Replica Cluster Variational Method: the Replica Symmetric solution for the 2D random bond Ising model

TL;DR

This paper develops and solves the Replica Symmetric equations within the Replica Cluster Variational Method for the 2D random bond Ising model, improving phase diagram predictions and aligning with inference algorithm behavior.

Contribution

It introduces a novel solution approach for the Replica Symmetric equations in the 2D random bond Ising model, enhancing phase diagram accuracy and applicability to glassy phases.

Findings

Improved estimates of spin-glass transition temperatures.

Accurate phase diagrams for square and triangular lattices.

Method for solving equations in the glassy phase.

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 tricritical 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.