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arXiv:1211.3695 · Physical Review B

Twisted quantum double model of topological phases in two dimensions

Yuting Hu, Yidun Wan, Yong-Shi Wu

Published Mar 11, 2013• 213 citations
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Abstract

We propose a discrete model---the twisted quantum double model---of 2D topological phases based on a finite group $G$ and a 3-cocycle $\ensuremath{\alpha}$ 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 ${D}^{\ensuremath{\alpha}}(G)$ of $G$. When $\ensuremath{\alpha}$ 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

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We propose a discrete model---the twisted quantum double model---of 2D topological phases based on a finite group GGG and a 3-cocycle \ensuremathα\ensuremath{\alpha}\ensuremathα over GGG. 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\ensuremathα(G){D}^{\ensuremath{\alpha}}(G)D\ensuremathα(G) of GGG. When \ensuremathα\ensuremath{\alpha}\ensuremathα is the trivial 3-cocycle, the model becomes Kitaev's quantum double model based on the finite group GGG, in which the elementary excitations are known to be classified by the quantum double D(G)D(G)D(G) of GGG. 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 6j6j6j symbols to the corresponding 3-cocycles.
We propose an exactly solvable lattice Hamiltonian model of topological phases in 3+13+13+1 dimensions, based on a generic finite group GGG and a 4-cocycle $\ensuremath{\omega}$ over GGG. We show that our model has topologically protected degenerate ground states and obtain the formula of its ground state degeneracy on the 3-torus. In particular, the ground state spectrum implies the existence of purely three-dimensional looplike quasiexcitations specified by two nontrivial flux indices and one charge index. We also construct other nontrivial topological observables of the model, namely the SL(3,Z)SL(3,\mathbb{Z})SL(3,Z) generators as the modular SSS and TTT matrices of the ground states, which yield a set of topological quantum numbers classified by $\ensuremath{\omega}$ and quantities derived from $\ensuremath{\omega}$. Our model fulfills a Hamiltonian extension of the (3+1)(3+1)(3+1)-dimensional Dijkgraaf-Witten topological gauge theory with a gauge group GGG. This work is presented to be accessible for a wide range of physicists and mathematicians.
Doubled topological phases introduced by Kitaev, Levin, and Wen supported on two-dimensional lattices are Hamiltonian versions of three-dimensional topological quantum field theories described by the Turaev-Viro state sum models. We introduce the latter with an emphasis on obtaining them from theories in the continuum. Equivalence of the previous models in the ground state is shown in case of the honeycomb lattice and the gauge group being a finite group by means of the well-known duality transformation between the group algebra and the spin network basis of lattice gauge theory. An analysis of the ribbon operators describing excitations in both types of models and the three-dimensional geometrical interpretation are given.
The recent article [arXiv:2307.12552] gave local topological order (LTO) axioms for a quantum spin system, showed they held in Kitaev's Toric Code and in Levin-Wen string net models, and gave a bulk boundary correspondence to describe bulk excitations in terms of the boundary net of algebras. In this article, we prove the LTO axioms for Kitaev's Quantum Double model for a finite group GGG. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G])(\mathsf{Hilb}(G),\mathbb{C}[G])(Hilb(G),C[G]) or (Rep(G),CG)(\mathsf{Rep}(G),\mathbb{C}^G)(Rep(G),CG) depending on whether the boundary cut is rough or smooth respectively. This allows us to make connections to work of Ogata on the type of the cone von Neumann algebras in the algebraic quantum field theory approach to topological superselection sectors. We show that the boundary algebras can also be calculated from a trivial GGG-symmetry protected topological phase (GGG-SPT), and that the gauging map preserves the boundary algebras. Finally, we compute the boundary algebras for the (3+1)D Quantum Double model associated to an abelian group.
Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters.
Abstract We derive a partial electric-magnetic (PEM) duality transformation of the twisted quantum double (TQD) model TQD(G, α) — discrete Dijkgraaf-Witten model — with a finite gauge group G, which has an Abelian normal subgroup N , and a three-cocycle α ∈ H3(G, U(1)). Any equivalence between two TQD models, say, TQD(G, α) and TQD(G′, α′), can be realized as a PEM duality transformation, which exchanges the N-charges and N-fluxes only. Via the PEM duality, we construct an explicit isomorphism between the corresponding TQD algebras Dα(G) and Dα′(G′) and derive the map between the anyons of one model and those of the other.
A bstract In this work, we employ the Tannaka-Krein reconstruction to compute the quantum double 𝒟(𝒢) of a finite 2-group 𝒢 as a Hopf monoidal category. We also construct a 3+1D lattice model from the Dijkgraaf-Witten TQFT functor for the 2-group 𝒢, generalizing Kitaev’s 2+1D quantum double model. Notably, the string-like local operators in this lattice model are shown to form 𝒟(𝒢). Specializing to 𝒢 = ℤ 2 , we demonstrate that the topological defects in the 3+1D toric code model are modules over 𝒟(ℤ 2 ).