Tight-binding electronic spectra on graphs with spherical topology. II. The effect of spin-orbit interaction

TL;DR

This paper analyzes how spin-orbit interactions affect electronic spectra on spherical graphs, providing analytical solutions for various polyhedral structures and revealing symmetries related to magnetic monopoles.

Contribution

It introduces a discretized spin-orbit Hamiltonian on polyhedral graphs and derives analytical energy spectra, uncovering symmetries and connections to magnetic monopole models.

Findings

Spectra are symmetric under exchange of coupling parameter and link length.

Analytical solutions obtained for all Platonic solids except C60.

At symmetric coupling, the model maps onto a magnetic monopole problem.

Abstract

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 $\exp({\rm i}\mu{\bm n}\cdot{\bm\sigma})$ living on the oriented links of the graph. For a given structure, the dimensionless coupling constant $\mu$ 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 $\mu$-dependence of all the energy levels is obtained analytically in closed form. Rather unexpectedly, the spectra are symmetric under the exchange $\mu\leftrightarrow\Theta-\mu$, where $\Theta$ is the common arc length of the links. For the symmetric point $\mu=\Theta/2$, 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 $\mu$ is investigated in all examples.