 Supersymmetry in Condensed Matter and Statistical Physics (5/5)
Konstantin B. Efetov
Ruhr-Universität Bochum et IPhT
Fri, Apr. 13th 2012, 10:00
Salle Claude Itzykson, Bât. 774, Orme des Merisiers
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)
Contact : lbervas 