String-net models with ZN fusion algebra

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 quantum group algebra. DOI: 10.1103/PhysRevB.86.235132