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arXiv:1204.0439 · Journal of Physics A Mathematical and Theoretical

Replica cluster variational method: the replica symmetric solution for the 2D random bond Ising model

Alejandro Lage-Castellanos, Roberto Mulet, Federico Ricci‐Tersenghi, Tommaso Rizzo

Published Mar 13, 2013• 20 citations
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Abstract

We present and solve the replica symmetric equations in the context of the replica cluster variational method for the 2D random bond Ising model (including the 2D Edwards–Anderson spin-glass model). First, we solve a linearized version of these equations to obtain the phase diagrams of the model on the square and triangular lattices. In both cases, the spin-glass transition temperatures and the multicritical point estimations improve largely over the Bethe predictions. Moreover, we show that this phase diagram is consistent with the behavior of inference algorithms on single instances of the problem. Finally, we present a method to consistently find approximate solutions to the equations in the glassy phase. The method is applied to the triangular lattice down to T = 0, also in the presence of an external field.

DOI
10.1088/1751-8113/46/13/135001

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We present and solve the replica symmetric equations in the context of the replica cluster variational method for the 2D random bond Ising model (including the 2D Edwards–Anderson spin-glass model). First, we solve a linearized version of these equations to obtain the phase diagrams of the model on the square and triangular lattices. In both cases, the spin-glass transition temperatures and the multicritical point estimations improve largely over the Bethe predictions. Moreover, we show that this phase diagram is consistent with the behavior of inference algorithms on single instances of the problem. Finally, we present a method to consistently find approximate solutions to the equations in the glassy phase. The method is applied to the triangular lattice down to T = 0, also in the presence of an external field.
We apply the real space renormalization group (RG) to the replica Hamiltonian of the two dimensional Ising Edwards–Anderson model to discuss the existence of the spin glass (SG) phase. We derive the RG equations under the replica symmetric (RS) ansatz and the resulting flow diagram indicates the existence of the SG phase. The critical exponent for the SG transition has a plausible value while that of the multicritical point (MCP) is a complex number. We consider that this failure at the MCP is due to not taking the RS breaking into account. Indeed we find that the RS breaking parameter is relevant both at the SG critical point and the MCP, indicating the nontriviality of the SG phase of this model.
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.
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.
We introduce a method for computing corrections to the Bethe approximation for spin models on arbitrary lattices. Unlike cluster variational methods, the new approach takes into account fluctuations on all length scales. The derivation of the leading correction is explained and applied to two simple examples: the ferromagnetic Ising model on d -dimensional lattices, and the spin glass on random graphs (both in their high-temperature phases). In the first case we rederive the well known Ginzburg criterion and the upper critical dimension. In the second, we compute finite-size corrections to the free energy.
A process that images or measures bond energies in the critical Ising model can be in distinct measurement ``phases'', depending on the precision of measurement. We study the transition into the strong-measurement phase using replica field theory (an epsilon expansion around six dimensions) and numerical simulations in two dimensions. The results reveal multiscaling of correlation functions at the critical point, and a striking enlarged symmetry of the replica description. This is an analog of the Nishimori phenomenon in the Ising spin glass, in a distinct replica limit. The enlarged symmetry is present microscopically for certain measurement protocols, but more generally can emerge in the infrared, and it fixes the exact value of the exponent for the Edwards-Anderson correlator both in 2D and near the upper critical dimension. We also examine the epsilon expansion for models with power-law interactions and/or long-range measurement.
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.