directly read from the source. Together with n-k parity symbols, i.e., a total of n/2 symbols, uniformly-distributed symbols in _p determine the phase of CQAM points within constellation shells. Our p-ary coded modulation suited for probabilistic shaping assumes that R_c ≥ 1/2, i.e., k ≥ n/2. Coding rates in the interval [0,1/2[ are less attractive for probabilistic amplitude shaping because, for small R_c, a constellation with equiprobable points already shows a rate that is too close to channel capacity in terms of signal-to-noise ratio. § NUMERICAL RESULTS Two typical values of p are considered in this section, namely p=7 and p=13. The target rate herein is expressed in bits per real dimension for both real and complex constellations. Gaps and gains are expressed in decibels. For probabilistic amplitude shaping via time sharing, signal-to-noise ratio gaps and gains are presented in Table <ref> for different values of the coding rate R_c. As discussed in Section <ref>, the effective gain due to shaping decreases at very high rate. A coding rate around 4/5 yields the highest effective gain. One of our perspectives is to analytically determine the optimal coding rate (or its range) from (<ref>). Tables <ref> includes results for CQAM shaping. As suggested in the previous section, CQAM radii are stretched to improve the range for P_MB(ν,x). Here, radius ρ_i of shell i is taken to be 1+(ρ_max-1) ((i-1)/(p-1))^β where ρ_max > ρ_out(∞). At R_c=2/3, optimized parameters are ρ_max=4.8 and β=0.76 for 7^2-CQAM and ρ_max=6.0 and β=0.80 for 13^2-CQAM. Square (p-ASK)^2 constellations are not valid for full PAS because they require time sharing, however we added them for comparison purpose. Shaping with 7^2-CQAM and 13^2-CQAM is about 0.1 dB from the additive white Gaussian noise channel capacity. 1 Bocherer2015 G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched Low-Density Parity-Check coded modulation,” , vol. 63, no. 12, pp. 4651–4665, Dec. 2015. Schulte2016 P. Schulte and G. Böcherer, “Constant composition distribution matching,” , vol. 62, no. 1, pp. 430–434, Jan. 2016. Kschischang1993 F. R. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for Gaussian channels,” , vol. 39, no. 3, pp. 913–929, May 1993. Gallager1968 R. G. Gallager, Information theory and reliable communication. New York: Wiley, 1968. Calderbank1987 A. R. Calderbank and N. J. A. Sloane, “New trellis codes based on lattices and cosets,” , vol. 33, no. 2, pp. 177–195, Mar. 1987. Forney1989a G. D. Forney and L.-F. Wei, “Multidimensional constellations – Part I: Introduction, figures of merit, and generalized cross constellations, , vol. 7, no. 6, pp. 877–892, Aug. 1989. Forney1989b G. D. Forney, “Multidimensional constellations – Part II: Voronoi constellations,” , vol. 7, no. 6, pp. 941–958, Aug. 1989. Calderbank1990 A. R. Calderbank and L. H. Ozarow, “Non-equiprobable signaling on the Gaussian channel,” , vol. 36, no. 4, pp. 726-740, Jul. 1990. Fortier1992 P. Fortier, A. Ruiz, and J. M. Cioffi, “Multidimensional signal sets through the shell construction for parallel channels,” , vol. 40, no. 3, pp. 500–512, Mar. 1992. Forney1992 G. D. Forney, “Trellis shaping,” , vol. 38, no. 2, pp. 281–300, Mar. 1992. Khandani1993 A. K. Khandani and P. Kabal, “Shaping multidimensional signal spaces – Part 1. Optimum shaping, shell mapping,” , vol. 39, no. 6, pp. 1799–1808, Nov. 1993. Laroia1994 R. Laroia, N. Farvardin, and S. A. Tretter, “On optimal shaping of multi-dimensional constellations,” , vol. 40, no. 4, pp. 1044–1056, Jul. 1994. Forney2000 G. D. Forney, M. D. Trott, and S.-Y. Chung, “Sphere-bound-achieving coset codes and multilevel coset codes,” , vol. 46, no. 3, pp. 820–850, May 2000. Urez2005 U. Erez, S. Litsyn, and R. Zamir, “Lattices which are good for (almost) everything,” , vol. 51, no. 10, pp. 3401–-3416, Oct. 2005. Pie2016 N. di Pietro and J. Boutros, “Leech constellations of construction-A lattices,” Sub. to IEEE Trans. Commun., Jan. 2017, [Online]. Available: http://arxiv.org/pdf/1611.04417v2.pdf. Gallager1963 R. G. Gallager, Low-Density Parity-Check codes. Cambridge, MA: MIT Press, 1963. Richardson2008 T. Richardson and R. Urbanke, Modern coding theory. Cambridge, U.K: Cambridge Univ. Press, 2008. Ling2014 C. Ling and J. C. Belfiore, “Achieving AWGN channel capacity with lattice Gaussian coding,” , vol. 60, no. 10, pp. 5918–5929, Oct. 2014. Mondelli2014 M. Mondelli, S. H. Hassani, and R. Urbanke, “How to achieve the capacity of asymmetric channels,” in Proc. Allerton Conf. Commun. Control Comput., Oct. 2014, pp. 789–796. Kramer2016 G. Kramer, “Probabilistic amplitude shaping applied to fiber-optic communication systems,” in Proc. Int. Symp. on Turbo Codes and Iterative Inf. Proc., Oct. 2016. Lentmaier2010 M. Lentmaier, A. Sridharan, D. J. Costello, and K. S. Zigangirov, “Iterative decoding threshold analysis for LDPC convolutional codes,” , vol. 56, no. 10, pp. 5274–5289, Oct. 2010. Arikan2009 E. Arikan, “Channel polarization: a method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” , vol. 57, no. 7, pp. 3051–3073, Jul. 2009. Kudekar2013 S. Kudekar, T. Richardson and R. Urbanke, “Spatially coupled ensembles universally achieve capacity under belief propagation,” , vol. 59, no. 12, pp. 7761–7813, Dec. 2013. Hartmann1976 C.R.P. Hartmann and L.D. Rudolph, “An optimum symbol-by-symbol decoding rule for linear codes,” , vol. 22, no. 5, pp. 514-517, Sep. 1976. Boutros2006 J. Boutros, A. Ghaith, and Y. Yuan-Wu, “Non-binary adaptive LDPC codes for frequency selective channels: code construction and iterative decoding,” in Proc. IEEE Inf. Theory Workshop, pp. 184-188, Chengdu, Oct. 2006. Yankov2014 M.P. Yankov, D. Zibar, K.J. Larsen, L.P.B. Christensen, and S. Forchhammer, “Constellation Shaping for Fiber-Optic Channels with QAM and High Spectral Efficiency,” IEEE Photon. Technol. Lett., vol. 26, no. 23, pp. 2407–2410, Dec. 2014. Buchali2015 F. Buchali, G. Böcherer, W. Idler, L. Schmalen, P. Schulte, F. Steiner, “Experimental Demonstration of Capacity Increase and Rate-Adaptation by Probabilistically Shaped 64-QAM,” in Proc. ECOC, Aug. 2015.

http://arxiv.org/abs/1701.08205v1

20170127215550

Bounds on curvature in regular graphs

math.CO

Consistent SPH Simulations of Protostellar Collapse and Fragmentation José M. Ramírez-Velasquez1 December 30, 2023 ===================================================================== We study the curvature-dimension inequality in regular graphs. We develop techniques for calculating the curvature of such graphs, and we give characterizations of classes of graphs with positive, zero, and negative curvature. Our main result is to compare the curvature-dimension inequality in these classes to the so-called Ollivier curvature. A consequence of our results is that in the case that the graph contains no subgraph isomorphic to either K_3 or K_2,3 these curvatures usually have the same sign, and we characterize the exceptions. § INTRODUCTION Recently there have been several attempts to translate the well-understood concept of curvature from Riemannian geometry to discrete spaces. In the continuous setting, the Bochner formula characterizes harmonic functions in terms of the curvature. Based on this, Bakry and Émery <cit.> developed the Curvature-Dimension inequality, which has since been adapted to define curvature in a discrete setting <cit.>. There have recently been many results on the spectral and isoperimetric properties of graphs under a bound on the CD-curvature <cit.><cit.> with further references therein. An alternate notion of discrete curvature is the Ollivier curvature, proposed by Y. Ollivier<cit.> and independently by Sammer <cit.> which has been further investigated at length, e.g. <cit.>. This curvature compares the minimum-transport distance of balls on a curved space to that of balls in Euclidean space. In this work we investigate the CD-curvature for regular graphs, and compare to the Ollivier curvature. It is known that these notions are not in general equivalent, indeed, in this work we give examples of graphs for which the curvature is positive in one notion but not the other. For graphs that are triangle-free and have no subgraph isomorphic to K_2,3, we develop rules for calculating both types of curvature, and observe that the signs of the curvatures are usually equivalent. We also characterize the classes of graphs for which the signs are not equivalent. We calculate the curvature for several examples of interest, including the graph of the state space of the random interchange process. In Section 2, we provide the definitions of both the curvature-dimension inequality and of Ollivier's curvature. In Sections 3 and 4, we investigate the curvature of regular graphs that have no subgraph isomorphic to either K_3. In addition we prove a number of short results related to these notions of curvature. § PRELIMINARIES §.§ Curvature-Dimension inequality The CD-inequality was introduced in <cit.> and several variations have been studied. Our definitions follow from <cit.> and we use the non-normalized Laplacian matrix. For a d-regular graph, the curvature differs from that explored by Chung, Lin, Yau <cit.> by a multiplicative factor of d; those definitions result from choosing a normalized Laplacian. If G = (V,E) is a locally finite graph and f,g:V→, then Γ (f,g)(x) = 1/2∑_y∼ xf(y)-f(x)g(y)-g(x), Γ f(x) :=Γ(f,f)(x), Δ f(x) = ∑_y∼ x f(y)-f(x), and Γ_2 f = 1/2ΔΓ f - Γ(f,Δ f). Let f:V→ and x∈ V. Because Δ f = Δ (f+c) and Γ f = Γ(f+c), we may (and will) assume that f(x) = 0. In that case straightforward manipulation (as in <cit.>) reveals a form for Γ_2f(x) that often simplifies computation: 2Γ_2f(x) = 1/2∑_u∼ v∼ x d(x,u) = 2f(u)-2f(v)^2 + ∑_v∼ xf(v)^2 + ∑_v∼ x4-d(x)-d(v)/2f^2(v) + ∑_Δ(x,v,u)[2f(v)-f(u)^2 + 1/2f^2(v) + f^2 (u)] , where Δ(x,v,u) means that x∼ v∼ u∼ v. We say that G satisfies the CD(ρ,∞) condition at x iff for all f:V→, Γ_2f(x)≥ρΓ f(x). Whether CD(ρ,∞) is satisfied for G at x is a local property; in particular, it depends only on the structure of edges that are incident to at least one neighbor of x. It is possible to remove all other edges without affecting the curvature at x. An obvious concern is to characterize the non-constant function f that minimizes Γ_2f(x)/Γ f(x). It is understood how to calculate the f(u) if d(x,u) = 2. A simple optimization reveals that, holding f(v) constant for all v∼ x, the value of f(u) minimizing Γ_2f(x) is f(u) = 2∑_v: x∼ v∼ u f(v)/#{v:x∼ v∼ u}. §.§ Ollivier curvature The second form of discrete curvature that we consider is the so-called Ollivier curvature. For probability measures μ,ν on V, the L_1 Wasserstein (i.e., minimum-transport) distance is W_1(μ,ν) = min_m ∫_V× V d(x,y) dm(x,y), where the minimum is taken over all probability measures m on V× V so that ∫_V m(x,y) dν(y) = μ(x) and ∫_V m(x,y) dμ(x) = ν(y). In plain words, we transport μ to ν by shifting a load of size m(x,y) along an (x,y)-geodesic, and W_1 minimizes the average distance transported over all choices of transport function m. Let G be a d-regular graph. For x∈ V, define a probability measure μ_x so that μ_x(v) = 12 if v = x 1/2d if v ∼ x 0 otherwise. If x,y∈ V and x∼ y, the curvature is κ(x,y) = 1-W_1(μ_x,μ_y). § K_2,3 AND TRIANGLE-FREE GRAPHS In this section we compare the CD curvature to Ollivier's curvature for regular graphs that contain no subgraphs isomorphic to either K_3 or K_2,3. If x∈ V and y,z∼ x, we say that y and z are linked if there is a vertex w≠ x so that y∼ w∼ z. We write y≈ z if y and z are linked and y≉z if not. If x,y∈ V and x∼ y, the non-linking number N_x(y) is the number of other neighbors of x that are not linked to y: |{w∼ x:w≠ y, w≉y}|. Let G be d-regular and have no subgraph isomorphic to either K_3 or K_2,3. Let N = max_y∼ x N_x(y). (i) If N = 0, then G satisfies CD(ρ,∞) at x iff ρ≤ 2 (G is positively curved at x.) (ii) If N = 1, then G satisfies CD(ρ,∞) at x iff ρ≤ 0. (G is flat at x.) (iii) If N ≥ 2, then G does not satisfy CD(0,∞) at x. (G is negatively curved at x.) Because G has no subgraph isomorphic to K_2,3, if neighbors y,z of x are linked, then they are linked by a unique vertex w, and the link w cannot be adjacent to any neighbor of x other than y and z. (i) If N=0, then G is locally isomorphic to the hypercube Ω_d. That is, there is a local isomorphism ϕ:V(G)→ V(Ω_d) with the property that Γ_2 f(x) = Γ_2 f∘ϕ(x) and Γ f(x) = Γ f∘ϕ(x) for all f:V→. It is well-known <cit.> that Ω_d has positive curvature, likewise G has positive curvature. (ii) To show that CD(ρ,∞) fails at x if ρ > 0, we describe a test-function f for which Γ_2 f (x) = 0: Let y,z be a pair of unlinked neighbors of x, set f(x) = 0, f(y) = 1, f(z) = -1. For any other neighbor of x, set f = 0. On the second-neighbors of x, set f to be twice the average of f evaluated on all neighbors of x adjacent to that vertex, so as to minimize Γ_2. Notice that y will be linked to d-2 neighbors of x(all besides z and itself) by d-2 distinct vertices, each with f = 1, and will have one neighbor that is not adjacent to any other neighbor of x, at which f = 2. Likewise, z will have d-2 neighbors with f = -1 and one neighbor with f = -2. And, each other neighbor of x will be adjacent to one vertex with f = 1 and one with f = -1: the vertices linking them to y and z respectively. With this function it is straightforward to see that 2Γ_2 f(x) = 0. We now show that for any function f, 2Γ_2f(x) ≥ 0, and therefore CD(ρ,∞) holds for any ρ≤ 0: Because there is a bijection between pairs of linked neighbors of x and the linking vertices, 1/2∑_u∼ v∼ x d(x,u) = 2f(u)-2f(v)^2 ≥1/2∑_y,z∼ x y≈ zf(y)-2f(w)^2 + f(z)-2f(w)^2 ≥∑_y,z∼ x y≈ zf(y)-f(z)^2, where w is the vertex linking y and z, and the sum is taken over all unordered pairs y,z of linked neighbors of x. Using this in the expression for Γ_2, we find 2Γ_2f(x) ≥∑_y,z∼ x y≈ zf(y)-f(z)^2+∑_y∼ xf(v)^2 + (2-d)∑_y∼ x f^2(v) = ∑_y∼ x (3-d + #{z∼ x:y≈ z})f^2(y) + ∑_y,z∼ x y≈ z2f(y)f(z) ≥∑_y∼ x f^2(y) + ∑_y,z∼ x y≈ z2f(y)f(z) ≥∑_y,z∼ x y≈ zf(y)-f(z)^2 ≥ 0, because each neighbor y of x is linked to at least d-2 of the d-1 other neighbors of x, and not linked to at most 1 other neighbor of x. (iii). Let y be a neighbor of x with N_x(y)>1. Define a test-function f: f(x) = 0, f(y) = d-1, otherwise if z∼ x, f(z) = -1, and choose the values of f on the second-neighbors of x to minimize Γ_2. It is straightforward to calculate that 2Γ_2f(x) = (∑_z∼ x y≈ zf(y)-f(z)^2) + (2-d)[(d-1)^2 + (d-1)] ≤ (d-3)d^2 + (2-d)(d^2 - d) = -2d < 0, and therefore CD(0,∞) fails at x. The bound on the first term is due to the fact that y is unlinked to at least 2 of the d-1 other neighbors of x. We give a similar characterization of the Ollivier curvature: Let G be d-regular and have no subgraph isomorphic to either K_3 or K_2,3. Let N = max_y∼ x N_x(y). (i) If N = 0, then κ(x,y) > 0. (ii) If N = 1, then κ(x,y) ≥ 0. (iii) If N ≥ 2, then κ(x,y) ≤ 0. (i) As in our previous result G must be locally isomorphic to Ω_d, it is well-known that κ(x,y) = 1d in this case. (ii) Let y,z be an unlinked pair of neighbors of x, and let v_1,…, v_d-2 be the other neighbors of x. For each v_i there is a w_i that links v_i and y, let u be the neighbor of y that is not in {x,w_1,…, w_d-2}. d(u,z)≤ 3 as u∼ y∼ x∼ z. To bound W_1(μ_x,μ_y) from above, consider the transfer of mass along paths x→ y, v_i → w_i and z→ u. This gives a bound of W_1(μ_x,μ_y) ≤ 1, and so κ(x,y)≥ 0. (iii) Let y be a neighbor of x with N_x(y)>1. We use a test-function f to bound the dual formulation of W_1, W_1(μ,ν) = max_f∈ Lip(1)∫ f dμ-∫ f dν. For a test-function, set f(x) = 0, f(y) = 1, f(v) = 0 if v is any other neighbor of x, f(w) = 1 if w links y to a neighbor of x, f(u) = 2 if u is any other neighbor of y, and f = 1 on every other vertex, so that f is 1-lipschitz. W_1(μ_x,μ_y)≥∫ f dμ_y -∫ f dμ_x = 1 + N-2/2d≥ 1, so κ(x,y) ≤ 0. κ > 0 is satisfied under hypothesis (ii) for G = C_5, or graphs with similar structure for which (in the language of the proof) d(u,z) = 2. κ = 0 is satisfied under hypothesis (iii) e.g. for the dodecahedral graph, again because d(u,z) = 2. We combine the previous results to show a relationship between the curvatures. Let G be d-regular and have no subgraph isomorphic to either K_3 or K_2,3. Let x∈ V. (i) If and only if there exists ρ > 0 so that G satisfies CD(ρ,∞) at x, then κ(x,y)>0 for all y∼ x. (ii) If G satisfies CD(0,∞) at x, then κ(x,y) ≥ 0 for all y∼ x. (iii) If G does not satisfy CD(0,∞) at x, then there is a vertex y∼ x for which κ(x,y)≤ 0. If there is a vertex y∼ x for which κ(x,y) < 0, then G does not satisfy CD(0,∞) at x. §.§ Examples We give examples of graphs with no subgraph isomorphic to K_3 or K_2,3, categorized by which subhypothesis of Theorems 3.1 and 3.2 they satisfy. Most of these graphs are vertex-transitive, so the curvature is identical at every vertex. As before, let v be a neighbor of x that is not linked to the largest number of other neighbors of x, and let N be that number. Common graphs with N=0, satisfying hypothesis (i) include * The hypercube Ω_d. Common graphs with N = 1, satisfying hypothesis (ii) include * The square lattice ^n where n≥ 1. * The cyclic graph C_k if k≥ 5. * Any product of the above graphs, or product of these graphs with a graph from the previous list. Common graphs with N > 1, satisfying hypothesis (iii) include * The infinite d-regular tree if d≥ 3. * Any d-regular graph with girth ≥ 5 if d≥ 3. * The graph of triangulations of an n-gon (n≥ 6), with edges representing the action of flipping one interior arc of the polygon. * S_n with edges corresponding to adjacent transpositions. §.§.§ Interchange Process Given an underlying graph H = ([n],F), the interchange process labels the vertices of H and at each step, we are allowed to exchange the labels of a pair of adjacent vertices. Let G be the graph of possible states with an edge between two states if we can move from one state to the other in a single step. In other words, G is the Cayley graph of the subgroup of S_n with generating set A = {(i,j):{i,j}∈ F}. G is always K_3-free, and will be K_2,3-free if and only if H is triangle-free. If x∈ V(G) and a,b∈ A, ax≈ bx iff a and b are not incident to the same vertex as edges of H. Because of this, we can easily determine the value of N for the interchange process in terms of H. * If H is a matching, G satisfies hypothesis (i). * If H is the disjoint union of paths with length ≤ 2 (at least one of which has length equal to 2), G satisfies hypothesis (ii). * If H has any vertex of degree ≥ 3 or any path of length ≥ 3, then G satisfies hypothesis (iii). § TRIANGLE-FREE REGULAR GRAPHS. In this section we compare the curvature-dimension inequality to the Ollivier curvature for regular triangle-free graphs. Unlike Section 3, we allow the graph to have K_2,3 as a subgraph. In this more general case our results are less complete. In both notions of discrete curvature we give characterizations of classes of graphs with positive curvature, but it remains open to give a full characterization of graphs with positive curvature. Let G = (V,E) be a triangle-free d-regular graph. Let u,w be two neighbors of x. The linkage of u and w is calculated by summing over all vertices z≠ x for which u∼ z∼ w: l(u,w) = ∑_z 1/|{y: x∼ y∼ z}| As discussed in Section 2, if G contains no subgraph isomorphic to K_2,3, then there can be at most one vertex z with z≠ x, u∼ z∼ w, which will have {y:x∼ y∑ z} = {u,w}. In this case l(u,w) = 1/2 and (as before) we say that u and w are linked. Let G be a triangle-free graph, and x∈ V(G). If for every pair u,w of neighbors of x, l(u,w)≥1/2, then G satisfies CD(ρ,∞) at x iff ρ≤ 2. It is known that any triangle-free graph G fails CD(ρ,∞) if ρ > 2 <cit.>. Recall that if f is the minimizer of Γ_2f(x)/Γ f(x) and d(x,z) = 2, f(z) = 2/# y: x∼ y∼ z∑_y:x∼ y∼ z f(y). With this minimization (and the assumption of a triangle-free graph), straightforward algebraic manipulation reveals a form for Γ_2 f: 2Γ_2 f(x) = ∑_y∼ x (3-d)f^2(y) + ∑_w,y∼ x2f(y) f(w) + ∑_w,y∼ x 2l(w,y) f(w)-f(y)^2 Because l≥ 1/2 we can bound this equation: 2Γ_2 f(x) ≥∑_y∼ x (3-d)f^2(y) + ∑_w,y∼ x2f(y) f(w) + ∑_w,y∼ xf(w)-f(y)^2 = ∑_y∼ x (3-d)f^2(y) + ∑_w,y∼ x2f(y) f(w) +∑_y∼ x (d-1)f^2(y)- ∑_w,y∼ x2f(y)f(w) = 2∑_y∼ xf^2(x) = 4Γ f(x), and therefore CD(2,∞) is satisfied. We prove a result regarding the Ollivier curvature of a related class of graphs: Let G be a triangle-free graph. If every pair of adjacent edges of G are contained within exactly one maximal complete bipartite induced subgraph whose parts have equal size, then κ(x,y) = 1/d for all pairs of neighbors x,y. Let x,y be a pair of neighboring vertices in G. The neighbors w of x with w≠ y are partitioned into S_1,...,S_k according to which maximal complete bipartite subgraph contains {xy} and {xw}. Similarly neighbors of y (other than x) are partitioned into T_1,...,T_k, so that {x}∪ T_i,{y}∪ S_i are the equally sized parts of a maximal complete bipartite graph. To minimize W_1(μ(x),μ(y)) requires shifting a mass of size |S_i|/2d from vertices of S_i to those in T_i and a mass of size 1/2-1/2d from x to y at a total cost of 1/2-1/2d+1/2d∑_i|S_i| = 1 - 2/2d, this proves the theorem. Observe that under the hypothesis of Theorem 4.2, a pair of vertices y,w that are neighbors of x will be contained within a copy of K_m,m for some value m≥ 2. In this case, l(w,y)≥ (m-1)1/m≥ 1/2, so the hypothesis of Theorem 4.1 is also satisfied, and such a graph will have positive curvature both in terms of the CD inequality and Ollivier curvature. Examples of common graphs: * The hypercube Ω_n, where neighbors w and y of x are linked by a unique vertex z, so that x,z,w,y are the vertices of a copy of K_2,2, and l(x,y) = 1/2. Ω_n satisfies CD(ρ,∞) iff ρ≤ 2 and has κ = 1/n. * G = K_n,n for n≥ 2, which has l(y,w) = n-1/n. K_n,n satisfies CD(ρ,∞) iff ρ≤ 2 and has κ = 1/n. * The Cayley graph of S_n generated by all interchanges (ij):i,j∈ [n]: If y = (ij)x and w=(ik)x, then x,w,y are contained within the copy of K_3,3 that also includes (jk)x,(ijk)x,(jik)x. In this case l(w,y) = 2/3. On the other hand if y = (ij)x and w = (kl)x, those vertices are contained within a square that includes x and (ij)(kl)x, and l(w,y) = 1/2. G satisfies K_n,n satisfies CD(ρ,∞) iff ρ≤ 2 and has κ = 1/n2 * If X = ([n],E) is a graph, the interchange process on X will always give an |E|-regular triangle-free graph. If X is a union of disjoint cliques, it is simple to see that G satisfies the hypotheses of Theorems 4.1 and 4.2. Indeed many of the previous examples are of this type: Ω_d corresponds to X being a perfect matching on 2d vertices, and S_n corresponds to X = K_n. * An (n,d,k)-incidence graph is a d-regular bipartite graph with partite sets A_1,A_2 of n elements each so that each pair of elements in A_i share k common neighbors in A_1-i. In order that such a graph exists, (n-1)k = d(d-1) must count the number of 2-paths starting (and not ending) at x. Without loss of generality x∈ A_1 and x has neighbors y,w∈ A_2, then y and w share k-1 other neighbors, each of which is adjacent to k neighbors of x. l(w,y) = k-1/k. If k≥ 2, an (n,d,k) incidence graph satisfies CD(ρ,∞) iff ρ≤ 2. § OTHER RESULTS §.§ zig-zag product The zig-zag product is defined <cit.> for graphs G_1,G_2, where * G_1 is regular with degree d = |V_2|, and for each a∈ V(G_1) the incident edges are indexed by V_2 - so that we can write a[x] for the unique neighbor of a that shares an edge labelled x with a. * G_2 is D-regular. * The vertex set of the zig-zag product is G_1× G_2. * If x∼ y∼ z is a 2-walk in G_2, then (a,x)∼ (a[y],z). The zig-zag product is a D^2-regular graph that inherits its expansion properties from G_1, which may have much larger degree. For this reason the zig-zag product is useful in generating expander graphs of bounded degree. The question arises whether the zig-zag product inherits the curvature properties from G_1 or G_2. In general this is not the case, we give an example of such a graph: If G_1 and G_2 are both abelian Cayley graphs, it is not in general true that the zig-zag product will have non-negative curavture: as a simple example if G_1 = Ω_d and G_2 = Z_d with vertices 1…, d labelled in order around the cycle, and x∈ V_1 with y = x[2], z = x[d], then (x,1) has four neighbors - (y,1), (y,3), (z,1), (z,d-1). (y,1) has neighbors (x,1),(x,3),(y[d],1),(y[d],d-1). (y,3) has neighbors (x,1),(x,3),(y[4],3),(y[4],5). (z,1) has neighbors (x,1),(x,d-1),(z[2],1),(z[2],3). (z,d-1) has neighbors (x,1), (x,d-1),(z[d-2],d-1),(z[d-2],d-3). G is K_3 and K_2,3-free, we examine which pairs of neighbors of (x,1) are not linked. We see that (y,3)≉(z,1), (y,3)≉(z,d-1), (y,1)≉(z,d-1). As such, N=2, and G satisfies hypothesis (iii) of Theorems 3.1-2. As such, G fails CD(0,∞), and simple calculation gives κ((x,1),(y,3)) = -14. §.§ Diameter bounds It is well-known that a positive lower bound on Ollivier curvature gives an upper bound on the diameter of the graph, according to the following argument developed from <cit.>. Assume for every pair of neighboring vertices x,y, κ(x,y)≥κ*>0. Let x_0,…,x_l be a geodesic path, then W_1(μ_x_0,μ_x_l)≤∑_i=1^l W_1(μ_x_i-1,μ_x_i) ≤ l(1-κ*), but, considering the 1-Lipschitz function f(y) = d(x_0,y), W_1(μ_x_0,μ_x_l)≥∫ f dμ_x_l-∫ fdμ_x_0≥ (l-12)-(12) = l-1. Thus l(1-κ*)≥ l-1, and 1≥ lκ*; as the diameter is achieved on a geodesic path, its length D is bounded above by D≤1/κ*. If G is a d-regular graph with positive curvature, then the diameter is bounded by D≤ 2d. For any x,z∈ V, μ_x(z) is an integer multiple of 12d. There is an optimal solution to the minimum-transport problem between μ_x and μ_y with the property that a multiple of 12d is transported along each path used - as each path has integer length, W_1(μ_x,μ_y) is a multiple of 12d. If x,y are neighbors and κ(x,y) > 0, then it must be κ(x,y) ≥12d. If x,y are not neighbors, then for any z∼ x along the shortest x-y path, κ(x,y)≥κ(x,z)≥12d. Thus κ* ≥12d, and D≤ 1/κ* ≤ 2d. If d is instead an upper bound on the degree of the vertex, then for any x,y,z,w, μ_x(z) and μ_y(w) are both multiples of 1/2d_xd_y, and the same argument follows with κ(x,y)≥1/2d_xd_y≥1/2d^2-2d and thus D≤ 2d^2-2d. This argument depends in the specifics on the definition of μ_x which is not universal, but for any of the other common definitions I am aware of a similar argument will follow. There is no infinite family of bounded-degree graphs with a lower bound on curvature of κ > 0. It is known that a planar graph with bounded degree on n vertices has spectral gap λ = O(1/n)<cit.>. The previous theorem tells us that among planar graphs with bounded degree, there is no infinite family that makes the bound κ = O(1/n) tight, indeed, κ≤ 0 for all but finitely many bounded degree graphs. In the case of graphs without bounded degree, it is true that κ = 1/n for the star graph with n leaves, which is planar. § ACKNOWLEDGEMENTS The author thanks Max Fathi and Prasad Tetali for valuable discussions and examples of graphs relevant to this topic. plainnat

http://arxiv.org/abs/1701.08050v2

20170127134032

Mixed Weyl semimetals and dissipationless magnetization control in insulators by spin-orbit torques

