Hartree-Fock-Bogoliubov Approximation in Models with Four-Fermion Interaction

An exactly solvable model with a four-fermion interaction, which is of interest in the theory of superconductivity, is considered. It is shown that an asymptotically exact solution can be constructed for this model using an approximating Hamiltonian method. A theorem is proved which enables one to calculate an asymptotically exact expression for the free energy in the thermodynamic limit under sufficiently general conditions imposed on the parameters of the model system in question. An approximate method of investigating general models with four-fermion interactions is proposed which combines the ideas of the approximating Hamiltonian scheme for the models with separable interactions with the Hartree-Fock-Bogolubov method based on the idea of self-consistency. The Bardeen-Cooper-Schriffer model, playing an important role in the theory of superconductivity, is considered as an illustration of efficiency of the proposed approach.