Dynamical Symmetries in Nanophysics 1 Dynamical Symmetries in Nanophysics

The main goal of this short review is to demonstrate the relevance of dynam-ical symmetry and its breaking to one of the most active fields in contemporary condensed matter physics, namely, impurity problems and the Kondo effect. It is intended to expose yet another facet of the existing deep and profound relations between quantum field theory and condensed matter physics. At the same time, the reader should not get the impression that this topics is limited to abstract entities and that it is governed by complicated and sophisticated mathematical structures. On the contrary, it will be shown that many of the concepts introduced below are experimentally relevant. It is one of the rare occasions where the parameters of a dynamical group (e.g the number n of a group SO(n)) can be determined experimentally. We elaborate on the role of dynamical symmetry in a special subdiscipline of condensed matter physics, which enters under the name of correlated im-purity problems. It is concerned with the physics which is exposed when the system is composed of strongly correlated localized electrons on the one hand, and itinerant electrons on the other hand. Experimentally, it can be realized in quantum dots and other artificial objects which can be controlled by exter-nal parameters such as magnetic fields and gate voltages. Recently, impurity problems have been shown to be realizable also on single molecules. The size of the pertinent impurities then reduces to nanometers and the relevant scientific activity is referred to as nano-science. At low temperatures, the most striking manifestation of impurity problems is the Kondo effect, which is present in bulk systems and, in the last few years, appears to be an indispensable ingredient in nano-science. The relevant Hamiltonian in impurity problem is a generalized Anderson Hamiltonian which, under certain conditions can be approximated by a gener-alized spin Hamiltonian encoding the exchange interactions between localize