The Stiefel--Whitney theory of topological insulators (original) (raw)

2016, arXiv: Mathematical Physics

We study the topological band theory of time reversal invariant topological insulators and interpret the topological mathbbZ_2\mathbb{Z}_2mathbbZ2 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 mathbbZ2\mathbb{Z}_2mathbbZ2. So the topological mathbbZ2\mathbb{Z}_2mathbbZ2 invariant will be understood by the Stiefel--Whitney theory, which detects the orientability of a principal mathbbZ2\mathbb{Z}_2mathbbZ2-bundle. Moreover, the relation between weak and strong topological insulators will be understood based on cobordism theory. Finally, the topological mathbbZ2\mathbb{Z}_2mathbbZ_2 invariant gives rise to a fully extended topological quantum field theory (TQFT).