Two-step condensation of interacting bosons in three-dimensional isotropic harmonic traps

The Hartree–Fock–Popov theory of interacting Bose particles in a uniform space is generalized to the interacting boson system in three-dimensional isotropic harmonic traps. At finite temperature T, we find that the Bose condensation of nonideal bosons in three-dimensional isotropic harmonic traps is the two-step condensation. In other words, for a fixed particle number there are two transition temperatures. The first transition temperature is the standard critical temperature . The second transition temperature is the critical temperature , which is smaller than and is determined by the minimum of the curve of condensate fraction versus temperature. The boson system undergoes a first-order phase transition from the normal state to the Bose condensed state. (paper: quantum statistical physics, condensed matter, integrable systems)