Shaffer, Daniel2020-10-262020-10-262020-08https://hdl.handle.net/11299/216832University of Minnesota Ph.D. dissertation. August 2020. Major: Physics. Advisors: Fiona Burnell, Rafael Fernandes. 1 computer file (PDF); xxi, 178 pages.The subject of this dissertation is at the intersection of two major fields of condensed matter physics: unconventional superconductivity (SC) and topological phases of matter. Both conventional and unconventional superconductors exhibit similar qualitative behavior: they pass currents with zero resistance and expel magnetic fields, both effects due to formation of a Cooper pair condensate. Broadly, an unconventional superconductor is simply one that is not described by the textbook Bardeen-Cooper-Schrieffer theory. There are at least three things that can make a superconductor unconventional: the pairing mechanism, the symmetry of the Cooper pair, and topology. In many unconventional superconductors the paring mechanism is thought to arise due to the 2D nature of the material that can exhibit strong quantum fluctuations. Unconventional pairing can lead to spin-triplet Cooper pairs with a non-zero orbital momentum, or even a non-zero total momentum, in which case they can realize the so-called pair density wave (PDW). Since magnetic fields align spins that can only form spin-triplet states, one possibility for realizing unconventional superconductors is to look for superconductors that survive in large magnetic fields. This is well-known to occur in systems with strong spin-orbit coupling (SOC) that is also well-known to lead to possible topological phases and topological superconductors in particular, which exhibit Majorana edge modes that may one day be useful for building a quantum computer. All of these elements come together in a family of monolayer materials with strong SOC known as the 1H transition metal dichalcogenides (TMDs) that have recently been found to be superconducting. The thesis of this dissertation is that they can indeed host interesting unconventional and topological SC phases. To show this, in Chapter 2 we analyze the possible symmetry breaking instabilities using the parquet renormalization group that has been successfully used in other unconventional superconductors. We find that Coulomb interactions can lead to unconventional SC and PDW. In Chapter 3, we explain what makes phases in general topological, and how the topology is restricted by their symmetry. In Chapter 4, we combine the results of the two previous chapter to study 1H-TMDs in a mean-field theory, and find unconventional topological phases. In Chapter 5 we study the PDW phase in more detail, and in Chapter 6 we conclude by looking at recent experimental data that indeed suggests that 1H-NbSe\(_2\) may be an unconventional superconductor, but perhaps not the one we anticipated in theory.ennon-centrosymmetricsuperconductivitytopologicaltransition metal dichalcogenidesThesis or Dissertation