QCD under extreme conditions. Inhomogeneous condensation

Almost 40 years after the first publication on the phase diagram of quantum chromodynamics (QCD) big progress has been made but many questions are still open. This work covers several aspects of low-energy QCD and introduces advanced methods to calculate selected parts of the QCD phase diagram. Spontaneous chiral symmetry breaking as well as its restoration is a major aspect of QCD. Two effective models, the Nambu-Jona-Lasinio (NJL) model and the linear σ-model, are widely used to describe the QCD chiral phase transition. We study the large-N_c behavior of the critical temperature T_c for chiral symmetry restoration in the framework of both models. While in the NJL model T_c is independent of N_c (and in agreement with the expected QCD scaling), the scaling behavior in the linear σ-model reads T_c ∝ N^1/2_c. However, this mismatch can be corrected: phenomenologically motivated temperature-dependent parameters or the extension with the Polyakov-loop renders the scaling in the linear σ-model compatible with the QCD scaling. The requirement that the chiral condensate which is the order parameter of the chiral symmetry is constant in space is too restrictive. Recent studies on inhomogeneous chiral condensation in cold, dense quark matter suggest a rich crystalline structure. These studies feature models with quark degrees of freedom. In this thesis we investigate the formation of the chiral density wave (CDW) in the framework of the so-called extended linear sigma model (eLSM) at high densities and zero temperature. The eLSM is a modern development of the linear σ-model which contains scalar, pseudoscalar, vector, as well as axial-vector mesons, and in addition, a light tetraquark state. The nucleon and its chiral partner are introduced as parity doublets in the mirror assignment. The model describes successfully the vacuum phenomenology and nuclear matter ground-state properties. As a result we find that an inhomogeneous phase in the form of a CDW dominates the high density regime. The formation of a homogeneous nuclear matter ground state depends on the parameters determined in the vacuum. However, even in the case of a homogeneous nuclear matter ground state the onset of the CDW is not higher than 5.04ρ₀, a density at which the eLSM is still applicable. Motivated by the rich structure that inhomogeneous condensation produces, and in order to study inhomogeneous condensation in a general framework we describe the finite-mode approach. Former limitations of the finite mode approach to 1+1 dimensions and only one condensate are successively overcome. Different error sources are analyzed and strategies to minimize them are outlined. First, the well-known analytic results for 1+1 dimensional models are reproduced in this purely numerical approach. Second, the finite-mode approach shows to be capable to describe up to four inhomogeneous condensates. Finally, the method is applied to the 3+1 dimensional NJL model. The famous inhomogeneous ''island'' as well as the inhomogeneous ''continent'' are reproduced. The continent persists for different constituent quark masses and different numbers of regulators. However, in contrast to previous findings the continent becomes thinner for increasing chemical potential.