Pedestrian index theorem a la Aharonov-Casher for bulk threshold modes in corrugated multilayer graphene

TL;DR

This paper extends the Aharonov-Casher index theorem to multilayer graphene, demonstrating the robustness of zero-energy modes and threshold states under gauge fields, with implications for topological and supersymmetric properties.

Contribution

It generalizes the Aharonov-Casher argument to multilayer graphene, showing the persistence of zero-modes and threshold states in complex gauge field environments.

Findings

Zero-modes are topologically protected in multilayer graphene.

Threshold modes remain degenerate at nonzero energy.

The spectrum's symmetry is preserved due to supersymmetry.

Abstract

Zero-modes, their topological degeneracy and relation to index theorems have attracted attention in the study of single- and bilayer graphene. For negligible scalar potentials, index theorems explain why the degeneracy of the zero-energy Landau level of a Dirac hamiltonian is not lifted by gauge field disorder, for example due to ripples, whereas other Landau levels become broadened by the inhomogenous effective magnetic field. That also the bilayer hamiltonian supports such protected bulk zero-modes was proved formally by Katsnelson and Prokhorova to hold on a compact manifold by using the Atiyah-Singer index theorem. Here we complement and generalize this result in a pedestrian way by pointing out that the simple argument by Aharonov and Casher for degenerate zero-modes of a Dirac hamiltonian in the infinite plane extends naturally to the multilayer case. The degeneracy remains, though at nonzero energy, also in the presence of a gap. These threshold modes make the spectrum asymmetric. The rest of the spectrum, however, remains symmetric even in arbitrary gauge fields, a fact related to supersymmetry. Possible benefits of this connection are discussed.