Topological and symmetry broken phases of Z_N parafermions in one dimension

TL;DR

This paper classifies and constructs all gapped phases of Z_N parafermions in one dimension, revealing a rich structure of symmetry-breaking and topological phases based on divisors of N.

Contribution

It provides a comprehensive classification and explicit realization of all gapped phases of Z_N parafermions, including symmetry-breaking and topological phases, in one-dimensional systems.

Findings

Multiple phases characterized by divisors of N

Purely topological phases occur for unitary divisors

Explicit models constructed for all symmetry-broken phases

Abstract

We classify the gapped phases of Z_N parafermions in one dimension and construct a representative of each phase. Even in the absence of additional symmetries besides parafermionic parity, parafermions may be realized in a variety of phases, one for each divisor n of N. The phases can be characterized by spontaneous symmetry breaking, topology, or a mixture of the two. Purely topological phases arise if n is a unitary divisor, i.e. if n and N/n are co-prime. Our analysis is based on the explicit realization of all symmetry broken gapped phases in the dual Z_N-invariant quantum spin chains.