Anomalous Dynamical Scaling and Bifractality in the One-dimensional Anderson Model

We investigate dynamical scaling properties of the one-dimensional tightbinding Anderson model with a weak diagonal disorder, by means of the spreading of a wave packet. In the absence of disorder, and more generally in the ballistic regime (t≪ ξ0 in reduced units, with ξ0 being the localisation length near the band centre), the wavefunction exhibits sharp fronts. These ballistic fronts yield an anomalous time dependence of the q-th moment of the local probability density, or dynamical participation number of order q, with a non-trivial exponent τ(q) for q > 2. This striking feature is interpreted as bifractality. A heuristic treatment of the localised regime (t≫ ξ0) demonstrates a similar anomalous scaling, but with ξ0 replacing time. The moments of the position of the particle are not affected by the fronts, and obey normal scaling. The crossover behaviour of all these quantities between the ballistic and the localised regime is described by scaling functions of one single variable x = t/ξ0. These predictions are confirmed by accurate numerical data, both in the normal and in the anomalous case. P.A.C.S.: 72.15.Rn, 71.23.An, 71.30.+h. S/98/041 To appear in Journal of Physics A T/98/053 (a) Email: [email protected] (b) Author to whom correspondence should be addressed. E-mail: [email protected] (⋆) UMR 6622 of C.N.R.S.