Continuous symmetry breaking along the Nishimori line

TL;DR

This paper proves continuous symmetry breaking in three-dimensional disordered models along the Nishimori line, extending understanding of phase transitions in complex spin systems with disorder.

Contribution

It establishes symmetry breaking for a class of disordered models on the Nishimori line using a novel approach based on group synchronization and gauge transformations.

Findings

Proves symmetry breaking in 3D disordered models on the Nishimori line.

Extends symmetry breaking results to models with spins in groups like $ ext{S}^1$, $SU(n)$, or $SO(n)$.

Utilizes a gauge transformation and a theorem on group synchronization, avoiding reflection positivity.

Abstract

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.