Phase randomness in the one-dimensional Anderson model

Abstract

The probability distribution $P(\ensuremath{\nu})$ of the phase $\ensuremath{\nu}$ which determines the scaling properties of the dimensionless resistance $\frac{R}{T}$ is analyzed for the one-dimensional Anderson model with diagonal disorder. For weak disorder ($\frac{W}{V}\ensuremath{\ll}1$) and $E\ensuremath{\ne}0$ we find that $P(\ensuremath{\nu})$ is initially sharply peaked for short chains but becomes uniform over $2\ensuremath{\pi}$ as $N$ increases. Moreover, the length at which $P(\ensuremath{\nu})$ becomes uniform is found to be independent of $\frac{W}{V}$ (as long as $\frac{W}{V}\ensuremath{\ll}1$); instead it appears to depend only on the distance in energy from band center, i.e., on the extent to which the electron's wavelength is incommensurate with the lattice. Thus, unlike the disordered Kronig-Penny model, there is a phase randomness length, shorter than the localization length but longer than the lattice spacing. $P(\ensuremath{\nu})$ is also studied in the limit of large disorder and analytic expressions for the mean and variance of the inverse localization length are derived for weak and strong disorder. Our results suggest that uniform phase randomness is indeed a property of most disordered 10 systems in the limit of weak disorder.