Studying Superfluid Transition of a Dilute Bose Gas by Conserving   Approximations
  We consider the Bose-Einstein transition of homogeneous weakly interacting spin-0 particles based on the normal-state Phi-derivable approximation. Self-consistent calculations of Green's function and the chemical potential with several approximate Phi's are performed numerically as a function of temperature near Tc, which exhibit qualitatively different results. The ladder approximation apparently shows a continuous transition with the prefactor c=2.94 for the transition-temperature shift given in terms of the scattering length a and density n. In contrast, the second-order, particle-hole, and fluctuation-exchange approximations yield a first-order transition. The fact that some standard Phi's predict a first-order transition challenges us to clarify whether or not the transition is really continuous. 
K. Tsutsui and T. Kita
cond-mat.quant-gas cond-mat.stat-mech cond-mat.supr-con