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 in a special quenched random environment 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 on both the disorder and the spin configurations due to Nishimori [Nis81, GHLDB85]. The proof does not use reflection positivity.