Conserving gapless mean-field theory for Bose-Einstein condensates

We formulate a conserving gapless mean-field theory for Bose-Einstein condensates on the basis of a Luttinger-Ward thermodynamic functional. It is applied to a weakly interacting uniform gas with density n and s-wave scattering length a to clarify its fundamental thermodynamic properties. It is found that the condensation here occurs as a first-order transition. The shift of the transition temperature AT, from the ideal-as result To is positive and given to the leading order by ATc = 2.33an(1/3) To, in agreement with a couple of previous estimates. The theory is expected to form a new theoretical basis for trapped Bose-Einstein condensates at finite temperatures.