Phan, Dan2023-09-192023-09-192023-05https://hdl.handle.net/11299/257079University of Minnesota Ph.D. dissertation. May 2023. Major: Physics. Advisor: Andrey Chubukov. 1 computer file (PDF); ix, 138 pages.In this dissertation, I study how superconductivity evolves away from the high-density weak-coupling limit. In recent years, there has been a resurgence of interest in superconductivity at low carrier density, propelled by experimental advances in a number of materials. These advances have led theorists to re-analyze the assumptions of traditional Bardeen-Cooper-Schrieffer (BCS) theory, and investigate how these assumptions break down at low densities. The first study I discuss in my thesis pertains to the calculation of the superconducting transition temperature Tc in a phonon-mediated superconductor. Taking the weak-coupling limit and working in two dimensions, I obtain Tc not only in the adiabatic limit where the Fermi energy is much larger than the phonon frequency, but also in the opposite anti-adiabatic limit. In doing so, I include Kohn-Luttinger-type corrections to the pairing interaction which must be included to obtain the correct prefactor for Tc. Afterwards, I turn to the issue of repulsive interactions and how they affect superconductivity at low density. In particular, I study a three-dimensional system with a Bardeen-Pines-like interaction in the low-density limit, where the chemical potential mu is much smaller than the phonon frequency. Parameterizing the strength of the repulsion by a dimensionless parameter f, I find that Tc approaches a nonzero value in the mu = 0 limit as long as f is below a certain threshold f*. In this limit, I find that Tc goes to zero as a power of f*-f, in contrast to the high-density limit, where Tc goes to zero exponentially quickly as f approaches f*. I then discuss my work investigating the Higgs (amplitude) mode, and how its dispersion, damping rate, and residue vary away from the high-density limit in a two-dimensional superconductor. I also study how the Higgs mode is affected by the long-range Coulomb interactions between electrons, finding that the Higgs mode is unaffected by the long-range Coulomb interaction in both two and three dimensions.enamplitudeBCSBECHiggslow-densitysuperconductivityThesis or Dissertation