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

## TL;DR

This paper analyzes the equilibrium states of C*-algebras derived from actions of congruence monoids on algebraic integer rings, revealing phase transitions and classifying states by type, extending previous work in number theory and operator algebras.

## Contribution

It generalizes the computation of KMS and ground states for these C*-algebras, including phase transition phenomena and type classification, beyond prior specific cases.

## Key findings

- Unique KMS$_eta$ states for $eta ext{ in }[1,2]$
- Phase transition at $eta=2$ with extremal states linked to class groups
- Further phase transition at $eta= ext{infinity}$ with non-KMS ground states

## 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.