Instabilitäten und spontane Symmetriebrechung in zweidimensionalen Fermisystemen
Since the discovery of the superconducting cuprates and their hitherto unknown electronic properties there has been a continuously growing interest in a theoretical treatment of suitable models of strongly correlated fermions which at least qualitatively reflect the essential aspects of these physical systems. The Hubbard model in two dimensions plays an important role in this context. The present thesis examines instabilities and possible symmetry broken phases of the two-dimensional repulsive Hubbard model with the help of numeric and semi analytic methods. While in the first part it is pointed out how such instabilities can be treated by a mean field theory, the second part of the work supplies the theoretical bases of the numeric procedures developed in part III. For the case of a pure nearest neighbour hopping it is shown that the second order contributions of perturbation theory generate an effective attraction between the electrons. The energy scale of the superconducting instabilities is also examined; it is shown that the vanHove singularity drastically enhances this critical scale at half filling. The investigation of a possible coexistence of two different kinds of spontaneous symmetry breaking forms the emphasis of the thesis: Lately accomplished renormalization group calculations showed (for suitably chosen parameters) the occurrence of a Pomeranchuk instability of the two-dimensional Hubbard model with nonvanishing next nearest neighbour hopping, which should give rise to a ground state that spontaneously breaks the model's lattice symmetry as well as its U(1) symmetry, i.e. the ground state could be characterized by two co-existing order parameters. For treatment of the problem of co-existing instabilities the ansatz of a renormalized perturbation theory is used, which had been suggested in order to avoid the occurrence of unphysical singularities arising within the framework of naive perturbation theory; these singularities are related to interaction-induced shifts of the Fermi surface, which cannot be captured by simply shifting the chemical potential. The attractive Hubbard model is also treated within the framework of renormalized perturbation theory; the well-known reduction of the superconducting order parameter with respect to its BCS value reported in earlier investigations can be confirmed. A spontaneously broken lattice symmetrie does not arise here, in contrast to the repulsive case.
info:eu-repo/classification/ddc/530, Physik, korrelierte Fermionen, Supraleitung, Hubbard-Modell
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