Theory of defects in Abelian topological states

TL;DR

This paper develops a comprehensive theory for classifying and understanding point and line defects in 2+1D Abelian topological states, revealing their topological properties, zero modes, and potential for exotic quantum phenomena.

Contribution

It introduces a classification scheme for gapped boundaries and defects using Lagrangian subgroups, and derives formulas for defect quantum dimensions and parafermion zero modes.

Findings

Classification of gapped boundaries via Lagrangian subgroups

Derivation of quantum dimensions for point defects

Connection to generalized parafermion chains and conformal field theories

Abstract

The structure of extrinsic defects in topologically ordered states of matter is host to a rich set of universal physics. Extrinsic defects in 2+1 dimensional topological states include line-like defects, such as boundaries between topologically distinct states, and point-like defects, such as junctions between different line defects. Gapped boundaries in particular can themselves be \it topologically \rm distinct, and the junctions between them can localize topologically protected zero modes, giving rise to topological ground state degeneracies and projective non-Abelian statistics. In this paper, we develop a general theory of point defects and gapped line defects in 2+1 dimensional Abelian topological states. We derive a classification of topologically distinct gapped boundaries in terms of certain maximal subgroups of quasiparticles with mutually bosonic statistics, called Lagrangian subgroups. The junctions between different gapped boundaries provide a general classification of point defects in topological states, including as a special case the twist defects considered in previous works. We derive a general formula for the quantum dimension of these point defects, a general understanding of their localized "parafermion" zero modes, and we define a notion of projective non-Abelian statistics for them. The critical phenomena between topologically distinct gapped boundaries can be understood in terms of a general class of quantum spin chains or, equivalently, "generalized parafermion" chains. This provides a way of realizing exotic 1+1D generalized parafermion conformal field theories in condensed matter systems.