Quantum spin probabilities at positive temperature are Hölder Gibbs probabilities

Abstract

We consider the KMS state associated to the Hamiltonian [Formula: see text] over the quantum spin lattice [Formula: see text] For a fixed observable of the form [Formula: see text] where [Formula: see text] is self-adjoint, and for positive temperature [Formula: see text] one can get a naturally defined stationary probability [Formula: see text] on the Bernoulli space [Formula: see text]. The Jacobian of [Formula: see text] can be expressed via a certain continued fraction expansion. We will show that this probability is a Gibbs probability for a Hölder potential. Therefore, this probability is mixing for the shift map. For such probability [Formula: see text] we will show the explicit deviation function for a certain class of functions. When decreasing temperature we will be able to exhibit the explicit transition value [Formula: see text] where the set of values of the Jacobian of the Gibbs probability [Formula: see text] changes from being a Cantor set to being an interval. We also present some properties for quantum spin probabilities at zero temperature (for instance, the explicit value of the entropy).