Topological phases in gapped edges of fractionalized systems

TL;DR

This paper classifies and demonstrates the variety of topological phases possible in gapped edges of fractionalized systems, specifically using parafermionic chains in fractional topological insulators.

Contribution

It introduces a new classification scheme for phases in parafermionic chains and confirms the realization of all predicted phases through numerical entanglement spectrum analysis.

Findings

Parafermions support topological and condensed phases.

All predicted phases are realizable in a $ u=1/3$ fractional topological insulator edge model.

The phases form a non-Abelian group with additional symmetries.

Abstract

Recently, it has been proposed that exotic one-dimensional phases can be realized by gapping out the edge states of a fractional topological insulator. The low-energy edge degrees of freedom are described by a chain of coupled parafermions. We introduce a classification scheme for the phases that can occur in parafermionic chains. We find that the parafermions support both topological symmetry fractionalized phases as well as phases in which the parafermions condense. In the presence of additional symmetries, the phases form a non-Abelian group. As a concrete example of the classification, we consider the effective edge model for a $\nu= 1/3$ fractional topological insulator for which we calculate the entanglement spectra numerically and show that all possible predicted phases can be realized.