Zero-energy states in corrugated bilayer graphene

TL;DR

This paper demonstrates the existence of zero-energy modes in corrugated bilayer graphene by applying the Atiyah-Singer index theorem, extending topological protection concepts from single-layer to bilayer systems.

Contribution

It extends the application of the Atiyah-Singer index theorem to bilayer graphene, proving the existence of zero-energy states in corrugated bilayer graphene.

Findings

Zero-energy modes are guaranteed in corrugated bilayer graphene.

Topological protection of zero-energy states is established for bilayer systems.

Theoretical proof extends known single-layer results to bilayer graphene.

Abstract

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 non-zero. This leads, in particular, to 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.