Duality and integer quantum Hall effect in isotropic 3D crystals

Abstract

We show here 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 (3D) periodic systems in magnetic fields $\Vec{B}$, also arise in the isotropic case unless $\Vec{B}$ points in high-symmetry directions. Accompanying integer quantum Hall conductivities $({\sigma }_{xy},{\sigma }_{yz},{\sigma }_{zx})$ can, surprisingly, take values $\propto (1,0,0),(0,1,0),(0,0,1)$ even for a fixed direction of $\Vec{B}$ unlike in the anisotropic case. We can intuitively explain the high-magnetic field spectra and the 3D QHE in terms of quantum mechanical hopping by introducing a ``duality'', which connects the 3D system in a strong $\Vec{B}$ with another problem in a weak magnetic field $(\propto 1/B)$.

7 pages, 6 figures

References in corpus (3)

• Phase Diagram for the Hofstadter butterfly and integer quantum Hall effect in three dimensions
• Wrapping Current versus Bulk Integer Quantum Hall Effect in Three Dimensions
• Berry phase and quantized Hall effect in three-dimension

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