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Physical review. B, Condensed matter

Pair theory of the Hubbard Hamiltonian

H. Barentzen

Published Oct 15, 1983• 7 citations
Publisher

Abstract

A detailed account is given of a recently developed pair theory of the Hubbard Hamiltonian for the case of a half-filled band. The resulting pair Hamiltonian is an exact reformulation of the original Hubbard Hamiltonian in terms of pair operators which satisfy elementary boson commutation relations. The Hamiltonian takes the familiar form of a sum of a quadratic part representing independent-pair energies and a quartic part representing pair interactions. The matrix elements entering the pair Hamiltonian are the pair energies and products of pair wave functions, which are formally given as the solutions of a certain eigenvalue equation. By the solving of the eigenvalue problem, explicit analytical expressions are obtained for both pair energies and wave functions for an arbitrary number of dimensions. For the case $M=1$, where $M$ is the number of spin-down electrons, pair interactions do not enter since there is only one electron-hole pair, and the exact energies of the Hubbard Hamiltonian are obtained. This will be explicitly verified for the one-dimensional case; here we demonstrate that the pair energies agree with the exact energies found by other authors for this case.

DOI
10.1103/physrevb.28.4143

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Journal of the Physical Society of Japan•Jan 15, 1985•39 citations
A detailed account is given of a recently developed pair theory of the Hubbard Hamiltonian for the case of a half-filled band. The resulting pair Hamiltonian is an exact reformulation of the original Hubbard Hamiltonian in terms of pair operators which satisfy elementary boson commutation relations. The Hamiltonian takes the familiar form of a sum of a quadratic part representing independent-pair energies and a quartic part representing pair interactions. The matrix elements entering the pair Hamiltonian are the pair energies and products of pair wave functions, which are formally given as the solutions of a certain eigenvalue equation. By the solving of the eigenvalue problem, explicit analytical expressions are obtained for both pair energies and wave functions for an arbitrary number of dimensions. For the case M=1M=1M=1, where MMM is the number of spin-down electrons, pair interactions do not enter since there is only one electron-hole pair, and the exact energies of the Hubbard Hamiltonian are obtained. This will be explicitly verified for the one-dimensional case; here we demonstrate that the pair energies agree with the exact energies found by other authors for this case.
A pair theory of the Hubbard Hamiltonian is presented for the case of a half-filled band. The original Hamiltonian is reformulated exactly in terms of pair operators which satisfy boson commutation relations. The pair energies and wave functions are obtained explicitly. In the one-dimensional case, these reduce to the results derived by Lieb and Wu for M=1M=1M=1. The pair Hamiltonian offers a better starting point for perturbation treatments or variational calculations.
A novel variational approach is outlined for treating electron correlation in many-particle systems and is applied to the one-band Hubbard model. The authors' treatment is based on the concept of fermion pairs and their representation in terms of ideal-boson operators. In this representation approximate eigen-states of the Hamiltonian are obtained in the form of an antisymmetrised product of pair orbitals. For the ground state of the Hubbard model these orbitals are then determined by the use of the variational principle. Because of the complexity of the resulting equation a simplified ansatz is used for the orbitals, which leads to the same energy expression as is obtained by the traditional Bogolubov method. The trial states used in both approaches are, however, quite different. They find reasonable qualitative agreement with well established results for the Hubbard model in one dimension even in those cases where the Bogolubov or related approaches fail. They are also able to show that in the atomic limit the Bogolubov energy bound is the best upper bound that can be obtained by using a single determinant of pair orbitals. Therefore, the only way to improve the ground-state energy in the atomic limit is by means of configuration interaction.
We have studied the pair susceptibilities for all possible pair wave functions that fit on a two-dimensional (2D) eight-site Hubbard cluster by exact diagonalization of the Hamiltonian. Band fillings corresponding to four and six electrons were studied (two or four holes in the half-filled band) for a wide range of Hubbard interaction strengths and temperatures. Our results show that all pairing susceptibilities are suppressed by the Hubbard repulsion. We have also carried out perturbation-theory calculations which show that the leading-order U2{U}^{2}U2 contributions to the d-wave pair susceptibility suppresses d-wave pairing over a significant temperature range. These results are consistent with recent Monte Carlo results and provide further evidence suggesting that the 2D Hubbard model does not exhibit superconductivity.
A model to describe electronic correlations in energy bands is considered. The model is a generalization of the conventional Hubbard model that allows for the fact that the wave function for two electrons occupying the same Wannier orbital is different from the product of single-electron wave functions. We diagonalize the Hamiltonian exactly on a four-site cluster and study its properties as a function of band filling. The quasiparticle weight is found to decrease and the quasiparticle effective mass to increase as the electronic band filling increases, and spectral weight in one- and two-particle spectral functions is transferred from low to high frequencies as the band filling increases. Quasiparticles at the Fermi energy are found to be more ``dressed'' when the Fermi level is in the upper half of the band (hole carriers) than when it is in the lower half of the band (electron carriers). The effective interaction between carriers is found to be strongly dependent on band filling becoming less repulsive as the band filling increases and attractive near the top of the band in certain parameter ranges. The effective interaction is most attractive when the single-hole carriers are most heavily dressed, and in the parameter regime where the effective interaction is attractive, hole carriers are found to ``undress,'' hence become more like electrons, when they pair. It is proposed that these are generic properties of electronic energy bands in solids that reflect a fundamental electron-hole asymmetry of condensed matter. The relation of these results to the understanding of superconductivity in solids is discussed.
The ground-state energy E and momentum distribution n k of the electrons are expanded from the atomic limit for the half-filled Hubbard model. The coefficients of expansion are represented by the spin correlation functions of the spin 1 / 2 Heisenberg model. Using the spin-wave theory, approximate values of coefficients are calculated for the square lattice and the simple cubic lattice. In one dimension, the theory shows a good agreement with the exact solution. An effective spin Hamiltonian for the half-filled Hubbard model with arbitrary hopping integrals is obtained up to the fifth order. It is shown that the fourth term contains four spin interactions.
Electron correlation is studied in the special case of a single reversed spin in an otherwise fully aligned band in the one-dimensional Hubbard model for the purpose of examining our previous theory which exactly takes account of the three-body correlation. The importance of the three-body correlation is demonstrated through a comparison of our results with those of exact theories. The ground-state energy and the single particle spectra are in good agreement with McGuire-Lieb-Wu's exact solution and also with exact numerical solutions for finite size systems over a wide range of the intraatomic Coulomb integral and the electron number. The properties of the ground state are studied in detail through the calculation of pair correlation functions.