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

Abstract

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. 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. The paper is presented in a way such that it is accessible to a wide range of physicists.

37 pages, Revtex4, one appendix added, typos correct, published in PRB

References in corpus (4)

• Topological Field Theory of Time-Reversal Invariant Insulators
• Characterizing topological order by studying the ground states of an infinite cylinder
• Zero modes of two-dimensional chiral p-wave superconductors
• String-Net Models with $Z_N$ Fusion Algebra

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