cond-mat.mtrl-sci

Reliable and energy efficient magnetization switching by electrically-induced spin-orbit torques is of crucial technological relevance for spintronic devices implementing memory and logic functionality. Here we predict that the strength of spin-orbit torques and the related interaction in topologically non-trivial magnetic insulators can exceed by far that of conventional metallic magnets. In analogy to the quantum anomalous Hall effect, we explain this extraordinary response in absence of longitudinal currents as a hallmark of magnetic monopoles in the electronic structure of systems that are interpreted most naturally within the framework of mixed Weyl semimetals. We thereby launch the effect of spin-orbit torque into the field of topology and reveal its crucial role in mediating the topological phase transitions arising due to the complex interplay between magnetization direction and momentum-space topology. The concepts presented here may be exploited to understand and utilize magneto-electric coupling phenomena in insulating ferromagnets and antiferromagnets. j.hanke@fz-juelich.de Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany Mixed Weyl semimetals and dissipationless magnetization control in insulators by spin-orbit torques Yuriy Mokrousov April 24, 2017 =================================================================================================== Progress in control and manipulation of the magnetization in magnetic materials is pivotal for the innovative design of future non-volatile, high-speed, low power, and scalable spintronic devices. The effect of spin-orbit torque (SOT) provides an efficient means of magnetization control by electrical currents in systems that combine broken spatial inversion symmetry and spin-orbit interaction <cit.>. These current-induced torques are believed to play a key role in the practical implementation of various spintronics concepts, since they were demonstrated to mediate the switching of single ferromagnetic layers <cit.> and antiferromagnets <cit.> via the exchange of spin angular momentum between the crystal lattice and the (staggered) collinear magnetization. Among the two different contributions to SOTs, the so-called anti-damping torques are of utter importance owing to the robustness of their properties with respect to details of disorder <cit.>. Only recently, the research on electrically-controlled magnetization switching started to reach out to topological condensed matter - for example, very efficient magnetization switching has been achieved lately in metallic systems incorporating topological insulators <cit.>. And although in latter cases a strong torque can be generated, the resulting electric-field response does not rely on the global topological properties of these trivial systems. The discovery of a quantized version of the anomalous Hall effect in magnetic insulators with non-trivial topology in momentum space <cit.> led to a revolution in forging new spintronic device concepts that utilize topology. On the other hand, moving the field of magnetization control by SOTs into the realm of topological spintronics would open bright avenues in exploiting universal arguments of topology for designing magneto-electric coupling phenomena in magnetic insulators. With this work, we firmly put the phenomenon of SOT on the topological ground. Employing theoretical techniques we investigate the origin and size of anti-damping SOTs and interaction (DMI) in prototypes of topologically non-trivial magnetic insulators, demonstrate that complex topological properties have a direct strong impact on the emergence and magnitude of SOT and DMI in various classes of magnetic insulators, and formulate intriguing perspectives for the electric-field control of magnetization in absence of longitudinal charge currents. Results Mixed Weyl semimetals and spin-orbit torque. In a clean sample, the anti-damping SOT T acting on the magnetization in linear response to the electric field E is mediated by the so-called torkance tensor τ, i.e., T=τ E <cit.> (see Fig. <ref>a,b). The Berry phase nature of the anti-damping SOT manifests in the fact that the tensor elements τ_ij are proportional to the mixed Berry curvature Ω^m̂ k_ij= ê_i · 2Im∑_n^occ⟨∂_m̂ u_ k n| ∂_k_j u_ k n⟩ of all occupied states <cit.>, which incorporates derivatives of lattice-periodic wave functions u_ k n with respect to both crystal momentum k and magnetization direction m̂. Here, ê_i denotes the ith Cartesian unit vector. Intimately related to the anti-damping SOT is the DMI <cit.>, crucial for the emergence of chiral domain walls and chiral skyrmions <cit.>, which can be quantified by the so-called spiralization tensor D reflecting the change of the free energy F due to chiral perturbations ∂_jm̂ according to F=∑_ijD_ijê_i·(m̂×∂_j m̂) <cit.>. Optimizing the efficiency of magnetization switching in spintronic devices by current-induced SOTs relies crucially on the knowledge of the microscopic origin of most prominent contributions to the electric-field response. To promote the understanding, it is rewarding to draw an analogy between the anti-damping SOT as given by Ω^m̂ k_ij and the intrinsic anomalous Hall effect as determined by the Berry curvature Ω^ k k_ij=2Im∑_n^occ⟨∂_k_i u_ k n|∂_k_j u_ k n⟩ <cit.>. Both Ω^ k k and Ω^m̂ k are components of a general curvature tensor Ω in the composite ( k,m̂) phase space combining crystal momentum and magnetization direction <cit.>. Band crossings, also referred to as magnetic monopoles in k-space, are known <cit.> to act as important sources or sinks of Ω^ k k. When transferring this concept to current-induced torques, crossing points in the composite phase space can be anticipated to give rise to a large mixed Berry curvature Ω^m̂ k, which in turn yields the dominant microscopic contribution to torkance and spiralization. Thus, materials providing such monopoles close to the Fermi energy can be expected to exhibit notably strong SOTs and DMI. In the field of topological condensed matter <cit.>, the recent advances in the realization of quantum anomalous Hall, or, Chern insulators have been striking <cit.>. These magnetic materials are characterized by a quantized value of the anomalous Hall conductivity and an integer non-zero value of the Chern number in k-space, 𝒞 = 1/(2π)∫Ω^ k k_xydk_x dk_y. On the other hand, topological semimetals have recently attracted great attention due to their exceptional properties stemming from monopoles in momentum space. Among these latter systems, magnetic Weyl semimetals host gapless low-energy excitations with linear dispersion in the vicinity of non-degenerate band crossings at generic k-points <cit.>, which are sources of Ω^ k k. Their conventional description in terms of the Weyl Hamiltonian can be formally extended to the case of what we call the mixed Weyl semimetal as described by H_W=v_x k_x σ_x + v_y k_y σ_y + v_θθσ_z, where σ=(σ_x,σ_y,σ_z) is the vector of Pauli matrices, and θ is the angle that the magnetization m̂=(sinθ,0,cosθ) makes with the z-axis. As illustrated in Fig. <ref>c, mixed Weyl semimetals feature monopoles in the composite phase space of k and θ, which are sources of the general curvature Ω. In analogy to conventional Weyl semimetals <cit.>, we can characterize the topology and detect magnetic monopoles by monitoring the flux of the mixed Berry curvature through planes of constant k_y as given by the integer mixed Chern number 𝒵=1/(2π)∫Ω^m̂ k_yxdθ dk_x, Fig. <ref>c. In the following, we show that a significant electric-field response near monopoles in mixed Weyl semimetals is invaluable in paving the road towards dissipationless magnetization control by SOTs <cit.>. Magnetically doped graphene. We begin with a tight-binding model of magnetically doped graphene <cit.>: H = -t ∑_⟨ ij ⟩α c_iα^† c_jα^† + t_so∑_⟨ ij ⟩αβê_z · (σ× d_ij) c_iα^† c_jβ^† + λ∑_iαβ (m̂·σ) c_iα^† c_iβ^† - λ_nl∑_⟨ ij ⟩αβ (m̂·σ) c_iα^† c_jβ^† , which is sketched in Fig. <ref>a. Here, c_iα^† (c_iα^) denotes the creation (annihilation) of an electron with spin α at site i, ⟨ ...⟩ restricts the sums to nearest neighbors, and the unit vector d_ij points from j to i. Besides the usual hopping with amplitude t, the first line in equation (<ref>) contains the Rashba spin-orbit coupling of strength t_so originating in the surface potential gradient of the substrate. The remaining terms in equation (<ref>) are the exchange energy due to the local (λ) and non-local (λ_nl) exchange interaction between spin and magnetization. Depending on m̂, the non-local exchange describes a hopping process during which the spin can flip. Supplementary Note 1 provides further details on the tight-binding model and its numerical solution. First, by monitoring the evolution of the mixed Chern number 𝒵 we demonstrate that the above model hosts a mixed Weyl semimetal state. Indeed, as shown in Fig. <ref>b, the topological index 𝒵 changes from -2 to 0 at a certain value of k_y, indicating thus the presence of a band crossing in composite phase space that carries a topological charge of +2. One of these monopoles appears near the K-point off any high-symmetry line if the magnetization is oriented in-plane along the x-direction (see Fig. <ref>d). The emergence of the quantum anomalous Hall effect <cit.>, Fig. <ref>c, over a wide range of magnetization directions can be understood as a direct consequence of the magnetic monopoles acting as sources of the curvature Ω^ k k. Correspondingly, for m̂ out of the plane, the system is a quantum anomalous Hall insulator. Moreover, large values of the mixed curvature Ω^m̂ k in the vicinity of the monopole are visible in the momentum-space distributions of torkance and spiralization in the insets of Figs. <ref>d and <ref>e, respectively. For an out-of-plane magnetization, the primary microscopic contribution to the effects arises from an avoided crossing along Γ K – a residue of the Weyl point in ( k,θ)-space. Since the expression for the mixed Berry curvature relies only on the derivative of the wavefunction with respect to one of the components of the Bloch vector, the symmetry between k_x and k_y in the distributions of torkance and spiralization is broken naturally (see Methods). As a consequence of the monopole-driven momentum-space distribution, the energy dependence of the torkance τ_yx, Fig. <ref>d, displays a decent magnitude of 0.1 ea in the insulating region (with a being the interatomic distance), and stays constant throughout the band gap. In contrast to the Chern numbers 𝒞 and 𝒵, the torkance τ_yx is, however, not guaranteed to be quantized to a robust value, i.e., the height of the torkance plateau in Fig. <ref>d is sensitive to fine details of the electronic structure such as magnetization direction and model parameters. Because of their intimate relation in the Berry phase theory <cit.>, the plateau in torkance implies a linear behavior of the spiralization D_yx within the gap, changing from 8mta/uc to -6mta/uc as shown in Fig. <ref>e, where “uc" refers to the in-plane unit cell containing two atoms. To provide a realistic manifestation of the model considerations above, we study from ab initio systems of graphene decorated by transition-metal adatoms, Fig. <ref>a. These systems, which exhibit complex spin-orbit mediated hybridization of graphene p states with d states of the transition metal, have by now become one of the prototypical material classes for realization of the quantum anomalous Hall effect <cit.>. Details on the first-principles calculations are provided in Supplementary Note 2. In the Chern insulator phase of these materials with magnetization perpendicular to the graphene plane, depending on the transition-metal adatom, both torkance and spiralization can reach colossal magnitudes that originate from mixed Weyl points. In the case of W in 4×4-geometry on graphene, for example, the torkance amounts to a huge value of τ_yx=-2.9 ea_0 (with a_0 being Bohr's radius), and the spiralization D_yx ranges from -5meVa_0/uc to 60meVa_0/uc, Fig. <ref>b-e, surpassing thoroughly the values obtained in metallic magnetic heterostructures <cit.> and non-centrosymmetric bulk magnets <cit.>. Since the details of the electronic structure can influence the value of the torkance in the gap, upon replacing W with other transition metals, the magnitude of SOT and DMI can be tailored in the gapped regions of corresponding materials according to our calculations. Functionalized bismuth film. Aiming at revealing pronounced magneto-electric coupling effects in magnetic insulators with larger band gaps as compared to the above examples, we turn to a semi-hydrogenated Bi(111) bilayer, Fig. <ref>a, which is a prominent example of functionalized insulators realizing non-trivial topological phases <cit.>. As we show, semi-hydrogenated Bi(111) bilayer is a mixed Weyl semimetal. For an out-of-plane magnetization, the system is a valley-polarized quantum anomalous Hall insulator <cit.> with a magnetic moment of 1.0 μ_B per unit cell, and it exhibits a large band gap of 0.18eV at the Fermi energy as well as a distinct asymmetry between the valleys K and K^', Fig. <ref>b. Analyzing the evolution of the mixed Chern number 𝒵 as a function of k_y in Fig. <ref>b, we detect two magnetic monopoles of opposite charge that emerge at the transition points between the topologically distinct phases with 𝒵=-1 and 𝒵=0. Alternatively, these crossing points and the monopole charges in the composite phase space could be identified by monitoring the variation of the momentum-space Chern number 𝒞 with magnetization direction. These monopoles occur at generic points near the valley K for θ=43^∘ (see Fig. <ref>e) and in the vicinity of the K^'-point for θ= 137^∘, respectively. The presence of such mixed Weyl points in the electronic structure drastically modifies the behavior of the general curvature Ω in their vicinity, as visible from the three-dimensional representation of Ω displayed in Fig. <ref>c,d. Revealing characteristic sign changes when passing through monopoles in composite phase space, the singular behavior of the Berry curvature underlines the role of the mixed Weyl points as sources or sinks of Ω. For an out-of-plane magnetization, the complex nature of the electronic structure in momentum space manifests in the quantization of 𝒞 to +1, Fig. <ref>e, which is primarily due to the pronounced positive contributions near K. Calculations of the energy dependence of the torkance and spiralization in the system, shown in Figs. <ref>f and <ref>g, reveal the extraordinary magnitudes of these phenomena of the order of 1.1 ea_0 for τ_yx and 50meVa_0/uc for D_yx, exceeding by far the typical magnitudes of these effects in magnetic metallic materials <cit.>. Proof of monopole-driven SOT enhancement. An important question to ask at this point is whether the colossal magnitude of the SOT in the insulators considered above can be unambiguously identified with the mixed Weyl semimetallic state. In the following, we answer this question by explicitly demonstrating the utter importance of the emergent mixed monopoles for driving pronounced magneto-electric response. First, by removing the mixed Weyl points from the electronic structure of the model (<ref>) via, e.g., including an intrinsic spin-orbit coupling term, we confirm that the electric-field response is strongly suppressed, which promotes the monopoles as unique origin of large SOT and DMI. Secondly, to verify this statement from the first-principles calculations, we analyze the electric-field response throughout the topologically trivial gaps above the Fermi level that are highlighted in Figs. <ref>b and <ref>b. Since these gaps do not exhibit the mixed Weyl points, we obtain a greatly diminished magnitude of the torkance τ_yx within these energy regions as apparent from Figs. <ref>f and <ref>d. Finally, we clearly demonstrate the key role of these special points by studying an illustrative example: a thin film of GaBi with triangular lattice structure, Fig. <ref>g. The initial system is a non-magnetic trivial insulator, on top of which we artificially apply an exchange field B=B_0(sinθ,0,cosθ), with the purpose of triggering a topological phase transition as a function of the exchange field strength, see Supplementary Note 4. When tuning the exchange field strength B_0 we carefully monitor the evolution of the system from a trivial magnetic insulator for |B_0|≤ 0.2eV to a mixed Weyl semimetal as indicated by the emergence of magnetic monopoles in the electronic structure. The latter phase is accompanied by the quantum anomalous Hall effect prominent for a finite range of directions θ, for instance, if B is perpendicular to the film plane, Fig. <ref>h,i. Comparing in Fig. <ref>f the electric-field response for these two distinct phases, we uniquely identify drastic changes in sign and magnitude of the torkance τ_yx with the transition from the trivial insulator to the mixed Weyl semimetal hosting monopoles near the Γ-point. This proves the crucial relevance of emergent monopoles in driving magneto-electric coupling effects in topologically non-trivial magnetic insulators. Discussion Remarkably, the magnetization switching via anti-damping torques in mixed Weyl semimetals can be utilized to induce topological phase transitions from a Chern insulator to a trivial magnetic insulator mediated by the complex interplay between magnetization direction and momentum-space topology in these systems as illustrated in Fig. <ref>a,b. In the case of the functionalized bismuth film, for instance, the material is a trivial magnetic insulator with a band gap of 0.25eV if the magnetization is oriented parallel to the film plane. Nevertheless, the resulting anti-damping torkance in this trivial state is still very large, and the DMI exhibits a strong variation within the gap, see Supplementary Note 3. We therefore motivate experimental search and realization of large magneto-electric response and topological phase transitions in quantum anomalous Hall systems fabricated to date <cit.>. Overall, mixed Weyl semimetals that combine exceptional electric-field response with a large band gap (such as, e.g., functionalized bismuth films) lay out extremely promising vistas in room-temperature applications of magneto-electric coupling phenomena for dissipationless magnetization control – a subject which is currently under extensive scrutiny (see, e.g., refs. Avci2016,Chu2008,Chiba2008). In contrast to the anti-damping SOT in magnetic metallic bilayers (such as Co/Pt) for which large spin-orbit interaction in the non-magnetic substrate is necessary for generating large spin Hall effect and large values of SOT <cit.>, the magnitude of the SOT in insulating phases of a mixed Weyl semimetal is driven by the presence of the mixed monopole rather than the spin-orbit strength itself. This opens perspectives in exploiting a strong magneto-electric response of weak-spin-orbit materials. In the examples that we considered here, the non-trivial topology of mixed Weyl semimetals leads to DMI changes over a wide range of values throughout the bulk band gap, implying that proper electronic-structure engineering enables us to tailor both strength and sign of the DMI in a given system, for instance, by doping or applying strain. Such versatility could be particularly valuable for the stabilization of chiral magnetic structures such as skyrmions in insulating ferromagnets. In the latter case, very large values of the anti-damping SOT arising in these systems would open exciting perspectives in manipulation and dynamical properties of chiral objects associated with minimal energy consumption by magneto-electric coupling effects. Generally, we would like to remark that magnetic monopoles in the composite phase space, which we discuss here, do not only govern the electric-field response in insulating magnets but are also relevant in metals, where they appear on the background of metallic bands. Ultimately, in analogy to the (non-quantized) anomalous Hall effect in metals, this makes the analysis of SOT and DMI in metallic systems very complex owing to competing contributions to these effects from various bands present at the Fermi energy. In addition, the electric-field strength in metals is typically much smaller, limiting thus the reachable magnitude of response phenomena as compared to insulators. At the end, we reveal the relevance of the physics discussed here for antiferromagnets (AFMs) that satisfy the combined symmetry of time reversal and spatial inversion. SOTs in such antiferromagnets are intimately linked with the physics of Dirac fermions, which are doubly-degenerate elementary excitations with linear dispersion <cit.>. In these systems, the reliable switching of the staggered magnetization by means of current-induced torques has been demonstrated very recently <cit.>. In analogy to the concept of mixed Weyl semimetals presented here, we expect that the notion of mixed Dirac semimetals in a combined phase space of crystal momentum and direction of the staggered magnetization vector will prove fruitful in understanding the microscopic origin of SOTs in insulating antiferromagnets. Following the very same interpretation that we formulated here for ferromagnets, monopoles in the electronic structure of AFMs can be anticipated to constitute prominent sources or sinks of the corresponding general non-Abelian Berry curvature, whose mixed band-diagonal components correspond to the sublattice-dependent anti-damping SOT, in analogy to the spin Berry curvature for quantum spin Hall insulators and Dirac semimetals <cit.>. Correspondingly, exploiting the principles of electronic-structure engineering for topological properties depending on the staggered magnetization could result in an advanced understanding and utilization of pronounced magneto-electric response in insulating AFMs. Methods Tight-binding calculations. The Hamiltonian (<ref>) is a generalization of the model in ref. Qiao2010, taking additionally into consideration arbitrary magnetization directions m̂ as well as the non-local exchange interaction. A brief description of its numerical solution is given in Supplementary Note 1. First-principles electronic structure calculations. Using the full-potential linearized augmented plane-wave code  <cit.>, we performed self-consistent density functional theory calculations of the electronic structure of the considered materials using the structural parameters of refs. Zhang2012 and Niu2015. The effect of spin-orbit coupling was treated within the perturbative second-variation scheme. Starting from the converged charge density, we constructed higher-dimensional Wannier functions <cit.> by employing our extension of the code <cit.>. We used these functions to generalize the Wannier interpolation <cit.> allowing us to evaluate efficiently anomalous Hall conductivity, torkance, and spiralization. Further details on the electronic structure calculations are given in Supplementary Note 2. Berry phase expressions for torkance and spiralization. In order to characterize the anti-damping SOTs, we evaluate within linear response the torkance <cit.> τ_ij = 2e/N_ kê_i ·∑_ k n^occ[m̂×Im⟨∂_m̂ u_ k n | ∂_k_j u_ k n⟩] , where N_ k is the number of k-points, and e>0 denotes the elementary positive charge. Similarly, the spiralization <cit.> is obtained as D_ij = ê_i/N_ kV·∑_ kn^occ[ m̂×Im⟨∂_m̂ u_ k n | h_ k n | ∂_k_j u_ k n⟩] , where h_ kn=H_ k+ℰ_ k n-2ℰ_F, H_ k is the lattice-periodic Hamiltonian with eigenenergies ℰ_ k n, ℰ_F is the Fermi level, and V is the unit cell volume. Code availability. The tight-binding code that supports the findings of this study is available from the corresponding authors on request. Data availability. The data that support the findings of this study are available from the corresponding authors on request. References 10 url<#>1urlprefixURL iblabel[1]#1. Chernyshov2009 authorChernyshov, A. et al. titleEvidence for reversible control of magnetization in a ferromagnetic material by means of spin–orbit magnetic field. journalNat. Phys. volume5, pages656–659 (year2009). Miron2010 authorMiron, I. M. et al. titleCurrent-driven spin torque induced by the Rashba effect in a ferromagnetic metal layer. journalNat. Mater. volume9, pages230–234 (year2010). Miron2011 authorMiron, I. M. et al. titleFast current-induced domain-wall motion controlled by the Rashba effect. journalNat. Mater. volume10, pages419–423 (year2011). Garello2013 authorGarello, K. et al. titleSymmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures. journalNature Nanotech. volume8, pages587–593 (year2013). Freimuth2014a authorFreimuth, F., authorBlügel, S. & authorMokrousov, Y. titleSpin-orbit torques in Co/Pt (111) and Mn/W (001) magnetic bilayers from first principles. journalPhys. Rev. B volume90, pages174423 (year2014). Miron2011a authorMiron, I. M. et al. titlePerpendicular switching of a single ferromagnetic layer induced by in-plane current injection. journalNature volume476, pages189–193 (year2011). Liu2012 authorLiu, L., authorLee, O., authorGudmundsen, T., authorRalph, D. & authorBuhrman, R. titleCurrent-induced switching of perpendicularly magnetized magnetic layers using spin torque from the spin Hall effect. journalPhys. Rev. Lett. volume109, pages096602 (year2012). Wadley2016 authorWadley, P. et al. titleElectrical switching of an antiferromagnet. journalScience volume351, pages587–590 (year2016). Mellnik2014 authorMellnik, A. R. et al. titleSpin-transfer torque generated by a topological insulator. journalNature volume511, pages449–451 (year2014). Haldane1988 authorHaldane, F. D. M. titleModel for a quantum Hall effect without landau levels: Condensed-matter realization of the "parity anomaly". journalPhys. Rev. Lett. volume61, pages2015–2018 (year1988). Yu2010 authorYu, R. et al. titleQuantized anomalous Hall effect in magnetic topological insulators. journalScience volume329, pages61–64 (year2010). Chang2013 authorChang, C.-Z. et al. titleExperimental observation of the quantum anomalous Hall effect in a magnetic topological insulator. journalScience volume340, pages167–170 (year2013). Freimuth2014 authorFreimuth, F., authorBlügel, S. & authorMokrousov, Y. titleBerry phase theory of Dzyaloshinskii–Moriya interaction and spin–orbit torques. journalJ. Phys.: Condens. Matter volume26, pages104202 (year2014). Kurebayashi2014 authorKurebayashi, H. et al. titleAn antidamping spin-orbit torque originating from the Berry curvature. journalNature Nanotech. volume9, pages211 – 217 (year2014). Dzyaloshinsky1958 authorDzyaloshinsky, I. titleA thermodynamic theory of weak ferromagnetism of antiferromagnetics. journalJ. Phys. Chem. Solids volume4, pages241–255 (year1958). Moriya1960 authorMoriya, T. titleAnisotropic superexchange interaction and weak ferromagnetism. journalPhys. Rev. volume120, pages91 (year1960). Neubauer2009 authorNeubauer, A. et al. titleTopological Hall effect in the a phase of MnSi. journalPhys. Rev. Lett. volume102, pages186602 (year2009). Kanazawa2011 authorKanazawa, N. et al. titleLarge topological Hall effect in a short-period helimagnet MnGe. journalPhys. Rev. Lett. volume106, pages156603 (year2011). Franz2014 authorFranz, C. et al. titleReal-space and reciprocal-space Berry phases in the Hall effect of Mn_1- xFe_xSi. journalPhys. Rev. Lett. volume112, pages186601 (year2014). Gayles2015 authorGayles, J. et al. titleDzyaloshinskii-Moriya interaction and Hall effects in the skyrmion phase of Mn_1-xFe_xGe. journalPhys. Rev. Lett. volume115, pages036602 (year2015). Nagaosa2010 authorNagaosa, N., authorSinova, J., authorOnoda, S., authorMacDonald, A. H. & authorOng, N. P. titleAnomalous Hall effect. journalRev. Mod. Phys. volume82, pages1539–1592 (year2010). Xiao2005 authorXiao, D., authorShi, J. & authorNiu, Q. titleBerry phase correction to electron density of states in solids. journalPhys. Rev. Lett. volume95, pages137204 (year2005). Freimuth2013 authorFreimuth, F., authorBamler, R., authorMokrousov, Y. & authorRosch, A. titlePhase-space Berry phases in chiral magnets: Dzyaloshinskii-Moriya interaction and the charge of skyrmions. journalPhys. Rev. B volume88, pages214409 (year2013). Fang2003 authorFang, Z. et al. titleThe anomalous Hall effect and magnetic monopoles in momentum space. journalScience volume302, pages92–95 (year2003). Hasan2010 authorHasan, M. Z. & authorKane, C. L. titleColloquium : Topological insulators. journalRev. Mod. Phys. volume82, pages3045–3067 (year2010). Qi2011 authorQi, X.-L. & authorZhang, S.-C. titleTopological insulators and superconductors. journalRev. Mod. Phys. volume83, pages1057–1110 (year2011). Xu2011 authorXu, G., authorWeng, H., authorWang, Z., authorDai, X. & authorFang, Z. titleChern semimetal and the quantized anomalous Hall effect in HgCr_2Se_4. journalPhys. Rev. Lett. volume107, pages186806 (year2011). Xu2015 authorXu, S.-Y. et al. titleDiscovery of a Weyl fermion semimetal and topological Fermi arcs. journalScience volume349, pages613–617 (year2015). Lv2015 authorLv, B. Q. et al. titleExperimental discovery of Weyl semimetal TaAs. journalPhys. Rev. X volume5, pages031013 (year2015). Gosalbez2015 authorGosálbez-Martínez, D., authorSouza, I. & authorVanderbilt, D. titleChiral degeneracies and Fermi-surface Chern numbers in bcc Fe. journalPhys. Rev. B volume92, pages085138 (year2015). Avci2016 authorAvci, C. O. et al. titleCurrent-induced switching in a magnetic insulator. journalNat. Mater. (year2016). Qiao2010 authorQiao, Z. et al. titleQuantum anomalous Hall effect in graphene from Rashba and exchange effects. journalPhys. Rev. B volume82, pages161414(R) (year2010). Thonhauser2011 authorThonhauser, T. titleTheory of orbital magnetization in solids. journalInt. J. Mod. Phys. B volume25, pages1429–1458 (year2011). Hanke2016 authorHanke, J.-P. et al. titleRole of Berry phase theory for describing orbital magnetism: From magnetic heterostructures to topological orbital ferromagnets. journalPhys. Rev. B volume94, pages121114(R) (year2016). Ding2011 authorDing, J., authorQiao, Z., authorFeng, W., authorYao, Y. & authorNiu, Q. titleEngineering quantum anomalous/valley Hall states in graphene via metal-atom adsorption: An ab-initio study. journalPhys. Rev. B volume84, pages195444 (year2011). Zhang2012 authorZhang, H., authorLazo, C., authorBlügel, S., authorHeinze, S. & authorMokrousov, Y. titleElectrically tunable quantum anomalous Hall effect in graphene decorated by 5d transition-metal adatoms. journalPhys. Rev. Lett. volume108, pages056802 (year2012). Acosta2014 authorAcosta, C. M., authorLima, M. P., authorMiwa, R. H., authorda Silva, A. J. R. & authorFazzio, A. titleTopological phases in triangular lattices of Ru adsorbed on graphene: Ab initio calculations. journalPhys. Rev. B volume89, pages155438 (year2014). Hu2015 authorHu, J., authorZhu, Z. & authorWu, R. titleChern half metals: A new class of topological materials to realize the quantum anomalous Hall effect. journalNano Lett. volume15, pages2074–2078 (year2015). Ren2016 authorRen, Y., authorQiao, Z. & authorNiu, Q. titleTopological phases in two-dimensional materials: a review. journalRep. Prog. Phys. volume79, pages066501 (year2016). Niu2015 authorNiu, C. et al. titleFunctionalized bismuth films: Giant gap quantum spin Hall and valley-polarized quantum anomalous Hall states. journalPhys. Rev. B volume91, pages041303 (year2015). Checkelsky2014 authorCheckelsky, J. G. et al. titleTrajectory of the anomalous Hall effect towards the quantized state in a ferromagnetic topological insulator. journalNat. Phys. volume10, pages731–736 (year2014). Kou2014 authorKou, X. et al. titleScale-invariant quantum anomalous Hall effect in magnetic topological insulators beyond the two-dimensional limit. journalPhys. Rev. Lett. volume113, pages137201 (year2014). Chang2015 authorChang, C.-Z. et al. titleHigh-precision realization of robust quantum anomalous Hall state in a hard ferromagnetic topological insulator. journalNat. Mater. volume14, pages473–477 (year2015). Chu2008 authorChu, Y.-H. et al. titleElectric-field control of local ferromagnetism using magnetoelectric multiferroic. journalNat. Mater. volume7, pages478–482 (year2008). Chiba2008 authorChiba, D. et al. titleMagnetization vector manipulation by electric fields. journalNature volume455, pages515–518 (year2008). Tang2016 authorTang, P., authorZhou, Q., authorXu, G. & authorZhang, S.-C. titleDirac fermions in an antiferromagnetic semimetal. journalNat. Phys. volume12, pages1100–1104 (year2016). Smejkal2017 author ŠŠmejkal, L., author ŽŽelezný, J., authorSinova, J. & authorJungwirth, T. titleElectric control of dirac quasiparticles by spin-orbit torque in an antiferromagnet. journalPhys. Rev. Lett. volume118, pages106402 (year2017). Murakami2007 authorMurakami, S. titlePhase transition between the quantum spin Hall and insulator phases in 3d: emergence of a topological gapless phase. journalNew J. Phys. volume9, pages356 (year2007). Murakami2008 authorMurakami, S. & authorKuga, S.-i. titleUniversal phase diagrams for the quantum spin Hall systems. journalPhys. Rev. B volume78, pages165313 (year2008). Yang2014 authorYang, B.-J. & authorNagaosa, N. titleClassification of stable three-dimensional Dirac semimetals with nontrivial topology. journalNat. Commun. volume5, pages4898 (year2014). fleur howpublishedSee <http://www.flapw.de>. Hanke2015 authorHanke, J.-P., authorFreimuth, F., authorBlügel, S. & authorMokrousov, Y. titleHigher-dimensional Wannier functions of multiparameter Hamiltonians. journalPhys. Rev. B volume91, pages184413 (year2015). Mostofi2014 authorMostofi, A. A. et al. titleAn updated version of wannier90: A tool for obtaining maximally-localised Wannier functions. journalComput. Phys. Commun. volume185, pages2309 – 2310 (year2014). Wang2006 authorWang, X., authorYates, J. R., authorSouza, I. & authorVanderbilt, D. titleAb initio calculation of the anomalous Hall conductivity by Wannier interpolation. journalPhys. Rev. B volume74, pages195118 (year2006). Yates2007 authorYates, J. R., authorWang, X., authorVanderbilt, D. & authorSouza, I. titleSpectral and Fermi surface properties from Wannier interpolation. journalPhys. Rev. B volume75, pages195121 (year2007). Acknowledgements We gratefully acknowledge computing time on the supercomputers JUQUEEN and JURECA at Jülich Supercomputing Center as well as at the JARA-HPC cluster of RWTH Aachen, and funding under the HGF-YIG programme VH-NG-513 and SPP 1538 of DFG. Author contributions J.-P.H. uncovered the mixed Weyl points as origin of large magneto-electric coupling effects through model considerations and first-principles calculations. J.-P.H. and Y.M. wrote the manuscript. All authors discussed the results and reviewed the manuscript. Additional information Competing financial interests. The authors declare no competing financial interests. Supplementary Material: Mixed Weyl semimetals and dissipationless magnetization control in insulators by spin-orbit torques Jan-Philipp Hanke, Frank Freimuth, Chengwang Niu, Stefan Blügel, and Yuriy Mokrousov Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany Supplementary Note 1 | Tight-binding model To arrive at the model Hamiltonian of the main text, the model in ref. Qiao2010 has been generalized to account for arbitrary magnetization directions m̂ and the non-local exchange interaction. We obtained a 4×4-matrix representation of the resulting Hamiltonian on the bipartite lattice of graphene by introducing four orthonormal basis states |N α⟩ that describe electrons with spin α={↑,↓} on the sublattice N={A,B}. Using Fourier transformations, we transformed this matrix to a representation H( k) in momentum space, which was subsequently diagonalized at every k-point to access the electronic and topological properties of the system. The model parameters t_so=0.3t, λ=0.1t, and λ_nl=0.4t were employed in this work. We chose the magnetization direction as m̂=(sinθ,0,cosθ) for a direct comparison between the model and the first-principles calculations. Supplementary Note 2 | First-principles electronic structure calculations Using the full-potential linearized augmented plane-wave code  <cit.>, we performed self-consistent density functional theory calculations of the electronic structure of (i) graphene decorated with W adatoms in 4×4-geometry, and (ii) a semi-hydrogenated Bi(111) bilayer. The structural and computational parameters of refs. Zhang2012 and Niu2015 were assumed in the respective cases. Starting from the converged charge density, the Kohn-Sham equations were solved on an equidistant mesh of 8×8 k-points [6×6 in case (i)] for 8 different magnetization directions m̂=(sinθ,0,cosθ), where the angle θ covers the unit circle once. Based on the resulting wave-function information in the composite phase space, we constructed a single set of higher-dimensional Wannier functions <cit.> (HDWFs) for each of the systems by employing our extension of the code <cit.>. In case (i), we generated 274 HDWFs out of 360 energy bands with the frozen window up to 4eV above the Fermi level, and in the case (ii), we extracted from 28 bands 14 HDWFs for a frozen window that extends to 2eV above the Fermi energy. We used the Wannier interpolation <cit.> that we generalized to treat crystal momentum and magnetization direction on an equal footing <cit.> in order to evaluate the Berry curvatures Ω^ k k and Ω^m̂ k. Taking into account the above parametrization of the magnetization direction by θ, we were thus able to access efficiently the anomalous Hall conductivity σ_ij, the torkance τ_yj, and the spiralization D_yj: σ_ij = e^2/h1/2π∫ 2Im∑_n^occ⟨∂ u_ k n/∂ k_i | ∂ u_ k n/∂ k_j⟩ dk_x dk_y , τ_yj = e ∫ 2Im∑_n^occ⟨∂ u_ k n/∂θ | ∂ u_ k n/∂ k_j⟩ dk_x dk_y , D_yj = 1/V∫Im∑_n^occ⟨∂ u_ k n/∂θ | h_ k n | ∂ u_ k n/∂ k_j⟩ dk_x dk_y , with the same definitions as in the main text. Convergence of these quantities was achieved using 1024× 1024 k-points in the Brillouin zone. We obtained the mixed Chern number 𝒵(k_y)=1/(2π)∫ 2Im∑_n^occ⟨∂_θ u_ k n | ∂_k_x u_ k n⟩ dθ dk_x by integrating the mixed Berry curvature on a uniform mesh of 1024 k_x-values and 512 angles θ in [0,2π). Supplementary Note 3 | Anisotropy with magnetization direction in semi-hydrogenated Bi bilayer In Supplementary Fig. 1, we show the dependence of anomalous Hall conductivity, torkance, and spiralization on the magnetization direction m̂=(sinθ,0,cosθ) in the semi-hydrogenated bismuth film. For general magnetization directions, both torkance and spiralization display also small non-zero components τ_yy and D_yy, respectively, since the shape of these response tensors is dictated by the crystal symmetries and not due to Onsager's reciprocity relations. When the Weyl point emerges in the electronic structure at θ=43^∘, the system undergoes a topological phase transition from a Chern insulator to a trivial magnetic insulator, which is accompanied by a jump in σ_xy and a similar drop of τ_ij. As apparent from Supplementary Figs. 1 and 2, the torkance τ_yx is still remarkably prominent in the regime of the trivial insulator, i.e., for θ>43^∘. Supplementary Note 4 | GaBi film with exchange field Using the code <cit.>, we performed self-consistent electronic structure calculations of an intrinsically non-magnetic GaBi film, employing the generalized gradient approximation and a plane-wave cut-off of 4.0 a_0^-1, where a_0 is Bohr's radius. The in-plane lattice constant was 8.5 a_0 and we chose a muffin tin radius of 2.45 a_0 for both atom species. Subsequently, we constructed 16 maximally-localized Wannier functions out of 32 energy bands with the frozen window extending up to 2eV above the Fermi level. Finally, in order to substantiate the predicted effect of monopole-driven spin-orbit torques, the exchange term B·σ was added to the corresponding tight-binding Hamiltonian, where σ is the vector of Pauli matrices and B=B_0(sinθ,0,cosθ) denotes the imposed exchange field. Supplementary Figures

http://arxiv.org/abs/1701.07530v1

20170126003724

Actively Calibrated Line Mountable Capacitive Voltage Transducer For Power Systems Applications

