Rammelmüller, L. (2016). Energetics and equal-time response of strongly-coupled fermions in one and two dimensions [Diploma Thesis, Technische Universität Wien]. reposiTUm. https://doi.org/10.34726/hss.2016.36234
strongly correlated electron systems; low-dimensional physics; quantum many-body numerical simulations
Strongly interacting Fermi gases constitute a very challenging and interesting area within many-body 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 ground-state 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 one-dimensional (1D) case analytically, there currently is no known method to extract results from two-dimensional (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 lattice-QCD 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 ground-state of 1D and 2D systems in chapters 3 and 4, respectively. We focus on equal-time 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.