The goal of these lectures is to present an introduction to the use of modern, Supersymmetryinspired tools in Condensed Matter and in Statistical Physics. par I will motivate why Grasmann variables are useful in the study of disordered metals. I will show how one develops the conventional diagrammatic technique, and why one encounters problems in applying it for most interesting systems. Then, I will show how anti-commuting variables help averaging over the disorder, and I will derive the super-matrix non-linear $sigma$-model. After that I will present how non-trivial problems of disordered systems have been attacked by using the $sigma$-model. Proceeding in this way I will discuss Anderson localization in one dimensional thick wires and in two dimensional films, and I will find the solution in high dimensionality or on the Bethe lattice. Then, I will present how the zero dimensional $sigma$-model can be useful for mesoscopic systems, and I will show that Random Matrix Theory is equivalent to the zero dimensional $sigma$-model. This equivalence establishes the connection between disordered mesoscopic systems and quantum chaos. I will show an extension of the conventional $sigma$-model to the ballistic one and an exact mapping onto a generalized $sigma$-model (super-bosonization formula). The tentative plan of the lectures follows. 1 – Disorder in normal metals. 2 – Grassmann variables and non-linear supermatrix $sigma$-model. 3 – Renormalization group for the $sigma$-model in 2 and 2 + $varepsilon$ dimensions. 4 – Solving one dimensional and high dimensional models. 5 – Zero dimensional $sigma$-model for small metal particles. Random Matrix Theory and the supersymmetry. Ballistic $sigma$-model and Superbosonization. (Cours organisés en collaboration avec l’Ecole Doctorale de Physique de la Région Parisienne – ED 107)

Ruhr-Universität Bochum et IPhT