Geometry-induced Monopole Magnetic Field and Quantum Spin Hall Effect

TL;DR

This paper explores how the geometry of a Möbius surface induces a monopole magnetic field affecting spin particles, leading to quantum spin Hall effects, with implications for designing nanodevices.

Contribution

It introduces a geometric gauge potential on a Möbius surface that acts as a monopole magnetic field, enabling control of spin Hall effects through geometry.

Findings

Geometric gauge potential acts as a monopole magnetic field.

Induces quantum spin Hall effects on Möbius surfaces.

Potential for designing nanodevices with tailored topologies.

Abstract

The geometric effects of two-dimensional curved systems have been an interesting topic for a long time. A M\"{o}bius surface is specifically considered. For a relativistic particle confined to the nontrivial surface, we give the effective Dirac equation in the thin-layer quantization formalism, and we find a geometric gauge potential that results from the rotation transformation of the local frame moving on M\"obius strip, and an effective mass that is from the rescaling transformation. Intriguingly, the geometric gauge potential can play a role of monopole magnetic field for the particles with spin, and which can produce quantum spin Hall effects. As potential applications, effective monopole magnetic fields and spin Hall phenomena can be generated and manipulated by designing the geometries and topologies of two-dimensional nanodevices.