Hermitian spin-orbit Hamiltonians on a surface in orthogonal curvilinear coordinates: a new practical approach

TL;DR

This paper develops a practical method to derive Hermitian spin-orbit Hamiltonians for particles confined to surfaces in orthogonal curvilinear coordinates, incorporating geometric effects and gauge fields, with explicit formulas for common geometries.

Contribution

It introduces a new approach that correctly accounts for the transverse momentum operator, geometric potentials, and gauge fields in deriving surface Hamiltonians with spin-orbit coupling.

Findings

Derived Hermitian Hamiltonians for SOC on surfaces of cylinders, spheres, and tori.

Showed geometric potential couples gauge fields with surface curvature.

Applied formalism to 3D Rashba SOC, obtaining explicit surface Hamiltonians.

Abstract

The Hermitian Hamiltonian of a spin one-half particle with spin-orbit coupling (SOC) confined to a surface that is embedded in a three-dimensional space spanned by a general Orthogonal Curvilinear Coordinate (OCC) is constructed. A gauge field formalism, where the SOC is expressed as a non-Abelian $SU(2)$ gauge field is used. A new practical approach, based on the physical argument that upon confining the particle to the surface by a potential, then it is the physical Hermitian momentum operator transverse to the surface, rather than just the derivative with respect to the transverse coordinate that should be dropped from the Hamiltonian.Doing so, it is shown that the Hermitian Hamiltonian for SOC is obtained with the geometric potential and the geometric kinetic energy terms emerging naturally. The geometric potential is shown to represent a coupling between the transverse component of the gauge field and the mean curvature of the surface that replaces the coupling between the transverse momentum and the gauge field. The most general Hermitian Hamiltonian with linear SOC on a general surface embedded in any 3D OCC system is reported. Explicit plug-and-play formulae for this Hamiltonian on the surfaces of a cylinder, a sphere and a torus are given. The formalism is applied to the Rashba SOC in three dimensions (3D RSOC) and the explicit expressions for the surface Hamiltonians on these three geometries are worked out.