The Full Set of KMS-States for Abelian Kitaev Models

TL;DR

This paper characterizes all KMS states of an abelian Kitaev model using groupoid $C^*$-algebra techniques, establishing their uniqueness for finite inverse temperature and identifying the ground state.

Contribution

It provides a complete description of the KMS states for abelian Kitaev models via groupoid $C^*$-algebra methods, including their uniqueness and ground state limit.

Findings

Full set of KMS states identified for all inverse temperatures.

Uniqueness of KMS states for $eta o ext{finite}$ established.

Limit at $eta o ext{infinity}$ matches the ground state.

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 $(c_H,\beta)$-KMS measures for this groupoid, we identify the full set of KMS states of the model and prove its uniqueness for $\beta \in [0,\infty)$. Furthermore, we show that its limit at $\beta \rightarrow \infty$ exists and coincides with the unique frustration-free ground state of the model.