Towards finite density properties and real-time dynamics in strongly correlated systems

In this thesis we study strongly correlated quantum systems in extreme temperature and density conditions. A prominent example is the quark-gluon plasma produced in ultra- relativistic heavy-ion collisions. Since the degrees of freedom in the systems of interest are strongly coupled the use of non-perturbative methods is required. Our working method is lattice field theory allowing for high precision ab-initio calculations of physical quan- tities based on Monte Carlo simulations. We study QCD-like theories at finite density. This introduces the sign problem to the lattice preventing the applicability of importance sampling algorithms. Promising methods to tackle the sign problem are the Complex Langevin evolution on the one hand and Lefschetz Thimbles on the other hand. In this work we propose novel algorithmic ideas to both approaches which are put to work in simple models with a sign problem as well as in lattice gauge theories. Moreover, we in- vestigate real-time properties of quantum systems at finite temperature which is necessary for the computation of transport coefficients. To that end we conduct lattice simulations in imaginary time which are aimed at improving the spectral reconstruction of thermal features. Moreover, we study simulations directly in real-time by means of the Complex Langevin evolution equipped with variable transformations.