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arXiv:1401.4074 · Nuclear Physics B

Generalized belief propagation for the magnetization of the simple cubic Ising model

Alessandro Pelizzola

Published Jan 15, 2014• 3 citations
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Abstract

A new approximation of the cluster variational method is introduced for the three-dimensional Ising model on the simple cubic lattice . The maximal cluster is, as far as we know, the largest ever used in this method. A message-passing algorithm, generalized belief propagation, is used to minimize the variational free energy . Convergence properties and performance of the algorithm are investigated. The approximation is used to compute the spontaneous magnetization , which is then compared to previous results. Using the present results as the last step in a sequence of three cluster variational approximations, an extrapolation is obtained which captures the leading critical behavior with a good accuracy.

DOI
10.1016/j.nuclphysb.2014.01.008

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A new approximation of the cluster variational method is introduced for the three-dimensional Ising model on the simple cubic lattice . The maximal cluster is, as far as we know, the largest ever used in this method. A message-passing algorithm, generalized belief propagation, is used to minimize the variational free energy . Convergence properties and performance of the algorithm are investigated. The approximation is used to compute the spontaneous magnetization , which is then compared to previous results. Using the present results as the last step in a sequence of three cluster variational approximations, an extrapolation is obtained which captures the leading critical behavior with a good accuracy.
The cluster variation method has been developed into a general theoretical framework for treating short-range correlations in many-body systems after it was first proposed by Kikuchi in 1951. On the numerical side, a message-passing approach called generalized belief propagation (GBP) was proposed by Yedidia, Freeman and Weiss about a decade ago as a way of computing the minimal value of the cluster variational free energy and the marginal distributions of clusters of variables. However the GBP equations are often redundant, and it is quite a non-trivial task to make the GBP iteration converges to a fixed point. These drawbacks hinder the application of the GBP approach to finite-dimensional frustrated and disordered systems. In this work we report an alternative and simple derivation of the GBP equations starting from the partition function. Based on this derivation we propose a natural and systematic way of removing the redundance of the GBP equations. We apply the simplified generalized belief propagation (SGBP) equations to the two-dimensional and the three-dimensional ferromagnetic Ising model and Edwards-Anderson spin glass model. The numerical results confirm that the SGBP message-passing approach is able to achieve satisfactory performance on these model systems. We also suggest that a subset of the SGBP equations can be neglected in the numerical iteration process without affecting the final results.
Abstract. The cluster variation method has been developed into a general theoretical framework for treating short-range correlations in many-body systems after it was first proposed by Kikuchi in 1951. On the numerical side, a message-passing approach called generalized belief propagation (GBP) was proposed by Yedidia, Freeman and Weiss about a decade ago as a way of computing the minimal value of the cluster variational free energy and the marginal distributions of clusters of variables. However the GBP equations are often redundant, and it is quite a non-trivial task to make the GBP iteration converges to a fixed point. These drawbacks hinder the application of the GBP approach to finite-dimensional frustrated and disordered systems. In this work we report an alternative and simple derivation of the GBP equations starting from the partition function expression. Based on this derivation we propose a natural and systematic way of removing the redundance of the GBP equations. We apply the simplified generalized belief propagation (SGBP) equations to the two-dimensional and the three-dimensional ferromagnetic Ising model and Edwards-Anderson spin glass model. The numerical results confirm that the SGBP message-passing approach is able to achieve satisfactory performance on these model systems. We also suggest that a subset of the SGBP equations can be neglected in the numerical iteration process without affecting the final results. 1.
The cluster-variation-Pade-approximant method is a recently proposed tool, based on the extrapolation of low- and high-temperature results obtained with the cluster-variation method, for the determination of critical parameters in Ising-like models. Here the method is applied to the three-dimensional simple cubic Ising model, and new results, obtained with an 18-site basic cluster, are reported. Other techniques for extracting nonclassical critical exponents are also applied and their results compared with those by the cluster-variation-Pade-approximant method.
Starting from a cluster variational method, and inspired by the correctness of the paramagnetic ansatz [at high temperatures in general, and at any temperature in the two-dimensional (2D) Edwards-Anderson (EA) model] we propose a message-passing algorithm--the dual algorithm--to estimate the marginal probabilities of spin glasses on finite-dimensional lattices. We use the EA models in 2D and 3D as benchmarks. The dual algorithm improves the Bethe approximation, and we show that in a wide range of temperatures (compared to the Bethe critical temperature) our algorithm compares very well with Monte Carlo simulations, with the double-loop algorithm, and with exact calculation of the ground state of 2D systems with bimodal and Gaussian interactions. Moreover, it is usually 100 times faster than other provably convergent methods, as the double-loop algorithm. In 2D and 3D the quality of the inference deteriorates only where the correlation length becomes very large, i.e., at low temperatures in 2D and close to the critical temperature in 3D.
The spontaneous magnetization of the three-dimensional Ising model is studied numerically by the Monte Carlo method. The values of squared magnetizations for finite simple-cubic lattices are extrapolated to those of the infinite system to estimate the values of spontaneous magnetization. The extrapolation parameter is the inverse of the system size. By assuming that the spontaneous magnetization is a simple pole function near the critical point, the values of the critical point K c and the exponent β of the spontaneous magnetization are estimated to be 0.221657(3) and 0.3205(6), respectively. If we assume that it is the form of simple pole function plus regular function, they are estimated to be 0.221651(4) and 0.3265(9), respectively. Therefore they are estimated to be 0.221654(6) and 0.324(4), respectively, from the mid values of these estimations.
We study the performance of different message passing algorithms in the two dimensional Edwards Anderson model. We show that the standard Belief Propagation (BP) algorithm converges only at high temperature to a paramagnetic solution. Then, we test a Generalized Belief Propagation (GBP) algorithm, derived from a Cluster Variational Method (CVM) at the plaquette level. We compare its performance with BP and with other algorithms derived under the same approximation: Double Loop (DL) and a two-ways message passing algorithm (HAK). The plaquette-CVM approximation improves BP in at least three ways: the quality of the paramagnetic solution at high temperatures, a better estimate (lower) for the critical temperature, and the fact that the GBP message passing algorithm converges also to non paramagnetic solutions. The lack of convergence of the standard GBP message passing algorithm at low temperatures seems to be related to the implementation details and not to the appearance of long range order. In fact, we prove that a gauge invariance of the constrained CVM free energy can be exploited to derive a new message passing algorithm which converges at even lower temperatures. In all its region of convergence this new algorithm is faster than HAK and DL by some orders of magnitude.