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Physical review. B, Condensed matter

Exact two-particle effective interaction and superconductivity in the two-level Hubbard model

R. Medina, Umberto Marini Bettolo Marconi, Sergio J Sciutto

Published Mar 1, 1989• 7 citations
Publisher

Abstract

A two-dimensional two-level Hubbard model with on-site repulsions only, is studied. By solving the two-particle problem, the effective interaction in the lower band is calculated exactly in the empty band limit. This interaction is found to consist of two terms: an on-site repulsion and a nearest-neighbor resonant superexchange. Within the Hartree-Fock approximation it is found that high-${T}_{c}$ s-wave superconductivity develops in a variety of situations. The validity of this model for the high-${T}_{c}$ Cu-O superconductors is discussed.

DOI
10.1103/physrevb.39.4277

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Physical review. B, Condensed matter•Feb 1, 1989•49 citations
A two-dimensional two-level Hubbard model with on-site repulsions only, is studied. By solving the two-particle problem, the effective interaction in the lower band is calculated exactly in the empty band limit. This interaction is found to consist of two terms: an on-site repulsion and a nearest-neighbor resonant superexchange. Within the Hartree-Fock approximation it is found that high-Tc{T}_{c}Tc​ s-wave superconductivity develops in a variety of situations. The validity of this model for the high-Tc{T}_{c}Tc​ Cu-O superconductors is discussed.
The authors study a simple model for superconductivity based on an extended Hubbard model with on-site repulsive and inter-site attractive interaction for arbitrary electron density. The two-dimensional square lattice d-wave pairing is most stable for a nearly half-filled band. p-wave and then extended s-wave pairing become stable upon decreasing band filling. Competition between the various anisotropic pairings and antiferromagnetism as well as the effect of next-nearest-neighbour hopping is discussed. The authors relate the present model to high-T c superconducting oxides.
Quasiparticle bands of the two-dimensional Hubbard model are calculated using the Roth two-pole approximation to the one-particle Green's function. Excellent agreement is obtained with recent Monte Carlo calculations, including an anomalous volume of the Fermi surface near half-filling, which can possibly be explained in terms of a breakdown of Fermi liquid theory. The calculated bands are very flat around the (\ensuremath{\pi},0) points of the Brillouin zone in agreement with photoemission measurements of cuprate superconductors. With doping there is a shift in spectral weight from the upper band to the lower band. The Roth method is extended to deal with superconductivity within a four-pole approximation allowing electron-hole mixing. It is shown that triplet p-wave pairing never occurs. A self-consistent solution with singlet ${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$-wave pairing is found and optimal doping occurs when the van Hove singularity, corresponding to the flat band part, lies at the Fermi level. Nearest-neighbor antiferromagnetic correlations play an important role in flattening the bands near the Fermi level and in favoring superconductivity. However, the mechanism for superconductivity is a local one, in contrast to spin-fluctuation exchange models. For reasonable values of the hopping parameter the transition temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ is in the range 10--100 K. The optimum doping ${\mathrm{\ensuremath{\delta}}}_{\mathit{c}}$ lies between 0.14 and 0.25, depending on the ratio U/t. The gap equation has a BCS-like form and 2${\mathrm{\ensuremath{\Delta}}}_{\mathrm{max}}$/kTc{\mathit{kT}}_{\mathit{c}}kTc​\ensuremath{\simeq}4.
In this paper, we analyze a simple extended Hubbard model with an intermediate on-site interaction (both repulsive and attractive) and a weak intersite attractive interaction. Following Hubbard decoupling approximations and introducing Hubbard subband operators, we obtain a generalized gap function for singlet s-wave pairing that explicitly depends on the Hubbard subband energies. For the on-site repulsive-interaction case, we find that the superconductivity is not destroyed in the intermediate-interaction regime, contrary to the prediction of a Hartree-Fock mean-field treatment. The essential consequence of the on-site repulsion is the formation of the Hubbard subbands separated by the Mott-Hubbard gap, and it is within these subbands that pairing induced by the intersite interaction occurs. For the attractive on-site interaction case, the on-site pairing amplitude is found to be proportional to the bandwidth, and the gap function has contributions from both on-site and intersite pairing. The relevance of the model to high-temperature superconductivity is discussed.
Extensive numerical studies have demonstrated that the two-dimensional single-band Hubbard model contains much of the key physics in cuprate high-temperature superconductors. However, there is no definitive proof that the Hubbard model truly possesses a superconducting ground state or, if it does, of how it depends on model parameters. To answer these longstanding questions, we study an extension of the Hubbard model including an infinite-range ddd-wave pair field term, which precipitates a superconducting state in the ddd-wave channel. Using exact diagonalization on 16-site square clusters, we study the evolution of the ground state as a function of the strength of the pairing term. This is achieved by monitoring the fidelity metric of the ground state, as well as determining the ratio between the two largest eigenvalues of the ddd-wave pair/spin/charge-density matrices. The calculations show a ddd-wave superconducting ground state in doped clusters bracketed by a strong antiferromagnetic state at half filling controlled by the Coulomb repulsion UUU and a weak short-range checkerboard charge ordered state at larger hole doping controlled by the next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$. We also demonstrate that negative ${t}^{\ensuremath{'}}$ plays an important role in facilitating ddd-wave superconductivity.
In this work, we study superconductivity in the strong coupling limit of the two-dimensional Hubbard model using a generalization of the Hubbard-I approximation. The results are compared with those obtained by Beenen and Edwards with the two-pole method of Roth, revealing a qualitative agreement between the two approaches. The effect of the hopping parameter t' between next-nearest neighbour sites on the critical temperature is considered. It is shown that the present approach reproduces the relation between t' and the maximum Tc in high temperature superconductors reported by Pavarini et al (2001 Phys. Rev. Lett. 87 047003).
An extended multiband Hubbard model of the two-dimensional CuO2{\mathrm{O}}_{2}O2​ planes of high-Tc{T}_{c}Tc​ superconductors is examined by exact diagonalization of the Hamiltonian for small clusters with periodic boundary conditions. Holes are shown to bind in the limit of large Cu-site and nearest-neighbor Cu-O Coulomb repulsion. A simple explanation of the source of the binding energy is given in this strong-coupling limit. The more realistic case of relatively weak nearest-neighbor repulsion is also examined, and suggestive evidence is presented that the effective attraction will also exist in the thermodynamic limit for this case.