Numerical Studies of Superconductivity and Charge-Density-Waves: Progress on the 2D Holstein Model and a Superconductor-Metal Bilayer

The problem of superconductivity has been central in many areas of condensed matter physics for over 100 years. Despite this long history, there is still no theory capable of describing both conventional and unconventional superconductors. Recent experimental observations such as the dilute superconductivity in SrTiO3 and near room-temperature superconductivity in hydride compounds under extreme pressure have renewed interest in electron-phonon systems. Adding to this is evidence that electron-phonon coupling may play a supporting role in unconventional systems like the cuprates and monolayer FeSe on SrTiO3. One way to make sense of these observations is to construct simple models that capture the essential physics. Among the models with electron-phonon interactions, the simplest and most studied is the two-dimensional Holstein model. It describes a single band of electrons that hop between sites on a square lattice and interact with atomic oscillators by coupling linearly to their displacements. This model gives rise to superconductivity and charge-density-wave order spanning different regions of doping. Surprisingly, even this model is not entirely understood. First, we present a comprehensive study of the Holstein model phase diagram using self-consistent many-body perturbation theory. We then discuss one potential avenue for accelerating non-perturbative quantum Monte Carlo simulations of electron-phonon models using artificial neural networks. Following these topics, we wrap up the electron-phonon-related part by discussing the importance of nonlinear interaction terms and moving beyond the Holstein model. The last problem of this dissertation revisits a proposal by Steve Kivelson. He hypothesized and later showed that coupling a superconductor with a large pairing scale but low phase stiffness to a metal raises the transition temperature (Tc). Expanding on previous work, we studied a more general case with a 2D negative-U Hubbard model coupled with a metallic layer via single-particle tunneling. Here, we use the dynamical cluster approximation to estimate Tc, finding it is maximal for finite tunneling values, thereby confirming Kivelson’s hypothesis in the general case. Collectively, the results in this dissertation shed new light on superconductivity in conventional systems and demonstrate a need to incorporate more aspects of real materials into models