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arXiv:2601.05515

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

Siddharth Vadnerkar

Published Jan 9, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2601.05515

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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 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.
We provide a superselection theory of symmetry defects in 2+1D symmetry enriched topological (SET) order in the infinite volume setting. For a finite symmetry group GGG with a unitary on-site action, our formalism produces a GGG-crossed braided tensor category GSecG\mathsf{Sec}GSec. This superselection theory is a direct generalization of the usual superselection theory of anyons, and thus is consistent with this standard analysis in the trivially graded component GSec1G\mathsf{Sec}_1GSec1​. This framework also gives us a completely rigorous understanding of symmetry fractionalization. To demonstrate the utility of our formalism, we compute GSecG\mathsf{Sec}GSec explicitly in both short-range and long-range entangled spin systems with symmetry and recover the relevant skeletal data.
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.
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.
We investigate the algebraic theory of symmetry-enriched topological (SET) order in (2+1)D bosonic topological phases of matter and its applications to topological quantum computing. Our goal is twofold: first, to demonstrate how an abstract categorical approach can be applied to understand phenomena in (2+1)D topological phases of matter, and second, to show how ideas from physics can be useful for categorification.After reviewing modular tensor categories (MTCs) and their role as algebraic theories of anyons in topological phases of matter, we recall their associated quantum representations and their interpretation as quantum gates for a topological quantum computer. Next we recall the characterization of SET order in terms of GGG-crossed braided extensions of MTCs and the mathematical formalism of topological quantum computing (TQC) with anyons and symmetry defects.We then apply modular tensor category theory to construct algebraic models of symmetry defects in multi-layer (2+1)D topological order with layer-exchange permutation symmetry. Our main result frames a correspondence between bilayer symmetry enriched topological order and monolayer topological order on surfaces with genus, illuminating a connection between quantum symmetry and topological order that first appeared in the work of condensed matter theorists Barkeshi, Ji, and Qian.
We describe the mathematical theory of topological quantum computing with symmetry defects in the language of fusion categories and unitary representations. Symmetry defects together with anyons are modeled by G-crossed braided extensions of unitary modular tensor categories. The algebraic data of these categories afford projective unitary representations of the braid group. Elements in the image of such representations correspond to quantum gates arising from exchanging anyons and symmetry defects in topological phases of matter with symmetry. We provide some small examples that highlight features of practical interest for quantum computing. In particular, symmetry defects show the potential to generate non-abelian statistics from abelian topological phases and to be used as as a tool to enlarge the set of quantum gates accessible to an anyonic device.