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We compute the KMS (equilibrium) states for the canonical time evolution onC*-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, andwe prove that it is a factor state of type III$_1$. There is a phase transitionat $\beta=2:$ 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 inwhich the fibers are extremal traces on certain group C*-algebras associatedwith the ideal classes. Moreover, in most cases, there is a further phasetransition at $\beta=\infty$ in the sense that there are ground states that arenot KMS$_\infty$ states. Our computation of KMS and ground states generalizesthe results of Cuntz, Deninger, and Laca for the full $ax+b$-semigroup over aring of integers, and our type classification generalizes a result of Laca andNeshveyev in the case of the rational numbers and a result of Neshveyev in thecase of arbitrary number fields.

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