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 $\mathbf{B},$ also arise in the isotropic case unless $\mathbf{B}$ points in high-symmetry crystallographic directions. Accompanying integer quantum Hall conductivities $({\sigma }_{\mathrm{xy}},{\sigma }_{\mathrm{yz}},{\sigma }_{\mathrm{zx}})$ can, surprisingly, take values $\propto \mathrm{}(1,0,0),(0,1,0),(0,0,1)\mathrm{}$ even for a fixed direction of $\mathbf{B}$ 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.