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

Anyons in infinite quantum systems: QFT in d=2+1 and the Toric Code

Pieter Naaijkens

Published Oct 1, 2007• 7 citations
OpenAlex

Abstract

Anyons, comprising a special type of particles that exhibit braid statistics,\nhave received considerable attention in the past decade, from physicists,\nmathematicians, and computer scientists. It turns out that so-called modular\ntensor categories naturally appear in the description of the key properties of\nanyons, which partly explains the interest of mathematicians in the subject.\nIn this thesis, we study the way such modular tensor categories emerge in two\nclasses of models, viz. quantum field theory in low-dimensional Minkowski\nspace-time, and quantum spin systems on an infinite spatial lattice. Although\nthe nature of these two classes of models is quite different, we show in this\nthesis that they may both be discussed in a similar (mathematically rigorous)\nframework. This approach is inspired by the work of Dpplicher, Haag and\nRoberts (DHR) in algebraic quantum field theory.\nThe DHR approach leads to braided tensor categories, which describe the charges\nin the theory (in our case, these are anyons). These categories need not be\nmodular. In this thesis we present a method to remove an obstruction for\nmodularity, in the case of a space-time of dimension three. We then study (a\nvariant of) Kitaev's toric code, regarded as a quantum spin model on an\ninfinite lattice. Using techniques inspired by the DHR programme, we show that\none can obtain a modular tensor category describing all relevant properties of\nthe anyons in this system. Finally, some generalisations to non-abelian models\nare discussed.

Related papers

The Category of Anyon Sectors for Non-Abelian Quantum Double Models

Alex Bols, Mahdie Hamdan, Pieter Naaijkens +1 more

Abstract We study Kitaev’s quantum double model for arbitrary finite gauge group in infinite volume, using an operator-algebraic approach. The quantum double model hosts anyonic excitations which can be identified with equivalence classes of ‘localized and transportable endomorphisms’, which produce anyonic excitations from the ground state. Following the Doplicher–Haag–Roberts (DHR) sector theory from AQFT, we organize these endomorphisms into a braided monoidal category capturing the fusion and braiding properties of the anyons. We show that this category is equivalent to $$\textbf{Rep}_f {{\mathcal {D}}}(G)$$ Rep f D ( G ) , the representation category of the quantum double of G . This establishes for the first time the full DHR structure for a class of 2d quantum lattice models with non-abelian anyons.

arXiv:2503.15611•Communications in Mathematical Physics•Dec 5, 2025•1 citation

An Operator-Algebraic Framework for Anyons and Defects in Quantum Spin Systems

Siddharth Vadnerkar

In this dissertation, we detail an operator algebraic approach to studying topological order in the infinite volume setting. We give a thorough and self-contained review of the DHR-style approach on quantum spin systems, which builds a category $\mathrm{\textbf{DHR}}$ of anyon sectors starting from microscopic lattice spin systems. In general, this category has the structure of a braided $\mathrm{C}^*$-tensor category. We will verify in full detail that $\mathrm{\textbf{DHR}}$ is the expected category in Kitaev's Quantum Double model, a paradigmatic model for studying topological order on the lattice. We will then extend the DHR-style analysis to systems in the presence of a global on-site symmetry, and introduce a category of symmetry defects, $G\mathsf{Sec}$, and show that it has the structure of a $G$-crossed braided $\mathrm{C}^*$-tensor category.

arXiv:2601.05515•Jan 9, 2026•0 citations

Anyon condensation and tensor categories

Liang Kong

Instead of studying anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relation between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. A more general situation is also studied in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin–Wen types of lattice models and some chiral topological phases.

arXiv:1307.8244•Nuclear Physics B•Jul 9, 2014•299 citations

A mathematical theory of anyon condensation

Liang Kong

Instead of constructing anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relations between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. More general situation is also discussed in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin-Wen types of lattice models and some chiral topological phases.

•Jul 31, 2013•4 citations

Witness algebra and anyon braiding

Andreas Blass, Yuri Gurevich

Abstract Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of the present framework necessary? The computations of associativity and braiding matrices can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.

arXiv:1807.10414•Mathematical Structures in Computer Science•Mar 1, 2020•2 citations

In and around abelian anyon models *

Liang Wang, Zhenghan Wang

Abstract Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.

