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Abstract

This thesis investigates fermionic ultracold atom systems in reduced dimensions using stochastic and lattice-based methods. One particular focus of the current work lies in exploring the role trapping potentials play in these systems. Including traps in lattice simulations poses a challenge, as they break translational symmetry, and is made possible via efficient sampling techniques. Another aspect we examine is the effect of population imbalances, which lead to a sign problem in Monte Carlo simulations. To mitigate this issue, we employ both complex Langevin and reweighting methods, analyzing their effectiveness and applicability. Notably, our approach to trapped systems matches experimental and theoretical benchmark results in one dimension perfectly but allows for significantly larger particle numbers and imbalances, for which we find signs of FFLO-type pairing. In two-dimensional untrapped systems, we encounter only mild sign problems and explore the normal phase of the BEC-BCS crossover regime. We offer falsifiable predictions for thermodynamic quantities. In addition to the results on fermionic systems, we introduce a novel normalizing flow architecture for upscaling field configurations. This architecture shows promise for reducing computational complexity and tackling the problem of critical slowing down in lattice simulations of all kinds.