Hartree–Fock analysis of the effects of long-range interactions on the Bose–Einstein condensation

Abstract

We consider a Bose gas with two-body Kac-like scaled interactions where is a given repulsive and integrable potential, while is a positive parameter which controls the range of the interactions and their amplitude at a distance r. Using the Hartree–Fock (HF) approximation we find that, at finite non-zero temperatures, the Bose–Einstein (BE) condensation is destroyed by the repulsive interactions when they are sufficiently long-range. More precisely, we show that for sufficiently small but finite the off-diagonal part of the one-body density matrix always vanishes at large distances. Our analysis sheds light on the coupling between critical correlations and long-range interactions, which might lead to the breakdown of the off-diagonal long-range order even beyond the HF approximation. Furthermore, our HF analysis shows the existence of a threshold value above which the BE condensation is restored. Since is an unbounded increasing function of the temperature this implies for a fixed , namely for a fixed scaled potential, that a condensate cannot form above some critical temperature whatever the value of the density.