Strongly correlated electron systems are the place of a strong competition between electronic localization and delocalization. Application of pressure or doping, i.e., of an adjustable parameter , permits to continuously modify the properties of the system, such as the degree of localization of the electrons. Generally, the localization of the electrons leads to an isolating fundamental state of antiferromagnetic nature, while their delocalization is accompanied by a paramagnetic Fermi liquid fondamental state. Typical examples are Mott isolating systems: such systems are composed of localized electrons which can become itinerent under pressure or by chemical doping, leading to a metallic state. Organics systems (some of them become superconducting at the borderline between their localized and itinerant regimes), as well as high-temperature superconductors and iron-based superconductors belong to this family of strongly correlated fermions.
In heavy-fermion systems, strong electronic correlations lead to a strong renormalization of the effective mass of the electrons (which explains the name "heavy fermions") and to a Fermi liquid behavior at low temperature (, where is the specific heat,
the temperature and the magnetic susceptibility, , where is the electrical resistivity, etc.). Due to the presence of f and conduction electrons, their physics is dominated by two phenomena :
the Kondo effect, which consists in an hybridization of f orbitals and conduction bands, and leads to the formation of a strongly renormalized Fermi liquid (the so-called heavy fermion regime).
the RKKY exchange interaction between two f electrons, via the conduction electrons, which leads to a localized magnetically ordered state of f electrons.
Heavy-fermion systems are also subject to a strong competition between electronic localization and delocalization. In this case, this competition concerns only one kind of electrons, the f-electrons.
Figure 1 : Generic phase diagram of heavy-fermion systems
The phase diagram of heavy-fermion systems, shown in Figure 1, results from the competition between Kondo and RKKY interactions . It can be controlled via an adjustable parameter , such as pressure or chemical doping, and is constituted of a generally antiferromagnetic phase and of a paramagnetic and Fermi-liquid regime. These two regimes are separated at T = 0 by a quantum phase transition, also called quantum critical point. Spectacular phenomena, such as the so-called "non-Fermi liquid regime"  or the development of a superconducting pocket  are observed in the neighbouring of the quantum phase transition of many heavy-fermion systems. The notion of critical magnetic fluctuations is at the heart of the heavy-fermion problem . The description of quantum criticality and of the related phenomena is now subject to speculations and controversies in the heavy-fermion community and a consensual understanding of these effects is still lacking.
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