physics.ins-det

Constructing Tame Supercuspidal Representations Jeffrey Hakim December 30, 2023 =============================================== A class of Actively Calibrated Line Mounted Capacitive Voltage Transducers (LMCVT) are introduced as a viable line mountable instrumentation option for deploying large numbers of voltage transducers onto the medium and high voltage systems. Active Calibration is shown to reduce the error of line mounted voltage measurements by an order of magnitude from previously published techniques. The instrument physics and sensing method is presented and the performance is evaluated in a laboratory setting. Finally, a roadmap to a fully deployable prototype is shown. § INTRODUCTION Modern smart grid infrastructure envisions extensive deployment of renewable generation, electric vehicles, storage and many other technologies. To enable these deployments, increased sensing and enhanced situational awareness is required. In distribution systems, voltage and current are the main quantities which must be monitored and deploying technologies to measure these in a large scale can be prohibitively expensive. Therefore, there is a need to develop inexpensive yet accurate sensor technology for the distribution system. This work presents a new sensing technology which aims to dramatically reduce the cost of accurate voltage sensing for distribution system. The state of the art in voltage measurement technology is either (1) expensive to deploy in large numbers or (2) suffers from errors making any practical use impossible. Accurate voltage sensing in a substation setting relies on Voltage Transformers or Capacitively Coupled Voltage Transducer (CCVT) <cit.> technology to step down the voltage. Inexpensive line mounted voltage relies on (1) electrostatic field and (2) capacitively coupled measurements which do not reach the required accuracy of metering quality sensing. This work proposes a solution to the limitations of line mounted voltage sensing. The main contributions of this work are the following: (1) A detailed physics model of Line Mounted Capacitive Voltage Transducers is presented with experimental validation. (2) The active calibration methodology is detailed along with experiments showing its effectiveness of reducing the voltage magnitude errors. (3) A research roadmap to enable a full deployable solution is given. The paper is organized as follows. Section <ref> reviews existing line mounted voltage measurement approaches. Section <ref> presents the basic line mounted capacitive voltage transducer. Section <ref> introduces the active calibration concept and discusses some of it's properties. Section <ref> describes the development of a practical prototype. Section <ref> presents in detail the signal processing methods required. Section <ref> presents experimental results proving the efficacy of the technology. Finally, Section <ref> addresses future challenges and opportunities for the proposed sensing technology. § LINE MOUNTED VOLTAGE MEASUREMENT METHODS This section reviews the main classes of line mounted voltage measurement methods and identifies their benefits and limitations. §.§ Electrostatic Field Measurement The electric field generated by an energized power line is measured inside the device at one or more points at varying distances from the line. The measured field is then used to reconstruct the line voltage based on a fixed mapping between the measured field and line voltage. Much of the research on electric field based measurements has focused on the MEMs transducer themselves, <cit.> <cit.>, <cit.> <cit.>. The fundamental limitation for the line mounted electrostatic sensors is the dependence on the physical arrangement of the conductor and the ground which dictates the observed electric field <cit.> <cit.>, <cit.>. This can change over time, but must be calibrated before sensor deployment typically in a laboratory setting. When the physical environment changes over time, the inferred voltage level will deviate from the true value. §.§ Capacitive Voltage Measurement The term capacitive voltage measurements enumerate a number of different configurations of traditional and non-traditional voltage measurements: 1 Line/Ground Mounted devices are electrically connected both to ground and the line. This method covers CCVTs which are installed in high voltage substations. 2 Ground Mounted devices are connected to ground and capacitively coupled to the energized line. This method is most often associated with "Capacitively Coupled" measurement and is considered non-contact since it does not need electrical connection to the energized line. The method has found uses in non-contact voltage instrumentation <cit.>. However, issues with multiple interfering power lines and large distance between the ground and the conductors have limited the accuracy. Some applications have been capable of estimating the power line phase arrangements <cit.>, even with large sensor errors. 3 Line Mounted devices are connected to the energized line and capacitively coupled to earth ground. This method requires electrical connection to the energized line, but does not require grounded connection. The paper focuses on this method since it is the only method that is line mountable, easy to deploy but also (as will be shown) allows for an active calibration procedure enabling high accuracy. There has been some attention paid to line mounted capacitively coupled voltage transducers for high voltage applications. Two previously published patents: <cit.> and <cit.> introduce 'body capacitive voltage measurement' similar to those shown here. In <cit.>, the authors state the use of a calibrated capacitor divider circuit for determining the voltage on the line. The device is a doughnut shaped conductive material with an identical charge amplifier circuit presented in Section <ref>. In <cit.>, the authors introduce a 'body capacitive' probe comprising of a fixed size sphere which hangs on the power line. They present the sensing circuitry to measure the accumulated charge on the device as well. Like <cit.>, the calibration of the probe capacitance is done offline. In <cit.> and <cit.>, the authors develop a similar understanding of the capacitive coupling and propose methods to track or mitigate changes in probe capacitance. In <cit.>, multiple conductors are used to mitigate the effect of nearby conductors. The results show a nominal voltage magnitude error of 1-12 % with 1-5 minute averaging periods. In <cit.> an algorithm is proposed to mitigate the nearby conductors and determine the height of the device from ground. A parametric model relating the unknown height of the device from ground is used along with long time captures to estimate the height of the device and the probe capacitance. Finally, the estimate of the probe capacitance is used to estimate the line voltage. From these the prior and proposed LMCVT technology can be classified in the following classes. 1 Offline Calibration estimates the voltage magnitude from a fixed mapping function which is determined beforehand in laboratory testing <cit.>, <cit.>. All electrostatic sensor based methods will fall into this category as well, since they rely on previously computed mapping function. 2 Online Passive Calibration tracks specific changes such as the height of the device or proximity to conductors by processing multiple passive sources <cit.>, <cit.>. The changes are tracked in an online manner so that the mapping function changes over time, however the task is performed by processing passive measurements. The commonality in both methods is that passive calibration requires (1) long captures (2) parametric models of disturbances and probe-to-ground capacitance. 3 Online Active Calibration uses active voltage injection onto the capacitive probe and then recover the perturbed value (depending on the probe to ground capacitance). Careful signal design and signal processing techniques track in real time changes to the probe capacitance and estimate the line voltage. The third method is what is proposed here to enable low cost high and medium voltage measurement technology. Unlike passive techniques, this method does not require parametric models, and can compute the capacitance directly via pilot signaling mechanism. § PHYSICAL MODEL OF LINE MOUNTED CAPACITIVE VOLTAGE TRANSDUCER Various properties of a body capacitive probe are modeled through the first principles of the devices physical operation. First the charge accumulation of an ideal conductor held at the high voltage while leads to the floating capacitor is introduced. Then, the effective capacitance of the system is discussed as well as interference effects on nearby conductors. §.§ Ideal Body Capacitive Probe Model Figure <ref> introduces an ideal body capacitive probe. The probe is a conducting sphere of radius r_probe. Although it is impractical for a final device to be a round sphere, this model is used since it's capacitance is simple to compute. Connected to the sphere is an ideal charge sensor which measures the charge that accumulates on the surface of the conductor. The model assumes that the voltage source and the sensor take infinitely small volume compared to the conducting sphere, therefore producing no electric field of its own. (the case of non-negligible voltage source is considered in <ref>). Furthermore, there is no voltage drop between the ideal charge sensor and the conducting sphere. Assume at time t ≤ t_0 the device is previously being uncharged and after t > t_0 the device is connected to the voltage source. Calculating the accumulated charge on the sphere necessary to maintaining a voltage of V_L leads to Q = 4 πϵ_0 r_probe V_L. This is the basic definition of the probe capacitance to ground, where C = 4 πϵ_0 r_probe. The capacitance follows the standard relationship Q = C V_L which leads to a very simple voltage transducer. If the value of C is known with certaintly, for example by building a probe with a spherical shape of known radius placed in free space, then measuring the waveform of Q(t) gives us the waveform for V_L(t). §.§ Ideal Probe with Power Line The proximity of the capacitive probe to the charging power line, or any other charged conductor will decrease the effective capacitance of the sphere. Consider Figure <ref> with the capacitive transducer but now attached between the device and the ideal voltage source is a power line. The center of the conducting sphere is at a distance of r_s from the center of the power line. Assume a switch connecting the probe to the voltage source is switched on at some time t_0 At t ≤ t_0 the cable is charged at V_L but disconnected from the probe. Given that the line is at V_L there is a non-zero electric potential at various points in the system, V(r, ϕ, z). Following the infinite length power line assumption, the voltage profile will have rotational symmetry as well as uniformity along the cable. Therefore, only V(r) needs to be considered. At the moment t > t_0 only Q = C (V_L - V(r_s) ) amount of electrons need to accumulate on the conductor surface in order to bring the device to line voltage V_L. The voltage at surface of the conductor due to the accumulated charge is V_L - V(r_s) while the contribution from the power line is V(r_s) leading to both the power line and the device being charged to V_L. This defines the effective capacitance C_p≜ Q/V_L which holds regardless of power line proximity. Since Q = C (V_L - V(r_s) ), the probe capacitance is now: C_p = C (1 - α(r)) where α(r_s) = V(r_s)/V_L is invariant to the actual voltage level and is computed from the system geometry. Therefore, introducing the power line and voltage source only reduces the capacitance seen by the ideal charge sensor. §.§ Ideal Probe with Coupling Interference Source Coupled measurements between nearby conductors is an important effect that must be considered. In a capacitive line sensor, the predominant form of interference is from crosstalk between the various lines. This work considers only a single interferer for now, but the results can be extended to multiple interfering power lines. Figure <ref>, shows the ideal body capacitor connected to the high voltage power line. This three conductor system, contains (1) the body capacitive probe v_1, (2) earth environment which is grounded and (3) the interferer v_2. In a multi conductor system, a capacitance matrix describes the electrostatic geometry <cit.>, <cit.>. Given the arrangement in Figure <ref> the full electrostatic environment is described by: [ [ q_1; q_2 ]] = [ c_11 + c_12 -c_12; - c_21 c_22 + c_12 ][ [ v_1; v_2 ]] Consider only the charge accumulation on probe q_1 since q_2 is of no interest. Recall the effective capacitance C_p is due to both the ground and the external environment. The cross term c_12 is the interference term C_I. This leads to v_1(t) = (c_11 + c_12) v_L(t) - c_12 v_2(t) = C_p v_L(t) - C_I v_I(t). § SENSING METHODOLOGY This section incorporates the physical model in Section <ref> to develop a circuit representation of a passive LMCVT device. Then the active calibration is introduced and it's practically implemented is shown. §.§ Circuit Model of Body Capacitive Sensor The circuit equivalent of the body capacitive probe with both proximity effect and interference is shown in Figure <ref>. The ideal charge sensor used in Section <ref> which measures the charge induced can be implemented in practice by an op-amp with feedback capacitor C_s. Assume the line voltage is v_L(t) = V_L cos(ω t + ϕ) and interference capacitance can be ignored, so C_I = 0 Calculate the output of op-amp (and high end of the differential ADC), V^ADC+(jω). Since it is a non-inverting operational amplifier with feedback amplifier Z_F and input impedance Z_IN the output voltage is V^ADC+(jω) = (1 + Z_F/ Z_IN) V^+(jω) = (1 + C_p/C_s) V_L(jω). Here, assume that the operational amplifier is ideal. In practice a low M Ω resistor is put in parallel with the capacitor to maintaining the leakage current of the device. The low end of the differential ADC is V^ADC-(jω) = V_L(jω). So the differential voltage measured at the input of the analog input is: V(jω) = V^ADC+(jω) - V^ADC-(jω) = (C_p/C_s) V_L(jω) The addition of an interference source can be done via superposition principle. Shorting the AC voltage source and the body capacitor, leads to a negative feedback amplifier which results in the final form: V(jω) = C_p/C_s V_L(jω) - C_I/C_s V_I(jω) or in time domain: v(t) = C_p/C_s v_L(t) - C_I/C_s v_I(t) §.§ Passively Calibrated LMCVT Given this circuit model, a passive LMCVT merely samples the signal in (<ref>). The digital signal v[n] is then used to estimate the probe capacitance and the line voltage. With offline calibration methods, Ĉ_p is assumed fixed and known. The line magnitude is then recovered via: V̂_L = (C_s/Ĉ_p) V̂, where V̂ is the magnitude of the digital waveform v[n]. Passive calibration techniques in <cit.> and <cit.>, propose using modeling assumptions on Ĉ_p and v(t) as well as long time captures to estimate Ĉ_p and finally V̂_L as before. §.§ Actively Calibrated LMCVT Active calibration is an alternative method where one or more out of band pilot signals close to 60 Hz be inserted between the line voltage and the charge sensing stage. This can be performed with a general DSP platform, as in Figure <ref>. In this situation, the device is energized at voltage level v_L(t), and the DAC output of pilot signal is v_P(t). The non-inverting terminal voltage in an active calibration system is v^+(t) = v_L(t) + v_P(t) vs. v^+(t) = v_L(t) in the passive case. From (<ref>), omitting the interfering power lines, the input of the differential measurement is v(t) = C_p/C_s(v_L(t) + v_P(t) ). Since the line voltage and the pilot signal are at different frequencies, the received line voltage signal (C_p/C_s) v_L(t) can be filtered, leaving only the term (C_p/C_s) v_P(t). The pilot signal is a known quantity, therefore, it is possible to use the known signal to recover C_p. The practical implementation can be performed in an onboard DSP platform. Active calibration eliminates the need of performing offline calibration for 'typical arrangements' of capacitive probes on transmission and distribution systems. In theory, a small out of band pilot signal can estimate the probe capacitance at very high voltages. Consider a 300 KV high voltage line, and an injected pilot signal of 10 V. Given the typical body capacitance of C_p = 20 pF, if the maximum desired the input magnitude is ± 5 V, then the feedback capacitor must be 600 nF. Also, in this situation, the amplitude of the pilot in the ADC is 167 μ V. This may be lower than the noise floor, but since the geometry in high voltage lines changes so infrequently averaging periods can be rather long. In comparison, for a distribution line with nominal voltage of 10 kV, the minimum pilot voltage is 5 mV which will be close to but higher than the noise floor. §.§ Active Calibration and Line-Mounted Voltage Sensing LMCVT technology is the only line mounted measurement technique that lends itself to active calibration. To understand why, consider both the electrostatic field technique and capacitive coupling method shown in Figure <ref>, <ref>. In the case of electrostatic field measurement, the Point of Measurement (P.O.M) is a point inside the device. The measured value E = f(V_L) depends on the physical configuration of the conductor and the remaining environment. Normally, some offline procedure is used to calibrate an inverse mapping V̂_L = f^-1(E), where f^-1(·) is fixed. An active calibration must have a voltage perturbation on the entire conductor, to measure a perturbed output f(V_p) since the field depends on the changed conductor interacting with the environment. In Figure <ref>, the perturbation voltage must be placed on the entire conductor. Alternatively, for capacitive coupling, the measured Q = g(V_L) depends on physical configuration of the capacitive material and the remaining environment. In this case, active calibration needs to only to inject a voltage perturbation onto the conductive material to measure a perturbed output g(V_p) since the accumulated charge is caused by the conductive materials interaction with the environment. This arrangement is feasible in a practical line mounted circuit. In Figure <ref>, the perturbation voltage can be placed between the conductor and the floating capacitor. § DEVICE PROTOTYPE Figure <ref> shows a line mountable prototype for evaluating the actively calibrated LMCVT technology. The prototype includes a number of commercial off the shelf (COTS) and custom hardware components (shown in Figure <ref>): * Analog front end for pilot injection and signal recovery. * NI-9223 ADC and NI cDAQ-9191 will provide a 100 KS/s, 12 bit ADC and wireless streaming. * High density LiPo Batteries, which allow up to 5 Hrs of continuous capture time. The Analog front end board consists of the following components (shown in Figure <ref>): * PCB mountable GPS with Pulse Per Second (PPS) output; * Low bias current op-amp and feedback capacitor for outputting the appropriate scaled down voltage signal; * Frequency Devices SPPOSC-02 Series Programmable Oscillator <cit.> which can be programmed to output pure and broadband pilot signals; * Bluetooth enabled microcontroller to control programmable oscillator; The device currently under development is shown in Figure <ref> and the analog front end in Figure <ref>. The modular design allows testing of the technology and upgrades of each component. The signal generator <cit.> is capable of generating frequencies from 400 Hz to 100 kHz from an internal DSP. § ACTIVE CALIBRATION: ALGORITHMS This section provides a high level architecture view of the tasks required for active calibration and detailed descriptions of each task block. §.§ Signal Processing Architecture Figure <ref> illustrates the general signal processing workflow devised to extract the line voltage from the received signal v[n]. For clarity, Appendix Table <ref> shows the different signals used in the architecture diagram. 1 Line and Pilot Signal Estimation: This step estimates α[k], ϕ[k] the line magnitude/phase and β[k] β_M[k], the pilot magnitude in a single cycle basis. Signal α[k] is used to recover line voltage, and β_1[k] β_M[k] is used to recover the probe capacitance. 2 Frequency Selector: Certain frequency bands can be periodically corrupted by interfering line sources. To ensure clear frequency access, the recovered signal must be deemed reliable, if not a reliable pilot frequency must be determined. This block controls the M pilot frequencies and magnitudes: f_p, m, V_p,m. 3 Probe Capacitance Estimator: Signals β_1[k] β_M[k] are used to reconstruct the probe to ground capacitance C_p. Ideally environmental changes are infrequent (on order of minutes or hours) so very accurate estimates can be made. 4 Environmental Change Detector: The recovered pilot signal β[k] is used to detect changes to the environment so that a portion of the system output is discarded. After disturbances, the previous estimate data are discarded. In the following sections we elaborate each of the subcomponents. Further work is needed to fully design optimal algorithms for these subcomponents. §.§ Sinusoid Estimation A Non Linear Least Squares (NLLS) Estimation procedure is applied to both v_L[n] and v_B[n]. This reduces to a standard least squares signal estimator for the signal amplitude and phase, <cit.> and <cit.>. The method is applied on a vector of length N, which is set by the sample rate and expected device output rate. The nominal device output rate is f_frame = 60 S/s, which corresponds to a single cycle frame length. The general least squares estimation problem is {θ̂, ω̂}= θ, ωmin∑^N_n ( f_θ, ω[n] - v[n] )^2. Given the sampled waveform v[n] and the fitting function f_θ, ω[n]. The fit function, f_θ, ω[n], is a sum of L sinusoids of various unknown frequency (ω), amplitude (α_l) and phase (θ), given by f_θ, ω[n] = ∑^L_l=1α_l sin ( w_l n + ϕ_l ). Assuming that the number of sinusoids can be determined from prior knowledge or pre-processing step (FFT analysis), the problem can be split into two separate subproblems: (1) ω known/θ unknown (2) ω, θ unknown. Both can be solved by the same computational steps: if ω is known, θ̂ can be calculated by the following least square analysis. Note the fit function can be rewritten as f_θ, ω[n] = ∑^L_l=1α_l cos(ϕ_l)_γ_lsin( w_l n ) + α_l sin(ϕ_l)_η_lsin( w_l n ) , where θ = [γ_1, η_1, , γ_L, η_L]^T is unknown. The inference problem in (<ref>) is non-linear in α_l and ϕ_l, but linear in γ_l, η_l, where γ_l = α_l cos(ϕ_l) and η_l = α_l sin(ϕ_l). The matrix F(ω, θ) = [ f_θ, ω[0], , f_θ, ω[N] ]^T is constructed, which is equivalent to F(ω, θ) = M(ω) θ, with the matrix M(ω) = [ [ cos(w_1 0) sin(w_1 0) cos(w_L 0) sin(w_L 0); ⋮ ⋮ ⋮ ⋮; cos(w_1 N) sin(w_1 N) cos(w_L N) sin(w_L N) ]] and 𝐯 = [ v[0], , v[N] ]^T. Assuming fixed frequency, the amplitude/phase component can be determined via linear least squares analysis: θ̂(ω) = θmin𝐯 - M(ω) θ^2. The least square solution is the following: θ̂(ω) = (M(ω)^TM(ω))^-1M(ω)^Tv. Finally given the γ_l, η_l values, the sinusoid amplitude and phase compute by: Amplitude_l = √(γ_l^2 + η_l^2 ) Phase_l = tan^ - 1( γ_l / η_l ). This formulation allows us to separate the estimation of the multiple sinusoid parameters from the estimation of the frequency of the line/pilot/harmonics. Since the minimization in (<ref>) depends on fixed pilot/harmonic frequencies ω, a second outer minimization can be performed to track each frequency. In <cit.>, the authors present a number of techniques to quickly estimate the frequency of various harmonics. This level of computation is required for tracking the pilot signal when it is surrounded by multiple harmonics. This has practical reason also due to ADC clock drift: actively tracking frequency improves the estimate output. §.§ Capacitance Estimation Assuming non-interfering pilot frequency, and no-environmental changes, the capacitance of the system is fixed in moderate timescale (minutes-hours) Depending on whether single or multiple frequencies are used in the estimation, different estimation procedures can be deployed. Here only a single pilot mechanism is discussd. Assume a constant C_p(t) = C_p over a short time horizon. In single pilot mode, the injected pilot signal is v_P(t) = V_P cos( 2 π f_1 t + ϕ_1 ). From (<ref>), the recovered signal can be represented with the following linear equation: β[k] = V_P/C_sC_p + w[k] for processing interval k = {1, , K }. Here, w[k] is additive noise with noise bandwidth coming from the bandpass filter in Figure <ref>. Over a long timeframe, where the probe capacitance is assumed fixed, and no environmental changes are detected. The least square estimate of the probe capacitance is then: Ĉ_P = C_s/V_P( 1/K∑_0^Kβ[k] ) . A similar least square method can be devised for multi-frequency pilot scheme, but is not explored here. §.§ Line Voltage Estimation Finally given the estimate of the probe capacitance, Ĉ_P[k], the line voltage can be estimated from V̂_L[k] = C_s/Ĉ_P[k]α[k]. Assuming that Ĉ_p[k] = Ĉ_p, ∀ k. The estimation interval of α[k] is set by the measurement output rate of the device. § ACTIVE CALIBRATION: EXPERIMENTS §.§ Experiment Setup The wall outlet voltage is connected to a variable transformer (variac) then to a step up transformer so that various test voltages can be generated (see appendix for further details). The maximum achievable voltage under this technique is 1.28 kV. This signal has a significant amount harmonic distortion. However, achieving high accuracy under this condition lends confidence of the pilot mechanism working in an actual distribution and transmission lines where out of band noise is common. In the following sections, tests at various voltages are performed, indicated as {T_0, T_1, T_9, T_10}. This corresponds to variac positions of { 0%, 10,  90 %, 100 %} of the maximum voltage output. The multimeter ground truth values are V_L, t, for t = {0, , 10 }, and given in Table <ref>, Column 1. §.§ Received Signal Figure <ref>, shows the typical voltage waveform v[n] in sample and frequency domain. It is clear that the injected pilot signal is observed in v[n], as shown in Figure <ref>. However, looking in the frequency domain, Figure <ref>, the signal is close in magnitude with respect to the various harmonics of the mains voltage. These harmonics can be caused by (1) harmonic distortion from the ADC (2) harmonics of the 60 Hz signal that is normally present in the system. Figure <ref> indicates that care must be taken in extracting the pilot amplitude. §.§ Line and Pilot Signal Estimation The captured waveform v[n] is low pass filtered to remove any high frequency components. The NLLS procedure is applied on v_L[n] under various test voltages. Figure <ref> shows the recovered amplitude time series in a single cycle capture window. In the case of tracking the line frequency, since the line magnitude is orders of magnitude larger than any harmonics, v[n] can be low pass filtered and tracked with a single sinusoid M=1 model. Note that the pilot magnitude, β[k] is estimated identically as α[k], using (<ref>). Figure <ref> show box plots of the ground truth (GT), controlled (C) and free space (NC) relative voltage magnitude captures. The experiments indicate that the received voltage magnitude are comparable in variation. This is very important since it indicated that under nominal conditions of fixed C_p, the variation between a traditional grounded instrument (GT), fixed floating capacitor (C), and open air float in capacitor (NC) are nearly identical. As shown in section <ref>, the largest source of error is the estimation error of the probe capacitance, and not the variation of the line signal. This makes sense since, the maximum variation of the line voltage α[k] is well within the ± 1% limits for full capture variation. §.§ Capacitance Estimation Figure <ref> shows the estimate of Ĉ_p under each experiment. Although the signals look very close to each other, an upward bias is evident (blue). A full scale estimate of the probe capacitance, C^⋆_p, is shown as well (horizontal-green). α̅_̅t̅ = κ V_L, t + e_t, C^⋆_p = C_s κ̂ This is estimated as from the experimental data by solving the following regression equation solving for κ in (<ref>). Then (<ref>) calculates the true capacitance. Here, V_L, t is the mean voltage recorded on the digital multimeter ground truth and α̅_̅t̅ = 1/K∑_k α_t[k]. Each variac test is an independent observation of (<ref>). The error of a single cycle (K=1) estimate, using (<ref>), is the following: σ(Ĉ_p) = ( C_s/V_P) σ_W. For the 1.2 kV test environment, C_s = 10 nF and V_P = 10 V, however additive variance is extremely small. To see why, the total additive noise on the signal is 3σ = 2 mV and since the signal is bandpass filtered, only a small portion of that initial noise will remain. Typically, σ(Ĉ_p) /Ĉ_p≈ 3.8 × 10^-4. So, the variation of the single cycle case, under moderate averaging intervals is fairly low. A more important source of error is bias in the estimate, as discussed in Section <ref>, leads to most of the error. §.§ Line Voltage Estimation Results Table <ref>, shows the performance attained in the experiment where the the line voltage magnitude is estimated via active calibration. Table <ref>, column 1, indicates the ground truth voltage which is measured by a digital multimeter measurement of the line signal. Columns 2 and 3 indicate the results of the pilot based estimation. The output of the device is a single cycle estimate of the line voltage V_L[k]. From this the single cycle relative errors: e[k] = V_L - V_L[k]/V_L for each test, can be computed. Therefore, following terms are reported on a single cycle basis: 1 error mean μ(e) = 1/K∑_k e[k], 2 standard deviation σ(e) = √(1/K∑_k (e[k] - μ(e))^2 ) 3 range: r(e) = ( min_k e[k], max_k e[k] ) The results indicate that the current prototype is able to reach metering quality, in half of the test cases. The mean relative error over all the tests is 0.72 %. This is because of the overall overestimate of the probe capacitance in each test as shown in Figure <ref> indicating a positive bias. However, this may be removed by a multi-frequency pilot mechanism or higher frequency pilots where there is less harmonic distortion on the signal. Columns 3 and 4 indicate the voltage estimation results when the full scale capacitance C^⋆_p is known. Clearly the mean bias over all tests is reduced so that it is close to 0 %. In the experiments however, it is difficult to calibrate one device over another due to lack of high resolution multimeters. Both digital multimeters and the LMCVT may very well have offsets on the order of 1%. This however, does not take away from the fact that in both cases, the maximum variation over all tests is quite small. In both cases, the mean error is less than 1 % in the higher voltage cases, where the effect of external interferers is minimal. These results can be compare to the passive calibration method published in <cit.>. This comparison is not exact since in <cit.>, 5 minute averages are used and the test voltage levels ranging from 5 kV to 25 kV. Following Row 2 of Table IV (HV_2) in <cit.>, the authors report μ(e) = 1.72%, r(e) = (0.1, 14.8)%. In comparison, the results in Table <ref>, Column 3 show single cycle tests where the mean error is μ(e) = 0.71 %. Additionally, the maximum errors are (excluding the lowest voltage level r(e) = (0.03, 1.79). Comparing the variation of errors, <cit.> reports σ(e) = 4.2%, while the active calibration procedure attains σ(e) = 0.05%. Although, the mean error decreases by 60 %, the maximum error decreases by 90 % while the standard deviation decreases by 98 %. It should be noted that this comparison is overly conservative since the reported values in Table <ref> are for single cycle estimates, while <cit.> reports 5 minute averaged values. §.§ Pilot Frequency Selector Since the line magnitude can be orders of magnitude larger than the injected pilot signal, the presence of harmonics or spurious signal a the exact pilot frequency can be a source of error. Care must be taken in deciding what frequency is chosen. An experiment is performed where two non-harmonic pilot frequencies were chosen, 3.2 kHz and 5 kHz, and are used for active calibration. In this case, the 3.2 kHz frequency contained energy from other sources, while the 5 kHz frequency has none. These are referred to as available and occupied pilot frequencies. In the reconstruction of the pilot frequency, there is a very clear difference between the two signals shown in Figure <ref>. In the experiment the least square estimate in (<ref>), is solved every 1/60 seconds. Using a sample rate of 55 kHz, this leads to ∼ 917 samples for estimation, leading to high accuracy in the NLLS procedure. Observing the two situations indicate a clear difference in each signal type. A clear channel frequency leads to constant receive pilot amplitude, assuming that the environment is fixed. In this case, the error is of very low variance, uncorrelated and gaussian. On the other hand, when the pilot frequency is occupied, the received amplitude is highly correlated with a variance an order of magnitude larger than in the clear channel case. There is likely to be confusion between interfering signals at a chosen pilot, and actual environmental changes that can lead to error if a single pilot signal in a fixed frequency is used. If multiple frequencies, or multiple pilots are used, this error can be averted. §.§ Capacitance Change Detection If the pilot frequency is clear of any interference sources, β[k] can be used to detect changes in the environment. At low enough signal frequencies it can be assumed that all frequencies will see the same C_P(t). Any environmental disturbance affecting the 60 Hz line can be distinguished from actual voltage changes since it will be seen on a clear pilot frequency. An experiment illustrating this was performed (see Appendix for image). A metallic pendulum was built and used to bring a grounded surface repeatedly close to the shell of the device, as what would happen in some fast changing environmental change on the line. A line voltage of 1.2 kV was applied on the main line, with a pilot signal of 10 V at 5 kHz. Again, a 1/60 second capture window was chosen for processing. Although this can be varied, it's implausible that changes in the physical environment will occur at a very high rate. Figure <ref> shows clearly that disturbances on both the 60 Hz line and 5 kHz pilot signal can be detected due to the sensitivity of the pilot signal. The frequency of the pendulum given by 1/2 π√(g/l)≈ 1.3 Hz matches closely to the frequency of detected oscillation. The output of the detector will be a bad data quality flag indicating accuracy compromise of the system. Since environmental disturbances do no constantly occur on a stable transmission lines, the impact of measurement continuity is likely to be minor. The variance of the pilot estimate in a fixed environment will be crucial to the performance of any change point detector. There is a tradeoff in computational resources dedicated to more refined pilot tracking; for example, (1) tracking and removing various harmonics, adaptively tracking and estimating their frequencies (3) length and bandwidth of bandpass filters. It is clear that the change detection problem and the capacitance tracking problem can be solved separately. However, provided better modeling of how probe capacitances behave in the real world, further joint optimization is possible but not explored here. § FUTURE WORK AND OPEN PROBLEMS In order to further improve the accuracy of such technology and enable full deployment, the following future areas must be explored: * Interference Decoupling: In the case of multi-conductor systems, there will be multiple coupled sources at similar voltage levels as well as measurements at each source. Work is needed to design methods and algorithms where multiple phasor measurements are used to decouple each voltage source in this situation. * Bias Correction via multi-Frequency Pilot Mechanism: As shown in Section <ref>, the offset in received pilot magnitude leads to an offset in the recovered probe capacitance as well as voltage estimate. A multi-frequency pilot can potentially remove this bias and improve the estimated voltages, and should be explored. * Adaptive Frequency Selection: Work on efficient frequency probing/selection algorithms must be done to ensure that the pilot frequencies used can be used for probe estimation. Figure <ref>, illustrates that simple time series testing can detect such changes properly. * Environmental Change Detection: Efficiently detecting environmental changes from the measured pilot signals to guarantee that any change in α[k] is in fact from line voltage change. § CONCLUSIONS AND FUTURE WORK This work presents a method of achieving high accuracy line mountable capacitive voltage transducers for high voltage applications. The method relies on computing the probe-to-ground capacitance in real time using out of band pilot signal injection. The concept is tested in experiments and a roadmap of a functional prototype is given. § TESTING ENVIRONMENT §.§ Low Voltage Test Environment The test environment we use is a metallic mesh cage connected to ground. Low voltage tests are performed are susceptible to 60 Hz interference at 120/240 V which is an order of magnitude larger than the line voltage. Therefore a metallic cage is useful in making an interference free environment. Consider prototyping a system with V_L = 10 V. Unfortunately, all nearby interference sources are scattered randomly in a real laboratory environment with V_I = 120 V. §.§ High Voltage Test Environment Figure <ref> shows the test setup used to generate a high voltage AC source. A variac connected to a step up transformer can generate a moderated AC signal for full scale voltage testing. § EXPERIMENTAL EVALUATION PHYSICAL MODEL OF BODY CAPACITIVE VOLTAGE SENSOR §.§ Verifying Probe Capacitance in Low Voltage Testing Here we demonstrate the basic body capacitive effect. First given a set of sheet metal plates of various sizes, we compute the theoretical capacitance of the probe with no charged device in proximity of the of the charged plate. This is done in COMSOL Multiphysics simulation environment using the electrostatic simulation package. The test arrangement of this is shown in Figure <ref>. To experimentally evaluate the charged plate with an ideal voltage source with no proximity effect reducing the effective capacitance we perform the following. * Place the voltage source (NI Hardware) and charge sensing device (metallic box) outside the grounded cage. * Connect the plate to a shielded cable which becomes unshielded at the entrance of the cage * Compute the total capacitance of the cable and plate body capacitance. This is done by measuring the rms input/output voltage (Figure <ref>). This leads to an experimental value of C_P. * Repeat step 3 with the cable alone. The difference in capacitances being that of the plate alone. Table <ref> shows that for larger plates, the simulated value matches very closely to the electrostatic simulation. However the simulation and experiment diverge for smaller plates. Regardless, the experiments validate the basic premise of the body capacitor model. Figure <ref> shows a linear relation between V_L and V_CAP for each plate, validating the model. §.§ Effective Capacitance We show a method of verifying the proposed model for the effect of power line proximity to the body capacitance of a probe. Figure <ref> shows a test setup inside the grounded cage. Like the test in Section <ref> the instrumentation is placed outside of the cage to prevent any unintentional charge to act on the probe. However, there is an external field from the copper power line in the experiment. The probes are copper wires set on at fixed distances from the power line. This is done to minimize measurement error in evaluating various distances. A similar procedure is conducted as in Section <ref> where the capacitance of the cable and the probe is computed first; then the capacitance of the cable alone. The difference of the two being the experimental effective capacitance under a charged (C_p) and uncharged (C_P) power line. An experimental value of α(r) = 1 - C_p/C_P is computed, where C_p and C_P are computed at each distance. A theoretical α(r_k) can be computed easily without computing the individual C_p or C_P since α(r) = V(r)/V_L. Given the simulation tool, we can evaluate V(r) from the geometry as shown in Figure <ref>. For simplicity we evaluate the CFD solution for the point voltage along the same position of the cable probe. An average of the solution vector is then V(r). Figure <ref> shows the results of measuring the quantity α(r) at the various locations of capacitive probe. The results show considerable agreement between the two models thus leading us to accept the model. The proximity effect is important to consider in the design of a final body capacitive probe. Notice that in location 4 which is only 4.3 cm from the line, α(r) is 0.8. Any probe that is closer to the power line will have an effective capacitance that is much smaller than in the uncharged case. Therefore, there is a minimum separation we need between the probe and the power line so as to have a value of C_p that works in practice. § PROBE CAPACITANCE DISTURBANCE EXPERIMENT SETUP Figure <ref> shows the pendulum test setup to generate a periodic disturbance that is easily detectable by the pilot signaling mechanism. unsrt

http://arxiv.org/abs/1701.08007v2

20170127104713

Role of the intraspecies scattering length in the Efimov scenario with large mass difference

