Pedestrian index theorem à la Aharonov-Casher for bulk threshold modes in corrugated multilayer grapheneJ. Kailasvuori
Institut für theoretische Physik, Freie Universität Berlin - Arnimallee 14, 14195 Berlin, Germany, EU and Max-Planck-Institut für Physik komplexer Systeme - Nöthnitzer Str. 38, 01187 Dresden, Germany, EU
received 28 May 2009; accepted in final form 12 August 2009; published August 2009
published online 15 September 2009
Zero-modes, their topological degeneracy and relation to index theorems have attracted attention in the study of single-layer and bilayer graphene. For negligible scalar potentials, index theorems can 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 non-zero 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.
73.21.Ac - Multilayers.
73.43.Cd - Theory and modeling.
11.30.-j - Symmetry and conservation laws.
© EPLA 2009