1301.7355 (Michael Levin)

Protected edge modes without symmetry    [PDF]

Michael Levin

We discuss the question of when a 2D topological phase 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 in systems with $K_H = 0$. We argue that these modes are protected, not by any kind of Hall response, but rather by the structure of the quasiparticle braiding statistics in the bulk. We show that some types of braiding statistics are compatible with a gapped edge, while others are fundamentally incompatible. More generally, we give a criterion for when an abelian topological phase that is built out of electrons and has $K_H = 0$ 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.

View original: http://arxiv.org/abs/1301.7355