Statistics of anomalously localized states at the center of band E = 0 in the onedimensional Anderson localization model
Abstract
We consider the distribution function P(ψ^{2}) of the eigenfunction amplitude at the centerofband (E = 0) anomaly in the onedimensional tightbinding chain with weak uncorrelated onsite disorder (the onedimensional Anderson model). The special emphasis is on the probability of the anomalously localized states (ALS) with ψ^{2} much larger than the inverse typical localization length ℓ_{0}. Using the recently found solution for the generating function Φ_{an}(u, ϕ) we obtain the ALS probability distribution P(ψ^{2}) at ψ^{2}ℓ_{0} ≫ 1. As an auxiliary preliminary step, we found the asymptotic form of the generating function Φ_{an}(u, ϕ) at u ≫ 1 which can be used to compute other statistical properties at the centerofband anomaly. We show that at moderately large values of ψ^{2}ℓ_{0}, the probability of ALS at E = 0 is smaller than at energies away from the anomaly. However, at very large values of ψ^{2}ℓ_{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(ψ^{2}) at small ψ^{2} ≪ ℓ^{1}_{0} and show that it is consistent with the exponential localization corresponding to the Lyapunov exponent found earlier by Kappus and Wegner.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 January 2013
 DOI:
 10.1088/17518113/46/2/025001
 arXiv:
 arXiv:1208.4789
 Bibcode:
 2013JPhA...46b5001K
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks;
 Mathematical Physics
 EPrint:
 25 pages, 9 figures