Commensurability effects in one-dimensional Anderson localization: Anomalies in eigenfunction statistics

Journal Article · Fri Jul 15 00:00:00 EDT 2011 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2011.02.009· OSTI ID:21583304

Kravtsov, V.E., E-mail: [e-mail] ^[1]; Landau Institute for Theoretical Physics, 2 Kosygina st., 117940 Moscow ^[2]; Yudson, V.I., E-mail: [e-mail] ^[3]

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1. Abdus Salam International Centre for Theoretical Physics, P.O. Box 586, 34100 Trieste (Italy)
2. (Russian Federation)
3. Institute for Spectroscopy, Russian Academy of Sciences, 142190 Troitsk, Moscow reg. (Russian Federation)

Highlights: > Statistics of normalized eigenfunctions in one-dimensional Anderson localization at E = 0 is studied. > Moments of inverse participation ratio are calculated. > Equation for generating function is derived at E = 0. > An exact solution for generating function at E = 0 is obtained. > Relation of the generating function to the phase distribution function is established. - Abstract: The one-dimensional (1d) Anderson model (AM), i.e. a tight-binding chain with random uncorrelated on-site energies, has statistical anomalies at any rational point f=(2a)/({lambda}{sub E}) , where a is the lattice constant and {lambda}{sub E} is the de Broglie wavelength. We develop a regular approach to anomalous statistics of normalized eigenfunctions {psi}(r) at such commensurability points. The approach is based on an exact integral transfer-matrix equation for a generating function {Phi}{sub r}(u, {phi}) (u and {phi} have a meaning of the squared amplitude and phase of eigenfunctions, r is the position of the observation point). This generating function can be used to compute local statistics of eigenfunctions of 1d AM at any disorder and to address the problem of higher-order anomalies at f=p/q with q > 2. The descender of the generating function P{sub r}({phi}){identical_to}{Phi}{sub r}(u=0,{phi}) is shown to be the distribution function of phase which determines the Lyapunov exponent and the local density of states. In the leading order in the small disorder we derived a second-order partial differential equation for the r-independent ('zero-mode') component {Phi}(u, {phi}) at the E = 0 (f=1/2 ) anomaly. This equation is nonseparable 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 computed moments I{sub m} = N< vertical bar {psi} vertical bar {sup 2m}> (m {>=} 1) for a chain of the length N {yields} {infinity} and found an essential difference between their m-behavior in the center-of-band anomaly and for energies outside this anomaly. Outside the anomaly the 'extrinsic' localization length defined from the Lyapunov exponent coincides with that defined from the inverse participation ratio ('intrinsic' localization length). This is not the case at the E = 0 anomaly where the extrinsic localization length is smaller than the intrinsic one. At E = 0 one also observes an anomalous enhancement of large moments compatible with existence of yet another, much smaller characteristic length scale.

• OSTI ID: 21583304
• Journal Information: Annals of Physics (New York), Vol. 326, Issue 7; Other Information: DOI: 10.1016/j.aop.2011.02.009; PII: S0003-4916(11)00038-8; Copyright (c) 2011 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; ISSN 0003-4916
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
AMPLITUDES
DE BROGLIE WAVELENGTH
DENSITY
DISTRIBUTION FUNCTIONS
EIGENFUNCTIONS
EXACT SOLUTIONS
INTEGRAL CALCULUS
LATTICE PARAMETERS
LYAPUNOV METHOD
MATRICES
ONE-DIMENSIONAL CALCULATIONS
PARTIAL DIFFERENTIAL EQUATIONS
QUADRATURES
RANDOMNESS
STATISTICS
SYMMETRY
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
EQUATIONS
FUNCTIONS
MATHEMATICAL SOLUTIONS
MATHEMATICS
PHYSICAL PROPERTIES
WAVELENGTHS