Tight binding spectrum of electrons on the sphere, subject to the field of: I. Magnetic charge, II. Electric charge. A surprising relation between I. and II
Tight binding quantum mechanical spectrum of an electron hopping on spherical graph and subject to radial magnetic or electric fields is elucidated ( think for example of an electron hopping on the atoms of the Fullerene).
In the first part, calculation of the spectrum in the presence of central magnetic charge g is studied. It should take into account the fact that g is quantized as g=(hc/2e)n where n is the monopole number. This restriction requires a meticulous determination of the phase factors appearing in the hopping matrix elements. Having solved this mathematical problem, the spectrum of the symmetric polytopes (tetrahedron, cube, octahedron, dodecahedron and icosahedron) is calculated analytically and shown to display a beautiful pattern, which is entirely distinct from that of the Hofstadter butterfly.
In the second part, the spectrum in the central field of an electric charge Q is calculated in the second part. The radial electric field induces Rashba type spin-orbit interaction on the hopping electron. These are constructed leading to the tight-binding form of the familiar atomic L.S interaction. The spectra of the five symmetric polytopes are calculated analytically as function of Q and display rich and beautiful patterns with some unexpected symmetries.
Finally, we expose a remarkable relation between the two seemingly distinct physical problems: The spectrum of the second system (electron in the field of central electric charge inducing spin-orbit interaction) is found to be identical with the spectrum of the first system (electron in the field of magnetic monopole) at n = 1. This means that it is principally possible to test the spectrum of an experimentally inaccessible system (magnetic monopole) in terms of an experimentally accessible one (electron subject to spin-orbit force induced by central electric charge ).