cond-mat.quant-gas

Physikalisches Institut, Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany Present address: American Physical Society, 1 Research Road, Ridge, New York 11961, USA. Department of Physics, Kansas State University, 116 Cardwell Hall, Manhattan, Kansas 66506, USA chgreene@purdue.edu Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana, 47907-2036, USA weidemueller@uni-heidelberg.de Physikalisches Institut, Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany Hefei National Laboratory for Physical Sciences at the Microscale and Department of Modern Physics, and CAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, Anhui 230026, China We experimentally and theoretically study the effect of the intraspecies scattering length onto the heteronuclear Efimov scenario, following up on our earlier observation of Efimov resonances in an ultracold Cs-Li mixture for negative [Pires et al., Phys. Rev. Lett. 112, 250404 (2014)] and positive Cs-Cs scattering length [Ulmanis et al., Phys. Rev. Lett. 117, 153201 (2016)]. Three theoretical models of increasing complexity are employed to quantify its influence on the scaling factor and the three-body parameter: a simple Born-Oppenheimer picture, a zero-range theory, and a spinless van der Waals model. These models are compared to Efimov resonances observed in an ultracold mixture of bosonic ^133Cs and fermionic ^6Li atoms close to two Cs-Li Feshbach resonances located at 843 G and 889 G, characterized by different sign and magnitude of the Cs-Cs interaction. By changing the sign and magnitude of the intraspecies scattering length different scaling behaviors of the three-body loss rate are identified, in qualitative agreement with theoretical predictions. The three-body loss rate is strongly influenced by the intraspecies scattering length. 34.50.Cx, 67.85.Pq, 31.15.ac, 21.45.-v Role of the intraspecies scattering length in the Efimov scenario with large mass difference Matthias Weidemüller December 30, 2023 ============================================================================================ § INTRODUCTION The Efimov scenario <cit.> addressing universal properties of three particles interacting via resonant forces has become one of the cornerstones of modern few-body quantum physics <cit.>. The hallmark of this bizarre effect is the manifestation of an infinite geometrical progression of three-body bound states, the Efimov states, that follow a discrete scaling law. Such series have been observed in experiments with homonuclear Bose <cit.>, three-component Fermi <cit.>, and heteronuclear Bose-Fermi <cit.> systems (see Ref. <cit.> for a recent review), to a good extent confirming one of the long-standing predictions: the universal law a_-^(n) =λ a_-^(n-1), where a_-^(n) is the position of the nth Efimov resonance that is described by the s-wave scattering length a, and λ is the universal scaling factor that depends only on quantum statistics, number of resonant interactions, and mass ratios of the three atoms <cit.>. The excited helium trimer ^4He_3, as observed in a molecular beam experiment, has recently been found to accurately obey the predictions for a universal Efimov trimer <cit.>. The universality in the Efimov scenario manifests itself at asymptotically large scattering lengths and thermal wavelengths. If the specific details of the short-range interactions are not resolved, their effect can be incorporated in a single quantity, the three-body parameter (3BP). In ultracold homonuclear systems that interact through pairwise -C_6/r^6 potentials the 3BP is connected to the molecular van der Waals (vdW) tails by an approximate, species-independent constant <cit.>. Apart from slight modifications due to short-range effects <cit.>, this general scaling constitutes an example of the so-called vdW universality <cit.> and can be seen as a precursor to a larger class of universal low-energy three-body observables that are governed by mutual power-law interactions at small particle separations <cit.>. Although the Efimov effect in ultracold heteronuclear gases is based on the same physical principles as in homonuclear systems, its signatures are much richer. The inclusion of an additional short-range length scale and interaction complicate the simple picture. For a mass imbalanced system consisting of two heavy bosonic atoms that resonantly interact with one lighter atom, the 3BP is predicted to depend not only on the pairwise vdW tails, but also on the mass ratio and the intraspecies scattering length between the two heavy atoms <cit.>. Furthermore, owing to denser trimer spectra <cit.> and a comparatively small |a_-^(0)|, Efimov states are found in the transition regime between the long- and short-range dominated potential parts <cit.>. Thus the spectra may contain contributions from both regimes. Additionally, in a real ultracold atomic system, the inter- and intraspecies interactions are generally not controlled independently and therefore deviations from the exact log-periodic scaling behavior are expected, reflecting in a period dependent scaling factor λ^(n) <cit.>. The experimental exploration of such scenarios, however, so far has been limited. To date, mainly isolated Efimov resonances were observed, as for K-Rb <cit.> and ^7Li-^87Rb <cit.> mixtures. In the ^6Li-^133Cs system successive Efimov resonances have been observed for attractive <cit.> and repulsive <cit.> CsCs interactions. In this paper we juxtapose the heteronuclear Efimov scenario for two heavy bosons B and one distinguishable particle X for positive and negative intraspecies scattering lengths. We discuss three theoretical models at different levels of complexity that can be used to solve the three-body Schrödinger equation, and show that the boson-boson scattering length critically modifies the three-body energy spectrum. The qualitative influence of the boson-boson scattering length and the finite-range effects is already revealed with a minimalistic hybrid Born-Oppenheimer (BO) approximation, treating the BX interaction as contact like and the BB interaction as hardcore van der Waals potential. A more quantitative approach is given by the universal zero-range theory in the hyperspherical adiabatic approximation <cit.>. This description also includes the universal BB dimer state for positive intraspecies interactions, which leads to a splitting of the traditional Efimov scenario into two Efimov branches. Efimov states with energy smaller than the dimer energy at the intraspecies unitarity do not connect to the three-body threshold and thus can lead to the absence of Efimov resonances in three-body recombination spectra. As the third model, we use the spinless van der Waals (vdW) theory, which models pairwise interactions with single-channel Lennard-Jones potentials <cit.>. The inclusion of realistic finite-range potentials allows one to quantitatively compare the theoretical results with the experimentally determined three-body loss spectra of Cs+Cs+Li recombination, and extract Efimov resonance positions. Finally, by comparing experimental three-body loss rates close to two different Cs-Li Feshbach resonances (FRs) we find two distinct scaling behaviors, in agreement with previous predictions <cit.>. Such distinctive scaling properties can be used, for example, to tune and significantly increase the three-body lifetime due to Cs-Cs-Li collisions, which is an important step towards studies of strongly interacting Bose-Fermi mixtures. This paper is structured as follows: In Sec. <ref> we give an overview of the theoretical models to explain the measurements of three-body recombination spectra in a heavy-heavy-light system. The experimental procedure for the investigation of three-body recombination in the Cs-Cs-Li system is given in Sec. <ref>. A comparison between experiment and theory for negative and positive intraspecies scattering length is given in Secs. <ref> and <ref>, respectively. The scaling behavior of three-body recombination near overlapping Feshbach resonances is analyzed in Sec. <ref>. § THEORETICAL MODELS The three-body wave function for two identical bosons B with mass m_B and one distinguishable particle X with mass m_X is determined by the three-body Schrödinger equation: [-ħ^2/m_B∇^2_𝐫- ħ^22m_B+m_X/4m_Bm_X∇^2_ρ+V_BB(r)+V_BX(|ρ+𝐫/2|)+V_BX(|ρ-𝐫/2|)]Ψ(ρ,𝐫)=EΨ(ρ,𝐫) , where ħ is the reduced Planck's constant, 𝐫 the vector between the two bosons B, and ρ the vector from the center of mass of atoms B to the X atom. V_BB/BX denote the intra- and interspecies interaction potentials, respectively. The corresponding scattering lengths are labeled a_BB and a_BX. Here, we employ three methods in order to solve the Schrödinger equation. §.§ Born-Oppenheimer approximation In order to get an intuitive understanding of the influence of the intraspecies scattering length and finite-range effects onto the heteronuclear Efimov effect we solve the three-body Schrödinger equation [Eq. (<ref>)] within the Born-Oppenheimer (BO) approximation for a large mass imbalance m_X/m_B ≪ 1 <cit.>. It is assumed that the motion of the light particle adapts almost immediately to the distance between the two heavy particles 𝐫 and therefore the total wave function Ψ(ρ,𝐫) is considered to be separable, Ψ(ρ,𝐫)=ψ(ρ;𝐫) ϕ(𝐫), where ϕ(𝐫) is the wave function of the two heavy particles and ψ(ρ;𝐫) the wave function of the light particle, which parametrically depends on 𝐫. Within the BO approximation the Schrödinger equation can be separated into two coupled equations. By solving the equation for the light particle the BO potential is obtained and the three-body problem is reduced to an effective two-body problem. The energy spectrum can be found by solving the equation for the heavy particles, [-ħ^2/m_B∇^2_𝐫 +V_BB(r)+E_𝐫] ϕ(𝐫)= E ϕ(𝐫), where E_𝐫 is the BO potential, which is well known to be E_𝐫=-ħ^2 W(1)^2/2 m_X r^2 for r≪ |a_BX| and contact interaction between B and X. Here W(1) is the Lambert-W function which is connected to the Efimov scaling factor s_0^2=m_B W(1)^2/2m_X -1/4. We model the intraspecies interaction potential with a hard-core van der Waals potential, V_BB(r)= ∞, r<r_0, -C_6,BB/r^6, r>r_0, where C_6,BB is the dispersion coefficient which depends on the details of the electronic configurations. This allows one to introduce characteristic length scale r_vdW and energy scale E_vdW <cit.>. The short-range cutoff r_0 is analytically connected to the scattering length a_BB <cit.>. We solve Eq. (<ref>) numerically for the case of a_BX→∞ and r> r_0 and find the wave functions and energies of the trimer states. We apply the BO approximation to the case of the Cs-Cs-Li system with a mass ratio of m_B/m_X=22.1. The Cs-Cs scattering length a_ is changed by tuning the short-range cutoff r_0 of the intraspecies potential <cit.>. The energy of four bound states is shown in Fig. <ref> in dependence of a_. The bound state energies (black lines in Fig. <ref>) follow the discrete scaling law of the Efimov effect, but are strongly influenced by the intraspecies interaction and develop a steplike behavior around |a_|≈ r_vdw^. For comparison we plot the two least bound states of the pure Cs_2 two-body hard-core vdW potential V_BB(r). For a≫ r_vdW^ the binding energy of the least bound state is given by the universal relation E_b∝ -1/a_^2. The binding energy of the second least bound state develops a gradual step at a_≈ r_vdW^, due to its qualitative change from a vdW-dominated into a halo state (red dashed lines in Fig. <ref>). Our BO model suggests a strong dependence of the Efimov state's energy and therefore also of the Efimov resonance positions a_-^(n) at the three-body threshold on the intraspecies scattering length a_, which will be further studied within the universal zero-range theory and the spinless van der Waals theory. §.§ Zero-range theory For a qualitative description of the influence of the intraspecies scattering length between the two heavy bosons on the Efimov scenario, let us employ the universal zero-range theory in the hyperspherical adiabatic approximation, where pairwise contact interactions are assumed as used in <cit.>. We introduce the relevant reduced mass factors: μ _BB=m_B/2, μ _BX=m_X m_B/m_X+m_B, μ = √(m_X m_B^2/2m_B+m_X). Relevant factors that relate to mass ratios are: d_BB=√(m_X(2m_B)/μ (2m_B+m_X)), d_BX=√(m_B(m_X+m_B)/ μ (2m_B+m_X)), and β _BB = arctan[ √(m_X(2m_B+m_X)/m_B^2)], β _BX = arctan[ √((2m_B+m_X)/m_X)] . The following functions are also required as additional preliminaries, before we can state the transcendental equation, that determines the adiabatic hyperspherical potentials, in a compact form: f(s,α )≡√(2/cosα)Γ(3/2)/ sinαP_s-1/2,-1/2(sinα ) , where Γ(z) is the gamma function and P_n,m is the associated Legendre function of the first kind and W(s) = πΓ (2+s/2)Γ (2-s/2)/2Γ (3/2)^2Γ (1+s/2)Γ (1-s/2), X(s) = Γ (2+s/2)Γ (2-s/2)/2Γ (3/2)^2f(s,β _BX), Y(s) = Γ (2+s/2)Γ (2-s/2)/2Γ (3/2)^2f(s,β _BB). Then the roots s(R), which determine the hyperradial potential curves U(R)=ħ ^2/2μs(R)^2-1/4/R^2 are solutions of the following transcendental equation: ( W(s)μ a_BB/μ _BBd_BB^3-R) ( μ a_BX/ μ _BXd_BX^3(W(s)-Y(s))-R) -2X(s)^2μ ^2a_BXa_BB/μ _BXd_BX^3μ _BBd_BB^3=0. Here R is the hyperradius defined by μ R^2=μ_BB r^2+2m_B m_X/2m_B+m_Xρ^2 . Note that in general there could be real solutions s(R) corresponding to hyperspherical potential curve energies higher than -ħ ^2/8μ R^2, which is the critical coefficient to support an infinite number of three-body bound states, and imaginary solutions for s(R) that correspond to potential curve energies below -ħ ^2/8μ R^2. The heavy-heavy-light adiabatic hyperspherical potential curves are shown in Fig. <ref> for the case of positive and negative intraspecies scattering length a_BB. Here, a mass ratio of m_B/m_X=22.1 and different values of a_BX<0 are assumed. If both scattering lengths are negative, the original Efimov scenario with an ∝ -1/R^2 hyperspherical potential is recovered for a_BX→∞, supporting an infinite number of bound states (Fig. <ref>, left panel). For finite interspecies scattering lengths a potential barrier at R≈ 2|a_BX| leads to quasi-bound states, generating three-body recombination resonances, when passing the dissociation threshold. In this case a three-body parameter is indispensable to regularize the energy spectrum. For positive intraspecies scattering lengths (a_BB>0) the situation is considerably altered by the existence of a dimer BB with binding energy E_BB (Fig. <ref>, right panel). The channel BB+X splits the potential curves into two Efimov branches. The lower branch, which is asymptotically connected to the BB+X channel, recovers the original Efimov scenario with a potential that is proportional to -1/R^2. In this case regularization with a three-body parameter is required to avoid a diverging binding energy of the ground state. Note that there is no potential barrier arising for finite values of a_BX. However, the upper Efimov branch with energies E>E_BB shows a different behavior. It exhibits a potential barrier at R≈ 2a_BB, roughly independent of a_BX. Therefore, even for resonant interactions (a_BX→∞), where the ∝ -1/R^2 potential is restored for large interpaticle separations, the energy spectrum is well defined and no regularization with a three-body parameter is necessary. For finite a_BX an effective potential barrier forms at R≈ 2|a_BX|, leading to recombination resonances. §.§ Spinless van der Waals theory Additionally we solve the three-body problem within the spinless vdW theory <cit.>. The approach consists of numerically solving the three-body Schrödinger equation in the full hyperspherical formalism, where the two-body interaction potentials V_BB/BX between equal bosons and the third particle separated by distance r_BB/BX are modeled by single-channel Lennard-Jones potentials with vdW tails. V_BB/BX(r_BB/BX)=-C_6,BB/BX/r_BB/BX^6[1-(r_c,BB/BX/r_BB/BX)^6] , where C_6,BB/BX are the dispersion coefficients. The scattering lengths a_BX and a_BB are reproduced by tuning the short-range cutoffs r_c,BB/BX. We assume that the hyperradial and hyperangular motions are approximately separable <cit.> and treat the hyperradius as adiabatic parameter. The Schrödinger equation is reduced to a set of coupled 1D equations <cit.>. We include the nonadiabatic couplings between the hyperspherical potentials and assume a J^Π=0^+ symmetry [The absence of the Efimov effect in higher partial waves for our mass ratio of m_B/m_X=22.1 <cit.>, leads to a substantial suppression of J>0 contributions to the recombination rate in the ultracold regime with k_B T≪ E_vdW. The suppression is estimated to be (k r_vdW)^2p_0≪ 1 for a_<0 and (k a_)^2p_0≪ 1 for the a_≈+190a_0 case <cit.>, where p_0>1 is a universal constant for the given mass ratio <cit.> and k the thermal wavevector.] (the dominant contribution at ultracold temperatures), where J is the total angular momentum and Π the total parity. This formalism is applied to the Cs-Cs-Li system and allows us to compare the theoretical predictions directly to our experimental observations. The interaction potentials are tuned such that the scattering lengths a_, a_, and their functional dependence a_(a_) are reproduced for the experimentally employed field ranges (see Fig. <ref>). The adiabatic hyperspherical potentials are plotted in Fig. <ref> for intraspecies scattering lengths of a_=-1500a_0 and a_=+200a_0 resembling the two cases of the 843 G and 889 G Cs-Li FRs. The interspecies scattering length a_=-400a_0 is tuned close to the value, where the ground-state Efimov resonance is expected <cit.>. For positive intraspecies interactions an avoided crossing originating from the coupling between the attractive three-body potential and a repulsive atom-dimer channel leads to an effectively repulsive Efimov potential. This prevents the scattering wavefunction from probing short-range parts of the potential and, consequently, recombination. Such an avoided crossing is not present for the case of a_=-1500a_0, where the original Efimov effect is recovered. The adiabatic potentials calculated by the spinless vdW theory closely resemble those by zero-range theory when R, |a_BB|, and |a_BX| are all significantly greater than r_vdW^ and r_vdW^. The universal condition |a_BB|, |a_BX| ≫ r_vdW^, and r_vdW^ is satisfied or approximately satisfied in both cases of the 843G and 889G Feshbach resonances; therefore, the zero-range theory is still helpful for understanding the scaling behavior of the three-body losses observed in our experiment, as discussed above. The quantitative determination of the observables, especially of the 3BP, requires the knowledge of the three-body dynamics when R is smaller than or comparable with the van der Waals lengths, which needs the spinless van der Waals theory to resolve. We calculate the energy-dependent three-body recombination rate from the S matrix <cit.> and perform thermal averaging <cit.>, assuming Boltzmann distributions with the experimentally determined temperatures. The width of the experimentally observed recombination features is reproduced by adding a heuristic hyperradial loss channel that assumes near unity loss at short distances, without modifying the resonance positions <cit.>. The obtained three-body recombination rates are plotted in Figs. <ref>(a) and <ref>(c) together with the energy spectrum of the three energetically lowest Efimov states [see Figs. <ref>(b) and <ref>(d)]. For the case of the 843 G Cs-Li FR, where a_≈ -1500a_0, the classical Efimov scenario is recovered [Fig. <ref>(b)]. By lowering a_ the Efimov states successively disappear by crossing the three-body scattering threshold. For positive intraspecies scattering length a_≈ + 190a_0, as it is the case for the 889 G Cs-Li FR, the Efimov sates split into two branches (Fig. <ref>(d)). While the lowest state predissociates into a Cs_2+Li state (blue dashed line) before reaching the three-body threshold and consequently does not lead to a recombination resonance at the three-body threshold, the higher-lying states recover the original Efimov scenario <cit.>. § COMPARISON WITH EXPERIMENT §.§ Cs-Li Feshbach resonances The Cs-Cs-Li system offers the unique possibility to study the influence of the intraspecies scattering length onto the heteronuclear Efimov scenario. The experimentally adjustable interspecies and intraspecies scattering lengths a_ and a_ are shown in Fig. <ref>. Two intermediately broad (s_res≈ 0.7 <cit.>) Cs-Li Feshbach resonances (FRs), located at approx. 843 G and 889 G, are well suited for the study of the heteronuclear Efimov scenario <cit.>. They are characterized by different sign and magnitude of the Cs-Cs scattering length a_: while close to the 843 G resonance the intraspecies scattering length is large and negative a_≈ -1500a_0, the 889 G resonance is characterized by a small and positive a_≈ +190a_0 <cit.>. By investigating the narrow resonances located at approximately 816 G, 889 G, and 943 G characterized by s_res≤ 0.03 <cit.>, the Cs-Cs-Li system offers the possibility to investigate the influence of the multichannel nature of Feshbach resonances onto the universal Efimov scenario <cit.>, which has been studied recently <cit.>. §.§ Experimental determination of L_3 Our experimental procedure is described in detail in Refs. <cit.>. In brief we prepare an ultracold mixture of fermionic ^6Li atoms in one of the two energetically lowest spin states |f,m_f⟩=|1/2,1/2⟩ or |1/2,-1/2⟩ and bosonic ^133Cs in the absolute ground state |3,3⟩. Here f and m_f refer to the total angular momentum and its projection. By usage of a bichromatic trapping scheme <cit.> we prepare samples of 1× 10^4 (7× 10^3) Cs (Li) atoms at temperatures as low as 120 nK. The spatial overlap of the two atomic clouds is approximately 45% and assumed to be constant within the investigated magnetic field range. The measured trapping frequencies are ω_Cs=2π× (5.7, 115,85) Hz and ω_Li=2π× (25, 160,180) Hz. The three-body recombination rates are measured in dependence of the external magnetic field analogous to our previous work <cit.>. We prepare the atomic mixture approximately 4 G away from the pole of the two broad interspecies Feshbach resonances at 843 G and 889 G. Within 150 ms we increase the dipole trap potential by 10% in order to stop residual plain evaporation and to let the magnetic field stabilize. This leads to a temperature increase on the order of 10%. The final magnetic field value is set by a fast ramp. After a variable hold time, both atomic species are imaged by high-field absorption imaging from which atom numbers and cloud sizes are deduced. The three-body loss coefficient L_3 for the loss of one Li and two Cs atoms is retrieved by numerically fitting the coupled rate equations, Ṅ_Li = -ℒ_1^LiN_Li -ℒ_3 N_LiN_Cs^2 , Ṅ_Cs = -ℒ_1^CsN_Cs - 2ℒ_3 N_LiN_Cs^2 - ℒ_3^CsN_Cs^3 , where ℒ_1^Li and ℒ_1^Cs are the one-body loss rates for each species in the trap and ℒ_3^Cs the Cs three-body loss rate, which are determined in independent single-species measurements under the same experimental conditions. Hence ℒ_3 and the initial atom numbers N_0,Li and N_0,Cs are the only fitting parameters. The three-body loss rate coefficient L_3 is obtained from ℒ_3 via modeled atomic density distributions. The error bars are obtained by bootstrapping and resemble one standard deviation of the resampled distribution. The systematic error for the determination of the absolute value of L_3 is estimated to be a factor of 3 (120 nK data <cit.>) and 0.8 (450 nK data <cit.>), respectively, and are mainly caused by uncertainties in the determination of atom cloud temperatures, densities, overlap, and trapping frequencies. Since no significant increase in temperature during the hold time is observed <cit.>, we neglect recombinational heating <cit.> in our analysis. The magnetic field stability is around 16 mG (one standard deviation) resulting from long-term drifts, residual field curvature along the long axis of the cigar-shaped trap, and calibration uncertainties. Due to intraspecies losses, the extraction of L_3 is limited to scattering lengths of |a_| ≳ 1000a_0 for our data at 120 nK <cit.> and |a_| ≳ 400a_0 for the data at the 889 G FR and temperature of 320 nK. The given loss rates in this region represent an upper bound of the actual L_3. §.§ Negative intraspecies scattering length The observed Cs-Cs-Li three-body recombination rates L_3 versus the scattering length a_ for the broad Cs-Li FR at 843 G at temperatures of 450 nK and 120 nK are shown in Fig. <ref>(a) together with the theoretical recombination rate from the spinless vdW theory. Here the intraspecies scattering length is in the range -1560a_0 ≲ a_≲ -1000a_0. The measured rates of different data sets have been scaled by numerical constants, which are extracted from the universal zero-range theory, but lie well within the experimental uncertainties <cit.>. Three Cs-Cs-Li recombination resonances are evident, while the two excited ones are located in the universal regime [inset of Fig. <ref>(a)]. The spinless vdW theory <cit.> is in excellent agreement with the experimental data. The theoretically calculated loss rate spectrum recovers not only the location of the two excited-state resonances, but also the position of the ground-state resonance. This is in stark contrast to the analysis with the universal zero-range theory <cit.>, where no agreement between theory and experiment was observed for the ground-state resonance <cit.>. Since the essential difference from the zero-range theory is the inclusion of the vdW length scales that determine the short-range behavior of realistic pairwise potentials, we conclude that the previously observed deviation of the CsCsLi ground-state resonance from the universal zero-range theory <cit.> predominantly originates from the vdW interaction. An excellent agreement is also found with the calculated energy spectrum [Fig. <ref>(b)], where the positions at which the three-body states become unbound [grayed areas in Figs. <ref>(a) and <ref>(b)] perfectly align with the maxima of the measured recombination rates. An important question is the extent of the influence of van der Waals forces on the scaling between consecutive Efimov resonances. Therefore, we extract Efimov resonance positions a_-^(n) and scaling factors λ^(n)=a^(n)_-/a^(n-1)_- by three different methods: first, the experimental resonance positions B_expt^(n) are obtained by fitting a Gaussian profile with linear background to the three-body loss rate L_3(B) and conversion to scattering length a^(n)_-,expt via the parametrization given in <cit.>, where the uncertainty consists of statistical, systematic, and conversion errors. Second, the whole L_3(B) spectrum is fitted with the universal zero-range theory <cit.> and the three-body parameter as well as the inelasticity parameter are extracted. The resonance positions a^(n)_-,zr are obtained by setting the temperature and inelasticity parameter in the theory equal to zero <cit.>. By this, finite temperature effects can be eliminated. Third, the calculated trimer energy spectrum from the spinless vdW theory is employed to extract a^(n)_-,vdW as the average value of the two numerical grid points, between which the three-body state merges with the scattering continuum [see grayed areas in Fig. <ref>(b)]. The uncertainty is given by one-half of the step size of the local grid. In this way the influence of finite temperature can be safely neglected, since it modifies the three-body recombination rates, but not the energy spectrum below the scattering continuum. For comparison, the resonance positions a^(n)_- and scaling factors λ^(n) obtained from all three methods are listed in Table <ref>. Remarkably, the experimental and vdW value of a_-^(0) agree very well, while the zero-range theory deviates and predicts a much larger value. This deviation can be attributed to short-range effects and was observed as a deviation between the zero-range theory and the measured recombination rates for the ground-state resonance <cit.>. Therefore, the scaling factor from the spinless vdW model is larger by about 4%, if compared to the zero-range theory with actual scattering lengths <cit.>, and larger by about 7%, if compared to the zero-range theory for noninteracting bosons <cit.>. The experiment gives an even larger value for λ^(1) due to the larger value of a_-,expt^(1) in comparison to the zero-range and vdW models. On the other hand, the vdW and zero-range theory predict a similar scaling factor between the first and second excited Efimov states, highlighting a behavior of the CsCsLi system that is independent of short-range effects. §.§ Positive intraspecies scattering length The measured three-body recombination rate spectra close to the 889 G Cs-Li FR (+180a_0 ≲ a_≲ +360a_0) at temperatures of 320 nK and 120 nK together with calculated recombination rates from the spinless vdW model are shown in Fig. <ref>(c). Here, only two Cs-Cs-Li Efimov resonances are observed, while the first recombination resonance is located at a_≈ -2000a_0. This is about a factor of seven larger than in the case of negative a_ [see Fig. <ref>(a)], which is consistent with the energy spectrum of the Efimov states [see Fig. <ref>(d)], where the most deeply bound state predissociates into a universal atom-dimer state before reaching the three-body continuum <cit.> and hence does not generate a resonance at the scattering threshold. This is consistent with the findings of Sec. <ref>, where the existence of a Feshbach dimer BB at positive intraspecies scattering lengths a_BB>0 leads to a splitting into two Efimov branches asymptotically connecting to the three-atom and atom-dimer channels. Therefore, we assign the first recombination feature to the first excited Efimov resonance. The calculated recombination spectra from the vdW model show qualitative agreement with the experimental observations. However, a shift between the experimental and theoretical resonance positions will require more detailed analysis and may be due to the multichannel nature of the employed FR <cit.> for tuning a_. It is shown that significant deviation of the Efimov resonance positions from the spinless vdW theory has been observed near a FR with s_res about 35 times smaller than those in our current cases <cit.>. However, the resonance positions obtained from the energy spectra (grayed areas) coincide very well with the experimental observations. Within the framework of an effective field theory corrections due to finite effective range and intraspecies scattering length on three-body recombination rates and Efimov resonance positions have been studied recently <cit.>. Similar to Sec. <ref> we extract resonance positions and scaling factors by three different methods. The experimental and vdW parameters are extracted as described previously. Here, a^(n)_-,zr are calculated within the zero range model described in Sec. <ref>. We find the energy levels of the upper branch adiabatic hyperspherical potential curves (see Fig. <ref>) and search for the values of a_ where their energy intersects zero. As opposed to the negative intraspecies scattering length case, there is no three-body parameter necessary. The positions and scaling factors are listed in Table <ref>. The experimentally determined position of the first excited Efimov resonance a_-^(1) close to the 889 G FR is shifted by ≈ 300a_0 with respect to the 843 G FR, while the position of the second excited resonance a_-^(2) is nearly unchanged. The resonance position a_-^(2) extracted by all three methods agree very well with each other, while for the first excited state the zero-range theory clearly deviates from the two other models. This deviation may be explained by neglect of nonadiabatic couplings in our single-channel zero-range theory. Additionally, the binding energy of the Cs_2 dimer is not reproduced in this model by disregard of effective range corrections. However, the quantitative influence of these limitations requires further detailed analysis. The scaling factor obtained from the zero-range theory agrees very well with the theoretically predicted scaling factor of 4.9 between consecutive CsCsLi Efimov resonances <cit.>. The obtained scaling factors from experiment and vdW theory of 3.8(1)(3)(3) and 4.0(3) between the two excited Efimov resonances clearly deviate from the universal zero-range theory. §.§ Scaling laws of three-body recombination rates The scaling behavior of three-body recombination in a heteronuclear system is expected to drastically depend on the sign and magnitude of the intra- and interspecies scattering lengths <cit.>. We study this behavior by comparing the three-body recombination spectra close to the 843 G and 889 G Cs-Li FRs for temperatures of 450 nK and 320 nK in Fig. <ref>. They feature power-law scaling behavior with the scattering lengths a_ and a_, which qualitatively agree with the expected scaling laws near overlapping Feshbach resonances <cit.>. In certain ranges of a_ each of the loss rate spectra corresponds to one of two distinctive cases of three-body scattering: the one near the 889 G FR can be characterized by | a_| ≫ a_≈ 190a_0 for which L_3 ∝ a_^4 is expected, whereas the one near the 843 G FR for small a_ approximately corresponds to the case of | a_| ≪| a_Cs| ≈ 1500a_0 with an expected scaling of L_3 ∝ a_^2 a_^2 <cit.>. The scaling laws can be explained by tunneling through effective three-body potential barriers within a simple WKB model. The power laws are displayed in the respective range in Fig. <ref> as a guide to the eye. Since the experimentally employed scattering lengths only approximately capture the inequalities imposed by the theory, especially in the latter case, the power laws can only approximately recover the behavior of the actual Cs-Li system. Close to the pole of the FR the theoretical scaling does not apply anymore due to the unitarity limit. The shown power law for the 843 G FR does not account for varying a_, which is tuned simultaneously with the magnetic field and changes by approximately a factor of 1.5 for the experimentally employed Feshbach resonance (see Fig. <ref>). For example, in the case of the 843 G Cs-Li Feshbach resonance, far away from the resonance the inequality | a_| ≪| a_| is fulfilled, whereas close to the pole of the resonance, the opposite is true, i.e., | a_| ≫| a_|. If a_ was a constant, this would lead to a qualitative change in the power law from L_3∝ a_^2 a_^2 to L_3∝ a_^4 for small to large scattering lengths, respectively. Such a transition in the present data is masked by finite-temperature and short-range effects. However, it might become observable in samples with further reduced temperature. Similar behavior can be found in the universal zero-range theory with finite intraspecies scattering length <cit.> and formalisms based on optical potentials <cit.>. The different observed power laws enable us to manipulate the three-body loss in a heteronuclear system. By choosing an appropriate FR, we can control the intraspecies scattering length between the heavy bosons and by this drastically influence the three-body loss rate. For scattering lengths a_≲ 500a_0 the Cs-Cs-Li losses are reduced by approximately two orders of magnitude when changing from large and negative to small and postive intraspecies scattering lengths, paving the way to produce long-lived strongly interacting Bose-Fermi mixtures. § CONCLUSION In summary, we have presented three theoretical methods at different levels of complexity to solve the three-body Schrödinger equation for two heavy identical bosons and one distinguishable particle and compared them to measurements of three-body recombination rates in a system of ultracold Cs and Li atoms. By detailed analysis we confirm the decisive influence of the intraspecies scattering length on the heteronuclear Efimov effect. The minimalistic hybrid Born-Oppenheimer model gives an intuitive understanding of the strong dependence of the energies on the intraspecies scattering length. Within this framework the binding energies of the three-body states for a_→∞ in dependence of the intraspecies scattering length a_ were calculated and a steplike behavior close to the Cs-Cs vdW length |a_|≈ r_vdW^ was predicted, analogous to calculations in the hyperspherical vdW model <cit.>, qualitatively explaining the experimentally observed change in the resonance position of the first excited Efimov state a_-^(1) between the two Cs-Li FRs. Within the hyperspherical adiabatic zero-range theory two qualitatively distinct cases of the heteronuclear Efimov scenario were shown. While for a_BB<0 the original Efimov scenario is observed, with an infinite number of log-periodically spaced three-body states, the existence of a weakly bound dimer for a_BB>0 splits the adiabatic hyperspherical potentials into two Efimov branches. Efimov states situated in the lower potential branch may not lead to Efimov resonances in the B+B+X channel as observed in our measured three-body recombination rate spectra. For the upper branch a universal potential barrier at R≈2a_BB makes the introduction of an artificial three-body parameter superfluous and allows for the determination of Efimov resonance positions. However, the applied adiabatic approximation and the neglect of effective range corrections may affect the observed deviations from the experimentally determined resonance positions. An encouraging level of understanding of the heteronuclear Efimov effect is provided by the spinless van der Waals theory in the adiabatic hyperspherical approximation, for both positive and negative intraspecies scattering lengths. This theory is compared to our measurements of three-body recombination rate spectra in an ultracold mixture of Cs and Li atoms at temperatures as low as 120 nK for two Cs-Li FRs, characterized by different sign and magnitude of the intraspecies scattering length a_. We find excellent agreement between experiment and theory on the negative side of a_. For positive intraspecies interactions a good agreement between the observed resonances and the energy spectrum is observed. However, a shift between experimental and theoretical three-body recombination rates demands further analysis and may be due to the multichannel character of the Feshbach resonance. No Efimov resonance which can be assigned to the ground Efimov state is observed at the Cs+Cs+Li threshold for positive a_. Before reaching the three-body continuum the most deeply bound state dissociates at the atom-dimer threshold. Away from the two Cs-Li FRs we observe power-law scalings of the three-body recombination rates, which can be attributed to two distinct cases of overlapping Feshbach resonances. We are able to suppress the three-body loss by up to two orders of magnitude via choosing a different Cs-Li Feshbach resonance and drastically increase the lifetime of the Bose-Fermi mixture. A predicted qualitative change in the scaling behavior of the three-body recombination rate close to the 843 G FR from L_3∝ a_^2 a_^2 to L_3∝ a_^4 might be observable in mixtures with further reduced temperature. We are grateful to P. Giannakeas, C. Zimmermann, D. Petrov, R. Grimm, and F. Ferlaino for fruitful discussions. This work is supported in part by the Heidelberg Center for Quantum Dynamics. S.H. acknowledges support by the IMPRS-QD. E.D.K. is indebted to the Baden-Württemberg Stiftung for the financial support of this research project by the Eliteprogramme for Postdocs. Y.W. acknowledges support by an AFOSR-MURI and Department of Physics, Kansas State University, and C.H.G. from the Binational Science Foundation, Grant No. 2012504 and the U.S. National Science Foundation Grant No. PHY-1607180. This work was supported in part by the Deutsche Forschungsgemeinschaft under Project No. WE2661/11-1 and Collaborative Research Centre "SFB 1225 (ISOQUANT)". 60 fxundefined [1] ifx#1 fnum [1] #1firstoftwo secondoftwo fx [1] #1firstoftwo secondoftwo noop [0]secondoftwo ref[1]@startlink#1@href href[1]#1@endlink anitize@url [0]` 12`$12`&12`#12`1̂2`_12`%12 startlink[1] endlink[0] rl [1]href #1 @bib@innerbibempty [Efimov(1970)]Efimov1970 author author V. Efimov, http://dx.doi.org/10.1016/0370-2693(70)90349-7 journal journal Phys. Lett. B volume 33, pages 563 (year 1970)NoStop [Efimov(1971)]Efimov1971 author author V. Efimov, @noop journal journal Sov. J. Nuc. Phys. volume 12, pages 589 (year 1971)NoStop [Efimov(1973)]Efimov1973 author author V. Efimov, http://dx.doi.org/10.1016/0375-9474(73)90510-1 journal journal Nucl. Phys. A volume 210, pages 157 (year 1973)NoStop [Efimov(1979)]Efimov1979 author author V. Efimov, @noop journal journal Sov. J. Nuc. Phys. volume 29, pages 546 (year 1979)NoStop [Braaten and Hammer(2006)]Braaten2006 author author E. Braaten and author H.-W. Hammer, http://dx.doi.org/10.1016/j.physrep.2006.03.001 journal journal Phys. Rep. volume 428, pages 259 (year 2006)NoStop [Ferlaino et al.(2011)Ferlaino, Zenesini, Berninger, Huang, Nägerl, and Grimm]Ferlaino2011 author author F. Ferlaino, author A. Zenesini, author M. Berninger, author B. Huang, author H.-C. Nägerl, and author R. Grimm, 10.1007/s00601-011-0260-7 journal journal Few-Body Syst. volume 51, pages 113 (year 2011)NoStop [Wang et al.(2013)Wang, D'Incao, and Esry]Wang2013 author author Y. Wang, author J. P. D'Incao, and author B. D. Esry, http://dx.doi.org/10.1016/B978-0-12-408090-4.00001-3 title Ultracold Few-Body Systems, edited by editor P. R. B. Ennio Arimondo and editor C. C. Lin, series Advances in Atomic, Molecular, and Optical Physics, Vol. volume 62 (publisher Academic Press, year 2013)NoStop [Frederico et al.(2012)Frederico, Delfino, Tomio, and Yamashita]Frederico2012 author author T. Frederico, author A. Delfino, author L. Tomio, and author M. Yamashita, http://dx.doi.org/10.1016/j.ppnp.2012.06.001 journal journal Prog. Part. Nucl. Phys. volume 67, pages 939 (year 2012)NoStop [Huang et al.(2014a)Huang, Sidorenkov, Grimm, and Hutson]Huang2014 author author B. Huang, author L. A. Sidorenkov, author R. Grimm, and author J. M. Hutson, 10.1103/PhysRevLett.112.190401 journal journal Phys. Rev. Lett. volume 112, pages 190401 (year 2014a)NoStop [Williams et al.(2009)Williams, Hazlett, Huckans, Stites, Zhang, and O'Hara]Williams2009 author author J. R. Williams, author E. L. Hazlett, author J. H. Huckans, author R. W. Stites, author Y. Zhang, and author K. M. O'Hara, 10.1103/PhysRevLett.103.130404 journal journal Phys. Rev. Lett. volume 103, pages 130404 (year 2009)NoStop [Pires et al.(2014a)Pires, Ulmanis, Häfner, Repp, Arias, Kuhnle, and Weidemüller]Pires2014 author author R. Pires, author J. Ulmanis, author S. Häfner, author M. Repp, author A. Arias, author E. D. Kuhnle, and author M. Weidemüller, 10.1103/PhysRevLett.112.250404 journal journal Phys. Rev. Lett. volume 112, pages 250404 (year 2014a)NoStop [Tung et al.(2014)Tung, Jiménez-García, Johansen, Parker, and Chin]Tung2014 author author S.-K. Tung, author K. Jiménez-García, author J. Johansen, author C. V. Parker, and author C. Chin, 10.1103/PhysRevLett.113.240402 journal journal Phys. Rev. Lett. volume 113, pages 240402 (year 2014)NoStop [Ulmanis et al.(2016)Ulmanis, Häfner, Pires, Kuhnle, Wang, Greene, and Weidemüller]Ulmanis2016b author author J. Ulmanis, author S. Häfner, author R. Pires, author E. D. Kuhnle, author Y. Wang, author C. H. Greene, and author M. Weidemüller, 10.1103/PhysRevLett.117.153201 journal journal Phys. Rev. Lett. volume 117, pages 153201 (year 2016)NoStop [Ulmanis et al.(2016a)Ulmanis, Häfner, Kuhnle, and Weidemüller]Ulmanis2016a author author J. Ulmanis, author S. Häfner, author E. D. Kuhnle, and author M. Weidemüller, 10.1093/nsr/nww018 journal journal Natl. Sci. Rev. volume 3, pages 174 (year 2016a), http://arxiv.org/abs/http://nsr.oxfordjournals.org/content/3/2/174.full.pdf http://nsr.oxfordjournals.org/content/3/2/174.full.pdf NoStop [D'Incao and Esry(2006a)]DIncao2006a author author J. P. D'Incao and author B. D. Esry, 10.1103/PhysRevA.73.030703 journal journal Phys. Rev. A volume 73, pages 030703 (year 2006a)NoStop [Kunitski et al.(2015)Kunitski, Zeller, Voigtsberger, Kalinin, Schmidt, Schöffler, Czasch, Schöllkopf, Grisenti, Jahnke, Blume, and Dörner]Kunitski2015 author author M. Kunitski, author S. Zeller, author J. Voigtsberger, author A. Kalinin, author L. P. H. Schmidt, author M. Schöffler, author A. Czasch, author W. Schöllkopf, author R. E. Grisenti, author T. Jahnke, author D. Blume, and author R. Dörner, 10.1126/science.aaa5601 journal journal Science volume 348, pages 551 (year 2015)NoStop [Berninger et al.(2011)Berninger, Zenesini, Huang, Harm, Nägerl, Ferlaino, Grimm, Julienne, and Hutson]Berninger2011 author author M. Berninger, author A. Zenesini, author B. Huang, author W. Harm, author H.-C. Nägerl, author F. Ferlaino, author R. Grimm, author P. S. Julienne, and author J. M. Hutson, 10.1103/PhysRevLett.107.120401 journal journal Phys. Rev. Lett. volume 107, pages 120401 (year 2011)NoStop [Roy et al.(2013)Roy, Landini, Trenkwalder, Semeghini, Spagnolli, Simoni, Fattori, Inguscio, and Modugno]Roy2013 author author S. Roy, author M. Landini, author A. Trenkwalder, author G. Semeghini, author G. Spagnolli, author A. Simoni, author M. Fattori, author M. Inguscio, and author G. Modugno, 10.1103/PhysRevLett.111.053202 journal journal Phys. Rev. Lett. volume 111, pages 053202 (year 2013)NoStop [Wild et al.(2012)Wild, Makotyn, Pino, Cornell, and Jin]Wild2012 author author R. J. Wild, author P. Makotyn, author J. M. Pino, author E. A. Cornell, and author D. S. Jin, 10.1103/PhysRevLett.108.145305 journal journal Phys. Rev. Lett. volume 108, pages 145305 (year 2012)NoStop [Gross et al.(2009)Gross, Shotan, Kokkelmans, and Khaykovich]Gross2009 author author N. Gross, author Z. Shotan, author S. Kokkelmans, and author L. Khaykovich, 10.1103/PhysRevLett.103.163202 journal journal Phys. Rev. Lett. volume 103, pages 163202 (year 2009)NoStop [Gross et al.(2010)Gross, Shotan, Kokkelmans, and Khaykovich]Gross2010 author author N. Gross, author Z. Shotan, author S. Kokkelmans, and author L. Khaykovich, 10.1103/PhysRevLett.105.103203 journal journal Phys. Rev. Lett. volume 105, pages 103203 (year 2010)NoStop [Huang et al.(2014b)Huang, O'Hara, Grimm, Hutson, and Petrov]Huang2014a author author B. Huang, author K. M. O'Hara, author R. Grimm, author J. M. Hutson, and author D. S. Petrov, 10.1103/PhysRevA.90.043636 journal journal Phys. Rev. A volume 90, pages 043636 (year 2014b)NoStop [Huang et al.(2015)Huang, Sidorenkov, and Grimm]Huang2015a author author B. Huang, author L. A. Sidorenkov, and author R. Grimm, 10.1103/PhysRevA.91.063622 journal journal Phys. Rev. A volume 91, pages 063622 (year 2015)NoStop [Wang et al.(2012a)Wang, D'Incao, Esry, and Greene]Wang2012 author author J. Wang, author J. P. D'Incao, author B. D. Esry, and author C. H. Greene, 10.1103/PhysRevLett.108.263001 journal journal Phys. Rev. Lett. volume 108, pages 263001 (year 2012a)NoStop [Naidon et al.(2014a)Naidon, Endo, and Ueda]Naidon2014 author author P. Naidon, author S. Endo, and author M. Ueda, 10.1103/PhysRevA.90.022106 journal journal Phys. Rev. A volume 90, pages 022106 (year 2014a)NoStop [Wang and Julienne(2014)]Wang2014a author author Y. Wang and author P. S. Julienne, doi:10.1038/nphys3071 journal journal Nat. Phys. volume 10, pages 768 (year 2014)NoStop [Wang et al.(2015)Wang, Julienne, and Greene]Wang2015b author author Y. Wang, author P. Julienne, and author C. H. Greene, in http://www.worldscientific.com/doi/abs/10.1142/9789814667746_0002 booktitle Annual Review of Cold Atoms and Molecules, Vol. volume 3, editor edited by editor K. W. Madison, editor K. Bongs, editor L. D. Carr, editor A. M. Rey, and editor H. Zhai (publisher World Scientific Publishing, year 2015) Chap. chapter 2, pp. pages 77–134, note also at: http://arxiv.org/abs/1412.8094v1NoStop [Naidon et al.(2014b)Naidon, Endo, and Ueda]Naidon2014a author author P. Naidon, author S. Endo, and author M. Ueda, 10.1103/PhysRevLett.112.105301 journal journal Phys. Rev. Lett. volume 112, pages 105301 (year 2014b)NoStop [Wang et al.(2012b)Wang, Wang, D'Incao, and Greene]Wang2012d author author Y. Wang, author J. Wang, author J. P. D'Incao, and author C. H. Greene, 10.1103/PhysRevLett.109.243201 journal journal Phys. Rev. Lett. volume 109, pages 243201 (year 2012b)NoStop [Ulmanis et al.(2016b)Ulmanis, Häfner, Pires, Werner, Petrov, Kuhnle, and Weidemüller]Ulmanis2016 author author J. Ulmanis, author S. Häfner, author R. Pires, author F. Werner, author D. S. Petrov, author E. D. Kuhnle, and author M. Weidemüller, 10.1103/PhysRevA.93.022707 journal journal Phys. Rev. A volume 93, pages 022707 (year 2016b)NoStop [D'Incao and Esry(2009)]DIncao2009a author author J. P. D'Incao and author B. D. Esry, 10.1103/PhysRevLett.103.083202 journal journal Phys. Rev. Lett. volume 103, pages 083202 (year 2009)NoStop [Hu et al.(2014)Hu, Bloom, Jin, and Goldwin]Hu2014 author author M.-G. Hu, author R. S. Bloom, author D. S. Jin, and author J. M. Goldwin, 10.1103/PhysRevA.90.013619 journal journal Phys. Rev. A volume 90, pages 013619 (year 2014)NoStop [Bloom et al.(2013)Bloom, Hu, Cumby, and Jin]Bloom2013 author author R. S. Bloom, author M.-G. Hu, author T. D. Cumby, and author D. S. Jin, 10.1103/PhysRevLett.111.105301 journal journal Phys. Rev. Lett. volume 111, pages 105301 (year 2013)NoStop [Barontini et al.(2009)Barontini, Weber, Rabatti, Catani, Thalhammer, Inguscio, and Minardi]Barontini2009 author author G. Barontini, author C. Weber, author F. Rabatti, author J. Catani, author G. Thalhammer, author M. Inguscio, and author F. Minardi, 10.1103/PhysRevLett.103.043201 journal journal Phys. Rev. Lett. volume 103, pages 043201 (year 2009)NoStop [Barontini et al.(2010)Barontini, Weber, Rabatti, Catani, Thalhammer, Inguscio, and Minardi]Barontini2010 author author G. Barontini, author C. Weber, author F. Rabatti, author J. Catani, author G. Thalhammer, author M. Inguscio, and author F. Minardi, 10.1103/PhysRevLett.104.059901 journal journal Phys. Rev. Lett. volume 104, pages 059901(E) (year 2010)NoStop [Maier et al.(2015)Maier, Eisele, Tiemann, and Zimmermann]Maier2015 author author R. A. W. Maier, author M. Eisele, author E. Tiemann, and author C. Zimmermann, 10.1103/PhysRevLett.115.043201 journal journal Phys. Rev. Lett. volume 115, pages 043201 (year 2015)NoStop [Ulmanis(2015)]Ulmanis2015b author author J. Ulmanis, title Universality and non-universality in the heteronuclear Efimov scenario with large mass imbalance, @noop Ph.D. thesis, school Ruperto-Carola-University of Heidelberg (year 2015)NoStop [Chin et al.(2010)Chin, Grimm, Julienne, and Tiesinga]Chin2010 author author C. Chin, author R. Grimm, author P. Julienne, and author E. Tiesinga, 10.1103/RevModPhys.82.1225 journal journal Rev. Mod. Phys. volume 82, pages 1225 (year 2010)NoStop [Gribakin and Flambaum(1993)]Gribakin1993 author author G. F. Gribakin and author V. V. Flambaum, 10.1103/PhysRevA.48.546 journal journal Phys. Rev. A volume 48, pages 546 (year 1993)NoStop [Flambaum et al.(1999)Flambaum, Gribakin, and Harabati]Flambaum1999 author author V. V. Flambaum, author G. F. Gribakin, and author C. Harabati, 10.1103/PhysRevA.59.1998 journal journal Phys. Rev. A volume 59, pages 1998 (year 1999)NoStop [Macek(1968)]Macek1968 author author J. Macek, http://stacks.iop.org/0022-3700/1/i=5/a=309 journal journal J. Phys. B: At. Mol. Phys. volume 1, pages 831 (year 1968)NoStop [Wang(2010)]Wang2010c author author Y. Wang, title Universal Efimov Physics in Three- and Four-Body Collisions, @noop Ph.D. thesis, school Kansas State University (year 2010)NoStop [Note1()]Note1 note The absence of the Efimov effect in higher partial waves for our mass ratio of m_B/m_X=22.1 <cit.>, leads to a substantial suppression of J>0 contributions to the recombination rate in the ultracold regime with k_B T≪ E_vdW. The suppression is estimated to be (k r_vdW)^2p_0≪ 1 for a_CsCs<0 and (k a_CsCs)^2p_0≪ 1 for the a_CsCs≈ +190a_0 case <cit.>, where p_0>1 is a universal constant for the given mass ratio <cit.> and k the thermal wavevector.Stop [Mehta et al.(2009)Mehta, Rittenhouse, D'Incao, von Stecher, and Greene]Mehta2009 author author N. P. Mehta, author S. T. Rittenhouse, author J. P. D'Incao, author J. von Stecher, and author C. H. Greene, 10.1103/PhysRevLett.103.153201 journal journal Phys. Rev. Lett. volume 103, pages 153201 (year 2009)NoStop [Suno et al.(2003)Suno, Esry, and Greene]Suno2003 author author H. Suno, author B. D. Esry, and author C. H. Greene, 10.1103/PhysRevLett.90.053202 journal journal Phys. Rev. Lett. volume 90, pages 053202 (year 2003)NoStop [Tung et al.(2013)Tung, Parker, Johansen, Chin, Wang, and Julienne]Tung2013 author author S.-K. Tung, author C. Parker, author J. Johansen, author C. Chin, author Y. Wang, and author P. S. Julienne, 10.1103/PhysRevA.87.010702 journal journal Phys. Rev. A volume 87, pages 010702 (year 2013)NoStop [Repp et al.(2013)Repp, Pires, Ulmanis, Heck, Kuhnle, Weidemüller, and Tiemann]Repp2013 author author M. Repp, author R. Pires, author J. Ulmanis, author R. Heck, author E. D. Kuhnle, author M. Weidemüller, and author E. Tiemann, 10.1103/PhysRevA.87.010701 journal journal Phys. Rev. A volume 87, pages 010701 (year 2013)NoStop [Pires et al.(2014b)Pires, Repp, Ulmanis, Kuhnle, Weidemüller, Tiecke, Greene, Ruzic, Bohn, and Tiemann]Pires2014a author author R. Pires, author M. Repp, author J. Ulmanis, author E. D. Kuhnle, author M. Weidemüller, author T. G. Tiecke, author C. H. Greene, author B. P. Ruzic, author J. L. Bohn, and author E. Tiemann, 10.1103/PhysRevA.90.012710 journal journal Phys. Rev. A volume 90, pages 012710 (year 2014b)NoStop [Ulmanis et al.(2015)Ulmanis, Häfner, Pires, Kuhnle, Weidemüller, and Tiemann]Ulmanis2015 author author J. Ulmanis, author S. Häfner, author R. Pires, author E. D. Kuhnle, author M. Weidemüller, and author E. Tiemann, http://stacks.iop.org/1367-2630/17/i=5/a=055009 journal journal New J. Phys. volume 17, pages 055009 (year 2015)NoStop [Berninger et al.(2013)Berninger, Zenesini, Huang, Harm, Nägerl, Ferlaino, Grimm, Julienne, and Hutson]Berninger2013 author author M. Berninger, author A. Zenesini, author B. Huang, author W. Harm, author H.-C. Nägerl, author F. Ferlaino, author R. Grimm, author P. S. Julienne, and author J. M. Hutson, 10.1103/PhysRevA.87.032517 journal journal Phys. Rev. A volume 87, pages 032517 (year 2013)NoStop [Johansen et al.(2017)Johansen, DeSalvo, Patel, and Chin]Johansen2017 author author J. Johansen, author B. J. DeSalvo, author K. Patel, and author C. Chin, http://dx.doi.org/10.1038/nphys4130 journal journal Nat Phys volume advance online publication (year 2017)NoStop [Weber et al.(2003)Weber, Herbig, Mark, Nägerl, and Grimm]Weber2003 author author T. Weber, author J. Herbig, author M. Mark, author H.-C. Nägerl, and author R. Grimm, 10.1103/PhysRevLett.91.123201 journal journal Phys. Rev. Lett. volume 91, pages 123201 (year 2003)NoStop [Petrov and Werner(2015)]Petrov2015 author author D. S. Petrov and author F. Werner, 10.1103/PhysRevA.92.022704 journal journal Phys. Rev. A volume 92, pages 022704 (year 2015)NoStop [Helfrich et al.(2010)Helfrich, Hammer, and Petrov]Helfrich2010 author author K. Helfrich, author H.-W. Hammer, and author D. S. Petrov, 10.1103/PhysRevA.81.042715 journal journal Phys. Rev. A volume 81, pages 042715 (year 2010)NoStop [Sørensen et al.(2012)Sørensen, Fedorov, Jensen, and Zinner]Sorensen2012 author author P. K. Sørensen, author D. V. Fedorov, author A. S. Jensen, and author N. T. Zinner, 10.1103/PhysRevA.86.052516 journal journal Phys. Rev. A volume 86, pages 052516 (year 2012)NoStop [Acharya et al.(2016)Acharya, Ji, and Platter]Acharya2016 author author B. Acharya, author C. Ji, and author L. Platter, 10.1103/physreva.94.032702 journal journal Physical Review A volume 94 (year 2016), 10.1103/physreva.94.032702NoStop [Helfrich and Hammer(2010)]Helfrich2010a author author K. Helfrich and author H.-W. Hammer, 10.1051/epjconf/20100302007 journal journal EPJ Web (year 2010), 10.1051/epjconf/20100302007NoStop [Mikkelsen et al.(2015)Mikkelsen, Jensen, Fedorov, and Zinner]Mikkelsen2015 author author M. Mikkelsen, author A. S. Jensen, author D. V. Fedorov, and author N. T. Zinner, 10.1088/0953-4075/48/8/085301 journal journal J. Phys. B: At., Mol. Opt. Phys. volume 48, pages 085301 (year 2015)NoStop [D'Incao and Esry(2006b)]DIncao2006 author author J. P. D'Incao and author B. D. Esry, 10.1103/PhysRevA.73.030702 journal journal Phys. Rev. A volume 73, pages 030702 (year 2006b)NoStop [Wang and Esry(2011)]Wang2011d author author Y. Wang and author B. D. Esry, http://stacks.iop.org/1367-2630/13/i=3/a=035025 journal journal New J. Phys. volume 13, pages 035025 (year 2011)NoStop

http://arxiv.org/abs/1701.07807v2

20170126183553

Private Information Retrieval from MDS Coded Data with Colluding Servers: Settling a Conjecture by Freij-Hollanti et al.

