Localization Protection and Symmetry Breaking in One-dimensional Potts Chains

Recent work on the 3-state Potts and $Z_3$ clock models has demonstrated that their ordered phases are connected by duality to a phase that hosts topologically protected parafermionic zero modes at the system's boundary. The analogy with Kitaev's example of the one-dimensional Majorana chain (similarly related by duality to the Ising model) suggests that such zero modes may also be stabilized in highly excited states by many-body localization (MBL). However, the Potts model has a non-Abelian $S_3$ symmetry believed to be incompatible with MBL; hence any MBL state must spontaneously break this symmetry, either completely or into one of its abelian subgroups ($Z_2$ or $Z_3$), with the topological phase corresponding to broken $Z_3$ symmetry. We therefore study the excited state phase structure of random three-state Potts and clock models in one dimension using exact diagonalization and real-space renormalization group techniques. We also investigate the interesting possibility of a direct excited-state transition between MBL phases that break either $Z_3$ or $Z_2$ symmetry, forbidden within Landau theory.