A theoretical study of strongly interacting superfluids and superconductors

In this thesis we study two different strongly interacting systems which show superfluidity and superconductivity respectively: (1) Cold atomic Fermi gases in BCS-BEC crossover and (2) High temperature superconductors.; In the first part of the thesis we study the effects of quantum fluctuations at T = 0 on the equation of state in the BCS-BEC crossover. We: (a) recover the Lee Yang Galitskii corrections in the BCS limit and the Lee Yang Huang theory of the interacting Bose gas in the BEC limit and (b) are able to account for the differences between the mean field and Quantum Monte Carlo equations of state at the strongly interacting unitary point.; We next study the evolution of the structure of a single vortex across the BCSBEC crossover using the Bogoliubov de-Gennes theory. Our main results are: (a) The Andreev bound states in the vortex core persists deep into the BEC regime and (b) The critical current is non-monotonic across the crossover with a peak at unitarity, showing that the unitary superfluid is the most robust superfluid.; In the second part of the thesis we study the effects of the strong local Coulomb repulsion on d-wave superconductivity in the proximity of a Mott insulating state. We find that the Mott correlations leave their signature in the striking particle-hole asymmetry in the tunneling spectra. Using variational wavefunctions for the superconducting (SC) state and a semi-analytic Gutzwiller approximation we gain insight into the magnitude and doping dependence of a variety of observables. This include: (a) The striking difference between the energy gap and SC order parameter. (b) The dispersion and spectral weight of low energy quasiparticle excitations. (c) The "underlying Fermi surface" in the SC state and violation of Luttinger's theorem. (d) The doping and temperature dependence of the superfluid stiffness at low temperatures. We compare our results with earlier variational Monte Carlo results and experimental data on hole doped high Tc cuprate superconductors.