Quantum Spin probabilities at positive temperature are Hölder Gibbs probabilities
We consider the KMS state associated to the Hamiltonian $H= σ^x \otimes σ^x$ over the quantum spin lattice $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2 \otimes ...$. For a fixed observable of the form $L \otimes L \otimes L \otimes ...$, where $L:\mathbb{C}^2 \to \mathbb{C}^2 $ is self adjoint, and for positive temperature $T$ one can get a naturally defined stationary probability $μ_T$ on the Bernoulli space $\{1,2\}^\mathbb{N}$. 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).