Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1412.6589 · Journal of Physics A Mathematical and Theoretical

On enriching the Levin–Wen model with symmetry

Liang Chang, Meng Cheng, Shawn Xingshan Cui, Yuting Hu, Wei Jin, Ramis Movassagh, Pieter Naaijkens, Zhenghan Wang, Amanda Young

Published Mar 2, 2015• 26 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/48/12/12ft01

Related papers

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.

arXiv:2401.13838•Jan 24, 2024•1 citation

Levin-Wen is a Gauge Theory: Entanglement from Topology

Kyle Kawagoe, Corey Jones, Sean Sanford +2 more

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.

Communications in Mathematical Physics•Oct 12, 2024•5 citations

Internal Levin-Wen models

Vincentas Mulevičius, Ingo Runkel B, Thomas Voß

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 .

arXiv:2309.05755•SciPost Physics•Sep 24, 2024•3 citations

Axiomatization of the Levin--Wen Wave Function

Zhengwei Liu, Zishuo Zhao

The Levin--Wen model provides a lattice realization of topological orders associated with a given unitary fusion category. A longstanding open problem is to characterize Levin--Wen ground-state wave functions intrinsically, without assuming a priori categorical symmetry data or a Hamiltonian. We address this by proposing six axioms on a family of wave functions defined on lattices at multiple scales. These axioms allow us to reconstruct the underlying unitary fusion category and prove that the resulting wave functions map to nonzero Levin--Wen ground-state vectors of the emergent category.

math-phCross-listedarXiv:2608.21109•Aug 24, 2026•0 citations

Classification of topological phases with finite internal symmetries in all dimensions

Liang Kong, Tian Lan, Xiao-Gang Wen +2 more

A bstract We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.

arXiv:2003.08898•Journal of High Energy Physics•Sep 1, 2020•70 citations

Generalized string-net model for unitary fusion categories without tetrahedral symmetry

Alexander Hahn, Ramona Wolf

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 $F$-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.

arXiv:2004.07045•Physical review. B./Physical review. B•Sep 24, 2020•69 citations
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.
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 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.
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 .
The Levin--Wen model provides a lattice realization of topological orders associated with a given unitary fusion category. A longstanding open problem is to characterize Levin--Wen ground-state wave functions intrinsically, without assuming a priori categorical symmetry data or a Hamiltonian. We address this by proposing six axioms on a family of wave functions defined on lattices at multiple scales. These axioms allow us to reconstruct the underlying unitary fusion category and prove that the resulting wave functions map to nonzero Levin--Wen ground-state vectors of the emergent category.
A bstract We develop a mathematical theory of symmetry protected trivial (SPT) orders and anomaly-free symmetry enriched topological (SET) orders in all dimensions via two different approaches with an emphasis on the second approach. The first approach is to gauge the symmetry in the same dimension by adding topological excitations as it was done in the 2d case, in which the gauging process is mathematically described by the minimal modular extensions of unitary braided fusion 1-categories. This 2d result immediately generalizes to all dimensions except in 1d, which is treated with special care. The second approach is to use the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk relation. This approach also leads us to a precise mathematical description and a classification of SPT/SET orders in all dimensions. The equivalence of these two approaches, together with known physical results, provides us with many precise mathematical predictions.
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.