Theory of defects in Abelian topological states

TL;DR

This work develops a comprehensive framework for extrinsic defects in Abelian 2+1D topological states by classifying gapped edges with Lagrangian subgroups and analyzing domain-wall junctions as point defects. Using folding, edge backscattering terms, and Wilson-line algebras, it derives a one-to-one correspondence between Lagrangian subgroups and gapped edges, and quantifies the non-Abelian structure via quantum dimensions and zero-mode algebras. The point defects are shown to realize generalized parafermion zero modes and map to genons on high-genus surfaces, enabling projective non-Abelian statistics through edge-tunneling protocols. The results illuminate critical phenomena between edge types through generalized parafermion spin chains and establish a broad, constructive approach to understanding and realizing exotic boundary physics in Abelian topological phases with potential realizations in condensed matter platforms.

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.

Figures (11)

• Figure 1: Examples of point defects studied previously in the literature. (a) Genons in bilayer systemsbarkeshli2012abarkeshli2013genon (b) domain walls between ferromagnetic and superconducting backscattering at the edge of a fractional quantum spin Hall (FQSH) state,clarke2013lindner2012cheng2012 (c) lattice dislocations in solvable models of $Z_N$ topological order.bombin2010you2012
• Figure 2: A line defect can be considered to be a domain wall between two kinds of topological phases, $A_1$ and $A_2$. By folding one side over onto the other, this can be mapped to an edge between $A_1 \times \bar{A_2}$, and the trivial gapped state, which we label "0". Under general conditions, the line defect will either host topologically protected gapless edge states or be fully gapped.
• Figure 3: (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 trivial gapped state, "0". (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 $0$.
• Figure 4: Depiction of the process $W_{\boldsymbol{m}}(\gamma)$, which creates a quasiparticle $\boldsymbol{m}$ at location $a$ on the edge with a local operator, the quasiparticle propagates along a path $\gamma$ in the bulk, and is annihilated at $b$ with a local operator.
• Figure 5: (a) Lattice model of $Z_N$ toric code with gapped boundary (red line) that corresponds to the Lagrangian subgroup generated by $(1,0)$ and $(0,N)$. The Wilson line (the orange dash line) shows that the quasi-particle $(1,0)$ and its anti-particle $(-1,0)$ can be created together from the vacuum and annihilated at different locations of the boundary. (b) The $Z_N$ toric code with gapped boundary (red line) that corresponds to the Lagrangian subgroup generated by $(r,0)$ and $(0,t)$. The Wilson line (the green dash line) shows that the quasi-particle $(0,t)$ and its anti-particle $(0,-t)$ can be created together from the vacuum and annihilated at different locations of the boundary.