Large Deviations for Quantum Spin probabilities at temperature zero

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.