Symmetry Classes of Disordered Fermions and Topological Insulators
Professor Martin Zirnbauer, Cologne University
Universal properties of disordered and chaotic quantum Hamiltonian systems can often be described by random matrix models. A key question in this context is that of symmetries and universality classes. In a 1962 paper known as the "Threefold Way" Freeman J. Dyson proved that, given an arbitrary group of unitary and antiunitary symmetries, every set of irreducible Hamiltonians commuting with these symmetries must be a set of Hermitian matrices with matrix elements that are either real numbers, or complex numbers, of quaternions. I will explain how to refine Dyson's threefold classi- fication scheme by the so-called "tenfold way" handling the case of disordered fermions. Developed in Koeln beginning in the mid-90s, the refined scheme encompasses noninteracting quasiparticles in disordered metals and superconductors as well as relativistic fermions in random gauge field backgrounds. The same scheme underlies a recent classification of topological insulators by Kitaev and others.