A note on the non-commutative Laplace-Varadhan integral lemma

TL;DR

This paper extends the Laplace-Varadhan integral lemma to non-commutative quantum systems, providing a variational formula for free energy involving non-commuting observables, thus advancing understanding of quantum lattice spin systems.

Contribution

It generalizes the Laplace-Varadhan lemma to non-commutative settings, offering a new variational principle for quantum free energy with non-commuting operators.

Findings

Proves a variational formula for quantum free energy involving non-commuting operators.

Extends previous results to more general quantum lattice systems.

Provides a non-commutative analogue of the classical Laplace-Varadhan asymptotic formula.

Abstract

We continue the study of the free energy of quantum lattice spin systems where to the local Hamiltonian $H$ an arbitrary mean field term is added, a polynomial function of the arithmetic mean of some local observables $X$ and $Y$ that do not necessarily commute. By slightly extending a recent paper by Hiai, Mosonyi, Ohno and Petz [9], we prove in general that the free energy is given by a variational principle over the range of the operators $X$ and $Y$. As in [9], the result is a noncommutative extension of the Laplace-Varadhan asymptotic formula.