We study deterministic power-law quantum hopping model with an amplitude J(r)∝−r−β and local Gaussian disorder in low dimensions d=1,2 under the condition d<β<3d∕2. 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 ν(E) decays into the tail region E<0 as simple exponential, ν(E)=ν0eE∕E0, while characteristic energy E0 varies smoothly across edge localization transition. We develop simple analytic theory which describes E0 dependence on power-law exponent β, 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=1,2; however localization length grows fast with energy decrease, contrary to the case of usual Schrodinger equation with local disorder.
- Anderson localization