Numerical Studies of Electron-Phonon Interactions and Non-equilibrium Phenomena in Strongly Correlated Systems

Strongly correlated systems account for many fascinating physics phenomena, from superconductivity (SC) to charge density wave (CDW) order. Finding a complete explanation for these phenomena, however, is very challenging: solving the problem on a mean-field level is useful but approximate, and treating interactions non-perturbatively is extremely hard because of the many-body nature of these systems. Therefore, numerical algorithms such as quantum Monte Carlo (QMC) and exact diagonalization (ED) that are able to solve strongly correlated systems exactly start to play a hugely important role. In this thesis, we first introduce in chapters 2 and 3 the model Hamiltonians and numerical methods employed to explore pairing and charge orders, then we present several numerical studies with a variety of focuses. Chapters 4 and 5 look at the effect of electron-phonon interactions on Dirac Fermions and charge order and related phase transitions. In chapter 6 we study the possibility of orbital-selective behavior of charge order, when there is inter-orbital hybridization and two distinct electron-phonon couplings.Chapter 7 displays an out-of-equilibrium study of pairing, charge and magnetic orders upon photoirradiation, where an enhancement of $d$-wave superconductivity is observed. Finally in the last chapter, we provide a summary and outlook of our work