Curvature effects in the surface states of topological materials

Chia-Hsin Chen^1*, Po-Hao Chou², Chung-Yu Mou¹

¹Center for Quantum Technology and Department of Physics, National Tsing Hua University, Hsinchu, Taiwan
²Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan

* Presenter:Chia-Hsin Chen, email:log800920@gmail.com

The distinct feature of topological materials is the presence of surface electronic states, which are gapless and can be described by the massless Dirac equation. We consider topological materials of different geometric shape. In particular, by considering the parallel transport of a spinor field on the curved surface, we show that a spin-connection must be introduced. The spin connection plays the role as a gauge field, which gives rise to an effective magnetic field. Furthermore, we show that the effective magnetic field is proportional to the Gauss curvature of the surface, which is a topological invariance upon integration over closed surfaces. The topological invariant is classified by the genus number of the closed surface and the spin of the quasi-particle in the surface state. We analyze effects of the effective magnetic field on electronic structure for surface states by computing Landau levels in geometries with lowest genus numbers (sphere and torus). The effect of curvature on the transport properties of electrons on the surface of the topological materials is also investigated. We show that a small bump on the surface of topological materials may generate spin Hall effect. Finally, we consider the situation when a topological insulator becomes superconducting on the surface. In this case, we find that due to the induced magnetic fields by surface curvature, vortices may spontaneously form on the surface without invoking any external magnetic fields.

Keywords: topological insulator, curvature, spin-connection, monopole