Quantum spin probabilities at positive temperature are Hölder Gibbs probabilities

Abstract

We consider the KMS state associated to the Hamiltonian H = σ x ⊗ σ x over the quantum spin lattice C 2 ⊗ C 2 ⊗ C 2 ⊗ .... For a fixed observable of the form L ⊗ L ⊗ L ⊗ ..., where L : C 2 → C 2 is self adjoint, and for positive temperature T one can get a naturally defined stationary probability µ T on the Bernoulli space {1, 2} N . The Jacobian of µ T 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 µ T 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 T c where the set of values of the Jacobian of the Gibbs probability µ T 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). arXiv:1805.01784v2 [math.DS] 16 May 2018Given a selfadjoint operator H acting on a finite dimensional complex Hilbert space H and a temperature T > 0, the density operatorwhere Z(T ) = Trace e − 1 T H , is called the KMS operator associated to the Hamiltonian H. It is usual to denote β = 1/T . A general reference on KMS operators and KMS states is [5].The set of linear operators acting on C 2 will be denoted by M 2 . We will call ω = ω n : M 2