On the Mott formula for the ac conductivity and binary correlators in the strong localization regime of disordered systems

Kirsch, Werner; Lenoble, O.; Pastur, L. А.

doi:10.1088/0305-4470/36/49/003

Cited by 17 publications

(22 citation statements)

References 22 publications

(65 reference statements)

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“…verges to the Poisson process on R. If we project it to the space axis, we have a result on the distribution of localization centers. To be precise, pick an interval J(⊂ R) and let In [11], they assume the Poisson distribution of localization centers and discuss the derivation of Mott's formula on the a.c. conductivity(rigorous derivation of Mott's formula is recently done by [12]).…”

Section: Assumption B (Minami's Estimate)mentioning

confidence: 99%

Distribution of Localization Centers in Some Discrete Random Systems

Nakano

2007

Rev. Math. Phys.

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As a supplement of our previous work [10], we consider the localized region of the random Schrödinger operators on l 2 (Z d ) and study the point process composed of their eigenvalues and corresponding localization centers. For the Anderson model we show that, this point process in the natural scaling limit converges in distribution to the Poisson process on the product space of energy and space. In other models with suitable Wegner-type bounds, we can at least show that limiting point processes are infinitely divisible. Mathematics Subject Classification (2000): 82B44, 81Q10where ν is the distribution of V ω (0) [15].(2)(Anderson localization) There is an open interval I ⊂ Σ such that with probability one, the spectrum of H ω on I is pure point with exponentially decaying eigenfunctions. I can be taken (i) I = Σ if λ is large enough, (ii) on the band edges, and (iii) away from the spectrum of the free Laplacian if λ sufficiently small (e.g., [7,20,1,2]).Recently, some relations between the eigenvalues and the corresponding localization centers are discussed [17]. It roughly implies,Hence the distribution of the localization centers are "thin" in space 1 .(Hence the localization centers are repulsive if the energies get closer.On the other hand, in [10], they study the "natural scaling limit" of the random measure in R d+1 (the product of energy and space) composed of the eigenvalues and eigenfunctions. The result there roughly implies that their distribution with eigenvalues in the order of L −d from the reference energy E 0 , and with eigenfunctions in the order of L from the origin, obey the Poisson law on R d+1 . This work can also be regarded as an extension of the work by Minami [16] who showed that the point process on R composed of the eigenvalues of H in the finite volume approximation converges to the Poisson process on R. To summarize, [17,10] imply that the eigenfunctions whose energies are in the order of L −d are non-repulsive while those in the order of L −2d are repulsive, which are consistent with Minami's result [16].The aim of this paper is to supplement [10] from a technical point of view : (i) to study the distribution of the localization centers which is technically different from what is done in [10], and (ii) to study what can be said for those models in which Minami's estimate and the fractional moment bound, which are the main tool in [10], are currently not known to hold.We set some notations. Notation :1 This result follows easily from the upper bound on the density of states. So in the Lifschitz tail region, we have |x(E)| ≥ (const.)e (const.)L d 2 2if |E − E 0 | ≤ L

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Section: Assumption B (Minami's Estimate)mentioning

confidence: 99%

Distribution of Localization Centers in Some Discrete Random Systems

Nakano

2007

Rev. Math. Phys.

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“…[12] proves ( * ) in a special model on one space dimension, with a complicated statement. On the other hand, the observation ( * ) was used by Mott in the study of the ac-conductivity of random systems whose mathematical study is done recently (Kirsch-Lenoble-Pastur, Klein-Lenoble-Müller [7,9]). The purpose of this paper is to obtain a quantitative statement of ( * ) which holds for almost surely.…”

Section: Introductionmentioning

confidence: 99%

The Repulsion Between Localization Centers in the Anderson Model

Nakano

2006

J Stat Phys

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“…The centers x A and x B of the pair of localized states are located outside the interval (x 1 ,x 2 ). The variables r A , r B , and r used in Eq (41). are the pairwise distances between the four points x A , x 1 , x 2 , and x B .…”

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Hybridization of wave functions in one-dimensional localization

Иванов

Skvortsov²,

Ostrovsky³

et al. 2012

Phys. Rev. B

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A quantum particle can be localized in a disordered potential, the effect known as Anderson localization. In such a system, correlations of wave functions at very close energies may be described, due to Mott, in terms of a hybridization of localized states. We revisit this hybridization description and show that it may be used to obtain quantitatively exact expressions for some asymptotic features of correlation functions, if the tails of the wave functions and the hybridization matrix elements are assumed to have log-normal distributions typical for localization effects. Specifically, we consider three types of one-dimensional systems: a strictly one-dimensional wire and two quasi-one-dimensional wires with unitary and orthogonal symmetries. In each of these models, we consider two types of correlation functions: the correlations of the density of states at close energies and the dynamic response function at low frequencies. For each of those correlation functions, within our method, we calculate three asymptotic features: the behavior at the logarithmically large "Mott length scale", the low-frequency limit at length scale between the localization length and the Mott length scale, and the leading correction in frequency to this limit. In the several cases, where exact results are available, our method reproduces them within the precision of the orders in frequency considered.Comment: 10 pages, 5 figures. Several references added, minor corrections corresponding to the journal versio

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On the Mott formula for the ac conductivity and binary correlators in the strong localization regime of disordered systems

Cited by 17 publications

References 22 publications

Distribution of Localization Centers in Some Discrete Random Systems

Distribution of Localization Centers in Some Discrete Random Systems

The Repulsion Between Localization Centers in the Anderson Model

Hybridization of wave functions in one-dimensional localization

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