Abstract
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 also arise in the isotropic case unless points in high-symmetry crystallographic directions. Accompanying integer quantum Hall conductivities can, surprisingly, take values even for a fixed direction of 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.
- Received 23 October 2002
DOI:https://doi.org/10.1103/PhysRevB.67.195336
©2003 American Physical Society