Abstract
Electron-phonon and electron-electron interactions are two of the most fundamental interactions in quantum many-body systems. They account for many fascinating phenomena including magnetic order, charger order, bond order, hightemperature superconductivity, and many more. The electron-electron interaction has been studied extensively in the Hubbard model, which adds an onsite electron-electron repulsion or attraction to the tight-binding model. On the other hand, the electron-phonon interaction is commonly studied in the Holstein model, which features an on-site potential modulated by the phonon displacement. An equally intriguing but less explored scenario is the Su-Schrieffer- Heeger (SSH) model, where the phonons modulate the electron hopping. Previously, the SSH model with full quantum dynamics of the phonons has been mostly studied in one dimension. In this thesis, we focus on the two-dimensional lattice. Using the determinant Monte Carlo method, we identify the quantum and thermal phase transitions in the half-filled square lattice SSH model and uncover a phonon bond order with wave vector q = (p, p). Building on the half-filled square lattice SSH model, we study the interplay between the electron-phonon and electron-electron interactions by adding the repulsive Hubbard interaction. The result is a clear-cut first-order antiferromagnetic to bond order wave phase transition with no intermediate phase. Lastly, we change the repulsive Hubbard interaction into an attractive one and evaluate the ground state phases away from half-filling. The ground state behaviors are drastically changed beyond the critical electron-phonon coupling strength.