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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 | Abstract: | The 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 | Services: | Forum | Review | PDF | Favorites |
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