An Index Theorem for Graphene

Abstract

We consider a graphene sheet folded in an arbitrary geometry, compact or with nanotube-like open boundaries. In the continuous limit, the Hamiltonian takes the form of the Dirac operator, which provides a good description of the low energy spectrum of the lattice system. We derive an index theorem that relates the zero energy modes of the graphene sheet with the topology of the lattice. The result coincides with analytical and numerical studies for the known cases of fullerene molecules and carbon nanotubes and it extend to more complicated molecules. Potential applications to topological quantum computation are discussed.

4 pages, 1 figure, new version to appear in IJMPB

References in corpus (13)

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• Fault-tolerant quantum computation by anyons
• Controlling Spin Exchange Interactions of Ultracold Atoms in Optical Lattices
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• Molecules of Fermionic Atoms in an Optical Lattice
• A Class of $P,T$-Invariant Topological Phases of Interacting Electrons
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• Production of a Fermi gas of atoms in an optical lattice
• Topological Phases in Graphitic Cones
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• Insulating Behavior of a Trapped Ideal Fermi Gas
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Cited by in corpus (14)

• Electronic States of Graphene Grain Boundaries
• Massless Dirac Fermions in a Square Optical Lattice
• The gapped state of a carbon mono-layer in periodic magnetic and electric fields
• Manifestations of topological effects in graphene
• Fermi velocity renormalization and dynamical gap generation in graphene
• Kaluza-Klein description of geometric phases in graphene
• Topological Aspects of Fermions on a Honeycomb Lattice
• Test of the Atiyah-Singer Index Theorem for Fullerene with a Superconducting Microwave Resonator
• Topological effective field theories for Dirac fermions from index theorem
• Bound-states and polarized charged zero modes in three-dimensional topological insulators induced by a magnetic vortex
• Berry phases and zero-modes in toroidal topological insulator
• Aharonov-Bohm-Coulomb Problem in Graphene Ring
• Quadratic fermionic interactions yield effective Hamiltonians for adiabatic quantum computing
• Proof of Atiyah-Singer Index Theorem by Canonical Quantum Mechanics