We study the Levin-Wen string-net model with a ${\mathbb{Z}}_{N}$ 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-$(N\ensuremath{-}1)/2$ model with ${\mathbb{Z}}_{N}$ gauge symmetry on a triangular lattice as an exact dual model of the string-net model with a ${\mathbb{Z}}_{N}$ type fusion algebra on a honeycomb lattice. This exact duality exists only when the spins are coupled to a ${\mathbb{Z}}_{N}$ gauge field living on the links of the triangular lattice. The ungauged ${\mathbb{Z}}_{N}$ lattice spin models are a class of quantum systems that bear symmetry-protected topological phases that may be classified by the third cohomology group ${H}^{3}({\mathbb{Z}}_{N},U(1))$ of ${\mathbb{Z}}_{N}$ . Our results apply also to any case where the fusion algebra is identified with a finite group algebra or a quantum group algebra.We study the Levin-Wen string-net model with a ${\mathbb{Z}}_{N}$ 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-$(N\ensuremath{-}1)/2$ model with ${\mathbb{Z}}_{N}$ gauge symmetry on a triangular lattice as an exact dual model of the string-net model with a ${\mathbb{Z}}_{N}$ type fusion algebra on a honeycomb lattice. This exact duality exists only when the spins are coupled to a ${\mathbb{Z}}_{N}$ gauge field living on the links of the triangular lattice. The ungauged ${\mathbb{Z}}_{N}$ lattice spin models are a class of quantum systems that bear symmetry-protected topological phases that may be classified by the third cohomology group ${H}^{3}({\mathbb{Z}}_{N},U(1))$ of ${\mathbb{Z}}_{N}$. Our results apply also to any case where the fusion algebra is identified with a finite group algebra or a quantum group algebra.

arXiv:1207.6169Physical Review BDec 19, 201236 citations

We study the Levin-Wen string-net model with a

{\mathbb{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-$(N\ensuremath{-}1)/2$ model with

{\mathbb{Z}}_{N}

gauge symmetry on a triangular lattice as an exact dual model of the string-net model with a

{\mathbb{Z}}_{N}

type fusion algebra on a honeycomb lattice. This exact duality exists only when the spins are coupled to a

{\mathbb{Z}}_{N}

gauge field living on the links of the triangular lattice. The ungauged

{\mathbb{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}({\mathbb{Z}}_{N},U(1))

{\mathbb{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.