BAND-EDGE LOCALIZATION AS INTERMITTENT CHAOS

Detailed statistical properties of the wave functions corresponding to the near-band-edge states, of the diagonal disorder Anderson model on a one-dimensional lattice are derived. The calculations are based on an exact map of the Anderson model to the problem of intermittency near a tangent bifurcation. Specifically, it is shown that a variable that equals the ratio of the wave function at neighboring sites satisfies a recursion relation similar to those investigated in the context of tangent bifurcations. As in the latter case one obtains intermittent dynamics of the pertinent variable, which is characterized by long laminar regions, followed by intermittent events and reinjection into a laminar region. Each intermittent event is shown to correspond to a node of the wave function. The wavelength of the wave function is thus space dependent, being determined by the number of steps (sites) between consecutive intermittent bursts. The correlation of the dynamical variables pertaining to different laminar regions is shown to vanish in the noiseless limit. It is shown that the amplitudes of the wave functions satisfy log-normal statistics whose parameters are computed and the corresponding phases obey normal statistics whose parameters are calculated as well. These properties are shown to be of a universal nature, to leading order in the strength of the noise. Beyond the results mentioned above our formulation reproduces in a straightforward manner the known values of the corresponding Lyapunov exponents and density of states and the appropriate scaling emerges in a natural way. Some physical implications of the above results are discussed.