Andreas Ludwig (UC Santa Barbara)

November 11, 2013

**"Topological Insulators and Superconductors"**

Topological Insulators (and Superconductors) are quantum phases of non-interacting Fermions which are electrical (or thermal) insulators in the bulk, but whose boundaries conduct electrical current (or heat) like a metal. These boundaries are very special and unusual conductors in that they are ("holographically") protected by the topological nature of the quantum state of the bulk material. The first examples of such phases were theoretically predicted less than a decade ago and were discovered experimentally shortly thereafter in two- and three-dimensional electronic systems. Topological Insulators have attracted significant interest as fundamentally novel electronic phases. They can be viewed as generalizations of the Two-Dimensional Integer Quantum Hall effect to systems in different dimensions and with different symmetry properties. For example, they can occur in three dimensions and in the absence of a time-reversal breaking magnetic field. - Here we give an introduction to these systems, and will explain the basic ideas underlying an exhaustive classification scheme of all Topological Insulators (and Superconductors), which links these systems of relevance for Condensed Matter laboratory experiments to a variety of general notions in Theoretical Physics and Mathematics.