Multiple five- and seven-membered rings in graphene surfaces have long-range effects which are very sensitive to their relative placement. The sensitivity is described by an “ n - m ” combination rule derived here. This rule classifies graphene cones by the localized fictitious gauge flux through their apices; leading to two classes of graphene cones with two apical pentagons. The local density of states vanishes at the Fermi energy in one class; but in the other; it is nonzero and decreases inversely with distance from the apex. Fictitious apical flux also affects the response of electrons to a magnetic field; leading to position dependent local degeneracy of Landau levels as well as anomalous Landau levels which are semiclassically linked to cyclotron orbits which encircle the apex multiple times before closing. We derive these results in a continuum theory; but confront them with numerical tight-binding computations; finding good agreement. We also discuss an intrinsic Aharonov-Bohm effect which persists in the ray-optics limit; and possibilities for producing different inhomogeneous effective fields by exploiting the conical shape.
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