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

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          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.

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          Green's Functions for a Particle in a One-Dimensional Random Potential

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            Superpotentials for a Generally Covariant Field Theory

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              Author and article information

              Journal
              23 August 2012
              Article
              10.1088/1751-8113/46/2/025001
              1208.4789
              31fd2734-6acf-4ae9-83b8-6ddc80a5d102

              http://arxiv.org/licenses/nonexclusive-distrib/1.0/

              History
              Custom metadata
              JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL v.46, 025001 (2013)
              25 pages, 9 figures
              cond-mat.dis-nn math-ph math.MP

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