Crisanti-Sommers formula and simultaneous symmetry breaking in multi-species spherical spin glasses

TL;DR

This paper derives the Crisanti-Sommers formula for multi-species spherical spin glasses and explores how replica symmetry breaking in one species influences others, revealing conditions for simultaneous symmetry breaking.

Contribution

It provides the Crisanti-Sommers variational representation for multi-species models and establishes conditions linking symmetry breaking across species.

Findings

RSB in one species implies RSB in others if they share quadratic interactions.

Symmetry breaking levels are identical across species in certain conditions.

External fields allow RSB regardless of interaction types.

Abstract

There is a rich history of expressing the limiting free energy of mean-field spin glasses as a variational formula over probability measures on $[0,1]$, where the measure represents the similarity (or "overlap") of two independently sampled spin configurations. At high temperatures, the formula's minimum is achieved at a measure which is a point mass, meaning sample configurations are asymptotically orthogonal up to a magnetic field correction. At low temperatures, though, a very different behavior emerges known as replica symmetry breaking (RSB). The deep wells in the energy landscape create more rigid structure, and the optimal overlap measure is no longer a point mass. The exact size of its support remains in many cases an open problem. Here we consider these themes for multi-species spherical spin glasses. Following a companion work in which we establish the Parisi variational formula, here we present this formula's Crisanti-Sommers representation. In the process, we gain new access to a problem unique to the multi-species setting. Namely, if RSB occurs for one species, does it necessarily occur for other species as well? We provide sufficient conditions for the answer to be yes. For instance, we show that if two species share any quadratic interaction, then RSB for one implies RSB for the other. Moreover, the level of symmetry breaking must be identical, even in cases of full RSB. In the presence of an external field, any type of interaction suffices.