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

TL;DR

The paper introduces the twisted quantum double (TQD) framework for 2D topological phases built from a finite group $G$ and a 3-cocycle $α∈H^3(G,U(1))$, unifying Kitaev’s quantum double as the trivial-$α$ case and connecting to Dijkgraaf–Witten theory and Levin–Wen dualities. Ground states form the twisted double $D^{α}(G)$, with topological data—ground-state degeneracy (GSD), modular $S$ and $T$ matrices, and topological spins—derived directly from $α$ and the associated twisted 2-cocycles $β_g$. The construction provides a Hamiltonian realization of a DW-type TQFT boundary and an orbifold CFT interpretation, enabling a cohomology-based classification of topological phases and explicit computation of invariants. Examples across abelian and nonabelian groups illustrate how different cohomology classes yield distinct or equivalent topological data, highlighting the roles of centralizers and projective representations in characterizing anyonic excitations.

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 $α$ 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 $D^α(G)$ of $G$. When $α$ 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.

Figures (9)

• Figure 1: A portion of a graph that represent the basis vectors in the Hilbert space. Each edge carries an arrow and is assigned a group element denoted by $[ab]$ with $a<b$.
• Figure 2: (a) The defining graph of the $3$--cocycle $\alpha([v_1v_2], [v_2v_3],[v_3v_4])$. (b) For $\alpha([v_1v_2], [v_2v_3],[v_3v_4])^{-1}$.
• Figure 3: The topology of the action of $A_{v_3}^g$.
• Figure 4: Triangulation of a torus, with $g,h\in G$.
• Figure 5: $\mathcal{S}$ and $\mathcal{T}$ transformations of a torus.

Theorems & Definitions (1)

• Proposition 1