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 , SU(n) or SO(n). Our proof is based on a theorem about group synchronization proved by Abbe et al. [Math. Stat. Learn. 1(3), 227–256 (2018)]. It also relies on a gauge transformation acting jointly on the disorder and the spin configurations due to Nishimori [Prog. Theor. Phys. 66(4), 1169–1181 (1981)]. The proof does not use reflection positivity. The correlation inequalities of Messager et al. [Commun. Math. Phys. 58(1), 19–29 (1978)] imply symmetry breaking for the classical XY model without disorder.
Notice the factor in the modified Gibbs weight which was not present in the classical XY model in (1.1). This is due to that fact that in this less symmetric case one needs to sum over oriented edges (equivalently one may also choose a prescribed direction for each edge and remove ). Both definitions match when u → ∞.