Classical Limits of Euclidean Gibbs States for Quantum Lattice Models

TL;DR

This paper investigates how Euclidean Gibbs states of quantum lattice models approach their classical counterparts as the particle mass increases, establishing weak convergence of Gibbs measures and order parameters.

Contribution

It introduces a rigorous framework for the convergence of quantum Gibbs states to classical states via weak convergence of Gibbs measures, including translation-invariant models.

Findings

Quantum Gibbs measures weakly converge to classical measures as mass tends to infinity.

Conditional Gibbs measures of quantum models approach those of classical models.

Convergence of order parameters for translation-invariant pair interactions.

Abstract

Models of quantum and classical particles on the d-dimensional cubic lattice with pair interparticle interactions are considered. The classical model is obtained from the corresponding quantum one when the reduced physical mass of the particle tends to infinity. For these models, it is proposed to define the convergence of the Euclidean Gibbs states, when the reduced mass tends to infinity, by the weak convergence of the corresponding Gibbs specifications, determined by conditional Gibbs measures. In fact it is proved that all conditional Gibbs measures of the quantum model weakly converge to the conditional Gibbs measures of the classical model. A similar convergence of the periodic Gibbs measures and, as a result, of the order parameters, for such models with the pair interactions possessing the translation invariance, has also been proven.