Strong-coupling models for the electronic structure of
La2CuO4 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.
Spin-only descriptions of the half-filled one-band Hubbard model are relevant for a wide range of Mott insulators. In addition to the usual Heisenberg exchange, many other types of interactions, including ring exchange, appear in the effective Hamiltonian in the intermediate coupling regime. In order to improve on the quantitative description of magnetic excitations in the insulating antiferromagnetic phase of copper-oxide (cuprate) materials, and to be consistent with band-structure calculations and photoemission experiments on these systems, we include second- and third-neighbor hopping parameters, ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, into the Hubbard Hamiltonian. A unitary transformation method is used to find systematically the effective Hamiltonian and any operator in the spin-only representation. The results include all closed four-hop electronic pathways in the canonical transformation. The method generates many ring exchange terms that play an important role in the comparison with experiments on
La2CuO4. Performing a spin-wave analysis, we calculate the magnon dispersion as a function of
U,
t, ${t}^{\ensuremath{'}}$, and ${t}^{\ensuremath{''}}$. The four parameters are estimated by fitting the magnon dispersion to the experimental results of Coldea et al. [Phys. Rev. Lett. 86, 5377 (2001)] for
La2CuO4. The ring exchange terms are found essential, in particular to determine the relative sign of ${t}^{\ensuremath{'}}$ and ${t}^{\ensuremath{''}}$, with the values found in good agreement with independent theoretical and experimental estimates for other members of the cuprate family. The zero-temperature sublattice magnetization is calculated using these parameters and also found to be in good agreement with the experimental value estimated by Lee et al. [Phys. Rev. B 60, 3643 (1999)]. We find a value of the interaction strength $U\ensuremath{\simeq}8t$ consistent with Mott insulating behavior.
Constrained density-functional theory (CDFT) has been used by several authors for determining model Hamiltonian parameters of high- superconductors and other transition metal compounds. These methods use the fact that the orbitals are well localized and can easily be split into atomic components. We generalize them to systems with relatively delocalized electrons, so that they can be used, for example, for determining parameters for a Hubbard model of conducting polymers (e.g., polyacetylene). We selected the molecule and a linear C chain as test cases. The results of the constrained density-functional calculations were fitted to Hartree-Fock model calculations of the Hubbard model. In the case of we also fitted the CDFT results to exact solutions of the Hubbard model. For we found a substantial discrepancy between the U -values from the exact and the Hartree-Fock model calculations, which shows that it is important to specify the approximations that are used in determining the model parameters. On the other hand, the CDFT calculations themselves lead to roughly uniquely defined model parameters.
We perform a low-energy reduction of the three-band Hubbard Hamiltonian (
H3b), keeping in the relevant Hilbert subspace not only local singlets (Zhang-Rice singlets), but also triplet states between Cu holes and O holes at the Wannier function of the same site, with
x2-
y2 symmetry. We solve exactly the resulting Hamiltonian
HT in a system of 2\ifmmode\times\else\texttimes\fi{}2 unit cells. From the analytical dependence of the parameters of
HT and the numerical results, one can see that the local triplet states can be practically neglected for finite O-Cu on-site energy difference \ensuremath{\Delta}, very large Cu on-site Coulomb repulsion
Ud, and O-O hopping
tpp=0. This fact is in contrast with the mapping of
H3b to a one-band model using nonorthogonal singlets, which is very accurate when the
Cu+ configuration can be neglected. Although the amount of local triplet states in the low-energy eigenstates is in general small, it increases with
tpp and for large
tpp it is necessary to introduce higher-order corrections in the one-band model to accurately represent the low-energy physics. In all cases even when local triplets are not important, the t-J model should be supplemented with other terms, to describe the lowest-energy levels. We also discuss briefly the effect of nonbonding O orbitals.
Various methods and approximation schemes are used to study many-electron interacting systems. Two important many-particle models, the Anderson model and the Hubbard model, and their electromagnetic properties have been investigated in many parameter regimes, and applied to physical systems. An Anderson single-impurity model Hamiltonian based calculation of the magnetic susceptibility is performed for YbN in the presence of crystal fields using an alteration of the Non-Crossing Approximation proposed by Zwicknagl et.al., incorporating parameters obtained from ab initio band structure calculations. It yields good agreement with experimental data. For the Anderson lattice model, a variational scheme which uses specific many-electron wavefunctions as basis is applied to both one- and two-dimensional systems represented by symmetric Anderson lattice Hamiltonians. Without much computational effort, the ground state energy is well approximated, especially in strong-coupling limit. Some electronic properties are examined using the variational ground state wavefunction. The one-dimensional Hubbard model has been solved exactly for small-size clusters by diagonalizing the Hamiltonian in the basis of many-electron Bloch states. The results for the energy spectrum and eigenfunctions of the ground state and low-lying excited states are presented. Also, mean field calculations of the two-dimensional single-band Hubbard model and Cu-O lattice model (three-band Hubbard model) are carried out for various physical quantities including the energy, occupation probability, staggered magnetization, momentum distribution Fermi surface and density of states, by using a projection operator formalism. To develop a systematic approach to solving many-electron problems, the many-particle partition function for the free electron gas system is explored using a cumulant expansion scheme. Starting from the ground state, the partition function can be approximated to any order in terms of excitation energy. Its application to interacting systems such as the Anderson model and the Hubbard model is briefly discussed.
We have performed numerical studies of multiband Hubbard models for the Cu-O planes of the high-
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.