Strongly correlated quantum many-body systems provide a major challenge in modern solid state physics. In many cases, these systems are intrinsically inhomogeneous, either due to interfaces or disorder, or an artificial confinement. Their description requires non-perturbative techniques which are capable of capturing both broken translational invariance and strong correlations. One very successful approach is provided by dynamical mean-field theory (DMFT) and its extensions to spatial inhomogeneity, based on mapping the system to an ensemble of effective quantum impurity models.

First, a well-defined and unbiased measure of the strength of correlations in quantum many-particle systems will be discussed, based on the relative von Neumann entropy computed from the local density operator of correlated and uncorrelated states. The usefulness of this general concept is demonstrated by quantifying correlations of interacting electrons in the Hubbard model and in a series of transition-metal oxides using dynamical mean-field theory.

I will then focus on the interplay of disorder and strong correlations, where statistical DMFT allows a description of Anderson- and Mott-insulating phases on the level of one-particle correlation functions. We have observed characteristic reentrant behaviour close to the insulating phases, indicating that competing disorder and interaction actually delocalize the Fermions. Within the paramagnetic state, Mott- and Anderson-insulating phases are found to be adiabatically connected. Recent extensions of these calculations, which include antiferromagnetic long-range order, have yielded a remarkably rich ground state phase diagram with paramagnetic and antiferromagnetic metallic phases in addition to the corresponding insulating states. For binary disorder, which can be realized in binary alloys, a new Mott transition at non-integer filling was found.