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

Siddharth Vadnerkar

TL;DR

The work develops an operator-algebraic framework to study topological order, focusing on anyons and symmetry defects in quantum spin systems. It builds the DHR-style category of anyon sectors on infinite lattices, proves its braided C*-tensor structure, and identifies it with the representation category of the quantum double Rep $D(G)$ in Kitaev’s quantum double model. It then extends the analysis to symmetry-enriched phases, introducing G-crossed braided C*-tensor categories for symmetry defects. A complete classification of Kitaev’s non-Abelian quantum double sectors is achieved by constructing explicit anyon states and demonstrating a one-to-one correspondence with irreps of $D(G)$, thereby unifying lattice models with the quantum-double framework. The results establish rigorous connections between operator algebras, tensor categories, and topological order, with potential implications for topological quantum computation and the study of SET phases in lattice systems.

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.

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

TL;DR

The work develops an operator-algebraic framework to study topological order, focusing on anyons and symmetry defects in quantum spin systems. It builds the DHR-style category of anyon sectors on infinite lattices, proves its braided C*-tensor structure, and identifies it with the representation category of the quantum double Rep in Kitaev’s quantum double model. It then extends the analysis to symmetry-enriched phases, introducing G-crossed braided C*-tensor categories for symmetry defects. A complete classification of Kitaev’s non-Abelian quantum double sectors is achieved by constructing explicit anyon states and demonstrating a one-to-one correspondence with irreps of , thereby unifying lattice models with the quantum-double framework. The results establish rigorous connections between operator algebras, tensor categories, and topological order, with potential implications for topological quantum computation and the study of SET phases in lattice systems.

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 of anyon sectors starting from microscopic lattice spin systems. In general, this category has the structure of a braided -tensor category. We will verify in full detail that 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, , and show that it has the structure of a -crossed braided -tensor category.
Paper Structure (215 sections, 260 theorems, 744 equations, 50 figures)

This paper contains 215 sections, 260 theorems, 744 equations, 50 figures.

Table of Contents

  1. History Of Topological Phases
  2. Discovery And Early Understanding
  3. Braided Tensor Categories And TQFTs
  4. Lattice Realizations Of Topological Order
  5. Operator Algebras And Topological Phases
  6. Continuum field theories
  7. Lattice systems
  8. Future
  9. The Physics Of Anyons
  10. Quantum lattice systems
  11. Anyons
  12. Strings and remote detectability
  13. Fusion
  14. Braiding
  15. Making anyons rigorous
  16. ...and 200 more sections

Key Result

Lemma 3.1.6

Let $F: {\mathcal{C}} \rightarrow {\mathcal{D}}$ be a functor for linear categories ${\mathcal{C}}, {\mathcal{D}}$. Then $F$ is linear if and only if it preserves direct sums, i.e., given $(X, \{\pi_i\}, \{\iota_i\})$ is a direct sum of $X_1,\cdots , X_n \in {\mathcal{C}}$, then $(F(X), \{F(\pi_i)\}

Figures (50)

  • Figure 1: Transporting localized excitations from a disk $R_1$ to a disk $R_2$ along a thin strip containing path $\gamma$.
  • Figure 2: The transport process from Figure \ref{['fig:particle transport']} viewed as a world-line in space-time.
  • Figure 3: For usual excitations, circling one around the other does not change the state at long distances.
  • Figure 4: One can freely deform a string (in red) as long as an anyon is not crossed. Local probes (drawn in green) supported away from anyons cannot see how the string is drawn.
  • Figure 5: Any loop around the hole must intersect the annulus, so loop processes can probe the enclosed charge.
  • ...and 45 more figures

Theorems & Definitions (443)

  • Definition 3.1.1
  • Definition 3.1.2
  • Definition 3.1.3
  • Definition 3.1.4
  • Definition 3.1.5
  • Lemma 3.1.6
  • Definition 3.1.7
  • Definition 3.1.8
  • Definition 3.1.9
  • Theorem 3.1.10
  • ...and 433 more