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arXiv:2503.15611 · Communications in Mathematical Physics

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

Alex Bols, Mahdie Hamdan, Pieter Naaijkens, Siddharth Vadnerkar

Published Dec 5, 2025• 1 citation
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Abstract

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.

DOI
10.1007/s00220-025-05492-2

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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.
This is the continuation of our study of the Levin-Wen model based on an arbitrary unitary fusion category C\mathcal{C}C on the infinite plane. The ground state of the Levin-Wen model hosts anyonic excitations whose fusion and braiding properties are captured by the associated braided C∗\rm C^*C∗-tensor category of superselection sectors SSS\mathsf{SSS}SSS. By constructing explicit isomorphisms between the fusion spaces of SSS\mathsf{SSS}SSS and those of the Drinfeld center Z(C)Z(\mathcal{C})Z(C), we show that these two categories have isomorphic FFF- and RRR-symbols. It follows that the full subcategory of finite sectors is unitarily braided monoidally equivalent to the Drinfeld center,  SSSf≃Z(C).\,\mathsf{SSS}_f \simeq Z(\mathcal{C}).SSSf​≃Z(C). This provides the first complete characterisation of the category of superselection sectors for a class of two-dimensional lattice models supporting anyons with non-integer quantum dimensions.
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.
Abstract We describe in a simple setting how to extract a braided tensor category from a collection of superselection sectors of a two-dimensional quantum spin system, corresponding to abelian anyons. We extract from this category its fusion ring as well as its F and R-symbols. We then construct the double semion state in infinite volume and extract the braided tensor category describing its semion, anti-semion, and bound state excitations. We verify that this category is equivalent to the representation category of the twisted quantum double Dϕ(Z2)\mathcal {D}^{\phi }(\mathbb {Z}_2)Dϕ(Z2​) D ϕ ( Z 2 ) .
We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute. As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.
We describe in a simple setting how to extract a unitary braided fusion category from a collection of superselection sectors of a two-dimensional quantum spin system, corresponding to abelian anyons. The structure of the unitary braided fusion category is given by F and R-symbols, which describe fusion and braiding of the anyons. We then construct the double semion state in infinite volume and extract the unitary braided fusion category describing its semion, anti-semion, and bound state excitations. We verify that this category corresponds to the representation category of the twisted quantum double of Z_2.