Stability of the replica-symmetric saddle-point in general mean-field spin-glass models

TL;DR

This paper analyzes the stability of the replica-symmetric saddle-point in mean-field spin-glass models, deriving the de Almeida-Thouless line for various models using representation theory and confirming previous results.

Contribution

It introduces a group-theoretic block-diagonalization of the Hessian to derive stability conditions for general spin-glass models, extending the understanding of the replica approach.

Findings

Derived the de Almeida-Thouless line for general models

Confirmed results for SK, Viana-Bray, and Lévy spin glasses

Provided a unified framework for stability analysis

Abstract

Within the replica approach to mean-field spin-glasses the transition from ergodic high-temperature behaviour to the glassy low-temperature phase is marked by the instability of the replica-symmetric saddle-point. For general spin-glass models with non-Gaussian field distributions the corresponding Hessian is a $2^n\times 2^n$ matrix with the number $n$ of replicas tending to zero eventually. We block-diagonalize this Hessian matrix using representation theory of the permutation group and identify the blocks related to the spin-glass susceptibility. Performing the limit $n\to 0$ within these blocks we derive expressions for the de~Almeida-Thouless line of general spin-glass models. Specifying these expressions to the cases of the Sherrington-Kirkpatrick, Viana-Bray, and the L\'evy spin glass respectively we obtain results in agreement with previous findings using the cavity approach.