Topological transitions and freezing in XY models and Coulomb gases with quenched disorder: renormalization via traveling waves

TL;DR

This paper introduces a novel renormalization group method using traveling wave solutions to analyze topological transitions and freezing phenomena in disordered XY models and Coulomb gases, revealing a glassy freezing transition driven by rare regions.

Contribution

Develops a new RG approach based on traveling wave solutions to study disorder-induced topological transitions in XY models and Coulomb gases, connecting to KPP equations and glassy freezing.

Findings

Disorder distribution broadens below freezing temperature.

Transition controlled by rare favorable defect regions.

Universality linked to front velocity in KPP equations.

Abstract

We study the two dimensional XY model with quenched random phases and its Coulomb gas formulation. A novel renormalization group (RG) method is developed which allows to study perturbatively the glassy low temperature XY phase and the transition at which frozen topological defects (vortices) proliferate. This RG approach is constructed both from the replicated Coulomb gas and, equivalently without the use of replicas, using the probability distribution of the local disorder (random defect core energy). By taking into account the fusion of environments (i.e charge fusion in the replicated Coulomb gas) this distribution is shown to obey a Kolmogorov's type (KPP) non linear RG equation which admits travelling wave solutions and exhibits a freezing phenomenon analogous to glassy freezing in Derrida's random energy models. The resulting physical picture is that the distribution of local disorder becomes broad below a freezing temperature and that the transition is controlled by rare favorable regions for the defects, the density of which can be used as the new perturbative parameter. The determination of marginal directions at the disorder induced transition is shown to be related to the well studied front velocity selection problem in the KPP equation and the universality of the novel critical behaviour obtained here to the known universality of the corrections to the front velocity. Applications to other two dimensional problems are mentionned at the end.