Supersymmetry approach to disordered quantum systems

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The rapid development of fabrication technology for small electronic structures allows us now tostudy systems with dimensions of a few nanometres (nanodevices). At these scales the wave nature of electronsbecomes important and so their behaviour is described by quantum mechanics. Another essential feature of many nanodevices is an inevitable presence of imperfections or disorder. Therefore the statistical description is most appropriate in this situation. Statistical properties of quantum systems containing different kind of disorder isthe subject of the theory of disordered quantum systems. Developed initially to study electronic properties of solids, it was applied later to other areas of physics, in which interplay between interference effects and disorder is essential. Among them, for instance, are light propagation in disordered media or physics of ultra cold atomic gases.One of the most powerful tools in the theory of disordered quantum systems is the field-theoretical approach.Originated in high-energy physics, the approach plays an increasingly important role in the modern condensed matter theory. In particular, its application to disordered quantum systems was extremely successful. In spite of this, a number of fundamental problems in the field resisted rigorous understanding.One of the central problems of that kind is the Anderson localization phenomenon. In 1958, P. W. Andersonconjectured that the diffusive propagation of an electron subject to a random potential can be completelysuppressed due to the destructive interference effects. Despite its long history and many important insights obtained, a rigorous theory of Anderson localization in dimensions D>1 is still lacking. The issue attracts a lot of interest from physicists and mathematicians. Indeed, it will be the subject of a forthcoming six-month programme ``Mathematics and Physics of Anderson localization: 50 Years After'' at the Isaac Newton Institutein Cambridge.The goal of this project is to develop novel field-theoretical techniques allowing to solve various problems in the field of quantum disordered systems. Application of those techniques to particular systems, such as, for example, the multidimensional Anderson model, should improve significantly our understanding of the properties of quantum disordered systems, both in the localized and the critical regimes.