Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Enriching the Levin-Wen model with Symmetry

Liang Chang, Meng Cheng, Shawn X. Cui, Yuting Hu, Wei Jin, Ramis Movassagh, Pieter Naaijkens, Zhenghan Wang, Amanda Young

TL;DR

This paper extends the Levin-Wen lattice construction from unitary fusion categories to unitary multi-fusion categories, enabling a rigorous lattice approach to symmetry protected and symmetry enriched topological phases in two dimensions. It provides explicit multi-fusion input data, analyzes the Mn case, and proves that the double $D(oldsymbol{ m M}_n)$ collapses to $oldsymbol{ m Vec}$, while revealing boundary degeneracies and zero topological entanglement entropy for closed surfaces. The authors then introduce a framework for incorporating on-site symmetry via group-graded half-labels, classify the resulting 6j-symbols through $H^3(G,U(1))$, and establish a de-equivariantization that connects $G$-symmetric LW models to traditional LW models coupled to a gauge action, thereby realizing $G$-SPTs and SETs. Collectively, the work offers a principled, solvable path to study symmetry fractionalization, defects, and gauging in 2D topological phases, with a concrete algebraic and lattice construction. The results lay groundwork for future rigorous explorations of symmetry phenomena in lattice topological phases and provide a bridge between multi-fusion categorical data and physical SPT/SET behavior.

Abstract

Symmetry protected and symmetry enriched topological phases of matter are of great interest in condensed matter physics due to new materials such as topological insulators. The Levin-Wen model for spin/boson systems is an important rigorously solvable model for studying $2D$ topological phases. The input data for the Levin-Wen model is a unitary fusion category, but the same model also works for unitary multi-fusion categories. In this paper, we provide the details for this extension of the Levin-Wen model, and show that the extended Levin-Wen model is a natural playground for the theoretical study of symmetry protected and symmetry enriched topological phases of matter.

On Enriching the Levin-Wen model with Symmetry

TL;DR

This paper extends the Levin-Wen lattice construction from unitary fusion categories to unitary multi-fusion categories, enabling a rigorous lattice approach to symmetry protected and symmetry enriched topological phases in two dimensions. It provides explicit multi-fusion input data, analyzes the Mn case, and proves that the double collapses to , while revealing boundary degeneracies and zero topological entanglement entropy for closed surfaces. The authors then introduce a framework for incorporating on-site symmetry via group-graded half-labels, classify the resulting 6j-symbols through , and establish a de-equivariantization that connects -symmetric LW models to traditional LW models coupled to a gauge action, thereby realizing -SPTs and SETs. Collectively, the work offers a principled, solvable path to study symmetry fractionalization, defects, and gauging in 2D topological phases, with a concrete algebraic and lattice construction. The results lay groundwork for future rigorous explorations of symmetry phenomena in lattice topological phases and provide a bridge between multi-fusion categorical data and physical SPT/SET behavior.

Abstract

Symmetry protected and symmetry enriched topological phases of matter are of great interest in condensed matter physics due to new materials such as topological insulators. The Levin-Wen model for spin/boson systems is an important rigorously solvable model for studying topological phases. The input data for the Levin-Wen model is a unitary fusion category, but the same model also works for unitary multi-fusion categories. In this paper, we provide the details for this extension of the Levin-Wen model, and show that the extended Levin-Wen model is a natural playground for the theoretical study of symmetry protected and symmetry enriched topological phases of matter.

Paper Structure

This paper contains 19 sections, 5 theorems, 50 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multi-fusion categories and their doubles
  3. Multi-fusion category
  4. Quantum Doubles
  5. Doubles of $n\times n$ $\mathbf{2}$-matrices $\mathcal{M}_n$
  6. Levin-Wen model for Multi-fusion Categories
  7. Levin-Wen Hamiltonian schema for unitary fusion categories
  8. Multi-fusion category extension of the Levin-Wen model
  9. Multi-fusion category extension of the Levin-Wen model
  10. The $n\times n$ $\mathbf{2}$-matrix $\mathcal{M}_n$ as input
  11. Degeneracy on a Disk
  12. Topological Entanglement Entropy
  13. Symmetry Enriching the Levin-Wen model
  14. Classification of $n\times n$ $\mathbf{2}$-matrices
  15. $G$-symmetric Hamiltonian Schema
  16. ...and 4 more sections

Key Result

Theorem 2.4

Let $\mathcal{C}=(\mathcal{C}_{ij})_{1\leq i,j\leq n}$ be an $n\times n$ indecomposable multi-fusion category. Then the quantum double $D(\mathcal{C})$ of $\mathcal{C}$ is equivalent to $D(\mathcal{C}_{ii})$ for any $1\leq i \leq n$. It follows that all $\mathcal{C}_{ii}$ are categorically Morita eq

Figures (6)

  • Figure 1: $\textrm{Hexagon Equations}$
  • Figure 2: $\textrm{Morphisms in } D(\mathcal{M}_n)$
  • Figure 3: A configuration of string types on a directed trivalent graph. The configuration (b) is treated the same as (a), with some of the directions of some edges reversed and the corresponding labels $j$ conjugated $j^*$.
  • Figure 4: (a). Disk with a loop boundary. (b). Double line representation for $\mathcal{L}^{Q=1}$.
  • Figure 5: Partition into subsystems $A$ and $B$ with the boundary along a dashed curve.
  • ...and 1 more figures

Theorems & Definitions (14)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4
  • proof
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Definition 4.1
  • Definition 4.2
  • ...and 4 more