We propose a discrete model—the twisted quantum double model—of 2D topological phases based on a finite group $G$ and a 3-cocycle $\alpha$ over $G$. The detailed properties of the ground states are studied, and we find that the ground-state subspace can be characterized in terms of the twisted quantum double $D^{\alpha }\left(G\right)$ of $G$. When $\alpha$ is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group $G$, in which the elementary excitations are known to be classified by the quantum double $D\left(G\right)$ of $G$. Our model can be viewed as a Hamiltonian extension of the Dijkgraaf-Witten topological gauge theories to the discrete graph case with gauge group being a finite group. We also demonstrate a duality between a large class of Levin-Wen string-net models and certain twisted quantum double models, by mapping the string-net $6j$ symbols to the corresponding 3-cocycles.

• Received 7 December 2012

DOI:https://doi.org/10.1103/PhysRevB.87.125114