Universal distributions and scaling in disordered systems

Cohen, Asaf; Roth, Yehuda; Shapiro, Boris

doi:10.1103/physrevb.38.12125

Cited by 104 publications

(112 citation statements)

References 36 publications

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“…With the use of the Migdal-Kadanoff renormalization, he studied the size dependence of the conductance distribution and he proved that the critical conductance distribution, p c (g), is universal, independent of the system length. The later works [103] however showed that the Migdal-Kadanoff renormalization overestimates the conductance fluctuations, and is therefore not suitable for a quantitative description of the critical conductance distribution.…”

Section: Statistical Properties Of the Conductance In The Critical Rementioning

confidence: 99%

Numerical analysis of the Anderson localization

Markoš¹

2006

Acta Physica Slovaca. Reviews and Tutorials

136

242

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The aim of this paper is to demonstrate, by simple numerical simulations, the main transport properties of disordered electron systems. These systems undergo the metal insulator transition when either Fermi energy crosses the mobility edge or the strength of the disorder increases over critical value. We study how disorder affects the energy spectrum and spatial distribution of electronic eigenstates in the diffusive and insulating regime, as well as in the critical region of the metal-insulator transition. Then, we introduce the transfer matrix and conductance, and we discuss how the quantum character of the electron propagation influences the transport properties of disordered samples. In the weakly disordered systems, the weak localization and anti-localization as well as the universal conductance fluctuation are numerically simulated and discussed. The localization in the one dimensional system is described and interpreted as a purely quantum effect. Statistical properties of the conductance in the critical and localized regimes are demonstrated. Special attention is given to the numerical study of the transport properties of the critical regime and to the numerical verification of the single parameter scaling theory of localization. Numerical data for the critical exponent in the orthogonal models in dimension 2 < d ≤ 5 are compared with theoretical predictions. We argue that the discrepancy between the theory and numerical data is due to the absence of the self-averaging of transmission quantities. This complicates the analytical analysis of the disordered systems. Finally, theoretical methods of description of weakly disordered systems are explained and their possible generalization to the localized regime is discussed. Since we concentrate on the one-electron propagation at zero temperature, no effects of electronelectron interaction and incoherent scattering are discussed in the paper.

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Section: Statistical Properties Of the Conductance In The Critical Rementioning

confidence: 99%

Numerical analysis of the Anderson localization

Markoš¹

2006

Acta Physica Slovaca. Reviews and Tutorials

136

242

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“…If the disorder is weak, then there is a universal relation between the mean and the variance of ln T (single parameter scaling). On the other hand, for strong disorder the two become independent of one another (two parameter scaling) [38].…”

Section: Strong Disordermentioning

confidence: 99%

Statistics of resonances in one-dimensional disordered systems

Gurevich¹,

Shapiro

2012

Lith. J. Phys.

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The paper is devoted to the problem of resonances in one-dimensional disordered systems. Some of the previous results are reviewed and a number of new ones is presented. These results pertain to different models (continuous as well as lattice) and various regimes of disorder and coupling strength. In particular, a close connection between resonances and the Wigner delay time is pointed out and used to obtain information on the resonance statistics.

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“…The two-parameter scaling is accepted to appear for strong disorder, where ∆ 2 is not an unambiguous function of f . 10 Interesting to note, the authors of Ref. 11 found two-parameter scaling also for weak disorder, namely for the Anderson 1D disorder at certain conditions.…”

Section: Introductionmentioning

confidence: 99%

Coherent resistance of a disordered one-dimensional wire: Expressions for all moments and evidence for non-Gaussian distribution

et al. 2003

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We study coherent electron transport in a one-dimensional wire with disorder modeled as a chain of randomly positioned scatterers. We derive analytical expressions for all statistical moments of the wire resistance ρ. By means of these expressions we show analytically that the distribution P (f ) of the variable f = ln(1 + ρ) is not exactly Gaussian even in the limit of weak disorder. In a strict mathematical sense, this conclusion is found to hold not only for the distribution tails but also for the bulk of the distribution P (f ).

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Universal distributions and scaling in disordered systems

Cited by 104 publications

References 36 publications

Numerical analysis of the Anderson localization

Numerical analysis of the Anderson localization

Statistics of resonances in one-dimensional disordered systems

Coherent resistance of a disordered one-dimensional wire: Expressions for all moments and evidence for non-Gaussian distribution

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