Spin textures in curved paths on a curved surface

Abstract

This study investigates the quantum dynamics of a spin-1/2 particle confined to a curved path from the dynamics of a two-dimensional curved thin-layer system incorporating spin connection contributions. We demonstrate that the geodesic curvature, normal curvature, and geodesic torsion govern the emergent non-Abelian gauge potential, while the geodesic and Gaussian curvatures govern the effective scalar potential in the Hamiltonian. The resulting spin precession dynamics induced by the gauge potential are analyzed with and without the adiabatic approximation. Under this approximation, the surface topology is linked to the rotation angle of spin orientation along a surface boundary and to the pseudo-magnetic flux. Spin texture evolution along helices illustrates distinct behaviors under geodesic versus non-geodesic propagation. Furthermore, the spin evolution along Viviani's curve exemplifies surface dependence. The curve's topology ensures closure of the spin direction and independence of the spin from the path direction. Our theory establishes a framework for spin-state manipulation via engineered nanostructured channels, enabling novel topological quantum control strategies.