Geometry-induced monopole magnetic field and quantum spin Hall effects

Abstract

For a relativistic particle confined to a M\"obius strip, the effective Dirac equation is first given in the thin-layer quantization formalism. We find that an effective gauge potential results from the rotation transformation of the adapted frame moving on the M\"obius strip, and that an effective mass results from the rescaling transformation determined by the metric tensor of the M\"obius strip. It is intriguing that the geometry-induced gauge potential can provide an effective monopole magnetic field for the particle with spin, and can induce quantum spin Hall effects. As potential applications, an effective monopole magnetic field and spin Hall effects can be generated and manipulated by designing the geometry of a two-dimensional nanodevice.