Replica Symmetry Breaking Instability in the 2DXYModel in a Random Field

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 $T<\mathrm{}{T}_c$ is unstable to RSB. The susceptibility $\chi$ associated to infinitesimal RSB perturbation in the high-temperature phase is found to diverge as $\chi \propto {(T-T_c)}^{-\gamma }$, when $T\to T_c^{+}$. This provides analytical evidence that RSB occurs in finite dimensional models. The physical consequences for the glass phase are discussed.