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.
Highlights
There is a need to understand the anti-ferromagnetic properties in the Hubbard model of strongly correlated electron systems
This is because a proper understanding of the procedure for obtaining the ground state energy and wave functions will lay a good foundation for the calculation of other low energy ground state properties of the interaction of higher number of electrons
Great effort was made in the review work of Dagotto (1994) to present several techniques that has been applied in the extensive study of highly correlated electron systems using the Hubbard model
Summary
There is a need to understand the anti-ferromagnetic properties in the Hubbard model of strongly correlated electron systems. According to Dagotto’s review paper, two methods were found to be self consistent; these were the mean free theories and the variation approximations, but there were no standard ways to judge if they describe the properties of the ground state or if they converge to excited states. Of these aforementioned techniques, the exact method provide the exact solution, because of the exponential growth of the Hilbert space with the lattice size, exact calculation becomes inadequate to handle relatively large clusters. This fueled the need to look for an adequate approximation method that can produce near accurate results
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