Interacting fermions in external fields, at finite temperature and non-zero density

In this thesis we investigate a certain class of low-dimensional fermionic quantum field theories with four-Fermi interactions. Such theories are commonly used as effective lowenergy models for quantum chromodynamics, the theory of strong interactions, and have applications in the realm of condensed-matter physics as well. Four-Fermi theories are well suited for the study of non-trivial phenomena within strongly-interacting quantum field theories. For our purposes, their most important properties are the notion of chiral symmetry as well as its spontaneous breakdown, which is why a major part of this thesis is devoted to their study. In our non-perturbative investigations, we mostly employ numerical lattice quantum field theory simulations, but we also present a brief analytical study performed in the mean-field limit. In particular, we are concerned with two major questions: On the one hand, we investigate a certain two-dimensional four-Fermi theory, the so-called chiral Gross-Neveu model, which has a continuous chiral symmetry group. We study the possibility of the model at finite temperature and density to exhibit inhomogeneous regions, where the order parameter for chiral symmetry breaking shows oscillatory behavior with the spatial coordinate. In the light of numerous no-go theorems, this is a non-trivial question. We find that remnant inhomogeneous structures can indeed be found even beyond the mean-field limit, albeit likely only on short scales. On the other hand, we consider a related theory with a discrete chiral group, referred to as the Gross-Neveu model, in three space-time dimensions. There, we shall be concerned with the influence of an external magnetic field on chiral symmetry and its spontaneous breaking. While mean-field approaches predict a rich phase structure for non-zero magnetic field and chemical potential, our simulations suggest that this is not the case in the full quantum theory. For all parameter values within the region of spontaneously broken chiral symmetry, we find the magnetic field to enhance the symmetry breaking even further, a phenomenon referred to as magnetic catalysis. In fact, a major goal of this thesis is to contrast the findings of our lattice simulations with existing mean-field results in order to understand to which extent the latter are capable of representing the full theory.