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Article

Tail states and unusual localization transition in low-dimensional Anderson model with power-law hopping

We study deterministic power-law quantum hopping model with an amplitude J(r)~r^{-\beta} and local Gaussian disorder in low dimensions d=1 or 2 under the condition d<\beta<3/2\d. We demonstrate unusual combination of exponentially decreasing density of the ”tail states” and localization–delocalization transition (as function of disorder strength W) pertinent to a small (vanishing in thermodynamic limit) fraction of eigenstates. In a broad range of parameters density of states  \nu(E) decays into the tail region  as simple exponential, \nu(E)=\nu_0\exp{E/E_0} , while characteristic energy E_0  varies smoothly across edge localization transition. We develop simple analytic theory which describes E_0  dependence on power-law exponent \beta, dimensionality d and W, and compare its predictions with exact diagonalization results. At low energies within the bare ”conduction band”, all eigenstates are localized due to strong quantum interference at d=1 or 2; however localization length grows fast with energy decrease, contrary to the case of usual Schrodinger equation with local disorder.