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Greener Journal of Science Engineering and Technological Research

Electron Interaction at Low Filling in Two Dimensions Using a Simplification of the Modified Lanczos Technique

E. O. Osafile, D.O. Ojuh, Joa Idiodi

Published Apr 20, 2013• 0 citations
Publisher

Abstract

In this paper, a detailed study of the dynamics of the behavior of two electrons in two dimensions is carried out. Over the years, the Hubbard model has been heavily used as one of the standard models for the study of strongly correlated electron systems. Despite this, the mechanism of interaction of these electrons in two- dimensional systems is yet to be properly understood. Based on this clarity problem, a modification of the standard Lanczos algorithm which is slightly different, simpler and more analytical than that presented by Dagotto in his review paper is used in a concerted attempt to produce relevant results for the correlated ground state energies and wave functions and the behavior of the interacting pairs is explained with the aid of the well known single band Hubbard model. Hence this study has been able to develop, stabilize and utilize a simplified modification of the standard Lanczos technique. From results obtained, for large and positive values of u, interaction is repulsive while it is attractive for negative values of u. The wave function can be readily obtained at the end of each step of iteration.

DOI
10.15580/gjsetr.2013.4.010613361

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In this paper, a detailed study of the dynamics of the behavior of two electrons in two dimensions is carried out. Over the years, the Hubbard model has been heavily used as one of the standard models for the study of strongly correlated electron systems. Despite this, the mechanism of interaction of these electrons in two- dimensional systems is yet to be properly understood. Based on this clarity problem, a modification of the standard Lanczos algorithm which is slightly different, simpler and more analytical than that presented by Dagotto in his review paper is used in a concerted attempt to produce relevant results for the correlated ground state energies and wave functions and the behavior of the interacting pairs is explained with the aid of the well known single band Hubbard model. Hence this study has been able to develop, stabilize and utilize a simplified modification of the standard Lanczos technique. From results obtained, for large and positive values of u, interaction is repulsive while it is attractive for negative values of u. The wave function can be readily obtained at the end of each step of iteration.
The major characteristic of the single-band Hubbard model (HM) is to redistribute electrons at a uniform lattice separation distance within the molecular lattice. Hence, it is only linearly dependent on lattice separations distance. Thus the single-band Hubbard model does not consider the lattice gradient encountered by interacting electrons as they hop from one lattice point to another. The linear dependence of the single-band HM only on lattice separations would certainly not provide a thorough understanding of the interplay between interacting electrons. Consequently, we have in this study developed a gradient Hamiltonian model to solve the associated defects pose by the limitations of the single band Hubbard model. Thus, we utilized the single-band HM and the gradient Hamiltonian model to study the behaviour of two interacting electrons on a two dimensional (2D) 9X9 square lattice. It is revealed in this study that the results of the ground-state energies produced by the gradient Hamiltonian model are more favourable when compared to those of the single-band Hubbard model. We have also shown in this work, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength.
In this work, we developed the unit step model as an approximate solution to the single-band Hubbard Hamiltonian to solve variationally strongly correlated interacting elections on a two-dimensional (2D) square lattice. We also showed primarily how to derive possible electronic states available for several 2D N x N square lattices, although, with special emphasis on a 2D 5 x 5 square lattice. The results emerging from our present study was compared with the results of Gutzwiller variational approach (GVA) and correlated variational approach (CVA), at the large limit of the Coulomb interaction strength (U/4t).The approximation to the Hubbard Hamiltonian study is actually necessary because of the strong limitation and difficulty pose by the Hubbard Hamiltonian as we move away from finite - size lattices to larger N - dimensional lattices. Thus this work has provided a means of overcoming the finite - size lattice defects as we pass on to a higher dimension. We have shown in this study, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength. This study reveals that when the Coulomb interaction is zero, that is, for free electron system (non-interacting), the variational parameters which describe the probability distribution of lattice electron system is the same. The spectra intensity increases with increase in the interaction strength and it decreases to zero when the interaction strength is made negatively large. Keywords : unit step Hamiltonian, Hubbard Hamiltonian, 3D cubic lattice, interaction strength, total energy, lattice separation.
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.
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.