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arXiv · 1604.02792

The Stiefel--Whitney theory of topological insulators

Ralph M. Kaufmann·Dan Li·Birgit Wehefritz-Kaufmann
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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).

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Ralph M. Kaufmann, Dan Li, Birgit Wehefritz-Kaufmann. 2016-04-11. The Stiefel--Whitney theory of topological insulators. https://arxiv.org/abs/1604.02792

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