$d$-dimensional spherical ferromagnets in random fields: Metastates, continuous symmetry breaking, and spin-glass features

TL;DR

This paper investigates the large-volume behavior of $d$-dimensional spherical ferromagnets in random fields, revealing complex metastate structures, spin-glass features, and symmetry breaking phenomena that depend on the dimension and scaling of the random fields.

Contribution

It provides a detailed analysis of metastates, overlap distributions, and symmetry breaking in spherical models with random fields across different dimensions and scaling regimes.

Findings

Metastates are supported on a continuum of product states.

In $d=2$, the set of Gibbs measure cluster points includes non-trivial mixtures.

The overlap distribution exhibits spin-glass features and non-self-averaging behavior in scaled models.

Abstract

We study the large-volume behavior of the spherical model for $d$-dimensional local spins, in the presence of $d$-dimensional random fields, for $d\geq 2$. We compare two models, one with volume-scaled random fields, and another one with non-scaled random fields, on the level of Aizenman-Wehr metastates, Newman-Stein metastates, as well as overlap distributions. We show that in $d\geq 2$ the metastates are fully supported on a continuity of random product states, with weights which we describe, for both models. For the non-scaled random fields, the set of a.s. cluster points of Gibbs measures contains these product states, but behaves differently in the 'recurrent' spin dimension $d=2$ where it also contains non-trivial mixtures of tilted measures. For the scaled model, moreover the overlap distribution displays spin-glass characteristics, as it is non-self averaging, and shows replica symmetry breaking, although it is ultrametric if and only if $d=1$. For $d\geq 2$ it oscillates chaotically on a set of continuous distributions for large volumes, while the limiting set contains only discrete distributions in $d=1$. Our results are based on concentration estimates, analysis of Gibbs measures in finite but large volumes, and the asymptotics of $d$-dimensional random walks and their spherical projections.