Abstract
We study the tight-binding model of non-interacting electrons on a two-dimensional square lattice within a strong magnetic field. The recursion method is applied to this problem, and the asymptotic behaviour of the continued fraction coefficients and the appropriate termination of the continued fractions are discussed. For an ordered system the local density of states at bulk sites can efficiently and accurately be calculated. For any rational number α=p/q of magnetic flux quanta per lattice site the spectral function is splitted in up toq different (Landau-like) subbands. At edge sites the gaps between the “Landau” subbands disappear. For a disordered system an appropriate termination of the continued fractions is more difficult. Nevertheless, reasonable results for the (bulk) density of states in the presence of disorder can be obtained by averaging over different system realizations. The corresponding result obtained within the coherent potential approximation (CPA) is in good agreement with the exact (averaged) density of states of the disordered system. It is shown that the broadening of each subband due to the disorder is considerably smaller than the disorder strength. The site off-diagonal matrix elements of the one-particle Green function can also be calculated and their unusual properties are discussed. Finally it is discussed, why a determination of the transport coefficients σ xx and σ xy from the Kubo formula was not yet possible within this method, not even within the CPA transport theory.
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Czycholl, G., Ponischowski, W. The recursion method for a two-dimensional electron system in a strong magnetic field. Z. Physik B - Condensed Matter 73, 343–356 (1988). https://doi.org/10.1007/BF01314273
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DOI: https://doi.org/10.1007/BF01314273