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On a C*-Diagonal Generated by the Toric Code

Danilo Polo Ojito, Emil Prodan

TL;DR

The paper analyzes the abelian subalgebra generated by the toric code's star and face operators inside the AF-algebra ${\mathcal{A}} \cong M_{2^\infty}$, proving it forms a $C^\ast$-diagonal ${\mathcal{C}}$ with Cantor spectrum. It establishes regularity and the unique extension property via LTQO and a family of locally representable symmetries, leading to a canonical conditional expectation $E: {\mathcal{A}} \to {\mathcal{C}}$. Using the Weyl groupoid framework and Krieger invariants, it shows ${\mathcal{C}}$ is equivalent to the standard diagonal ${\mathcal{D}}=C^*\{\sigma_e^z\}$, via an AF-groupoid isomorphism that preserves the diagonal structure. The result implies the toric-code diagonal is not special beyond automorphism equivalence, highlighting the role of groupoid invariants in classifying $C^\ast$-diagonals in gapped topological models. This bridges quantum spin models with operator-algebraic diagonals, illustrating that canonical diagonal data suffice to capture the toric code's diagonal structure.

Abstract

We study the abelian sub-C*-algebra of the CAR algebra generated by the start and face opertors of Kitaev's toric code. We show that it is a C*-diagonal equivalent to the canonical diagonal of the CAR algebra.

On a C*-Diagonal Generated by the Toric Code

TL;DR

The paper analyzes the abelian subalgebra generated by the toric code's star and face operators inside the AF-algebra , proving it forms a -diagonal with Cantor spectrum. It establishes regularity and the unique extension property via LTQO and a family of locally representable symmetries, leading to a canonical conditional expectation . Using the Weyl groupoid framework and Krieger invariants, it shows is equivalent to the standard diagonal , via an AF-groupoid isomorphism that preserves the diagonal structure. The result implies the toric-code diagonal is not special beyond automorphism equivalence, highlighting the role of groupoid invariants in classifying -diagonals in gapped topological models. This bridges quantum spin models with operator-algebraic diagonals, illustrating that canonical diagonal data suffice to capture the toric code's diagonal structure.

Abstract

We study the abelian sub-C*-algebra of the CAR algebra generated by the start and face opertors of Kitaev's toric code. We show that it is a C*-diagonal equivalent to the canonical diagonal of the CAR algebra.
Paper Structure (4 sections, 14 theorems, 45 equations)

This paper contains 4 sections, 14 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. The toric code model: A short review
  3. The $C^\ast$-diagonal generated by the toric code
  4. Equivalence class of the $C^\ast$-diagonal

Key Result

Theorem 2.2

If a frustration-free net of proper projections $\{Q_\Lambda\}$ satisfies the LTQO property, then the net converges to a minimal projection in the double dual of ${\mathcal{A}}$. Consequently, there exists a unique frustration-free ground state $\omega$, which is moreover pure, and it is explicitly

Theorems & Definitions (27)

  • Definition 2.1: DET1
  • Theorem 2.2: DET1
  • Theorem 2.3
  • Proposition 2.4: KitaevAOP2003
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 17 more