# Localization-protected order in spin chains with non-Abelian discrete symmetries

## TL;DR

This paper investigates the non-equilibrium phase structure of a disordered three-state quantum Potts chain with non-Abelian symmetry, revealing multiple MBL phases, symmetry breaking, and topological order at strong disorder.

## Contribution

It demonstrates the existence of two distinct symmetry-breaking MBL phases and a stable topological phase with parafermionic zero modes in a non-Abelian symmetric spin chain.

## Key findings

- Two broken-symmetry MBL phases at strong disorder
- Existence of a topological phase with parafermionic zero modes
- An infinite-randomness critical point between phases

## Abstract

We study the non-equilibrium phase structure of the three-state random quantum Potts model in one dimension. This spin chain is characterized by a non-Abelian $D_3$ symmetry recently argued to be incompatible with the existence of a symmetry-preserving many-body localized (MBL) phase. Using exact diagonalization and a finite-size scaling analysis, we find that the model supports two distinct broken-symmetry MBL phases at strong disorder that either break the ${\mathbb{Z}_3}$ clock symmetry or a ${\mathbb{Z}_2}$ chiral symmetry. In a dual formulation, our results indicate the existence of a stable finite-temperature topological phase with MBL-protected parafermionic end zero modes. While we find a thermal symmetry-preserving regime for weak disorder, scaling analysis at strong disorder points to an infinite-randomness critical point between two distinct broken-symmetry MBL phases.