Fullerenes, Zero-modes, and Self-adjoint Extensions

TL;DR

This paper investigates the electronic properties of graphene cones with pentagon defects, revealing how short-range lattice effects influence bound states and zero modes through self-adjoint boundary conditions in the Dirac equation.

Contribution

It introduces a detailed analysis of self-adjoint extensions for the Dirac operator near pentagon defects, highlighting the importance of short-range effects in bound state formation.

Findings

Deficiency indices of the radial Dirac operator are (2,2).

Four-parameter family of boundary conditions determines bound states.

Distribution of pentagons affects zero modes and bound states.

Abstract

We consider the low-energy electronic properties of graphene cones in the presence of a global Fries-Kekul\'e 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.