Tight-binding electronic spectra on graphs with spherical topology. II. The effect of spin-orbit interaction
Abstract
Description
This is the second of two papers devoted to tight-binding electronic spectra on graphs with the topology of the sphere. We investigate the problem of an electron subject to a spin-orbit interaction generated by the radial electric field of a static point charge sitting at the center of the sphere. The tight-binding Hamiltonian considered is a discretization on polyhedral graphs of the familiar form ${\bm L}\cdot{\bm S}$ of the spin-orbit Hamiltonian. It involves SU(2) hopping matrices of the form $\exp({\rm i}μ{\bm n}\cdot{\bmσ})$ living on the oriented links of the graph. For a given structure, the dimensionless coupling constant $μ$ is the only parameter of the model. An analysis of the energy spectrum is carried out for the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron and icosahedron) and the C$_{60}$ fullerene. Except for the latter, the $μ$-dependence of all the energy levels is obtained analytically in closed form. Rather unexpectedly, the spectra are symmetric under the exchange $μ\leftrightarrowΘ-μ$, where $Θ$ is the common arc length of the links. For the symmetric point $μ=Θ/2$, the problem can be exactly mapped onto a tight-binding model in the presence of the magnetic field generated by a Dirac monopole, studied recently. The dependence of the total energy at half filling on $μ$ is investigated in all examples.
25 pages, 15 figures, 12 tables. Various kinds of minor improvements
25 pages, 15 figures, 12 tables. Various kinds of minor improvements