Twisted quantum double model of topological phases in two dimensions

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(G) 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(G) 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. DOI: 10.1103/PhysRevB.87.125114