Strong electronic correlations pose one of the biggest challenges to solid state theory. Recently developed methods that address this problem by starting with the local, eminently important correlations of dynamical mean field theory (DMFT) are reviewed. In addition, nonlocal correlations on all length scales are generated through Feynman diagrams, with a local two-particle vertex instead of the bare Coulomb interaction as a building block. With these diagrammatic extensions of DMFT long-range charge, magnetic, and superconducting fluctuations as well as (quantum) criticality can be addressed in strongly correlated electron systems. An overview is provided of the successes and results achieved, mainly for model Hamiltonians, and an outline is given of future prospects for realistic material calculations.

The calculation of the electronic properties of materials is an important task of solid-state theory, albeit particularly difficult if electronic correlations are strong, e.g., in transition metals, their oxides and in f-electron systems. The standard approach to material calculations, the density functional theory in its local density approximation (LDA), incorporates electronic correlations only very rudimentarily and fails if the correlations are strong. Encouraged by the success of dynamical mean field theory (DMFT) in dealing with strongly correlated model Hamiltonians, physicists from the bandstructure and the many-body communities have joined forces and developed a combined LDA + DMFT method recently. Depending on the strength of electronic correlations, this new approach yields a weakly correlated metal as in the LDA, a strongly correlated metal or a Mott insulator. This approach is widely regarded as a breakthrough for electronic structure calculations of strongly correlated materials. We review this LDA + DMFT method and also discuss alternative approaches to employ DMFT in electronic structure calculations, e.g., by replacing the LDA part with the so-called GW approximation. Different methods to solve the DMFT equations are introduced with a focus on those that are suitable for realistic calculations with many orbitals. An overview of the successful application of LDA + DMFT to a wide variety of materials, ranging from Pu and Ce, to Fe and Ni, to numerous transition metal oxides, is given.

We develop a diagrammatic approach with local and nonlocal self-energy diagrams, constructed from the local irreducible vertex. This approach includes the local correlations of dynamical mean-field theory and long-range correlations beyond. It allows us, for example, to describe (para)magnons and weak localization effects—in strongly correlated systems. As a first application, we study the interplay between nonlocal antiferromagnetic correlations and the strong local correlations emerging in the vicinity of a Mott-Hubbard transition.

The theoretical understanding of the spin-orbit coupling (SOC) effects at LaAlO${}_{3}$/SrTiO${}_{3}$ interfaces and SrTiO${}_{3}$ surfaces is still in its infancy. We perform first-principles density-functional-theory calculations and derive from these a simple tight-binding Hamiltonian, through a Wannier function projection and group theoretical analysis. We find striking differences to the standard Rashba theory for spin-orbit coupling in semiconductor heterostructures due to multiorbital effects: By far the biggest SOC effect is at the crossing point of the $xy$ and $yz$ (or $zx$) orbitals, and around the $$\Gamma${}$ point a Rashba spin splitting with a cubic dependence on the wave vector $\stackrel{P\vec}{k}$ is possible.

Describes a fully-automatic three-dimensional (3-D)-segmentation technique for brain magnetic resonance (MR) images. By means of Markov random fields (MRF's) the segmentation algorithm captures three features that are of special importance for MR images, i.e., nonparametric distributions of tissue intensities, neighborhood correlations, and signal inhomogeneities. Detailed simulations and real MR images demonstrate the performance of the segmentation algorithm. In particular, the impact of noise, inhomogeneity, smoothing, and structure thickness are analyzed quantitatively. Even single-echo MR images are well classified into gray matter, white matter, cerebrospinal fluid, scalp-bone, and background. A simulated annealing and an iterated conditional modes implementation are presented.