Kasturirangan, Saumitran2023-04-132023-04-132023-02https://hdl.handle.net/11299/253719University of Minnesota Ph.D. dissertation. February 2023. Major: Physics. Advisor: Fiona Burnell. 1 computer file (PDF); ix, 166 pages.We consider how disorder affects the electronic transport and localization in the vicinity of a topological phase transition in quasi-1D. This is studied for one of the most elementary examples of a topological insulator, the SSH chain. At the topological phase transition, the addition of disorder that respects the chiral symmetry of the system keeps it at a critical point. The electronic wavefunctions at zero energy are not exponentially localized, as one would expect in 1D. The conventional Fokker-Planck approach governing the evolution of transport statistics and the associated single parameter scaling, breaks down in describing the crossover of statistics from class BDI to class AI. We show that a second parameter, the product of energy and relaxation time, is required to capture this crossover. This is demonstrated using data collapse of numerically obtained transport. The regimes of transport behavior are characterized and appear to be universal. These results are used to study zigzag graphene nanoribbons, which is a topological semi-metal. The edge states have a power-law dispersion depending on the width and it is shown that the system is at a topological multicritical point. Upon adding hopping disorder, the transport, density of states, and localization length all obey the same behavior as the SSH chain at criticality, when re-scaled. The edge states are found to be energetically stable and remain close to the boundary. However, they are localized at any non-zero energy. We consider the implications of an out-of-plane field on superconductivity in monolayer NbSe$_2$. We find that the strong Ising spin-orbit coupling arising from the broken inversion symmetry, results in mixing the singlet and triplet components of the superconducting gap. On increasing the magnetic field strength, is it possible to find a gapless superconductor with Bogoliubov Fermi surfaces if the triplet pairing is sufficiently large.enCritical phenomenaDisorderQuantum TransportSpin-orbit couplingSuperconductivityTopological phasesThesis or Dissertation