Critical Hamiltonians on one-dimensional disordered lattices

We focus on tight-binding Hamiltonians on a regular one-dimensional lattice with non-random long-range inter-site coupling J(mn) = J/\m - n\(mu) and random uncorrelated site energies. Within the model the localization-delocalization transition occurs at one of the energy band edges provided 1 <mu <3/2. Using the model we demonstrate that the ratio of the first two momenta of the participation number distribution for the critical states is a size invariant parameter at some value of the disorder magnitude Delta(c). We claim that the invariance manifests the transition. We find that Delta(c) not equal 0 at 1 <mu <3/2, suggesting that the system undergoes the localization-delocalization transition with respect to disorder magnitude. At mu greater than or equal to 3/2, all states are localized. (C) 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.