This thesis covers a variety of topics connected to disordered quantum systems. The quantum Hall effect serves as the archetype of a topological quantum material. The properties of its critical point at the quantum Hall plateau transition is investigated. The critical exponents found are consistent with universality of the quantum Hall critical fixed point. Moreover, quantum Hall critical wave functions appearing at the surfaces of three dimensional topological insulators are studied with respect to their stability upon introducing perturbations. It is found that they can be localized and stabilized depending on the presence or absence of spectral flow in the underlying Hamiltonian.
The second part of the thesis covers toy models of ergodicity in the realm of many-particle physics and disorder. Both spectral statistics and the finite-time dynamics of finite-size systems are studied. A pre-thermal plateau can be identified which crosses over to the asymptotic thermal behavior after a cross-over time scale which is identified as a function of the coupling strength of the electron-electron interaction.

In dieser Dissertation werden diverse Themen im Bereich der ungeordneten Quantensysteme betrachtet. Diese reichen von kritischen Eigenschaften des Quanten-Hall Effekts zu stabilen und instabilen ausgedehnten Zuständen auf den Oberflächen dreidimensionaler topologischer Isolatoren. Im zweiten Teil der Arbeit werden Vielteilchen-Systeme diskutiert. Insbesondere wird ein prä-thermales Plateau in der Thermalisierungsdynamik eines Modells von Ergodizität und Integrabilität identifiziert. Dieses Regime geht nach einer Thermalisierungszeitskala über in die asymptotische ergodische Phase.