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In this work we investigate (within a tight-binding model) the quantum mechanical problem of an electron confined to move on a two-dimensional surface with spherical topology (e.g the C60 Fullerene). An analysis of the spectra is carried out for the five Platonic solids and the Fullerene.
In the first part, the spectrum is elucidated as a function of a radial magnetic field produced by a central magnetic charge (a quantized Dirac monopole). A rich pattern of Landau levels with unusual multiplicity structure is exposed and the distinction between the effect of magnetic field on the Fullerene and on a graphene sheet is emphasized. In the second part, the electron is subject to a Rashba type spin-orbit interaction generated by the radial electric field of a central static point charge. The tight-binding Hamiltonian is a discretization of the familiar atomic spin-orbit term L.S. As a function of the spin orbit coupling constant the spectra display a set of remarkable symmetries which combine physical dynamics and spherical geometry. At one symmetry point the problem can be exactly mapped onto that of an electron in the field of a Dirac monopole, studied in the first part. This means that the spectrum of an inaccessible system (electron in the field of a magnetic charge) can be measured in a system which is experimentally accessible (electron in the field of an electric charge). |
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