Statistics of anomalously localized states at the center of band E=0 in the one-dimensional Anderson localization model

TL;DR

This paper analyzes the probability distribution of eigenfunction amplitudes at the band center in a one-dimensional Anderson model, revealing unique localization properties of anomalously localized states at E=0.

Contribution

It provides a detailed analytical derivation of the distribution function for anomalously localized states at the band center, including asymptotic behaviors and comparison with energies away from the anomaly.

Findings

At moderate amplitudes, ALS probability at E=0 is lower than away from the anomaly.

At very large amplitudes, ALS are more probable at E=0 than away from it.

The small amplitude behavior aligns with exponential localization characterized by known Lyapunov exponents.

Abstract

We consider the distribution function $P(|\psi|^{2})$ of the eigenfunction amplitude at the center-of-band (E=0) anomaly in the one-dimensional tight-binding chain with weak uncorrelated on-site disorder (the one-dimensional Anderson model). The special emphasis is on the probability of the anomalously localized states (ALS) with $|\psi|^{2}$ much larger than the inverse typical localization length $\ell_{0}$. Using the solution to the generating function $\Phi_{an}(u,\phi)$ found recently in our works we find the ALS probability distribution $P(|\psi|^{2})$ at $|\psi|^{2}\ell_{0} >> 1$. As an auxiliary preliminary step we found the asymptotic form of the generating function $\Phi_{an}(u,\phi)$ at $u >> 1$ which can be used to compute other statistical properties at the center-of-band anomaly. We show that at moderately large values of $|\psi|^{2}\ell_{0}$, the probability of ALS at E=0 is smaller than at energies away from the anomaly. However, at very large values of $|\psi|^{2}\ell_{0}$, the tendency is inverted: it is exponentially easier to create a very strongly localized state at E=0 than at energies away from the anomaly. We also found the leading term in the behavior of $P(|\psi|^{2})$ at small $|\psi|^{2}<< \ell_{0}^{-1}$ and show that it is consistent with the exponential localization corresponding to the Lyapunov exponent found earlier by Kappus and Wegner and Derrida and Gardner.