Large Deviations for Quantum Spin probabilities at temperature zero

TL;DR

This paper studies large deviations for certain quantum spin observables at zero temperature, revealing non-Gibbsian stationary measures with explicit free energy, and establishing a large deviation principle.

Contribution

It provides the first analysis of large deviations for quantum spin systems at zero temperature with non-Gibbsian measures and explicit free energy calculations.

Findings

Stationary measure is ergodic but not mixing.

Measure is not Gibbsian but Jacobian takes only two values.

Large deviation principle holds for a class of functions.

Abstract

We consider certain self-adjoint observables for 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 $, and for the zero temperature limit one can get a naturally defined stationary probability $\mu$ on the Bernoulli space $\{1,2\}^\mathbb{N}$. This probability is ergodic but it is not mixing for the shift map. It is not a Gibbs state for a continuous normalized potential but its Jacobian assume only two values almost everywhere. Anyway, for such probability $\mu$ we can show that a Large Deviation Principle is true for a certain class of functions. The result is derived by showing the explicit form of the free energy which is differentiable.