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

TL;DR

This paper introduces the twisted quantum double model for 2D topological phases, extending Kitaev's model with a 3-cocycle twist, and explores its properties, dualities, and relation to topological gauge theories.

Contribution

It formulates a new twisted quantum double model based on finite groups and 3-cocycles, generalizing existing models and connecting to Dijkgraaf--Witten theories and string-net models.

Findings

Ground states characterized by twisted quantum double D^{α}(G)

Model reduces to Kitaev's quantum double when α is trivial

Establishes duality with Levin-Wen string-net models

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.