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

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 ${\Phi }_{an}(u,\varphi )$ 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 ${\Phi }_{an}(u,\varphi )$ 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.

25 pages, 9 figures

References in corpus (2)

• Band-center anomaly of the conductance distribution in one-dimensional Anderson localization
• Scaling and the center of band anomaly in a one-dimensional Anderson model with diagonal disorder

Cited by in corpus (2)

• Disordered Josephson junction chains: Anderson localization of normal modes and impedance fluctuations
• Wavefunction extreme value statistics in Anderson localization