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Physical review. B, Condensed matter

dx2−y2 superconductivity in a model of correlated fermions

Alexander Nazarenko, Adriana Moreo, Elbio R. Dagotto, José A. Riera

Published Jul 1, 1996• 45 citations
Publisher

Abstract

A two-dimensional fermionic model with attractive interactions is discussed. Its form is motivated by the phenomenology of the underdoped high-${T}_{c}$ cuprates, as discussed in previous literature. The exact solution to the two-particle problem leads to a bound state in the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ subspace. Numerical techniques show that the model has strong ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlations in the ground state at low fermionic density. Within a self-consistent random phase approximation diagrammatic study, the model has a $d$-wave superconducting (SC) phase. The density dependence of the critical temperature is calculated. We argue that in the context of $d$-wave SC this model fulfills the role that the attractive on-site Hubbard model has played for $s$-wave SC. We also show that another candidate, the attractive $t\ensuremath{-}U\ensuremath{-}V$ model, is actually not useful as a toy model for $d$-wave SC for a variety of reasons.

DOI
10.1103/physrevb.54.r768

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A two-dimensional fermionic model with attractive interactions is discussed. Its form is motivated by the phenomenology of the underdoped high-Tc{T}_{c}Tc​ cuprates, as discussed in previous literature. The exact solution to the two-particle problem leads to a bound state in the dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ subspace. Numerical techniques show that the model has strong dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​ pairing correlations in the ground state at low fermionic density. Within a self-consistent random phase approximation diagrammatic study, the model has a ddd-wave superconducting (SC) phase. The density dependence of the critical temperature is calculated. We argue that in the context of ddd-wave SC this model fulfills the role that the attractive on-site Hubbard model has played for sss-wave SC. We also show that another candidate, the attractive t\ensuremath−U\ensuremath−Vt\ensuremath{-}U\ensuremath{-}Vt\ensuremath−U\ensuremath−V model, is actually not useful as a toy model for ddd-wave SC for a variety of reasons.
A Model of Correlated Fermions with dx2−y2d_{x^2-y^2}dx2−y2​ Superconductivity
Motivated by the phenomenology of the high-Tc cuprates, a two dimensional fermionic model with attractive interactions is here discussed. The exact solution to the two particle problem leads to a bound state in the dx2−y2d_{x^2 - y^2}dx2−y2​ subspace. Numerical techniques suggest that the model has dx2−y2d_{x^2 - y^2}dx2−y2​ superconductivity (SC) in the ground state at low fermionic density. Within a self-consistent RPA diagrammatic study, the density dependence of the critical temperature is calculated. We argue that in the context of d-wave SC this model fulfills the role that the attractive on-site Hubbard model has played for s-wave SC. We also show that another candidate, the attractive ``t-U-V'' model, which has d-wave SC at the mean-field level is actually not useful as a realization of this family of condensates for a variety of reasons.
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.
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.
Supercomputing correlations in the two dimensional t-J model at zero temperature are evaluated using numerical techniques. At the fermionic density 〈n〉\ensuremath{\sim}1/2, strong signals of dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 superconductivity were observed in the ground state. These conclusions are based on a study of static pairing correlations, the Meissner effect, and flux quantization as indicators of superconductivity. A tentative phase diagram of the two dimensional t-j model is presented.
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.
Inhomogeneities on various length scales suggesting phase separation seem to be a generic feature of hole-doped high-temperature superconducting cuprates. Also firmly established is the d-wave symmetry of the superconducting state. The interplay of superconductivity and phase separation has been noted in previous mean-field studies of correlated electron models. To further explore this issue in a more rigorous approach we have conducted quantum Monte Carlo calculations of a two-dimensional Hubbard model with nearest-neighbor attractive interaction. For a vanishing on-site repulsion, the result agrees with that of Micnas, Ranninger, and Robaszkiewicz [Phys. Rev. B 39, 11 653 (1989)], and indicates that for a given strength of the attractive interaction, homogeneous d-wave superconducting phase can exist only above a critical doping concentration, below which phase separation occurs. A finite on-site repulsion modifies the phase separation behavior. A stronger d-wave pairing correlation is seen in the inhomogeneous phase compared to the homogeneous phase.