| In this dissertation, we describe two projects, organized into two chapters, which comprise the study of condensed matter systems using self-consistent mean-field theories.;In the first chapter, we study the exchange constants of MnV2O 4 using linear response, based on the magnetic force theorem (MFT), and the LSDA+U approximation of DFT in the LMTO basis. We obtain the exchanges for three different orbital orderings of the Vanadium atoms of the spinel. We then map the exchange constants to a Heisenberg model with single-ion anisotropy and solve for the spin-wave excitations in the non-collinear, low temperature phase of the spinel. The single-ion anisotropy parameters are obtained from an atomic multiplet exact-diagonalization program, taking into effect the crystal-field (CF) splitting and the spin-orbit coupling (SOC). We find good agreement between the spin-waves of one of our orbital ordered setups with previously reported experimental spin-waves as determined by neutron-scattering. We can therefore determine the correct orbital order (OO) from various proposals.;In the second chapter, we show that the presence of a spin-dependent random potential in a superconductor or a superfluid atomic gas leads to distinct transitions at which the energy gap and average order parameter vanish, generating an intermediate gapless superfluid phase, in marked contrast to the case of spin-symmetric randomness where no such gapless superfluid phase is seen. By allowing the pairing amplitude to become inhomogeneous, the gapless superconducting phase persists to considerably higher disorder compared to the much earlier prediction of Abrikosov-Gor'kov. The low-lying excited states are located predominantly in regions where the pairing amplitude vanishes and coexist with the superfluid regions with a finite pairing. Our results are based on inhomogeneous Bogoliubov-de Gennes (BdG) mean field theory for a two dimensional attractive Hubbard model with spin-dependent disorder.;Further, the finite temperature phase diagram for the 2D attractive fermion Hubbard model with spin-dependent disorder is also considered within BdG mean field theory. Three types of disorder are studied. In the first, only one species is coupled to a random site energy; in the second, the two species both move in random site energy landscapes which are of the same amplitude, but different realizations; and finally, in the third, the disorder is in the hopping rather than the site energy. For all three cases we find that, unlike the case of spin-symmetric randomness, where the energy gap and average order parameter do not vanish as the disorder strength increases, a critical disorder strength exists separating distinct phases. In fact, the energy gap and the average order parameter vanish at distinct transitions, Vcgap and Vc op, allowing for a gapless superconducting (gSC) phase. The gSC phase becomes smaller with increasing temperature, until it vanishes at a temperature T*. |
