Novel approaches to the description of one-dimensional correlated fermions

In this thesis, we study two different aspects of one-dimensional (1D) interacting spinless fermions in equilibrium and (mostly) at zero temperature. In the first part, we consider an extension of the Luttinger liquid (LL) theory. In the second part, the phases of two microscopic models are examined with the functional renormalization group (FRG).LL universality implies that, in the scaling limit, the correlation functions of many gapless 1D models are equal to those of the exactly solvable Tomonaga-Luttinger (TL) model. Only two parameters have to be determined for complete knowledge of the low-energy properties. In the TL model, the dispersion relation of the fermions is assumed to be linear. Taking into account a nonlinear dispersion to understand correlation functions beyond the scaling limit is a great challenge. Guided by the result from a first order perturbative calculation of the dynamic structure factor (DSF), the nonlinear Luttinger liquid (NLLL) phenomenology was developed. In this, an exactly solvable mobile-impurity model Hamiltonian was suggested to be the appropriate low-energy effective model. It predicts power law behavior in correlation functions such as the DSF or the single-particle spectral function. Here, we provide a calculation of the second order perturbative correction of the DSF, and show that the result is compatible with this prediction. Furthermore, we carefully analyze the construction of the mobile-impurity model Hamiltonian. Several approximations must be implemented, and we assess their justifications. In particular, we find that it is impossible to keep the full momentum dependence of the interaction potential. This is discussed in the context of known results of the TL model, where similar approximations can lead to artifacts in the single-particle spectral function.In the second part, we examine a tight-binding model with nearest and next-nearest neighbor interactions using the recently developed extended coupled ladder approximation (eCLA) FRG. The phase diagram of the model is known, and there are phase transitions at intermediate and strong interactions. We can, therefore, study the performance of the eCLA FRG in this parameter regime. We find that the eCLA captures phase transitions from a metallic phase to charge ordered phases at dominant nearest or next-nearest neighbor interactions. This is not possible within a first order FRG scheme. However, a bond order phase known to be present in the phase diagram is absent in our analysis.We finally investigate the Aubry-André model using the eCLA FRG. This model, which features a quasirandom on-site potential, is known to exhibit a phase transition from an ergodic to a localized phase, both in the noninteracting and in the interacting case (many-body localization). We use different observables in an attempt to pinpoint the critical on-site potential strength. For this, we derive flow equations for a composite operator within the eCLA in order to calculate the bipartite fluctuations. We also examine the one-particle density matrix. In the eCLA FRG, we do not have access to states with a finite energy density which are usually used in localization studies. We, therefore, stop the flow at a finite flow parameter, or consider a finite temperature. Unfortunately, we do not find an observable that we can calculate with the eCLA FRG that allows us to distinguish appropriately between the ergodic and the localized phase