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29 March 2024
 
  » arxiv » 1011.1480

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Commensurability effects in one-dimensional Anderson localization: anomalies in eigenfunction statistics
V.E.Kravtsov ; V.I.Yudson ;
Date 5 Nov 2010
AbstractThe one-dimensional (1d) Anderson model (AM), i.e. a tight-binding chain with uncorrelated Gaussian disorder in the on-site energies, has statistical anomalies at any rational point $f=2a/lambda_{E}$, where $a$ is the lattice constant and $lambda_{E}$ is the de Broglie wavelength. We develop a regular approach to anomalous eigenfunction statistics at such commensurability points. The approach is based on an exact integral transfer-matrix equation for a generating function $Phi(u,phi;r)$ ($u$ and $phi$ have a meaning of the squared amplitude and phase of eigenstates, $r$ is the position of the observation point). This generating function can be used to compute local statistics of normalized eigenfunctions of 1d AM at any disorder and to address the problem of higher-order anomalies at $f=p/q$ with $q>2$. However, in this paper we concentrate at the principle (center-of-band) anomaly at $E=0$ ($f=1/2$).
In the leading order in the small disorder we have derived a second-order partial differential equation for the $r$-independent ("zero-mode") component $Phi(u, phi)$ at the $E=0$ anomaly. This equation is non-separable in variables $u$ and $phi$. Yet, we show that due to a hidden symmetry, it is integrable and we construct an exact solution for $Phi(u, phi)$ explicitly in quadratures. Using this solution we have computed moments $I_{m}=<|psi|^{2m}>$ ($m>1$) of the wave function distribution for a very long chain and found an essential difference between their $m$-behavior in the center-of-band and for energies outside the region of anomaly. Outside the anomaly the function $I_{m}$ is determined by only one parameter $l_{0}$ while more parameters are needed to describe $ I_{m}$ at $E=0$.
Source arXiv, 1011.1480
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