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 not equal 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 two-dimensional boson systems.