Replica Symmetry Breaking Instability in the 2D XY Model in a Random Field.

Abstract

We study the 2D vortex-free $\mathrm{XY}$ model in a random field, a model for randomly pinned flux lines in a plane. We construct controlled renormalization group recursion relations which allow for replica symmetry breaking (RSB). The fixed point previously found by Cardy and Ostlund in the glass phase $T<{T}_{c}$ is unstable to RSB. The susceptibility $\ensuremath{\chi}$ associated to infinitesimal RSB perturbation in the high-temperature phase is found to diverge as $\ensuremath{\chi}\ensuremath{\propto}{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, when $T\ensuremath{\rightarrow}{T}_{c}^{+}$. This provides analytical evidence that RSB occurs in finite dimensional models. The physical consequences for the glass phase are discussed.