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:2303.07291

Anyon condensation in the string-net models

Chien‐Hung Lin, F. J. Burnell

Published Mar 13, 2023• 0 citations
arXivHTMLPDF

Abstract

We study condensation of abelian bosons in string-net models, by constructing a family of Hamiltonians that can be tuned through any such transition. We show that these Hamiltonians admit two exactly solvable, string-net limits: one deep in the uncondensed phase, described by an initial, uncondensed string net Hamiltonian, and one deep in the condensed phase, described by a final, condensed string net model. We give a systematic description of the condensed string net model in terms of the uncondensed string net and the data associated with the condensing abelian bosons. Specifically, if the uncondensed string net is described by a fusion category $\mathcal{C}$, we show how the string labels and fusion data of the fusion category $\mathcal{\tilde{C}}$ describing the condensed string net can be obtained from that of $\mathcal{C}$ and the data describing the string oeprators that create the condensing boson. This construction generalizes previous approaches to anyon condensation in string nets, by allowing the condensation of arbitrary abelian bosons, including chiral bosons in string nets constructed from (for example) Chern-Simons theories, which describe time-reversal invariant bilayer states. This gives a method for obtaining the full data for string nets without explicit time-reversal symmetry from such bilayer models. We illustrate our approach with several examples.

DOI
10.48550/arxiv.2303.07291

Related papers

Anyon condensation in string-net models

Chien‐Hung Lin, F. J. Burnell

We study the condensation of Abelian bosons in string-net models by constructing a family of Hamiltonians that can be tuned through any such transition. We show that these Hamiltonians admit two exactly solvable, string-net limits: one deep in the uncondensed phase, described by an initial, uncondensed string-net Hamiltonian, and one deep in the condensed phase, described by a final, condensed string-net model. We give a systematic description of the condensed string-net model in terms of the uncondensed string-net and the data associated with the condensing Abelian bosons. Specifically, if the uncondensed string-net is described by a fusion category $\mathcal{C}$, we show how the string labels and fusion data of the fusion category $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{C}}$ describing the condensed string-net can be obtained from that of $\mathcal{C}$ and the data describing the string operators that create the condensing boson. This construction generalizes previous approaches to anyon condensation in string-nets by allowing the condensation of arbitrary Abelian bosons, including chiral bosons in string-nets constructed from, for example, Chern-Simons theories, which describe time-reversal invariant bilayer states. This gives a method for obtaining the full data for string-nets without explicit time-reversal symmetry from such bilayer models. We illustrate our approach with several examples.

Physical review. B./Physical review. B•Sep 13, 2024•6 citations

Nonabelian anyon condensation in 2+1d topological orders: A string-net model realization

Yu Zhao, Yidun Wan

A bstract We develop a comprehensive framework for realizing anyon condensation of topological orders within the string-net model by constructing a Hamiltonian that bridges the parent string-net model before and the child string-net model after anyon condensation. Our approach classifies all possible types of bosonic anyon condensation in any parent string-net model and identifies the basic degrees of freedom in the corresponding child models. The Kogut-Susskind lattice gauge theory model is a special case of our model if the full degrees of freedom of the model are truncated from SU(2) representations to quantum group SU(2) k . Compared with the traditional UMTC perspective of topological orders, our method offers a finer categorical description of anyon condensation at the microscopic level. We also explicitly represent relevant UMTC categorical entities characterizing anyon condensation through our model-based physical quantities, providing practical algorithms for calculating these categorical data.

arXiv:2409.05852•Journal of High Energy Physics•May 20, 2025•2 citations

Generalizations and limitations of string-net models

Chien‐Hung Lin, Michael Levin

We ask which topological phases can and can not be realized by exactly soluble string-net models. We answer this question for the simplest class of topological phases, namely, those with Abelian braiding statistics. Specifically, we find that an Abelian topological phase can be realized by a string-net model if and only if (i) it has a vanishing thermal Hall conductance and (ii) it has at least one Lagrangian subgroup, a subset of quasiparticles with particular topological properties. Equivalently, we find that an Abelian topological phase is realizable if and only if it supports a gapped edge. We conjecture that the latter criterion generalizes to the non-Abelian case. We establish these results by systematically constructing all possible Abelian string-net models and analyzing the quasiparticle braiding statistics in each model. We show that the low-energy effective field theories for these models are multicomponent $\text{U}(1)$ Chern-Simons theories, and we derive the $K$-matrix description of each model. An additional feature of this work is that the models we construct are more general than the original string-net models, due to several new ingredients. First, we introduce two new objects $\ensuremath{\gamma},\ensuremath{\alpha}$ into the construction which are related to ${\mathbb{Z}}_{2}$ and ${\mathbb{Z}}_{3}$ Frobenius-Schur indicators. Second, we do not assume parity invariance. As a result, we can realize topological phases that were not accessible to the original construction, including phases that break time-reversal and parity symmetry.

arXiv:1402.4081•Physical Review B•May 21, 2014•118 citations

Anyon Condensation In Symmetry-Enriched Topological Phases: $G$-Grading of Multifusion Categories

