A Correlation Estimate for Quantum Many-Body Systems at Positive Temperature

TL;DR

This paper introduces a new inequality providing lower bounds on interaction energies in quantum many-body systems at positive temperature, with applications to high-density free energy expansions for jellium.

Contribution

It develops a rigorous inequality akin to first order perturbation theory for quantum systems at positive temperature, applicable to both fermions and bosons.

Findings

Derived a lower bound on two-body interaction expectations.

Proved high density free energy expansion for jellium.

Applicable to Bose-Einstein condensation above transition temperature.

Abstract

We present an inequality that gives a lower bound on the expectation value of certain two-body interaction potentials in a general state on Fock space in terms of the corresponding expectation value for thermal equilibrium states of non-interacting systems and the difference in the free energy. This bound can be viewed as a rigorous version of first order perturbation theory for many-body systems at positive temperature. As an application, we give a proof of the first two terms in a high density (and high temperature) expansion of the free energy of jellium with Coulomb interactions, both in the fermionic and bosonic case. For bosons, our method works above the transition temperature (for the non-interacting gas) for Bose-Einstein condensation.