|Date:||Thu, October 8, 2020|
|Place:||online via zoom (please write the organizer for the meeting id and passcode)|
Abstract: In disordered Quantum Systems all translational and rotational symmetries are broken and the disorder can only be statistically defined by its distribution function in an ensemble of random systems. Accordingly, all physical properties are fully described only by their distribution functions, such as the distribution of level spacings or wave function intensities. These are determined by the remaining symmetries such as Time Reversal Symmetry, Spin Symmetry, Particle- Hole Symmetry and Parity. This maps the problem of Disordered Quantum Systems on the problem of random Hermitian matrices. Following the classification of symmetric spaces by mathematician Élie Cartan one finds that these can be divided into 10 symmetry classes, which are now all known to be realised in physical systems such as disordered metal particles, quantum chaotic billiards or in the high excitation spectra of nuclei, where all symmetry classes are realized by imposing different discrete symmetries.
Here, we review how we can thereby study disordered quantum systems by the dynamics on a special set of highly symmetric manifolds, the Riemannian symmetric spaces and the corresponding supersymmetric spaces. In particular we focus on the information one can extract by considering averaged autocorrelation functions of spectral determinants (ASD). We review how these are mapped in the functional intergal representation on the problem of the dynamics on compact symmetric spaces which allows us to study noninteracting disordered systems from quantum chaos to Anderson localisation with rigourous methods, albeit with the warning that the obtained results are only valid for the ASD which yields only limited although nontrivial information on the disordered system. We conclude with a review and an outlook to open problems, including the Integer Quantum Hall transition, the Anderson delocalisation transition of noninteracting particles and the theory of disordered interacting many body systems. In particular we argue that the ASD may be a useful guide and yields us useful insights towards the nonperturbative solution of these problems.