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Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy

Tian Lan, Juven Wang, Xiao-Gang Wen

TL;DR

By studying many examples, this work finds evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls, including closed 2-manifolds and open 2- manifolds with gapped boundaries.

Abstract

Gapped domain walls, as topological line defects between 2+1D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which 2+1D topological orders must have gapless edge modes, namely which 1+1D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix $\mathcal W$, whose entries are the fusion-space dimensions $\mathcal W_{ia}$, to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.

Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy

TL;DR

By studying many examples, this work finds evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls, including closed 2-manifolds and open 2- manifolds with gapped boundaries.

Abstract

Gapped domain walls, as topological line defects between 2+1D topologically ordered states, are examined. We provide simple criteria to determine the existence of gapped domain walls, which apply to both Abelian and non-Abelian topological orders. Our criteria also determine which 2+1D topological orders must have gapless edge modes, namely which 1+1D global gravitational anomalies ensure gaplessness. Furthermore, we introduce a new mathematical object, the tunneling matrix , whose entries are the fusion-space dimensions , to label different types of gapped domain walls. By studying many examples, we find evidence that the tunneling matrices are powerful quantities to classify different types of gapped domain walls. Since a gapped boundary is a gapped domain wall between a bulk topological order and the vacuum, regarded as the trivial topological order, our theory of gapped domain walls inclusively contains the theory of gapped boundaries. In addition, we derive a topological ground state degeneracy formula, applied to arbitrary orientable spatial 2-manifolds with gapped domain walls, including closed 2-manifolds and open 2-manifolds with gapped boundaries.

Paper Structure

This paper contains 17 sections, 42 equations, 3 figures, 2 tables.

Table of Contents

  1. Sec. 1. Equivalence between tunneling matrices and Lagrangian subgroups for Abelian topological orders
  2. Sec. 2. Examples
  3. I. Gapped boundaries of toric code phase: 2 types
  4. II. Gapped domain walls between two toric codes: 6 types
  5. III. Gapped boundary of double-semion phase: 1 type
  6. IV. Gapped domain walls between double-semion and toric code phases: 2 types
  7. V. Gapped boundary of doubled Fibonacci phase: 1 type
  8. VI. Gapped domain walls between doubled Fibonacci and toric code phases: 2 types
  9. VII. Gapped boundary of doubled Ising phase: 1 type
  10. VIII. Gapped domain walls between doubled Ising and toric code phases: 3 types
  11. IX. Gapped boundaries of $D(S_3)$ phase: 4 types
  12. X. Gapped domain walls between $D(S_3)$ and toric code: 12 types
  13. XI. Gapped boundaries of $D(D_4)$ phase: 11 types
  14. XII. Gapped boundaries of $D(Q_8)$ phase: 6 types
  15. XIII. Gapped boundaries of $D^{\omega_3{[3d]}}(Z_2{}^3)$ phase: 5 types
  16. ...and 2 more sections

Figures (3)

  • Figure 1: (a)(b) Tunneling channels. (c) Separated domain walls $\mathcal{W}^{(1)}$ and $\mathcal{W}^{(2)}$. (d) Composite domain wall $\mathcal{W}^{(2)}\mathcal{W}^{(1)}$.
  • Figure 2: Computing GSD by tensor contraction: Cut a complicated manifold (e) into simple segments, add oriented skeletons and anyon indices. Associate the segments with: (a) a cylinder with $\delta_{ab}$, (b) a domain wall with its tunneling matrix $\mathcal{W}_{ia}$, (c) a pair of pants with the fusion tensor $\mathcal{N}_{ij}^k$ and (d) a cap with $\delta_{1u}$. Finally, contract all the tensors.
  • Figure 3: Some 2-manifolds with gapped domain walls.