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Annales Henri Poincaré

The Double Semion State in Infinite Volume

Alex Bols, Boris Kjær, Alvin Moon

Published Jul 2, 2024• 2 citations
Publisher

Abstract

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 $$\mathcal {D}^{\phi }(\mathbb {Z}_2)$$ D ϕ ( Z 2 ) .

DOI
10.1007/s00023-024-01445-y

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arXiv:2601.05515•Jan 9, 2026•0 citations
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 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.
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.
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 dissertation discusses some properties of topologically ordered states as they appear in the setting of infinite quantum spin systems. We begin by studying the set of infinite volume ground states for Kitaev's abelian quantum double models. We show that states describing a single excitation in the bulk are infinite volume ground states, that is, local perturbations cannot remove the charge. The single excitations states, which are inequivalent for distinct charges, give a complete characterization of the sector theory for the set of ground states. Furthermore, any pure ground state is equivalent to some single excitation ground state. We proceed to study the stability of charges that are classified by certain representations of the algebra of observables. We introduce a new superselection criterion selecting almost localized and transportable ∗*∗-endomorphisms with respect to a vacuum state. Equivalence classes of representations satisfying the superselection criterion form different charged superselection sectors of the system. We show that if the vacuum state satisfies certain locality conditions then the superselection structure will be a braided tensor C∗C^*C∗-category. Further, this superselection structure is stable up to deformations by a quasi-local dynamics. This result is then applied to show that the anyon structure of the abelian quantum double models is stable under local perturbations.
The article arXiv:2410.21454 constructed a braided W∗\mathrm{W^*}W∗-tensor category of superselection sectors associated to a net of von Neumann algebras on a suitably geometric poset using localized and transportable endomorphisms. In this article, we construct an equivalent tensor product and braiding using the Connes fusion tensor product without localizing, using ideas from arXiv:1812.04470 for conformal nets. As an application, we prove that the category of superselection sectors associated to the 2D braided quantum spin system constructed from a unitary braided fusion category B\mathcal{B}B in arXiv:2506.19969 is equivalent to Hilb(B)op\mathsf{Hilb}(\mathcal{B})^{\rm op}Hilb(B)op, the opposite of the unitary ind-completion of B\mathcal{B}B, as a braided W∗\mathrm{W^*}W∗-tensor category.
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.