The Stiefel--Whitney theory of topological insulators

Abstract

We study the topological band theory of time reversal invariant topological insulators and interpret the topological ${\mathbb{Z}}_2$ invariant as an obstruction in terms of Stiefel--Whitney classes. The band structure of a topological insulator defines a Pfaffian line bundle over the momentum space, whose structure group can be reduced to ${\mathbb{Z}}_2$. So the topological ${\mathbb{Z}}_2$ invariant will be understood by the Stiefel--Whitney theory, which detects the orientability of a principal ${\mathbb{Z}}_2$-bundle. Moreover, the relation between weak and strong topological insulators will be understood based on cobordism theory. Finally, the topological ${\mathbb{Z}}_2$ invariant gives rise to a fully extended topological quantum field theory (TQFT).

31 pages