Free Energy Difference Fluctuations in Short-Range Spin Glasses

TL;DR

This paper demonstrates that free energy difference fluctuations in short-range Ising spin glasses follow a Gaussian distribution with variance scaling with volume, confirming a longstanding conjecture and shedding light on the nature of the spin glass phase.

Contribution

Introduces a new metastate framework to analyze free energy differences, proving Gaussian fluctuation behavior and variance scaling in spin glasses, confirming Fisher and Huse's conjecture.

Findings

Free energy difference fluctuations are Gaussian-like.

Variance of fluctuations scales with the volume.

Results apply to pure, mixed, and ground states under certain conditions.

Abstract

It is generally believed (but not yet proved) that Ising spin glasses with nearest-neighbor interactions have a phase transition in three and higher dimensions to a low-temperature spin glass phase, but the nature of this phase remains controversial, especially whether it is characterized by multiple incongruent Gibbs states. Of particular relevance to this question is the behavior of the typical free energy difference restricted to a finite volume between two such putative Gibbs states, as well as the nature of the fluctuations of their free energy difference as the couplings within the volume vary. In this paper we investigate these free energy difference fluctuations by introducing a new kind of metastate which classifies Gibbs states through their edge overlap values with a reference Gibbs state randomly chosen from the support of the periodic boundary condition (PBC) metastate. We find that the free energy difference between any two incongruent pure states, regardless of the details of how they're organized into mixed states within the PBC metastate, converges to a Gaussian (or Gaussian-like) distribution whose variance scales with the volume, proving a decades-old conjecture of Fisher and Huse. The same conclusion applies, though with some additional restrictions, to both mixed Gibbs states and ground states. We discuss some implications of these results.