The Hartree–Fock–Bogolyubov Approximation in the Models with Four-Fermion Interaction

An exactly solvable model with pairwise four-fermion interaction is considered, which is of interest in the theory of superconductivity. It is shown that one can construct an asymptotically exact solution for this model by using the method of approximating Hamiltonians. A theorem is proved that allows one to calculate, with asymptotic accuracy in the thermodynamic limit, the density of free energy under sufficiently general conditions imposed on the parameters of a model system. An approximate method is proposed for the investigation of general models with four-fermion interaction. This method is based on the idea of constructing a certain approximating Hamiltonian and allows one to investigate the dynamic properties of the above models. It combines the approach, conventional for the method of approximating Hamiltonians, to the investigation of models with separable interaction and the Hartree–Fock scheme of approximate calculations that is based on the concept of self-consistency. As an illustration of the efficiency of the approach proposed, the BCS model is considered, which plays an important role in the theory of superconductivity.