Quantum percolation in power-law diluted chains

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 appear for α<1.5. In the regime of 1.5<α<2.0 there is no truly extended states even in the presence of a spanning cluster. Instead, a phase of critical wave-functions sets up.