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arXiv:cond-mat/9809187 · Physical review. B, Condensed matter

dx2−y2 superconductivity in a generalized Hubbard model

Liliana Arrachea, Armando A. Aligia

Published Jan 1, 1999• 31 citations
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Abstract

We consider an extended Hubbard model with nearest-neighbor correlated hopping and next-nearest-neighbor hopping ${t}^{\ensuremath{'}}$ obtained as an effective model for cuprate superconductors. Using a generalized Hartree-Fock BCS approximation, we find that for high enough ${t}^{\ensuremath{'}}$ and doping, antiferromagnetism is destroyed and the system exhibits d-wave superconductivity. Near optimal doping we consider the effect of antiferromagnetic spin fluctuations on the normal self-energy using a phenomenological susceptibility. The resulting superconducting critical temperature as a function of doping is in good agreement with experiment.

DOI
10.1103/physrevb.59.1333

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We consider an extended Hubbard model with nearest-neighbor correlated hopping and next-nearest-neighbor hopping t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ obtained as an effective model for cuprate superconductors. Using a generalized Hartree-Fock BCS approximation, we find that for high enough t\ensuremath′{t}^{\ensuremath{'}}t\ensuremath′ and doping, antiferromagnetism is destroyed and the system exhibits d-wave superconductivity. Near optimal doping we consider the effect of antiferromagnetic spin fluctuations on the normal self-energy using a phenomenological susceptibility. The resulting superconducting critical temperature as a function of doping is in good agreement with experiment.
Based on the known phenomenology of high-Tc cuprates and the available numerical calculations of the t − J model, a two-dimensional effective fermionic model with the nearest neighbor attraction is proposed. Numerical calculations suggest that the model has the dx2−y2 superconductivity (SC) in the ground state at a low fermionic density. We argue that this model captures the important physics of the dx2−y2 superconducting correlations found earlier in the t − J model by the exact diagonalization approach. Within a self-consistent RPA diagrammatic study, the density and the coupling strength dependence of the critical temperature is calculated. We also investigate the influence of the impurities on our results and show that the suppression of the superconductivity is insignificant, when the retardation effects are accounted for as opposed to the Hartree–Fock approximation.
Using numerical diagonalization of a $4\ifmmode\times\else\texttimes\fi{}4$ cluster, we calculate on-site s, extended-s,s,s, and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlation functions (PCF's) in an effective generalized Hubbard model for the cuprates, with nearest-neighbor correlated hopping and next-nearest-neighbor hopping ${t}^{\ensuremath{'}}.$ The vertex contributions to the PCF's are significantly enhanced, relative to the ${t\ensuremath{-}t}^{\ensuremath{'}}\ensuremath{-}U$ model. The behavior of the PCF's and their vertex contributions, and signatures of anomalous flux quantization, indicate superconductivity in the d-wave channel for moderate doping and in the s-wave channel for high doping and small U.
We show in a mean-field calculation that phase diagrams remarkably similar to those recently proposed for the cuprates arise in simple microscopic models of interacting electrons near half-filling. The models are extended Hubbard models with nearest-neighbor interaction and correlated hopping. The underdoped region of the phase diagram features ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ density-wave (DDW) order. In a certain regime of temperature and doping, DDW order coexists with antiferromagnetic order. For larger doping, it coexists with ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ superconductivity. While phase diagrams of this form are robust, they are not inevitable. For other reasonable values of the coupling constants, drastically different phase diagrams are obtained. We comment on implications for the cuprates.
The Hubbard model is an iconic model in quantum many-body physics and has been intensely studied, especially since the discovery of high-temperature cuprate superconductors. Combining the complementary capabilities of two computational methods, we found superconductivity in both the electron- and hole-doped regimes of the two-dimensional Hubbard model with next-nearest-neighbor hopping. In the electron-doped regime, superconductivity was weaker and was accompanied by antiferromagnetic Néel correlations at low doping. The strong superconductivity on the hole-doped side coexisted with stripe order, which persisted into the overdoped region with weaker hole-density modulation. These stripe orders varied in fillings between 0.6 and 0.8. Our results suggest the applicability of the Hubbard model with next-nearest hopping for describing cuprate high–transition temperature ( T c ) superconductivity.
In order to study the material dependence of the correlation between stripes and dx2-y2-wave superconductivity (SC) among cuprate superconductors, we have performed a variational Monte Carlo simulation on a two-dimensional Hubbard model with first-, second- and third-nearest-neighbor hopping terms (t, t’ and t”, respectively) by using the Gutzwiller-Jastrow-type wave function taking account of three-body correlations on a triangle, which is efficient for frustrated systems. The condensation energies of the vertical stripes and the homogeneous SC were computed as a function of t’/t with a fixed relation t”= –t’/2. We have found that, at the nearly optimal doping with moderately strong on-site Coulomb interaction, the state of maximum condensation energy switches from the vertical stripes to the homogeneous dSC as |t’/t| increases.
With high-Tc cuprates in mind, properties of correlated dx2-y2-wave superconducting (SC) and antiferromagnetic (AF) states are studied for the Hubbard (t-t'-U) model on square lattices, using a variational Monte Carlo method. We employ simple trial wave functions including only crucial parameters, such as a doublon-holon binding factor indispensable to describe correlated SC and normal states as doped Mott insulators. U/t, t'/t and δ(doping rate) dependence of relevant quantities are systematically calculated. As U/t increases, a sharp crossover of SC properties occurs at U_co/t \sim 10 from a conventional BCS type to a kinetic-energy-driven type for any t'/t. As δdecreases, U_co/t is smoothly connected to the Mott transition point at half filling. For U/t\lsim 5, steady superconductivity corresponding to the cuprates is not found, whereas the d-wave SC correlation function Pd^\infty rapidly increases for U/t\gsim 6 and becomes maximum at U=U_co. Comparing the δdependence of Pd^\infty with experimentally observed dome-shaped Tc and condensation energy, we find that the effective value of UUU for the cuprates should be larger than the band width, for which the t-J model is valid. Analyzing the kinetic energy, we reveal that for U>U_co only doped holes (electrons) become charge carriers, which will make a small Fermi surface (hole pocket), but for UU_co is determined by two factors, i.e., the AF spin correlation, which creates singlet pairs (pseudogap), and the charge mobility dominated by Mott physics.