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arXiv:1312.0230 · Physical Review E

Bethe free-energy approximations for disordered quantum systems

Indaco Biazzo, Abolfazl Ramezanpour

Published Jun 26, 2014• 3 citations
arXivHTMLPDF

Abstract

Given a locally consistent set of reduced density matrices, we construct approximate density matrices which are globally consistent with the local density matrices we started from when the trial density matrix has a tree structure. We employ the cavity method of statistical physics to find the optimal density matrix representation by slowly decreasing the temperature in an annealing algorithm, or by minimizing an approximate Bethe free energy depending on the reduced density matrices and some cavity messages originated from the Bethe approximation of the entropy. We obtain the classical Bethe expression for the entropy within a naive (mean-field) approximation of the cavity messages, which is expected to work well at high temperatures. In the next order of the approximation, we obtain another expression for the Bethe entropy depending only on the diagonal elements of the reduced density matrices. In principle, we can improve the entropy approximation by considering more accurate cavity messages in the Bethe approximation of the entropy. We compare the annealing algorithm and the naive approximation of the Bethe entropy with exact and approximate numerical simulations for small and large samples of the random transverse Ising model on random regular graphs.

DOI
10.1103/physreve.89.062137

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Given a locally consistent set of reduced density matrices, we construct approximate density matrices which are globally consistent with the local density matrices we started from when the trial density matrix has a tree structure. We employ the cavity method of statistical physics to find the optimal density matrix representation by slowly decreasing the temperature in an annealing algorithm, or by minimizing an approximate Bethe free energy depending on the reduced density matrices and some cavity messages originated from the Bethe approximation of the entropy. We obtain the classical Bethe expression for the entropy within a naive (mean-field) approximation of the cavity messages, which is expected to work well at high temperatures. In the next order of the approximation, we obtain another expression for the Bethe entropy depending only on the diagonal elements of the reduced density matrices. In principle, we can improve the entropy approximation by considering more accurate cavity messages in the Bethe approximation of the entropy. We compare the annealing algorithm and the naive approximation of the Bethe entropy with exact and approximate numerical simulations for small and large samples of the random transverse Ising model on random regular graphs.
A local and distributive algorithm is proposed to find an optimal trial wave function minimizing the Hamiltonian expectation in a quantum system. To this end, the quantum state of the system is connected to the Gibbs state of a classical system with the set of couplings playing the role of variational parameters. The average energy is written within the replica-symmetric approximation, and the optimal parameters are obtained by a heuristic message-passing algorithm based on the Bethe approximation. The performance of this approximate algorithm depends on the structure and quality of the trial wave functions, starting from a classical system of isolated elements, i.e., mean-field approximation, and improving on that by considering the higher-order many-body interactions. The method is applied to some disordered quantum Ising models in transverse fields, and the results are compared with the exact ones for small systems.
We propose the variational quantum cavity method to construct a minimal energy subspace of wavevectors that are used to obtain some upper bounds for the energy cost of the low-temperature excitations. Given a trial wavefunction we use the cavity method of statistical physics to estimate the Hamiltonian expectation and to find the optimal variational parameters in the subspace of wavevectors orthogonal to the lower-energy wavefunctions. To this end, we write the overlap between two wavefunctions within the Bethe approximation, which allows us to replace the global orthogonality constraint with some local constraints on the variational parameters. The method is applied to the transverse Ising model and different levels of approximations are compared with the exact numerical solutions for small systems.
We study two free energy approximations (Bethe and plaquette-CVM) for the Random Field Ising Model in two dimensions. We compare results obtained by these two methods in single instances of the model on the square grid, showing the difficulties arising in defining a robust critical line. We also attempt average case calculations using a replica-symmetric ansatz, and compare the results with single instances. Both, Bethe and plaquette-CVM approximations present a similar panorama in the phase space, predicting long range order at low temperatures and fields. We show that plaquette-CVM is more precise, in the sense that predicts a lower critical line (the truth being no line at all). Furthermore, we give some insight on the non-trivial structure of the fixed points of different message passing algorithms. A study of the Monte Carlo dynamics for an arbitrary sample shows that GBP states are very well correlated with states that are attractors of the stochastic dynamics.
The inverse Ising problem consists in inferring the coupling constants of an Ising model given the correlation matrix. The fastest methods for solving this problem are based on mean-field approximations, but which one performs better in the general case is still not completely clear. In the first part of this work, I summarize the formulas for several mean- field approximations and I derive new analytical expressions for the Bethe approximation, which allow to solve the inverse Ising problem without running the Susceptibility Propagation algorithm (thus avoiding the lack of convergence). In the second part, I compare the accuracy of different mean field approximations on several models (diluted ferromagnets and spin glasses) defined on random graphs and regular lattices, showing which one is in general more effective. A simple improvement over these approximations is proposed. Also a fundamental limitation is found in using methods based on TAP and Bethe approximations in presence of an external field.
We propose a message-passing algorithm to compute the Hamiltonian expectation with respect to an appropriate class of trial wave functions for an interacting system of fermions. To this end, we connect the quantum expectations to average quantities in a classical system with both local and global interactions, which are related to the variational parameters and use the Bethe approximation to estimate the average energy within the replica-symmetric approximation. The global interactions, which are needed to obtain a good estimation of the average fermion sign, make the average energy a nonlocal function of the variational parameters. We use some heuristic minimization algorithms to find approximate ground states of the Hubbard model on random regular graphs and observe significant qualitative improvements with respect to the mean-field approxi- mation.
Anderson localization - the suppression of wave propagation by interference in a sufficiently disordered medium - is one of the most remarkable phenomena in condensed matter physics. These lecture notes provide a pedagogical introduction to Anderson localization on the Bethe lattice, an infinite tree with fixed connectivity and no loops, and one of the few settings in which the transition can be studied exactly. The problem is important for two main reasons. First, the Anderson model on the Bethe lattice provides a mean-field, infinite-dimensional limit in which the transition point and critical behavior can be determined exactly. Second, it provides a simplified description of the high-dimensional configuration spaces of interacting quantum systems, and thus a useful toy model for many-body localization. We focus on the Green's function formalism and the cavity method. After recalling standard diagnostics of localization based on transport, spectral statistics, and eigenfunction moments, we show that they are encoded in the statistics of the Green's functions, and that the full distribution of the local density of states is the natural order parameter of the transition. We derive the exact self-consistent cavity equations on the Bethe lattice, use them to characterize the two phases and critical behavior - including an analytic solution at large connectivity - and describe population-dynamics algorithms that solve them with high precision. We explain how an exponentially large correlation volume and the localized nature of the critical point account for the strong finite-size effects on random regular graphs, which led to a long debate over a possible intermediate non-ergodic phase. Finally, we discuss the mapping onto directed polymers in random media, interpreting localization as a freezing transition dominated by a few rare resonant paths.