Quantum Spin probabilities at positive temperature are H\"older Gibbs probabilities

TL;DR

This paper demonstrates that quantum spin probabilities at positive temperature are Gibbs probabilities with H"older potentials, exhibiting phase transition behavior and explicit deviation functions, linking quantum states to classical Gibbs measures.

Contribution

It establishes that quantum spin probabilities at positive temperature are Gibbs measures with H"older potentials and analyzes their phase transition and deviation properties.

Findings

Probabilities are Gibbs with H"older potentials

Explicit transition temperature T_c identified

Deviation function explicitly computed

Abstract

We consider the KMS state associated to the Hamiltonian $H= \sigma^x \otimes \sigma^x$ 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 $\mu_T$ on the Bernoulli space $\{1,2\}^\mathbb{N}$. The Jacobian of $\mu_T$ can be expressed via a certain continued fraction expansion. We will show that this probability is a Gibbs probability for a H\"older potential. Therefore, this probability is mixing for the shift map. For such probability $\mu_T$ 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 $\mu_T$ 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).