In this thesis, three examples of fermionic systems in two dimensions in which spatial modulations either arise spontaneously or are externally imposed are studied using a combination of numerical and analytical techniques. First, the effects of Landau level mixing on charge density wave (CDW) formation in quantum Hall systems is studied. It is shown that the tendency towards CDW formation in quantum Hall systems survives in the presence of Landau level mixing. A Landau free energy theory is then developed using the Hartree-Fock approximation to describe the possible CDW states that may arise when mixing is strong enough to bring two energy levels near degeneracy. The possible orderings that can take place for the specific example of Rashba spin-orbit coupling as a basis for Landau level mixing are then considered. Second, the effect of disorder on quantities reflecting superconducting tendencies in the checkerboard Hubbard model is studied using exact diagonalization calculations. The results suggest that spatial modulations in the hopping parameters of the Hubbard model lead to a robustness to disorder for $d$-wave superconductivity. Third, a tight-binding model whose low energy excitations are relativistic fermions with two different Fermi velocities is introduced. It is shown numerically that these birefringent fermions have the novel property that there exist fractionalized zero-modes that break vortex-anti-vortex symmetry for an appropriately chosen topological defect. In each system, spatial modulations in the real-space properties of the system have a non-trivial effect on the observed physics.
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Thesis advisor: Kennett, Malcolm
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