Topological insulators and K-theory

Abstract

We analyze the topological ${\mathbb{Z}}_2$ invariant, which characterizes time reversal invariant topological insulators, in the framework of index theory and K-theory. The topological ${\mathbb{Z}}_2$ invariant counts the parity of generalized Majorana zero modes, which can be interpreted as an analytical index. As we show, it fits perfectly into a mod 2 index theorem, and the topological index provides an efficient way to compute the topological ${\mathbb{Z}}_2$ invariant. Finally, we give a new version of the bulk-boundary correspondence which yields an alternative explanation of the index theorem and the topological ${\mathbb{Z}}_2$ invariant. Here the boundary is not the geometric boundary of a probe, but an effective boundary in the momentum space.

57 pages, 8 figures

Cited by in corpus (2)

• The $K$-theoretic bulk-edge correspondence for topological insulators
• T-duality simplifies bulk-boundary correspondence: some higher dimensional cases