Abstract

We generalize the two-dimensional Landau level problem to three dimensions and above for both the relativistic and non-relativistic cases. For the 3D case filled with fermions, the systems are topological insulators belong to the Z2-class. The non-relativistic case is a problem of spin-1/2 fermions coupling to the Aharanov-Casher SU(2) gauge field. They exhibit flat Landau levels in which the orbital angular momentum and spin are coupled with a fixed helicity. In spite of the intrinsic spatial inhomogeneity, magnetic translations can be defined for the highest weight states of the total angular momentum. Each Landau level contributes one branch of gapless helical Dirac channel to the surface spectra, whose topological properties belong to the $\mathbb{Z}_{2}$-class. This problem can be further generalized to Dirac fermions, which is essentially a square root problem of the previous version. Each zero energy Landau level state is a half-fermion mode.