Single-parameter scaling in one-dimensional Anderson localization: Exact analytical solution

The variance of the Lyapunov exponent is calculated exactly in the one-dimensional Anderson model with random site energies distributed according to the Cauchy distribution. We derive an exact analytical criterion for the validity of single-parameter scaling in this model. According to this criterion, states with energies within the conduction band of the underlying nonrandom system satisfy single-parameter scaling when the disorder is small enough. At the same time, single-parameter scaling is not valid for states close to band boundaries and those outside of the original spectrum, even in the case of small disorder. The results obtained are applied to the Kronig-Penney model with the potential in the form of periodically positioned d functions with random strengths. We show that an increase in disorder can restore single-parameter scaling behavior for states within the band gaps.