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Physical Review B

Particle-particle response function as a probe for electronic correlations in the p−d Hubbard model

S. Ugenti, Michele Cini, G. Seibold, J. Lorenzana, Enrico Perfetto, Gianluca Stefanucci

Published Aug 27, 2010• 9 citations
Publisher

Abstract

We discuss and compare different approximations to the particle-particle response function in the $p\text{\ensuremath{-}}d$ (three-band) Hubbard model for the ${\text{CuO}}_{2}$ plane of superconducting cuprates. Besides the relevance for understanding the role of correlations in high-${T}_{c}$ superconductors, the interest in the ${\text{CuO}}_{2}$ plane is due to the presence of three incompletely filled valence bands. The bare ladder approximation (BLA) was employed long ago in the context of Auger core-valence-valence spectroscopy of late transition metals while the time-dependent (TD) Gutzwiller approximation (GA) is a much more sophisticated and recent development. The validity of both is assessed by comparing with exact-diagonalization results from a finite six-site cluster. We find that for standard parameter sets TDGA and BLA yield two-hole spectra in excellent agreement with the exact ones. Although the interaction is comparable to the kinetic energy, the system is far from the extreme Mott limit often assumed in cuprates, where the Mott insulating character is completely local. In order to identify possible fingerprints of the extreme Mott regime we artificially reduce the bandwidth. We find that the BLA breaks down while the TDGA keeps near the exact results. Our findings provide a simple criterion to identify doped and undoped extreme Mott insulators.

