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On the order of BEC transition in weakly interacting gases predicted by mean-field theory

Predictions from Hartree-Fock (HF), Popov (P), Yukalov-Yukalova (YY) and $t$-matrix approximations regarding the thermodynamics from the normal to the BEC phase in weakly interacting Bose gases are considered. By analyzing the dependence of the chemical potential $μ$ on temperature $T$ and particle density $ρ$ we show that none of them predicts a second-order phase transition as required by symmetry-breaking general considerations. In this work we find that the isothermal compressibility $κ_{T}$ predicted by these theories does not diverge at criticality as expected in a true second-order phase transition. Moreover the isotherms $μ=μ(ρ,T)$ typically exhibit a non-singled valued behavior in the vicinity of the BEC transition, a feature forbidden by general thermodynamic principles. This behavior can be avoided if a first order phase transition is appealed. The facts described above show that although these mean field approximations give correct results near zero temperature they are endowed with thermodynamic anomalies in the vicinity of the BEC transition. We address the implications of these results in the interpretation of current experiments with ultracold trapped alkali gases.