UC Davis

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.