Duality and integer quantum Hall effect in isotropic 3D crystals

Details Duality and integer quantum Hall effect in isotropic 3D crystals Duality and integer quantum Hall effect in isotropic 3D crystals

2002-11-13

arxiv.org/abs/cond-mat/0211250v1

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

7 pages, 6 figures

Scientific paper

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 $(\propto 1/B)$.

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