Curvature effects in surface states of topological materials

In recent years, researchers have revealed the pivotal role that curvature plays in 3D topological materials. For instance, in the context of the spherical topological insulators, curvature-induced magnetic monopole effects provide the basis for the emerging Landau levels. Another noteworthy example can be found in the realm of the spherical topological superconductors, where the same curvature effects result in the generation of uniform magnetic fields, leading to the formation of vortices. These instances collectively underscore the profound impact of surface curvature in topological materials, manifesting as the production of magnetic fields. The surface states of topological insulators can be effectively described using the two-dimensional massless Dirac equation. We consider the parallel transport of the Dirac spinor field on the curved surface and show that the spin-connection must be introduced. The spin connection plays the role as a gauge field, which gives rise to an effective magnetic field. 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. Furthermore, we analyze the effect of curvature on transport properties of electrons on the surface of the topological materials. In particular, it is shown that surface roughness on the surface of topological insulators generally generate the spin Hall effect.