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arXiv:0907.2670 · Physical Review B

Mapping Kitaev’s quantum double lattice models to Levin and Wen’s string-net models

Oliver Buerschaper, M. Aguado

Published Oct 28, 2009• 120 citations
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Abstract

We exhibit a mapping identifying Kitaev's quantum double lattice models explicitly as a subclass of Levin and Wen's string-net models via a completion of the local Hilbert spaces with auxiliary degrees of freedom. This identification allows one to carry over to these string-net models the representation-theoretic classification of the excitations in quantum double models as well as define them in arbitrary lattices and provides an illustration of the abstract notion of Morita equivalence. The possibility of generalizing the map to broader classes of string nets is considered.

DOI
10.1103/physrevb.80.155136

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We exhibit a mapping identifying Kitaev's quantum double lattice models explicitly as a subclass of Levin and Wen's string-net models via a completion of the local Hilbert spaces with auxiliary degrees of freedom. This identification allows one to carry over to these string-net models the representation-theoretic classification of the excitations in quantum double models as well as define them in arbitrary lattices and provides an illustration of the abstract notion of Morita equivalence. The possibility of generalizing the map to broader classes of string nets is considered.
We provide a simple expression for the ground states of arbitrary string-net models in the form of local tensor networks. These tensor networks encode the data of the fusion category underlying a string-net model and thus represent all doubled topological phases of matter in the infrared limit according to Levin and Wen [Phys. Rev. B 71, 045110 (2005)]. Furthermore, our construction highlights the importance of the fat lattice equivalence between lattice and continuum descriptions of string-net models.
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.
The recent article by Jones et al. [arXiv:2307.12552 (2023)] 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 G. We identify the boundary nets of algebras with fusion categorical nets associated to (Hilb(G),C[G]) or (Rep(G),CG) depending on whether the boundary cut is rough or smooth, respectively. This allows us to make connections to the work of Ogata [Ann. Henri Poincaré 25, 2353–2387 (2024)] 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 G-symmetry protected topological phase (G-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.
We propose a discrete model---the twisted quantum double model---of 2D topological phases based on a finite group GGG and a 3-cocycle $\ensuremath{\alpha}$ 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 ${D}^{\ensuremath{\alpha}}(G)$ of GGG. When $\ensuremath{\alpha}$ 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.
Abstract Levin-Wen string-net models provide a construction of (2+1)D topologically ordered phases of matter with anyonic localized excitations described by the Drinfeld center of a unitary fusion category. Anyon condensation is a mechanism for phase transitions between (2+1)D topologically ordered phases. We construct an extension of Levin-Wen models in which tuning a parameter implements anyon condensation. We also describe the classification of anyons in Levin-Wen models via representation theory of the tube algebra, and use a variant of the tube algebra to classify low-energy localized excitations in the condensed phase.
We study the Levin-Wen string-net model with a ZN{\mathbb{Z}}_{N}ZN​ type fusion algebra. Solutions of the local constraints of this model correspond to ZN{Z}_{N}ZN​ gauge theory and double Chern-Simons theories with quantum groups. For the first time, we explicitly construct a spin-$(N\ensuremath{-}1)/2$ model with ZN{\mathbb{Z}}_{N}ZN​ gauge symmetry on a triangular lattice as an exact dual model of the string-net model with a ZN{\mathbb{Z}}_{N}ZN​ type fusion algebra on a honeycomb lattice. This exact duality exists only when the spins are coupled to a ZN{\mathbb{Z}}_{N}ZN​ gauge field living on the links of the triangular lattice. The ungauged ZN{\mathbb{Z}}_{N}ZN​ lattice spin models are a class of quantum systems that bear symmetry-protected topological phases that may be classified by the third cohomology group H3(ZN,U(1)){H}^{3}({\mathbb{Z}}_{N},U(1))H3(ZN​,U(1)) of ZN{\mathbb{Z}}_{N}ZN​. Our results apply also to any case where the fusion algebra is identified with a finite group algebra or a quantum group algebra.