DOI
10.1103/physrevb.82.075137

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We discuss and compare different approximations to the particle-particle response function in the p\ensuremath-dp\text{\ensuremath{-}}dp\ensuremath-d (three-band) Hubbard model for the CuO2{\text{CuO}}_{2}CuO2​ plane of superconducting cuprates. Besides the relevance for understanding the role of correlations in high-Tc{T}_{c}Tc​ superconductors, the interest in the CuO2{\text{CuO}}_{2}CuO2​ plane is due to the presence of three incompletely filled valence bands. The bare ladder approximation (BLA) was employed long ago in the context of Auger core-valence-valence spectroscopy of late transition metals while the time-dependent (TD) Gutzwiller approximation (GA) is a much more sophisticated and recent development. The validity of both is assessed by comparing with exact-diagonalization results from a finite six-site cluster. We find that for standard parameter sets TDGA and BLA yield two-hole spectra in excellent agreement with the exact ones. Although the interaction is comparable to the kinetic energy, the system is far from the extreme Mott limit often assumed in cuprates, where the Mott insulating character is completely local. In order to identify possible fingerprints of the extreme Mott regime we artificially reduce the bandwidth. We find that the BLA breaks down while the TDGA keeps near the exact results. Our findings provide a simple criterion to identify doped and undoped extreme Mott insulators.
We calculate the photoemission and inverse photoemission spectra of holes in copper-oxygen planes, the characteristic structural unit of high-temperature superconductors. The computations are based on the extended Hubbard or Emery model in the limit of infinitely strong Coulomb repulsion Ud{\mathit{U}}_{\mathit{d}}Ud​ on copper sites. In order to evaluate the corresponding one-particle correlation functions the projection technique is applied, which is especially suitable for strongly correlated systems. Thereby the hole dynamics is restricted to a subspace of relevant operators, within which it can be treated exactly. In contrast to independent-particle approximations the excitation spectra show an energy gap at half-filling, which disappears when the system is doped with additional holes. Besides the insulator-to-metal transition the strong correlations lead also to a significant shift of spectral weight to states close to the Fermi energy when the hole concentration increases. Our results are in good agreement with the ones of exact diagonalization studies of the (CuO2{\mathrm{CuO}}_{2}CuO2​)4{)}_{4})4​ cluster with periodic boundary conditions.
On the basis of exact diagonalizations, a comparative study of two-particle optical and magnetic, as well as single-particle, excitations is presented for a two-dimensional (2D) multiorbital Hubbard model. For reasonable parameter sets appropriate for the cuprate superconductors, the single-particle excitations display strongly correlated states related to the Zhang-Rice Cu-O singlet construction. These states define the gap (to the upper Hubbard band) at half-filling and become partially occupied by doping holes in our 2\ifmmode\times\else\texttimes\fi{}2 unit-cell system. The optical results, which are the first quantitative calculations performed for realistic parameters of the three-band Hubbard model, clearly show three allowed optical transitions: (i) itinerant motion of the Cu-O singlets, having (for doping concentrations x\ensuremath{\ne}0) a spectral Drude distribution around \ensuremath{\omega}=0 with spectral weight proportional to x; (ii) unbinding of the O hole from the Cu spin in the singlet. This gives, in particular, a strong absorption peak due to singlet\ensuremath{\rightarrow}nonbonding oxygen transitions, again with relative weight \ensuremath{\sim}x.It is roughly centered at \ensuremath{\omega}\ensuremath{\sim}JKondo{\mathit{J}}_{\mathrm{Kondo}}JKondo​, where JKondo{\mathit{J}}_{\mathrm{Kondo}}JKondo​ is the Kondo coupling and \ensuremath{\Delta} the bare Cu-O charge-transfer energy. It is this singlet unbinding that results in the by far dominant absorption structure between Drude and higher-energy Cu-O transitions and not the often discussed ``mid-infrared'' absorption due to transitions between different singlet states. (iii) Cu-O charge-transfer processes at energy \ensuremath{\sim}\ensuremath{\Delta}+Upd{\mathit{U}}_{\mathit{p}\mathit{d}}Upd​. They show a pronounced excitonic effect due to the p-d interaction Upd{\mathit{U}}_{\mathit{p}\mathit{d}}Upd​ and have a reduced spectral weight shifted to higher energies for increased dopings. Findings (i)--(iii) are in general accordance with recent experimental data. Our study of the low-energy absorption is complemented with a numerical scaling analysis of the Drude weight in 1D, where, in particular, we find an interesting violation of Lenz's law for 4n-site Hubbard rings. Finally, the magnetic structure factor is calculated for the 2D case. For finite doping it contains a peak at 2JKondo{\mathit{J}}_{\mathrm{Kondo}}JKondo​, which should be detectable in experiment.
We evaluate numerically several superconducting correlation functions in a generalized t−Jt-Jt−J model derived for hole-doped CuO2_22​ planes. The model includes a three-site term t′′t''t′′ similar to that obtained in the large UUU limit of the Hubbard model but of opposite sign for realistic O-O hopping. For realistic parameters we obtain strong evidence of superconductivity of predominantly dx2−y2d_{x^2-y^2}dx2−y2​ character. The ground state has a large overlap with a very simple resonating-valence-bond wave function with off-diagonal long-range order. This function reproduces the main features of the magnetic and superconducting correlation functions.
Using density matrix renormalization group calculations, we compare results obtained for the $t\ensuremath{-}J,$ one-band Hubbard, and three-band Hubbard models of a two-leg CuO ladder. Spin and charge gaps, pair binding energies, and effective pair hoppings are calculated for a wide range of parameters. All three models have an insulating state at a filling corresponding to one hole per Cu site. For physically relevant parameters their spin gaps are similar in size but they exhibit quite different charge gaps. We find that the binding energy of a pair of doped holes is of the order of the undoped ladder spin gap for all three models. The main difference between the models is the size of the effective pair hopping, which is significantly larger in the three-band model with parameters appropriate for CuO materials than in the other two models.
The effective two-band $t\ensuremath{-}J$ Hamiltonian for holes in Cu-O planes of cuprates is obtained from the three-band Hubbard Hamiltonian. In this model a doped oxygen hole together with two neighboring copper holes is treated as a composite spin-12\frac{1}{2}21​ three-sites particle. The model is studied by exact diagonalization in a Cu16O32{\mathrm{Cu}}_{16}{\mathrm{O}}_{32}Cu16​O32​ square cluster. Both binding energy and hole-hole correlation function indicate that two holes bind together for J/t>0.11J/t>0.11J/t>0.11 and are predominantly located on the next-nearest oxygen sites in the same Cu-O chain. Electron-phonon interaction is studied in the adiabatic phonons approximation. Copper (\ensuremath{\pi},0) ``half-breathing'' mode is shown to couple strongly to doped holes and increase the hole-hole correlation as well as the binding energy. These results are compared with experimental data.
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