Protected Edge Modes without Symmetry

We discuss the question of when a gapped two-dimensional electron system without any symmetry has a protected gapless edge mode. While it is well known that systems with a nonzero thermal Hall conductance, $K_H\ne 0$, 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 $\mathcal{M}$ such that (1) all the quasiparticles in $\mathcal{M}$ have trivial mutual statistics, and (2) every quasiparticle that is not in $\mathcal{M}$ has nontrivial mutual statistics with at least one quasiparticle in $\mathcal{M}$. 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 two-dimensional boson systems.

Popular Summary

Insulating materials, by their very definition, cannot conduct an electric current in their interior. However, some two-dimensional insulators have the unusual property that they conduct an electric current at their edges. This conduction occurs through modes that propagate along the edge of the insulator like waves moving on the surface of the ocean. What is particularly interesting is that in some insulators these edge modes are guaranteed to be present, independent of the detailed structure of the boundary: The modes are somehow “protected” by some fundamental properties of the material inside.

What exactly protects these edge modes? In many cases, the stability of these modes depends deeply on symmetry. For example, the edge modes of the recently discovered “topological insulators” are only stable if time-reversal symmetry is not broken. Another mechanism that can protect edge modes is chirality: Some insulators have edge modes that only move in the clockwise and counterclockwise directions, and this chiral structure guarantees the stability of these modes.

In this paper, we show that there is a third (and final) mechanism that can protect edge modes of two-dimensional insulators or, more generally, two-dimensional electron systems. It has long been known that some two-dimensional electron systems have particle-like excitations that are neither bosons nor fermions. Such particles are said to have “fractional statistics.” Here, we show that certain types of fractional statistics can protect an edge mode, just as symmetry and chirality do. More generally, we present and prove a simple criterion that determines which types of fractional statistics lead to protected edge modes and which types do not.