Organiser: Martin Zirnbauer (Cologne)
Training Course Theme
When a single-particle quantum Hamiltonian system is subjected to a disorder potential, it is expected on physical grounds that a transition from localised to extended energy eigenstates takes place as a function of the disorder strength. Such a transition should be accompanied by a characteristic change in the energy spectrum: if the disorder is large enough for Anderson localisation to occur, the random Schrödinger operator is known to have dense point spectrum; on the other hand, if the disorder is weak and the space dimension larger than d = 2, then one expects the existence of absolutely continuous spectrum.
Giving a mathematical proof of this conjectured scenario, and clarifying the nature of the spectrum and the eigenfunctions at the transition point or in d = 2, remains an important and outstanding problem of mathematical physics. Many features of the scenario are believed to extend to a broader class of quantum systems including, most prominently, those exhibiting transitions of Quantum Hall type.
This training course is mainly directed at researchers in early stages of their careers. Its aim is to provide the participants with an introduction to the subject, by exposing them to ideas, terminology and analytical techniques of the rigorous as well as the heuristic kind. Methods used in the study of Anderson localisation by mathematicians and by theoretical physicists will be reviewed by experts from both communities. Reviewing the state of the art for both disciplines will hopefully help to bridge the existing language gap between the communities and create an environment conducive to fruitful collaboration between physicists and mathematicians during the rest of the program.
Tentative list of topics to be covered
- phenomenology of Anderson localisation (T. Spencer)
- introduction to the spectral theory of random Schrödinger operators (L. Pastur)
- introduction to supermatrix techniques and the nonlinear σ-model (Y. Fyodorov)
- rigorous techniques for 1D and quasi 1D systems (I. Goldsheid)
- rigorous methods in the statistical mechanics of phase transitions (D. Brydges)
- critical phenomena in two-dimensional disordered systems (A. Ludwig)