Geometry-induced Monopole Magnetic Field and Quantum Spin Hall Effect

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 provide quantum spin Hall effects. As potential applications, effective monopole magnetic fields and spin Hall phenomena can be generated and adjusted by designing the geometries and topologies of two-dimensional curved quantum devices.