Topological phases of parafermionic chains with symmetries

TL;DR

This paper classifies topological phases of parafermionic chains with a modified time reversal symmetry, revealing a richer structure and emergent Majorana modes, especially distinguishing phases based on whether m is odd or even.

Contribution

It provides a detailed topological classification of parafermionic chains under modified time reversal symmetry, highlighting new subclasses and the emergence of Majorana modes.

Findings

For odd m, phases split into two subclasses with Kramers doublets.

For even m, phases split into four subclasses due to emergent Majorana fermions.

Emergent Majorana zero modes appear in systems with fermions or bosons.

Abstract

We study the topological classification of parafermionic chains in the presence of a modified time reversal symmetry that satisfies ${\cal T}^2=1 $. Such chains can be realized in one dimensional structures embedded in fractionalized two dimensional states of matter, e.g. at the edges of a fractional quantum spin Hall system, where counter propagating modes may be gapped either by back-scattering or by coupling to a superconductor. In the absence of any additional symmetries, a chain of $\mathbb{Z}_m$ parafermions can belong to one of several distinct phases. We find that when the modified time reversal symmetry is imposed, the classification becomes richer. If $m $ is odd, each of the phases splits into two subclasses. We identify the symmetry protected phase as a Haldane phase that carries a Kramers doublet at each end. When $m $ is even, each phase splits into four subclasses. The origin of this split is in the emergent Majorana fermions associated with even values of $m$. We demonstrate the appearance of such emergent Majorana zero modes in a system where the constituents particles are either fermions or bosons.