Ground State Excitations and Energy Fluctuations in Short-Range Spin Glasses

TL;DR

This paper investigates the stability and structure of ground states in short-range spin glasses, proving uniqueness of metastates in 2D and ruling out certain excitations predicted by replica symmetry breaking.

Contribution

It demonstrates the non-existence of space-filling critical droplets in ground states and shows that certain excitations cannot occur, advancing understanding of spin glass ground state properties.

Findings

Metastates in 2D are unique and supported on a single pair of ground states.

Certain space-filling excitations with bounded energy do not exist in any dimension.

Critical droplets with positive density boundaries are absent in ground states from coupling-independent boundary conditions.

Abstract

We study the stability of ground states in the Edwards-Anderson Ising spin glass in dimensions two and higher against perturbations of a single coupling. After reviewing the concepts of critical droplets, flexibilities and metastates, we show that, in any dimension, a certain kind of critical droplet with space-filling (i.e., positive spatial density) boundary does not exist in ground states generated by coupling-independent boundary conditions. Using this we show that if incongruent ground states exist in any dimension, the variance of their energy difference restricted to finite volumes scales proportionally to the volume. This in turn is used to prove that a metastate generated by (e.g.) periodic boundary conditions is unique and supported on a single pair of spin-reversed ground states in two dimensions. We further show that a type of excitation above a ground state, whose interface with the ground state is space-filling and whose energy remains O(1) independent of the volume, as predicted by replica symmetry breaking, cannot exist in any dimension.