Mean-Field Equations for Spin Models with Orthogonal Interaction Matrices

TL;DR

This paper analyzes metastable states in orthogonal interaction matrix spin models, deriving mean-field equations and calculating the number of solutions, revealing extensive configurational entropy in certain temperature ranges.

Contribution

It derives mean-field equations for these models and computes the number of solutions, providing new insights into their metastable states and entropy behavior.

Findings

Mean-field equations derived for orthogonal interaction matrix spin models.

Number of solutions computed for random case, showing extensive entropy.

Replica calculation confirms analytical solutions for TAP states.

Abstract

We study the metastable states in Ising spin models with orthogonal interaction matrices. We focus on three realizations of this model, the random case and two non-random cases, i.e.\ the fully-frustrated model on an infinite dimensional hypercube and the so-called sine-model. We use the mean-field (or {\sc tap}) equations which we derive by resuming the high-temperature expansion of the Gibbs free energy. In some special non-random cases, we can find the absolute minimum of the free energy. For the random case we compute the average number of solutions to the {\sc tap} equations. We find that the configurational entropy (or complexity) is extensive in the range $T_{\mbox{\tiny RSB}}<T<T_{\mbox{\tiny M}}$. Finally we present an apparently unrelated replica calculation which reproduces the analytical expression for the total number of {\sc tap} solutions.