Zero modes of various graphene configurations from the index theorem

Abstract

In this article we consider a graphene sheet that is folded in various compact geometries with arbitrary topology described by a certain genus, $g$. While the Hamiltonian of these systems is defined on a lattice one can take the continuous limit. The obtained Dirac-like Hamiltonian describes well the low energy modes of the initial system. Starting from first principles we derive an index theorem that corresponds to this Hamiltonian. This theorem relates the zero energy modes of the graphene sheet with the topology of the compact lattice. For $g=0$ and $g=1$ these results coincide with the analytical and numerical studies performed for fullerene molecules and carbon nanotubes while for higher values of $g$ they give predictions for more complicated molecules.

7 pages, 1 figure. Proceedings of the Graphene Conference, MPI PKS Dresden, September 2006

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• Test of the Atiyah-Singer Index Theorem for Fullerene with a Superconducting Microwave Resonator
• Dirac fermions on a disclinated flexible surface
• On topological aspects of 2D graphene like materials
• On geometric delusions of hexagonal structures