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The Kondo problem occupies a central chapter in condensed matter physics, with a long history in dilute magnetic alloys and valence-fluctuating systems. Originally observed some 70 years ago as a minimum in the resistivity of dilute magnetic alloys, the Kondo effect has evolved in time into a paradigmatic example for strong electronic correlations in condensed matter physics. It pertains to the many-body screening of an impurity spin by the surrounding conduction electrons, leading to the formation of a strong scattering center at low temperatures. Besides the dramatic effect on the resistivity of otherwise pure metals, the Kondo effect is manifested in anomalous enhancements of thermodynamic and dynamic properties such as the specific heat, magnetic susceptibility, and thermopower to name a few. Over the past 40 years, the Kondo effect has played a pivotal role in the development of the field of strongly correlated electron systems. Many of the basic concepts and notions of the field have either been conceived or significantly advanced in the Kondo arena. Notable examples are the renormalization-group ideas of Anderson [1, 2] and Wilson [3]. Nearly all techniques of modern many-body physics have been applied to the problem, which continues to serve as an important testing ground for new approaches.

The Kondo effect has become a hallmark of coherent electron transport in a variety of nanostructures ranging from lithographically-defined semiconductors [1] to carbon nanotubes [2] and molecules [3, 4]. Kondo first introduced a phenomenological Hamiltonian [5] to describe how localized spin couples antiferromagnetically with strength to spins s of electrons in the surrounding reservoir.

There are numerous models in the literature of condensed matter theory, whose significance for achieving progress in our understanding of nature goes far beyond the original aim of explaining specific experimental observations. One may mention in this context the Bardeen-Cooper-Schrieffer’s explanation of the nature of electron pairing in superconductors, the Ginzburg-Landau equation intended for describing critical uctuations, the concept of selflocalization of excitations in a perfect crystal formulated by Deigen, Pekar and Toyozawa and various other seminal ideas. The explanation offered by J. Kondo for the puzzling shallow minimum in the temperature dependent resistivity of metals doped by magnetic impurities [1] is one of the most salient examples of this kind of scenario. To explain it, consider first Kondo’s original idea, which was formulated within the framework of a well-established Hamiltonian describing exchange interaction between an impurity spin located on a given site and the spin density pertaining to a Fermi sea of conduction electrons at this site. The latter is defined by the Fourier transform of the itinerant spin projected on the impurity site , namely .

Research interest in controllable two-level systems, which have been enthusiastically called quantum bits or qubits, has grown enormously during the last decade. Behind a huge burst of activity in this field stands an idea of what is possible in principle but extremely difficult to achieve instrumentally - the fascinating idea of quantum computing. The very principle of quantum superposition allows many operations to be performed on a quantum computer in parallel, while an ordinary ‘classical’ computer, however fast, can only handle one operation at a time. The enthusiasm is not held back by the fact that exploiting quantum parallelism is by no means straightforward, and there exist only a few algorithms (e.g., [1, 2]) for which the quantum computer (if ever built) would offer an essential improvement in comparison with its ‘classical’ counterpart. Even if other uses of quantum computing prove limited (which might or might not be the case), its existence would most certainly lead to a breakthrough in simulations of real physical many-particle systems.

Recent advances in nanotechnology make it possible to fabricate ultra small artificial physical systems like quantum dot, quantum interferometer, quantum wire, etc. in which quantum effects are experimentally observable. Both from the perspective of fundamental physics or potential applications, these artificial systems have generated a lot of excitement as they enabled the realization of a remarkable variety of physical phenomena such as the quantum Hall effect, ballistic transport, Aharonov-Bohm effect, universal conductance uctuation, Kondo effect [1] etc. arising out of the quantum effects. Among such artificial systems, the nanoscopic carbon systems like carbon nanotubes [2–4] and nanographite [5–7] have received enormous attention not only for their intriguing form, but also for their unusual physical properties. In these systems, the geometry of sp carbon networks crucially affects the electronic states near Fermi surface [8–10]. Studies with scanning tunneling microscopy and spectroscopy have confirmed the connection between the electronic states of single wall carbon nanotubes (SWCN) and their geometry [11, 12].

The simplest thermoelectric system is a closed loop made with two different metals connected in the form of junctions at both ends. In 1822–1823 Seebeck discovered an electric current owing through the loop when the junctions are kept at different temperatures, the so-called Seebeck effect. This system then works as a thermoelectric power generator. Since the electric current flows through both metals of the closed loop system, in each metallic branch there exists a voltage difference between its two ends. Let us consider the simpler case of one conductor in which a relevant electric field associated to is created by a gradient ∇ of the temperature . The above mentioned experiment suggests a relation = ∇, (1) where is the Seebeck coefficient or the thermopower.

Although this volume mainly concerns nanoscopic systems, this article is devoted to an intermediate range, between the nanoscopic and macroscopic scales, the so-called mesoscopic regime [1]. In this regime, the system to be considered may be large compared to the mean free path of the electrons. Disorder plays then a very important role and, in the so-called diffusive regime, the interplay between disorder and quantum interference effects is crucial. This is the main subject of this review article. Here, electronic interactions will be treated as a perturbation, in contrast with the topics discussed in the other chapters where the electronic correlations may play the most important role. I will try to present some personal points of view in order to describe these well-known signatures of phase coherence like weak localization or universal conductance uctuations. The goal here is to avoid technicalities as most as possible. The last part concerns the effect of electron-electron interactions.

The aim of this article is to provide a very general and pedestrian introduction to the notion of persistent currents in mesoscopic systems. Thus, this review is not exhaustive and mainly addressed to non-experts. Step by step, it will be shown that a single-particle picture is insufficient to explain the magnitude and sign of the measured persistent currents. The main idea would be to analyze whether the interplay between disorder and electron-electron interaction could eventually explain this discrepancy. This review gives the opportunity to emphasize the challenges raised by the experiments. Indeed, it will be shown that the amplitude and sign of the measured currents remain until now an open issue.

An emerging tendency of modern material science is to propose and investigate systems containing smaller and smaller structures. These smaller structures approach the so-called mesoscopic or nanoscopic regimes in which quantum effects become much more relevant for the behavior of these materials. This situates mesoscopic physics at the interface of statistical physics and quantum physics. The mesoscopic systems are very much smaller than the large-scale objects and they often have unusual physical and chemical properties. The study of mesoscopic systems provides a clear understanding of the behavior of a material as it goes from a few atoms to large visible and tangible objects.

The quantum Hall effect is one of the most remarkable phenomena in condensed matter physics discovered in the second half of the 20th century. In 1980 Klaus von Klitzing [1], who was investigating the magneto-galvanometric properties of the two-dimensional electron gas in high quality Silicon MOSFET in presence of high magnetic field, observed plateaus at integer multiples of the fundamental conductance quantum = in the Hall conductance and a vanishing longitudinal resistivity at a very low temperature (≈ 1 K). This phenomena is known as Integer Quantum Hall Effect (IQHE) (see Fig. 1).