Gapless mean-field theory of Bose-Einstein condensates

We present a topical review of the development of finite-temperature field theories of Bose-Einstein condensation in weakly interacting atomic gases. We highlight the difficulties in obtaining a consistent finite-temperature theory that has a gapless excitation spectrum in accordance with Goldstone's theorem and which is free from both ultraviolet and infrared divergences. We present results from the two consistent theories developed so far. These are the Hartree-Fock-Bogoliubov theory within the Popov approximation and a many-body T-matrix approach which we have termed gapless-Hartree-Fock-Bogoliubov (GHFB). Comparison with the available experimental results is made and the remaining difficulties are highlighted.