Structural analysis of Gibbs states and metastates in short-range classical spin glasses: indecomposable metastates, dynamically-frozen states, and metasymmetry

TL;DR

This paper investigates the complex structure of Gibbs states in short-range classical spin glasses, proving decomposition properties, stability under perturbations, and organization of pure states, with implications for understanding metastates and their dynamics.

Contribution

It introduces a comprehensive analysis of metastates, proving their decomposition into indecomposable components, stochastic stability, and organization of pure states, advancing the theoretical understanding of spin glass phases.

Findings

Any metastate can be decomposed into indecomposable metastates.

All Gibbs states from an indecomposable metastate are macroscopically similar.

Pure states follow a Poisson-Dirichlet distribution or organize as an ultrametric space.

Abstract

We consider short-range classical spin glasses, or other disordered systems, consisting of Ising spins. For a low-temperature Gibbs state in infinite size in such a system, for given random bonds, it is controversial whether its decomposition into pure states will be trivial or non-trivial. We undertake a general study of the overall structure of this problem, based on metastates, which are essential to prove the existence of a thermodynamic limit. A metastate is a probability distribution on Gibbs states, for given disorder, that satisfies certain covariance properties. First, we prove that any metastate can be decomposed as a mixture of indecomposable metastates, and that all Gibbs states drawn from an indecomposable metastate are alike macroscopically. Next, we consider stochastic stability of a metastate under random perturbations of the disorder, and prove that any metastate is stochastically stable. Using related methods and older results, we prove that if the pure-state decomposition of any Gibbs states drawn from an indecomposable metastate is countably infinite, then the weights in the decomposition follow a Poisson-Dirichlet distribution with a fixed value of the single parameter describing such distributions, and also that if the overlap takes a finite number of values, then the pure states are organized as an ultrametric space, as in $k$-RSB. Dynamically-frozen states play a role in the analysis of Gibbs states drawn from a metastate, either as states or as parts of states. Using a mapping into real Hilbert space, we prove further results about Gibbs states, and classify them into six types. Metastate-average states are studied, and can be related to states arising dynamically at long times after a quench from high temperature, under some conditions.