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.

Electron-phonon interactions play an important role in understanding the properties of many materials, and give rise to the non-trivial correlation effects, explaining the emergence of a variety of ordered phases. This dissertation presents a Hybrid Quantum Monte Carlo (HQMC) method for simulating quantum electron-phonon models. The details of this approach are explained within the context of applying it to the widely studied Holstein model. We show that by achieving a computational cost that scales near linearly with lattice size, HQMC enables the simulation of system sizes a full order of magnitude larger than previously possible. Moreover, by using intelligently constructed global updates that significantly reduce autocorrelation times, we are able to simulate models with phonon frequencies in the adiabatic limit that are physically relevant to real materials. We then study the emergence of Charge Density Wave (CDW) order in several distinct Holstein models. The first includes electron hoppings (kinetic energies), which are different in the x and y directions, describing a lattice to which strain has been applied. We also extend previous investigations of CDW order in the square Holstein model to a cubic lattice,providing the first determination of the critical temperature in three dimensions.

We explore the emergence of a variety of quantum phases of matter by performing computational studies of several model Hamiltonians. We begin by introducing the Holstein Hamiltonian which describes the electron-phonon interaction on a lattice, and present Determinant Quantum Monte Carlo (DQMC) simulations which reveal the subtle interplay between superconductivity and charge density wave order, focusing on the doped square lattice. We perform a finite-size scaling analysis of pair susceptibility data to accurately determine critical transition temperatures in this model. A recently developed Hybrid Monte Carlo (HMC) algorithm is used to explore charge ordering on the kagome lattice, where we discover a long-ranged charge ordered phase. We then discuss integer-spin Kitaev honeycomb models, and present numerical studies of their thermodynamic behavior. We illustrate the sensitivity of the quantum spin liquid phase to single-ion anisotropy, and discuss the rich variety of thermodynamic behavior in these models. The disordered Ising antiferromagnet on the triangular lattice is also analyzed using transfer matrix calculations and classical MonteCarlo techniques. Our focus here is on the robustness of residual entropy plateaus to disorder and other perturbations, and discuss the relevance of these results to experimental systems. Finally, we present a study of the triangular lattice Hubbard model in a magnetic field, focusing on large U/t at temperatures beyond the exchange parameter J = 4t^2/U. Motivated by recent experiments on triangular lattice compound LCSO, we compare our numerical results for magnetization and entropy to experimental data.

Electron-electron and electron-phonon interactions in strongly correlated systems have attracted considerable attention over the last several decades. The Hubbard, Holstein and Su-Schrieffer-Heeger(SSH) models are iconic models to capture the physics of these interactions and explore their rich and interesting phases caused by them, i.e. Ferromagnetism (FM), Anti-ferromagnetism (AF), Charge Density Wave (CDW), Bond Ordered Wave (BOW), Superconductivity(SC) and other exotic phases. Chapter 1 gives an introduction to the physics related to electron-electron and electron-phonon interactions. Then the paradigmatic models, the Hubbard, Holstein and SSH which describe these interactions are introduced in Chapter 2. Some basic properties of these models are discussed as well. Then we review the methods we use, Mean Field Theory (MFT), Determinant and Langevin quantum Monte Carlo (QMC) to study these systems in Chapter 3. Chapter 4 and Chapter 7 consider the effect of non-uniform hopping on a honeycomb lattice and a layered square lattice on Holstein electron-phonon couplings. In Chapter 5, we investigate the interplay of the flat band of a Lieb lattice and Holstein phonons. Chapter 6 considers both electron-electron and electron-phonon interactions and explores the competition between them and possible intermediate exotic phases. Finally Chapter 8 gives a summary and proposes some possible directions for future research.