Strongly interacting Fermi gases constitute a very challenging and interesting area within manybody physics and have received a tremendous amount of theoretical and experimental attention. Especially in the last decade, when it became possible to probe low dimensional, ultracold atomic gases experimentally, many new approaches were developed to understand the physics of reduced dimensionality. In this work, we set out to characterize the groundstate of interacting Fermi gases in one and two spatial dimensions. We calculate quantities across a wide range of interaction strengths and particle numbers, in order to characterize the crossover from few to many body physics. Although numerous methods exist to treat aspects of the onedimensional (1D) case analytically, there currently is no known method to extract results from twodimensional (2D) systems in such a way. We therefore need to address this problem numerically and choose to treat the problem by means of Quantum Monte Carlo (QMC) methods. Specifically, we calculate quantities on the lattice, using an auxiliary field decomposition, closely related to methods typically used in latticeQCD calculations. In the first part of this work, we introduce the physics of Fermi gases. Furthermore, we provide an overview of the necessary knowledge and definitions needed to understand this work. In the second chapter, the concept of stochastic integration is introduced. Starting at the basics of Monte Carlo integration, we arrive at the specific algorithms used in this work. Subsequently, we present results for the groundstate of 1D and 2D systems in chapters 3 and 4, respectively. We focus on equaltime density matrices as well as energetics in both cases. Finally, we conclude our work in the last chapter and point out possibilities to extend our research in the future.
