Strong Electron Correlations in the High Critical Temperature Cuprates and Roton-Vortex Interactions in Superfluid HELIUM-4.

In the first part of the dissertation, a quasiparticle theory for doped holes in a quantum antiferromagnetic background is formulated within the non-crossing approximation for the hole self-energy, by employing the correct quasiparticle pole structure in the spectral density function. The t -J model Hamiltonian is derived from the Hubbard model in the large U limit and on the constraint of no double occupancy at the same site. The spin degrees of freedom are treated in linear spin-wave theory, while the charge degrees of freedom are described as spinless fermions. Dyson's equation for the hole Green's function is solved numerically by introducing a lattice in momentum space. The evolution of the spectral density function, density of states, and the momentum distribution function for holes are studied as a function of the hole-magnon coupling constant t. We have found that the hole spectral function is significantly renormalized due to the strong interactions with spin waves. The momentum distribution shows a sharp drop at the Fermi energy, and Luttinger's theorem is verified in the t-J model. The same physical quantities are examined in the presence of hole-phonon interactions. It is observed that the spectral density function is more renormalized and phonon peaks become more pronounced right above and below the quasiparticle pole, as the hole-phonon coupling strength g increases. The sharp drop in the momentum distribution function is preserved, even if the holes strongly interact with phonons. And the vertex corrections due to hole-phonon interactions are found to be small, in contrast to an anticipation that it might be quite large because of the small hole bandwidth of order 2J. The optical conductivity is computed based on the lowest order and next lowest order diagrams. We found that the lowest order diagram gives a contribution to the optical conductivity that dominates over those from the next lowest order diagrams, due to the special nature of the hole-magnon interaction vertices. The major features in the spectral density function are reflected in the corresponding optical conductivity. In the second part of the dissertation, a first -order perturbation treatment of the roton-vortex interaction in superfluid He^4 is developed from a new mathematical identity by using the fractional difference between the momentum of a roton and the momentum at the bottom of the roton minimum as a small expansion parameter. The resulting analytic expressions for the distribution of transverse momentum transfer, in terms of elliptic integrals, are shown to be in excellent agreement with recent computer-simulation results of Samuel and Donnelly and are convenient for computing the total momentum transfer.