Variational approach to interfaces in random media: negative variances and replica symmetry breaking

TL;DR

This paper investigates the limitations of Gaussian variational approximations for interfaces in random media, revealing negative variances that are resolved through infinite order replica symmetry breaking, akin to phenomena in spin glass models.

Contribution

It demonstrates that incorporating infinite order replica symmetry breaking eliminates negative variances in variational studies of interfaces in random media.

Findings

Gaussian variational approximation leads to negative variance

Infinite order replica symmetry breaking results in zero variance

Analogy with spin glass models and negative entropies

Abstract

A Gaussian variational approximation is often used to study interfaces in random media. By considering the 1+1 dimensional directed polymer in a random medium, it is shown here that the variational Ansatz typically leads to a negative variance of the free energy. The situation improves by taking into account more and more steps of replica symmetry breaking. For infinite order breaking the variance is zero (i.e. subextensive). This situation is reminiscent of the negative entropies in mean field spin glass models, which were also eliminated by considering infinite order replica symmetry breaking.