Phase transitions in XY models along the Nishimori line

Abstract

We examine phase transitions of the classical XY model and a special class of disordered XY models. We start by showing that continuous symmetry cannot be broken in one and two dimensions for the classical case using the Mermin-Wagner theorem. The main goal is to prove that continuous symmetry breaking occurs at low temperatures on a lattice in three and higher dimensions along the Nishimori line, focusing on the case of a finite domain and free boundary conditions. The result is based on ideas from group synchronization. It relies on a special gauge symmetry of the Nishimori line, which implies the factorisation of local energy correlations after averaging over the disorder. We make use of the theory of unpredictable paths with exponential intersection tails. The presented proof of long-range order does not use reflection positivity. We also show that if continuous symmetry breaks along the Nishimori line, this implies continuous symmetry breaking for the classical XY model without disorder. By comparing the spin-spin correlation of the XY model to that of the Ising model, we prove exponential decay of correlation on the Nishimori line in the high temperature phase.