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String-Net Models with $Z_N$ Fusion Algebra

Ling-Yan Hung, Yidun Wan

TL;DR

This work analyzes Levin–Wen string-net models with a $Z_N$ fusion algebra, showing how local constraints give rise to $Z_N$ gauge theory and related topological quantum field theories. It constructs an explicit duality between a honeycomb lattice string-net with $ rak{F}_{bZ_N}$ fusion data and a spin model on the dual triangular lattice, including gauging to connect the full Hilbert spaces. The authors classify and discuss the $6j$-symbols ($F$ and $G$) for $bZ_N$ fusion, relate rescaling redundancies to group cohomology $H^3(G,U(1))$, and demonstrate how gauging the dual spin model yields an SPT–LRE correspondence, extended beyond the $N=2$ case. The results provide concrete mappings between SPT phases and corresponding string-net topological orders and outline future challenges, such as quasi-particle structure and non-group fusion algebras.

Abstract

We study the Levin-Wen string-net model with a $Z_N$ type fusion algebra. Solutions of the local constraints of this model correspond to $Z_N$ gauge theory and double Chern-simons theories with quantum groups. For the first time, we explicitly construct a spin-$(N-1)/2$ model with $Z_N$ gauge symmetry on a triangular lattice as an exact dual model of the string-net model with a $Z_N$ type fusion algebra on a honeycomb lattice. This exact duality exists only when the spins are coupled to a $Z_N$ gauge field living on the links of the triangular lattice. The ungauged $Z_N$ lattice spin models are a class of quantum systems that bear symmetry-protected topological phases that may be classified by the third cohomology group $H^3(Z_N,U(1))$ of $Z_N$. Our results apply also to any case where the fusion algebra is identified with a finite group algebra or a quantusm group algebra.

String-Net Models with $Z_N$ Fusion Algebra

TL;DR

This work analyzes Levin–Wen string-net models with a fusion algebra, showing how local constraints give rise to gauge theory and related topological quantum field theories. It constructs an explicit duality between a honeycomb lattice string-net with fusion data and a spin model on the dual triangular lattice, including gauging to connect the full Hilbert spaces. The authors classify and discuss the -symbols ( and ) for fusion, relate rescaling redundancies to group cohomology , and demonstrate how gauging the dual spin model yields an SPT–LRE correspondence, extended beyond the case. The results provide concrete mappings between SPT phases and corresponding string-net topological orders and outline future challenges, such as quasi-particle structure and non-group fusion algebras.

Abstract

We study the Levin-Wen string-net model with a type fusion algebra. Solutions of the local constraints of this model correspond to gauge theory and double Chern-simons theories with quantum groups. For the first time, we explicitly construct a spin- model with gauge symmetry on a triangular lattice as an exact dual model of the string-net model with a type fusion algebra on a honeycomb lattice. This exact duality exists only when the spins are coupled to a gauge field living on the links of the triangular lattice. The ungauged lattice spin models are a class of quantum systems that bear symmetry-protected topological phases that may be classified by the third cohomology group of . Our results apply also to any case where the fusion algebra is identified with a finite group algebra or a quantusm group algebra.

Paper Structure

This paper contains 21 sections, 2 theorems, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Review of the fundamentals
  3. A note on group cohomology classification of $6j$-symbols
  4. $\mathfrak{F}_{\mathbb{Z}_N}$ String--Net Models
  5. $\mathfrak{F}_{\mathbb{Z}_N}$ $6j$-symbols and quantum dimensions: General Properties
  6. The Magnetic Flux Operators
  7. The Hamiltonian
  8. The duality between $Z_{N}$ string--net models and (SPT) spin models
  9. Turning the strings states into spin states
  10. Branching rules and the dual spin model on a triangular lattice
  11. Gauging the dual spin model and counting of states
  12. The action of operators $B_p$
  13. $\mathfrak{F}_{\mathbb{Z}_2}$ spin and string--net models
  14. $\mathbb{Z}_3$ spin models and string--net models
  15. Discussions and Outlook
  16. ...and 6 more sections

Key Result

Theorem 1

For a $\mathfrak{F}_{\mathbb{Z}_N}$ string--net model, if $N\in 2\mathbb{Z}+1$, all the quantum dimensions $d_i$ of the model satisfy $d_i\equiv1$. If $N\in 2\mathbb{Z}$, $d_i \equiv 1$ for $i\in 2\mathbb{Z}_N$, and $d_j=d_k$, $\forall j,k\in 2\mathbb{Z}_N+1$.

Figures (2)

  • Figure 1: A string--net vertex.
  • Figure 2: Duality

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2