Nianrui Fu, Siyuan Wang, Yu Zhao +1 more

Although anyon condensation is a standard mechanism for relating topological orders, anyon condensation in symmetry-enriched topological (SET) phases is more intricate because the condensate must also be compatible with the global symmetry. We study symmetry-preserving anyon condensation in SET phases described by the enlarged Hu-Geer-Wu (HGW) string-net model with multifusion-category input data. We show that a $G$-preserving condensation is characterized by a compatible grading of the input multifusion category, and that this grading constructs the multifusion-category input of the child SET phase. To make this construction concrete, we consider the case where the relevant data come from a finite group extension $E$ of $G$ by $N$: an $E$-graded fusion category induces a $G$-SET input by passing to the quotient symmetry $G$, and the resulting input naturally carries a compatible $N$-grading that implements the further condensation inside the SET phase while preserving $G$. We illustrate the construction using three quantum-double examples: the trivial extension $\mathbb{Z}_2\times\mathbb{Z}_2$, the non-Abelian semidirect product $S_3$, and the nontrivial central extension $\mathbb{Z}_4$. The $\mathbb{Z}_4$ example further shows that symmetry fractionalized anyons do not obstruct symmetry-preserving condensation once the condensate is treated as a physical coherent state.

arXiv:2607.24740•Jul 27, 2026•0 citations

Generalized string-net models: A thorough exposition

Chien‐Hung Lin, Michael Levin, F. J. Burnell

We describe how to construct generalized string-net models, a class of exactly solvable lattice models that realize a large family of two-dimensional topologically ordered phases of matter. The ground states of these models can be thought of as superpositions of different ``string-net configurations,'' where each string-net configuration is a trivalent graph with labeled edges, drawn in the $xy$ plane. What makes this construction more general than the original string-net construction is that, unlike the original construction, tetrahedral reflection symmetry is not assumed, nor is it assumed that the ground-state wave function $\mathrm{\ensuremath{\Phi}}$ is ``isotropic'': i.e., in the generalized setup, two string-net configurations ${X}_{1},{X}_{2}$ that can be continuously deformed into one another can have different ground-state amplitudes $\mathrm{\ensuremath{\Phi}}({X}_{1})\ensuremath{\ne}\mathrm{\ensuremath{\Phi}}({X}_{2})$. As a result, generalized string-net models can realize topological phases that are inaccessible to the original construction. In this paper, we provide a more detailed discussion of ground-state wave functions, Hamiltonians, and minimal self-consistency conditions for generalized string-net models than what exists in the previous literature. We also show how to construct string operators that create anyon excitations in these models, and we show how to compute the braiding statistics of these excitations. Finally, we derive necessary and sufficient conditions for generalized string-net models to have isotropic ground-state wave functions on the plane or the sphere, a property that may be useful in some applications.

arXiv:2012.14424•Physical review. B./Physical review. B•May 26, 2021•93 citations

A lattice model for condensation in Levin-Wen systems

Jessica Christian, David Green, Peter Huston +1 more

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.

