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The category of anyon sectors for non-abelian quantum double models

Alex Bols, Mahdie Hamdan, Pieter Naaijkens, Siddharth Vadnerkar

TL;DR

We analyze Kitaev's quantum double model on the plane for a finite group $G$ using the DHR framework and cone Haag duality. By constructing localized, transportable amplimorphisms from ribbon operators and linking them to endomorphisms of an auxiliary algebra, we explicitly realize the anyon sectors as a braided $C^*$-tensor category. We prove that the finite-amplitude sector categories $\mathbf{Amp}_f$ and $\mathbf{DHR}_f$ are braided monoidally equivalent to $\mathbf{Rep}_f \mathcal{D}(G)$, thereby giving a complete non-abelian anyon classification for the quantum double model. This establishes, in a lattice-model setting, the full DHR structure for non-abelian anyons and shows the resulting category is a unitary modular tensor category, with clear connections to ribbon operators and topological order in 2D.

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 $\mathrm{Rep}_f \mathcal{D}(G)$, 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.

The category of anyon sectors for non-abelian quantum double models

TL;DR

We analyze Kitaev's quantum double model on the plane for a finite group using the DHR framework and cone Haag duality. By constructing localized, transportable amplimorphisms from ribbon operators and linking them to endomorphisms of an auxiliary algebra, we explicitly realize the anyon sectors as a braided -tensor category. We prove that the finite-amplitude sector categories and are braided monoidally equivalent to , thereby giving a complete non-abelian anyon classification for the quantum double model. This establishes, in a lattice-model setting, the full DHR structure for non-abelian anyons and shows the resulting category is a unitary modular tensor category, with clear connections to ribbon operators and topological order in 2D.

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 , the representation category of the quantum double of . This establishes for the first time the full DHR structure for a class of 2d quantum lattice models with non-abelian anyons.

Paper Structure

This paper contains 41 sections, 33 theorems, 65 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Setup and main result
  3. The quantum double model and its ground state
  4. Cone algebras, Haag duality, and the allowed algebra
  5. Categories of amplimorphisms and endomorphisms
  6. Main result
  7. Braided -tensor structure of Amp and DHR
  8. C*-structure, direct sums, and subobjects
  9. Direct sums and subobjects of amplimorphisms
  10. Direct sums and subobjects in DHR
  11. Braided C*-tensor structure of Amp and DHR
  12. Monoidal structure
  13. Braiding
  14. Equivalence of Amp and DHR
  15. Reduction to unital amplimorphisms
  16. ...and 26 more sections

Key Result

Theorem 1

Let $\pi_0$ be the GNS representation of the frustration free ground state of the quantum double model for a finite group $G$ defined on the plane and assume that it satisfies Haag duality. Then the category of representations satisfying eq:sselect is braided monoidally equivalent to $\mathbf{Rep}_f

Figures (7)

  • Figure 1: Snapshot of $\Gamma^E$. The edges are all oriented toward the right.
  • Figure 2: An example of the braiding setup. The arrow represents the forbidden direction.
  • Figure 3: Braiding positive ribbon operators, both having the same starting site $s_0$.
  • Figure 4: The finite ribbon $\overline{\xi_m}$ is a bridge from ribbon $\rho_m'$ to $\rho_m$.
  • Figure 5: The finite ribbon $\xi_{L,n}$ is a bridge from $\rho$ to $\rho_L$, and $\xi_{R,m}$ is a bridge from $\rho$ to $\rho_R$.
  • ...and 2 more figures

Theorems & Definitions (44)

  • Theorem : Informal
  • Theorem 2.1
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • ...and 34 more