Continuous symmetry breaking along the Nishimori line

Abbe, Massouli\'e, Montanari, Sly and Srivastava proved in [AMM+18] that for any compact matrix Lie group $G$, group synchronization holds on $\mathbb{Z}^d$ when $d\geq 3$ and the ambient noise is sufficiently low. They suggest that synchronization with noisy data implies that long-range order holds for spin $O(n)$ models with a special quenched random disorder called the Nishimori line. In this paper we prove continuous symmetry breaking for disordered models whose spins take values in $\mathbb{S}^1$, $SU(n)$ or $SO(n)$ along the Nishimori line at low temperature for $d\geq 3$. The proof is based on [AMM+18] and a gauge transformation acting jointly on the disorder and the spin configurations due to Nishimori [Nis81, ON93]. The proof does not use reflection positivity. Using the correlation inequalities of [MMSP78], our results imply symmetry breaking for the $XY$ model without disorder.