arXiv:2303.04711•Journal of High Energy Physics•Sep 11, 2023•13 citations
We study condensation of abelian bosons in string-net models, by constructing a family of Hamiltonians that can be tuned through any such transition. We show that these Hamiltonians admit two exactly solvable, string-net limits: one deep in the uncondensed phase, described by an initial, uncondensed string net Hamiltonian, and one deep in the condensed phase, described by a final, condensed string net model. We give a systematic description of the condensed string net model in terms of the uncondensed string net and the data associated with the condensing abelian bosons. Specifically, if the uncondensed string net is described by a fusion category C\mathcal{C}C, we show how the string labels and fusion data of the fusion category C~\mathcal{\tilde{C}}C~ describing the condensed string net can be obtained from that of C\mathcal{C}C and the data describing the string oeprators that create the condensing boson. This construction generalizes previous approaches to anyon condensation in string nets, by allowing the condensation of arbitrary abelian bosons, including chiral bosons in string nets constructed from (for example) Chern-Simons theories, which describe time-reversal invariant bilayer states. This gives a method for obtaining the full data for string nets without explicit time-reversal symmetry from such bilayer models. We illustrate our approach with several examples.
We study the condensation of Abelian bosons in string-net models by constructing a family of Hamiltonians that can be tuned through any such transition. We show that these Hamiltonians admit two exactly solvable, string-net limits: one deep in the uncondensed phase, described by an initial, uncondensed string-net Hamiltonian, and one deep in the condensed phase, described by a final, condensed string-net model. We give a systematic description of the condensed string-net model in terms of the uncondensed string-net and the data associated with the condensing Abelian bosons. Specifically, if the uncondensed string-net is described by a fusion category C\mathcal{C}C, we show how the string labels and fusion data of the fusion category $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathcal{C}}$ describing the condensed string-net can be obtained from that of C\mathcal{C}C and the data describing the string operators that create the condensing boson. This construction generalizes previous approaches to anyon condensation in string-nets by allowing the condensation of arbitrary Abelian bosons, including chiral bosons in string-nets constructed from, for example, Chern-Simons theories, which describe time-reversal invariant bilayer states. This gives a method for obtaining the full data for string-nets without explicit time-reversal symmetry from such bilayer models. We illustrate our approach with several examples.
A bstract We develop a comprehensive framework for realizing anyon condensation of topological orders within the string-net model by constructing a Hamiltonian that bridges the parent string-net model before and the child string-net model after anyon condensation. Our approach classifies all possible types of bosonic anyon condensation in any parent string-net model and identifies the basic degrees of freedom in the corresponding child models. The Kogut-Susskind lattice gauge theory model is a special case of our model if the full degrees of freedom of the model are truncated from SU(2) representations to quantum group SU(2) k . Compared with the traditional UMTC perspective of topological orders, our method offers a finer categorical description of anyon condensation at the microscopic level. We also explicitly represent relevant UMTC categorical entities characterizing anyon condensation through our model-based physical quantities, providing practical algorithms for calculating these categorical data.
We ask which topological phases can and can not be realized by exactly soluble string-net models. We answer this question for the simplest class of topological phases, namely, those with Abelian braiding statistics. Specifically, we find that an Abelian topological phase can be realized by a string-net model if and only if (i) it has a vanishing thermal Hall conductance and (ii) it has at least one Lagrangian subgroup, a subset of quasiparticles with particular topological properties. Equivalently, we find that an Abelian topological phase is realizable if and only if it supports a gapped edge. We conjecture that the latter criterion generalizes to the non-Abelian case. We establish these results by systematically constructing all possible Abelian string-net models and analyzing the quasiparticle braiding statistics in each model. We show that the low-energy effective field theories for these models are multicomponent U(1)\text{U}(1)U(1) Chern-Simons theories, and we derive the KKK-matrix description of each model. An additional feature of this work is that the models we construct are more general than the original string-net models, due to several new ingredients. First, we introduce two new objects $\ensuremath{\gamma},\ensuremath{\alpha}$ into the construction which are related to Z2{\mathbb{Z}}_{2}Z2​ and Z3{\mathbb{Z}}_{3}Z3​ Frobenius-Schur indicators. Second, we do not assume parity invariance. As a result, we can realize topological phases that were not accessible to the original construction, including phases that break time-reversal and parity symmetry.
Anyon Condensation In Symmetry-Enriched Topological Phases: GGG-Grading of Multifusion Categories
Although anyon condensation is a standard mechanism for relating topological orders, anyon condensation in symmetry-enriched topological (SET) phases is more intricate because the condensate must also be compatible with the global symmetry. We study symmetry-preserving anyon condensation in SET phases described by the enlarged Hu-Geer-Wu (HGW) string-net model with multifusion-category input data. We show that a GGG-preserving condensation is characterized by a compatible grading of the input multifusion category, and that this grading constructs the multifusion-category input of the child SET phase. To make this construction concrete, we consider the case where the relevant data come from a finite group extension EEE of GGG by NNN: an EEE-graded fusion category induces a GGG-SET input by passing to the quotient symmetry GGG, and the resulting input naturally carries a compatible NNN-grading that implements the further condensation inside the SET phase while preserving GGG. We illustrate the construction using three quantum-double examples: the trivial extension Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​, the non-Abelian semidirect product S3S_3S3​, and the nontrivial central extension Z4\mathbb{Z}_4Z4​. The Z4\mathbb{Z}_4Z4​ example further shows that symmetry fractionalized anyons do not obstruct symmetry-preserving condensation once the condensate is treated as a physical coherent state.
We describe how to construct generalized string-net models, a class of exactly solvable lattice models that realize a large family of two-dimensional topologically ordered phases of matter. The ground states of these models can be thought of as superpositions of different ``string-net configurations,'' where each string-net configuration is a trivalent graph with labeled edges, drawn in the xyxyxy plane. What makes this construction more general than the original string-net construction is that, unlike the original construction, tetrahedral reflection symmetry is not assumed, nor is it assumed that the ground-state wave function $\mathrm{\ensuremath{\Phi}}$ is ``isotropic'': i.e., in the generalized setup, two string-net configurations X1,X2{X}_{1},{X}_{2}X1​,X2​ that can be continuously deformed into one another can have different ground-state amplitudes $\mathrm{\ensuremath{\Phi}}({X}_{1})\ensuremath{\ne}\mathrm{\ensuremath{\Phi}}({X}_{2})$. As a result, generalized string-net models can realize topological phases that are inaccessible to the original construction. In this paper, we provide a more detailed discussion of ground-state wave functions, Hamiltonians, and minimal self-consistency conditions for generalized string-net models than what exists in the previous literature. We also show how to construct string operators that create anyon excitations in these models, and we show how to compute the braiding statistics of these excitations. Finally, we derive necessary and sufficient conditions for generalized string-net models to have isotropic ground-state wave functions on the plane or the sphere, a property that may be useful in some applications.
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.