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These two "magic terms" comprise the physics which we investigate and which is commonly called Condensed Matter Physics.

Electrons are **fermions** with half-integer spin. They come in two species, distinguished by their spin state. The Fermi statistics controls the behavior of the electrons to a large extent through the Pauli exclusion principle: two fermions cannot be in exactly the same quantum mechanical state. The momentum states of an electronic system with a finite density will therefore constitute a Fermi sea – with each state in the Fermi sea filled by up to two electrons. In most metals they form a "Landau Fermi liquid" in which excitations near the Fermi surface dominate the electronic properties.

Electrons are **charged particles**. Hence they interact among themselves and with the lattice. It proved to be a formidable problem to find solutions for such an interacting many-body system. Even now, after many decades of research and progress in this field, these systems remain absolutely fascinating: we still cannot fathom all the consequences of a strong Coulomb interaction on the state of matter.

The scope of the physics of "strongly correlated electrons" is nowadays enormous. It extends to itinerant magnetism, correlation-induced metal-insulator transitions, unconventional superconductivity, and fractional quantum Hall effect. Also in nanostructured systems an increasing number of investigations is focused on interaction effects: strong local interactions can easily dominate since screening may be suppressed on this microscopic scale.

A part of the Coulomb interaction can be accounted for by introducing an effective mass and a finite lifetime of electronic states. These "dressed" electrons are called quasiparticles if they fulfill certain requirements, expected of well-behaved particles. The quasiparticles are the constituents of Fermi liquids, an ingenious concept introduced by Lev D. Landau in the 1950s which has thoroughly shaped our understanding of metals. It may be seen as a fortunate coincidence that physicists for a long time had to deal only with metals which indeed behave as Landau Fermi liquids, and the electronic properties of many of these metals are well understood.

A large class of microscopic electronic models identifies the respective low-lying excited states as quasiparticles, and Landau's Fermi liquid theory applies. This is certainly true for weakly interacting electrons in three space dimensions. However, "weak interaction" is not necessarily a prerequisite for Fermi liquid theory. Helium-3, for example, is a three-dimensional system of strongly interacting fermions, and Fermi liquid theory is valid down to the transition into the superfluid phases. On the other hand, interacting one-dimensional electrons never form quasiparticle states in the Landau sense. Actually, the failure of Fermi liquid theory in certain electronic systems is not directly related to the interaction strength. It is associated with the much more subtle aspects of scattering properties and phase space: if low-energy excitations impose a rearrangement of sufficiently many states in the Fermi sea, quasiparticles cannot be set up and Fermi liquid theory does not apply. Recently, with the discovery of the high-temperature superconductors, much attention was drawn to measurements which question the Fermi liquid properties of these cuprate systems in the normal, non-superconducting state. Since the charge transport in these cuprates is very anisotropic, that is, mostly within copper-oxygen layers, theoretical models will have to address the issue if strongly correlated two-dimensional electrons have quasiparticle character or rather behave incoherently.

Besides charge, electrons are characterized by their respective spin and orbital states. In the insulating phase the charge is localized but spin and orbital degree of freedom may still fluctuate and move through the lattice. One of the basic models for strongly correlated electrons is the Hubbard model which embodies the two opposing characters of narrow band electrons: electron hopping (with energy scale *t*) supports the itinerant, metallic character, whereas a local on-site Coulomb interaction *U* may drive the electrons into an insulating state. The low-energy excitations of the insulator are (antiferromagnetic) spin excitations. The energy scale of these excitations is *J~t ^{2}/U*, the Heisenberg exchange coupling between nearest neighbor sites. The ground state of the two-dimensional Heisenberg model is antiferromagnetic and the excitations lowest in energy are spin waves. The ground state of the one-dimensional Heisenberg model is paramagnetic with strong singlet correlations and the excitations were called "spinons" which may be visualized approximately as broken singlets. The crossover from one to two dimensions is still being investigated. It is realized in nature by lattices with a ladder structure, whereby the ladder may have an arbitrary number of legs. Spin-ladders are being investigated with analytical and numerical methods by our group.

If those electrons, which dominate the low-energy physics of a metal, can live locally in two (or more) orbitals we have to include this "orbital degree of freedom" in the interpretation of experimental data and in the microscopic modelling. In the manganites charge, spin and orbital degrees of freedom are to be considered on an equal footing. Apart from these purely electronic degrees of freedom, also a strong coupling of electrons to lattice distortions has to be accounted. The colossal magnetoresistance in doped manganites probably results from a complex interplay between these degrees of freedom. There are so many involved aspects to the cooperative behavior of these degrees of freedom that the manganites will stay a topic of research for many more years and more surprises to come.

Reduced dimensionality strongly influences the electronic properties of matter. Spin and charge fluctuations are quite different in two-dimensional systems as compared to their three-dimensional relatives because quantum fluctuations become more pronounced the lower the dimensionality. The physics of reduced dimensionality not only plays a decisive role in quasi two-dimensional or quasi one-dimensional systems such as the layered cuprate perovskites in the high-T_{c} compounds or the ladder systems, enhanced quantum fluctuations are also crucial for the electronic behavior at surfaces and interfaces. High resolution scanning probe microscopys, as developed and advanced at our chair, may soon become a unique tool to investigate local fluctuations of spin and charge at surfaces to a certain extent directly. Theoretical work on this exciting topic is in progress.

Interfaces of correlated electronic systems are a tremendous challenge in solid state physics. On one side, it is now technically feasible to produce quasi-controlled barriers with refined properties between normal metals and also superconductors. On the other side, interfaces determine the transport through granular superconductors in a seemingly uncontrolled way. However, recently it has been possible to tune the grain boundaries, either by controlling the angle between crystallographic axes in adjacent grains, or doping the interface between grains with charge carriers - a break-through having been accomplished in the experimental groups at the chair. In order to achieve a thorough understanding of electronic transport through cuprate grain boundaries one not only has to investigate lattice reconstruction at the interface and then inspect tunneling of band-electrons through these interfaces, one also will have to consider cooperative effects such as the the localization of electronic charge and spin at the interface due to disorder and interaction effects and its feedback on transport through the grain boundary.

It sounds involved? Yes indeed, the adventure of exploring highly correlated electronic systems has just begun with frontiers still beyond our comprehension.

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- 19.12.2017