Fullerenes, Zero-modes, and Self-adjoint Extensions

Abstract

We consider the low-energy electronic properties of graphene cones in the presence of a global Fries-Kekulé Peierls distortion. Such cones occur in fullerenes as the geometric response to the disclination associated with pentagon rings. It is well known that the long-range effect of the disclination deficit-angle can be modelled in the continuum Dirac-equation approximation by a spin connection and a non-abelian gauge field. We show here that to understand the bound states localized in the vicinity of a pair of pentagons one must, in addition to the long-range topological effects of the curvature and gauge flux, consider the effect the short-range lattice disruption near the defect. In particular, the radial Dirac equation for the lowest angular-momentum channel sees the defect as a singular endpoint at the origin, and the resulting operator possesses deficiency indices $(2,2)$. The radial equation therefore admits a four-parameter set of self-adjoint boundary conditions. The values of the four parameters depend on how the pentagons are distributed and determine whether or not there are zero modes or other bound states.

20 pages LaTeX, 7 figures

References in corpus (6)

• Electron fractionalization in two-dimensional graphenelike structures
• Topological Phases in Graphitic Cones
• Electron fractionalization for two-dimensional Dirac fermions
• Manifestations of topological effects in graphene
• The continuum gauge field-theory model for low-energy electronic states of icosahedral fullerenes
• Graphene with geometrically induced vorticity

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• Effect of topology on the critical charge in graphene
• Aharonov-Bohm Radiation of Fermions
• Fermionic edge states and new physics
• An artificial boundary approach for short-ranged interactions
• A perturbative method for resolving contact interactions in quantum mechanics
• Fermionic zero modes in the vortex field in arbitrary dimensions and index of Dirac operator with Majorana-like interaction