The distribution of the reflection phase of disordered conductors

The reflection coefficients of one-dimensional disordered conductors is asymptotically a pure phase, when the sample length is much larger than the localisation length. The authors study the stationary distribution of this reflection phase for tight-binding chains with diagonal disorder, i.e. random site potentials. They derive the Dyson integral equation obeyed by this distribution and relate it to the Lyapunov exponent and the integrated density of states. This general approach allows them to consider then more specifically two types of disorder. In the case of a binary alloy, where the potentials take two values, the distribution of the reflection phase is shown to have generically an infinity of power-law singularities related to the occurrence of periodic patterns of potentials. This phenomenon produces divergences in the distribution when the strength of disorder exceeds some energy-dependent threshold. They also obtain an exact expression for the distribution of the reflection phase for a symmetric exponential distribution of potentials. This non-trivial solvable model allows them to examine analytically various properties of the phase distribution, such as the anomalies which occur at weak disorder.