Phase Distribution in a Disordered Chain and the Emergence of a Two-parameter Scaling in the Quasi-ballistic to the Mildly Localized Regime

TL;DR

This paper investigates how the phase distribution of reflection coefficients in a disordered 1D system influences resistance behavior, revealing a transition from two-parameter to one-parameter scaling as disorder increases.

Contribution

It uncovers the evolution of phase distribution and resistance scaling in disordered chains, highlighting a crossover from two-parameter to one-parameter scaling regimes.

Findings

Stationary phase distribution is typically double-peaked, becoming single-peaked at strong disorder.

Mean and variance of log(1+R_4) diverge with different exponents in the quasi-ballistic regime.

Exponents for divergence become equal and unity as system length approaches infinity.

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 $R_4$. The stationary ($L \to \infty$) 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 quasi-ballistic 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 $log(1+R_4)$ diverge independently with length with different exponents. As $L \to \infty$, these two exponents become identical (unity). Obviously, these facts imply two relevant parameters for scaling in the quasi-ballistic to the mildly localized regime finally crossing over to one-parameter scaling in the strongly localized regime.