Levin–Wen String-Net

Steven H. Simon

Published Sep 29, 2023• 0 citations

Abstract

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.

DOI: 10.1093/oso/9780198886723.003.0033

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Levin–Wen String-Net

Steven H. Simon

Published Sep 29, 2023• 0 citations

Publisher

Abstract

DOI: 10.1093/oso/9780198886723.003.0033

Related papers

A lattice model for condensation in Levin-Wen systems

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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.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.

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Levin–Wen models are a class of two-dimensional lattice spin models with a Hamiltonian that is a sum of commuting projectors, which describe topological phases of matter related to Drinfeld centres. We generalise this construction to lattice systems internal to a topological phase described by an arbitrary modular fusion category \mathcal{C} 𝒞 . The lattice system is defined in terms of an orbifold datum \mathbb{A} 𝔸 in \mathcal{C} 𝒞 , from which we construct a state space and a commuting-projector Hamiltonian H_{\mathbb{A}} H 𝔸 acting on it. The topological phase of the degenerate ground states of H_{\mathbb{A}} H 𝔸 is characterised by a modular fusion category \mathcal{C}_\mathbb{A} 𝒞 𝔸 defined directly in terms of \mathbb{A} 𝔸 . By choosing different \mathbb{A} 𝔸 ’s for a fixed \mathcal{C} 𝒞 , one obtains precisely all phases which are Witt-equivalent to \mathcal{C} 𝒞 . As special cases we recover the Kitaev and the Levin–Wen lattice models from instances of orbifold data in the trivial modular fusion category of vector spaces, as well as phases obtained by anyon condensation in a given phase \mathcal{C} 𝒞 .Levin–Wen models are a class of two-dimensional lattice spin models with a Hamiltonian that is a sum of commuting projectors, which describe topological phases of matter related to Drinfeld centres. We generalise this construction to lattice systems internal to a topological phase described by an arbitrary modular fusion category \mathcal{C} 𝒞 . The lattice system is defined in terms of an orbifold datum \mathbb{A} 𝔸 in \mathcal{C} 𝒞 , from which we construct a state space and a commuting-projector Hamiltonian H_{\mathbb{A}} H 𝔸 acting on it. The topological phase of the degenerate ground states of H_{\mathbb{A}} H 𝔸 is characterised by a modular fusion category \mathcal{C}_\mathbb{A} 𝒞 𝔸 defined directly in terms of \mathbb{A} 𝔸 . By choosing different \mathbb{A} 𝔸 ’s for a fixed \mathcal{C} 𝒞 , one obtains precisely all phases which are Witt-equivalent to \mathcal{C} 𝒞 . As special cases we recover the Kitaev and the Levin–Wen lattice models from instances of orbifold data in the trivial modular fusion category of vector spaces, as well as phases obtained by anyon condensation in a given phase \mathcal{C} 𝒞 .

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Kitaev Quantum Double Model

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Abstract One generalization of the toric code is the so-called Kitaev Quantum Double model which realizes the Drinfeld double of a group. In this model edges of a lattice are directed and labeled with elements of a group G which then give us a planar diagram algebra. We consider mainly the “untwisted” case where the F-matrices (or equivalently 3-cocyles) are trivial. This generalizes the toric code where the group is Z2. Again we have vertex and plaquette operators, and a code space which satisfies both. As with the toric code, the ground state is topological although here (depending on the group G) the quasiparticles may be nonabelian. We turn to a continuum model for a more detailed analysis of the ground state space and the quasiparticles using the tube algebra, which, using some amount of group theory, is easily tractable. We discuss the so-called quasiparticle “ribbon operators” which generalize the error strings of the toric code. We relate the Kitaev quantum double model to lattice gauge theory and briefly discuss its generalization to higher dimensions.Abstract One generalization of the toric code is the so-called Kitaev Quantum Double model which realizes the Drinfeld double of a group. In this model edges of a lattice are directed and labeled with elements of a group G which then give us a planar diagram algebra. We consider mainly the “untwisted” case where the F-matrices (or equivalently 3-cocyles) are trivial. This generalizes the toric code where the group is Z2. Again we have vertex and plaquette operators, and a code space which satisfies both. As with the toric code, the ground state is topological although here (depending on the group G) the quasiparticles may be nonabelian. We turn to a continuum model for a more detailed analysis of the ground state space and the quasiparticles using the tube algebra, which, using some amount of group theory, is easily tractable. We discuss the so-called quasiparticle “ribbon operators” which generalize the error strings of the toric code. We relate the Kitaev quantum double model to lattice gauge theory and briefly discuss its generalization to higher dimensions.

