KMS states on $C^*$-algebras associated to higher-rank graphs

TL;DR

This paper analyzes KMS states on higher-rank graph C*-algebras, revealing how their structure varies with temperature and identifying conditions for unique or multiple equilibrium states.

Contribution

It extends the study of KMS states to higher-rank graphs, using Perron-Frobenius theory for commuting matrices to characterize state simplices at different temperatures.

Findings

For large inverse temperatures, the KMS state simplex has dimension d-1.

A critical temperature _c exists where the KMS state is unique when the subgroup is dense.

The structure of KMS states depends on the dynamics and temperature, with phase transitions at _c.

Abstract

Consider a higher-rank graph of rank k. Both the Cuntz-Krieger algebra and the Toeplitz-Cuntz-Krieger algebra of the graph carry natural gauge actions of the torus T^k, and restricting these gauge actions to one-parameter subgroups of T^k gives dynamical systems involving actions of the real line. We study the KMS states of these dynamical systems. We find that for large inverse temperatures \beta, the simplex of KMS_\beta states on the Toeplitz-Cuntz-Krieger algebra has dimension d one less than the number of vertices in the graph. We also show that there is a preferred dynamics for which there is a critical inverse temperature \beta_c: for \beta larger than \beta_c, there is a d-dimensional simplex of KMS states; when \beta=\beta_c and the one-parameter subgroup is dense, there is a unique KMS state, and this state factors through the Cuntz-Krieger algebra. As in previous studies for k=1, our main tool is the Perron-Frobenius theory for irreducible nonnegative matrices, though here we need a version of the theory for commuting families of matrices.