arXiv:2004.12048•Journal of Physics A Mathematical and Theoretical•Nov 2, 2020•69 citations
Anyons, comprising a special type of particles that exhibit braid statistics,\nhave received considerable attention in the past decade, from physicists,\nmathematicians, and computer scientists. It turns out that so-called modular\ntensor categories naturally appear in the description of the key properties of\nanyons, which partly explains the interest of mathematicians in the subject.\nIn this thesis, we study the way such modular tensor categories emerge in two\nclasses of models, viz. quantum field theory in low-dimensional Minkowski\nspace-time, and quantum spin systems on an infinite spatial lattice. Although\nthe nature of these two classes of models is quite different, we show in this\nthesis that they may both be discussed in a similar (mathematically rigorous)\nframework. This approach is inspired by the work of Dpplicher, Haag and\nRoberts (DHR) in algebraic quantum field theory.\nThe DHR approach leads to braided tensor categories, which describe the charges\nin the theory (in our case, these are anyons). These categories need not be\nmodular. In this thesis we present a method to remove an obstruction for\nmodularity, in the case of a space-time of dimension three. We then study (a\nvariant of) Kitaev's toric code, regarded as a quantum spin model on an\ninfinite lattice. Using techniques inspired by the DHR programme, we show that\none can obtain a modular tensor category describing all relevant properties of\nthe anyons in this system. Finally, some generalisations to non-abelian models\nare discussed.
Abstract We study Kitaev’s quantum double model for arbitrary finite gauge group in infinite volume, using an operator-algebraic approach. The quantum double model hosts anyonic excitations which can be identified with equivalence classes of ‘localized and transportable endomorphisms’, which produce anyonic excitations from the ground state. Following the Doplicher–Haag–Roberts (DHR) sector theory from AQFT, we organize these endomorphisms into a braided monoidal category capturing the fusion and braiding properties of the anyons. We show that this category is equivalent to RepfD(G)\textbf{Rep}_f {{\mathcal {D}}}(G)Repf​D(G) Rep f D ( G ) , the representation category of the quantum double of G . This establishes for the first time the full DHR structure for a class of 2d quantum lattice models with non-abelian anyons.
In this dissertation, we detail an operator algebraic approach to studying topological order in the infinite volume setting. We give a thorough and self-contained review of the DHR-style approach on quantum spin systems, which builds a category DHR\mathrm{\textbf{DHR}}DHR of anyon sectors starting from microscopic lattice spin systems. In general, this category has the structure of a braided C∗\mathrm{C}^*C∗-tensor category. We will verify in full detail that DHR\mathrm{\textbf{DHR}}DHR is the expected category in Kitaev's Quantum Double model, a paradigmatic model for studying topological order on the lattice. We will then extend the DHR-style analysis to systems in the presence of a global on-site symmetry, and introduce a category of symmetry defects, GSecG\mathsf{Sec}GSec, and show that it has the structure of a GGG-crossed braided C∗\mathrm{C}^*C∗-tensor category.
Instead of studying anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relation between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. A more general situation is also studied in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin–Wen types of lattice models and some chiral topological phases.
Instead of constructing anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relations between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) A in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of A-modules in C. More general situation is also discussed in this paper. We will also show how to determine such algebra A from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin-Wen types of lattice models and some chiral topological phases.
Abstract Topological quantum computation employs two-dimensional quasiparticles called anyons. The generally accepted mathematical basis for the theory of anyons is the framework of modular tensor categories. That framework involves a substantial amount of category theory and is, as a result, considered rather difficult to understand. Is the complexity of the present framework necessary? The computations of associativity and braiding matrices can be based on a much simpler framework, which looks less like category theory and more like familiar algebra. We introduce that framework here.
Abstract Anyon models are algebraic structures that model universal topological properties in topological phases of matter and can be regarded as mathematical characterization of topological order in two spacial dimensions. It is conjectured that every anyon model, or mathematically unitary modular tensor category, can be realized as the representation category of some chiral conformal field theory, or mathematically vertex operator algebra/local conformal net. This conjecture is known to be true for abelian anyon models providing support for the conjecture. We reexamine abelian anyon models from several different angles. First anyon models are algebraic data for both topological quantum field theories and chiral conformal field theories. While it is known that each abelian anyon model can be realized by a quantum abelian Chern–Simons (CS) theory and chiral conformal field theory, the construction is not algorithmic. Our goal is to provide such an explicit algorithm for a K -matrix in CS theory and a positive definite even one for a lattice conformal field theory. Secondly anyon models and chiral conformal field theories underlie the bulk-edge correspondence for topological phases of matter. But there are interesting subtleties in this correspondence when stability of the edge theory and topological symmetry are taken into consideration. Therefore, our focus is on the algorithmic reconstruction of extremal chiral conformal field theories with small central charges. Finally we conjecture that a much stronger reconstruction holds for abelian anyon models: every abelian anyon model can be realized as the representation category of some non-lattice extremal vertex operator algebra generalizing the moonshine realization of the trivial anyon model.