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arXiv:1207.6169 · Physical Review B

String-net models with ZN fusion algebra

Ling-Yan Hung, Yidun Wan

Published Dec 19, 2012• 36 citations
arXivHTMLPDF

Abstract

We study the Levin-Wen string-net model with a ${\mathbb{Z}}_{N}$ type fusion algebra. Solutions of the local constraints of this model correspond to ${Z}_{N}$ 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 ${\mathbb{Z}}_{N}$ gauge symmetry on a triangular lattice as an exact dual model of the string-net model with a ${\mathbb{Z}}_{N}$ type fusion algebra on a honeycomb lattice. This exact duality exists only when the spins are coupled to a ${\mathbb{Z}}_{N}$ gauge field living on the links of the triangular lattice. The ungauged ${\mathbb{Z}}_{N}$ lattice spin models are a class of quantum systems that bear symmetry-protected topological phases that may be classified by the third cohomology group ${H}^{3}({\mathbb{Z}}_{N},U(1))$ of ${\mathbb{Z}}_{N}$. Our results apply also to any case where the fusion algebra is identified with a finite group algebra or a quantum group algebra.

DOI
10.1103/physrevb.86.235132

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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.
We introduce a class of 2d2d2d quantum spin systems based on a unitary spherical fusion category with a chosen object. These models have frustration-free, reflection positive, but generally non-commuting local interactions, generalizing the Levin-Wen string-net models. Based on the theory developed in arXiv:2510.20662, we explicitly realize the Temperley-Lieb-Jones algebra, and more generally fusion spin chains as their boundary algebras.
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.
A bstract We present a systematic framework for constructing exactly-solvable lattice models of symmetry-enriched topological (SET) phases based on an enlarged version of the string-net model. We also gauge the global symmetries of our SET models to obtain string-net models of pure topological phases. Without invoking externally imposed onsite symmetry actions, our approach promotes the string-net model of a pure topological order, specified by an input unitary fusion category F \mathcal{F} F F , to an SET model, specified by a multifusion category together with a set of isomorphisms. Two complementary construction strategies are developed in the main text: (i) promotion via outer automorphisms of F \mathcal{F} F F and (ii) promotion via the Frobenius algebras of F \mathcal{F} F F . The global symmetries derived via these two strategies are intrinsic to topological phases and are thus termed blood symmetries, as opposed to adopted symmetries, which can be arbitrarily imposed on topological phases. We propose the concept of symmetry-gauging family of topological phases, which are related by gauging their blood symmetries. With our approach, we construct the first explicit lattice realization of a nonabelian-symmetry-enriched topological phase — the S 3 symmetry-enriched ℤ 2 × ℤ 2 quantum-double phase. The approach further reveals the role of local excitations in SET phases and establishes their symmetry constraints.
We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.
We construct exactly solvable models for a wide class of symmetry-enriched topological (SET) phases. Our construction applies to two-dimensional (2D) bosonic SET phases with finite unitary on-site symmetry group GGG and we conjecture that our models realize every phase in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: If we couple them to a dynamical lattice gauge field with gauge group GGG, the resulting gauge theories are equivalent to string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a Z2{\mathbb{Z}}_{2}Z2​ symmetry which exchanges the eee and mmm type anyons. We further illustrate our construction with a number of additional examples.