UC Davis

This thesis develops and applies advanced Monte Carlo tools to explore both classical spin models and quantum many-body systems, and introduces a flexible software framework for determinant quantum Monte Carlo (DQMC) in Julia. The organization is as follows:First, in Chapter 1, we survey Markov-chain Monte Carlo (MCMC) and DQMC methods as the core computational engines of our research. In Chapter 2, we apply classical MCMC to a modified Potts model inspired by quantum synthetic-dimension experiments on collections of Rydberg atoms at low temperature, in which the manifold of rotational states can formally be viewed as an additional spatial dimension. We identify, using traditional correlation functions, two finite-temperature transitions and a “sheet” phase that mirrors its quantum counterpart, and independently locate the phase boundaries using Machine Learning (ML) techniques, showing intriguing physics of the model and the power of ML techniques for exploring emergent phases.Chapter 3 presents SmoQyDQMC.jl, a new Julia package implementing state-of-the-art DQMC with hybrid-Monte-Carlo sampling of phonon fields for Hubbard and electron–phonon Hamiltonians on arbitrary lattice geometries. We showcase its performance and extensibility for on-site interactions, dispersive/anharmonic phonons, and global and “checkerboard” updates, providing a versatile tool for correlated-electron simulations.In Chapter 4, we employ SmoQyDQMC.jl to study a two-layer heterostructure: an attractive-Hubbard sheet coupled to a metallic layer. By mapping out the superconducting transition temperature Tc and superfluid stiffness as functions of interlayer hybridization, we confirm a nonmonotonic behavior of Tc with the coupling between the layers—pushing forward a strategy introduced by the Kivelson group in which hybridization of interacting and non-interacting fermions serves as a route to achieving high superconducting transition temperatures. Specifically, our simulations reveal enhanced phase coherence arising predominantly from the proximity of the metallic layer to the correlated layer as giving rise to robust pairing.Finally, we summarize these contributions and outline future work: integrating Bayesian optimization and machine-learning–guided searches to navigate large parameter spaces, extending DQMC to longer-range and multiband interactions, and leveraging GPU acceleration within the Julia ecosystem. Together, these studies advance both the methodology and physics of strongly correlated and synthetic-dimension systems. Preliminary results on the layer resolved superfluid density tensor are also shown in this chapter, as a work in progress to further illustrate the origin of the enhanced superfluid density studied in Chapter 4. Chapters 2 and 3 are based on published work: Physical Review A109, 013303 (2023) and SciPostPhysCodeb.29-r0.3, respectively. Chapter 4 is based on arXiv:2501.15428 and has been submitted for publication in The Physical Review.