Enriched classification of parafermionic gapped phases with time reversal symmetry

TL;DR

This paper classifies one-dimensional parafermionic gapped phases with time reversal symmetry, revealing an enriched structure characterized by three indices and identifying various topological and symmetry-breaking phases.

Contribution

It introduces a detailed classification scheme for parafermionic phases with TR symmetry, including new indices and distinctions between phases with odd and even parameters.

Findings

Classified parafermionic phases with TR symmetry using three indices.

Identified new symmetry enriched topological and SSB phases.

Determined topological degeneracies from entanglement spectra.

Abstract

Based on the recently established parafermionic matrix product states, we study the classification of one-dimensional gapped phases of parafermions with the time reversal (TR) symmetry satisfying $T^{2}=1$. Without extra symmetry, it has been found that $\mathbb{Z}_{p}$ parafermionic gapped phases can be classified as topological phases, spontaneous symmetry breaking (SSB) phases, and a trivial phase, which are uniquely labelled by the divisors $n$ of $p$. In the presence of TR symmetry, however, the enriched classification is characterized by three indices $n$, $\kappa $ and $\mu $, where $\kappa \in \mathbb{Z}_{2}$ denotes the linear or projective TR actions on the edges, and $\mu \in \mathbb{Z}_{2}$ indicates the commutation relations between the TR and (fractionalized) charge operator. For the $\mathbb{Z}_{r}$ symmetric parafermionic ground states, where $r=p$ for trivial or topological phases, and $r=p/n$ for SSB phases, the original gapped phases with odd $r$ are divided into two phases, while those phases with even $r$ are further separated into four phases. The gapped parafermionic phases with the TR symmetry include the symmetry protected topological phases, symmetry enriched topological phases, and the SSB coexisting symmetry protected topological phases. From analyzing the structures and symmetries of their reduced density matrices of those resulting topological phases, we can obtain the topological protected degeneracies of their entanglement spectra.