Antiferromagnetism, ferromagnetism and superconductivity in the Hubbard model with repulsion

The temperature Green's functions method is applied for the investigation the Hubbard method with repulsive interaction as a toy model for the High-Temperature Superconductivity (HTSC). The existence of the antiferromagnetic state in the case of the half-filled band is demonstrated. It turns out that the superconducting state with the $p$-pairing exists in the three-dimensional Hubbard model in the low density limit of electrons (or holes). The maximal possible critical temperature for the superconducting transition is achieved if the saddle point of the fermion energy function (Van-Hove point) lies on the Fermi surface. The type of Cooper pairing for this case may be called a superposition of pairings with odd angular momenta, and one can consider this pairing as the $p$-pairing in the first approximation. The problem of completition between superconductivity and ferromagnetism is investigated. It is shown that the superconducting state becomes more preferrable for $u/t\gtrsim8$.