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

We consider a Bose gas with two-body Kac-like scaled interactions V $\gamma$ (r) = $\gamma$ 3 v($\gamma$r) where v(x) is a given repulsive and integrable potential, while $\gamma$ is a positive parameter which controls the range of the interactions and their amplitude at a distance r. Using the Hartree-Fock approximation we find that, at finite non-zero temperatures, the Bose-Einstein condensation is destroyed by the repulsive interactions when they are sufficiently long-range. More precisely, we show that for $\gamma$ 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 Hartree-Fock approximation. Furthermore, our Hartree-Fock analysis shows the existence of a threshold value $\gamma$ 0 above which the Bose-Einstein condensation is restored. Since $\gamma$ 0 is an unbounded increasing function of the temperature this implies for a fixed $\gamma$, namely for a fixed scaled potential, that a condensate cannot form above some critical temperature whatever the value of the density.