Table of Contents:
In this work, we employ the Gutzwiller wave function approach, to approximate the ground-state of a Hubbard model with px-py orbitals on a square lattice. The Gutzwiller variational ground state starts from the independent-particle picture where the electrons are distributed over all lattice sites to optimize the kinetic energy. This statistical distribution leads to atomic configurations that are energetically unfavorable for the Hubbard interaction. In the Gutzwiller wave function, the weight of such configurations is reduced with the help of the Gutzwiller correlator. In this way, we can include local correlations into the ground state of the noninteracting system. As in standard many-body theory, the evaluation of expectation values requires the calculation of diagrams to infinite order. The Gutzwiller correlator permits the setup of a diagrammatic formalism in such a way that, in the infinite dimensional limit, the scaling of the single-particle density matrix leads to a cancellation of all nontrivial diagrams. However, the evaluation of the action of the Gutzwiller correlator in infinite dimensions neglects important spatial correlations of the density matrix. Therefore, we derive the diagrammatic expansion of a general multi-band model in finite dimensions. In finite dimensions, the evaluation of the Gutzwiller wave function on a square lattice requires the evaluation of all diagrams. We compute the ground-state energy up to and including two internal vertices. As applications, we adress (i) the ferromagnetic phase transitions as a function of the band-filling, and (ii) the Fermi surface deformations induced by the interaction. We confirm preliminary findings that ferromagnetism is a phenomena of strongly correlated electrons. In the Gutzwiller wave function, the ferromagnetic order is strongly suppressed so that much larger interaction strength are needed than predicted by the Hartree-Fock solution. Moreover, the regions in parameter space where non-saturated ferromagnetism occurs are much broader in Gutzwiller theory. Moreover, we find that correlation-induced deformations of the Fermi surface can be substantial and that they can even change the Fermi surface topology. This shows that a simple application of the Fermi liquid theory or any other theory which starts from the noninteracting Fermi surface will not be suitable to describe the system properly.