# Low temperature spin glass fluctuations: expanding around a spherical approximation

## TL;DR

This paper develops an expansion method around the spherical approximation to better understand low-temperature spin glass fluctuations, focusing on corrections that approach the local Ising constraint.

## Contribution

It introduces a systematic expansion of the replicated free energy functional around the spherical approximation using a constraint-field and Legendre Transform.

## Key findings

- First correction to the spherical approximation analyzed
- Provides a framework to approach local Ising constraints
- Enhances understanding of spin glass stability near zero temperature

## Abstract

The spin glass behavior near zero temperature is a complicated matter. To get an easier access to the spin glass order parameter $Q(x)$ and, at the same time, keep track of $Q_{ab}$, its matrix aspect, and hence of the Hessian controlling stability, we investigate an expansion of the replicated free energy functional around its ``spherical'' approximation. This expansion is obtained by introducing a constraint-field and a (double) Legendre Transform expressed in terms of spin correlators and constraint-field correlators. The spherical approximation has the spin fluctuations treated with a global constraint and the expansion of the Legendre Transformed functional brings them closer and closer to the Ising local constraint. In this paper we examine the first contribution of the systematic corrections to the spherical starting point.