Exact solution for eigenfunction statistics at the center-of-band anomaly in the Anderson localization model

TL;DR

This paper provides an exact analytical solution for the eigenfunction statistics at the center-of-band anomaly in the 1D Anderson localization model, revealing violations of one-parameter scaling and new length scales.

Contribution

It derives an exact expression for the statistical moments of wavefunctions at the anomaly, advancing understanding of localization phenomena.

Findings

Exact expression for moments of wavefunctions at E=0

Violation of one-parameter scaling near the anomaly

Emergence of an additional length scale at E≈0

Abstract

An exact solution is found for the problem of the center-of-band ($E=0$) anomaly in the one-dimensional (1D) Anderson model of localization. By deriving and solving an equation for the generating function $\Phi(u,\phi)$ we obtained an exact expression in quadratures for statistical moments $I_{q}=\langle |\psi_{E}({\bf r})|^{2q}\rangle$ of normalized wavefunctions $\psi_{E}({\bf r})$ which show violation of one-parameter scaling and emergence of an additional length scale at $E\approx 0$.