Beyond the Mean Field Approximation for Spin Glasses

TL;DR

This paper advances the understanding of spin glasses by extending the Bethe-Peierls approximation within the replica framework, providing analytical estimates of key thermodynamic quantities beyond the mean field level.

Contribution

It introduces a novel approach that incorporates cluster interactions and effective replica interactions, improving upon traditional mean field approximations for spin glasses.

Findings

Analytic estimates of internal energy in d dimensions

Critical temperature predictions for spin glasses

Enhanced modeling of interactions within clusters

Abstract

We study the d-dimensional random Ising model using a Bethe-Peierls approximation in the framework of the replica method. We take into account the correct interaction only inside replicated clusters of spins. Our ansatz is that the interaction of the borders of the clusters with the external world can be described via an effective interaction among replicas. The Bethe-Peierls model is mapped into a single Ising model with a random gaussian field, whose strength (related to the effective coupling between two replicas) is determined via a self-consistency equation. This allows us to obtain analytic estimates of the internal energy and of the critical temperature in d dimensions.