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This thesis deals with several aspects of non-perturbative calculations in low-dimensional quantum field theories. It is split into two main parts:
The first part focuses on method development and testing. Using exactly integrable QFTs in zero spacetime dimensions as toy models, the need for non-perturbative methods in QFT is demonstrated. In particular, we focus on the functional renormalization group (FRG) as a non-perturbative exact method and present a novel fluid-dynamic reformulation of certain FRG flow equations. This framework and the application of numerical schemes from the field of computational fluid dynamics (CFD) to the FRG is tested and benchmarked against exact results for correlation functions. We also draw several conclusions for the qualitative understanding and interpretation of renormalization group (RG) flows from this fluid-dynamic reformulation and discuss the generalization of our findings to realistic higher-dimensional QFTs.
The topics discussed in the second part are also manifold. In general, the second part of this thesis deals with the Gross-Neveu (GN) model, which is a prototype of a relativistic QFT. Even though being a model in two spacetime dimensions, it shares many features of realistic models and theories for high-energy particle physics, but also emerges as a limiting case from systems in solid state physics. Especially, it is interesting to study the model at non-vanishing temperatures and densities, thus, its thermodynamic properties and phase structure.
First, we use this model to test and apply our findings of the first part of this thesis in a realistic environment. We analyze how the fluid-dynamic aspects of the FRG realize themselves in the RG flow of a full-fledged QFT and how we profit from this numeric framework in actual calculations. Thereby, however, we also aim at answering a long-standing question: Is there still symmetry breaking and condensation at non-zero temperatures in the GN model, if one relaxes the commonly used approximation of an infinite number of fermion species and works with a finite number of fermions? In short: Is matter (in the GN model) in a single spatial dimension at non-zero temperature always gas-like?
In general, we also use the GN model to learn about the correct description of QFTs at non-zero temperatures and densities. This is of utmost relevance for model calculations in low-energy quan- tum chromodynamics (QCD) or other QFTs in medium and we draw several conclusions for the requirements for stable calculations at non-zero chemical potential.