Wang, Hanteng2021-10-132021-10-132021-08https://hdl.handle.net/11299/224945University of Minnesota Ph.D. dissertation. August 2021. Major: Physics. Advisor: Alex Kamenev. 1 computer file (PDF); vi, 153 pages.In this thesis, we use analytical and numerical methods to shed light on the interplay between disorder and interactions in discorded systems. We investigate existence of replica off-diagonal solutions in the field-theoretical description of Majorana version of Sachdev-Ye-Kitaev (SYK) model. We conclude that all our numerical results are in a quantitative agreement with the theory based on the replica-diagonal saddle point plus Schwarzian and massive Gaussian fluctuations, which indicate a non-Fermi liquid phase rather than glassy phase in SYK. Besides the Majorana version of SYK model, We also investigate a possibility of having a superconducting off-diagonal long-range order and a pseudogap phase in SYK model with spin-1/2 fermions attracted by Hubbard interaction. We figure out the SYK + Hubbard model is approaching a certain generalization of the integrable Richardson model at large Hubbard term and exists a quantum phase transition described by synchronization effect in a quantum version of the Kuramoto model at small Hubbard term. The thesis also include the investigation of localization- delocalization transition in SYK4 + SYK2 model, and a non-ergodic extend phase could be found in certain parameter space. This indicate two extensive states in Hilbert space of SYK4 + SYK2 model, are not necessarily have level repulsion with each, which is similar to the case of single-particle Anderson transition on a Cayley tree lattice. At last, we demonstrate our attempt to the diabatic iterative version of the quantum optimization, with an addition reference Hamiltonian during annealing process. By choosing certain optimal trajectory in phase space, in one iteration of a four-stage annealing protocol, one may search a better solution compare to previous inputted ansatz (i.e. reference state) within power-law rather than exponentially long time with a high probability. This algorithm may overcome the issue of the exponentially small gap at the end of the annealing procedure of the standard “forward” quantum annealing due to many-body localization effect, and find out an outcome of the optimization problem with desired accuracy within a power-law annealing time.enThesis or Dissertation