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Physical review. B, Condensed matter

Derivation of a one-band Hubbard model for CuO planar materials

Silvia B. Bacci, Eduardo R. Gagliano, Richard M. Martin, James F. Annett

Published Oct 1, 1991• 47 citations
Publisher

Abstract

We have performed numerical studies of multiband Hubbard models for the Cu-O planes of the high-${\mathit{T}}_{\mathit{c}}$ materials. For the range of realistic parameters, all low-energy (\ensuremath{\le}2 eV) charge and spin excitations are described by the single-band (possibly extended) Hubbard model with U\ensuremath{\simeq}W, where W is the bandwidth.

DOI
10.1103/physrevb.44.7504

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We have performed numerical studies of multiband Hubbard models for the Cu-O planes of the high-Tc{\mathit{T}}_{\mathit{c}}Tc​ materials. For the range of realistic parameters, all low-energy (\ensuremath{\le}2 eV) charge and spin excitations are described by the single-band (possibly extended) Hubbard model with U\ensuremath{\simeq}W, where W is the bandwidth.
Key questions for theory of high temperature superconductors involve the nature of the electronic system: the minimum essential Hamiltonian and the low energy states of such Hamiltonians. To attempt to answer these questions we are carrying out studies using constrained density functional (CDF) methods to identify the relevant electronic states and realistic values of the parameters, combined with exact calculations on the reduced model Hamiltonians. The CDF calculations lead to a 3-band Hubbard model for Cu-O planes with parameters in general agreement with high energy spectroscopies and low energy magnetic excitations. We describe a modified Lanczos method to carry out large many-body calculations and give a representative result, where we find the spectra are described by the 1-band Hubbard model in regimes which imply spin interactions beyond the usual Heisenberg form.
The so-called three-band Hubbard model is generally believed to be a good model for fermions on a CuO 2 lattice and to contain key features allowing to understand high-temperature superconductivity in the cuprates. A simpler and more popular model is the one-band Hubbard model. In this thesis,an extended one-band Hubbard model is derived from the three-band model. First, some mathematical background is given, as well as an introduction to Hubbard-like models. Then, to derive the extended one-band Hubbard model, the CuO 2 lattice is divided into clusters of one Cu and two O sites, and a variant of the Feshbach method is used to replace each such cluster by a single lattice site. To nd a suitable one-band model, all Hamiltonian matrix elements are matched between the two models. Finally, dierent sets of three-band parameters are considered, and it is studied how this aects the parameters in the extended one-band Hubbard model.
We calculate the properties of the two-band Hubbard model using the dynamical cluster approximation. The phase diagram resembles the generic phase diagram of the cuprates, showing a strong asymmetry with respect to electron- and hole-doped regimes, in agreement with experiment. Asymmetric features are also seen in one-particle spectral functions and in the charge, spin, and ddd-wave pairing-susceptibility functions. We address the possible reduction of the two-band model to a low-energy single-band one, as it was suggested by Zhang and Rice. Comparing the two-band Hubbard model properties with the single-band Hubbard model ones, we have found similar low-energy physics provided that the next-nearest-neighbor hopping term ${t}^{\ensuremath{'}}$ has a significant value $({t}^{\ensuremath{'}}∕t\ensuremath{\approx}0.3)$. The parameter ${t}^{\ensuremath{'}}$ is the main culprit for the electron-hole asymmetry. However, a significant value of ${t}^{\ensuremath{'}}$ cannot be provided in a strict Zhang and Rice [Phys. Rev. B 37, R3759 (1988); 41, 7243 (1990)] picture where the extra holes added into the system bind to the existing Cu holes forming local singlets. We note that by considering approximate singlet states, such as plaquette states, reasonable values of ${t}^{\ensuremath{'}}$, which capture qualitatively the physics of the two-band model, can be obtained. We conclude that a single-band $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U$ Hubbard model captures the basic physics of the cuprates concerning superconductivity, antiferromagnetism, pseudogap, and electron-hole asymmetry, but is not suitable for a quantitative analysis or to describe physical properties involving energy scales larger than about 0.5 eV.
Strong-coupling models for the electronic structure of La2{\mathrm{La}}_{2}La2​CuO4{\mathrm{CuO}}_{4}CuO4​ are derived from the local-density-functional results in two successive stages of renormalization. First, a three-band Hubbard model is derived with parameters explicitly calculated from first principles using a constrained density-functional approach and a mean-field fit to the Cu-O pd\ensuremath{\sigma} bands. Second, exact diagonalization studies of finite clusters within the three-band Hubbard model are used to select and map the low-energy spectra onto effective one-band Hamiltonians, e.g., the Heisenberg, one-band Hubbard, or ``t-t'-J'' model. At each stage, calculated observables are in quantitative agreement with experiment.
The Hubbard model is one of the most widely studied many-body models for interacting electrons in solids. Here we present a study of the Extended Hubbard model, which provides a more realistic description of real materials by including some longer-range Coulomb interactions. We discuss the derivation of the model, numerical methods for many-body systems, the ground state of the one-dimensional model, and an application of the model to organic charge-transfer solids. Our discussion of numerical methods includes details of using the Constrained Path Monte Carlo procedure to treat extended interactions and possible extension of the method for imaginary times. The chapter on the one-dimensional model focuses on the region V>U, where previous work had suggested the possibility of superconductivity. Our results show that instead phase separation is prevalent in that parameter regime. The second application is to model the organic charge-transfer salts, which are intermediate between one and two dimensions. Here we find a coexistence of charge and spin ordering that helps to explain the results of recent experiments on these materials.
Strong coupling diagram technique equations are derived for hole Green’s functions of the three-band Hubbard model, which describes Cu–O planes of high-[Formula: see text] cuprates. The equations are self-consistently solved in the approximation, in which the series for the irreducible part in powers of the oxygen–copper hopping constant is truncated to two lowest-order terms. For parameters used for hole-doped cuprates, the calculated energy spectrum consists of lower and upper Hubbard subbands of predominantly copper nature, oxygen bands with a small admixture of copper states and the Zhang–Rice states of mixed nature, which are located between the lower Hubbard subband and oxygen bands. The spectrum contains also pseudogaps near transition frequencies of Hubbard atoms on copper sites.