Phase Distribution in a Disordered Chain and the Emergence of a Two-Parameter Scaling in the Quasiballistic to the Mildly Localized Regime

Abstract

We study the phase distribution of the complex reflection coefficient in different configurations as a disordered 1D system evolves in length, and its effect on the distribution of the 4-probe resistance R4. The stationary (L → ∞) phase distribution is almost always strongly non-uniform and is in general double-peaked with their separation decaying algebraically with growing disorder strength to finally give rise to a single narrow peak at infinitely strong disorder. Further in the length regime where the phase distribution still evolves with length (i.e. in the quasiballistic to the mildly localized regime), the phase distribution affects the distribution of the resistance in such a way as to make the mean and the variance of log (1+R4) diverge independently with length with different exponents. As L → ∞, these two exponents become identical (unity). Obviously, these facts imply two relevant parameters for scaling in the quasiballistic to the mildly localized regime finally crossing over to one-parameter scaling in the strongly localized regime.