Integer quantum Hall effect in isotropic three-dimensional crystals

We show that a series of energy gaps as in Hofstadter’s butterfly, which have been shown to exist by Koshino et al. [Phys. Rev. Lett. 86, 1062 (2001)] for anisotropic three-dimensional periodic systems in magnetic fields B, also arise in the isotropic case unless B points in high-symmetry crystallographic directions. Accompanying integer quantum Hall conductivities (σ_xy, σ_yz, σ_zx) can, surprisingly, take values ∝ (1, 0, 0), (0, 1, 0), (0, 0, 1) even for a fixed direction of B unlike in the anisotropic case. We also propose here to intuitively explain the spectra and the quantization of the Hall conductivity in terms of the quantum-mechanical hopping between semiclassical orbits in the momentum space, which is usually ignored for weak magnetic fields.