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arXiv:cond-mat/0402386 · Physical Review B

Superconductivity in the quasi-two-dimensional Hubbard model

Xin-Zhong Yan

Published Mar 29, 2005• 17 citations
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Abstract

On the basis of spin-pairing-fluctuation-exchange approximation, we study the superconductivity in the quasi-two-dimensional Hubbard model. The integral equations for the Green's function are self-consistently solved by numerical calculation. Solutions for the order parameter, London penetration depth, density of states, and transition temperature are obtained. Some of the results are compared with the experiments for the cuprate high-temperature superconductors. Numerical techniques are presented in detail. With these techniques, the amount of numerical computations can be greatly reduced.

DOI
10.1103/physrevb.71.104520

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On the basis of spin-pairing-fluctuation-exchange approximation, we study the superconductivity in the quasi-two-dimensional Hubbard model. The integral equations for the Green's function are self-consistently solved by numerical calculation. Solutions for the order parameter, London penetration depth, density of states, and transition temperature are obtained. Some of the results are compared with the experiments for the cuprate high-temperature superconductors. Numerical techniques are presented in detail. With these techniques, the amount of numerical computations can be greatly reduced.
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.
A progress report on the development of numerical and rigorous solutions of the effective two-band two-dimensional Hubbard model of the high-temperature superconductivity in cuprates [N.M. Plakida et al., PRB 51, 16599 (1995)], within the equation of motion technique of the thermodynamic Green function approach, is given. Four important consequences of the rigorous mean-field solution are worth mentioning: (1) Occurrence, both in the normal and superconducting states, of the spin-charge separation, conjectured by P.W. Anderson on intuitive grounds. (2) Origin of the static d-wave pairing in the hopping singlet conduction. (3) Complete suppression of the s-type superconducting pairing by the crystal symmetry in square two- dimensional lattices and occurrence of a residual s-type term in rectangular lattices, in agreement with the experiment. (4) The energy band hybridization in the superconducting state points to an overall displacement of the energy spectrum toward lower energies, as pointed by very precise optical measurements.
An interplay between the superconductivity and spin density wave (SDW) is studied from the perturbative point of view based on the two dimensional Hubbard model. It is shown that the d-pairing superconductivity is induced by spin fluctuations near the SDW boundary. The superconducting transition temperature ( T c ) is found to be remarkably suppressed by the renormalization through the electron self-energy, and to be sensitive to the band parameters due to nesting of the Fermi surface and the presence of the van Hove singularity as well as to the Fermi energy and the on-site interaction. As a consequence T c is enhanced by the next-nearest-neighbour hopping integral. The relation between the present result and experiments of high T c superconductors is discussed.
Results of a conserving fluctuation exchange calculation for the superconducting state of a two-dimensional Hubbard model are presented. Self-consistent solutions for the full momentum- and frequency-dependent gap, renormalization, and frequency shift parameters have been obtained. The resulting frequency- and momentum-dependent gap is found to have dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 symmetry. Below Tc{\mathit{T}}_{\mathit{c}}Tc​, the maximum gap opens rapidly to a low temperature 2\ensuremath{\Delta}(0)/kTc{\mathit{kT}}_{\mathit{c}}kTc​ value of order 10. We find that the effective interaction arising from the exchange of spin and charge fluctuations is stabilized by self-energy feedback effects as the superconducting gap opens.
A systematic diagrammatic expansion for Gutzwiller-wave functions (DE-GWF) is formulated and used for the description of superconducting (SC) ground state in the two-dimensional Hubbard model with electron-transfer amplitudes ttt (and ${t}^{\ensuremath{'}}$) between nearest (and next-nearest) neighbors. The method is numerically very efficient and allows for a detailed analysis of the phase diagram as a function of all relevant parameters (UUU, $\ensuremath{\delta}$, ${t}^{\ensuremath{'}}$) and a determination of the kinetic-energy driven pairing region. SC states appear only for substantial interactions, $U/t\ensuremath{\gtrsim}3$, and for not too large hole doping, $\ensuremath{\delta}\ensuremath{\lesssim}0.32$ for ${t}^{\ensuremath{'}}=0.25t$; this upper critical doping value agrees well with experiment for the cuprate high-temperature superconductors. We also obtain other important features of the SC state: (i) the SC gap at the Fermi surface resembles ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave only around the optimal doping and the corrections to this state are shown to arise from the longer range of the pairing; (ii) the nodal Fermi velocity is almost constant as a function of doping and agrees quantitatively with the experimental results; (iii) the SC transition is driven by the kinetic-energy lowering for low doping and strong interactions.
A microscopic theory of electronic spectrum and superconducting pairing within the Hubbard model is formulated.The Dyson equation for the normal and anomalous Green functions in terms of the Hubbard operators is derived by applying the Mori-type projection technique.The self-energy is evaluated in the noncrossing approximation for electron scattering on spin and charge fluctuations induced by kinematic interaction for Hubbard operators.Numerical results for electron dispersion in the strong correlation limit are presented.Superconducting pairing mediated by antiferromagnetic exchange and spin fluctuations is discussed.