Classification of Topological Defects in Abelian Topological States

Abstract

In this paper we propose the most general classification of point-like and line-like extrinsic topological defects in (2+1)-dimensional Abelian topological states. We first map generic extrinsic defects to boundary defects, and then provide a classification of the latter. Based on this classification, the most generic point defects can be understood as domain walls between topologically distinct boundary regions. We show that topologically distinct boundaries can themselves be classified by certain maximal subgroups of mutually bosonic quasiparticles, called Lagrangian subgroups. We study the topological properties of the point defects, including their quantum dimension, localized zero modes, and projective braiding statistics.

Figures (3)

• Figure 1: (a) Schematic picture of bilayer system with a pair of genons.barkeshli2012abarkeshli2013genon (b) is the same figure as (a) with part of the system removed, to see the branch-cut line clearly.
• Figure 2: (a) A domain wall between two different kinds of gapped edges separating topological phases $A_1$ and $A_2$. By folding $A_2$ over, this can be mapped to a domain wall on the boundary separating $A_1 \times \bar{A}_2$ and the vacuum. (b) A junction where multiple gapped edges meet is also a possible type of point defect. On an infinite plane, by applying the folding trick multiple times, this can also be mapped to a domain wall on the boundary separating a topological phase and the vacuum.
• Figure 3: (a) The non-contractible loops in the bilayer system with $4$ genons. Loop $a$ is in the upper (blue) layer and loop $b$ runs from upper layer to the lower (orange) layer across the branch-cut lines. After folding, the genons become domain wall between different gapped boundaries, and the non-contractible loops become Wilson lines that terminate on the boundaries. (b) The general Wilson lines in a system with boundary defects and point defects. Lines $a$ ($b$) defines the unitary operator $W_{\bf m}(a)$ ($W_{\bf m'}(b)$) which correspond to adiabatic motion of bosonic quasiparticle ${\bf m}({\bf m'})$ along the paths $a(b)$, respectively.