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Levin-Wen is a gauge theory: entanglement from topology

Kyle Kawagoe, Corey Jones, Sean Sanford +2 more

We show that the Levin-Wen model of a unitary fusion category $\mathcal{C}$ is a gauge theory with gauge symmetry given by the tube algebra $\operatorname{Tube}(\mathcal{C})$ . In particular, we define a model corresponding to a $\operatorname{Tube}(\mathcal{C})$ symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case $\mathcal{C}=\mathsf{Hilb}(G,ω)$ , we show how our procedure reduces to the twisted gauging of a trivial $G$ -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.We show that the Levin-Wen model of a unitary fusion category $\mathcal{C}$ is a gauge theory with gauge symmetry given by the tube algebra $\operatorname{Tube}(\mathcal{C})$. In particular, we define a model corresponding to a $\operatorname{Tube}(\mathcal{C})$ symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case $\mathcal{C}=\mathsf{Hilb}(G,ω)$, we show how our procedure reduces to the twisted gauging of a trivial $G$-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.

arXiv:2401.13838Jan 24, 20241 citation

Sector Theory of Levin-Wen Models I : Classification of Anyon Sectors

Alex Bols, Boris Kjær

We classify the irreducible anyon sectors of Levin-Wen models over an arbitrary unitary fusion category $\mathcal{C}$ , showing that they are in one-to-one correspondence with equivalence classes of simple objects of the Drinfeld center $Z(\mathcal{C})$ . We achieve this by making explicit how the Levin-Wen Hamiltonian stabilizes subspaces isomorphic to state spaces of the corresponding Turaev-Viro TQFT, and developing a detailed understanding of these state spaces on punctured disks. In particular, we construct Drinfeld insertion operators on such spaces which can move anyons between the punctures, and can change their fusion channels. Using these Drinfeld insertions, we construct explicit string operators that excite anyons above the ground state. The fusion and braiding properties of these anyons will be analysed in a companion paper.We classify the irreducible anyon sectors of Levin-Wen models over an arbitrary unitary fusion category $\mathcal{C}$, showing that they are in one-to-one correspondence with equivalence classes of simple objects of the Drinfeld center $Z(\mathcal{C})$. We achieve this by making explicit how the Levin-Wen Hamiltonian stabilizes subspaces isomorphic to state spaces of the corresponding Turaev-Viro TQFT, and developing a detailed understanding of these state spaces on punctured disks. In particular, we construct Drinfeld insertion operators on such spaces which can move anyons between the punctures, and can change their fusion channels. Using these Drinfeld insertions, we construct explicit string operators that excite anyons above the ground state. The fusion and braiding properties of these anyons will be analysed in a companion paper.

arXiv:2511.21521Nov 26, 20250 citations

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.

We show that the Levin-Wen model of a unitary fusion category

\mathcal{C}

is a gauge theory with gauge symmetry given by the tube algebra

\operatorname{Tube}(\mathcal{C})

. In particular, we define a model corresponding to a

\operatorname{Tube}(\mathcal{C})

symmetry protected topological phase, and we provide a gauging procedure which results in the corresponding Levin-Wen model. In the case

\mathcal{C}=\mathsf{Hilb}(G,ω)

, we show how our procedure reduces to the twisted gauging of a trivial

G

-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.

We classify the irreducible anyon sectors of Levin-Wen models over an arbitrary unitary fusion category

\mathcal{C}

, showing that they are in one-to-one correspondence with equivalence classes of simple objects of the Drinfeld center

Z(\mathcal{C})

. We achieve this by making explicit how the Levin-Wen Hamiltonian stabilizes subspaces isomorphic to state spaces of the corresponding Turaev-Viro TQFT, and developing a detailed understanding of these state spaces on punctured disks. In particular, we construct Drinfeld insertion operators on such spaces which can move anyons between the punctures, and can change their fusion channels. Using these Drinfeld insertions, we construct explicit string operators that excite anyons above the ground state. The fusion and braiding properties of these anyons will be analysed in a companion paper.