The Full Set of KMS-States for Abelian Kitaev Models

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 $\beta \in [0,\mathrm{\infty })$. Furthermore, we show that its limit at $\beta \to \mathrm{\infty }$ exists and coincides with the unique frustration-free ground state of the model.