We prove continuous symmetry breaking in three dimensions for a special class of disordered models described by the Nishimori line. The spins take values in a group such as $\mathbb{ S}^1$, $SU(n)$ or $SO(n)$. Our proof is based on a theorem about group synchronization proved by Abbe, Massouli\'e, Montanari, Sly and Srivastava [AMM+18]. It also relies on a gauge transformation acting jointly on the disorder and the spin configurations due to Nishimori [Nis81, GHLDB85]. The proof does not use reflection positivity. The correlation inequalities of [MMSP78] imply symmetry breaking for the classical $XY$ model without disorder.
Categories: math.PR math-ph math.MP math.ST stat.TH