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We study the non-equilibrium phase structure of the three-state randomquantum Potts model in one dimension. This spin chain is characterized by anon-Abelian $D_3$ symmetry recently argued to be incompatible with theexistence of a symmetry-preserving many-body localized (MBL) phase. Using exactdiagonalization and a finite-size scaling analysis, we find that the modelsupports two distinct broken-symmetry MBL phases at strong disorder that eitherbreak the ${\mathbb{Z}_3}$ clock symmetry or a ${\mathbb{Z}_2}$ chiralsymmetry. In a dual formulation, our results indicate the existence of a stablefinite-temperature topological phase with MBL-protected parafermionic end zeromodes. While we find a thermal symmetry-preserving regime for weak disorder,scaling analysis at strong disorder points to an infinite-randomness criticalpoint between two distinct broken-symmetry MBL phases.

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