Vertices and vortices in high Tc superconductors

Two different aspects of high T_c superconductivity is studied in two independent parts of this thesis. In the first part we study Landau's Fermi liquid theory and nearly antiferromagnetic Fermi liquid theory (NAFL) in 2D. We that inflection points on the Fermi surface provide 2D singularities which have to be taken seriously in Fermi liquid considerations. We then focus on self-consistency of NAFL in calculations base on Eliashberg formalism. We calculate the first vertex correction for an optimally doped system and show that it is of the same order of magnitude as the bare vertex. Migdal's theorem is therefore not valid and the Eliashberg formalism is not applicable. The same conclusion is obtained even after inclusion of the quasiparticle residue Z to the calculation. In part II of the thesis, we establish a method to study vortex lattice properties of d-wave superconductors based on a generalization of the London model. The method has the advantage of simplicity as well as having very few free parameters (one at most) compared to other methods. The generalized London free energy is obtained from an s-d mixing Ginsburg-Landau free energy and also from the microscopic theory of Gorkov. The generalized London equation is found to be non-analytic at very low temperatures because of the nodes on the superconducting gap. We present the results of our calculations for some measurable quantities, such as the vortex lattice geometry and the effective penetration depth (as defined in muSR experiments) with comparison to experiments.