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

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.

8 pages, 1 figure

References in corpus (6)

• Spin-orbit coupling in curved graphene, fullerenes, nanotubes, and nanotube caps
• Quantum mechanics on curved 2D systems with electric and magnetic fields
• Non-Abelian spin-orbit gauge: Persistent spin helix and quantum square ring
• Spin dynamics in rolled-up two dimensional electron gases
• Gauge symmetry breaking and topological quantization for the Pauli Hamiltonian
• Gauge Covariance and Spin Current Conservation in the Gauge Field Formulation of Systems with Spin-Orbit Coupling

Cited by in corpus (10)

• Geometric influences of a particle confined to a curved surface embedded in three-dimensional Euclidean space
• Position-dependent mass effects in the electronic transport of two-dimensional quantum systems
• Hamiltonian for a particle in a magnetic field on a curved surface in orthogonal curvilinear coordinates
• Hermitian and Gauge-Covariant Hamiltonians for a particle in a magnetic field on Cylindrical and Spherical Surfaces
• Spin-polarized transport in helical membranes due to spin-orbit coupling
• The self-adjoint toroidal dipole operator in nanostructures
• Spin Force and Torque in Non-relativistic Dirac Oscillator on a Sphere
• n-Qubit Operations on Sphere and Queueing Scaling Limits for Programmable Quantum Computer
• Quantum motion of a spinless particle in curved space: A viewpoint of scattering theory
• Pure Gauge Spin-Orbit Couplings