Replica symmetry breaking in multi-species Sherrington-Kirkpatrick model

TL;DR

This paper extends the AT line criterion for replica symmetry breaking to multi-species SK models, deriving explicit thresholds and establishing a novel non-asymptotic condition for symmetry breaking in these complex systems.

Contribution

It introduces the first non-asymptotic symmetry breaking condition for multi-species spin glasses, specifically analyzing the two-species case with explicit AT temperature thresholds.

Findings

Derived explicit AT temperature threshold for two-species SK model.

Established a non-asymptotic symmetry breaking condition.

Showed coincidence with replica symmetric critical point criteria.

Abstract

In the Sherrington-Kirkpatrick (SK) and related mixed $p$-spin models, there is interest in understanding replica symmetry breaking at low temperatures. For this reason, the so-called AT line proposed by de Almeida and Thouless as a sufficient (and conjecturally necessary) condition for symmetry breaking, has been a frequent object of study in spin glass theory. In this paper, we consider the analogous condition for the multi-species SK model, which concerns the eigenvectors of a Hessian matrix. The analysis is tractable in the two-species case with positive definite variance structure, for which we derive an explicit AT temperature threshold. To our knowledge, this is the first non-asymptotic symmetry breaking condition produced for a multi-species spin glass. As possible evidence that the condition is sharp, we draw further parallel with the classical SK model and show coincidence with a separate temperature inequality guaranteeing uniqueness of the replica symmetric critical point.