Quantum percolation in power-law diluted chains

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We investigate the quantum percolation problem in a diluted chain with long-range hopping amplitudes. Each bond is activated with probability p(r) = p₁/rα, where r is the distance between two sites and α characterizes the range of the interactions. The average participation ratio of all eigenstates is used as a measure of the wave-functions localization length. We found that, above a quantum percolation threshold p₁⁽q⁾, true extended states appears for α< 1.5. In the regime of 1.5 < α<2.0 there is no trully extended states even in the presence of a spanning cluster. Instead, a phase of critical wave-functions sets up.