Supercritical Phase Transitions from Number Theory




Schulz, Tyler

Journal Title

Journal ISSN

Volume Title



We classify the KMSβ states of the right ax + b C*-dynamical system of N in the supercritical range β ∈ (0, 1], thus completing the classification of KMSβ states initiated in [19]. We show that the simplex of KMSβ states is affinely isomorphic to the simplex of subconformal measures on the circle. We then provide explicit formulas for the extremal subconformal measures and corresponding KMSβ states in terms of classical arithmetic functions. For β ∈ (0, 1], our measures are parameterized by the compact space N×∪{∞}, and in particular, demonstrate phase transition at each value of β, a novel feature among C*-dynamical systems related to number theory. Another new feature of the right ax + b system is the existence of equivariant quotients, corresponding to the quotient rings Z/mZ for m ∈ N×. We provide a classification of the KMSβ states of the quotient C*-dynamical systems, and show that the quotient systems exhibit spontaneous symmetry-breaking with respect to the group of units (Z/mZ)∗. We then use this action to compute the type of the high-temperature KMSβ states with parameter belonging to N× ⊆ N× ∪ {∞}



Number theory, Quantum statistical mechanics, Equilibrium states, Phase transition, Operator algebras