Features of a spin glass in the random field Ising model

TL;DR

This paper demonstrates that a weak random external field in the finite-dimensional random field Ising model induces mean-field spin glass properties such as replica symmetry breaking and ultrametricity at subcritical temperatures.

Contribution

It shows that short-range disordered systems can exhibit key mean-field spin glass features under specific weak external fields, bridging the gap between finite-dimensional and mean-field models.

Findings

Site overlap exhibits one step of replica symmetry breaking

Overlap distribution is non-self-averaging

Overlap has Parisi ultrametric property

Abstract

A longstanding open question in the theory of disordered systems is whether short-range models, such as the random field Ising model or the Edwards-Anderson model, can indeed have the famous properties that characterize mean-field spin glasses at nonzero temperature. This article shows that this is at least partially possible in the case of the random field Ising model. Consider the Ising model on a discrete $d$-dimensional cube under free boundary condition, subjected to a very weak i.i.d. random external field, where the field strength is inversely proportional to the square-root of the number of sites. It turns out that in $d\ge 2$ and at subcritical temperatures, this model has some of the key features of a mean-field spin glass. Namely, (a) the site overlap exhibits one step of replica symmetry breaking, (b) the quenched distribution of the overlap is non-self-averaging, and (c) the overlap has the Parisi ultrametric property. Furthermore, it is shown that for Gaussian disorder, replica symmetry does not break if the field strength is taken to be stronger than the one prescribed above, and non-self-averaging fails if it is weaker, showing that the above order of field strength is the only one that allows all three properties to hold. However, the model does not have two other features of mean-field models. Namely, (a) it does not satisfy the Ghirlanda-Guerra identities, and (b) it has only two pure states instead of many.