cs.IT

Private Information Retrieval from MDS Coded Data with Colluding Servers: Settling a Conjecture by Freij-Hollanti et al. Hua Sun and Syed A. Jafar ============================================================================================================================== Hua Sun (email: huas2@uci.edu) and Syed A. Jafar (email: syed@uci.edu) are with the Center of Pervasive Communications and Computing (CPCC) in the Department of Electrical Engineering and Computer Science (EECS) at the University of California Irvine. Private Information Retrieval from MDS Coded Data with Colluding Servers: Settling a Conjecture by Freij-Hollanti et al. Hua Sun and Syed A. Jafar ============================================================================================================================== A (K, N, T, K_c) instance of the MDS-TPIR problem is comprised of K messages and N distributed servers. Each message is separately encoded through a (K_c, N) MDS storage code. A user wishes to retrieve one message, as efficiently as possible, while revealing no information about the desired message index to any colluding set of up to T servers. The fundamental limit on the efficiency of retrieval, i.e., the capacity of MDS-TPIR is known only at the extremes where either T or K_c belongs to {1,N}. The focus of this work is a recent conjecture by Freij-Hollanti, Gnilke, Hollanti and Karpuk which offers a general capacity expression for MDS-TPIR. We prove that the conjecture is false by presenting as a counterexample a PIR scheme for the setting (K, N, T, K_c) = (2,4,2,2), which achieves the rate 3/5, exceeding the conjectured capacity, 4/7. Insights from the counterexample lead us to capacity characterizations for various instances of MDS-TPIR including all cases with (K, N, T, K_c) = (2,N,T,N-1), where N and T can be arbitrary. § INTRODUCTION Private Information Retrieval (PIR) is the problem of retrieving one out of K messages from N distributed servers (each stores all K messages) in such a way that any individual server learns no information about which message is being retrieved. The rate of a PIR scheme is the ratio of the number of bits of the desired message to the total number of bits downloaded from all servers. The supremum of achievable rates is the capacity of PIR. The capacity of PIR was shown in <cit.> to be C_ = (1+1/N+1/N^2+⋯+1/N^K-1)^-1 The capacity of several variants of PIR has also since been characterized in <cit.>. The focus of this work is on a recent conjecture by Freij-Hollanti, Gnilke, Hollanti and Karpuk (FGHK conjecture, in short) in <cit.> which offers a capacity expression for a generalized form of PIR, called MDS-TPIR. MDS-TPIR involves two additional parameters: K_c and T, which generalize the storage and privacy constraints, respectively. Instead of replication, each message is encoded through a (K_c, N) MDS storage code, so that the information stored at any K_c servers is exactly enough to recover all K messages. Privacy must be preserved not just from each individual server, but from any colluding set of up to T servers. MDS-TPIR is a generalization of PIR, because setting both T=1 and K_c=1 reduces MDS-TPIR to the original PIR problem for which the capacity is already known (see (<ref>)). The capacity of MDS-TPIR is known only at the degenerate extremes – when either T or K_c takes the value 1 or N. If either T or K_c is equal to N then by analogy to the single server setting it follows immediately that the user must download all messages, i.e., the capacity is 1/K. If K_c=1 or T=1, then the problem specializes to TPIR, and MDS-PIR, respectively. The capacity of TPIR (K_c=1) was shown in <cit.> to be C_ = (1+T/N+T^2/N^2+⋯+T^K-1/N^K-1)^-1 The capacity of MDS-PIR (T=1) was characterized by Banawan and Ulukus in <cit.>, as C_ = (1+K_c/N+K_c^2/N^2+⋯+K_c^K-1/N^K-1)^-1 It is notable that K_c and T play similar roles in the two capacity expressions. The capacity achieving scheme of Banawan and Ulukus <cit.> improved upon a scheme proposed earlier by Tajeddine and Rouayheb in <cit.>. Tajeddine and Rouayheb also proposed an achievable scheme for MDS-TPIR for the T=2 setting. The scheme was generalized by Freij-Hollanti et al. <cit.> to the (K, N, T, K_c) setting, T+K_c≤ N, where it achieves the rate 1 - T+K_c -1/N. Remarkably, the rate achieved by this scheme does not depend on the number of messages, K. In support of the plausible asymptotic (K→∞) optimality of their scheme, and based on the intuition from existing capacity expressions for PIR, MDS-PIR and TPIR, Freij-Hollanti et al. conjecture that if T+K_c≤ N, then the capacity of MDS-TPIR is given by the following expression. FGHK Conjecture <cit.>: C^_ =( 1 + T+K_c -1/N + ⋯ + (T+K_c -1)^K-1/N^K-1)^-1 The conjecture is appealing for its generality and elegance as it captures all four parameters, K, N, T, K_c in a compact form. T and K_c appear as interchangeable terms, and the capacity expression appears to be a natural extension of the capacity expressions for TPIR and MDS-PIR. Indeed, the conjectured capacity recovers the known capacity of TPIR if we set K_c=1 and that of MDS-PIR if we set T=1. However, in all non-degenerate cases where T,K_c∉{1,N}, the capacity of MDS-TPIR, and therefore the validity of the conjecture is unknown. In fact, in all these cases the problem is open on both sides, i.e., the conjectured capacity expression is neither known to be achievable, nor known to be an outer bound. The lack of any non-trivial outer bounds for MDS-TPIR is also recently highlighted in <cit.>. This intriguing combination of plausibility, uncertainty and generality of the FGHK conjecture motivates our work. Our contribution is summarized next. §.§.§ Summary of Contribution As the main outcome of this work, we disprove the FGHK conjecture. For our counterexample, we consider the setting (K, N, T, K_c) = (2, 4, 2, 2) where the data is stored using the (2,4) MDS code (x,y)→ (x,y,x+y,x+2y). The conjectured capacity for this setting is 4/7. We show that the rate 3/5 > 4/7 is achievable, thus disproving the conjecture. As a converse argument, we show that no (scalar or vector) linear PIR scheme can achieve a rate higher than 3/5 for this MDS storage code subject to T=2 privacy. The insights from the counterexample lead us to characterize the exact capacity of various instances of MDS-TPIR. This includes all cases with (K,N,T,K_c)=(2,N,T,N-1), where N and T can be arbitrary. The capacity for these cases turns out to be C = N^2-N/2N^2-3N+T Note that this is the information theoretic capacity, i.e., for K=2 messages, no (N-1,N) MDS storage code and no PIR scheme (linear or non-linear) can beat this rate, which is achievable with the simple MDS storage code (x_1, x_2,⋯, x_N-1)→ (x_1, x_2, ⋯, x_N-1, ∑_i=1^N-1 x_i) and a linear PIR scheme. The general capacity expression for MDS-TPIR remains unknown. However, we are able to show that it cannot be symmetric in K_c and T, i.e., the two parameters are not interchangeable in general. Also, between K_c and T the capacity expression does not consistently favor one over the other. These findings are illustrated by the following four cases for which the capacity is settled. [ 4c(K,N,T,K_c); 2-5 (2,4,2,3) (2,4,3,2) (2,4,1,3) (2,4,3,1); 6/11 4/7 4/7 4/7; <cit.> <cit.>; ] The first two columns show that the capacity is not symmetric in K_c and T, since switching their values changes the capacity. The first two columns also suggest that increasing K_c hurts capacity more than increasing T. However, considering columns 3 and 4 as the baseline where the capacities are equal, and comparing the drop in capacity from column 3 to column 1 when T is increased, versus no change in capacity from column 4 to column 2 when K_c is increased shows the opposite trend. Therefore, neither T nor K_c is consistently dominant in terms of the sensitivity of capacity to these two parameters. Finally, taking an asymptotic view of capacity of MDS-TPIR, we show that if T+K_c>N, then the capacity collapses to 0 as the number of messages K→∞. This is consistent with the restriction of T+K_c≤ N that is required by the achievable scheme of Freij-Hollanti et al. whose rate does not depend on K. Notation: For n_1, n_2 ∈ℤ, define the notation [n_1: n_2] as the set {n_1, n_1+1,⋯, n_2}, A_n_1:n_2 as the vector (A_n_1, A_n_1+1, ⋯, A_n_2), and S(n_1 : n_2,:) as the submatrix of a matrix S formed by retaining only the n_1^th to the n_2^th rows. The notation X ∼ Y is used to indicate that X and Y are identically distributed. The cardinality of a set ℐ is denoted as |ℐ|. The determinant of a matrix S is denoted as |S|. For an index set ℐ = {i_1, ⋯, i_n} such that i_1 < ⋯ < i_n, the notation A_ℐ represents the vector (A_i_1, ⋯, A_i_n). (V_1; V_2; ⋯; V_n) refers to a matrix whose i^th row vector is V_i, i ∈ [1:n]. § PROBLEM STATEMENT Consider[While the problem statement is presented in its general form, we will primarily consider cases with K=2 messages in this paper (outer bounds for larger K are presented in Section <ref>).] K independent messages W_1, ⋯, W_K∈𝔽_p^L× 1, each represented as an L× 1 vector comprised of L i.i.d. uniform symbols from a finite field 𝔽_p for a prime p. In p-ary units, H(W_1) = ⋯ = H(W_K) = L H(W_1, ⋯, W_K) = H(W_1) + ⋯ + H(W_K) There are N servers. The n^th server stores (W_1n, W_2n, ⋯, W_Kn), where W_kn∈𝔽^L/K_c× 1 represents L/K_c symbols from W_k, k ∈ [1:K]. H(W_kn | W_k) = 0,  H(W_kn) = L/K_c We require the storage system to satisfy the MDS property, i.e., from the information stored in any K_c servers, we can recover each message, i.e., []   H(W_k | W_k 𝒦_c) = 0, ∀𝒦_c ⊂ [1:N], |𝒦_c| = K_c Let us use ℱ to denote a random variable privately generated by the user, whose realization is not available to the servers. ℱ represents the randomness in the strategies followed by the user. Similarly, 𝒢 is a random variable that determines the random strategies followed by the servers, and whose realizations are assumed to be known to all the servers and to the user. The user privately generates θ uniformly from [1:K] and wishes to retrieve W_θ while keeping θ a secret from each server. ℱ and 𝒢 are generated independently and before the realizations of the messages or the desired message index are known, so that H(θ, ℱ, 𝒢, W_1, ⋯, W_K) = H(θ) + H(ℱ) + H(𝒢) + H(W_1) + ⋯ + H(W_K) Suppose θ = k. In order to retrieve W_k, k ∈ [1:K] privately, the user privately generates N random queries, Q_1^[k], ⋯, Q_N^[k]. H(Q_1^[k], ⋯, Q_N^[k] | ℱ) = 0, ∀ k ∈ [1:K] The user sends query Q_n^[k] to the n^th server, n ∈ [1:N]. Upon receiving Q_n^[k], the n^th server generates an answering string A_n^[k], which is a function of the received query Q_n^[k], the stored information W_1n, ⋯, W_Kn and 𝒢, H(A_n^[k] | Q_n^[k], W_1n, ⋯, W_Kn, 𝒢) = 0 Each server returns to the user its answer A_n^[k].[If the A_n^[k] are obtained as inner products of query vectors and stored message vectors, then such a PIR scheme is called a linear PIR scheme.] From all the information that is now available at the user (A_1:N^[k], Q_1:N^[k], ℱ,𝒢), the user decodes the desired message W_k according to a decoding rule that is specified by the PIR scheme. Let P_e denote the probability of error achieved with the specified decoding rule. To protect the user's privacy, the K strategies must be indistinguishable (identically distributed) from the perspective of any subset 𝒯⊂ [1:N] of at most T colluding servers, i.e., the following privacy constraint must be satisfied. []  (Q_𝒯^[k], A_𝒯^[k], 𝒢, W_1𝒯, ⋯, W_K𝒯) ∼ (Q_𝒯^[k'], A_𝒯^[k'], 𝒢, W_1𝒯, ⋯, W_K𝒯), ∀ k, k' ∈ [1:K], ∀𝒯⊂ [1:N], |𝒯| = T The PIR rate characterizes how many bits of desired information are retrieved per downloaded bit and is defined as follows. R = L/D where D is the expected value of the total number of bits downloaded by the user from all the servers. A rate R is said to be ϵ-error achievable if there exists a sequence of PIR schemes, indexed by L, each of rate greater than or equal to R, for which P_e → 0 as L →∞. Note that for such a sequence of PIR schemes, from Fano's inequality, we must have []  o(L) = 1/L H(W_k|A_1:N^[k], Q_1:N^[k], ℱ, 𝒢) (<ref>)= 1/L H(W_k|A_1:N^[k], ℱ, 𝒢) where o(L) represents a term whose value approaches zero as L approaches infinity. The supremum of ϵ-error achievable rates is called the capacity C.[Alternatively, the capacity may be defined with respect to zero error criterion, i.e., the supreme of zero error achievable rates where a rate R is said to be zero error achievable if there exists (for some L) a PIR scheme of rate greater than or equal to R for which P_e = 0. ] § SETTLING THE CONJECTURE Our main result, which settles the FGHK conjecture, is stated in the following theorem. For the MDS-TPIR problem with K = 2 messages, N = 4 servers, T = 2 privacy and the (K_c, N) = (2,4) MDS storage code (x,y)→ (x,y,x+y,x+2y), a rate of 3/5 is achievable. Since the achievable rate exceeds the conjectured capacity of 4/7 for this setting, the FGHK conjecture is false. Proof: We present a scheme that achieves rate 3/5. We assume that each message is comprised of L=12 symbols from 𝔽_p for a sufficiently[ It suffices to choose p=349 for Theorem <ref>. In general, the appeal to large field size, analogous to the random coding argument in information theory, is made to prove the existence of a scheme, but may not be essential to the construction of the PIR scheme. To underscore this point, Section <ref> includes some examples of MDS-TPIR capacity achieving schemes over small fields. ] large prime p. Define a∈𝔽_p^6×1 as the 6× 1 vector (a_1;a_2;⋯; a_6) comprised of i.i.d. uniform symbols a_i∈𝔽_p. Vectors b, c, d are defined similarly. Messages W_1, W_2 are defined in terms of these vectors as follows. W_1 = ( a; b) W_2 = ( c; d) §.§ Storage Code The storage is specified as (W_11,W_12, W_13, W_14) = ( a, b, a +b, a+2b) (W_21,W_22, W_23, W_24) = ( c, d, c +d, c+2d) Recall that W_kn is the information about message W_k that is stored at Server n. Thus, Server 1 stores ( a,c), Server 2 stores ( b,d), Server 3 stores ( a+b, c+d), and Server 4 stores ( a+2b, c+2d). In particular, each server stores 6 symbols for each message, for a total of 12 symbols per server. Any two servers store just enough information to recover both messages, thus the MDS storage criterion is satisfied. §.§ Construction of Queries The query to each server Q_n^[k] is comprised of two parts, denoted as Q_n^[k](W_1), Q_n^[k](W_2). Each part contains 3 row vectors, also called query vectors, along which the server should project its corresponding stored message symbols. Q_n^[k] = (Q_n^[k](W_1), Q_n^[k](W_2)) In preparation for the construction of the queries, let us denote the set of all full rank 6 × 6 matrices over 𝔽_p as 𝒮. The user privately chooses two matrices, S and S', independently and uniformly from 𝒮. Label the rows of S as V_1, V_2, V_3, V_4, V_5, V_6, and the rows of S' as U_0, U_1, U_2, U_3, U_4, U_5. Define 𝒱_1={V_1, V_2, V_3}, 𝒰_1={U_0, U_6, U_8} 𝒱_2={V_1, V_4, V_5}, 𝒰_2={U_0, U_7, U_9} 𝒱_3={V_2, V_4, V_6}, 𝒰_3={U_0, U_1, U_3} 𝒱_4={V_3, V_5, V_6}, 𝒰_4={U_0, U_2, U_4} U_6, U_7, U_8, U_9 are obtained as follows. U_6 = U_1 + U_2, U_7 = U_1 + 2U_2 U_8 = U_3 + U_4, U_9 = U_3 + 2U_4 As a preview of what we are trying to accomplish, we note that for Server n∈[1:4], 𝒱_n will be used as the query vectors for desired message symbols, while 𝒰_n will be used as query vectors for undesired message symbols. Since K_c=2, the same query vector V_i sent to two different servers will recover 2 independent desired symbols. Each V_i, i∈[1:6], is used exactly twice, so all queries for desired symbols will return independent information for a total of 12 independent desired symbols. On the other hand, for undesired symbols note that U_0 is used as the query vector to all 4 servers, but because K_c=2, it can only produce 2 independent symbols, i.e., 2 of the 4 symbols are redundant. The dependencies introduced via (<ref>),(<ref>) are carefully chosen to ensure that the queries along U_1, U_2, U_6, U_7 will produce only 3 independent symbols. Similarly, the queries along U_3,U_4, U_8,U_9 will produce only 3 independent symbols. Thus, all the queries for the undesired message will produce a total of only 8 independent symbols. The 12 independent desired symbols and 8 independent undesired symbols will be resolved from a total of 12+8=20 downloaded symbols, to achieve the rate 12/20=3/5. To ensure T=2 privacy, the 𝒰_i and 𝒱_i queries will be made indistinguishable from the perspective of any 2 colluding servers. The key to the T=2 privacy is that any 𝒱_n, 𝒱_n', n≠ n' have one element in common. Similarly, any 𝒰_n, 𝒰_n', n≠ n' also have one element in common. This is a critical aspect of the construction. Next we provide a detailed description of the queries and downloads for message W_k, k ∈ [1:2], both when W_k is desired and when it is not desired. To simplify the notation, we will denote W_k = ( x; y). Note that when k = 1, ( x; y) = ( a; b) and when k = 2, ( x; y) = ( c; d). §.§.§ Case 1. W_k is Desired The query sent to Server n is a 3× 6 matrix whose rows are the 3 vectors in 𝒱_n. The ordering of the rows is uniformly random, i.e., : Q_n^[k](W_k) = π_n (𝒱_n), n∈[1:4] For a set 𝒱={V_i_1,V_i_2,V_i_3}, π_n (𝒱) is equally likely to return any one of the 6 possibilities: (V_i_1; V_i_2; V_i_3), (V_i_1; V_i_3; V_i_2), (V_i_2; V_i_1; V_i_3), (V_i_2; V_i_3; V_i_1), (V_i_3; V_i_1; V_i_2) and (V_i_3; V_i_2; V_i_1). The π_n are independently chosen for each n∈[1:4]. After receiving the 3 query vectors Q_n^[k](W_k), Server n projects its stored W_kn symbols along these vectors. This creates three linear combinations of W_kn symbols (denoted as A_n^[k](W_k)). A_n^[k](W_k) = Q_n^[k](W_k) W_kn Define k^c=3-k as the complement of k, i.e., k^c=1 if k=2 and vice versa. The answers A_n^[k] to be sent to the user will be constructed eventually by combining A_n^[k](W_k) and A_n^[k](W_k^c), since separately sending these answers will be too inefficient. The details of this combining process will be specified later. Next we note an important property of the construction. Desired Symbols Are Independent: We show that if the user can recover A_1:4^[k](W_k) from the downloads, then he can recover all 12 symbols of W_k. From A_1:4^[k](W_k) the user recovers the 12 symbols V_1 x, V_2 x, V_3 x, V_1 y, V_4 y, V_5 y, V_2( x+y), V_4( x+y), V_6( x+y), V_3( x+2y), V_5( x+2y), V_6( x+2y). From these 12 symbols, he recovers V_i x and V_i y for all i∈[1:6]. Since S = (V_1; V_2; V_3; V_4; V_5; V_6) has full rank (invertible) and the user knows V_1:6, he recovers all symbols in x and y (thus W_k). §.§.§ Case 2. W_k is Undesired Similarly, the query sent to Server n is a 3× 6 matrix whose rows are the 3 vectors in 𝒰_n. The ordering of the rows is uniformly random for each n, and independent across all n∈[1:4]. : Q_n^[k^c](W_k) = π_n'(𝒰_n), n∈[1:4] Each server projects its stored W_kn symbols along the 3 query vectors to obtain, A_n^[k^c](W_k) = Q_n^[k^c](W_k) W_kn Interfering Symbols Have Dimension 8: A_1:4^[k^c](W_k) is comprised of U_0 x, U_6 x, U_8 x, U_0 y, U_7 y, U_9 y, U_0( x+y), U_1( x+y), U_3( x+y), U_0( x+2y), U_2( x+2y), U_4( x+2y). We now show that these 12 symbols are dependent and have dimension only 8.[Equivalently, the joint entropy of these 12 variables, conditioned on U_0:9 is only 8 p-ary units.] Because of (<ref>) and (<ref>), we have U_0 x + U_0 y = U_0 ( x+ y) U_0 x + 2U_0 y = U_0 ( x+2 y) U_6 x + U_7 y - U_1 ( x+ y) = U_2 ( x+2 y) U_8 x + U_9 y - U_3 ( x+ y) = U_4 ( x+2 y) Thus, of the 12 symbols recovered from A_1:4^[k^c](W_k), at least 4 are linear combinations of the remaining 8. It follows that A_1:4^[k^c](W_k) contains no more than 8 dimensions. The number of dimensions is also not less than 8 because, the following 8 undesired symbols (two symbols from each server) are independent, :  U_0 x, U_6 x = (U_1+U_2) x :  U_0 y, U_9 y = (U_3 + 2U_4) y :  U_1 ( x + y), U_3 ( x + y) :  U_2 ( x + 2 y), U_4 ( x + 2 y) To see that the 8 symbols are independent, we add 4 new symbols (U_1 x, U_3 y, U_5 x, U_5 y) such that from the 12 symbols, we can recover all 12 undesired symbols (S' x, S' y). Since the 4 new symbols cannot contribute more than 4 dimensions, the original 8 symbols must occupy at least 8 dimensions. §.§ Combining Answers for Efficient Download Based on the queries, each server has 3 linear combinations of symbols of W_1 in A_n^[k](W_1) and 3 linear combinations of symbols of W_2 in A_n^[k](W_2) for a total of 12 linear combinations of desired symbols and 12 linear combinations of undesired symbols across all servers. However, recall that there are only 8 independent linear combinations of undesired symbols. This is a fact that can be exploited to improve the efficiency of download. Specifically, we will combine the 6 queried symbols (i.e., the 6 linear combinations) from each server into 5 symbols to be downloaded by the user. Intuitively, 5 symbols from each server will give the user a total of 20 symbols, from which he can resolve the 12 desired and 8 undesired symbols. The following function maps 6 queried symbols to 5 downloaded symbols. ℒ(X_1, X_2, X_3, Y_1, Y_2, Y_3) = (X_1, X_2, Y_1, Y_2, X_3 + Y_3) Note that the first four symbols are directly downloaded and only the last symbol is mixed. The desired and undesired symbols are combined to produce the answers as follows. A_n^[k] = ℒ(C_n A_n^[k](W_1), C_n A_n^[k](W_2)) where C_n are deterministic 3× 3 matrices, that are required to satisfy the following two properties. Denote the first 2 rows of C_n as C_n. P1. All C_n must have full rank. P2. For all (3!)^4 distinct realizations of π_n', n ∈ [1:4], the 8 linear combinations of the undesired message symbols that are directly downloaded (2 from each server), C_1 A_1^[k](W_k^c), C_2 A_2^[k](W_k^c), C_3 A_3^[k](W_k^c), C_4 A_4^[k](W_k^c) are independent. As we will prove in the sequel, it is not difficult to find matrices that satisfy these properties. In fact, these properties are `generic', i.e., uniformly random choices of C_n matrices will satisfy these properties with probability approaching 1 as the field size approaches infinity. The appeal to generic property will be particularly useful as we consider larger classes of MDS-TPIR settings. Those (weaker) proofs apply here as well. However, for the particular setting of Theorem <ref>, based on a brute force search we are able to strengthen the proof by presenting the following explicit choice of C_n, n∈[1:4] which satisfies both properties over 𝔽_349. C_1 = ( [ 1 2 3; 6 5 4; 0 0 1 ]),  C_2 = ( [ 1 7 3; 11 9 8; 0 0 1 ]),  C_3 = ( [ 1 10 8; 7 5 4; 0 0 1 ]),  C_4 = ( [ 1 3 5; 12 9 3; 0 0 1 ]) Property P1 is trivially verified. Property P2 is verified by considering one by one, all of the 6^4 distinct realizations of π_n', n∈[1:4]. To show how this is done, let us consider one case here. Suppose the realization of the permutations is such that π_1'(𝒰_1) = (U_0, U_6, U_8) π_2'(𝒰_2) = (U_0, U_9, U_7) π_3'(𝒰_3) = (U_1, U_3, U_0) π_4'(𝒰_4) = (U_2, U_4, U_0) then we have (C_1 A_1^[k](W_k^c); ⋯; C_4 A_4^[k](W_k^c)) = ([ 1 2 0 -3 0 3 0 3; 6 5 0 -4 0 4 0 4; 0 -3 1 7 3 0 3 0; 0 -8 11 9 8 0 8 0; 8 0 8 0 1 10 0 0; 4 0 4 0 7 5 0 0; 5 0 10 0 0 0 1 3; 3 0 6 0 0 0 12 9 ])_≜𝒞( [ U_0 x; U_6 x; U_0 y; U_9 y; U_1 ( x+ y); U_3 ( x+ y); U_2 ( x+2 y); U_4 ( x+2 y) ]) The determinant of 𝒞 over 𝔽_349 is 321. Since the determinant is non-zero, all of its 8 rows are linearly independent. Note that the test for property P2 does not depend on the realizations of U_i vectors. To see why this is true, note that the 8 linear combinations of ( x, y) in the rightmost column vector of (<ref>) are linearly independent. Therefore, if 𝒞 is an invertible matrix then the 8 directly downloaded linear combinations on the LHS of (<ref>) are also independent (have joint entropy 8 p-ary units, conditioned on U_0:9). At this point the construction of the scheme is complete. All that remains now is to prove that the scheme is correct, i.e., it retrieves the desired message, and that it is T=2 private. §.§ The Scheme is Correct (Retrieves Desired Message) As noted previously, the first 4 variables in the output of the ℒ function are obtained directly, i.e., C_1 A_1^[k](W_1), C_2 A_2^[k](W_1), C_3 A_3^[k](W_1), C_4 A_4^[k](W_1) and C_1 A_1^[k](W_2), C_2 A_2^[k](W_2), C_3 A_3^[k](W_2), C_4 A_4^[k](W_2) are all directly recovered. By property P2 of C_n, C_1 A_1^[k](W_k^c), C_2 A_2^[k](W_k^c), C_3 A_3^[k](W_k^c), C_4 A_4^[k](W_k^c) are linearly independent. Since the user has recovered 8 independent dimensions of interference, and interference only spans 8 dimensions, all interference is recovered and eliminated. Once the interference is eliminated, since C_n matrices have full rank, the user is left with 12 independent linear combinations of desired symbols, from which he is able to recover the 12 desired message symbols. Therefore the scheme is correct. §.§ The Scheme is Private (to any T=2 Colluding Servers) To prove that the scheme is T = 2 private (refer to (<ref>)), it suffices to show that the queries for any 2 servers are identically distributed, regardless of which message is desired. Since each query is made up of two independently generated parts, one for each message, it suffices to prove that the query vectors for a message (say W_k) are identically distributed, regardless of whether the message is desired or undesired, (Q_n_1^[k](W_k), Q_n_2^[k](W_k)) ∼(Q_n_1^[k^c](W_k), Q_n_2^[k^c] (W_k)),  ∀ n_1, n_2 ∈ [1:4], n_1 < n_2 Note that (Q_n_1^[k](W_k), Q_n_2^[k](W_k)) = (π_n_1(𝒱_n_1), π_n_2(𝒱_n_2) ) (Q_n_1^[k^c](W_k), Q_n_2^[k^c](W_k)) = (π_n_1' (𝒰_n_1), π_n_2' (𝒰_n_2) ) Therefore, to prove (<ref>) it suffices to show the following. ( V_i_1, V_i_2, V_i_3, V_i_4, V_i_5) ∼( U_0, U_j_1, U_j_2, U_j_3, U_j_4) where 𝒱_n_1 = {V_i_1, V_i_2, V_i_3}, 𝒱_n_2 = {V_i_1,V_i_4,V_i_5}, 𝒰_n_1 = {U_0, U_j_1, U_j_2}, 𝒰_n_2 = {U_0, U_j_3,U_j_4}. Because S is uniformly chosen from the set of all full rank matrices, we have (V_i_1, V_i_2, V_i_3, V_i_4, V_i_5) ∼ (V_1, V_2, V_3, V_4, V_5) Next we note that there is a bijection between (U_0, U_j_1, U_j_2, U_j_3, U_j_4) ↔ (U_0, U_1, U_2, U_3, U_4) This is because (U_0, U_j_1, U_j_2, U_j_3, U_j_4) always includes U_0, two terms out of U_1, U_2, U_6, U_7 and two terms out of U_3, U_4, U_8, U_9. But from any two terms of U_1, U_2, U_6, U_7 there is a bijection to U_1, U_2, and from any two terms of U_3,U_4,U_8, U_9 there is a bijection to U_3, U_4. Now since S' = (U_0; U_1; U_2; U_3; U_4; U_5) is picked uniformly from 𝒮, conditioned on any feasible value of U_5, (U_0, U_1, U_2, U_3, U_4) is uniformly distributed over all possible values that preserve full rank for S'. Since (U_0, U_j_1, U_j_2, U_j_3, U_j_4) spans the same space as (U_0, U_1, U_2, U_3, U_4), they have the same set of feasible values. The bijection between them then means that (U_0, U_j_1, U_j_2, U_j_3, U_j_4) is also uniformly distributed over all possibilities that preserve full rank for S', conditioned on any feasible U_5. That means (U_0, U_j_1, U_j_2, U_j_3, U_j_4) ∼ (U_0, U_1, U_2, U_3, U_4) Finally, we note that S and S' are identically distributed, so we have (V_1, V_2, V_3, V_4, V_5) ∼ (U_0, U_1, U_2, U_3, U_4) Combining (<ref>), (<ref>) and (<ref>), we arrive at (<ref>) and (<ref>). §.§ Rate achieved is 3/5 The rate achieved is 12/20 = 3/5, because we download 20 symbols in total (5 from each server) and the desired message size is 12 symbols. § OPTIMALITY OF RATE 3/5 We presented a scheme that achieves the rate 3/5 for the setting (K, N, T, K_c) = (2, 4, 2, 2) with the MDS storage code (x,y)→(x,y,x+y,x+2y). But is the scheme optimal? i.e., is the rate 3/5 the highest rate possible for this setting? To settle this question we need an upper bound. So far the best information theoretic upper bound that we are able to prove is 8/13[Remarkably, 8/13 can be shown to be the capacity if the colluding sets of servers are restricted to servers {1,2},{2,3}, {3,4},{4,1} (see Section <ref>).] (see Section <ref>), which leaves the information theoretic capacity open for this setting. However, let us define the notion of “linear capacity" as the highest rate that can be achieved by any (scalar or vector) linear PIR scheme. It turns out that we are able to settle the linear capacity. For the MDS-TPIR problem with (K, N, T, K_c) = (2, 4, 2, 2) and the MDS storage code (x,y)→(x,y,x+y,x+2y), the linear capacity is 3/5. Proof: Since the achievability of 3/5 has already been shown, we are left to prove the converse, i.e., the upper bound. Let a, b, c, d∈𝔽_p^L/2× 1 be i.i.d. uniform L/2× 1 vectors over 𝔽_p. Without loss of generality, the MDS storage code for message W_k is represented as follows. W_1 = ( a; b) W_2 = ( c; d) and the storage is specified as (W_11,W_12, W_13, W_14) = ( a, b, a +b, a+2b) (W_21,W_22, W_23, W_24) = ( c, d, c +d, c+2d) The scheme is linear so that the download from each server consists of linear combinations of the stored symbols of both messages. Furthermore, without loss of generality, we assume that the scheme is symmetric[Any scheme can be made symmetric, e.g., by repeating the original scheme for each of the N! permutations of the servers to retrieve a correspondingly expanded message of length L'=N! L.] and the download from each server is comprised of d≤ L/2 independent symbols from each message. Therefore, the downloads can be expressed as A_n^[k] = V_1n^[k] W_1n+V_2n^[k] W_2n, ∀ n ∈ [1:4], k∈[1:2] (V_1n^[k]) = (V_2n^[k])=d where V_in^[k] are D/4× L/2 matrices that may be chosen randomly by the user (functions of ℱ). Clearly we must have 4d≥ L otherwise the L symbols of the desired message cannot be recovered. Define ϵ≥ 0 such that 4d = L(1+ϵ) Without loss of generality, let us assume henceforth that W_2 is the desired message. For the next set of arguments, we focus only on the downloads corresponding to W_2, i.e., set all W_1 symbols to 0. Further, let us use the notation 𝖵 to represent the row span of the matrix V. The symbols downloaded from Server n along 𝖵⊂𝖵^[2]_2n, are called redundant if they can be expressed as linear combinations of symbols downloaded from other servers, i.e., they contribute no new information. H(V W_2n|V^[2]_2n_1 W_2n_1, V^[2]_2n_2 W_2 n_2, V^[2]_2n_3W_2n_3,ℱ, V) =0 where n, n_1, n_2, n_3 are distinct indices in [1:4]. Note that we download no more than a total of L(1+ϵ) (possibly dependent) symbols of W_2 from all 4 servers, from which we must be able to decode all L independent symbols of W_2. Therefore, we cannot have more than ϵ L redundant symbols. Therefore, for any V that satisfies (<ref>) we must have (𝖵) ≤ ϵ L Next, let us consider the pairwise overlap between 𝖵^[2]_2i and 𝖵^[2]_2j, i < j, i, j ∈ [1:4]. By the symmetry of the scheme, there exist V_ij, ∀ i,j∈[1:4], i≠ j, and α≥ 0 such that 𝖵_ij = 𝖵^[2]_2i∩𝖵^[2]_2j,  (𝖵_ij) = α d The following lemma formalizes the intuition that the overlaps α must be small enough to ensure that we have enough independent symbols to recover W_2. 3α d ≤ d + 2ϵ L α ≤ 1/3 + 8/3( ϵ/1+ϵ) Proof: First, we show that (𝖵_12∩𝖵_13) ≤ϵ L For any vector v ∈𝖵_12∩𝖵_13 (note that v belongs simultaneously to 𝖵_21^[2], 𝖵_22^[2], 𝖵_23^[2]), the symbol v W_23 (downloaded from Server 3) is redundant because it is a linear combination of downloads from servers 1 and 2, v( c+d) = v c+v d ∴ v W_23 = v W_21 + v W_22 ⇒ H(v W_23|V^[2]_21 W_21, V^[2]_22 W_2 2,ℱ,v) = 0 From (<ref>) and (<ref>), we have (<ref>). Second, we show that ((𝖵_12∪𝖵_13) ∩𝖵_14) ≤ϵ L Consider any vector v ∈𝖵_12. Because v belongs to both 𝖵_21^[2] and 𝖵_22^[2], we have downloaded v W_21=v c and v W_22=v d from servers 1 and 2. Similarly, for any vector v' ∈𝖵_13, we have downloaded v' W_21=v' c and v' W_23 = v'( c+d)=v' W_21 + v' W_22 (from servers 1 and 3), from which we can recover v' W_21=v' c and v' W_22=v' d. Consider now any vector v^* ∈ (𝖵_12∪𝖵_13) ∩𝖵_14. Suppose v^* = h_1 v + h_2 v', v ∈𝖵_12, v' ∈𝖵_13 for constants h_1, h_2. The symbol v^* W_24=v^*( c+2d) (downloaded from Server 4) is redundant because it is a linear combination of downloads from servers 1, 2 and 3, v^* W_24 = (h_1 v + h_2 v')( c+2d) = h_1 v c + 2h_1 v d + h_2 v' c + 2h_2 v' d = h_1 v W_21 + 2h_1 v W_22 + h_2 v' W_21 + 2h_2 v' W_22 ⇒   H(v^* W_24 |V^[2]_21 W_21, V^[2]_22 W_2 2, V^[2]_23 W_23,ℱ, v^*) = 0 From (<ref>) and (<ref>), we have (<ref>). Next, consider (𝖵_12∪𝖵_13). (𝖵_12∪𝖵_13)    = (𝖵_12) + (𝖵_13) - (𝖵_12∩𝖵_13) ≥ 2α d - ϵ L   ( (<ref>)(<ref>)) Finally, consider (𝖵_12∪𝖵_13∪𝖵_14). d = (𝖵_21^[2]) ≥(𝖵_12∪𝖵_13∪𝖵_14) = (𝖵_12∪𝖵_13) + (𝖵_14) - ( (𝖵_12∪𝖵_13) ∩𝖵_14) ≥  2α d- ϵ L + α d - ϵ L    ( (<ref>)(<ref>)(<ref>)) ⇒ 3α d ≤ d + 2ϵ L We now proceed to complete the converse. D +o(L)L ≥ H(A_1:4^[1] | ℱ, 𝒢) + o(L)L (<ref>)= H(A_1:4^[1], W_1 | ℱ, 𝒢) (<ref>)= H(W_1) + H(A_1^[1] | W_1, ℱ, 𝒢) + H(A_2:4^[1] | W_1, A_1^[1]ℱ, 𝒢) ≥ H(W_1) + H(A_1^[1] | W_1, ℱ, 𝒢) + H(A_3:4^[1] | W_1, W_21, A_1^[1], ℱ, 𝒢) (<ref>)(<ref>)(<ref>)= H(W_1) + H(A_1^[1] | W_1, ℱ, 𝒢) + H(A_3:4^[1] | W_1, W_21, ℱ, 𝒢) (<ref>)(<ref>)= H(W_1) + H(A_1^[2] | W_1, ℱ, 𝒢) + H(A_3:4^[2] | W_1, W_21, ℱ, 𝒢) (<ref>)(<ref>)= H( a,b) + H(V_21^[2] c| ℱ) + H(V_23^[2]( c+d),V_24^[2]( c+2d) | c, ℱ) = H( a,b) + H(V_21^[2] c| ℱ) + H(V_23^[2] d,2V_24^[2] d | ℱ) (<ref>)= L + (𝖵_21^[2]) + (𝖵_23^[2]∪𝖵_24^[2]) (<ref>)(<ref>)= L + d + 2d - α d (<ref>)≥ L + (3- 1/3 - 8/3( ϵ/1+ϵ))(1+ϵ)L/4 = 5L/3 Letting L→∞, we have R = L/D ≤ 3/5. § CAPACITY OF A CLASS OF MDS-TPIR INSTANCES Building upon the insights from the achievable scheme and linear converse presented in the previous sections, we are able to settle the information theoretic capacity of a non-trivial class of MDS-TPIR instances. For the class of MDS-TPIR instances with (K,N, T, K_c)=(2,N,T,N-1), with arbitrary T,N, the capacity is C = N^2 - N/2N^2 - 3N + T. The case T=N is trivial because if all servers collude then the situation is equivalent to the single database scenario, i.e., it is optimal to download everything, and the capacity is 1/K=1/2. For the remaining cases, T<N, and the proof of converse is presented in Section <ref>. The proof of achievability for T=2 setting appears in Section <ref> where we present a scheme with zero error. The proof of achievability for T>2 settings appears in Section <ref> where we present a scheme with vanishing probability of error. The remainder of this section presents two examples (one with T=2 and one with T=3) to illustrate the key ideas. §.