Protected edge modes without symmetry

TL;DR

This paper establishes conditions under which 2D gapped electron systems without symmetry can have protected edge modes, highlighting the role of fractional statistics even when thermal Hall conductance is zero.

Contribution

It provides a new criterion for the existence of gapped edges in systems with abelian anyons and zero thermal Hall conductance, using three different theoretical approaches.

Findings

Certain fractional statistics are compatible with gapped edges.

A specific criterion involving mutual statistics determines gapped edge possibility.

The criterion is derived via microscopic, braiding, and conformal field theory methods.

Abstract

We discuss the question of when a gapped 2D electron system without any symmetry has a protected gapless edge mode. While it is well known that systems with a nonzero thermal Hall conductance, $K_H \neq 0$, support such modes, here we show that robust modes can also occur when $K_H = 0$ -- if the system has quasiparticles with fractional statistics. We show that some types of fractional statistics are compatible with a gapped edge, while others are fundamentally incompatible. More generally, we give a criterion for when an electron system with abelian statistics and $K_H = 0$ can support a gapped edge: we show that a gapped edge is possible if and only if there exists a subset of quasiparticle types $M$ such that (1) all the quasiparticles in $M$ have trivial mutual statistics, and (2) every quasiparticle that is not in $M$ has nontrivial mutual statistics with at least one quasiparticle in $M$. We derive this criterion using three different approaches: a microscopic analysis of the edge, a general argument based on braiding statistics, and finally a conformal field theory approach that uses constraints from modular invariance. We also discuss the analogous result for 2D boson systems.