Phase transitions on C*-algebras from actions of congruence monoids on rings of algebraic integers

Abstract

We compute the KMS (equilibrium) states for the canonical time evolution on C*-algebras from actions of congruence monoids on rings of algebraic integers. We show that for each $\beta \in [1,2]$, there is a unique KMS${}_{\beta }$ state, and we prove that it is a factor state of type III${}_1$. There is a phase transition at $\beta =2:$ For each $\beta \in (2,\infty ]$, the set of extremal KMS${}_{\beta }$ states decomposes as a disjoint union over a quotient of a ray class group in which the fibers are extremal traces on certain group C*-algebras associated with the ideal classes. Moreover, in most cases, there is a further phase transition at $\beta =\infty$ in the sense that there are ground states that are not KMS${}_{\infty }$ states. Our computation of KMS and ground states generalizes the results of Cuntz, Deninger, and Laca for the full $ax+b$-semigroup over a ring of integers, and our type classification generalizes a result of Laca and Neshveyev in the case of the rational numbers and a result of Neshveyev in the case of arbitrary number fields.

Revised version, accepted for publication in Int. Math. Res. Not. IMRN; 34 pages