Uniqueness of ground state in the Edwards-Anderson spin glass model

TL;DR

This paper rigorously proves the uniqueness of the ground state in the Edwards-Anderson spin glass model across all dimensions under certain conditions, implying no replica symmetry breaking at zero temperature.

Contribution

It establishes a rigorous proof of ground state uniqueness in the Edwards-Anderson model and clarifies differences from mean field models at zero temperature.

Findings

Ground state is unique in all dimensions for almost all interactions.

Replica symmetry breaking does not occur at zero temperature.

Site- and bond-overlap are concentrated at maximal values.

Abstract

It is proven rigorously that the ground state in the Edwards-Anderson spin glass model is unique in any dimension for almost all continuous random exchange interactions under a condition that a single spin breaks the global ${\mathbb Z}_2$ symmetry. This theorem implies that replica symmetry breaking does not occur at zero temperature. The site- and bond-overlap are concentrated at their maximal values. It is argued that behaviors of short range spin glass models are much different from those of mean field spin glass models near zero temperature. Errata have been attached at the final page.