Superconductive states in the two-dimensional repulsive Hubbard model

Investigation of the two-dimensional repulsive Hubbard model is continued for the case when the Fermi level is close to one of the saddle Van-Hove points of the quasiparticle energy function. The Bethe–Salpeter equation for the two-particle scattering amplitude and the system of Dyson–Gor'kov equations for normal and anomalous Green's functions axe considered. Closeness of the Van-Hove point and the Fermi level allows to simplify the investigation. A new method suggested for evaluation the kernel of investigated equations is suggested.
It is shown, that the nontrivial Cooper pairing (the superposition of pairings with odd angular momenta) arises if and only if the Fermi level is close to one of the Van-Hove points. The superconductive phase transition temperature is maximal for some special mutual location of the Fermi level and the Van-Hove point. Two different superconductive solutions (modes) are found which are antisymmetric functions in the momentum representation. These two modes coexist in some narrow interval of the doping parameter around the value of this parameter corresponding to the case of intersection the Van-Hove point and the Fermi level. Values of the energy gap are different for these two modes (two different gap values were observed in some experiments).