Abstract
It is shown that in the tight-binding Anderson model on the Bethe lattice the exponential decay rate of the Green function can be obtained for arbitrary energies and arbitrary disorder. Analytical results in the case of the Lorentzian distribution of site energies are presented. As an application of these results, a criterion for the localized region on the corresponding regular lattice on the level of the Bethe-Peierls approximation is proposed. Our criterion yields exact results in the one-dimensional limit and yields correct band edges for the hypercubic lattice in the vanishing limit of disorder. The mobility edge trajectory obtained by our criterion is given by an elliptic curve in the case of the Lorentzian distribution and its shape is found to be in qualitatively good agreement with that obtained by the finite-size scaling method in the three-dimensional system.
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