Integer quantum Hall effect in isotropic 3D systems

We study whether the quantum Hall effect (QHE) predicted to occur in three-dimensional (3D) anisotropic systems by Koshino et al. [Phys. Rev. Lett. 86 (2001) 1062] can also exist in isotropic 3D crystals from two different limits. We show, in both the tight-binding model and the weak potential limit, that energy gaps, accompanied by the QHE, arise universally unless the magnetic field points in high-symmetry crystallographic directions. The quantized Hall topological integers (σxy,σyz,σzx) in 3D have been obtained for each of the gaps.