Pseudo-magnetic field and effective spin-orbit interaction for a spin-1/2 particle confined to a curved surface

TL;DR

This paper derives how curvature-induced pseudo-magnetic fields and spin-orbit interactions affect a spin-1/2 particle on curved surfaces, enabling spin separation, with applications demonstrated on cylindrical geometries.

Contribution

It introduces a formalism linking surface curvature to pseudo-magnetic fields and spin-orbit effects, providing new insights into spin manipulation on curved surfaces.

Findings

Pseudo-magnetic field proportional to Gaussian curvature.

Effective spin-orbit interaction determined by Weingarten tensor.

Potential for spin separation based on surface geometry.

Abstract

By considering the spin connection, we deduce the effective equation for a spin-1/2 particle confined to a curved surface with the non-relativistic limit and in the thin-layer quantization formalism. We obtain a pseudo-magnetic field and an effective spin-orbit interaction generated by the spin connection. Geometrically, the pseudo-magnetic field is proportional to the Gaussian curvature and the effective spin-orbit interaction is determined by the Weingarten curvature tensor. Particularly, we find that the pseudo-magnetic field and the effective spin-orbit interaction can be employed to separate the electrons with different spin orientations. All these results are demonstrated in two examples, a straight cylindrical surface and a bent one.