Ê I nst i t ut e of P hysi cs, Si l esian Uni versity Uni werstytec ka 4, 40-007 Ka to wi ce, Pol an d ( Recei ved December 14, 2001)W e consider t he extended H ubb ard mo del for the single cubic lattice and re w rite it in the form of interacting dimers , using the exact solutio n of the dimer problem .W e anal ytical l y deri ve the second quanti zation form of the dimer H amiltoni an elimina tin g from the consideratio ns uno ccupie d dimer energy levels in the large U limit (it is the only assumption ).T he resulting dimer H amiltonian w ritten with the use of the Hubbard op erators and spin operators contains three terms, visuali zin g explicitl y comp eting magnetic interactions (f erromagnetic, antif erromagnetic) as a generaliz atio n of the t { J mo del.T he presented, nonp erturbati ve metho d, can in principl e b e applied to the cluster of any size (e.g. one central atom and z its nearest neighb ours).T he use of the pro j ection technique can further b e appli ed in the case of a crystal to obtain the second quantization form of the extended H ubbard mo del for the sc lattice in the large U limit.
The recently introduced concept of exact mean-field theories is applied to the extended simplified Hubbard model, which is a one-band extended Hubbard model (including nearest-neighbor interaction) where only the \ensuremath{\downarrow} electrons can hop. An exact mean-field Hamiltonian is derived for the model on a Bethe lattice in the limit of a large coordination number (Z\ensuremath{\rightarrow}\ensuremath{\infty}). The mean-field Hamiltonian is used to calculate the Green functions and the free energy at half-filling. This exact solution is analyzed analytically at small and large values of the on-site (U) and nearest-neighbor (V) interaction strengths. One finds that a phase transition occurs at sufficiently low temperatures for all U,V>0. The low-temperature phase is a charge-density wave for V>1/2U and a spin-density wave for V1/2U. Special attention is paid to (i) the critical temperature as a function of U and V, (ii) the order parameter as a function of temperature, and (iii) the density of states. The density of states shows that the low-temperature phase is an insulator for all U,V provided that V/U\ensuremath{\ne}1/2, while the high-temperature phase is a paramagnetic metal for all V if U2, and a paramagnetic insulator if U>2. Many detailed results are given.
We find the eigenvalues $E_α $ and eigenvectors
∣Eα> of the extended Hubbard 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, deeply hidden in the orginal form of the considered Hamiltonian. 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 Kulik-Pedan, 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. 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 high-
TC superconductors) where a strong competition between magnetism and superconductivity takes place.