§ Example: Capacity achieving scheme for (K, N, T, K_c)=(2, 4, 2,3) Let us present a scheme that achieves the rate 6/11, which is the capacity for this setting according to Theorem <ref>. As evident from the description below, the scheme builds upon the ideas that were introduced for Theorem <ref>. §.§.§ Message and Storage Code Let each message be comprised of L=N(N-1) = 12 independent symbols from a sufficiently large finite field 𝔽_p. Define a∈𝔽_p^4 × 1 as the vector (a_1; a_2; a_3; a_4) comprised of i.i.d. uniform symbols a_i ∈𝔽_p. Vectors b, c, d, e, f are defined similarly. Messages W_1, W_2 are defined in terms of these vectors as follows. W_1 = ( a; b; c) W_2 = ( d; e; f) The (N-1,N) = (3,4) MDS storage code is specified as follows. (W_11, W_12, W_13, W_14) = ( a, b, c, a+b+c) (W_21, W_22, W_23, W_24) = ( d, e, f, d+e+f) Note that each server stores 4 symbols for each message and any three serves store just enough information to recover both messages (MDS property is satisfied). §.§.§ Construction of Queries The query to each server consists of 6 vectors, the first three for W_1 (denoted as Q_n^[k](W_1)) and the last three for W_2 (denoted as Q_n^[k](W_2)). The queries and downloads for W_k, k ∈ [1:2] are described next. We denote W_k = ( x; y; z). When k = 1, ( x; y; z) = ( a; b; c) and when k = 2, ( x; y; z) = ( d; e; f). Denote the set of all full rank 4 × 4 matrices over 𝔽_p as 𝒮_4. The user privately chooses two matrices S, S', independently and uniformly from 𝒮_4. Label the rows of S as V_1, V_2, V_3, V_4, and the rows of S' as U_1, U_2, U_1, U_2. Define the following sets [ 𝒱_1 = { V_2, V_3, V_4},; 𝒱_2 = {V_1, V_3, V_4},; 𝒱_3 = {V_1, V_2, V_4},; 𝒱_4 = {V_1, V_2, V_3 }, ] [ 𝒰_1 = {U_1, U_2, U_1}; 𝒰_2 = {U_1, U_2, U_2}; 𝒰_3 = {U_1, U_2, U_3}; 𝒰_4 = {U_1, U_2, U_4}; ] where U_3, U_4 are obtained as follows. U_3 = U_1 + U_2, U_4 = U_1 + 2U_2 A preview of the scheme is as follows. For Server n ∈ [1:4], the vectors in 𝒱_n are for the desired message and the vectors in 𝒰_n are for the undesired message. Since K_c = N-1 = 3, and each query vector V_i is used no more than three times, all queries for the desired message will return independent symbols for a total of 12 desired symbols. For the undesired message, the same query vector U_1 is used 4 times such that only 3 independent symbols are produced. Similarly the 4 uses of U_2 produce only 3 independent symbols. Thus all queries for the undesired message will produce at most 6 + 4 = 10 independent undesired symbols. The 12 independent desired symbols and 10 undesired symbols will be resolved from a total of 12 + 10 = 22 downloaded symbols, to achieve the rate 12/22 = 6/11. Privacy is ensured by the observation that any 𝒱_n, 𝒱_n', n≠ n' have two elements in common and similarly any 𝒰_n, 𝒰_n', n≠ n' have two elements in common. We now proceed to the details. When W_k is desired, we have ∀ n ∈ [1:4], n: Q_n^[k](W_k) = π_n (𝒱_n), A_n^[k](W_k) = Q_n^[k](W_k) W_kn. Desired Symbols Are Independent: From A_1:4^[k](W_k), the user can recover the 12 symbols V_2 x, V_3 x, V_4 x, V_1 y, V_3 y, V_4 y, V_1 z, V_2 z, V_4 z, V_1( x+ y+ z), V_2( x+ y+ z), V_3( x+ y+ z) and therefore all 12 symbols (x; y; z) of W_k, since S = (V_1;V_2;V_3;V_4) has full rank. When W_k is undesired, we have ∀ n ∈ [1:4], n: Q_n^[k^c](W_k) = π_n' (𝒰_n), A_n^[k^c](W_k) = Q_n^[k^c](W_k) W_kn. Interfering Symbols Are Dependent and Have Dimension at most 10: Consider the interfering symbols along the common vectors U_1, U_2. Note that U_1 x + U_1 y + U_1 z = U_1 ( x + y + z) U_2 x + U_2 y + U_2 z = U_2 ( x + y + z) Since at least 2 interfering symbols are linear combinations of the rest, the 12 interfering symbols cannot have more than 10 dimensions, i.e., their joint entropy is no more than 10 in p-ary units. §.§.§ Combining Answers, Correctness and Rate The combining process and correctness proof are similar to that in Theorem <ref>. The difference is that in Theorem <ref>, we find the explicit choice of combining matrices, here we will only prove the existence of combining matrices over a sufficiently large field. The details are deferred to the general proof in Section <ref>. We repeat the above query construction two times independently such that each server has 6 × 2 = 12 symbols (6 in W_1 and 6 in W_2). These 12 symbols at each server are combined to 11 downloaded symbols, A_n^[k] and it is ensured that we can decode all interfering symbols and then extract the desired symbols. Thus, the rate achieved is 6/11. §.§.§ Privacy Proof The privacy proof is virtually identical to that in Theorem <ref>, so the details are deferred to the general proof in Section <ref>. §.§ Example: Capacity achieving scheme for (K, N, T, K_c)=(2, 4,3,3) Let us present a scheme that achieves the rate 12/23, which is the capacity for this setting according to Theorem <ref>. The key distinction of this T = 3 case with the T=2 case presented in the previous section is that permutations of the query vectors are no longer enough to ensure the privacy. So we will resort to sending the space spanned by the query vectors instead of the query vectors themselves. Furthermore, instead of guaranteeing zero-error, we will only show that the probability of error can be made arbitrarily small by choosing a sufficiently large message size. §.§.§ Message and Storage Code The message construction and storage code are the same as when T=2. Let each message be comprised of L=N(N-1) = 12 independent symbols from a sufficiently large finite field 𝔽_p. Define a∈𝔽_p^4 × 1 as the vector (a_1; a_2; a_3; a_4) comprised of i.i.d. uniform symbols a_i ∈𝔽_p. Vectors b, c, d, e, f are defined similarly. Messages W_1, W_2 are defined in terms of these vectors as follows. W_1 = ( a; b; c) W_2 = ( d; e; f) The (N-1,N) = (3,4) MDS storage code is specified as follows. (W_11, W_12, W_13, W_14) = ( a, b, c, a+b+c) (W_21, W_22, W_23, W_24) = ( d, e, f, d+e+f) §.§.§ Construction of Queries The query to each server consists of two vector spaces, one for W_1 (span of the rows of Q_n^[k](W_1)) and one for W_2 (span of the rows of Q_n^[k](W_2)). The queries and downloads for W_k, k ∈ [1:2] are described next. We denote W_k = ( x; y; z). When k = 1, ( x; y; z) = ( a; b; c) and when k = 2, ( x; y; z) = ( d; e; f). Denote the set of all full rank 4 × 4 matrices over 𝔽_p as 𝒮_4. The user privately chooses two matrices S, S', independently and uniformly from 𝒮_4. Label the rows of S as V_1, V_2, V_3, V_4, and the rows of S' as U_1, U_1, U_2, U_3. Define the following sets [ 𝒱_1 = { V_2, V_3, V_4},; 𝒱_2 = {V_1, V_3, V_4},; 𝒱_3 = {V_1, V_2, V_4},; 𝒱_4 = {V_1, V_2, V_3 }, ] [ 𝒰_1 = {U_1, U_1, U_2} = {U_1, U_1, U_2 }; 𝒰_2 = {U_1, U_3, U_4} = {U_1, U_3, U_1+U_2 }; 𝒰_3 = {U_1, U_5, U_6} = {U_1, U_1+U_3, U_2+U_3 }; 𝒰_4 = {U_1, U_7, U_8} = {U_1, U_1+U_2+U_3, U_1+2U_2+2U_3 } ] where U_1, ⋯, U_8 are the rows of U, obtained as follows. U = P (U_1; U_2; U_3) ([ U_1; U_2; U_3; U_4; U_5; U_6; U_7; U_8 ]) = ( [ 1 0 0; 0 1 0; 0 0 1; 1 1 0; 1 0 1; 0 1 1; 1 1 1; 1 2 2 ])([ U_1; U_2; U_3 ]) A preview of the scheme is as follows. For Server n ∈ [1:4], the span of 𝒱_n is the query space for the desired message and the span of 𝒰_n is the query space for the undesired message. Since K_c = N-1 = 3, and each query vector V_i is used no more than three times, all queries for the desired message will return independent symbols for a total of 12 desired symbols. For the undesired message, the same query vector U_1 is used 4 times such that only 3 independent symbols will be produced. Thus all queries for the undesired message will produce at most 3 + 8 = 11 independent undesired symbols. The 12 independent desired symbols and 11 undesired symbols will be resolved from a total of 12 + 11 = 23 downloaded symbols, to achieve the rate 12/23. Privacy is ensured by choosing P in such a way that it allows a bijective mapping between the 𝒰_n or 𝒱_n spaces that may be observed by any set of up to T = 3 colluding servers. The bijection shows that the queries for both desired and undesired messages are uniformly distributed, and therefore indistinguishable. While a specific P is chosen for this example, there are many choices of P that will work. In fact, P only needs to be sufficiently generic, so as the field size grows, almost all choices of P will work. We now proceed to the details. When W_k is desired, we have ∀ n ∈ [1:4], n: Q_n^[k](W_k) = 𝔹 (𝒱_n), A_n^[k](W_k) = Q_n^[k](W_k) W_kn. where 𝔹(𝒱) represents the reduced row echelon form of a matrix whose rows are the elements of 𝒱. The reduced row echelon form ensures that the queries reveal only the space spanned by the corresponding V_i vectors to each server, and not directly the V_i vectors themselves. Desired Symbols Are Independent: From A_1:4^[k](W_k), we can recover the 12 symbols of W_k. Note that because the user knows V_1:4, from A_1:4^[k](W_k) he can recover the projections along V_i. For example, the row reduced echelon form for 𝒱_1 is a change of basis operation that can be represented as 𝔹(𝒱_1)=B_1(V_2;V_3;V_4) for some invertible matrix B_1. Since the user knows B_1, he can multiply A_1^[k](W_k) with (B_1)^-1 as follows B_1^-1A_1^[k](W_k) = B_1^-1Q_n^[k](W_k) W_k1 = B_1^-1B_1(V_2;V_3;V_4) x = (V_2 x; V_3 x; V_4 x) Thus, from A_1:4^[k](W_k) the user recovers the 12 symbols V_2 x, V_3 x, V_4 x, V_1 y, V_3 y, V_4 y, V_1 z, V_2 z, V_4 z, V_1( x+ y+ z), V_2( x+ y+ z), V_3( x+ y+ z) and therefore all 12 symbols (x; y; z) of W_k, since S = (V_1;V_2;V_3;V_4) has full rank. When W_k is undesired, we have ∀ n ∈ [1:4], n: Q_n^[k^c](W_k) = 𝔹 (𝒰_n), A_n^[k^c](W_k) = Q_n^[k^c](W_k) W_kn. Interfering Symbols Are Dependent and Have Dimension at most 11: Consider the interfering symbols along the common vector U_1. Note that U_1 x + U_1 y + U_1 z = U_1 ( x + y + z) Since at least 1 interfering symbol is a linear combination of the rest, the 12 interfering symbols cannot have more than 11 dimensions, i.e., their joint entropy is no more than 11 in p-ary units. §.§.§ Combining Answers, Correctness and Rate The combining process and correctness proof are similar to that in Theorem 1 except that the combining matrices C_n are chosen in a uniformly random manner now (so the matrices are no longer deterministic). We will show in Section <ref> that independent and uniformly random choices of C_n are enough to guarantee that as the field size approaches infinity, i.e., p→∞, the probability of error, P_e→ 0. The reasoning for the rate calculation is as follows. We repeat the above query construction four times independently such that each server has 6 × 4 = 24 symbols (12 in W_1 and 12 in W_2). These 24 symbols at each server are combined to 23 downloaded symbols, A_n^[k] and it is ensured that we can almost surely decode all interfering symbols and then extract the desired symbols. Thus, the rate achieved is 12/23. §.§.§ Privacy Proof Since the privacy proof is a bit more involved now, let us use this example to introduce the key ideas. To show that the scheme is private to any T = 3 colluding servers, it suffices to show that the queries for W_k for any T = 3 servers are identically distributed, regardless of which message is desired. Consider 3 distinct indices i , j , l, i < j <l in [1:4], we require (Q_i^[k](W_k), Q_j^[k](W_k), Q_l^[k](W_k) ) ∼ (Q_i^[k^c](W_k), Q_j^[k^c](W_k), Q_l^[k^c](W_k) ) ⟺    (𝔹(𝒱_i), 𝔹(𝒱_j), 𝔹(𝒱_l) ) ∼ ( 𝔹(𝒰_i), 𝔹(𝒰_j), 𝔹(𝒰_l) ) Note that (𝔹(𝒱_i), 𝔹(𝒱_j), 𝔹(𝒱_l) ) = (𝔹({V_m, V_j, V_l}), 𝔹({V_m, V_i, V_l}), 𝔹({V_m, V_i, V_j}) ) where m ∉{i,j,l}, m ∈ [1:4]. To prove (<ref>), we wish to transform the spaces on the RHS to the form that is the same as (<ref>). To this end, we first compute the vectors that lie in the span of both 𝔹(𝒰_i) and 𝔹(𝒰_j), i < j. Note that the matrix P is designed such that except U_1, we have only one such vector (up to scaling), denoted as U_{i,j}. U_{i,j} are computed explicitly as follows. Further, we fix the scaling factor such that the U_{i,j} vector is unique. U_{1,2} = U_1 + U_2 U_{1,3} = U_1 - U_2 U_{1,4} = U_1 U_{2,3} = U_1 + U_2 + 2U_3 U_{2,4} = U_1 + U_2 + U_3 U_{3,4} = U_2 + U_3 It is easy to verify that U_{i,j}, U_{i,l}, U_{j,l}, i,j,l ∈ [1:4], i < j < l, are linearly independent, i.e., (i,j,l) = (1,2,3) (U_{1,2}; U_{1,3} ; U_{2,3}) = (U_1 + U_2; U_1 - U_2; U_1 + U_2 + 2U_3) = 3 (i,j,l) = (1,2,4) (U_{1,2}; U_{1,4}; U_{2,4}) = (U_1 + U_2; U_1; U_1 + U_2 + U_3) = 3 (i,j,l) = (1,3,4) (U_{1,3}; U_{1,4}; U_{3,4}) = (U_1 - U_2; U_1; U_2 + U_3) = 3 (i,j,l) = (2,3,4) (U_{2,3}; U_{2,4}; U_{3,4}) = (U_1 + U_2 + 2U_3; U_1 + U_2 + U_3; U_2 + U_3) = 3 As a result, we may equivalently represent Q_i^[k^c](W_k) as Q_i^[k^c](W_k) = 𝔹(𝒰_i) = 𝔹({U_1, U_{i,j}, U_{i,l}}), ∀ i,j,l ∈ [1:4], i ≠ j, i≠ l, j ≠ l Note that equipped with this representation, ( 𝔹(𝒰_i), 𝔹(𝒰_j), 𝔹(𝒰_l) ) is now of the same form as (𝔹(𝒱_i), 𝔹(𝒱_j), 𝔹(𝒱_l) ) and we are now ready to prove the privacy condition (<ref>). (<ref>) ⟺ (𝔹({V_m, V_j, V_l}), 𝔹({V_m, V_i, V_l}), 𝔹({V_m, V_i, V_j}) ) ∼ ( 𝔹({U_1, U_{i,j}, U_{i,l}}), 𝔹({U_1, U_{i,j}, U_{j,l}}), 𝔹({U_1, U_{i,l}, U_{j,l}}) ) Therefore, it suffices to show the following. (V_m, V_i, V_j, V_l) ∼ (U_1, U_{j,l}, U_{i,l}, U_{i,j}) Because S is uniformly chosen from the set of all full rank matrices, we have (V_m, V_i, V_j, V_l) ∼ (V_1, V_2, V_3, V_4) Based on (<ref>), there is a bijection between (U_1, U_{j,l}, U_{i,l}, U_{i,j}) ↔ (U_1, U_1, U_2, U_3) Now since S' = (U_1; U_1; U_2; U_3) is uniform in all full rank matrices, the above bijection then means that (U_1; U_{j,l}; U_{i,l}; U_{i,j}) is also uniform in all full rank matrices, i.e., (U_1, U_{j,l}, U_{i,l}, U_{i,j}) ∼ (U_1, U_1, U_2, U_3) Finally, note that S and S' have the same distribution, so we have (V_1, V_2, V_3, V_4) ∼ (U_1, U_1, U_2, U_3) Therefore, from (<ref>), (<ref>) and (<ref>), we have proved (<ref>) and (<ref>). § CONCLUSION We settle a conjecture on the capacity of MDS-TPIR by Freij-Hollanti et al. <cit.> by constructing a scheme that beats the conjectured capacity for one particular instance of MDS-TPIR. The rate achieved by the new scheme is shown to be the best possible rate that can be achieved by any linear scheme for the same MDS storage code. The insights from the achievability and converse arguments allow us to characterize the capacity of a class of MDS-TPIR instances. Through another counterexample, we are also able to prove that the capacity expression cannot be symmetric in T and K_c parameters, i.e., these parameters cannot be interchangeable in general. Nevertheless, the general capacity expression for MDS-TPIR remains unknown. § APPENDIX §.§ Examples of Optimal Schemes over Small Fields To highlight that the assumption of large field size (which was made convenience) may not be essential, in this section, we provide two examples of explicit MDS-TPIR capacity achieving schemes over small fields. §.§.§ Example 1 Consider the MDS-TPIR instance with (K,N,T,K_c) = (2,3,2,2). Note that the capacity of this setting is 6/11, as established in Theorem <ref>. We provide an alternative achievable scheme for rate 6/11. In particular, the scheme operates over the binary field and the upload is 4 bits per server (the query to each server takes values in a set with cardinality 2^4 = 16). We assume that each message is L = 6 bits. Denote a_1, ⋯, a_6, b_1, ⋯, b_6 as 12 i.i.d. uniform bits, a_i, b_i ∈𝔽_2. Messages W_1, W_2 are defined in terms of these bits as follows. W_1 = (a_1; a_2; a_3; a_4; a_5; a_6), W_2 = (b_1; b_2; b_3; b_4; b_5; b_6) The storage is specified as : W_11 = (a_1; a_2; a_3), W_21 = (b_1; b_2; b_3) : W_12 = (a_4; a_5; a_6), W_22 = (b_4; b_5; b_6) : W_13 = (α_1; α_2; α_3), W_23 = (β_1; β_2; β_3) where α_1, α_2, α_3, β_1, β_2, β_3 are obtained as follows. α_1 = a_1 + a_2 + a_5, β_1 = b_1 + b_2 + b_5 α_2 = a_1 + a_3 + a_6, β_2 = b_1 + b_3 + b_6 α_3 = a_2 + a_4 + a_6, β_3 = b_2 + b_4 + b_6 Further define α_4 = α_1 + α_2 + α_3 = a_3 + a_4 + a_5 β_4 = β_1 + β_2 + β_3 = b_3 + b_4 + b_5 Note that each server stores 3 bits of each message and the storage at any 2 servers is just enough to recover both messages (MDS storage property is satisfied). Define a function that maps 4 input bits to 3 output bits as follows. ℒ_3 (X_1, X_2, Y_1, Y_2) = (X_1+Y_2, X_2+Y_2, Y_1+Y_2) We now describe the PIR scheme. ℱ is a uniform random variable in [1:16]. Depending on the value of ℱ and the desired message index θ∈ [1:2], the user's query is specified by Table 1. The double-quotes notation around a random variable represents the query about its realization. Note that the queries to Server 1 and Server 2 are the same, regardless of the value of θ and the query to Server 3 is a deterministic function of that to Server 1 and Server 2. We show that the scheme is both correct and private. The schemes is correct because our scheme satisfies the important property (P1) that from the answers A_1^[k], A_2^[k], we always know one undesired bit in A_3^[k] and then we can extract the 2 desired bits in A_3^[k] (because if any 1 of the 4 input bits of the ℒ_3 function is known, the remaining 3 input bits can be solved from the 3 output bits). Combining these 2 desired bits with the other 4 desired bits (2 from Server 1 and 2 from Server 2), we obtain the desired message (easy to verify that these 6 bits are independent). The property (P1) is easy to verify. For example, consider k=1 and ℱ = 8. From A_1^[1], A_2^[1], we obtain b_1, b_3, b_4, b_5, from which we further obtain β_4 = b_3 + b_4 + b_5 and β_4 appears in A_3^[1]. The scheme is private because it is easy to verify that for any 2 servers, the queries are identically distributed no matter which message is desired and then the privacy condition (<ref>) is satisfied. The scheme downloads 4 bits from Server 1, 4 bits from Server 2 and 3 bits from Serve 3. It retrieves 6 desired message bits. Therefore the rate is 6/11. §.§.§ Example 2 Consider the MDS-TPIR instance with (K,N,T,K_c) = (2,4,3,2). The capacity of this setting turns out to be 4/7. The rate can not be more than 4/7 because the capacity of TPIR with (K,N,T) = (2,4,3) is 4/7 <cit.> and reducing K_c from 2 to 1 can not hurt. We provide an achievable scheme for rate 4/7. In particular, the scheme operates over the finite field 𝔽_13 and the upload is 6 bits per server (the query to each server takes values in a set with cardinality 2^6 = 64). We assume that each message is L = 4 symbols. Denote a_1, a_2, a_3, a_4, b_1, b_2, b_3, b_4 as 8 i.i.d. uniform symbols, a_i, b_i ∈𝔽_13. Messages W_1, W_2 are defined in terms of these symbols as follows. W_1 = (a_1; a_2; a_3; a_4), W_2 = (b_1; b_2; b_3; b_4) The storage is specified as : W_11 = (a_1; a_2), W_21 = (b_1; b_2) : W_12 = (a_3; a_4), W_22 = (b_3; b_4) : W_13 = (α_1; α_2), W_23 = (β_1; β_2) : W_14 = (α_3; α_4), W_24 = (β_3; β_4) where α_1, α_2, α_3, α_4, β_1, β_2, β_3, β_4 are obtained as follows. α_1 = 3a_1 + 2a_2 + 4a_3 + a_4, β_1 = 3b_1 + 2b_2 + 4b_3 + b_4 α_2 = 2a_1 +3a_2 + a_3 + 4a_4, β_2 = 2b_1 +3b_2 + b_3 + 4b_4 α_3 = 3a_1 + 12a_2 + 4a_3 + 6a_4, β_3 = 3b_1 + 12b_2 + 4b_3 + 6b_4 α_4 = 12a_1 + 3a_2 + 6a_3 + 4a_4, β_4 = 12b_1 + 3b_2 + 6b_3 + 4b_4 Note that each server stores 2 symbols of each message and the storage at any 2 servers is just enough to recover both messages (MDS storage property is satisfied). We now describe the PIR scheme. ℱ is a uniform random variable in [1:64]. The user's query is uniform over 64 choices and is specified by Table 2. Note that the queries to servers 1, 2 and 3 are the same, regardless of the value of θ and the query to Server 4 is a deterministic function of that to servers 1, 2 and 3. The key to the scheme is that the undesired symbol downloaded from Server 4 is known from that downloaded from servers 1, 2 and 3, while desired symbols are all independent. This observation is formalized in the following lemma. For all values of i_1, i_2, i_3, i_4, i_4', j_1, j_2, j_3, j_4, j_4' in Table 2, we have (a_i_1, a_i_2, α_i_3, α_i_4) = 4, (a_i_1, a_i_2, α_i_3, α_i_4') = 3 (b_j_1, b_j_2, β_j_3, β_j_4) = 3, (b_j_1, b_j_2, β_j_3, β_j_4') = 4 Lemma <ref> is proved by brute force, i.e., verifying (<ref>) and (<ref>) hold for each case. We show that the scheme is both correct and private. The schemes is correct because as Lemma <ref> has proved, the 4 undesired symbols only have dimension 3 and it is easy to see that the 3 undesired symbols in answers from the first 3 servers have dimension 3. Therefore, from the answers A_1^[k], A_2^[k], A_3^[k], we always know the undesired symbol in A_4^[k]. Subtracting the undesired symbol out from A_4^[k], we obtain the desired symbol interference freely. Lemma <ref> has proved that the 4 desired symbols are independent such that we can recover the desired message. The scheme is private because it is easy to verify that for any 3 servers, the queries are identically distributed no matter which message is desired and then the privacy condition (<ref>) is satisfied. The scheme downloads 2 symbols from Server 1, Server 2 and Server 3 each, and 1 symbol from Server 4. It retrieves 4 desired message symbols. Therefore the rate is 4/7. Let us conclude this example with the observation that this MDS-TPIR instance with (K,N,T,K_c) = (2,4,3,2) is not covered by Theorem <ref>, but we were still able to find its capacity. Let us also note that we are able to cast this example into a similar framework as Theorem <ref> and prove the existence of PIR schemes that achieve the same capacity for the (x,y)→(x,y,x+y,x+2y) MDS storage code, subject to the assumption of a sufficiently large finite field. The details are repetitive, and therefore omitted. However, we believe this example may provide useful insights for further generalizations. §.§ Achievability Proof for Theorem <ref> when T = 2 The proof for the general setting (arbitrary N) follows the same route as the N = 4 example presented earlier. We assume that each message is comprised of L=N(N-1) independent symbols from a sufficiently large finite field 𝔽_p. §.§.§ Storage Code The (N-1,N) MDS storage code is as follows. W_kn ∈ 𝔽_p^N × 1, k ∈ [1:2], n∈[1:N] W_k = (W_k1; W_k2; ⋯; W_k(N-1)) ∈𝔽_p^L × 1 W_kN = W_k1 + W_k2 + ⋯ + W_k(N-1) §.§.§ Construction of Queries The query to each server consists of 2(N-1) vectors, the first N-1 vectors for W_1 (Q_n^[k](W_1)) and the last N-1 vectors for W_2 (Q_n^[k](W_2)). The queries and downloads for W_k, k ∈ [1:2] are described next. Denote the set of all full rank N × N matrices over 𝔽_p as 𝒮_N. The user privately chooses two matrices S, S', independently and uniformly from 𝒮_N. Label the rows of S as V_1, ⋯, V_N, and the rows of S' as U_1, ⋯, U_N-2, U_1, U_2. Define ∀ n ∈ [1:N] 𝒱_n = {V_1, ⋯, V_n-1, V_n+1, ⋯, V_N} 𝒰_n = {U_1, ⋯, U_N-2, U_n} where U_n, n ∈ [1:N] are the rows of U, obtained as follows. U = _N × 2 (U_1; U_2) where _N × 2 is an N× 2 matrix such that any two of its rows are linearly independent. When W_k is desired, we have ∀ n, n: Q_n^[k](W_k) = π_n (𝒱_n), A_n^[k](W_k) = Q_n^[k](W_k) W_kn. Desired Symbols Are Independent: From A_1:N^[k](W_k), we can recover all N(N-1) symbols of W_k. This is easily seen because the storage is an (N-1, N) MDS code, no query dimension is repeated more than N-1 times and the matrix S has full rank. When W_k is undesired, we have ∀ n, n: Q_n^[k^c](W_k) = π_n' (𝒰_n), A_n^[k^c](W_k) = Q_n^[k^c](W_k) W_kn. Interfering Symbols Are Dependent and Have Dimension at most N(N-1) - (N-2): Consider the interfering symbols along the common vectors U_i, i ∈ [1:N-2]. Note that U_i W_k1 + ⋯ + U_i W_k(N-1) = U_i W_kN Therefore (N-2) interfering symbols are linear combinations of the other N^2 - 2N + 2 symbols. §.§.§ Combining Answers for Efficient Download Based on the queries, each server has 2(N-1) symbols, N-1 in W_1, A_n^[k](W_1) and N-1 in W_2, A_n^[k](W_2) for a total of L = N(N-1) desired symbols and L = N(N-1) undesired symbols. Note that there are at most N^2 - 2N + 2 ≜ I independent undesired symbols. Exploiting this fact, we will combine the 2(N-1) queried symbols from each server into (I+L)/N symbols to be downloaded by the user. Intuitively, (L+I)/N symbols from each server will give the user a total of L+I symbols, from which he can resolve the L desired and I undesired symbols. Define the following function that maps 2L/N ∈ℤ_+ input symbols to (L+I)/N ∈ℤ_+ output symbols. ℒ^*(X_1, X_2, ⋯, X_L/N, Y_1, Y_2, ⋯, Y_L/N) = (X_1, ⋯, X_I/N, Y_1, ⋯, Y_I/N, X_I/N+1 + Y_I/N+1, ⋯, X_L/N + Y_L/N) We formalize the combining process in the following lemma. Suppose each server has L/N desired symbols and L/N undesired symbols. Across all servers, the L desired symbols are independent, while the L undesired symbols have dimension at most I, i.e., all L undesired symbols can be expressed as linear combinations of symbols in s, where s is a set of I symbols. Further, each server contains I/N distinct symbols in s. The desired and undesired symbols are combined to produce the answers as follows. A_n^[k] = ℒ^*(C_n A_n^[k](W_1), C_n A_n^[k](W_2)) where C_n are deterministic L/N × L/N matrices, that are required to satisfy the following two properties. Denote the first I/N rows of C_n as C_n. P1. All C_n have full rank. P2. For all (N-1)!^N distinct realizations of π_n', n ∈ [1:N], the I symbols of the undesired message that are directly downloaded (I/N from each server), C_1 A_1^[k](W_k^c), C_2 A_2^[k](W_k^c), ⋯, C_N A_N^[k](W_k^c) are independent in variables in s. Then we have the following claim. Claim. The C_n satisfying the two required properties exist over 𝔽_p for a sufficiently large prime p.[In fact, the properties are generic, i.e., they are satisfied by almost all matrices over large fields.] This proof of existence will use Schwartz-Zippel lemma <cit.> about the roots of a polynomial. The variables for the polynomial are the coefficients of the C_n matrices. Let us start with an arbitrary choice of π_n', n ∈ [1:N]. Since all A_n^[k](W_k^c) can be expressed in terms of the I symbols in the vector s with constant coefficients, we can express (C_1 A_1^[k](W_k^c); ⋯; C_N A_N^[k](W_k^c)) = 𝒞_I× I s Now consider the polynomial given by the determinant of 𝒞. This is not the zero polynomial[A polynomial is a zero polynomial if all its coefficients are zero.] because we can easily assign values to C_n to make 𝒞 = I, the identity matrix. This is because the queried symbols from each server include I/N distinct symbols in s. Next do the same for every realization of π_n', n ∈ [1:N]. As there are N permutations involved, and each can take (N-1)! different values, so we have a total of (N-1)!^N different possibilities. We will consider each of them separately. Each time we find a different 𝒞, which gives us a different non-zero polynomial. Next consider the determinant of each C_n. This gives us another N non-zero polynomials. For each of these (N-1)!^N + N polynomials, Schwartz-Zippel lemma guarantees that a uniformly random choice of C_n produces a non-zero evaluation with high probability over a large field (probability approaching 1 as p →∞). Since the intersection of finite number of high probability events is also a high probability event, there must exist a realization of C_n over a large field for which all (N-1)!^N + N polynomials simultaneously evaluate to non-zero values, i.e., a realization that satisfies both properties. Hence, the claim is true. Next we prove that the scheme retrieves the desired message, and that it is T private. §.§.§ The Scheme is Correct (Retrieves Desired Message) Note that from (<ref>), independent undesired message symbols distribute evenly across the databases, such that Lemma <ref> applies. Note that the first 2I/N variables in the output of the ℒ^* function are obtained directly, i.e., C_1 A_1^[k](W_1), C_2 A_2^[k](W_1), ⋯, C_N A_N^[k](W_1) and C_1 A_1^[k](W_2), C_2 A_2^[k](W_2), ⋯, C_N A_N^[k](W_2) are all directly recovered. By property P2 of C_n, C_1 A_1^[k](W_k^c), C_2 A_2^[k](W_k^c), ⋯, C_N A_N^[k](W_k^c) are linearly independent. Since we have recovered I independent dimensions of interference, and interference only spans at most I dimensions, all interference is recovered and eliminated. Further, since the L desired symbols are independent and since the C_n matrices have full rank, the user is able to recover the L desired message symbols after the interference symbols are recovered and subtracted from the downloaded equations. Therefore the scheme is correct with zero error. §.§.§ The Scheme is Private (to any T=2 Colluding Servers) To prove that the scheme is T = 2 private (refer to (<ref>)), it suffices to show that the queries for any 2 servers are identically distributed, regardless of which message is desired. Since each query is made up of 2(N-1) vectors, N-1 for each message and the vectors for W_1 and the vectors for W_2 are generated independently, it suffices to prove that the vectors for one message (say W_k) are identically distributed, i.e., (Q_n_1^[k](W_k), Q_n_2^[k](W_k)) ∼(Q_n_1^[k^c](W_k), Q_n_2^[k^c] (W_k)),  ∀ n_1, n_2 ∈ [1:4], n_1 < n_2 Note that (Q_n_1^[k](W_k), Q_n_2^[k](W_k)) = (π_n_1(𝒱_n_1), π_n_2(𝒱_n_2) ) (Q_n_1^[k^c](W_k), Q_n_2^[k^c](W_k)) = (π_n_1' (𝒰_n_1), π_n_2' (𝒰_n_2) ) Therefore, to prove (<ref>) it suffices to show the following. ( V_1, ⋯, V_i_n_1-1, V_i_n_1+1, ⋯, V_i_n_2-1, V_i_n_2 +1, ⋯, V_N, V_i_n_2, V_i_n_1) ∼( U_1, ⋯, U_N-2, U_n_1, U_n_2) Because S is uniformly chosen from the set of all full rank matrices, we have ( V_1, ⋯, V_i_n_1-1, V_i_n_1+1, ⋯, V_i_n_2-1, V_i_n_2 +1, ⋯, V_N, V_i_n_2, V_i_n_1) ∼ S Recall that S = (V_1, ⋯, V_N). Next we note that there is a bijection between ( U_1, ⋯, U_N-2, U_n_1, U_n_2) ↔ S' because of (<ref>) so that there is a bijection between U_n_1, U_n_2 and U_1, U_2. Recall that S' = ( U_1, ⋯, U_N-2, U_1, U_2 ). Now as S' is uniform over all full rank matrices, ( U_1, ⋯, U_N-2, U_n_1, U_n_2) is also uniform over all full rank matrices, ( U_1, ⋯, U_N-2, U_n_1, U_n_2) ∼ S' Finally, we note that S and S' are identically distributed, so we have S ∼ S' Combining (<ref>), (<ref>) and (<ref>), we arrive at (<ref>) and (<ref>). §.§.§ Rate Achieved is (N^2 - N)/(2N^2 - 3N + 2) The rate achieved is (N^2 - N)/(2N^2 - 3N + 2), because we download 2N^2 - 3N + 2 symbols in total and the desired message size is N(N-1) symbols. §.§ Achievability Proof of Theorem <ref> when T >2 The proof for the general setting follows the same route as the N = 4, T = 3 example presented earlier. We assume that each message is comprised of L=N(N-1) independent symbols from a sufficiently large finite field 𝔽_p. §.§.§ Storage Code The (N-1,N) MDS storage code is as follows. W_kn ∈ 𝔽_p^N × 1, k ∈ [1:2], n∈[1:N] W_k = (W_k1; W_k2; ⋯; W_k(N-1)) ∈𝔽_p^L × 1 W_kN = W_k1 + W_k2 + ⋯ + W_k(N-1) §.§.