  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.
