Supercritical phase transition on the Toeplitz algebra of $\mathbb N^\times \ltimes \mathbb Z$
Marcelo Laca, Tyler Schulz
TL;DR
This work characterizes the KMS equilibrium states for the Toeplitz C*-algebra of the semidirect product ${\mathbb N^\times}\!\ltimes \mathbb Z$, under a natural dynamics, across all temperatures. The authors reduce the problem to analyzing $β$-subconformal measures on the unit circle, proving a Bauer-simplex structure for $β\in(0,1]$ with extremal states $ψ_{β,n}$ ($n\in{\mathbb N^\times}$) and $ψ_{β,∞}$, while recovering the known high-temperature uniqueness for $β>1$. They provide explicit atomic measures $ν_{β,n}$ and the Lebesgue measure $ν_{β,∞}$ that parametrize extremal KMS states, prove the uniqueness of the nonatomic case, and establish a spontaneous symmetry-breaking mechanism linked to cyclotomic Galois groups via equivariant quotients and connections to the Bost–Connes system. The paper further extends the framework to modular quotients and to the Toeplitz system for ${\mathbb N^\times}\!\ltimes (\mathbb Q/\mathbb Z)$, describing KMS states as projective limits over finite quotients and linking the structure to class-field theory through a detailed analysis of symmetry and type. Overall, the results illuminate a rich phase structure for Toeplitz monoid C*-algebras with number-theoretic dynamics and establish deep ties to Bost–Connes-type phenomena and K-theoretic/arithmetic invariants.
Abstract
We study the high-temperature equilibrium for the C*-algebra $\mathcal T (\mathbb N^\times \ltimes \mathbb N)$ recently considered by an Huef, Laca and Raeburn. We show that the simplex of KMS$_β$ states at each inverse temperature $β$ in the critical interval $(0,1]$ is a Bauer simplex whose space of extreme points is homeomorphic to $\mathbb N \sqcup\{\infty\}$. This is in contrast to the uniqueness of equilibrium at high temperature observed in previously considered systems arising from number theory. We also show that quotients of our system exhibit spontaneous symmetry-breaking by finite cyclotomic Galois groups and establish their connection to the Bost-Connes phase transition.