Day-Roberts, Ezra2022-03-172022-03-172022-01https://hdl.handle.net/11299/226653University of Minnesota Ph.D. dissertation. January 2022. Major: Physics. Advisors: Rafael Fernandes, Turan Birol. 1 computer file (PDF); viii, 112 pages.The relative strength of the electronic kinetic energy with respect to the electron-electron interaction controls what type of ground state the electronic system has. For small interaction strengths one has Fermi liquids that can be modeled in terms of quasi-electrons. In this weak interaction regime there are instabilities of the Fermi liquid that drive transitions to states like ferromagnets, antiferromagnets, and superconductors. When electron-electron interactions dominate one finds novel ground states such as Mott insulators. In this thesis I study three systems characterized by different interaction strengths. First, I solve the free-particle problem of a tight-binding model on the Penrose quasicrystal. Despite its simplicity, system has a macroscopic fraction of exactly zero-energy states (ZES). I show that all of these states derive from a staggered sublattice mismatch that forms self-similar domains over long distances. This explains the protection of ZES from magnetic fields and any perturbation that preserves the nearest-neighbor structure of the lattice. To do this I construct a Real Space Renormalization Group type approach that shows that all zero energy states come from this staggered mismatch. Second, I study the ferromagnetic transition in Fe$_{x}$Co$_{1-x}$S$_2$ where interactions are weak. I use Density Functional Theory (DFT) to compare the magnetic transition induced by chemical doping and electrostatic gating and find that the latter requires larger numbers of added carriers to induce ferromagnetism due to the absence of Co states near the Fermi level. I derive a tight-binding model from Maximally Localized Wannier Functions and use this to argue that this is a Stoner type transition. Finally, I investigate the Mott insulating state in NiS$_2$ where interactions are strong. I use Dynamical Mean Field Theory combined with DFT to treat the Mott state, which is not captured by DFT alone. I obtain a temperature/carrier density phase diagram for the insulator-metal transition and compare it with isovalent selenium substitution. A metal is obtained in both cases for sufficient doping, but under electrostatic gating there is significant incoherent weight at the Fermi level while selenium substitution yields well defined bands.enThesis or Dissertation