Notes on topological insulators

Dan Li, Ralph M. Kaufmann, Birgit Wehefritz-Kaufmann

Published 2015-01-13Version 1

This paper is a survey of the $\mathbb{Z}/\mathbb{Z}_2$-valued invariants of topological insulators in condensed matter physics. The $\mathbb{Z}$-valued topological invariant was originally called the TKNN invariant in physics, which has been fully understood as the first Chern number. The $\mathbb{Z}_2$ invariant is more mysterious, we will devote our efforts to reviewing its equivalent descriptions from different point of views. We emphasize that both invariants are realizations of the Atiyah--Singer index theorem in condensed matter physics. The topological K-theory also plays an important role in the classification of topological insulators with different symmetries.