Approaches to the Hubbard Model

This thesis analyzes several theoretical approaches to the one band Hubbard model in hopes of extracting selected physical quantities in limits most closely corresponding to real materials. Along the way, three rather remarkable theorems of a much broader scope are proven. It is hoped that these may be of general interest in a variety of related physical and mathematical disciplines. In chapter one, the well-known mean field theory developed by Affleck and Marston is studied in the presence of a magnetic field. Through a rather straightforward numerical procedure, phase diagrams in t/delta ^ace are generated as a function of field. The results of this study are then extended to a magnetic susceptibility calculation and to the analysis of the phase diagram of fan alternate mean field theory, the "generalized flux phases" proposed by Anderson. Several interesting properties and symmetries of the solutions are then briefly discussed. In chapter two, the Gutzwiller projector is analyzed both analytically and numerically, with the results being used to calculate the momentum density function for a trial wavefunction also proposed by Anderson. Two of the above mentioned theorems are developed in this chapter, the one prescribing the expansion of a general restricted sum in terms of its related unrestricted sums, and the other presenting the exact diagonilization of a component of the projector which is equivalent through a U(1) gauge transformation to the total spin operator. In chapter three, we discuss the exact solutions to the one dimensional Hubbard model first derived by Lieb and Wu. From their large U limiting behavior, we extract the phonon scattering matrix elements and first order single particle energies for some finite systems. The third potentially general theorem, which related charge determinants with an arbitrary number of "gaps" between their rows to a comparatively simple function of the corresponding van der Monde determinants, is proven here.