The repulsion of bodies carrying electrical charges of equal sign due to Coulomb force is certainly basic knowledge in physics. In contrast, the assumption of noncorrelated, i.e. essentially non-interacting electrons is well established throughout many areas of solid state physics. An excellent example for this is semiconductor physics which is at the heart of the ever more omnipresent electronics: its basic rules rely on the consideration of individual, non-interacting electrons. In the light of the comparatively strong Coulomb interaction, the application of a theory ignoring these electronic interactions—the so-called theory of a free electron gas—might be questioned at first. However, the key here is the screening of the positively charged ions (left behind by the delocalized electrons) which form the crystal lattice.

Beyond this there are, however, also materials the properties of which cannot be described by the theory of a free electron gas. Such discrepancies become specifically obvious at very low temperatures when the impact of thermal excitations is significantly reduced. Phenomena resulting from any electronic interaction can only be studied if the thermal energy (described by k_BT) is small compared to the relevant energy scale of the interaction under consideration. The ever growing interest—not only of physicists—in such phenomena based on electronic correlations is fueled by the fact that the properties of the whole ensemble of interacting entities (for us it’s the electrons) may, in some cases, be of a completely new quality that is neither related to nor expected from the properties of the individual building blocks; just as the psyche of a human being cannot be explained simply from the molecules he is made of. The occurrence of such a new quality is recently referred to as emerging behavior. A typical example is superconductivity which originates from an attractive interaction of the electrons mediated by vibrations of the crystal lattice, i.e., the phonons. The electrons form Cooper pairs. These ideas are at the base of BCS theory [Bardeen, Cooper and Schrieffer 1957] called after its founders John Bardeen, Leon Cooper, and Robert Schrieffer. Nowadays, a number of phenomena driven by strong electronic correlations is known ranging from colossal magnetoresistance (which will be discussed in detail elsewhere) to fractional quantum Hall effect.

Heavy-fermion metals are characterized by a dramatic increase of the effective mass of the charge carriers at low temperatures which may reach up to a thousand times the mass of a free electron. This is brought about by a magnetic interaction—the so-called Kondo effect—which couples the free electrons to the local magnetic moments (and hence, to those electrons that are fixed to the crystal lattice) such that the latter magnetic moments are effectively screened. The properties of these metals can often be described, according to Landau, by considering quasi-particles made up of the electrons and their interactions instead of the mere electrons within the free electron gas. In case of this description being applicable it is referred to as Landau Fermi-liquid (LFL) behavior.

But heavy-fermion metals are good for even more surprises. Today, more than twenty representatives of this class of materials are known to be superconductors whose properties, however, fail to obey the predictions of BCS theory. This extraordinary behavior pertains not only to the mechanism leading to the formation of the Cooper pairs but also to the symmetry of the so-called superconducting order parameter. The latter can be related to a dependence of certain superconducting material properties on the measurement direction with respect to the crystal lattice. These materials are usually termed unconventional superconductors. It should be noted that the recently exceptionally attractive copper oxide high-temperature superconductors belong to this class of materials.

One of the most intriguing phenomena that fascinate experimental physicists as well as theoreticians alike is the so-called quantum critical behavior. It is observed if the material undergoes a continuous phase transition (e.g., from a magnetically ordered phase into an LFL one, with the latter representing an ordered state in k-space) at absolute zero temperature. In most of these materials an antiferromagnetic interaction between the local magnetic moments is found; typically it is the so-called RKKY interaction which is also mediated—just like the aforementioned Kondo interaction—via the conduction electrons. Hence, these two interactions are in direct competition. The relative strength of these two competing interactions can be tuned by experimental parameters such as chemical doping, pressure and magnetic field. In case of this competition being adequately balanced the above discussed phase transition at T = 0 can be brought about by a well-directed change of these experimental parameters. Even though this quantum phase transition at T = 0 is not directly accessible to experiment it affects the finite temperature properties of the material in specific ways if investigated sufficiently close to the quantum critical point (QCP) at which the phase transition takes place in phase space. Such a continuous transition of two competing ordered states occurs if the corresponding fluctuations become critical and hence, such quantum phase transitions are a direct consequence of Heisenberg’s uncertainty principle.

Experimentally, quantum critical behavior and unconventional superconductivity are often found in close vicinity in the very same material. This led to the speculation that the electronic interaction mechanism in this class of superconductors could be magnetic in nature, i.e., the formation of Cooper pairs might not be mediated by phonons but rather by antiferromagnetic fluctuations.