We explore how the superconductivity arising from the on-site electron-electron repulsion changes when the repulsion is made long-ranged, 1∕r-like interaction by introducing an extended Hubbard model with the repulsion extended to distant (12th) neighbors. With a simplified fluctuation-exchange approximation, we have found for the square lattice that: (i) As the band filling becomes dilute enough, the charge susceptibility becomes comparable with the spin susceptibility, where p- and then s-wave pairings become relatively dominant, in agreement with the result for the electron gas by Takada, while (ii) the d wave, which reflects the lattice structure, dominates well away from the half-filling. All of these can be understood in terms of the spin and charge structures along with the shape and size of the Fermi surface.
In order to explore how superconductivity arises when charge fluctuations and spin fluctuations coexist, we have obtained a phase diagram against the off-site repulsion V and band filling n for the extended, repulsive Hubbard model on the square lattice with the fluctuation exchange approximation. We have found the existence of (i) a transition between dxy and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing symmetries, (ii) f-pairing in between the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and CDW phases for intermediate 0.5<n<1.0 and large V, and (iii) for anisotropic cases the pairing symmetry changing, in agreement with a previously proposed ``generic phase diagram,'' as $d\ensuremath{\rightarrow}f\ensuremath{\rightarrow}s$ when V (hence the charge fluctuations) are increased. All these are consequences of the structure in the charge and spin susceptibilities, which have peaks habitating at common or segregated positions in k space.
The leading superconducting instabilities of the two-dimensional extended repulsive one-band Hubbard model within spin-fluctuation pairing theory depend sensitively on electron density, band, and interaction parameters. We map out the phase diagrams within a random-phase-approximation spin- and charge-fluctuation approach, and find that while B1g (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) and B2g (dxy) pairing dominates in the absence of repulsive longer-range Coulomb interactions VNN, the latter induces pairing in other symmetry channels, including, e.g., A2g (g-wave), nodal A1g (extended s-wave), or nodal Eu (p-wave) spin-triplet superconductivity. At the lowest temperatures, transition boundaries in the phase diagrams between symmetry-distinct spin-singlet orders generate complex time-reversal symmetry broken superpositions. By contrast, we find that boundaries between singlet and triplet regions are characterized by first-order transitions. Finally, motivated by recent photoemission experiments, we have determined the influence of an additional explicitly attractive nearest-neighbor interaction, VNN<0, on the superconducting gap structure. Depending on the electronic filling, such an attraction boosts Eu (p-wave) spin-triplet or B1g (${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$) spin-singlet ordering.
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=0s-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.
We study the square-lattice extended Hubbard model with on-site U and nearest-neighbor V interactions by exact diagonalization. We show that nonequilibrium quench dynamics can help determine the equilibrium phase transition boundaries, which agree with the calculations of the fidelity metric, dynamical structure factor, and correlation function. At half filling, the phase diagrams in the strong-coupling regime include spin density wave and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave superconductivity at large positive U, charge density wave (extended s∗-wave superconductivity) at large positive (negative) V, and s-wave superconductivity at large negative U with vanishing V. The energies of different particle sectors also help determine the phase separation region. With carrier doping, charge fluctuation results in a strong competition between different orders, making it more difficult to identify the leading instability on finite-size clusters. Nevertheless, the more exotic p-wave superconducting pairing is found to be enhanced when the system is heavily overdoped by 37.5 holes, especially in the interaction parameter range relevant to the cuprate superconductors.
The one-band extended Hubbard model in two dimensions near band-filling 21 is solved in the fluctuation exchange approximation, including the long-range ( 1/r) part of the Coulomb interaction, up to 4th-neighbor distance. Our results suggest that ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ pairing in the Hubbard model is robust against the inclusion of long-range Coulomb interactions with moderate 1st-neighbor repulsion strength V1. ${d}_{{x}^{2}{\ensuremath{-}y}^{2}}$ pairing is suppressed only at large V1 ( $\ensuremath{\gtrsim}0.25U--0.4U$), due to incipient charge density wave instabilities.