Variational Studies on the Hubbard Model within the FHNC Theory

The Fermi Hypernetted Chain (FH NC) theory has been applied to variationally study the ground state properties of the Hubbard model. First of all, we found that the commonly used FHNC/n scheme is not efficient in the case of correlated wave functions with either strong quantum spin fluctuations or strong on-site correlations. In order to overcome this difficulty, we have developed a FH NC scheme, denoted as FH NC-n scheme, which has been found to be superior to FHNC/n in treating Jastrow correlated spin density wave (SDW) functions. In the case of the one-dimensional Hubbard model, the results are in good agreement with the available variational Monte Carlo (VMC) results. For the peculiar type of the on-site correlation operator, we propose a Gutzwiller Hypernetted Chain (GHNC) theory, which sums up all set of FHNC diagrams without violation of the Pauli principle. Subdiagrams in the GH NC theory are equivalent to diagrams in the perturbation theory of the Hubbard model and the G H N C theory can be also regarded as a unified scheme to study various issues investigated by Vollhardt and coworkers for the Gutzwiller correlated wave functions. Compared with the work of Vollhardt and coworkers, the GH NC theory is more easily applicable to higher dimensions and can provide better numerical results and goes beyond Random Phase approximation self-consistently. Moreover, based on the GHNC theory, one can incorporate an additional Jastrow operator for inter-site correlation on the correlated Gutzwiller wave functions (JGCWF), which is handled diagrammatically in the FHNC/GHNC theory. Compared with the FHNC theory, the GHNC theory is efficiently used in the whole range of the coupling constant as far as the VMC simulation. Based on the version of the G H N C theory, the CB F theory can be implemented for the Hubbard model. Finally, from existing results on the asymptotic behavior of one-dimensional Hubbard model and t - J model, we discuss long range behavior of the Jastrow correlations in connection with the exponent behavior of momentum distribution and the correlation functions for JGCWF. In summary, these are the main new results contained in this thesis: 1. A new FHNC-n scheme is developed for correlated SDW functions. The results are in good agreement with VMC results. 2. A G H N C theory is constructed to evaluate the binding energy and various correlation functions for Gutzwiller correlated wave functions ( GCWF) and also to provide a very general framework to interpret various issues studied by Vollhardt's group. Exact solutions to the GHNC theory are found in the one- and infinite- dimensional cases. It is very important that the GHNC theory self-consistently incorporates 1/ D312 correction in the evaluation of the physical quantities. The GHNC-ns scheme can be efficiently applied in the two- and three- dimensional cases for the Hubbard model. 3. A new Jastrow-Gutzwiller wave function is proposed for the Hubbard model, which significantly improves the critical behavior of the Gutzwiller wave function.