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Communications in Mathematical Physics

Levin-Wen is a Gauge Theory: Entanglement from Topology

Kyle Kawagoe, Corey Jones, Sean Sanford, David Green, David Penneys

Published Oct 12, 2024• 5 citations
Publisher

Abstract

Abstract We show that the Levin-Wen model of a unitary fusion category $${\mathcal {C}}$$ C is a gauge theory with gauge symmetry given by the tube algebra $${\text {Tube}}({\mathcal {C}})$$ Tube ( C ) . In particular, we define a model corresponding to a $${\text {Tube}}({\mathcal {C}})$$ Tube ( C ) symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case $${\mathcal {C}}=\textsf{Hilb}(G,\omega )$$ C = Hilb ( G , ω ) , we show how our procedure reduces to the twisted gauging of a trival G-SPT to produce the Twisted Quantum Double. We further provide an example which is outside the bounds of the current literature, the trivial Fibbonacci SPT, whose gauge theory results in the doubled Fibonacci string-net. Our formalism has a natural topological interpretation with string diagrams living on a punctured sphere. We provide diagrams to supplement our mathematical proofs and to give the reader an intuitive understanding of the subject matter.

DOI
10.1007/s00220-024-05144-x

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Abstract We show that the Levin-Wen model of a unitary fusion category C{\mathcal {C}}C C is a gauge theory with gauge symmetry given by the tube algebra Tube(C){\text {Tube}}({\mathcal {C}})Tube(C) Tube ( C ) . In particular, we define a model corresponding to a Tube(C){\text {Tube}}({\mathcal {C}})Tube(C) Tube ( C ) symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case C=Hilb(G,ω){\mathcal {C}}=\textsf{Hilb}(G,\omega )C=Hilb(G,ω) C = Hilb ( G , ω ) , we show how our procedure reduces to the twisted gauging of a trival G-SPT to produce the Twisted Quantum Double. We further provide an example which is outside the bounds of the current literature, the trivial Fibbonacci SPT, whose gauge theory results in the doubled Fibonacci string-net. Our formalism has a natural topological interpretation with string diagrams living on a punctured sphere. We provide diagrams to supplement our mathematical proofs and to give the reader an intuitive understanding of the subject matter.
We show that the Levin-Wen model of a unitary fusion category C\mathcal{C}C is a gauge theory with gauge symmetry given by the tube algebra Tube⁡(C)\operatorname{Tube}(\mathcal{C})Tube(C). In particular, we define a model corresponding to a Tube⁡(C)\operatorname{Tube}(\mathcal{C})Tube(C) symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case C=Hilb(G,ω)\mathcal{C}=\mathsf{Hilb}(G,ω)C=Hilb(G,ω), we show how our procedure reduces to the twisted gauging of a trivial GGG-SPT to produce the Twisted Quantum Double. We further provide an example which is outside the bounds of the current literature, the trivial Fibonacci SPT, whose gauge theory results in the doubled Fibonacci string-net. Our formalism has a natural topological interpretation with string diagrams living on a punctured sphere. We provide diagrams to supplement our mathematical proofs and to give the reader an intuitive understanding of the subject matter.
Abstract The Levin-Wen model is a general construction that uses an input planar diagram algebra to construct a Hamiltonian with a topologically ordered ground state (the Drinfeld double of the input category). The ground state (code) space consists of a sum over all diagrams weighted by the evaluation of the diagram. We work through the example of the doubled Fibonacci model. Here the input category is the Fibonacci fusion rules and F-matrices. We work with continuum diagrams and construct the full tube algebra of the model. We identify the quasiparticle types to include both left- and right-handed Fibonacci anyons. In the case that the input category admits a modular braiding, much of the analyis simplifies and the output is two copies of the input category of opposite chiralities (both left- and right-handed). We finally turn to the construction of the explicit Hamiltonian on the lattice, and construction of quasiparticle string (or ribbon) operators. We return to the reconsider the toric code and doubled semion models as simple examples.
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.
Symmetry protected (SPT) and symmetry enriched topological (SET) phases of matter are of great interest in condensed matter physics due to new materials such as topological insulators. The Levin–Wen (LW) model for spin/boson systems is an important rigorously solvable model for studying 2D topological phases. The input data for the LW model is a unitary fusion category, but the same model also works for unitary multi-fusion categories. In this paper, we provide the details for this extension of the LW model, and show that the extended LW model is a natural playground for the theoretical study of SPT and SET phases of matter.
Topological orders are a class of phases of matter that beyond the Landau symmetry breaking paradigm. The two (spatial) dimensional (2d) topological orders have been thoroughly studied. It is known that they can be fully classified by a unitary modular tensor category (UMTC) and a chiral central charge c. And a class of 2d topological orders whose boundary are gappable can be systematically constructed by Levin-Wen model whose ground states are string-net condensed states. Previously, the three spatial dimensional topological orders have been classified based on their canonical boundary described by some special unitary fusion 2-category, 2VecGω2\mathcal{V}ec_G^ω2VecGω​ or an EF 2-category. However, a lattice realization of a three spatial dimensional topological orders with both canonical boundary and arbitrary boundaries are still lacking. In this paper, we construct a 3d membrane-net model based on spherical fusion 2-category, which can be used to systematically study all general 3d topological order with gapped boundary. The partition function and lattice Hamiltonian of the membrane-net model is constructed based on state sum of spherical fusion 2-category. We also construct the 3d tube algebra of the membrane-net model to study excitations in the model. We conjecture all intrinsic excitations in membrane-net model have a one-to-one correspondence with the irreducible central idempotents (ICI) of the 3d tube algebra. We also provide a universal framework to study mutual statistics of all excitations in 3d topological order through 3d tube algebra. Our approach can be straightforwardly generalized to arbitrary dimension.
The Levin-Wen model of string-net condensation explains how topological phases emerge from the microscopic degrees of freedom of a physical system. However, the original construction is not applicable to all unitary fusion categories since some additional symmetries for the FFF-symbols are imposed. In particular, the so-called tetrahedral symmetry is not fulfilled by many interesting unitary fusion categories. In this paper, we present a generalized construction of the Levin-Wen model for arbitrary multiplicity-free unitary fusion categories that works without requiring these additional symmetries. We explicitly calculate the matrix elements of the Hamiltonian and, furthermore, show that it has the same properties as the original one.