Author Description

Electron-electron and electron-phonon interactions play fundamental roles in condensed matter physics. Strong correlations among electrons and between electrons and phonons lead to beautiful emergent phenomena both in materials and in the models used to describe them. Unfortunately, the complexity induced from the combination of interactions and large numbers of degrees of freedom makes analytically solving these models very difficult, even when greatly simplified. As a consequence, many important questions in many-body physics remain open. For example, the discoveries of charge density wave (CDW) in the pseudogap phase of the unconventional high-temperature cuprate superconductors motivate on-going research on electron-phonon interactions and its effects on the off-diagonal long-range order (ODLRO). In conventional superconductors, the attractive interaction between electrons which is mediated by the electron-phonon interaction is essential for the formation of Cooper pairs. However, if the electron-phonon interaction is sufficiently strong, charge order emerges near commensurate filling to compete with superconductivity. In this thesis, we use a combination of numerical and analytical methods to understand this sort of interplay between different types of order in the microscopic and macroscopic behavior of many-body systems. In Chapter 1, we introduce the Hubbard and Holstein Hamiltonians and the some of the exotic phases and phase transitions which they describe. We also build up some of the connections between numerical solutions of these models and experimental results for superconducting, charge, and spin order. In Chapter 2 and 3, we set up the frameworks of quantum Monte Carlo (QMC) algorithms and machine learning (ML) methods. We show how to translate a quantum-mechanical problem into an algorithm with analytical analysis encoded in it, which can be widely applied to various models and physics. In Chapter 4 and 5, we quantitatively determine the phase diag