One-dimensional interacting fermionic systems. A study of geometry, topology and symmetry in synthetic quantum matter

Recent years have seen a tremendous step forward in the manipulation of ultracold atomic gases. Laser cooling techniques give access to the quantum regime and experiments have reached a level, at which control and measurement of individual atoms is possible. Sophisticated laser schemes provide optical lattices with different geometries, and allow the precise tuning of interactions and the realization of new concepts such as artificial gauge fields. On the one hand, cold atoms can be used as quantum simulators to study condensed matter systems by mapping the relevant degrees of freedom of the original system to the experimentally better accessible setup. On the other hand, combining different experimental features permits the design of new phases of matter, so-called synthetic quantum matter, which may or may not exist outside the experimental environment. At the same time, concepts from quantum information are pushing the frontier of our understanding of quantum phases as a whole. The concept of entanglement revolutionizes the description of quantum many-body states by replacing (Hilbert space) wave functions with intuition-charged tensor networks. Taking these tensors as elementary building blocks makes it possible to explain phenomena like topology in a bottom-up approach. Moreover, tensor networks permit an efficient description of quantum many-body states and are therefore exploited for numerical simulations. In this thesis, we focus on one-dimensional fermionic systems and explore the influence of different ingredients such as interactions, internal degrees of freedom, artificial gauge fields, symmetries and differently-shaped trapping potentials. While the individual effects might be well understood, the combination of these factors offers new exciting physics. Theoretical research on such systems is encouraged by the forthcoming realizability in the above-mentioned experiments. For our investigation, we employ - besides analytical and perturbative approaches - tensor network methods as a numerical means. In particular, we study three instances of exotic one-dimensional fermionic systems: (i) a Creutz-Hubbard ladder model with a competition between interactions and topological features; we lay out the complete phase diagram and explain the (topological) phase transitions through effective theories; (ii) a ring-shaped system with a similar microscopic ladder architecture, which can be understood as an effective theory of relativistic massless fermions; for these Weyl fermions, we explore the current response to external fields; we find that in certain regimes, the interactions enhance the diamagnetic current flowing along the ring; (iii) a fermionic multi-component gas in a harmonic trap interacting through SU(N)-symmetric contact potentials; we offer a pedagogic understanding of the symmetry and establish a link to the magnetization and to the experimentally accessible momentum distribution of the energy eigenstates.