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.
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