§ Construction of Queries The query to each server consists of two vector spaces, one for W_1 (span of the rows of Q_n^[k](W_1)) and one for W_2 (span of the rows of Q_n^[k](W_2)). The queries and downloads for W_k, k ∈ [1:2] are described next. Denote the set of all full rank N × N matrices over 𝔽_p as 𝒮_N. The user privately chooses two matrices S, S', independently and uniformly from 𝒮_N. Label the rows of S as V_1, ⋯, V_N, and the rows of S' as U_1, ⋯, U_N-T, U_1, ⋯, U_T. Define ∀ n ∈ [1:N] 𝒱_n = {V_1, ⋯, V_n-1, V_n+1, ⋯, V_N} 𝒰_n = {U_1, ⋯, U_N-T, U_(n-1)(T-1)+1, ⋯, U_n(T-1)} where U_1, ⋯, U_N(T-1) are the rows of U, obtained as follows. U = P (U_1; ⋯; U_T) P is a deterministic N(T-1) × T matrix that is chosen in such a way that it allows a bijective mapping between the 𝒰_n or 𝒱_n spaces that may be observed by any set of up to T colluding servers. Intuitively, the only requirement on this matrix is that it is sufficiently `generic', so that almost all N(T-1) × T matrices over large finite fields are acceptable. Here unlike the previous example where we explicitly construct the matrix P, we will specify (later) the properties of this matrix and prove that such a matrix exists. When W_k is desired, we have ∀ n, n: Q_n^[k](W_k) = 𝔹 (𝒱_n), A_n^[k](W_k) = Q_n^[k](W_k) W_kn. Desired Symbols Are Independent: From A_1:N^[k](W_k), we can recover all N(N-1) symbols of W_k. This is easily seen because the storage is an (N-1, N) MDS code and the matrix S has full rank. When W_k is undesired, we have ∀ n, n: Q_n^[k^c](W_k) = 𝔹 (𝒰_n), A_n^[k^c](W_k) = Q_n^[k^c](W_k) W_kn. Interfering Symbols Are Dependent and Have Dimension at most N(N-1) - (N-T): Consider the interfering symbols along the common vectors U_i, i ∈ [1:N-T]. Note that U_i W_k1 + ⋯ + U_i W_k(N-1) = U_i W_kN Therefore (N-T) interfering symbols are linear combinations of the other N^2 - 2N + T symbols. §.§.§ Combining Answers for Efficient Download The idea of combining is the same as the T = 2 setting. That is, we will combine the 2(N-1) queried symbols from each server into (2N^2-3N+T)/N = (L+I)/N symbols to be downloaded by the user. We will use the same combining function ℒ^* defined in (<ref>). The difference lies in the combining matrices C_n. For T = 2, C_n are deterministic and the scheme has zero-error, while here C_n are random and the scheme has ϵ-error, with ϵ approaching zero as the message size approaches infinity. The combining process is described in the following lemma, which corresponds to Lemma <ref> (with differences brought by random C_n accounted). Suppose each server has L/N desired symbols and L/N undesired symbols from 𝔽_p. Across all servers, the L desired symbols are independent, while the L undesired symbols have dimension at most I, i.e., all L undesired symbols can be expressed as linear combinations of symbols in s, where s is a set of I symbols. Further, each server contains I/N distinct symbols in s. The desired and undesired symbols are combined to produce the answers as follows. A_n^[k] = ℒ^*(C_n A_n^[k](W_1), C_n A_n^[k](W_2)) where C_n are random L/N × L/N matrices, that are required to satisfy the following two properties. Denote the first I/N rows of C_n as C_n. P1. All C_n are full rank. P2. The I symbols of the undesired message that are directly downloaded (I/N from each server), C_1 A_1^[k](W_k^c), C_2 A_2^[k](W_k^c), ⋯, C_N A_N^[k](W_k^c) are independent in variables in s. Then the following claim must be true. Claim. The probability that C_n, n∈[1:N] with each element chosen independently and uniformly over 𝔽_p, satisfy the two required properties, approaches 1 as p→∞. Proof: Without loss of generality, we assume that I/N is an integer. There is no loss of generality because if I/N is not an integer, we may repeat the scheme a number of times (say M) such that IM/N becomes an integer. The proof relies on Schwartz-Zippel lemma <cit.> about the roots of a polynomial. The variables for the polynomial are the coefficients of the C_n matrices. Consider an arbitrary realization of the query spaces 𝒰_n. Generate uniformly random C_n, independent of 𝒰_n. Given 𝒰_n, n∈[1:N], since all A_n^[k](W_k^c) can be expressed in terms of the I symbols of the vector s with constant coefficients, we can express (C_1 A_1^[k](W_k^c); ⋯; C_N A_N^[k](W_k^c)) = 𝒞_I × I s Now consider the polynomial given by the determinant of 𝒞. This is not the zero polynomial because we can easily assign values to C_n to make 𝒞 = I, the identity matrix. This is because each server contains I/N distinct symbols in s. By the Schwartz-Zippel lemma, a non-zero polynomial evaluates to a non-zero value with probability approaching 1 as the field size p increases and C_n are chosen uniformly over 𝔽_p. Therefore Property P2 is satisfied with high probability. Next consider the determinant of each C_n. This gives us another N non-zero polynomials. When we choose C_n uniformly, the determinant of C_n is not zero almost surely for large p, so that C_n have full rank and Property P1 is satisfied with high probability. Now, because Property P1 and P2 are each satisfied with probability approaching 1, the probability that the two are simultaneously satisfied also approaches 1 (union bound). Since this is true conditioned on every possible realization of 𝒰_n, n∈[1:N], it is also true unconditionally. Next we prove that the scheme retrieves the desired message, and that it is T private. §.§.§ The Scheme is Correct (Retrieves Desired Message) Note that from (<ref>), independent undesired message symbols distribute evenly across the databases, such that Lemma <ref> applies. Note that the first 2I/N variables in the output of the ℒ^* function are obtained directly, i.e., C_1 A_1^[k](W_1), C_2 A_2^[k](W_1), ⋯, C_N A_N^[k](W_1) and C_1 A_1^[k](W_2), C_2 A_2^[k](W_2), ⋯, C_N A_N^[k](W_2) are all directly recovered. By property P2 of C_n, C_1 A_1^[k](W_k^c), C_2 A_2^[k](W_k^c), ⋯, C_N A_N^[k](W_k^c) are linearly independent with probability approaching 1 as p→∞. Since we have recovered I independent dimensions of interference, and interference only spans at most I dimensions, all interference is recovered and eliminated. Further, since the L desired symbols are independent and since the C_n matrices have full rank, the user is able to recover the L desired message symbols after the interference symbols are recovered and subtracted from the downloaded equations. Therefore the scheme is correct with a probability of error ϵ that approaches 0 as the field size p approaches infinity. Note that since each message is comprised of L independent and uniformly random symbols in 𝔽_p, as p approaches infinity, the size of each message also approaches infinity. So, given any ϵ>0, we can find a sufficiently large p, and a correspondingly large message size value such that the probability of error of the scheme described above, is less than ϵ. §.§.§ The Scheme is Private (to any T Colluding Servers) To prove that the scheme is T private (refer to (<ref>)), it suffices to show that the queries for any T servers are identically distributed, regardless of which message is desired. Since each query is made up of two vector spaces, one for each message and the two vector spaces are generated independently, it suffices to prove that the query spaces for one message (say W_k) are identically distributed whether it is desired or undesired. Consider an index set 𝒯 = {i_1, i_2, ⋯, i_T}⊂ [1:N] such that i_1 < i_2 < ⋯ < i_T. For all 𝒯, we require (Q_i_1^[k](W_k), ⋯, Q_i_T^[k](W_k) ) ∼ (Q_i_1^[k^c](W_k), ⋯, Q_i_T^[k^c](W_k) ) ⟺    (𝔹(𝒱_i_1), ⋯, 𝔹(𝒱_i_T) ) ∼ ( 𝔹(𝒰_i_1), ⋯, 𝔹(𝒰_i_T) ) Note that (𝔹(𝒱_i_1), 𝔹(𝒱_i_2), ⋯, 𝔹(𝒱_i_T) ) = (𝔹({V_𝒯^c, V_i_2, ⋯, V_i_T}), 𝔹({V_𝒯^c, V_i_1, V_i_3, ⋯, V_i_T}), 𝔹({V_𝒯^c, V_i_1, ⋯, V_i_T-1}) ) Next we transform the spaces on the RHS of (<ref>) to the form that is the same as (<ref>). To do this, we require the matrix P to satisfy the following properties. P1. For all 𝒯^* ={j_1,j_2,⋯,j_T-1}⊂ [1:N], |𝒯^*| = T-1, j_1<j_2<⋯ <j_T-1, there exists a function m_𝒯^*(P) that returns a non-zero vector which lies simultaneously in the spans of each of P_j_t≜ P((j_t-1)(T-1)+1:j_t(T-1),:), t∈[1:T-1]. Note that m_𝒯^*(P) is a 1 × T row vector that only depends on P (it does not depend on U). P2. For each 𝒯 = {i_1, i_2, ⋯, i_T}⊂ [1:N], the vectors m_𝒯^*(P), ∀𝒯^* ⊂𝒯, |𝒯^*| = T-1 (found in P1) are linearly independent. Equivalently, we require the following T × T matrix to have full rank. P_𝒯≜ (m_{i_[1:T]/{T}}(P); m_{i_[1:T]/{T-1}}(P); ⋯; m_{i_[1:T]/{1}}(P)) Claim. The P satisfying the two required properties exists over 𝔽_p for a sufficiently large p. Similar to the proof of existence of C_n matrices presented earlier, this proof of existence will use Schwartz-Zippel lemma <cit.> about the roots of a polynomial. The variables for the polynomial are the coefficients of the P matrix. Since P is a N(T-1) × T matrix, we have a total of NT(T-1) variables. Define a set 𝒫 that is comprised of all non-zero polynomials with NT(T-1) variables of P as its variables, and coefficients from 𝔽_p. We first consider Property P1. Recall that there are NT-1 choices for 𝒯^*. Let us start with an arbitrary choice of 𝒯^* = {j_1, j_2, ⋯, j_T-1} such that j_1 < j_2 < ⋯ < j_T-1. The required non-zero vector m_𝒯^*(P) is found as follows. m_𝒯^*(P) = H_1 P_j_1 = H_2 P_j_1 = ⋯ = H_T-1 P_j_T-1 ⇒ [ [ H_1 H_2 ⋯ H_T-1 ]] [ [ P_j_1 P_j_1 ⋯ P_j_1; -P_j_2 0 ⋯ 0; ⋮ -P_j_3 ⋱ 0; 0 0 0 -P_j_T-1 ]]_≜ P_𝒥 = [ [ 0 0 ⋯ 0 ]] where P_j_t, t ∈ [1:T-1] are (T-1) × T matrices, 0 is the (T-1) × T matrix with all elements equal to 0 and P_𝒥 is a (T-1)^2 × T(T-2) matrix. Note that the left null space of P_𝒥 is exactly of one dimension if P_𝒥 has full rank. Consider the matrix P_𝒥^*, which is a square matrix formed by the last T(T-2) rows of P_𝒥. We claim that the determinant of P_𝒥^* is a non-zero polynomial, i.e., |P_𝒥^*|∈𝒫. This is because we can identify a specific choice of P_j_t such that |P_𝒥^*| is not zero, as follows. We set P_j_t to be the matrix obtained by inserting an all zero column as the (T+1- t)^th column of the (T-1) × (T-1) identity matrix I_T-1. Equivalently, this means that P_j_t U = (U_1; ⋯; U_T-t; U_T+2-t; ⋯; U_T), t ∈ [1:T-1] Since U_1, ⋯, U_T are independent, m_𝒯^*(P) U can only be some scaled version of the U_1 vector. This means that P_𝒥^* has full rank (which is also easily verified by plugging the vaules of P_j_t in P_𝒥^*). Therefore, |P_𝒥^*|∈𝒫. To make m_𝒯^*(P) a function, i.e., to remove ambiguity due to scaling factors, let us normalize the vector [H_1, ⋯, H_T-1] by its first element, h, such that this vector is unique (scaling is fixed). Note that h∈𝒫 because if we use the same special choice of P_j_t as above, we find that h = 1 (non-zero). With normalized [H_1, ⋯, H_T-1], we obtain m_𝒯^*(P). Note that each element of m_𝒯^*(P) also belongs to 𝒫. Now do the same for every possible choice of 𝒯^*. There are NT-1 possibilities. We will consider each of them separately. Each time we obtain different |P_𝒥^*|, h ∈𝒫 and find a different m_𝒯^*(P). Putting all of these together, we have a set of 2 NT-1 non-zero polynomials. Next consider Property P2. Similarly, we consider all choices of 𝒯 separately. For each choice of 𝒯 = {i_1, i_2, ⋯, i_T} such that i_1 < i_2 ⋯ < i_T, we consider the determinant of P_𝒯. This determinant polynomial is non-zero because we may set P_i_t, t ∈ [1:T] to be the matrix obtained by inserting an all zero column as the (T+1- t)^th column of I_T-1, such that the common vector m_𝒯^*(P), ∀𝒯^* ∈𝒯, |𝒯^*| = T-1 can be computed explicitly P_i_t U = (U_1; ⋯, U_T-t; U_T+2-t⋯; U_T) m_{i_[1:T]/{t}}(P) = e_T+1 - t, ∀ t ∈ [1:T] where e_i represents the 1 × T unit row vector with a 1 in the i^th location and 0 at all other locations. Therefore, P_𝒯 is an identity matrix and the determinant is 1 (non-zero). With all choices of 𝒯, we have another NT non-zero polynomials. By Schwartz-Zippel lemma, as the field size grows, for each of the polynomials mentioned above, a uniform choice of P produces a non-zero evaluation with probability approaching 1. By the union bound, the probability that all polynomials simultaneously produce a non-zero value also approaches 1. In particular, for a sufficiently large field this probability is not zero, so there must exist a P matrix that satisfies both properties. Because of the two properties, we may equivalently represent Q_i_t^[k^c](W_k), t ∈ [1:T] as Q_i_t^[k^c](W_k) = 𝔹(𝒰_i) = 𝔹({U, U_{i_[1:T]/{1}}, ⋯, U_{i_[1:T]/{t-1}}, U_{i_[1:T]/{t+1}}, ⋯, U_{ i_[1:T]/{T}}}), We are now ready to prove the privacy condition (<ref>). (<ref>) ⟺ (𝔹({V_𝒯^c, V_i_2, ⋯, V_i_T}), 𝔹({V_𝒯^c, V_i_1, V_i_3, ⋯, V_i_T}), 𝔹({V_𝒯^c, V_i_1, ⋯, V_i_T-1}) ) ∼ ( 𝔹({U, U_{i_[1:T]/{2}}, ⋯, U_{i_[1:T]/{T}}}), 𝔹({U, U_{i_[1:T]/{1}}, U_{i_[1:T]/{3}}, ⋯, U_{i_[1:T]/{T}}}),  ⋯, 𝔹({U, U_{i_[1:T]/{1}}, ⋯, U_{ i_[1:T]/{T-1}}}) ) Therefore, it suffices to show the following. (V_𝒯^c, V_i_1, V_i_2, ⋯, V_i_T) ∼ (U, U_{i_[1:T]/{1}}, U_{i_[1:T]/{2}}, ⋯, U_{i_[1:T]/{T}}) Because S is uniformly chosen from the set of all full rank matrices, we have (V_𝒯^c, V_i_1, V_i_2, ⋯, V_i_T) ∼ (V_1, V_2, ⋯, V_N) Because of Property P2, there is a bijection between (U, U_{i_[1:T]/{1}}, U_{i_[1:T]/{2}}, ⋯, U_{i_[1:T]/{T}}) ↔ (U, U) Now since S' = (U; U) is uniform in all full rank matrices, the bijection implies that (U, U_{i_[1:T]/{1}}, U_{i_[1:T]/{2}}, ⋯, U_{i_[1:T]/{T}}) is also uniform in all full rank matrices, i.e., (U, U_{i_[1:T]/{1}}, U_{i_[1:T]/{2}}, ⋯, U_{i_[1:T]/{T}}) ∼ (U, U) Finally, note that S and S' have the same distribution, so we have (V_1, V_2, ⋯, V_N) ∼ (U, U) Therefore, from (<ref>), (<ref>) and (<ref>), we have proved (<ref>) and (<ref>). §.§.§ Rate Achieved is (N^2 - N)/(2N^2 - 3N + T) The rate achieved is (N^2 - N)/(2N^2 - 3N + T), because we download 2N^2 - 3N + T symbols in total and the desired message size is N(N-1) symbols. §.§ Converse for Arbitrary K In this section, we consider the information theoretic converse of MDS-TPIR, for two scenarios, one with (K, N, T, K_c) = (K, 4, 2, 2) and the other with (K, N, T, K_c) such that N < T + K_c. For both scenarios, we provide outer bounds that hold for arbitrary K. Let us start with two useful lemmas that hold for arbitrary K, N, T, K_c. For all 𝒯⊂ [1:N], |𝒯| = T and k, k' ∈ [1:K], (A_𝒯^[k], W_1, ⋯, W_K, ℱ, 𝒢) ∼ (A_𝒯^[k'], W_1, ⋯, W_K, ℱ, 𝒢) Proof: From (<ref>), we know that Q_𝒯^[k]∼ Q_𝒯^[k']. Combining with (<ref>), we have H(Q_𝒯^[θ] | ℱ) = 0 From (<ref>), we have I(θ; W_1, ⋯, W_K, ℱ, 𝒢) = 0 (<ref>)⟹ I(θ; W_1, ⋯, W_K, ℱ, 𝒢, Q_𝒯^[θ]) = 0 (<ref>)(<ref>)(<ref>)⟹ I(θ; W_1, ⋯, W_K, ℱ, 𝒢, A_𝒯^[θ]) = 0 ⟹ (A_𝒯^[k], W_1, ⋯, W_K, ℱ, 𝒢) ∼ (A_𝒯^[k'], W_1, ⋯, W_K, ℱ, 𝒢) For all 𝒦_c = {n_1, n_2, ⋯, n_K_c}⊂ [1:N], H(A_𝒦_c^[1] | W_1, ℱ, 𝒢) = ∑_n ∈𝒦_c H(A_n^[1] | W_1, ℱ, 𝒢) Proof: From (<ref>) and (<ref>), we know that for any K_c servers, the stored information is independent. H(W_k𝒦_c) = ∑_n ∈𝒦_c H(W_k n), ∀ k ∈ [1:K] (<ref>)⟹ H(W_2𝒦_c, ⋯, W_K𝒦_c | W_1, ℱ, 𝒢) = ∑_n ∈𝒦_c∑_k=2^K H(W_k n | W_1, ℱ, 𝒢) As answers are functions of the storage, the answers from any K_c servers are independent as well. Consider two arbitrary subsets of 𝒦_c that have no overlap, 𝒦_1, 𝒦_2 ⊂𝒦_c, 𝒦_1 ∩𝒦_2 = ∅. I(A_𝒦_1^[1]; A_𝒦_2^[1] | W_1, ℱ, 𝒢) ≤ I(A_𝒦_1^[1]; A_𝒦_2^[1], W_2𝒦_2, ⋯, W_K𝒦_2 | W_1, ℱ, 𝒢) (<ref>)(<ref>)= I(A_𝒦_1^[1]; W_2𝒦_2, ⋯, W_K𝒦_2 | W_1, ℱ, 𝒢) (<ref>)(<ref>)≤ I(W_2𝒦_1, ⋯, W_K𝒦_1 ; W_2𝒦_2, ⋯, W_K𝒦_2 | W_1, ℱ, 𝒢) (<ref>)= 0 Using (<ref>) repeatedly, we obtain (<ref>). Next we proceed to the two scenarios. To highlight the parameter K, in this section, the capacity C and the download cost D are denoted as C(K) and D(K), respectively. §.§.§ (K, N, T, K_c) = (K, 4, 2, 2) For the setting with (K, N, T, K_c) = (K, 4, 2, 2), we obtain a recursive upper bound that holds for arbitrary K. This result is stated in the following theorem. For the class of MDS-TPIR instances with (K,N, T, K_c)=(K,4,2,2), with arbitrary K, the following recursive relation on the capacity outer bound C(K) ≥ C(K) holds. C(K) ≤ ( 1 + 3/8( 1/C(K-1)) + (1 - (2/3)^K-1) 3/4)^-1, ∀ K ≥ 2 C(1) = 1 Consider an MDS-TPIR instance with (K,N, T, K_c)=(K,4,2,2). When K = 1, C(1) = 1 is a trivial bound on C(1). Next we consider K ≥ 2. Define C(K) = L/H(A_1:4^[1] | ℱ, 𝒢) C(K-1) = L/H(A_1:4^[2] | W_1, ℱ, 𝒢) C(K) is a valid outer bound on C(K), since C(K) = L/H(A_1:4^[1] | ℱ, 𝒢) ≥ L/D(K) = C(K) Similarly, C(K-1) is a valid outer bound on C(K-1). Now, substituting (<ref>) and (<ref>) to (<ref>), we have H(A_1:4^[1] | ℱ, 𝒢)/L ≥ 1 + 3/8 H(A_1:4^[2] | W_1, ℱ, 𝒢)/L + (1 - (2/3)^K-1) 3/4 We proceed to prove (<ref>). To simplify the notation, we define (W_1i, W_2i, ⋯, W_Ki) = W_*i, i ∈ [1:N]. H(A_1:4^[1] | ℱ, 𝒢) (<ref>)= H(A_1:4^[1], W_1 | ℱ, 𝒢) + o(L) L (<ref>)= H(W_1) + H(A_1^[1] | W_1, ℱ, 𝒢) + H(A_2:4^[1] | W_1, A_1^[1], ℱ, 𝒢) + o(L) L ≥ H(W_1) + H(A_1^[1] | W_1, ℱ, 𝒢) + H(A_3:4^[1] | W_1, W_*1, A_1^[1], ℱ, 𝒢) + o(L) L (<ref>)(<ref>)(<ref>)= L + H(A_1^[1] | W_1, ℱ, 𝒢) + H(A_3:4^[1] | W_1, W_*1, ℱ, 𝒢) + o(L) L (<ref>)(<ref>)= L + H(A_1^[2] | W_1, ℱ, 𝒢) + H(A_3:4^[2] | W_1, W_*1, ℱ, 𝒢) + o(L) L Advancing the databases indices, from (<ref>), we have H(A_1:4^[1] | ℱ, 𝒢) ≥ L + H(A_1^[2] | W_1, ℱ, 𝒢) + H(A_2:3^[2] | W_1, W_*1, ℱ, 𝒢) + o(L) L Adding (<ref>) and (<ref>), we have H(A_1:4^[1] | ℱ, 𝒢) + o(L) L ≥ L + H(A_1^[2] | W_1, ℱ, 𝒢) + 1/2( H(A_3:4^[2] | W_1, W_*1, ℱ, 𝒢) + H(A_2:3^[2] | W_1, W_*1, ℱ, 𝒢) ) ≥ L + H(A_1^[2] | W_1, ℱ, 𝒢) + 1/2( H(A_2:4^[2] | W_1, W_*1, ℱ, 𝒢) + H(A_3^[2] | W_1, W_*1, ℱ, 𝒢) ) (<ref>)(<ref>)(<ref>)= L + H(A_1^[2] | W_1, ℱ, 𝒢) + 1/2 H(A_3^[2] | W_1, ℱ, 𝒢) + 1/2 H(A_2:4^[2] | W_1, W_*1, ℱ, 𝒢) where we use the sub-modular property of entropy functions to obtain (<ref>). Now consider the term H(A_2:4^[2] | W_1, W_*1, ℱ, 𝒢). This corresponds to the total download for the setting where we have 3 servers (servers 2, 3 and 4), K-1 messages (W_2, W_3, ⋯, W_K), each message is of length L/2 and the MDS code is fully replicated (conditioning on W_*1, each other server contains the other half information of entropy L/2 about each message), i.e., the TPIR setting. W_2 is the desired message. As the capacity of this TPIR setting is 1/3(1 - (2/3)^K-1)^-1 <cit.>, we have H(A_2:4^[2] | W_1, W_*1, ℱ, 𝒢) ≥ 3(1 - (2/3)^K-1) L/2 Substituting back to (<ref>) and advancing database indices, we have ∀ i, j ∈ [1:4], i ≠ j, H(A_1:4^[1] | ℱ, 𝒢) + o(L) L ≥ L + H(A_i^[2] | W_1, ℱ, 𝒢) + 1/2 H(A_j^[2] | W_1, ℱ, 𝒢) + (1 - (2/3)^K-1) 3L/4 Adding (<ref>) for all i, j ∈ [1:4], we have H(A_1:4^[1] | ℱ, 𝒢) + o(L) L ≥ L + 1/4∑_i=1^4 H(A_i^[2] | W_1, ℱ, 𝒢) + 1/8∑_j=1^4 H(A_j^[2] | W_1, ℱ, 𝒢) + (1 - (2/3)^K-1) 3L/4 ≥ L + 3/8 H(A_1:4^[2] | W_1, ℱ, 𝒢) + (1 - (2/3)^K-1) 3L/4 Normalizing both sides by L, we arrive at (<ref>). Two observations from the converse argument are listed below. * When we set K=2, we obtain the information theoretic bound 8/13. C(2) ≤ C(2) (<ref>)≤ (1 + 3/8 × 1/C(1) + (1-2/3) × 3/4)^-1 (<ref>)= (1 + 3/8 × 1 + (1-2/3) × 3/4)^-1 = 8/13 * As K→∞, the capacity upper bound converges to 5/14. Since the MDS-TPIR scheme of Freij-Hollanti et al. <cit.> achieves the rate 1/4 for this setting as K→∞, we note that the asymptotic optimality of the scheme remains open. §.§.§ (K, N, T, K_c) with N < T + K_c For the setting with (K, N, T, K_c) and N < T + K_c, we obtain a recursive upper bound that holds for arbitrary K. This result is stated in the following theorem. For the class of MDS-TPIR instances (K,N, T, K_c) such that N < T + K_c, with arbitrary K, N, T, K_c, the following recursive relation on the capacity outer bound C(K) ≥ C(K) holds. C(K) ≤ ( 1 + N-T/N( 1/C(K-1)) + (K-1)(1-N-T/K_c) )^-1, ∀ K ≥ 2 C(1) = 1 Therefore, for constant N, T, K_c, when K →∞, C(K) decreases linearly with K such that downloading everything (rate 1/K) is order optimal. Consider an MDS-TPIR instance (K,N, T, K_c) such that N < T + K_c. When K = 1, C(1) = 1 is a trivial bound on C(1). Next we consider K ≥ 2. Define C(K) = L/H(A_1:N^[1] | ℱ, 𝒢) C(K-1) = L/H(A_1:N^[2] | W_1, ℱ, 𝒢) C(K) is a valid outer bound on C(K), since C(K) = L/H(A_1:N^[1] | ℱ, 𝒢) ≥ L/D(K) = C(K) Similarly, C(K-1) is a valid outer bound on C(K-1). Now, substituting (<ref>) and (<ref>) to (<ref>), we have H(A_1:N^[1] | ℱ, 𝒢)/L≥ 1 + N-T/NH(A_1:N^[2] | W_1, ℱ, 𝒢)/L + (K-1)(1-N-T/K_c) We proceed to prove (<ref>). Consider an index set 𝒩⊂ [1:N] with cardinality |𝒩| = N-T < K_c. Denote the complement of 𝒩 as 𝒩^c. H(A_1:N^[1] | ℱ, 𝒢) (<ref>)= H(A_1:N^[1], W_1 | ℱ, 𝒢) +o(L) L (<ref>)= H(W_1) + H(A_𝒩^[1] | W_1, ℱ, 𝒢) + H(A_𝒩^c^[1] | W_1, A_𝒩^[1], ℱ, 𝒢) +o(L) L (<ref>)(<ref>)≥ L + ∑_n ∈𝒩 H(A_n^[1] | W_1, ℱ, 𝒢) + H(A_𝒩^c^[1] | W_1, W_*𝒩, A_𝒩^[1], ℱ, 𝒢) +o(L) L (<ref>)(<ref>)(<ref>)= L + ∑_n ∈𝒩 H(A_n^[1] | W_1, ℱ, 𝒢) + H(A_𝒩^c^[1] | W_1, W_*𝒩, ℱ, 𝒢) +o(L) L (<ref>)(<ref>)= L + ∑_n ∈𝒩 H(A_n^[2] | W_1, ℱ, 𝒢) + H(A_𝒩^c^[2] | W_1, W_*𝒩, ℱ, 𝒢) +o(L) L (<ref>)(<ref>)(<ref>)= L + ∑_n ∈𝒩 H(A_n^[2] | W_1, ℱ, 𝒢) + H(A_1:N^[2] | W_1, W_*𝒩, ℱ, 𝒢) +o(L) L (<ref>)≥ L + ∑_n ∈𝒩 H(A_n^[2] | W_1, ℱ, 𝒢) + H(A_1:N^[2], W_2 | W_1, W_*𝒩, ℱ, 𝒢) + o(L) L ≥ L + ∑_n ∈𝒩 H(A_n^[2] | W_1, ℱ, 𝒢) + H(W_2 | W_1, W_*𝒩, ℱ, 𝒢) + H(A_1:N^[2] | W_1, W_2, W_*𝒩, ℱ, 𝒢) + o(L) L (<ref>)(<ref>)(<ref>)= L + ∑_n ∈𝒩 H(A_n^[2] | W_1, ℱ, 𝒢) + L(K_c - |𝒩|)/K_c + H(A_1:N^[2] | W_1, W_2, W_*𝒩, ℱ, 𝒢)+ o(L) L (<ref>)(<ref>)(<ref>)= L + ∑_n ∈𝒩 H(A_n^[2] | W_1, ℱ, 𝒢) + L(K_c - N + T)/K_c + H(A_𝒩^c^[2] | W_1, W_2, W_*𝒩, ℱ, 𝒢) + o(L) L To bound the term H(A_𝒩^c^[2] | W_1, W_2, W_*𝒩, ℱ, 𝒢), we repeat (<ref>) to (<ref>) for messages W_3, ⋯, W_K. This gives us H(A_1:N^[1] | ℱ, 𝒢) ≥ L + ∑_n ∈𝒩 H(A_n^[2] | W_1, ℱ, 𝒢) + L(K-1)(1-N-T/K_c) + o(L) L Consider (<ref>) for all subsets of [1:N] that have exactly N-T elements and average over all such subsets. We have H(A_1:N^[1] | ℱ, 𝒢) ≥ L + 1/NN-T∑_𝒩: |𝒩| = N -T∑_n ∈𝒩 H(A_n^[2] | W_1, ℱ, 𝒢) + L(K-1)(1-N-T/K_c) + o(L) L ≥ L + N-T/N H(A_1:N^[2] | W_1, ℱ, 𝒢) + L(K-1)(1-N-T/K_c) + o(L) L Letting L →∞ and normalizing by L, we have proved (<ref>) and (<ref>). Based on Theorem <ref> the following observations are relevant. * When we set K=2, K_c = N-1, we obtain the information theoretic bound for Theorem <ref>, i.e., (N^2 - N)/(2N^2 - 3N + T). C(2) ≤ C(2) (<ref>)≤ (1 + N-T/N×1/C(1) + (2-1)(1-N-T/N-1) )^-1 (<ref>)= (1 + N-T/N× 1 + T-1/N-1)^-1 = N^2 - N/2N^2 - 3N + T * As K→∞, Theorem <ref> shows that the capacity decays as 1/K, so that it converges to 0. As a sanity check, we note that indeed, the MDS-TPIR scheme of Freij-Hollanti et al. <cit.>, which does not depend on the number of messages K, does not apply when N<T+K_c. Thus, in this case the asymptotic optimality as K→∞ is trivially settled. §.§ Restricted Colluding Sets Recall that for the setting of our counterexample, i.e., (K, N, T, K_c)=(2,4,2,2), while the linear capacity is settled, the information theoretic capacity remains open. In particular, the best information theoretic capacity upper bound that we were able to obtain is 8/13. To gain insights into the potential tightness of this bound, here we look into the capacity of this setting with restricted colluding sets, a line of inquiry recently initiated by Tajeddine et al. in <cit.>. Our motivation for studying restricted colluding sets comes from the following observation. Consider TPIR, for which the capacity is known <cit.>. The TPIR formulation allows the possibility that any set of up to T servers may collude. However, suppose we relax the privacy constraint, by allowing only collusions between cyclically contiguous servers, i.e., the colluding servers must belong to the set of servers indexed {n, n+1, ⋯, n+T-1} for some n∈[1:N], with the indices interpreted modulo N. Because of the symmetry that is still maintained across servers, it is readily verified that the converse proof for TPIR in <cit.> still goes through unchanged. Thus, even though the restriction on colluding sets to cyclically contiguous servers relaxes the privacy constraint, it does not affect the capacity of TPIR. This leads us to question if a similar property might hold for MDS-TPIR. If so, then we could gain insights into the capacity of MDS-TPIR by imposing similar restrictions on the colluding sets. This line of thought leads us to two somewhat contrasting observations, that are presented in the following two subsections. §.§.§ (K, N, T,K_c) = (2,4,2,2) with Cyclically Adjacent Colluding Sets Our first observation is in favor of the tightness of the upper bound 8/13. Indeed, if colluding sets were restricted to cyclically contiguous sets then 8/13 is the capacity for the MDS-TPIR setting (K, N, T, K_c)=(2,4,2,2). This observation is summarized in a bit more detail next. For our counterexample we considered the MDS-TPIR setting (K, N, T, K_c) = (2,4,2,2) where any 2 servers may collude. Suppose, now we restrict the colluding sets of servers to cyclically adjacent pairs, i.e., any one of {1,2}, {2,3}, {3,4}, {4,1}. Essentially we have relaxed the privacy constraint by eliminating the possibilities that Server 1 might collude with Server 3, or that Server 2 might collude with Server 4. For this setting, we show that the capacity is 8/13. The converse is similar to that with T = 2, presented in Section <ref>. (<ref>) holds with restricted colluding sets when K = 2, because we are left with only K - 1 = 1 message. All other steps follow similarly because the assumption of symmetry across servers holds under cyclically adjacent colluding sets. As a result, the capacity upper bound of 8/13 (refer to (<ref>)) holds here. Next, we summarize the achievable scheme. The message construction and the storage code are specified as follows. W_kn∈𝔽_p^4× 1, k ∈ [1:2], n ∈ [1:4] W_k = (W_k1; W_k2) ∈𝔽_p^8 × 1 W_k3 = W_k1 + W_k2, W_k4 = W_k1 + 2W_k2 The construction of queries is similar to that with T= 2 in Section <ref>. The query to each server Q_n^[k] is comprised of two parts, Q_n^[k](W_1), Q_n^[k](W_2). Each part contains 2 row vectors, along which the server should project its corresponding stored message symbols. To generate the query vectors, the user privately chooses two matrices, S = (V_1;V_2; V_3; V_4) and S' = (U_0; U_1; U_2; U_3), independently and uniformly from 𝒮_4, the set of all full rank 4 × 4 matrices over 𝔽_p. Define 𝒱_1={V_1, V_2}, 𝒰_1={U_0, U_1+U_2} 𝒱_2={V_2, V_3}, 𝒰_2={U_0, U_1+2U_2} 𝒱_3={V_3, V_4}, 𝒰_3={U_0, U_1} 𝒱_4={V_4, V_1}, 𝒰_4={U_0, U_2} Independent random orderings of the rows in 𝒱_n are the queries to Server n for the desired message and independent random orderings of the rows in 𝒰_n are the queries to Server n for the undesired message. The rate achieved is 8/13 because the 8 desired symbols along the V_i vectors are all independent and the 8 undesired symbols occupy only 5 dimensions (the 4 symbols along U_0 contribute only 2 independent dimensions and the remaining 4 symbols contribute only 3 independent dimensions). Privacy follows from the observation that for each cyclically adjacent colluding set of servers, say Server 1 and Server 2, the sets 𝒱_1,𝒱_2 intersect in one of their elements, as do the sets 𝒰_1,𝒰_2, and both are otherwise uniformly random, thus making the distinction of 𝒰,𝒱 invisible to the colluding servers. Note that this scheme is not private to the non-adjacent colluding servers, say Server 1 and Server 3, because, 𝒱_1,𝒱_3 contain no common vectors, while 𝒰_1,𝒰_3 do share a common vector. The remaining details are virtually identical to the settings already covered in Section <ref> and Section <ref> and are omitted. §.§.§ Disjoint Colluding Sets of T Servers Each Our second observation provides a counterpoint to the first observation. The first observation favored the tightness of 8/13 bound based on the insight originating from TPIR, that certain restrictions on colluding sets may not affect capacity. The second observation challenges this viewpoint by showing that insights from TPIR do not carry over to MDS-TPIR. Consider again the TPIR problem. Suppose T divides N, i.e., mT=N for some m∈ℤ_+, and we partition the N servers into the m disjoint sets of T elements each: 𝒯_1={1,2,⋯, T}, 𝒯_2={T+1,T+2,⋯, 2T}, ⋯, 𝒯_m={(m-1)T+1, (m-1)T+2, ⋯, N}. Further, suppose we relax the privacy constraint and allow collusions between only those servers that belong to the same 𝒯_i, i∈[1:m]. Then, note that the TPIR problem with restricted colluding sets becomes equivalent to the PIR problem with N/T=m servers.[This is because storage is fully replicated, so that each disjoint set of T colluding servers may be equivalently replaced with 1 server.] However, the capacity of PIR with N/T servers is the same as the capacity of TPIR with N servers. Therefore, relaxing the privacy constraint by restricting the colluding sets to disjoint sets of cardinality T each, in the manner described above, does not affect the capacity of TPIR. However, as we will show next, the same is not true for MDS-TPIR. Consider MDS-TPIR with (K, N, T, K_c)=(2,4,3,2), where any T=2 of the N=4 servers may collude. From Theorem <ref> we know that the capacity of this setting is 6/11. However, now suppose we partition the servers into disjoint sets 𝒯_1={1,2}, 𝒯_2={3,4}, each of cardinality T=2. Now we allow collusions only between servers in the same 𝒯_i set, i.e., Server 1 can only collude with Server 2, while Server 3 can only collude with Server 4. Then, in contrast to TPIR where such a restriction on colluding sets does not affect the capacity, we now show that with these restricted colluding sets, the capacity of MDS-TPIR changes — it increases from 6/11 to 4/7. The converse for rate 4/7 is trivial, because the rate can not be higher than that of MDS-PIR with (K, N, K_c) = (2,4,3), where privacy needs to be ensured only to each individual server. From <cit.>, we know that the capacity of MDS-PIR with (K, N, K_c) = (2,4,3) is 4/7. Therefore, the upper bound follows. Next, we consider the achievable scheme. Each message consists of 12 symbols. The storage code is specified as follows. W_kn∈𝔽_p^4× 1, k ∈ [1:2], n ∈ [1:4] W_k = (W_k1; W_k2; W_k3) ∈𝔽_p^12 × 1 W_k4 = W_k1 + W_k2 + W_k3 The query to each server Q_n^[k] is comprised of vectors in 𝒱_n and 𝒰_n, given as follows. 𝒱_1={V_1, V_3, V_5}, 𝒰_1={U_0, U_1, U_2} 𝒱_2={V_1, V_3, V_5}, 𝒰_2={U_0, U_1, U_2} 𝒱_3={V_2, V_4, V_6}, 𝒰_3={U_0, U_1, U_2} 𝒱_4={V_2, V_4, V_6}, 𝒰_4={U_0, U_1,U_2} where S = (V_1;V_2; V_3; V_4; V_5; V_6) and S' = (U_0; U_1; U_2; U_3; U_4; U_5) are independent and uniform from the set of all full rank 6 × 6 matrices. The rate achieved is 12/(12+9) = 4/7 because the 12 desired symbols along the V_i vectors are all independent and the 12 undesired symbols occupy only 9 dimensions (the symbols along each U_i, i∈{0,1,2}, occupy only K_c = 3 dimensions). Privacy follows from the observation that for either colluding set {1,2} or {3,4}, the vectors in 𝒱 and 𝒰 are both the same. The remaining details can be filled in based on Section <ref> and Section <ref> and are omitted. In light of the two contrasting observations, the tightness of the 8/13 upper bound, as well as the general impact of restricted colluding sets on the capacity of MDS-TPIR remain intriguing open problems for future work. For readers interested in the latter problem, we conclude this section with two simple examples of such capacity characterizations. §.§.§ Examples of Capacity of MDS-TPIR under Restricted Colluding Sets As usual in this section, we will omit details of achievability arguments that follow directly from Section <ref> and Section <ref>. Example 1 Consider the setting (K, N, K_c)=(2,4,2) and let the restricted colluding sets be {1,2}, {3,4}. Alternatively, let the restricted colluding sets be {1,2}, {3}, {4}. In either case, the capacity is 2/3, same as that of MDS-PIR with (K, N, K_c) = (2, 4, 2) <cit.> so that the converse is implied. The scheme that achieves rate 4/6= 2/3 is as follows. W_kn∈𝔽_p^2× 1, k ∈ [1:2], n ∈ [1:4] W_k = (W_k1; W_k2) ∈𝔽_p^4 × 1 W_k3 = W_k1 + W_k2, W_k4 = W_k1 + 2W_k2 𝒱_1={V_1}, 𝒰_1={U_0} 𝒱_2={V_1}, 𝒰_2={U_0} 𝒱_3={V_2}, 𝒰_3={U_0} 𝒱_4={V_2}, 𝒰_4={U_0} where S = (V_1;V_2) and S' = (U_0; U_1) are independently and uniformly chosen from the set of all full rank 2 × 2 matrices. Example 2 Suppose (K, N, K_c) = (2, 3, 2) and the colluding sets are either {1,2},{2,3}. Alternatively, suppose the colluding sets are {1,2}, {3}. In both cases, the capacity is 4/7. The scheme that achieves rate 4/7 is as follows. W_kn∈𝔽_p^2× 1, k ∈ [1:2], n ∈ [1:3] W_k = (W_k1; W_k2) ∈𝔽_p^4 × 1 W_k3 = W_k1 + W_k2 𝒱_1={V_1}, 𝒰_1={U_0} 𝒱_2={V_1, V_2}, 𝒰_2={U_0, U_1} 𝒱_3={V_2}, 𝒰_3={U_0} where S = (V_1;V_2) and S' = (U_0; U_1) are independent and uniformly chosen from the set of all full rank 2 × 2 matrices over 𝔽_p. For the converse, consider (<ref>). Plugging in K = 2, K_c = 2, 𝒩 = {3}, N = 3, we have D ≥ H(A_1:3^[1] | ℱ, 𝒢) ≥ L + H(A_3^[2] | W_1, ℱ, 𝒢) + L/2 + o(L) L Note that (<ref>) still holds when |𝒩| = K_c. Plugging in K = 2, K_c = 2, 𝒩 = {1,2}, N = 3, we have D ≥ H(A_1:3^[1] | ℱ, 𝒢) ≥ L + H(A_1^[2] | W_1, ℱ, 𝒢) + H(A_2^[2] | W_1, ℱ, 𝒢) + o(L) L Adding the two inequalities above, we have 2D ≥ 5L/2 + H(A_1^[2], A_2^[2],A_3^[2] | W_1, ℱ, 𝒢) + L/2 + o(L) L (<ref>)≥ 5L/2 + H(W_2 | W_1, ℱ, 𝒢) + L/2 + o(L) L (<ref>)(<ref>)= 7L/2 + o(L) L Normalizing by L and taking limits as L approaches infinity, gives us the upper bound on the rate L/D as 4/7, which completes the converse. IEEEtran

