Parafermionic edge zero modes in Z_n-invariant spin chains

TL;DR

This paper demonstrates the existence of exact parafermionic edge zero modes in Z_n-invariant spin chains with broken parity and time-reversal symmetry, extending topological edge mode concepts beyond free-fermion models.

Contribution

It proves the presence of exact parafermionic edge zero modes in strongly interacting Z_n spin chains with chiral interactions, generalizing topological edge modes beyond free-fermion systems.

Findings

Parafermionic edge zero modes exist in certain Z_n spin chains with chiral interactions.

These modes are absent in ferromagnetic and antiferromagnetic cases.

Breaking spatial-parity and time-reversal symmetries is essential for their existence.

Abstract

A sign of topological order in a gapped one-dimensional quantum chain is the existence of edge zero modes. These occur in the Z_2-invariant Ising/Majorana chain, where they can be understood using free-fermion techniques. Here I discuss their presence in spin chains with Z_n symmetry, and prove that for appropriate coupling they are exact, even in this strongly interacting system. These modes are naturally expressed in terms of parafermions, generalizations of fermions to the Z_n case. I show that parafermionic edge zero modes do not occur in the usual ferromagnetic and antiferromagnetic cases, but rather only when the interactions are chiral, so that spatial-parity and time-reversal symmetries are broken.