Twisted Quantum Double Model of Topological Phases in Two--Dimension

Understand

We propose a new 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.