Conserving Gapless Mean-Field Theory for Bose-Einstein Condensates

TL;DR

This paper develops a conserving gapless mean-field theory for Bose-Einstein condensates using a Luttinger-Ward functional, revealing a first-order transition and a positive shift in transition temperature, aligning with previous estimates.

Contribution

It introduces a new conserving gapless mean-field framework for BECs based on a Luttinger-Ward functional, providing insights into thermodynamic properties and transition behavior.

Findings

Condensation occurs as a first-order transition.

Positive shift in transition temperature proportional to scattering length.

Agreement with previous estimates for the transition temperature shift.

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 $\Delta T_c$ 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 previous estimates. The theory is expected to form a new theoretical basis for trapped Bose-Einstein condensates at finite temperatures.