Conserving Gapless Mean-Field Theory for Bose–Einstein Condensates

Abstract

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 ΔT_c from the ideal-gas result T₀ is positive and given to the leading order by ΔT_c=2.33an^1⁄3T₀, 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.