Failure of single-parameter scaling of wave functions in Anderson localization

TL;DR

This paper investigates the statistical distribution of wavefunction amplitudes in Anderson localization, revealing deviations from single-parameter scaling theory and identifying invariant skewness properties across different dimensions.

Contribution

It introduces a transfer-matrix based method to analyze wavefunction distributions and demonstrates the failure of single-parameter scaling in 2D systems through empirical data.

Findings

Effective localization lengths grow with distance in 2D, contradicting single-parameter scaling.

Distributions exhibit negative skewness invariant under rescaling.

Skewness magnitude increases with distance, indicating non-Gaussian behavior.

Abstract

We show how to use properties of the vectors which are iterated in the transfer-matrix approach to Anderson localization, in order to generate the statistical distribution of electronic wavefunction amplitudes at arbitary distances from the origin of $L^{d-1} \times \infty$ disordered systems. For $d=1$ our approach is shown to reproduce exact diagonalization results available in the literature. In $d=2$, where strips of width $ L \leq 64$ sites were used, attempted fits of gaussian (log-normal) forms to the wavefunction amplitude distributions result in effective localization lengths growing with distance, contrary to the prediction from single-parameter scaling theory. We also show that the distributions possess a negative skewness $S$, which is invariant under the usual histogram-collapse rescaling, and whose absolute value increases with distance. We find $0.15 \lesssim -S \lesssim 0.30$ for the range of parameters used in our study, .