Griffiths-type theorems for short-range spin glass models

TL;DR

This paper extends Griffiths' theorems to short-range spin glass models, linking overlap distribution broadening, free energy non-differentiability, and replica symmetry breaking.

Contribution

It proves that nonzero overlap variance implies free energy non-differentiability and replica symmetry breaking in short-range spin glasses, generalizing classical results.

Findings

Nonzero overlap variance implies free energy non-differentiability.

Non-differentiability implies replica symmetry breaking.

Results apply to both short-range and long-range spin glass models.

Abstract

We establish relations between different characterizations of order in spin glass models. We first prove that the broadening of the replica overlap distribution indicated by a nonzero standard deviation of the replica overlap $R^{1,2}$ implies the non-differentiability of the two-replica free energy with respect to the replica coupling parameter $\lambda$. In $\mathbb Z_2$ invariant models such as the standard Edwards-Anderson model, the non-differentiability is equivalent to the spin glass order characterized by a nonzero Edwards-Anderson order parameter. This generalization of Griffiths' theorem is proved for any short-range spin glass models with classical bounded spins. We also prove that the non-differentiability of the two-replica free energy mentioned above implies replica symmetry breaking in the literal sense, i.e., a spontaneous breakdown of the permutation symmetry in the model with three replicas. This is a general result that applies to a large class of random spin models, including long-range models such as the Sherrington-Kirkpatrick model and the random energy model.