The exploration of the fermionic features of strongly interacting quantum field theories is of fundamental importance for our understanding of the basic principles of physics. The solutions to the Dirac equation, the quark wave functions, surprisingly show a localization behaviour at low energies, which is very similar to the behaviour of electrons in disordered solids. Firstly, we will study this localization phenomenon in quantum chromodynamics (QCD) on the lattice using a mixed lattice action of twisted mass Wilson fermions as sea quarks and overlap fermions as valence quarks, due to their chiral properties. We diagonalize the overlap Dirac operator for a wide temperature range in the chirally restored phase and extract the delocalization energy, the "mobility edge", which exhibits a linear temperature dependence and vanishes at the chiral transition. We identify an analogy between the Polyakov loop distribution and the disordered potential from the Anderson model of condensed matter, as the low-lying modes are localized in sinks of the local Polyakov line.We then study the topological features of finite temperature QCD and the effect of gradient flow on topological invariants. We find, that the discrepancy between the gluonic and the fermionic definition are mostly remedied at finite flow time and that localized quark modes carry topological charge. We also scout for local selfdual objects within the gauge configurations, such as instantons, calorons, or dyons.Furthermore we study two-colour quantum chromodynamics (QC2D) at finite chemical potential with staggered quarks and show that the correct continuum limit of this particular discretization method is maintained. The anti-unitary symmetry of the Dirac operator is restored when the discretization artifacts from the bulk phase are reduced. We map the diquark onset and present the eigenmodes of the staggered operator at finite chemical potential. We observe that finite mu both counteracts localization and level repulsion.We conclude this work with an investigation of QC2D with overlap sea quarks on a background of a pure gauge theory at finite temperature. We also map the temperature dependence of the mobility edge in this theory and find a linear behaviour at large temperatures, similar to QCD, but also observe the presence of some curvature term close to the deconfinement temperature. We close with the observation that the Polyakov loop plays the role of a disorder term here as well, thus causing localization.
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