We treat two different problems in condensed matter physics. The rst concerns nuclear magnetic resonance (NMR) in optimally-doped YBCO in the mixed state. We show that the line shape is broadened due to the fact that the Knight shift becomes position-dependent in the mixed state. We also identify a second mechanism, in which a pair of spin up quasiparticles is emitted or absorbed, by which the nuclear spins can relax in the presence of a magnetic field, and show that this second mechanism dominates at low temperatures. We then compare our results to experimental data on O-17 NMR and show that it is possible to explain the data without invoking the presence of antiferromagnetic correlations in the vortex cores. In fact, we show that the effects of such correlations on the O-17 relaxation rates are suppressed in the mixed state, as they are in the normal state. The second problem concerns the electronic phases of bilayer graphene at half filling. Using finite-temperature weak-coupling RG methods, we are able to analytically determine all possible outcomes of the RG ow equations for the nine coupling constants. From this, we are able to determine all of the possible leading instabilities that the system may exhibit as its temperature is lowered. We find that the full phase diagram exhibits a very rich structure, with many different possible instabilities. We then specialize to the case of finite range density-density interactions. We introduce such an interaction into the microscopic tight-binding model and show how it can be related to the coupling constants in the lowenergy effective theory, and apply these results to determine the leading instabilities of the system as a function of the range of the interaction. We consider two forms of the interaction, both motivated by experimental setups, namely a potential like that produced by an electron in the presence of an infinite conducting plate, and like that produced by an electron situated exactly halfway between two infinite conducting plates. We nd that the system is unstable to an antiferromagnetic phase for short-ranged interactions and towards a nematic phase, which breaks the rotational symmetry of the lattice, in agreement with previous work. While the antiferromagnetic phase is gapped, the nematic phase is gapless. Motivated by the fact that we find an instability towards an antiferromagnetic phase for short-ranged interactions and by experimental data that suggests the presence of a gap, we then turn our attention to an investigation of the antiferromagnetic phase in the presence of an applied magnetic field. This is done within the framework of variational mean field theory. We find that, at low fields, the antiferromagnetic order parameter (B) ô (0) B2. At higher elds, for which !c is larger than about 2&Delta(0), we find that &Delta(B) = &omega_c=[ln(&omega_c/&Delta(0)) + C], where C = 0.67 and omega_c = eB=m^* c is the cyclotron frequency. We also determine the energy gap for creating electron-hole excitations in the system, and, at high elds, we nd it to be a!c + 2(B), where a is a non-universal, interaction-dependent, constant.
