Conserving Gapless Mean-Field Theroy for Bose-Einstein Condensates

A new conserving gapless mean-field theory for Bose-Einstein condensates is constructed based on a Luttinger-Ward thermodynamic functional. It is applied to a weakly interacting uniform gas with the density $n$ and the $s$-wave scattering length $a$ to clarify its basic thermodynamic properties. It is found that the condensation here occurs as a first-order transition. The shift of the transition temperature from the ideal-gas result $T_{0}$ is positive and given to the leading order by $\Delta T_c \!=\! 2.33a n^{1/3}T_0$, in agreement with a couple of preceding estimates. The theory is expected to form a new theoretical basis for the trapped Bose-Einstein condensates at finite temperatures.