Phase transition on Exel crossed products assocaited to dilation matrices

TL;DR

This paper analyzes the phase transition phenomena in the KMS states of Exel crossed products associated with dilation matrices, revealing a unique state at a critical temperature and a phase transition in the Toeplitz extension.

Contribution

It computes KMS states for Exel crossed products linked to dilation matrices and identifies a phase transition in the Toeplitz extension at a specific inverse temperature.

Findings

Unique KMS state at inverse temperature log|det A|

Phase transition in the Toeplitz extension at the same temperature

KMS states form a simplex of probability measures for higher temperatures

Abstract

An integer matrix $A\in M_d(\Z)$ induces a covering $\sigma_A$ of $\T^d$ and an endomorphism $\alpha_A:f\mapsto f\circ \sigma_A$ of $C(\T^d)$ for which there is a natural transfer operator $L$. In this paper, we compute the KMS states on the Exel crossed product $C(\T^d)\rtimes_{\alpha_A,L}\N$ and its Toeplitz extension. We find that $C(\T^d)\rtimes_{\alpha_A,L}\N$ has a unique KMS state, which has inverse temperature $\beta=\log|\det A|$. Its Toeplitz extension, on the other hand, exhibits a phase transition at $\beta=\log|\det A|$, and for larger $\beta$ the simplex of KMS$_\beta$ states is isomorphic to the simplex of probability measures on $\T^d$.