Condensate theory versus pairing theory for degenerate Bose systems

The statistical variational principle for the thermodynamic potential is used to investigate the quantum statistically degenerate phase of an interacting Bose system. Starting with a general mean-field trial Hamiltonian, we find a set of integral equations that has two different types of solution: (a) the condensate sssssstheory with a finite c-number part for the lowest momentum state operator; (b) the pairing theory with zero c-number part. The condensate theory is identical to the self-consistent Hartree-Fock-Bogoliubov approximation of the Beliaev equations, whereas the pairing theory is identical to the theory proposed by Evans and Imry. We solve the resulting integral equations for both theories numerically and show that—independent of the special form of interacting potentials—the condensate theory gives the lower free energy and the higher transition temperature. Therefore we conclude that the condensate theory describes the thermodynamically stable state rather than the pairing theory.