Theory of defects in Abelian topological states

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 linelike defects, such as boundaries between topologically distinct states, and pointlike defects, such as junctions between different line defects. Gapped boundaries in particular can themselves be topologically 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 and 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.