Hofstadter butterfly and integer quantum Hall effect in three dimensions

Details Hofstadter butterfly and integer quantum Hall effect in three dimensions Hofstadter butterfly and integer quantum Hall effect in three dimensions

2000-06-26

arxiv.org/abs/cond-mat/0006393v1

Physics

Condensed Matter

Mesoscale and Nanoscale Physics

4 pages, 6 figures, RevTeX, uses epsf.sty,multicol.sty

Scientific paper

10.1103/PhysRevLett.86.1062

For a three-dimensional lattice in magnetic fields we have shown that the hopping along the third direction, which normally tends to smear out the Landau quantization gaps, can rather give rise to a fractal energy spectram akin to Hofstadter's butterfly when a criterion, found here by mapping the problem to two dimensions, is fulfilled by anisotropic (quasi-one-dimensional) systems. In 3D the angle of the magnetic field plays the role of the field intensity in 2D, so that the butterfly can occur in much smaller fields. The mapping also enables us to calculate the Hall conductivity, in terms of the topological invariant in the Kohmoto-Halperin-Wu's formula, where each of $\sigma_{xy}, \sigma_{zx}$ is found to be quantized.

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