Tight-binding electronic spectra on graphs with spherical topology. I. The effect of a magnetic charge

This is the first of two papers devoted to tight-binding electronic spectra on graphs with the topology of the sphere. In this work the one-electron spectrum is investigated as a function of the radial magnetic field produced by a magnetic charge sitting at the center of the sphere. The latter is an integer multiple of the quantized magnetic charge of the Dirac monopole, that integer defining the gauge sector. An analysis of the spectrum is carried out for the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron and icosahedron), the C$_{60}$ fullerene, and two families of polyhedra, the diamonds and the prisms. Except for the fullerene, all the spectra are obtained in closed form. They exhibit a rich pattern of degeneracies. The total energy at half filling is also evaluated in all the examples as a function of the magnetic charge.