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

Abstract

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 H mn=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< 3 2 . We demonstrate that at some value of the magnitude of disorder Δ c, 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 Δ c. This value is finite for 1<μ< 3 2 implying the existence of the transition, while in the marginal case (at μ= 3 2 ) the intersection point is at Δ c=0 implying localization of all the eigenstates.