A Relation between Disorder Chaos and Incongruent States in Spin Glasses on ${\mathbb Z}^d$

TL;DR

This paper establishes a theoretical link between disorder chaos and incongruent ground states in spin glasses, providing bounds and relations that support long-standing conjectures in the field.

Contribution

It derives lower bounds on energy variance differences and connects disorder chaos with the existence of incongruent ground states in the Edwards-Anderson model.

Findings

Disorder chaos is necessary for variance to be less than volume.

A relation between disorder chaos scale and critical droplet size is established.

Supports the conjecture linking droplet theory and incongruence absence.

Abstract

We derive lower bounds for the variance of the difference of energies between incongruent ground states, i.e., states with edge overlaps strictly less than one, of the Edwards-Anderson model on ${\mathbb Z}^d$. The bounds highlight a relation between the existence of incongruent ground states and the absence of edge disorder chaos. In particular, it suggests that the presence of disorder chaos is necessary for the variance to be of order less than the volume. In addition, a relation is established between the scale of disorder chaos and the size of critical droplets. The results imply a long-conjectured relation between the droplet theory of Fisher and Huse and the absence of incongruence.