A geometrical picture for finite dimensional spin glasses

TL;DR

This paper proposes a geometric framework for understanding finite-dimensional spin glasses, suggesting a coexistence of droplet-like behavior at small scales and mean field characteristics at the system size, based on low energy sponge-like excitations.

Contribution

It introduces a geometric perspective that reconciles droplet models with mean field theory in finite-dimensional spin glasses through system-size excitations.

Findings

Existence of sponge-like low energy excitations at system size

Coexistence of droplet behavior and mean field features

System-size excitations can have constant energy without destroying the spin glass phase

Abstract

A controversial issue in spin glass theory is whether mean field correctly describes 3-dimensional spin glasses. If it does, how can replica symmetry breaking arise in terms of spin clusters in Euclidean space? Here we argue that there exist system-size low energy excitations that are sponge-like, generating multiple valleys separated by diverging energy barriers. The droplet model should be valid for length scales smaller than the size of the system (theta > 0), but nevertheless there can be system-size excitations of constant energy without destroying the spin glass phase. The picture we propose then combines droplet-like behavior at finite length scales with a potentially mean field behavior at the system-size scale.