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

TL;DR

This paper develops an operator algebraic framework to analyze topological order, anyons, and defects in quantum spin systems, providing a rigorous categorical approach that extends to symmetric systems.

Contribution

It introduces a detailed operator algebraic method to construct and analyze categories of anyon sectors and symmetry defects in quantum spin systems.

Findings

Verified the DHR category matches Kitaev's Quantum Double model

Extended the framework to include symmetry defects with G-crossed braided categories

Provided a rigorous foundation for topological order analysis in infinite volume 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.