Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons

TL;DR

This paper analyzes a system of weakly interacting trapped bosons at positive temperature, demonstrating that the free energy can be approximated by a Hartree functional and showing that repulsive interactions lower the critical temperature for Bose-Einstein condensation.

Contribution

The study extends the Hartree energy functional to positive temperatures and establishes the relationship between microscopic and semiclassical free energies, revealing the impact of interactions on critical temperature.

Findings

Free energy approximated by Hartree functional

Bose-Einstein condensation occurs if and only if in the semiclassical limit

Repulsive interactions decrease the critical temperature

Abstract

We consider a system of N trapped bosons with repulsive interactions in a combined semiclassical mean-field limit at positive temperature. We show that the free energy is well approximated by the minimum of the Hartree free energy functional - a natural extension of the Hartree energy functional to positive temperatures. The Hartree free energy functional converges in the same limit to a semiclassical free energy functional, and we show that the system displays Bose-Einstein condensation if and only if it occurs in the semiclassical free energy functional. This allows us to show that for weak coupling the critical temperature decreases due to the repulsive interactions.