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Classification of Gapped Domain Walls of Topological Orders in 2+1 dimensions: A Levin-Wen Model Realization

Yanyan Chen, Siyuan Wang, Yu Zhao, Yuting Hu, Yidun Wan

TL;DR

This work develops a systematic lattice construction for gapped domain walls (GDWs) between 2+1D Levin-Wen topological orders by gluing two LW models along compatible open edges. GDWs are classified by the triple $(A_1,A_2,\eta)$, with domain-wall excitations described by generalized bimodules $A_1\overset{\eta}{-}A_2$, and the folding trick shows GDWs map to gapped boundaries described by Frobenius algebras in the folded phase, linking to the established GB classification. The authors substantiate the framework with explicit analyses in toric code and doubled Ising phases and explore walls between Ising and toric code, including $e$-$m$ exchange walls and condensation-driven walls. The results illuminate a unified picture of GDWs and GBs, clarify the role of Morita versus physical equivalence, and suggest avenues for studying domain-wall fusion, junctions, and higher-dimensional generalizations in topological phases.

Abstract

This paper introduces a novel systematic construction of gapped domain walls (GDWs) within the Levin-Wen (LW) model. By gluing two LW models along their open sides in a compatible way, we achieve a complete GDW classification by subsets of bulk input data, which encompass the classifications in terms of bimodule categories. A generalized bimodule structure is introduced to capture domain-wall excitations. Furthermore, we demonstrate that folding along any GDW yields a gapped boundary (GB) described by a Frobenius algebra of the input UFC for the folded model, thus bridging our GDW classification and the GB classification in \cite{hu2018boundary}.

Classification of Gapped Domain Walls of Topological Orders in 2+1 dimensions: A Levin-Wen Model Realization

TL;DR

This work develops a systematic lattice construction for gapped domain walls (GDWs) between 2+1D Levin-Wen topological orders by gluing two LW models along compatible open edges. GDWs are classified by the triple , with domain-wall excitations described by generalized bimodules , and the folding trick shows GDWs map to gapped boundaries described by Frobenius algebras in the folded phase, linking to the established GB classification. The authors substantiate the framework with explicit analyses in toric code and doubled Ising phases and explore walls between Ising and toric code, including - exchange walls and condensation-driven walls. The results illuminate a unified picture of GDWs and GBs, clarify the role of Morita versus physical equivalence, and suggest avenues for studying domain-wall fusion, junctions, and higher-dimensional generalizations in topological phases.

Abstract

This paper introduces a novel systematic construction of gapped domain walls (GDWs) within the Levin-Wen (LW) model. By gluing two LW models along their open sides in a compatible way, we achieve a complete GDW classification by subsets of bulk input data, which encompass the classifications in terms of bimodule categories. A generalized bimodule structure is introduced to capture domain-wall excitations. Furthermore, we demonstrate that folding along any GDW yields a gapped boundary (GB) described by a Frobenius algebra of the input UFC for the folded model, thus bridging our GDW classification and the GB classification in \cite{hu2018boundary}.

Paper Structure

This paper contains 32 sections, 93 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Motivations and Sketch of Our Approach
  3. Exactly solvable LW model with GDWs
  4. Definition of -bimodules
  5. Hilbert Space and Hamiltonian
  6. Exact solvability and condition on joining functions
  7. Ground states and GDW Excitations
  8. GDWs in the Toric code phase
  9. and : Realizing e-m exchange
  10. GDWs in the doubled Ising phase
  11. GDWs between the doubled Ising and Toric code phases
  12. GDW-GB Correspondence
  13. Conclusions and Outlook
  14. Review of the LW model
  15. The original LW model
  16. ...and 17 more sections

Figures (5)

  • Figure 1: The gluing process. (a) Part of the lattice of the $\mathcal{C}_1$-model and that of the $\mathcal{C}_2$-model. Their open sides are characterized by $A_1$ and $A_2$, repectively. (b) A GDW constructed by gluing the two lattice along their open sides via the joining function $\eta$.
  • Figure 2: The tailed LW model. The primary and secondary vertices are highlighted in the lattice.
  • Figure 3: Two extended LW models (red and blue) with input fusion categories $\mathcal{C}_1$ and $\mathcal{C}_2$, joint by a DW (light grey). The joining points, bearing the algebraic actions of $A_1$ and $A_2$, are characterized by the $A_1\space \mathrel{\mathop{-}\limits^{ \hbox{\ex@ $\eta$}}}\space A_2$-bimodules $M$.
  • Figure 4: Fold the lattice for the ground states along the middle (dotted) line of the GDW. Then the DOFs at the gluing points in the GDW becomes those at the endpoints of tails (colored purple) in the GB. Since the DOFs at the endpoints of the tails are fixed by $M_0$, we can omit it when discussing about the ground states.
  • Figure 5: The extended LW model by enlarging the Hilbert space