When the Fermi level lies in a gap, the Hall conductivity of three-dimensional electrons in a periodic potential is expressed in a topologically invariant form with a set of three integers. If the magnetic fluxes through the three independent areas of the periodic lattice are rational numbers, one obtains a Diophantine equation relating these numbers and the integers which characterize the Hall conductivity.
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Kohmoto et al. (1992) studied this question.
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