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Physical review. B, Condensed matter

Exact projection-operator formalism for highly correlated systems: Mean-field results for the CuO2 lattice

A. J. Fedro, Yu Zhou, T. C. Leung, B. N. Harmon, Sunil Kumar Sinha

Published Dec 1, 1992• 15 citations
Publisher

Abstract

We present an exact formulation of the single-particle Green's functions for the two-dimensional three-band Hubbard model of the ${\mathrm{CuO}}_{2}$ lattice based on a projection-operator formalism of Mori. The model is general and incorporates the various known local-density-approximation (LDA) interaction parameters. In this paper we present only the mean-field results. The inclusion of self-energy effects will be presented elsewhere. Already, in the mean-field approximation, we find that properties such as hole occupations and magnetic moments compare remarkably well with those from quantum Monte Carlo (MC) calculations. The densities of states are also calculated for various hole dopings, again using the parameters from the LDA calculations. Finally the detailed effect of hopping between different O p orbitals, are also presented.

DOI
10.1103/physrevb.46.14785

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We present an exact formulation of the single-particle Green's functions for the two-dimensional three-band Hubbard model of the CuO2{\mathrm{CuO}}_{2}CuO2​ lattice based on a projection-operator formalism of Mori. The model is general and incorporates the various known local-density-approximation (LDA) interaction parameters. In this paper we present only the mean-field results. The inclusion of self-energy effects will be presented elsewhere. Already, in the mean-field approximation, we find that properties such as hole occupations and magnetic moments compare remarkably well with those from quantum Monte Carlo (MC) calculations. The densities of states are also calculated for various hole dopings, again using the parameters from the LDA calculations. Finally the detailed effect of hopping between different O p orbitals, are also presented.
We present a mean-field calculation of the two-dimensional single-band Hubbard model based on a Mori-projection-operator formalism. Various physical properties including the energy, effective hopping matrix element, local moment, momentum distribution function, and Fermi surface are calculated. The results are compared with those of the Hubbard-I approximation, the four-boson mean-field theory of Kotliar and Rockenstein (SBMF), and quantum Monte Carlo simulations. Systematic improvement over the Hubbard-I approximation is seen, and good agreements are obtained between our calculation and the SBMF theory.
Following previous work on Hubbard and Anderson models, we introduce a simplified three-band Hubbard model for CuO2 planes of high-Tc superconductors. This allows for exact evaluation of single-particle Green's functions within the dynamical mean field approach. Keeping information about the lattice structure through the appropriate bare density of states, we study the existence of magnetic solutions, and their stability with respect to variations of the doping fraction, interaction parameters, and temperature. Normal-state properties of known superconducting cuprates are reproduced, and we show that the model has room for substantial enhancement of N?el temperatures.
Most of the basic ideas of the density-functional theory (DFT) are shown to be unrelated to the fact that the particle density is used as the basic variable. After presenting the general formalism and a simple example unrelated to densities, we discuss various approaches to density-functional theory on a lattice. One has proven useful for the understanding of various fundamental issues of the DFT. The exact Kohn-Sham band gap and the band gap in the local-density approximation (LDA) are compared to the exact results for a one-dimensional model. As the interacting homogeneous system can be solved exactly no further approximation is needed to formulate the LDA.
The single-band Hubbard Hamiltonian [1] is the simplest model of interacting electrons in a lattice.It was developed by John Hubbard, who introduced the model to take into account the local electronic correlations in narrow energy bands.The model has a kinetic term that allows hopping of electrons between atomic sites and a term that considers the on-site interaction between electrons.This model is largely used in solid state physics to study magnetic properties of solids, insulator-metal transition (Mott transition) and high temperature superconductors.However, this is a problem that has exact analytical solution in a very few limiting cases and must be treated approximately or numerically.In this work, we propose a new methodology to analytically solve the Hubbard Hamiltonian, mapping it into a two-site model (the Hubbard dimer).To obtain the Green's function for the lattice, we employed the cumulant expansion technique, using as a 'seed' the exact two-site Green's function (the atomic method) [2].From the Green's function, we obtained the density of states [DOS] and the occupation numbers as functions of the external parameters of the model, and compared the results with the Hubbard I approximation [1].
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.
Single-particle excitation spectra of the two-dimensional Hubbard model on the square lattice near half filling and at zero temperature are investigated on the basis of the self-consistent projection operator method. The method guarantees a high accuracy of the spectra with high energy and high momentum resolutions. It takes into account long-range intersite correlations as well as the strong on-site correlations. Effects of nonlocal excitations are clarified by comparing the results with those of the single-site approximation. The calculated spectra verify the quantum Monte-Carlo results for finite temperatures. The spectra at the Fermi level yield a hole-like Fermi surface in the underdoped region and an electron-like Fermi surface in the overdoped region. From a numerical analysis of the momentum dependent effective mass and self-energy, it is concluded that a marginal Fermi-liquid like state persists even at finite doping concentrations in the strongly correlated region because a van Hove singularity is pinned to the Fermi surface. It is also found that a kink structure appears in the quasiparticle energy band in the same region. The kink is shown to be caused by a mixing between the quasiparticle band and an excitation band with strong short-range antiferromagnetic correlations. These results suggest an explanation for some of the unusual properties of the normal state in high-TcT_{\rm c}Tc​ cuprates.