Implicit ladder summation in the Hartree-Fock-Bogoliubov approach

The fully variational Hartree-Fock-Bogoliubov approach for bosons is studied in the limit of zero-range forces in two and three dimensions. The equation of state obtained in two dimensions is expressed in a parametric form. It is shown that the $\Lambda$ potential permits to perform an implicit summation of the ladder diagrams without leaving the variational scheme, restoring thus the consistency of this approximation.