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.
Highlights
In two dimensions, some quantum many-body systems with a bulk energy gap have the property that they support gapless edge modes that are extremely robust
We show that, in general, this intuition is incorrect: We find that systems with KH 1⁄4 0 can have protected edge modes—if they support quasiparticle excitations with fractional statistics
We show that a gapped edge is possible if and only if there exists a set of quasiparticle ‘‘types’’ M satisfying two properties: (1) The particles in M have trivial mutual statistics: eimm0 1⁄4 1 for any m, m0 2 M
Summary
Some quantum many-body systems with a bulk energy gap have the property that they support gapless edge modes that are extremely robust. We show that, in general, this intuition is incorrect: We find that systems with KH 1⁄4 0 can have protected edge modes—if they support quasiparticle excitations with fractional statistics. We will derive the above criterion using three different approaches: a microscopic edge analysis, a general argument based on quasiparticle braiding statistics, and an argument that uses constraints from modular invariance. These derivations are complementary to one another.
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