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arXiv:cond-mat/9403076 · Physical Review Letters

Bogoliubov quasiparticle excitations in the two-dimensionalt-Jmodel

Y. Ohta, Tomohiro Shimozato, R. Eder, Sadamichi Maekawa

Published Jul 11, 1994• 67 citations
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Abstract

Using a proposed numerical technique for calculating anomalous Green's functions characteristic of superconductivity, we show that the low-lying excitations in a wide parameter and doping region of the two-dimensional t-J model are well described by the picture of dressed Bogoliubov quasiparticles in the BCS pairing theory. The pairing occurs predominantly in the ${\mathit{d}}_{\mathit{x}}^{2}$-${\mathit{y}}^{2}$-wave channel and the energy gap has a size ${\mathrm{\ensuremath{\Delta}}}_{\mathit{d}}$\ensuremath{\simeq}0.15J--0.27J between quarter and half fillings. Opening of the superconducting gap in the photoemission and inverse-photoemission spectra is demonstrated.

DOI
10.1103/physrevlett.73.324

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Using a proposed numerical technique for calculating anomalous Green's functions characteristic of superconductivity, we show that the low-lying excitations in a wide parameter and doping region of the two-dimensional t-J model are well described by the picture of dressed Bogoliubov quasiparticles in the BCS pairing theory. The pairing occurs predominantly in the dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2-wave channel and the energy gap has a size \ensuremathΔd{\mathrm{\ensuremath{\Delta}}}_{\mathit{d}}\ensuremathΔd​\ensuremath{\simeq}0.15J--0.27J between quarter and half fillings. Opening of the superconducting gap in the photoemission and inverse-photoemission spectra is demonstrated.
An exact-diagonalization technique on finite-size clusters is used to study the ground state and excitation spectra of the two-dimensional effective-fermion model, a fictious model of hole quasiparticles derived from numerical studies of the two-dimensional $t\ensuremath{-}J$ model at low doping. We show that there is actually a reasonable range of parameter values where the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing of holes occurs and the low-lying excitation can be described by the picture of Bogoliubov quasiparticles in the BCS pairing theory. The gap parameter of a size ${\ensuremath{\Delta}}_{d}\ensuremath{\simeq}0.13|V|$ (where VVV is the attractive interaction between holes) is estimated at low doping levels. The paired state gives way to the state of clustering of holes for some stronger attractions.
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.
Low-lying one-particle anomalous excitations are studied for Gutzwiller-projected strongly correlated BCS states. It is found that the one-particle anomalous excitations are highly coherent, and the numerically calculated spectrum can be reproduced quantitatively by a renormalized BCS theory, thus strongly indicating that the nature of low-lying excitations described by the projected BCS states is essentially understood within a renormalized Bogoliubov quasiparticle picture. This finding resembles the well-known fact that a Gutzwiller-projected Fermi gas is a Fermi liquid. The present results are consistent with numerically exact calculations of the two-dimensional $t\text{\ensuremath{-}}J$ model as well as recent photoemission experiments on high-TC{T}_{\mathrm{C}}TC​ cuprate superconductors.
An exact-diagonalization technique on small clusters is used to study low-lying excitations and superconductivity in the two-dimensional negative-U Hubbard model. We first calculate the Bogoliubov-quasiparticle spectrum, condensation amplitude, and coherence length as functions of the coupling strength U/t, thereby working out how the picture of Bogoliubov quasiparticles in the BCS superconductors is affected by increasing the attraction. We then define the Cooper-pair operator as a spatially extended composite boson and make a variational evaluation of its internal structure. We thereby calculate the single-particle spectral function of the Cooper pair and obtain the dispersion relation for its translational motion. The dynamical density correlation function of the pairs is also calculated. We thus demonstrate the applicability of our numerical method for gaining insight into low-lying excitations of models for the intermediate coupling superconductivity relevant to cuprate materials.
The quasiparticle and spin excitation spectra in the 2D Hubbard model are calculated self-consistently as functions of momentum on the real frequency axis. Near half filling we obtain in the normal state a pseudogap in the density of states. The pairing state has ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ symmetry. Below Tc{T}_{c}Tc​, gaps open rapidly in the frequency spectra of the gap function, the quasiparticle scattering rate, and the density of states. The scattering rate and spin-lattice relaxation rate drop abruptly below Tc{T}_{c}Tc​. Our results are consistent with experiments on ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Cu}{\mathrm{O}}_{4}$.
A new superconducting phase is found in the two-dimensional t - J model near half-filling within the mean-field theory and Gutzwiller approximation. This state has a spatial dependence of order parameters which is interpreted as a mixing of the extended s -wave component with the finite total momentum in addition to the d x 2 - y 2 -wave symmetry. The origin of this peculiar pairing is investigated. There is no node in the gap function in the new state and possible experimental consequences are discussed.