Recent experimental progress in realizing and controlling two-dimensional quantum materials of bilayer graphene or transition metal dichacogenides has enabled the investigation of a vast class of strongly correlated states of matter, including correlated insulators, strongly interacting mixtures of bosons and fermions, and fractional quantum anomalous Hall states. We develop theoretical descriptions for novel probes to detect unconventional quantum phases in these systems. Furthermore, we investigate how phases with topological order, such as fractional quantum Hall states and quantum spin liquids can be realized and tested in these systems. On the conceptual level, we also study how unconventional symmetries can be used to characterize states with topological order and predict their entanglement structure using two-dimensional Tensor Network States.

Recent conceptional and technical progress makes it possible to prepare and explore strongly-correlated non-equilibrium quantum states of matter. The tremendous level of control and favorable time scales achieved in experiments with synthetic quantum matter, such as ultracold atoms, trapped ions, or superconducting qubits renders these systems as ideal candidates to explore non-equilibrium quantum dynamics. Furthermore, very powerful experimental techniques have also been developed to study dynamic processes in condensed matter systems. We develop both analytical and numerical techniques to explore the far-from-equilibrium quantum dynamics of these systems and study fundamental questions including thermalization in closed quantum systems, emergent phenomena in periodically driven Floquet systems, dynamic phase transitions, intertwined order far from equilibrium, and the competition between coherence and dissipation.