Supercritical Phase Transitions from Number Theory
Date
2023-04-28
Authors
Schulz, Tyler
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Abstract
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× ∪ {∞}
Description
Keywords
Number theory, Quantum statistical mechanics, Equilibrium states, Phase transition, Operator algebras