Cui, Tianbai2019-12-112019-12-112019-08https://hdl.handle.net/11299/209049University of Minnesota Ph.D. dissertation. August 2019. Major: Physics. Advisor: Rafael Fernandes. 1 computer file (PDF); xvi, 140 pages.Strongly correlated materials offer promising prospects for numerous applications, from superconductivity to quantum information processing. The exotic electronic properties arise from the collective behavior due to strong electron-electron correlation. This leads to the complex phase diagram of strongly correlated materials consists of multiple distinct yet intertwined electronic orders, for examples spin density-wave, charge density-wave, nematic order, and superconductivity. Most theoretical studies of this delicate balance between different electronic orders in strongly correlated systems assume disorder is absent and equilibrium is reached, which sometimes makes comparison with experiments challenging. In this thesis, I will surpass these assumptions to show how disorder dramatically changes the way electronic orders develop, and also demonstrate that non-equilibrium perturbations enable us to understand different dynamics in various timescales and to search for new physical behaviors which are absent in equilibrium. In particular, I will discuss the rare region effect in inhomogeneous systems and show how it changes the critical behaviors of nematic and magnetic quantum phase transitions. I will propose a self-consistent perturbative approach to study the dynamics of the superconducting gap at the picosecond time scales after driven out of equilibrium. Using this approach, I will show that the dynamics of the multi-band superconductor is distinct from the single-band conventional superconductors. I will also elaborate on the damping and relaxation effects on the gap dynamics within the electronic system at picosecond time scales.enDisorder effectsnematic orderNon-equilibriumQuantum phase transitionsStrongly correlated systemsSuperconductivityThesis or Dissertation