String-Net Models with $Z_N$ Fusion Algebra

TL;DR

This paper explores string-net models with $Z_N$ fusion algebra, establishing an exact duality with $Z_N$ gauge theories and constructing a specific spin model on a triangular lattice, revealing new insights into topological phases.

Contribution

It explicitly constructs a spin-$rac{N-1}{2}$ model with $Z_N$ gauge symmetry as an exact dual to the string-net model on a honeycomb lattice, a novel result.

Findings

Established an exact duality between string-net models and $Z_N$ gauge theories.

Constructed a specific spin model on a triangular lattice with $Z_N$ gauge symmetry.

Linked the models to symmetry-protected topological phases classified by $H^3(Z_N,U(1))$.

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.