# Symmetry Classes of Disordered Fermions and Topological Insulators

## Professor Martin Zirnbauer, Cologne University

**Date:**
June 1, 2011
16:15

**Place:** Seminarraum A348/349, Fakultät für Physik der LMU München, Theresienstr. 37, 80333 München

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.