Spin Connections for Nonrelativistic Electrons on Curves and Surfaces

TL;DR

This paper develops a theory describing how spin connections influence nonrelativistic electrons on curved surfaces, revealing universal spin-dependent magnetic fields that affect electronic and magnetic properties without relying on relativistic effects.

Contribution

It introduces explicit expressions for spin connections on curved geometries derived from three-dimensional space, highlighting their universal presence and effects on electron spin phenomena.

Findings

Spin connections act as spin-dependent magnetic fields on electrons.

Curvature induces spin Hall effects and Dzyaloshinskii--Moriya interactions.

Spin connections influence orbital physics on curved surfaces.

Abstract

We propose a basic theory of nonrelativistic spinful electrons on curves and surfaces. In particular, we discuss the presence 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 reduction from the three-dimensional flat space. The spin connections act on electrons as spin-dependent magnetic fields, which have been known as `pseudomagnetic fields' in the context of, e.g., 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 kinds of their spinorial degrees of freedom and their dispersion relations. We discuss that the curvature effects via spin connections will induce the spin Hall effect and induce the Dzyaloshinskii--Moriya interactions between magnetic moments on curved surfaces, without relying on the relativistic spin-orbit couplings. We also mention the importance of spin connections on orbital physics of electrons on curved geometry.