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The Full Set of KMS-States for Abelian Kitaev Models

Danilo Polo Ojito, Emil Prodan

Abstract

We first prove that the subalgebra $\mathcal{C}$ generated by the vertex and face operators of an abelian Kitaev model is a $C^\ast$-diagonal of the UHF algebra $\mathcal{A}$ of quasilocal observables. This gives us access to the Weyl groupoid $\mathcal{G}_\mathcal{C}$ associated with the $C^\ast$-inclusion $\mathcal{C} \hookrightarrow \mathcal{A}$, which supplies a valuable presentation of $\mathcal{A}$ as a groupoid $C^\ast$-algebra where the dynamics of the model are generated by a groupoid 1-cocycle $c_H$. Making appeal to the notion of $(c_H,β)$-KMS measures for this groupoid, we identify the full set of KMS states of the model and prove its uniqueness for $β\in [0,\infty)$. Furthermore, we show that its limit at $β\rightarrow \infty$ exists and coincides with the unique frustration-free ground state of the model.

The Full Set of KMS-States for Abelian Kitaev Models

Abstract

We first prove that the subalgebra generated by the vertex and face operators of an abelian Kitaev model is a -diagonal of the UHF algebra of quasilocal observables. This gives us access to the Weyl groupoid associated with the -inclusion , which supplies a valuable presentation of as a groupoid -algebra where the dynamics of the model are generated by a groupoid 1-cocycle . Making appeal to the notion of -KMS measures for this groupoid, we identify the full set of KMS states of the model and prove its uniqueness for . Furthermore, we show that its limit at exists and coincides with the unique frustration-free ground state of the model.

Paper Structure

This paper contains 10 sections, 19 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $C^\ast$-Diagonals Generated by Abelian Kitaev Models
  3. Abelian Kitaev models
  4. The commutative algebra of local interactions
  5. The $C^\ast$-diagonal of an Abelian Kitaev model
  6. The Weyl Groupoids of Abelian Kitaev Models
  7. The complete set of KMS states
  8. KMS states
  9. KMS measures
  10. The set of KMS states

Key Result

Proposition 2.1

One has

Figures (1)

  • Figure 2.1: Black arrows represent the lattice $\mathcal{L}=\mathbb{Z}^2$ with its chosen orientation, while the dual lattice $\tilde{\mathcal{L}}$ is depicted by red arrows.

Theorems & Definitions (47)

  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7: DET1
  • ...and 37 more