A Wave Packet Approach to Interacting Fermions

In this thesis, we study the breakdown of the Fermi liquid state in cuprate superconductors using the renormalization group (RG). We seek to extend earlier work on the crossover from the Fermi liquid state to the pseudo gap phase based on RG flows in the so-called saddle point regime. Progress in the derivation of effective models for the conjectured spin liquid state has been hindered, by the difficulties involved in solving the strong coupling low energy Hamiltonian. We tackle the problem by introducing an orthogonal wave packet basis, the so-called Wilson-Wannier (WW) basis, that can be used to interpolate between the momentum space and the real space descriptions. We show how to combine the WW basis with the RG, such that the RG is used to eliminate high-energy degrees of freedom, and the remaining strongly correlated system is solved approximately in the WW basis. We exemplify the approach for different one-dimensional model systems, and find good qualitative agreement with exact solutions even for very simple approximations. Finally, we reinvestigate the saddle point regime of the two-dimensional Hubbard model. We show that the anti-nodal states are driven to an insulating spin-liquid state with strong singlet pairing correlations, thus corroborating earlier conjectures.