Magnetism and superconductivity within extended Hubbard model for a dimer. Exact results

We consider the extended Hubbard model containing intrasite and intersite Coloumb interactions. This model we apply to a dimer as an interisting and nontrivial example of two interacting ions which possesses exact, analytical solution. We find the eigenvalues Eα and eigenvectors |Eα〉 of the dimer and we represent each part Eα|Eα〉〈Eα| of the dimer Hamiltonian (α = 1, 2, ..., 16) in the second quantizations with the use of the Hubbard and spin operators. This procedure gives a review of all competitive, intrinsic interactions together with their exactly calculated coupling constants expressed by the model parameters. These competitive interactions, deeply hidden in the orginal form of the extended Hubbard model, make an evidence that the model is extremely complex because it describes a competition between magnetism and superconductivity. Among competitive interactions we can find ferromagnetic and antiferromagnetic interactions (also in the form appearing in the t − J model), hopping of the Cooper pairs between different dimer lattice sites (similar as in KulikPedan, Penson-Kolb models), as well as, intersite Cooper pair interactions. We plot several coupling constants of these interactions vs Coulomb intrasite interaction U to show that the competition between them strongly depends on the model parameters. The thermodynamical activity of each particular interaction, belonging to a given energy level, depends, however, on the occupation of this level. When e.g. a given dimer energy level is empty the corresponding intrinsic interaction belonging to this level is completely passive (irrelevant). From the presented review of interactions it is evident that the extended Hubbard model is capable to describe the properties of superconductors with magnetic ordering (including also highTC superconductors) where a strong competition between magnetism and Corresponding author. E-mail: [email protected]