PHYSICAL REVIEW B VOLUME 48, NUMBER 17 1 NOVEMBER 1993-I Anomalous diffusion in two-dimensional Anderson-localization dynamics Patrick Sebbah, Didier Sornette, and Christian Vanneste Laboratoire de Physique de la Matiere Condensee, Universite de Nice — Sophia Antipolis, Botte Postale 71, Pare Valrose, 06108 Nice Cedex 2, France (Received 19 January 1993; revised manuscript received 5 August 1993) Extensive numerical simulations of wave packets and pulses in two-dimensional (2D) random systems exhibit a subdi6'usion at intermediate times, shown to be linked to the fractal structure of 2D eigenstates. The mean-square pulse width ( ~r~ ) scales as t ", with 0 ~ v —, ' being a continuous function of the disor- der strength. Good agreement is found between numerical values of v and weak-localization predictions. At very long times, the subdiffusive regime crosses over to localization with long power-law asymptotics. The propagation of waves in arbitrary systems can be characterized essentially by three quantities: the spec- trum, the spatial eigenmodes, and the time-dependent response. In Anderson localization of waves by disorder, the first two quantities have been by far the most studied. In this way, the hallmarks of wave localization are the appearance of spatially localized eigenmodes and a pure point spectrum. ' More generally, Anderson localization is usually described in terms of specific properties of the stationary wave at a given frequency. On the other hand, much less attention has been devoted to the dynamical properties of wave localization, for example, the manner in which waves become localized as a function of time from an initial propagating state. Since both wave and Schrodinger equations are linear, localization dynamics can be formally deduced from the knowledge of the fre- quency dependence of the stationary localization prob- lem. ' However, the interest in investigating directly lo- calization dynamics is that it can exhibit more clearly certain striking features of the phenomenon in that it provides a genuine real space-time representation. Fur- thermore, localization dynamics determine many impor- tant properties of the energy wave transport in random systems, such as transient injection transmissions or currents, hopping kinetics, pulse transmission, and rejections. It may also be particularly e%cient for testing the scaling theory of localization as well as estimating critical exponents when using optimized algorithms. In this paper, we explore some salient aspects of the transient dynamics of two-dimensional localization, using a model introduced initially to compute the stationary transmission in random media and recently extended for calculating the time-dependent behavior of a wave in a random system. Instead of the usual diffusive behavior expected at scales I, ~ ~r~ ~ g, where l, is the mean free path and g the localization length, we observe a subdiffusive dynamics in which the mean square width ( ~r~ ) of an initial narrow pulse launched at t =0 scales as t, with an exponent v decreasing continuously from when the disorder increases. This remarkable phenomenon ' is observed by studying large systems (512X 512) over very long times (t — 10 to 10 in units of the inverse bandwidth). The subdiffusion saturates at long times to a localized regime and the asymptotic con- vergence of ( ~ r ~ ) is shown to be compatible with ( r(+ ~ ) ~ ) — ( ~r(t) ~ ) — t '. These results allow us to refine the standard picture of localization according to which a wave packet first diffuses at short times before realizing it is trapped as in an effective cavity of size given by the localization length. It shows in fact the subdiffusive nature of the transport at intermediate times followed by an extremely slow convergence to the fully localized regime. These results are compared quantita- tively with weak-localization computations. It is shown how these results are related to the continuous depen- dence on disorder strength of the fractal dimensions of eigenstates in two dimensions. ' The numerical simulations ha. ve been performed using the "wave-automaton" model. ' A scalar wave pulse (amplitude and phase) propagates in discrete time steps, taken as unity, at constant unit speed along the links of length unity of a square discrete lattice of size L XL. In most of the computations reported here, L =512. A.t each node is placed a different scatterer, which is mathematically characterized by a 4X4 scattering uni- tary and symmetric S matrix. Each S matrix transforms the four fields impinging upon the scatterer along its four links into four outgoing fields. In the case of isotropic scatterers, each S matrix is parametrized by two variables 2a (the phase taken by the wave during the scat- tering process) and 0 ~ 5 ~ ~ [which controls the transport scattering cross section o. , of the scatterer: o. , =2sin (5/2)]. We have checked that in all our simu- lations in double precision, the energy is perfectly con- served at all times to within a precision better than 10 We take a fixed value for 5 for all nodes and a parameter a which fluctuates randomly from node to node in the in- terval [ — b, a, +b, a] (phase disorder and same scattering strength for all nodes). The strength of the disorder is quantified by the mean free path I, defined as the average length over which a Bloch wave of wave vector K sees its initial intensity decrease by a factor e '. In the small 6 and Aa limit, it is given approximately by 1, — 6[sin K cos (6/2)]/[sin (5/2)(ba) ]. l, decreases as the disorder (b a) and the scattering strength (5) in- creases. 0163-1829/93/48(17)/12506(5)/$06. 00 48 12 506 1993 The American Physical Society