Spin Connections for Nonrelativistic Electrons on Curves and Surfaces

Abstract

We propose a basic theory of nonrelativistic spinful electrons on curves and surfaces. In particular, we discuss the existence and effects of spin connections, which describe how spinors and vectors couple to the geometry of curves and surfaces. We derive explicit expressions of spin connections by performing simple dimensional reductions from three-dimensional flat space. The spin connections act on electrons as spin-dependent magnetic fields, which are known as "pseudomagnetic fields" in the context of, for example, graphenes and Dirac/Weyl semimetals. We propose that these spin-dependent magnetic fields are present universally on curves and surfaces, acting on electrons regardless of the nature of their spinorial degrees of freedom or their dispersion relations. We discuss that, via the spin connections, the curvature effects will cause the spin Hall effect, and induce the Dzyaloshinskii-Moriya interactions between magnetic moments on curved surfaces, without relying on relativistic spin-orbit couplings. We also note the importance of spin connections on the orbital physics of electrons in curved geometries.