Universal parameter at the Anderson transition on a one-dimensional lattice with non-random long-range coupling

We study numerically the localization–delocalization transition in a class of one-dimensional tight-binding Hamiltonians H with non-random power-law inter-site coupling Hmn=J/|m−n|μ and random on-site energy. This model is critical with respect to the magnitude of disorder at one of the band edges, provided 1<μ<32. We demonstrate that at some value of the magnitude of disorder Δc, interpreted as the critical one, the ratio of the standard deviation to the mean of the participation number distribution is a size-invariant parameter: all curves of this ratio versus the magnitude of disorder, plotted for different system sizes, have a joint intersection point at Δc. This value is finite for 1<μ<32 implying the existence of the transition, while in the marginal case (at μ=32) the intersection point is at Δc=0 implying localization of all the eigenstates.