Two-band hopping model is treated on a two-dimensional square lattice. The atoms are located at the corners and the middles of the edges of the squares. In addition to the strongly overlapping orbitals of the atoms, there are extra orbitals at the corners, which are weakly hybridized. The assumption is made that the Fermi level is inside the broad band and is very near to the narrow band formed by the extra orbitals. The Hamiltonian is Hubbard type, but the off-diagonal part of the two-site interaction is kept also where one creation or annihilation operator acts on the extra orbital and the others on one of its neighbors. The weak coupling is enhanced by the on-site Coulomb repulsion at the corners, which enhancement is a power function of the ratio of the broad band width and the narrow band position measured from the Fermi level. That enhancement is obtained by summation of logarithmic Kondo-type corrections of orbital origin, which reflects the formation of a ground state of new type with strong orbital and spin correlations. Interaction between the particles of the broad band is generated by processes with one heavy and one light particle in the intermediate state. That interaction strongly depends on the momenta of the particles. The generated interaction can result in attractive interaction, but particular parity of the different orbitals are required in the given approximation. The model clearly demonstrates that repulsive interaction can result in superconductivity.
Our previous mean-field study concerning superconducting pairings in the extended Hubbard model with on-site repulsive and intersite attractive interaction and arbitrary electron density is extended to include the effects of nearest-neighbor exchange (
J) and correlated hopping (
K) interaction. Detailed numerical analysis of
s-,
p-, and
d-wave pairing solutions is performed for the two-dimensional square lattice with nearest- (
t) and next-nearest- (
t2) neighbor hopping, and the resulting mutual stability phase diagrams are given as a function of band filling and the interaction parameters. Antiferromagnetic exchange can enhance both the
d-wave and the extended
s-wave pairing. If it prevails over the repulsive intersite density-density interaction, such a term can be a leading mechanism for superconductivity. However, its effects strongly depend on the band filling and the
tt2 ratio, giving in particular for
t2=0 s-wave (
d-wave) pairing as most favorable in the low (high) density limit, but increasing
t2 can even reverse this tendency. The correlated hopping term which breaks the electron-hole symmetry affects
s-wave pairing mostly. A condensation transition (phase separation), possible in the presence of the attractive density-density interaction, is also analyzed within the random-phase approximation, and stability conditions of such an electron droplet phase with respect to other types of ordering are determined as a function of the band filling for the two-dimensional square lattice.
A model to describe electronic correlations in energy bands is considered. The model is a generalization of the conventional Hubbard model that allows for the fact that the wave function for two electrons occupying the same Wannier orbital is different from the product of single-electron wave functions. We diagonalize the Hamiltonian exactly on a four-site cluster and study its properties as a function of band filling. The quasiparticle weight is found to decrease and the quasiparticle effective mass to increase as the electronic band filling increases, and spectral weight in one- and two-particle spectral functions is transferred from low to high frequencies as the band filling increases. Quasiparticles at the Fermi energy are found to be more ``dressed'' when the Fermi level is in the upper half of the band (hole carriers) than when it is in the lower half of the band (electron carriers). The effective interaction between carriers is found to be strongly dependent on band filling becoming less repulsive as the band filling increases and attractive near the top of the band in certain parameter ranges. The effective interaction is most attractive when the single-hole carriers are most heavily dressed, and in the parameter regime where the effective interaction is attractive, hole carriers are found to ``undress,'' hence become more like electrons, when they pair. It is proposed that these are generic properties of electronic energy bands in solids that reflect a fundamental electron-hole asymmetry of condensed matter. The relation of these results to the understanding of superconductivity in solids is discussed.
The Hubbard model with occupation-dependent hopping rate exhibits superconductivity in a wide range of parameters within mean-field (BCS) theory. Here we study pair binding energies in this model for small clusters by exact diagonalization of the Hamiltonian. The model is defined by on-site and nearest-neighbor repulsions U and V, and occupation-dependent hopping rate t(n)=t+n\ensuremath{\Delta}t. We present results for one-dimensional chains and for two-dimensional square lattices of sizes 4\ifmmode\times\else\texttimes\fi{}4, 6\ifmmode\times\else\texttimes\fi{}6, and 8\ifmmode\times\else\texttimes\fi{}8. As a function of carrier density n the pair binding energy first increases and then decreases, and vanishes beyond a critical density. BCS results for the pair binding energy are found to be in remarkably good agreement with the exact results. In particular, BCS theory accurately reproduces the range of interaction parameters where pair binding exists in the exact solution. The model is found to exhibit no tendency to phase separation or clustering of more than two particles even for parameters giving rise to strong pair binding, in contrast to other tight-binding models where pairing occurs.
We study the superconducting order in a two-dimensional square lattice Hubbard model with weak repulsive interactions, subject to a Zeeman field and weak Rashba spin-orbit interactions. Diagonalizing the noninteracting Hamiltonian leads to two separate bands, and by deriving an effective low-energy interaction we find the mean field gap equations for the superconducting order parameter on the bands. Solving the gap equations just below the critical temperature, we find that superconductivity is caused by Kohn-Luttinger-type interaction, while the pairing symmetry of the bands is indirectly affected by the spin-orbit coupling. The dominating attractive momentum channel of the Kohn-Luttinger term depends on the filling fraction
n of the system, and it is therefore possible to change the momentum dependence of the order parameter by tuning
n. Moreover,
n also determines which band has the highest critical temperature. Rotating the magnetic field changes the momentum dependence from states that for small momenta reduce to a chiral ${p}_{x}\ifmmode\pm\else\textpm\fi{}i{p}_{y}$ type state for out-of-plane fields, to a nodal
p-wave-type state for purely in-plane fields.
The major characteristic of the single-band Hubbard model (HM) is to redistribute electrons at a uniform lattice separation distance within the molecular lattice. Hence, it is only linearly dependent on lattice separations distance. Thus the single-band Hubbard model does not consider the lattice gradient encountered by interacting electrons as they hop from one lattice point to another. The linear dependence of the single-band HM only on lattice separations would certainly not provide a thorough understanding of the interplay between interacting electrons. Consequently, we have in this study developed a gradient Hamiltonian model to solve the associated defects pose by the limitations of the single band Hubbard model. Thus, we utilized the single-band HM and the gradient Hamiltonian model to study the behaviour of two interacting electrons on a two dimensional (2D) 9X9 square lattice. It is revealed in this study that the results of the ground-state energies produced by the gradient Hamiltonian model are more favourable when compared to those of the single-band Hubbard model. We have also shown in this work, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength.
We consider an extended one-dimensional Hubbard model in which the magnitude of the hopping between two sites for particles with given spin depends on the occupation of the states with opposite spin at both sites. Diagonalizing exactly finite-size chains, and using known results of conformal field theory we delimit the regions of parameters for which two particles bind and the pair superconducting correlation functions are the dominant ones at large distances. For Coulomb repulsion U smaller than a critical density-dependent value
Uc and any density, there are ranges of the ratios of the hopping parameters for which the superconducting fluctuations dominate. At half filling, for parameters within this range, a transition from a regime with dominant superconducting correlations to an insulating state takes place as a function of U. We also study the model for parameters near a recently found exactly solvable limit in which the number of doubly occupied sites is conserved.