Andreas Schnyder, MPI-Stuttgart
A topological insulator is an insulator that has exotic metallic states on its boundary when placed next to vacuum. Similarly, a topological superconductor is a fully gapped superconductor that has delocalized Andreev surface states. These conducting boundaries are due to topological invariants, which cannot change as long as a the bulk gap remains finite. The integer quantum Hall effect, the quantum spin Hall insulators, and the chiral (p$_x$+i p$_y$) superconductor are well-known examples of topological insulators and superconductors, respectively. In this talk we show that topological insulators and superconductors can be classified within an elegant mathematical framework, which can be viewed, in a sense, as an extension of Bloch's band theory. The result of this exhaustive classification scheme is that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a $\mathbb{Z}$ or a $\mathbb{Z}_2$ topological invariant. An important outcome of the this classification scheme is the prediction of new topological phases of matter. Notable examples are superfluid ${}3$He B, or the time-reversal invariant topological singlet superconductor. We construct lattice models that realize these new topological states and argue that some of the non-centrosymmetric superconductors might be examples of nontrivial topological superconductors.