Anomalous dynamical scaling and bifractality in the one-dimensional Anderson model

We investigate dynamical scaling properties of the one-dimensional tight-binding Anderson model with weak diagonal disorder, by means of the spreading of a wavepacket. In the absence of disorder, and more generally in the ballistic regime (t much less than xi(0) in reduced units, with xi(0) being the localization length near the band centre), the wavefunction exhibits sharp fronts. These ballistic fronts yield an anomalous time dependence of the qth moment of the local probability density, or dynamical participation number of order q, with a non-trivial exponent tau(q) for q > 2. This striking feature is interpreted as bifractality. A heuristic treatment of the localized regime (t much greater than xi(0)) demonstrates a similar anomalous scaling, but with xi(0) 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 localized regime is described by scaling functions of one single variable x = t/xi(0). These predictions are confirmed by accurate numerical data, both in the normal and in the anomalous case.