http://arxiv.org/abs/1701.07801v1

20170126181225

Spekkens' toy model in all dimensions and its relationship with stabilizer quantum mechanics

quant-ph

]Spekkens' toy model in all dimensions and its relationship with stabilizer quantum mechanics 1University College London, Physics and Astronomy department, Gower St, London WC1E 6BT, UK lorenzo.catani.14@ucl.ac.uk Spekkens' toy model is a non-contextual hidden variable model with an epistemic restriction, a constraint on what an observer can know about reality. The aim of the model, developed for continuous and discrete prime degrees of freedom, is to advocate the epistemic view of quantum theory, where quantum states are states of incomplete knowledge about a deeper underlying reality. Many aspects of quantum mechanics and protocols from quantum information can be reproduced in the model. In spite of its significance, a number of aspects of Spekkens' model remained incomplete. Formal rules for the update of states after measurement had not been written down, and the theory had only been constructed for prime-dimensional, and infinite dimensional systems. In this work, we remedy this, by deriving measurement update rules, and extending the framework to derive models in all dimensions, both prime and non-prime. Stabilizer quantum mechanics is a sub-theory of quantum mechanics with restricted states, transformations and measurements. First derived for the purpose of constructing error correcting codes, it now plays a role in many areas of quantum information theory. Previously, it had been shown that Spekkens' model was operationally equivalent in the case of infinite and odd prime dimensions. Here, exploiting known results on Wigner functions, we extend this to show that Spekkens' model is equivalent to stabilizer quantum mechanics in all odd dimensions, prime and non-prime. This equivalence provides new technical tools for the study of technically difficult compound-dimensional stabilizer quantum mechanics. [ Lorenzo Catani1 and Dan E. Browne1 ====================================== § INTRODUCTION A long tradition of research, starting from the famous “EPR paper” <cit.>, has consisted of analysing quantum theory in terms of hidden variable models, with the aim of obtaining a more intuitive understanding of it. This has led to some crucial results in foundation of quantum mechanics, namely Bell's and Kochen-Specker's no-go theorems <cit.><cit.>. Nowadays a big question is whether to interpret the quantum state according to the ontic view, i.e. where it completely describes reality, or to the epistemic view, where it is a state of incomplete knowledge of a deeper underlying reality which can be described by the hidden variables. In 2005, Robert Spekkens <cit.> constructed a non-contextual hidden variable model to support the epistemic view of quantum mechanics. The aim of the model was to replace quantum mechanics by a hidden variable theory with the addition of an epistemic restriction (i.e. a restriction on what an observer can know about reality). The first version of the model <cit.> was developed in analogy with quantum bits (qubits), with 2-outcome observables. Despite the simplicity of the model, it was able to support many phenomena and protocols that were believed to be intrinsically quantum mechanical (such as dense coding and teleportation). Spekkens' toy model has influenced much research over the years: e.g. people provided a new notation for it <cit.>, studied it from the categorical point of view <cit.>, used it for quantum protocols <cit.>, exploited similar ideas to find a classical model of one qubit <cit.>, and tried to extend it in a contextual framework <cit.>. Also Spekkens' toy model addresses many key issues in quantum foundations: whether the quantum state describes reality or not, finding a derivation of quantum theory from intuitive physical principles and classifying the inherent non-classical features. A later version of the model <cit.>, which we will call Spekkens' Theory (ST), introduced a more general and mathematically rigorous formulation, extending the theory to systems of discrete prime dimension, where dimension refers to the maximum number of distinguishable measurement outcomes of observables in the theory, and continuous variable systems. Spekkens called these classical statistical theories with epistemic restrictions as epistricted statistical theories. By considering a particular epistemic restriction that refers to the symplectic structure of the underlying classical theory, the classical complementarity principle, theories with a rich structure can be derived. Many features of quantum mechanics are reproduced there, such as Heisenberg uncertainty principle, and many protocols introduced in the context of quantum information, such as teleportation. However, as an intrinsically non-contextual theory, it cannot reproduce quantum contextuality (and the related Bell non-locality), which, therefore arises as the signature of quantumness. Indeed, for odd prime dimensions and for continuous variables, ST was shown to be operationally equivalent to sub-theories of quantum mechanics, which Spekkens called quadrature quantum mechanics. In the finite dimensional case quadrature quantum mechanics is better known as stabilizer quantum mechanics (SQM). The latter is a sub-theory of quantum mechanics developed for the description and study of quantum error correcting codes <cit.> but subsequently playing a prominent role in many important quantum protocols. In particular, many studies of quantum contextuality can be expressed in the framework of SQM, including the GHZ paradox <cit.> and the Peres-Mermin square <cit.><cit.>. This exposes a striking difference between odd and even dimensional SQM. Even-dimensional SQM contains classical examples of quantum contextuality while odd-dimensional SQM exhibits no contextuality at all, necessary for its equivalence with Spekkens' Theory. While developed for qubits, SQM was rapidly generalised to systems of arbitrary dimension, <cit.>. However, for non-prime dimensions SQM remains poorly characterised and little studied (recent progress in this was recently reported in <cit.>). In spite of its importance, there remain some important aspects of Spekkens' Theory which have not yet been characterised and studied. First of all, all prior work on ST have only considered systems where the dimension is prime. Furthermore, while Spekkens' recent work strengthens the mathematical foundations of the model <cit.>, one key part of the theory has not yet been described in a general and rigorous way. These are the measurement update rules, the rules which tell us how to update a state after a measurement has been made. In prior work, these rules, and the principles behind them have been described but not formalised. In this paper, we complete this step, deriving a formal description of the measurement rules for prime-dimensional ST. Having done so, we now have a fully formal description of the model, which can be used as a basis to generalise it. We do so, generalising the framework from prime-dimensions to arbitrary dimensions and finding that it is the measurement update rule, where the richer properties of the non-prime dimension can be seen, which provides the key to this generalisation. Having developed ST for all finite dimensions, we then focus on the general odd-dimensional case, and prove that in all odd-dimensional cases Spekken's Theory is equivalent to Stabilizer Quantum Mechanics. The bridge between SQM and ST is given by Gross' theory (GT) of discrete Wigner function <cit.>. Unlike most other studies, Gross' treatment considered both prime and non-prime cases in its original formulation. To summarise the contributions of this paper, we provide a compete formulation of ST in all discrete dimensions, even and odd, endowed with the updating rules for sharp measurements both for prime and non-prime dimensional systems. We extend the equivalence between ST and SQM via Gross' Wigner functions to all odd dimensions, and find the measurement updating rules also for the Wigner functions. The above equivalence allows us to shed light onto a complete characterisation of SQM in non-prime dimensions. Finally the incredibly elegant analogy between the three theories in odd dimensions: ST, SQM and GT, is depicted in terms of their updating rules. The remainder of the paper is structured as follows. In section 1 we precisely and concisely describe the original framework of Spekkens' theory, in particular we define ontic and epistemic states, observables and the rule to obtain the outcome of the measurement of an observable given a state. In section 2 and 3 we state and prove the updating rules in Spekkens' theory respectively for prime and non-prime dimensional systems. We prove these in two steps: first considering the case in which the state and measurement commute, and then the more general (non-commuting) case. The mathematical difference between the set of integers modulo d, for d prime and non-prime, results in having two levels of observables: the fundamental ones - the fine graining observables - and the ones that encode some degeneracy - the coarse-graining observables. The latter are problematic and are only present in the non-prime case. This is the reason why we need a different formulation in the two cases. The updating rules for the coarse graining observables will need a step in which the coarse-graining observables are written in terms of fine graining ones. In section 4 we state the equivalence of ST and SQM via Gross' Wigner functions in all odd dimensions. We also express the already found updating rules in terms of Wigner functions and we use them to depict the elegant analogies between these three theories. The paper ends with a discussion of the possible applications of our achievements and with a summary of the main results. § SPEKKENS' THEORY We start by reviewing and introducing Spekkens' theory for prime-dimensional systems. We take a slightly different approach to <cit.> and <cit.>. ST is a hidden variable theory, where the hidden variables are points in a phase space. The state of the hidden variables is called the ontic state. In Spekkens' model the ontic state is hidden and can never be known by an experimenter. The experimenter's best description of the system is the epistemic state, representing a probability distribution over the points in phase space. For a single d-dimensional system, a phase space can be defined via the values of two conjugate fiducial variables, which we label X and P, in analogy to position and momentum. X and P can each take any value between 0 and d-1, and a single ontic state of the system is specified by a pair (x,p), where x is the value of X and p is the value of P. This phase space is equivalent to the space ℤ_d^2. In figure <ref> three examples of epistemic states of one trit (d=3) are depicted, where X and P are represented by the rows and columns in the phase space ℤ_3. A collection of n systems is described by n pairs of independent conjugate variables X_j and P_j, with j∈0,…,n-1 a label indexing the systems. The phase space, denoted by Ω, is simply the cartesian product of single system phases spaces and thus Ω≡(ℤ_d)^2n.[The dimension d is any positive number, and we will not, in general, restrict it to odd or even, prime or non-prime, unless specified.] The ontic state of the n-party system represents a set of values for each fiducial observables X_j and P_j. In other words, an ontic state is denoted by a point in the phase space λ∈Ω. We call X_j and P_j observables because they correspond to measurable quantities, and assume that these observables are sufficient to uniquely define the ontic state. We can refer to Ω as a vector space where the ontic states are vectors (bold characters) whose components (small letters) are the values of the fiducial variables: λ=(x_0,p_0,x_1,p_1,…,x_n-1,p_n-1). Not only are the fiducial variables important for defining the state space, they also generate the set of all general observables in the theory. A generic observable, denoted by Σ, is defined by any linear combination of fiducial variables: Σ=∑_m(a_mX_m+b_mP_m), where a_m,b_m∈ℤ_d and m∈0,…, n-1. The observables inhabit the dual space Ω^*, which is isomorphic to Ω itself. Therefore we can define them as vectors, in analogy with ontic states, Σ=(a_0,b_0,a_1,b_1,…,a_n-1,b_n-1). The formalism provides a simple way of evaluating the outcome σ of any observable measurement Σ given the ontic state λ, i.e. by computing their inner product: σ=Σ^Tλ=∑_j(a_jx_j+b_jp_j), where all the arithmetic is over ℤ_d. Spekkens' theory gains its special properties, and in particular, its close analogy with stabilizer quantum mechanics via the imposition of an epistemic restriction, a restriction on what an observer can know about the ontic state of a system. The observer's best description is called the epistemic state, which is represented by a probability distribution p(λ) over Ω (figure <ref>). The epistemic restriction of ST is called classical complementarity principle and it states that two observables can be simultaneously measured only when their Poisson bracket is zero. This is motivated by Stabilizer Quantum Mechanics, since it captures the condition for two observables in SQM to commute. We shall adopt the quantum terminology here, and say that if the Poisson bracket between two observables is zero they commute. This can be simply recast in terms of the symplectic inner product: ⟨Σ_1,Σ_2⟩≡Σ_1^TJΣ_2=0, where J= ⊕_j=1^n[ 0 1; -1 0 ]_j is the symplectic matrix. Note that each observable Σ_j partitions Ω into d subsets, each of the form (span{Σ_j})^⊥+w, where w is any ontic state such that Σ_j^T·w=σ_j. Let us now consider sets of variables that can be jointly known by the observer. Such variables commute, and represent a sub-space of Ω known as an isotropic subspace. We denote the subspace of the known variables as V=span{Σ_1,…,Σ_n}⊆Ω, where Σ_i denotes one of the generators (commuting observables) of V. Sets of known commuting variables are important as these define the epistemic states within the theory. In particular, we can define an epistemic state by the set of variables V that are known by the observer and also the values σ_1,…,σ_n that these variables take. This means that Σ_j^T·w=σ_j, where w∈ V is an ontic state that evaluates the known observables. We will call w a representative ontic state for the epistemic state. More precisely we can state the following theorem. EpistemicTheoremProposition The set of ontic states consistent with the epistemic state described by (V,w) is V^⊥+w, where the perpendicular complement of V is, by definition, V^⊥={a∈Ω | a^T b=0 ∀ b∈ V }. Let us start by considering the set of ontic states λ such that Σ_j^Tλ=0 ∀ j. By definition of perpendicular complements, the ontic states λ belong to V^⊥. If we consider an ontic state w such that Σ_j^T w=σ_j, then Σ_j^T (λ+w)=σ_j. Therefore the ontic states consistent with the epistemic state associated to (V,w) are the ones of the kind λ+w, i.e. the ones belonging to V^⊥+w. Note that the presence of w≠0 simply implies a translation, that is why we can also call it shift vector. By assumption the probability distribution associated to the epistemic state (V,w) is uniform (indeed we expect all possible ontic states to be equiprobable), so the probability distribution of one of the possible ontic states in the epistemic state (V,w) is P_(V,w)(λ)=1/d^nδ_V^⊥+w(λ), where the delta is equal to one only if λ∈ V^⊥+w (note this means that the theory is a possibilistic theory). In figure <ref> we specify the subspaces V and V^⊥ in three different examples of epistemic states of one trit. We can sum up our approach to Spekkens' model as follows: * Start from the intuitive (physically justified) formula (<ref>) that relates observables Σ_j, ontic states λ and outcomes σ_j. * Epistemic restriction: the compatible observables are the ones whose symplectic inner product is zero. * Compute the shift vector w. This allows us to shift back the set of points λ to obtain a subspace. * The set of ontic states compatible with the epistemic state (V,w) is V^⊥+w, where V is the isotropic subspace spanned by the observables Σ_j (the set of known variables). We say that this approach is physically intuitive because we start with equation (<ref>), which is physically motivated and states, observables and the corresponding outcomes are defined in terms of it. Equation (<ref>) also allows us to see that the shift comes from the need to recover the subspace structure. § UPDATING RULES - PRIME DIMENSIONAL CASE The formulation of ST in <cit.>, made for prime (and infinite) dimensional systems and described in the previous section, does not provide a full treatment of the transformative aspect of measurements, i.e. how the epistemic state has to be updated after a measurement procedure. In the following we will provide a proper formalization of it, and in the next section we will generalise the formalism to all dimensions, non-prime too. The set of integers modulo d shows different features depending on d being prime or not. In particular in the non-prime case it is not always possible to uniquely define the inverse of a number. The consequences of this will directly affect the updating rules. In particular the possible observables sometimes will not show full spectrum: some outcomes will not be possible because they would derive from arithmetics involving numbers with not well-defined inverses. This will divide the set of possible observables in two categories depending on whether they have full spectrum or not. We start from the prime case where problematic observables are not present because inverses always exist. Like in quantum theory, duality in the description of states and measurements characterises ST. This means that we can represent the elements of a measurement Π in an epistemic-state way, (V_Π,r), where we can go from one element of the measurement to the other by simply shifting the representative ontic vector r (see figure <ref>). In ST the measurement process corresponds to the process of learning some information (aka asking questions) about the ontic state of the system. According to the classical complementarity principle only the observables that are compatible (i.e. Poisson-commute) with the state of the system can be learned (jointly knowable). This means that the state after measurement will be given by the generators of the state before the measurement and the generators of the measurement which are compatible with it.[As an abuse of language we here talk of generators of a state meaning the orthogonal basis set that generates the subspace of known variables associated with the state.] It is then fundamental to understand how compatible sets of ontic states (the isotropic subspaces of known variables V and their perpendicular V^⊥) change when independent observables are added and removed from the set of known variables V. §.§ Adding and removing generators to/from V * Let us start with the case of adding a generator Σ' to the set of generators of V=span{Σ_1,…,Σ_n}. We assume that Σ' is linear independent with respect to the set spanned by the Σ_j. Let us see what happens to V^⊥. The subspace V after the addition becomes V'=V⊕ span{Σ'}. By definition the direct sum of two subspaces A⊕ B returns a subspace such that for each a∈ A and b∈ B, the sum a+b belongs to A⊕ B. The direct sum of two subspaces is a subspace. We are interested in the orthogonal complement of a direct sum. It is well known that (A⊕ B)^⊥=A^⊥∩ B^⊥. This means that by adding a generator to V, its perpendicular V^⊥ is given by V'^⊥=V^⊥∩ (span{Σ'})^⊥. Note that V'^⊥ is smaller than V^⊥. * We now analyse what happens if we remove a generator, say Σ_n, from the set of generators of V. This means that now V'=span{Σ_1,…, Σ_n-1}. The set V^⊥ is clearly contained in V'^⊥, since any vector orthogonal to all elements of V must also be orthogonal to all elements of V'. By definition, the set V'^⊥ is composed by all the ontic states λ such that Σ_j^T λ=0 for all j<n, but Σ_n^T λ≠ 0. This means that we need to remove the constraint Σ_n^T λ = 0 to enlarge V^⊥ to V'^⊥, i.e. we simply need to add the ontic states λ'=cγ to V^⊥, where c∈ℤ_d≠0 and γ is a vector such that Σ_n^Tγ=1. Indeed this implies that Σ_n^T(λ+λ')=Σ_n^T(λ+cγ)=0+c≠ 0. In prime dimensions γ uniquely exists and it corresponds to k^-1Σ_n, where k=Σ_n^TΣ_n. Indeed the inverse of an integer k∈ℤ_d≠ 0 always uniquely exists if d is a prime number. The formula for V'^⊥ then reads V'^⊥=⋃_c (V^⊥+ ck^-1Σ_n)≡⋃_w_n∈ V_n(V^⊥+w_n) = V^⊥⊕ V_n, where the addition of + w_n means that the whole set V^⊥ is shifted by w_n, and V_n=span{Σ_n}. The previous trick in general works as follows. Given the ontic state λ, the observable Σ and the outcome σ associated with them, i.e. Σ^Tλ=σ, then it is possible to shift the value σ by a constant k such that Σ_n^TΣ_n=k, by only adding Σ itself to the ontic state: Σ^T(λ+Σ)=σ +Σ^TΣ=σ + k. Note that the above identity allows us to change the value of the outcome associated with an ontic state by a constant factor (that we can also choose) without affecting any commuting observable (in this case Σ). §.§ Measurement updating rules We now want to find the updating rules for the state (V,w) of a prime dimensional system when we perform a measurement (V_Π,r) on it. We will consider V_Π being spanned by the generators denoted as Σ'_j. The representative ontic vector associated to the measurement, r, is such that, by definition, Σ'_j^Tr=σ'_j, where the σ'_j are the outcomes associated with the measurement. The subspace of known variables V can be written in terms of the sets generated by the generators Poisson-commuting with all the Σ'_j, V_commute, and non-commuting ones, V_other. According to this definition V_commute will always be a subspace. We cannot state the same for V_other, since the null vector does not belong to it. For this reason we augment V_other with the null vector in order to create a subspace. This implies that we can decompose V as V=V_commute⊕ V_other. We can also prove the following lemma. LemmaNonCommutingLemma The subspace V_other has dimension m, where m is the number of non-commuting generators of the measurement with the state. Let us initially assume the measurement to consist only of one non-commuting generator Σ', so m=1. Let us prove the lemma by contradiction. Let u,v be two orthogonal non-zero elements of V_other. Note that, by definition of a subspace, if u,v∈ V_other, also a linear combination of u,v has to belong to V_other. By definition u,v do not commute with Σ'. Therefore we can write Σ'^T J u=a, Σ'^T J v=b, where a,b≠0. In particular there will exist a constant c∈ℤ_d such that a-bc=0. This implies that Σ'^T J (u-cv)=0. Hence the linear combination (u-cv) belongs to V_commute. This is a contradiction, therefore V_other has dimension 1. From the same reasoning, in the case of m non-commuting generators of the measurement, the subspace V_other has dimensions at maximum equal to m. Let us assume now that the dimension of V_other is m-1. This is not possible because it would mean that, for example, Σ'_m-1 can be written as a linear combination of Σ'_0,…,Σ'_m-2. However this is not the case because, by definition of basis set, all the generators are linearly independent. Therefore V_other has dimension m. We will now provide the updating rules both for V and w in two steps: first considering the state and measurement to commute, and then the general (non-commuting) case. TheoremTheorem Commuting case. The epistemic state (V,w) after a measurement (V_Π,r) that commutes with it, i.e. their generators all Poisson commute, is described by the epistemic state (V',w') such that V'^⊥=(V^⊥+w-w')∩(V_Π^⊥+r-w'), where w' is given by equation w' = w+∑_iΣ'_i^T(r-w)γ_i, where Σ'_i are the generators of the measurement Π and γ_i is such that Σ'_i^Tγ_i=1. When the state and measurement commute we have to add the generators of the measurement to the set of generators of V, as we have seen in the previous subsection <ref> (learning stage). Therefore the updating rule for the subspace V is (equation (<ref>)) V→ V'=V⊕ span{Σ'_0,Σ'_1,…Σ'_i,…}= V⊕ V_Π. In terms of perpendicular subspaces this implies that V'^⊥=V^⊥∩ V_Π^⊥. Let us initially assume the measurement to consist only of one generator Σ'. Let us recall that the outcome associated with Σ' is σ'. We assume w is not compatible with this outcome, i.e. Σ'^Tw=σ'+x, for some shift x∈ℤ_d, and we want to find w' such that Σ'^Tw'=σ'. The identity (<ref>) we used in the previous section does the job. More precisely, w'=w-xγ, where the vector γ is such that Σ'^Tγ=1. The above expression can be also written as w'= w-k^-1xΣ', where k=Σ'^TΣ'. The inverse of k always exists because we are in the prime dimensional case. Without referring to x we can restate the updating rule for the representative ontic vector as w→w+k^-1(σ'-Σ'^Tw)Σ'= w+k^-1Σ'^T(r-w)Σ'. Note that if we consider more than one generator of the measurement, we simply have to sum over all those generators in the second term. This immediately follows from considering the whole measurement Π as a sequence of measurements given by each generator Σ'_i and apply every time the rule (<ref>). We state again that the above formula always holds for prime dimensional systems. We cannot claim the same in non-prime dimensions. The correct updating rule for the subspace V'^⊥ is found by combining the updating rules for V and w as in (<ref>). This correction simply sets the subspaces to the same origin in order to correctly compute their intersection, as schematically shown in figure <ref>. At the end we obtain for the epistemic state (V',w') that V'^⊥+w'=(V^⊥+w)∩(V_Π^⊥+r). We recall that the probability associated to each ontic state consistent with the epistemic state is uniform, i.e. given by P(V',w') =1/|V'^⊥+w'|=1/|V'^⊥| = 1/|(V^⊥+w)∩(V_Π^⊥+r)|, where |·| indicates the size of the subspace. Figure <ref> shows a basic example of theorem <ref>. NonCommutingTheorem[Theorem]Theorem Non-commuting case. The epistemic state (V,w) after a measurement (V_Π,r) that does not commute with it, i.e. some of the generators do not Poisson commute with the state, is described by the epistemic state (V',w') such that V'^⊥=(V_commute^⊥+w-w')∩(V_Π^⊥+r-w'), where V_commute^⊥ is given by V_commute^⊥=V^⊥⊕ V_other. The representative ontic vector w' is given by w' = w+∑_iΣ'_i^T(r-w)γ_i, where Σ'_i are the generators (even the non-commuting ones) of the measurement Π and γ_i is such that Σ'_i^Tγ_i=1. Let us assume that Σ'_j, for j∈{0,…,m-1}, do not commute with the generators of V. In addition to the learning stage of the previous commuting case, we also have a removal stage of the disturbing part of the measurement. We have already seen that we can split the subspace V in V=V_commute⊕ V_other, where V_other is generated, from lemma <ref>, by all the Σ'_j, for j∈{0,…,m-1}. Therefore we can reduce to the commuting case if we only consider V_commute instead of the whole V. The updating rule for the subspace V then becomes V→ V' =V_commute⊕ span{Σ'_0,Σ'_1,…Σ'_i,…} =V_commute⊕ V_Π. In terms of the perpendicular subspaces note that we can both write V'^⊥=(V^⊥⊕ V_other)∩ V_Π^⊥, and V'^⊥=V_commute^⊥∩ V_Π^⊥, from the usual property that the perpendicular of a direct sum is the intersection of the perpendicular subspaces. The updating rule for the representative ontic vector is the same as in the previous case (equation (<ref>)). The correct updating rule for the subspace V'^⊥ is found by combining the updating rules for V and w as in the previous case (<ref>), where V^⊥ is replaced by V^⊥_commute. At the end we obtain for the epistemic state (V',w') that V'^⊥+w'=(V_commute^⊥+w)∩(V_Π^⊥+r). Figure <ref> shows a basic example of theorem <ref>. § UPDATING RULES - NON PRIME DIMENSIONAL CASE It is quite common in studies of discrete theories, like Spekkens' model and SQM, to only consider the prime dimensional case because of the particular features of the set of integers modulo d, ℤ_d, when d is non-prime, like the impossibility of uniquely define inverses of numbers. For example in our present case, figure <ref> shows the peculiar properties of the observable 3X in d=6, which has not full spectrum of outcomes. The general formulation of Spekkens' model of section <ref> does not change; not even the rules for calculating the probabilities of outcome and the updating of the state after a reversible evolutions (which are present in <cit.>). The new formulation we provide affects the observables and the related measurements updating rules. More precisely our issue, as already noticed, regards the updating-rule formula (<ref>) and (<ref>) for the shift vector w' and the subspace V^⊥_commute, which do not always hold when the dimension d is non-prime. In fact the vector γ_i such that Σ'_i^Tγ_i=1 does not always exist in that case. On the other hand, in prime dimensions, it always uniquely exists because γ_i=k_i^-1Σ'_i and the inverse of the integer k_i=Σ'_i^TΣ'_i always uniquely exists. Unlike the original formulation due to Spekkens, we will now characterise Spekkens' model in non-prime dimensions. In particular we characterise which are the observables that are problematic in the above sense - the coarse-graining observables, like 3X in d=6 - and we then find the updating rules for a state subjected to the measurement of such observables by rewriting them in terms of non-problematic observables - the fine-graining observables. In the next subsection we assume single-system observables (i.e. of the kind Σ'=aX+bP, a,b∈ℤ_d) in order to soften the notation and facilitate the comprehension. This will bring more easily to the updating rules even in the most general case of many systems (subsection <ref>). In this case we recall, without making any reference to the quantity k^-1, but just in terms of the vector γ, the updating rule for the shift vector w', w'=w-xγ, where, as usual, x=-Σ'^T(r-w), and the expression for V^⊥_commute, V^⊥_commute=⋃_c(V^⊥+cγ). §.§ Coarse-graining and fine-graining observables We define a fine-graining observable as an observable that has full spectrum, i.e. it can assume all the values in ℤ_d. On the contrary a coarse-graining observable has not full spectrum. FineTheorem[LemmaNonCommuting]Lemma An observable O_fg has full spectrum, i.e. it is a fine-graining observable, if and only if it has the following form, O_fg=a'X+b'P, where a',b'∈ℤ_d are such that they do not share any integer factor or power factor of d. On the contrary a coarse-graining observable is written as O_cg=aX+bP=D(a'X+b'P), where a',b' ∈ℤ_d are again such that they do not share any integer factor or power factor of d and D is a factor shared by a,b∈ℤ_d. More precisely the factor D is called degeneracy and it is defined as D=D_1^n_1· D_2^n_2·… , where D_1,D_2,… are different integer factors of d shared by a and b, and n_1,n_2,… are the maximum powers of these factor such that they can still be grouped out from a and b. We take the maximum powers because we want the remaining part, a'X+b'P, to not share any common integer factor or power factor of d between a' and b'. In this way we can associate a fine-graining observable to a coarse graining one by simply dropping the degeneracy D from the latter. Let us first prove that an observable of the kind (<ref>), O_fg=a'X+b'P, is a full spectrum one. This can be proven by using Bezout's identity <cit.>: let a' and b' be nonzero integers and let D be their greatest common divisor. Then there exist integers X and P such that aX+bP=D. In our case the greatest common divisor D is equal to one, since a',b' are coprime. [ It could be that a',b' share a factor which is not a factor of d. In this case the argument follows identically as if they were coprime.] Therefore we have proven that there exist values of the canonical variables X,P∈ℤ_d such that O_fg=a'X+b'P=1. In order to reach all the other values of the spectrum we simply need to multiply both X and P in the previuos equation by j∈ℤ_d. We now prove the converse, i.e. that a full spectrum observable implies it to be written as (<ref>). We prove this by seeing that an observable written as (<ref>) has not full spectrum, i.e. we negate both terms of the reverse original implication. Proving the latter is straightforward, since the multiplication modulo d between an arbitrary quantity and a factor D, which is given by powers of integer factors of d, gives as a result a multiple of D. Since the multiples of D do not cover the whole ℤ_d, then any observable of the form (<ref>) has not full spectrum. [Multiples of D do not cover the whole spectrum of ℤ_d because D has not an inverse D^-1 (it is not coprime with d) and so we cannot obtain the whole values σ of ℤ_d by simply finding X,P such that a'X+b'P=D^-1σ.] Since an observable of the form (<ref>) is an observable that cannot be written as (<ref>) by definition, we obtain that a full spectrum observable implies the observable to be written as (<ref>). Given lemma <ref> we have got the expressions (<ref>) and (<ref>) for coarse-graining and fine-graining observables. We want now to prove the following lemma to ensure that fine-graining observables are characterised by precisely defined updating rules. GammaTheorem[LemmaNonCommuting]Lemma The vector γ in the updating rule (<ref>) for the shift vector w' and in the equation (<ref>) for the subspace V^⊥_commute exists if and only if the observable is a fine-graining one. Let us prove that if we have a fine graining observable the vector γ exists. In our case Σ'=(a',b') and, by definition of a',b' (as usual defined for fine-graining observables) and full spectrum, we can always find a vector γ=(γ_a,γ_b) such that Σ'^Tγ=a'γ_a + b'γ_b equals 1. Let us prove the converse. We now have the vector γ such that Σ'^Tγ=aγ_a + bγ_b=1, where the coefficients a,b∈ℤ_d define our observable aX+bP=σ. We want to prove that σ can achieve all the values of ℤ_d. Since Σ'^Tγ=1 we can set the values of (X,P) as equal to (γ_a,γ_b) in order to reach the value σ=1. We can now achieve all the other values of the spectrum by simply redefining γ as γ̃=c γ, where c assumes all the values in ℤ_d. The above lemma <ref> should convince us that in order to find the updating rules in the presence of a coarse graining observable, it is appropriate to decompose it in terms of fine-graining observables. Let us assume that our coarse-graining observable is O_cg=aX+bP=D(a'X+b'P)=σ, and the associated isotropic subspace and representative ontic vector are (V_cg,r_cg). To this observable we can associate D̅ different fine-graining observables O_fg=a'X+b'P=σ_j, where j∈0,…,D̅-1. The quantity D̅ is the degeneracy D without the powers n_1,n_2,…, i.e. D̅=D_1· D_2 ·…. Indeed the powers n_1,n_2,… simply represent multiplicities associated to each corresponding fine-graining observable. The associated isotropic subspaces and representative ontic vectors are (V_fg,r^(j)_fg), where V_fg=span{(a',b')} (see figure <ref>). By definition the perpendicular isotropic subspaces are V^⊥_cg ={v=(v_a,v_b)∈Ω | v_a a + v_b b =D(v_a a' + v_b b')=0 mod(d)} V^⊥_fg={v' =(v'_a,v'_b)∈Ω | v'_a a' + v'_b b'=0 mod(d)}. It is clear that V^⊥_cg⊃ V^⊥_fg and we can therefore construct V^⊥_cg as V^⊥_cg = ⋃^D̅-1_j=0(V^⊥_fg+v_j) =V^⊥_fg⊕ V_D, where the subspace V_D provides all the vectors that we need to combine with the vectors of V^⊥_fg to reach the whole V^⊥_cg. We call the subspace V_D the degeneracy subspace because it encodes the degeneracy of V_cg with respect to V_fg . It has dimension 1 and size D̅. This is consistent with the fact that the dimensions of V^⊥_cg and V^⊥_fg are respectively 2 and 1. The sizes are respectively D̅· d and d. The size of V^⊥_fg is d because it is always a maximally isotropic subspace and its dimension is 1 because from one generator we get all the other vectors of the subspace by multiplication with j∈ℤ_d. The dimension V^⊥_cg is 2 because it cannot be 1 (it would be the same subspace as V^⊥_fg) and it cannot be greater than 2 since also the whole phase space Ω=ℤ^2_d has dimension 2. In order to know the size of V^⊥_cg we need to count all the jv, where j∈{0,1,…,D̅-1}, that means D̅· d. Therefore it can be written as V_D=span{v}, and all its D̅ vectors are of the kind v_j=j v. The above reasoning easily extends to the case of n systems, where the dimensions are dim(V^⊥_cg)=2n, dim(V^⊥_fg)=n, dim(V_D)=n, and the sizes are |V^⊥_cg|=D̅^nd^n, |V^⊥_fg|= d^n, |V_D|=D̅^n. We can now prove that V_D is a vector space. The definition of V_D is V_D ={v∈Ω | αw + βv=t, where w∈ V^⊥_fg, α,β∈ℤ_d, t∈ V^⊥_cg}. To see that it is a vector space we just need to see that (0,0) belongs to V_D and that V_D is closed under addition and multiplication, i.e. under linear combinations. The null vector belongs to V_D because in the definition (<ref>) we would remain with αw=t, where w∈ V^⊥_fg and V^⊥_fg⊂ V^⊥_cg. Let us imagine that we have two vectors v,z∈ V_D. Is the vector γv+ δz, where γ,δ∈ℤ_d, still belonging to V_D? It is easy to see that if we apply the definition (<ref>) we would get αw+β(γv+ δz), which can be rewritten as ( αw+βγv)+ (0·w+ βδz), where each of the two terms in parenthesis belong to V^⊥_cg, and therefore the whole expression belongs to it too. We now define the shift vectors r^(j)_fg in terms of r_cg and see that we can encode the degeneracy expressed by V_D in there. The idea is schematically depicted in figure <ref>. Given the shift vector associated to the coarse-graining observable r_cg, the shift vectors r^(j)_fg associated to the corresponding fine-graining observables are of the kind r^(j)_fg=r_cg+v_j, where v_j∈ V_d and are therefore of the kind jv, where j∈{0,…,D̅-1}. This implies that if we assume the outcome associated to the coarse-graining observable to be σ_cg, i.e. Σ^T_cgr_cg=σ_cg, where Σ_cg=(a,b), then the outcomes associated to the fine graining-observables are Σ^T_fgr^(j)_fg=Σ^T_fg(r_cg+jv)=σ_cg/D+jC, where C is the anti-degeneracy and it is defined as a non-zero number belonging to ℤ_d such that D· C=0 mod(d). The idea is that the vector v∈ V_D is such that Σ^T_fgv=C≠ 0, so it does not belong to V^⊥_fg, but it does belong to V^⊥_cg, since D· C=0 mod(d). An easy way to find one of the possible v is to calculate it as CΣ_fg, where Σ_fg is the generator of V_fg. In this way we know that Dv=0, but v does not belong to V^⊥_fg, i.e. v_a a' + v_b b'≠ 0 because Σ_fg is not in V_fg^⊥. It is important to notice that equation (<ref>) implies that not all the outcomes are allowed for the fine-graining observables associated to the coarse-graining one; they are allowed only when the ratio σ_cg/D exists. Figure <ref> also explains this fact. §.§ Measurement updating rules Let us assume to have n systems and to measure the coarse-graining observable O_cg=a_1X_1+b_1P_1+… +a_nX_n+b_nP_n=D(a'_1X_1+b'_1P_1+… +a'_nX_n+b'_nP_n)=σ_cg, with corresponding isotropic subspace of known variables V_cg and shift vector r_cg, on the state ρ=α_1X_1+β_1P_1+… +α_nX_n+β_nP_n=σ, with corresponding isotropic subspace of known variables V=span{Σ_1,…,Σ_n} and shift vector w. The idea in order to find the updating rules for the state after measurement, the subspace of known variable V' and the representative ontic vector w' is to compute the updating rule of the initial state ρ with the fine-graining observables that are associated to the coarse graining observable O_cg, i.e. O^(j)_fg=a'_1X_1+b'_1P_1+… +a'_nX_n+b'_nP_n=σ^(j)_fg (indeed we know that the updating rules are valid for them from theorem <ref>), and then combine them together. More precisely, the following theorem holds. FinalbisTheorem[Theorem]Theorem The epistemic state (V,w) after a coarse-graining measurement (V_cg,r_cg) is described by the epistemic state (V',w') such that V'^⊥=⋃^D̅-1_j=0 [(V_commute^⊥+w-w')∩(V^⊥_fg+r^(j)_fg-w')], where the shift vector w' is the shift vector deriving from the updating rule of the state after the measurement of the fine-graining observable O^(j)_fg, w'=w'_j=w+∑_i=0^nΣ'^T_i(r^(j)_fg-w)γ_i, where the vectors γ_i are defined such that Σ'^T_iγ_i=1, and Σ'_i are the n generators of the subspace V_fg associated to the fine-graining observable O^(j)_fg. The subspace V_commute^⊥ is given by the original V after having removed the non-commuting part, i.e. equation (<ref>), V_commute^⊥=⋃^d_c=1(V^⊥+∑_l=N+1^ncγ_l)=V^⊥⊕_l=N+1^n V_l = V^⊥⊕ V_other, where γ_l is such that Σ_l^Tγ_l=1 and V_l are the subspaces spanned by the (n-N) non-commuting generators Σ_l. Obviously if the state and measurement commute, then V_commute^⊥=V^⊥. The above theorem tells us that the way we combine the updating subspaces of the state with each individual fine-graining observables is through their union. This result is clear in terms of schematic diagrams (figure <ref>). The updated shift vector is just one of the updated shift vectors of the state with the fine-graining observables, because the information needed to update the shift vector of the state is encoded in just one of the fine-graining shift vectors. The degeneracy includes a meaningless multiplicity in the coarse-graining shift vector, and therefore every fine-graining observable can do the job of correctly updating the shift vector of the state. Actually every combination of the shift vectors w'_j can do the job, apart from the ones that sum to 0 mod(d), like ∑_j=0^D̅-1w'_j. Note also that in the definition of V^⊥_commute the vector γ_l is, in general, degenerate. This is not a problem because any degenerate value of γ_l brings to the same subspace V^⊥_commute, since by definition its role is to add the vectors λ'=cγ_l to V^⊥ such that Σ_n^Tλ'≠0. We find the expression for the updated subspace V'^⊥ by simply reusing the already found formulas (<ref>) and (<ref>) of the prime-dimensional case and substituting V^⊥_Π with V^⊥_cg and r with r_cg, V'^⊥=(V_commute^⊥+w-w')∩(V^⊥_cg+r_cg-w'). If we now consider the decomposition of V^⊥_cg+r_cg as in (<ref>) and (<ref>), we obtain V'^⊥=(V_commute^⊥+w-w')∩[∪^D̅-1_j=0 (V^⊥_fg+r^(j)_fg)-w']. Since the intersection of a union is the union of the intersections, we have proven the first part of the theorem, V'^⊥=⋃^D̅-1_j=0 [(V_commute^⊥+w-w')∩(V^⊥_fg+r^(j)_fg-w')]. The second part of the proof regards w' being equal to any of the w'_j. Because of the degeneracy, any w'_j is equivalent to the others (with different value of j) in order to provide us with w', indeed it is possible to find one from another just by adding a vector v∈ V_D. The latter can be proven as follows. For simplicity let us assume to be in the case n=1 and that v is the generator of V_D. We know that, by the definition of state after measurement of a fine-graining observable, the updated shift vector w'_j is such that Σ^T_fgw'_j=σ_cg/D+jC=σ^(j)_fg, where C=Σ^T_fgv is the antidegeneracy (equation (<ref>)). It is straightforward to see that if we add v to w'_j, we get w'_j+v=w'_j+1, indeed Σ^T_fg(w'_j+v)=σ_cg/D+(j+1)C=σ^(j+1)_fg. Figure <ref> shows a basic example of theorem <ref>. § EQUIVALENCE OF SPEKKENS' THEORY AND SQM IN ALL ODD DIMENSIONS In <cit.> it has been shown that SQM and Spekkens' toy model are two operationally equivalent theories in odd prime dimensions via Gross' theory of discrete non-negative Wigner functions. We have generalised Spekkens' model to all discrete dimensions. The above equivalence does not hold in even dimensions, but we will now see that it holds in all odd dimensions. We will also state the equivalence in terms of the updating rules, where all its elegance arises. We recall that SQM and Gross' theory of non-negative Wigner functions are equivalent in all odd dimensions <cit.>. §.§ SQM - updating rules Stabilizer quantum mechanics is a subtheory of quantum mechanics where we only consider common eigenstates of tensors of Pauli operators, unitaries belonging to the Clifford group, and Pauli measurements <cit.>. We can always write a stabilizer state ρ as ρ=1/𝒩ρ_1·ρ_2·…·ρ_N, where 𝒩=Tr[ρ_1·ρ_2·…·ρ_N], j ∈{1,…, N≤ n}, n is the number of qudits and ρ_j=(𝕀_d+g_j+g_j^2+…+g_j^d-1), where g_j is a stabilizer generator, more precisely a Weyl operator: Ŵ(λ)=χ(pq)Ŝ(q)B̂(p), where χ(pq)=e^2π i/dpq, q,p are the coordinates of the phase space point λ=(q,p), and Ŝ,B̂ are respectively the shift and boost operators (generalised Pauli operators) and the arithmetics is modulo d, Ŝ(q)=∑_q'∈ℤ_d|q'-q⟩⟨q'| B̂(p)=∑_q∈ℤ_dχ(pq)|q⟩⟨q|. When considering more than one qudit, the Weyl operator is given by the tensor product of the single Weyl operators. We can write the stabilizer state ρ in a more compact way as ρ = 1/𝒩∏_j^n∑_i^d-1 g_j^i. However we will mostly use the following notation in terms of stabilizer generators, ρ→⟨ g_1,…, g_N⟩ . We now analyse the updating rules for the state ρ under the stabilizer measurement Π, Π→⟨ p_1,…, p_M⟩ where p_k is a stabilizer generator of Π and k ∈{1,…, M≤ n}. We analyse the updating rules first in the commuting case ([ρ,Π]=0) and then in the general case. * For non-disturbing (commuting) measurements, the state after measurement ρ' is given by adding the stabilizer generators of the measurement Π and the state ρ, unless some generators coincide. In the latter case we obviously count them only once. ρ' →⟨ g_1,g_2,…, g_N, p_1,p_2,…, p_M⟩, where we have here considered the case in which no generators coincide. This formula means that the state ρ' is now ρ'= 1/𝒩∏_j^N^*∑_i^d-1 r_j^i, where N^*=N+M and r_j is a stabilizer generator of ρ', i.e. it is either a valid (commuting) generator g_j or p_j. In the case where e.g. F generators coincide, then N^*=N+M-F. * For disturbing (non-commuting) measurements (the most general case) the idea is that if we remove the non-commuting factors ρ_j from the state ρ, i.e. [ρ_j,Π]≠ 0, this case reduces to the previous commuting one. We assume the state ρ to have only one non-commuting factor, say ρ_N, which corresponds to the stabilizer generator g_N. The state after measurement ρ' is given by removing the non-commuting generator and adding the remaining ones of the state and measurement, unless some generators coincide. In the latter case we obviously count them only once. ρ' →⟨ g_1,g_2,…, g_N-1, p_1,p_2,…, p_M⟩, where we have here considered the case in which no generators coincide. This formula means that the state ρ' is now ρ'=1/𝒩∏_j^N^*∑_i^d-1 r_j^i, where N^*=N+M-1 and r_j is a stabilizer generator of ρ', i.e. it is either a valid (commuting) generator g_j or p_j. In the case where e.g. F generators coincide, then N^*=N+M-1-F. To sum up, in the commuting case we add generators of state and measurement to obtain the state after measurement. In the non-commuting case we remove the non-commuting generator of the state and add all the others as in the commuting case. This structure is perfectly analogue to Spekkens' updating rules, which are just motivated by the classical complementarity principle. §.§ Gross' Wigner functions - updating rules Gross theory. In Gross' theory the Wigner function of a state ρ in a point of the phase space λ∈Ω is given by W_ρ(λ)=Tr[Â(λ)ρ], where Â(λ) is the phase point operator associated to each point λ, Â(λ)=1/d^n∑_λ'∈Ωχ(⟨λ,λ'⟩)Ŵ(λ'), where Ŵ(λ) are the Weyl operators defined in equation (<ref>). Note that the normalisation is such that Tr[Â(λ)]=1. We recall that a stabilizer state is a joint eigenstate of a set of commuting Weyl operators. Two Weyl operators commute if and only if the corresponding phase-space points a,a' have vanishing symplectic inner product: [Ŵ(a),Ŵ(a')]=0 if and only if ⟨a,a'⟩=a^TJa'=0. This result derives from the product rule of Weyl operators: Ŵ(a)Ŵ(a')=χ(⟨a,a'⟩)Ŵ(a+a'). From this result, the sets of commuting Weyl operators, and, as a consequence, the stabilizer states, are parametrized by the isotropic subspace M of Ω. More precisely, for each M and each w∈Ω we can define a stabilizer state (Gross construction) ρ_M,w as the projector onto the joint eigenspace spanned by {Ŵ(a):a∈ M}, where Ŵ(a) has eigenvalue χ(⟨w,a'⟩). The Wigner function associated to the state ρ_M,w is always positive (necessary and sufficient condition in odd dimensions) and it is of the kind W_(m,w)(λ)=1/d^nδ_M^C+w(λ), where M^C is the symplectic complement of M. Moreover the transformations that preserve the positivity of the Wigner functions are the Clifford unitaries. Gross' theory of non-negative Wigner functions is a faithful way of representing SQM. Equivalence of ST and GT. The Wigner function (<ref>) has the same form of the probability distribution (<ref>) associated to the epistemic state (V,w) in Spekkens' theory. More precisely, they are equivalent if we assume M=JV,[Note that the action of J is simply to map a variable into its conjugated.] indeed this transformation implies that V^⊥=M^C. The equivalence between Gross' theory and Spekkens theory, using the symplectic matrix J as the bridge, also extends in terms of transformations and measurement statistics <cit.>. This equivalence also implies the equivalence between Spekkens' theory and SQM in odd dimensions. Therefore we can see the description based on known variables (Spekkens) and the description based on Wigner functions (Gross) as two equivalent descriptions of stabilizer quantum mechanics in odd dimensions. We will now translate the already found updating rules of ST into Gross' Wigner functions. Updating rules. Let us consider a stabilizer state ρ=ρ_1·ρ_2·…·ρ_n, where n is the number of qudits (odd prime dimensions), and a measurement Π on the stabilizer state Π=Π_1·Π_2 …·Π_m, where, in general, m≤ n. Let us assume m=n in order to consider "total" measurements (not only to a part of the state). CommutingTheorem[Theorem]Theorem Commuting case. Let us assume the state and measurement to commute, i.e. [ρ,Π]=0. The Wigner function of the state after measurement is W_ρ'(λ)= 1/N W_ρ(λ)R_Π(λ), where λ∈Ω and R_Π denotes the Wigner function (also called response function) associated with the measurement Π. The normalisation factor N is N=∑_λ∈ΩW_ρ(λ)R_Π(λ). We rewrite the formula (<ref>) by replacing the Wigner functions with their definition in terms of Spekkens' subspaces, δ_λ,V'^⊥+w'=δ_λ,V^⊥+w·δ_λ,V_Π^⊥+r. The proof is straightforward. The RHS is one if and only if both the deltas are one; this means that λ has to belong simultaneously to V^⊥+w and V_Π^⊥+r, i.e. λ∈ (V^⊥+w)∩ (V_Π^⊥+r). If we recall equation (<ref>) (and figure <ref>), we see that (V^⊥+w)∩(V_Π^⊥+r)=(V'^⊥)+w', and we can conclude that the RHS of equation (<ref>) is one if and only if the LHS is one. At this point we can insert the normalisation factors on the RHS and the LHS. These guarantee that ∑_λ∈ΩW_ρ'(λ)=1 and the uniformity as expected. In the commuting case the updating rule in SQM consists of the addition of the stabilizer generators of state and measurement (equation (<ref>)). In ST the updating rule consists of the intersection of the perpendicular isotropic subspaces (equation (<ref>)). In GT addition and intersection translate into the product of the Wigner functions (equation (<ref>)). In particular this stage consists of introducing zeros to the Wigner function in correspondence of the addition of generators to the subspace of known variables V (and so removing generators from the subspace V^⊥). We will call this process - where we learn information about the state - the localization stage. MainTheorem[Theorem]Theorem Non-commuting case. Let us assume the measurement, in general, not to commute with the state, i.e. [ρ,Π]≠ 0. The Wigner function of the state after measurement is W_ρ'(λ)= 1/N∑_t∈ V_otherW_ρ(λ - t)R_Π(λ), where λ∈Ω, V_other is the set spanned by the non-commuting generators of Spekkens' subspace V associated to the state ρ. The normalisation factor N is N=∑_λ∈Ω∑_t∈ V_otherW_ρ(λ - t)R_Π(λ). Note that we could have stated the theorem in terms of stabilizer generators instead of Spekkens' generators. The former being related to the latter as follows, g_j=Ŵ(J^-1Σ_j), where J is the usual symplectic matrix, Σ_j are Spekkens' generators and g_j the corresponding stabilizer generators. The relation (<ref>) follows from the relation between ST and GT previously described, where the bridge between the two formulations is given by the matrix J. In general the state after measurement in quantum mechanics (up to a normalization) is ρ'=ΠρΠ. If [ρ,Π]=0 then ρ'=ρΠ. In order to simplify the proof, let us assume the case of only one non-commuting generator, say ρ_n. In the present case we know, from the structure of SQM and Spekkens' updating rules (adding the commuting factors between state and measurement and removing the non-commuting ones), that the state after measurement is ρ'=ρ^*Π, where ρ^*=ρ_1·…ρ_n-1. This means that we can write the state after measurement as a product of two commuting terms: ρ^* and Π. Therefore we can write the Wigner function of ρ' according to the product rule for the commuting case (equation (<ref>)): W_ρ'(λ)=1/NW_ρ^*(λ)R_Π(λ), where N=∑_λW_ρ^*(λ)R_Π(λ). We want now to prove that equation (<ref>) is equal to the latter. This means we want to prove the following: W_ρ'(λ)=∑_t∈ V_otherW_ρ(λ - t)R_Π(λ)=W_ρ^*(λ)R_Π(λ). We can simplify the terms R_Π(λ), thus getting ∑_t∈ V_otherW_ρ(λ - t)=W_ρ^*(λ). At this point, in order to prove the above theorem, we rewrite the formula (<ref>) by replacing the Wigner functions with their definition, i.e. Kronecker deltas, ∑_t∈ V_otherδ_λ-t,V^⊥+w=δ_λ,V_commute^⊥+w, where V_commute^⊥ = V^⊥⊕ V_other. Note that we have removed the response function of the measurement. This also implies that we do not have to change w, because we have only modified V^⊥ into V^⊥_commute and w' is not affected. We now want to see that the LHS of equation (<ref>) is different from zero exactly when the RHS is. The LHS is different from zero when at least one t∈ V_other is such that λ-t∈ V^⊥+w. The latter corresponds to λ∈ V^⊥+w + t. This means that λ∈ V^⊥⊕ V_other+w, i.e. λ∈ V_commute^⊥+w, which is precisely what makes the RHS different from zero. In the most general non-commuting case, in addition to the localization stage, in SQM we also have to remove the non-commuting generators from the state (equation (<ref>)). In ST this consists of the union and shifts in the perpendicular subspace (equation (<ref>)). In GT removal and union translate into the averaging out of the Wigner function (equation (<ref>)). In particular this stage consists of introducing ones to the Wigner function in correspondence of the removal of generators from the subspace of known variables V (and so adding generators to the subspace V^⊥). We can think of this process as the one where, after having learned some information in the localization stage, we need to forget something, otherwise we would get too much information about the ontic state, which is forbidden by the classical complementarity principle. This also explains why non-commuting measurements are also called disturbing measurements. We will call this forgetting-part of the process the randomization stage. Finally note that the general-case formula (<ref>) reduce to the product rule (<ref>) in the commuting case. Figure <ref> summarises the updating rules in the three theories in prime dimensions. In the non-prime dimensional case, we can rephrase all the reasonings already done in ST in terms of Wigner functions. LemmaCoarse[LemmaNonCommuting]Lemma The Wigner function W_cg(λ) of the coarse-graining observable O_cg=a_1X_1+b_1P_1+… +a_nX_n+b_nP_n=D(a'_1X_1+b'_1P_1+… +a'_nX_n+b'_nP_n)=σ_cg, can be written in terms of the Wigner functions W^(j)_fg(λ) of the associated fine graining observables O^(j)_fg=a'_1X_1+b'_1P_1+… +a'_nX_n+b'_nP_n=σ^(j)_fg as W_cg(λ)=1/D̅∑_j=0^D̅-1W^(j)_fg(λ). First of all the normalisation factor 1/D̅ is due to the fact that we are adding D̅ Wigner functions, each of them having a normalisation factor of 1/d, since they are Wigner functions of maximally isotropic subspaces (of dimension d). The proof of the rest of the formula is straightforward. According to the definition of Wigner functions, we need to prove that δ_V^⊥_cg+r_cg∝∑_jδ_V^⊥_fg+r^(j)_fg. From the decomposition of the isotropic subspaces and shift vectors in Spekkens' model, equations (<ref>) and (<ref>), we already know that V^⊥_cg+r_cg=V^⊥_fg⊕ V_D +r_cg = V^⊥_fg + ∑_j=0^D̅-1(r_cg+jv), which exactly proves that the RHS of (<ref>) is one if and only if the LHS is one. From the above construction and theorem <ref> we can immediately write the Wigner function of a stabilizer state after a coarse-graining measurement, thus generalising theorem <ref>. FinalWignerTheorem[Theorem]Theorem The Wigner function of the state ρ of n-qudit systems, where the dimension d is a non-prime intger, after the (non-commuting) measurement Π is given by W_ρ'(λ)= 1/N1/D̅∑_t∈ V_other∑^D̅-1_j=0W_ρ(λ - t)R^(j)_fg(λ), where λ∈Ω, V_other is the set spanned by the non-commuting generators of Spekkens' subspace V associated to the state ρ. The response function of the j-th fine-graining measurement is denoted by R^(j)_fg. The normalisation factor N is N=∑_λ∈Ω∑_t∈ V_otherW_ρ(λ - t)R_Π(λ), where R_Π(λ)=1/D̅∑^D̅-1_j=0R^(j)_fg(λ). We just need to apply lemma <ref> to the response function of the coarse graining measurement of theorem <ref>. Figure <ref> summarises the updating rules in ST and Gross' theory in prime and non-prime dimensions. § DISCUSSION The importance of completing Spekkens' theory with updating rules to determine the state after a sharp measurement relies on the possible applications of this theory for future works. In particular we think it is interesting to characterise ST in terms of its computational power, i.e. exploring which are the quantum computational schemes that can be represented by ST. As an example ST can be used as a non-contextual hidden variable model to represent the classically simulable part of some state-injection schemes, thus witnessing non-contextuality and also contributing to the aim of proving that contextuality is necessary for quantum speed-up in such schemes. The result about the equivalence between ST and SQM and the associated updating rules in prime and non-prime odd dimensions can provide a powerful new way to use and analyse SQM in non-prime dimensions, about which almost nothing is known. For example we are now facilitated to state, given a set of commuting Pauli operators, whether the joint eigenstate that they represent is pure. In non-prime dimensions the latter issue is not trivial because for coarse-graining observables the number of independent generators is not equal to the number of observables. However, from our construction to decompose coarse-graining into fine-graining observables, we know that the number of independent generators is equal to the number of fine-graining observables. Therefore if the set of commuting Pauli operators has the number of independent generators that equals the number of fine-graining observables, then the state is pure. Indeed fine-graining observables are associated to pure states. In addition, in the field of quantum error correction it could be interesting to study if the coarse-graining observables have any usefulness. Finally, the enforced equivalence of SQM, ST and Gross' theory in odd dimensions can be exploited to address a given problem from different perspectives, where, depending on the cases, one theory can be more appropriate than another. An example is the already mentioned one of addressing protocols based on SQM with Spekkens theory instead of SQM or Wigner functions. § CONCLUSION Spekkens' toy model is a very powerful model which has led to meaningful insights in the field of quantum foundations and that seems to have interesting applications in the field of quantum computation. We have extended it from prime to arbitrary dimensional systems and we have derived measurement updating rules for systems of prime dimensions when the state and measurement commute, equations (<ref>)(<ref>), when they do not, equations (<ref>)(<ref>), and for systems of non-prime dimensions (theorem <ref>). These results directly derive from the basic axiom of the theory: the classical complementarity principle. The latter characterises a structure for the updating rules which is the same as in stabilizer quantum mechanics: the state after measurement is composed by the generators of the measurement and the compatible (i.e. commuting) generators of the original state. Spekkens showed the equivalence between SQM and ST in odd prime dimensions via Gross' Wigner functions. We have extended this result to all odd dimensions and we have translated the updating rules of ST in terms of Wigner functions (theorems <ref>, <ref>, <ref>). We stress again that Spekkens' model and our measurement updating rules hold in all dimensions, in even dimensions too. However the equivalence between ST and SQM only holds in odd dimensions. The main reason is that SQM in even dimensions shows contextuality, while ST does not. One of the main future challenges is to find a hidden variable toy model which is also equivalent to qubit SQM. We treat the problem with systems of non-prime dimensions, which arises from the problem of defining an inverse in ℤ_d, by decomposing the problematic (coarse-graining) observables in terms of the non-problematic (fine-graining) ones. This approach naturally suggests the form of the updating rules. By comparing the updating rules in the three mentioned theories we highlight the beauty and the elegance of this equivalence, where addition and removal of generators in SQM correspond to intersection and union in ST and product and randomization in GT. This correspondence is schematically depicted, for the prime-dimensional case, in table <ref>. The non-prime case correspondence is represented in table <ref>. We believe that the fresh perspective gained by moving from one theory to another can give powerful new tools for new insights in the field of quantum computation. § ACKNOWLEDGMENTS We would like to thank Misja Steinmetz for suggesting Bezout's identity. This work was supported by EPSRC Centre for Doctoral Training in Delivering Quantum Technologies [EP/L015242/1]. § REFERENCES 10 EPR A. Einstein, B. Podolsky and N. Rosen Can Quantum-Mechanical Description of Physical Reality be Considered Complete?, Phys. Rev. 47 (10): 777-780 (1935). Bell J. S. Bell On the problem of hidden variables in quantum mechanics, Rev. Mod. Phys. 38, 447-452 (1966). Kochen S. Kochen and E.P. Specker The problem of hidden variables in quantum mechanics, J. Math. Mech. 17, 59-87 (1967). Spek1 R. W. Spekkens Evidence for the epistemic view of quantum states: A toy theory, Phys. Rev. A 75, 032110 (2007). Spek2 R. W. Spekkens Quasi-quantization: classical statistical theories with an epistemic restriction, arXiv:1409.5041 (2014). GottesPhD D. Gottesman Stabilizer Codes and Quantum Error Correction, PhD thesis,1862 (1997). GottesKnill D. Gottesman The Heisenberg representation of quantum computers, arXiv:quant-ph/9807006 (1998). GHZ D. Greenberger, M. Horne, A. Shimony, A. Zeilinger Bell's theorem without inequalities, Am. J. Phys. 58 (12): 1131(1990). Mermin N. D. Mermin Simple unified form for the major no-hidden-variables theorems, Phys. Rev. Lett. 65, 3373-3376 (1990). Peres A.Peres Incompatible results of quantum measurements, Phys. Lett. A 151, 107-108 (1990). Rauss R. Raussendorf, D.E. Browne, H.J. Briegel Measurement-based quantum computation on cluster states Phys. Rev. A 68, 022312 (2003). Bravij-Kitaev S. Bravyi, A. Kitaev Universal quantum computation with ideal Clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005). Gottes2 D. Gottesman Stabilizer codes with prime power qudits, invited talk at Caltech IQIM seminar (Pasadena, California). Howard M. Howard, J. Wallman, V. Veitch, J. Emerson Contextuality supplies the magic for quantum computation, Nat. 510, 351- 355 (2014). Gross D. Gross Hudson's theorem for finite-dimensional quantum systems, J. Math. Phys. 47, 122107 (2006). Wootters W. K. Wootters A Wigner-Function Formulation of Finite-State Quantum Mechanics, Annals of Physics 176,1 (1987). FerrieEmerson V. Veitch, C. Ferrie, D. Gross, J. Emerson Negative quasi-probability as a resource for quantum computation, New J. Phys. 14, 113011 (2012). WallmanBartlet J. J. Wallman, S. D. Bartlett Non-negative subtheories and quasiprobability representations of qubits, Phys. Rev. A 85, 062121 (2012). Pusey M. F. Pusey Stabilizer notation for Spekkens' toy theory, Found. Phys. 42, 688 (2012). Coecke B. Coecke, B. Edwards, R.W. Spekkens Phase Groups and the Origin of Non-locality for Qubits, Electron. Notes Theor. Comput., 270, 29 (2011). Damian L. Disilvestro, D. Markham Quantum Protocols within Spekkens' Toy Model, QPL, Glasgow 2016. Pawel P. Blasiak Quantum cube: A toy model of a qubit, Phys. Lett. A, 377, 847-850 (2013). Larsson J. Larsson A contextual extension of Spekkens' toy model, AIP Conf. Proc. 1424, 211 (2012). Bezout E. Bezout Theorie generale des equations algebriques Paris, France: Ph.D. Pierres (1779).