The study of topological electron states started with the 2D quantum Hall effect in the external magnetic fields which arises from the Landau level quantization of the cyclotron motion of electrons. However, the usual LLs crucially relies on the planar structure, and it is not obvious to identify its 3D counterpart. On the other hand, the current study of 3D topological insulators is largely confined to lattice systems with Bloch-wave bands. In this talk, we present the simple Hamiltonians for 3D topological insulators based on LLs in the 3D flat space for both Schroedinger and Dirac fermions. They possess the full 3D rotational symmetry NOT the layered construction of 2D LLs along the z-direction. Apparently, magnetic fields do not work which break rotational symmetry. For the 3D LLs of spin-1/2 Schroedinger fermions, we employ the SU(2) Aharanov-Casher potential. The resultant Hamiltonian is equivalent to a 3D harmonic oscillator with spin-orbit coupling. Each filled LL gives rise to a 2D helical Dirac Fermi surface at open surface, which demonstrates its Z2 topological nature. The 3D Dirac LL Hamiltonian is a square-root problem of the above Schroedinger one, which can be viewed as the Dirac equation defined in the phase space. These 3D LLs can be systematically generalized to an arbitrary high dimension. Due to the elegant analytical properties of LLs and the flatness of LL spectra, we expect high dimensional LLs will further facilitate the future study of high dimensional fractional topological states.

**Ref:**

1) Yi Li, Kenneth Intriligator, Yue Yu, Congjun Wu, Isotropic Landau levels of Dirac fermions in high dimensions Phys. Rev. B 85, 085132 (2012).

2) Yi Li, Congjun Wu, Three Dimensional Topological Insulators with Landau Levels, arXiv:1103.5422 .

3) Yi Li, Xiangfa Zhou, Congjun Wu, 2D and 3D topological insulators with isotropic and parity-breaking Landau levels, arXiv:1112.5907 .