Quantum corrections to the semiclassical Hartree-Fock theory of a harmonically trapped Bose gas

Using the phase-space expansion of the thermodynamical distribution functions we provide a general and systematic method for including effects beyond the local-density approximation to the semiclassical Hartree-Fock theories. We illustrate the method by applying it to the case of a strictly two-dimensional, harmonically trapped Bose gas. Thereby, we address the ambiguous prediction of the Hartree-Fock approximation, namely, whether a fixed number of trapped atoms undergoes Bose-Einstein condensation or not. We also investigate the dependence of the critical temperature on the interaction strength.