Dirac Landau levels for surfaces with constant negative curvature

Abstract

Studies of the formation of Landau levels based on the Schr\"odinger equation for electrons constrained to curved surfaces have a long history. These include as prime examples surfaces with constant positive and negative curvature, the sphere [Phys. Rev. Lett. 51, 605 (1983)] and the pseudosphere [Annals of Physics 173, 185 (1987)]. Now, topological insulators, hosting Dirac-type surface states, provide a unique platform to experimentally examine such quantum Hall physics in curved space. Hence, extending previous work we consider solutions of the Dirac equation for the pseudosphere for both, the case of an overall perpendicular magnetic field and a homogeneous coaxial, thereby locally varying, magnetic field. For both magnetic-field configurations, we provide analytical solutions for spectra and eigenstates. For the experimentally relevant case of a coaxial magnetic field we find that the Landau levels split and show a peculiar scaling $\propto B^{1/4}$, thereby characteristically differing from the usual linear $B$ and $B^{1/2}$ dependence of the planar Schr\"odinger and Dirac case, respectively. We compare our analytical findings to numerical results that we also extend to the case of the Minding surface.