Self-consistent t-matrix theory of the Hartree-Fock-Bogoliubov approximation for Bose-Einstein-condensed systems

We present a self-consistent t-matrix theory for Bose-Einstein-condensed systems within the Hartree-FocBogoliubov (HFB) approximation. Using the Lippmann-Schwinger equation for a t matrix describing the collision between two particles via an interparticle potential, we derive a set of equations for the normal and anomalous self-energies in the HFB approximation expressed in terms of the t matrix. These equations are solved for a hard-sphere potential. A result is then obtained which is valid over the full range of density, reducing to the exact expressions at low densities and to the Brueckner-Sawada theory at high densities. The spectrum is gapless and linear in small momentum, but does not have any roton minimum in the large-momentum region even for high densities such as those of He-4.