Zero-energy states in corrugated bilayer graphene

Anomalous quantum Hall effects in single-layer and bilayer graphene are related with nontrivial topological properties of electron states (Berry phases $\pi$ and $2\pi$, respectively). It was known that the Atiyah-Singer index theorem guarantees, for the case of the single layer, existence of zero-energy states for the case of inhomogeneous magnetic fields assuming that the total flux is nonzero. This leads, in particular, to the appearance of midgap states in corrugated graphene and topologically protects zero-energy Landau level in corrugated single-layer graphene. Here we apply this theorem to the case of bilayer graphene and prove the existence of zero-energy modes for this case.