Statistics of energy levels and eigenfunctions in disordered systems
TL;DR: In this article, a review of recent developments in the theory of fluctuations and correlations of energy levels and eigenfunction amplitudes in diffusive mesoscopic samples is presented, with emphasis on low-dimensional (quasi-1D and 2D) systems.
About: This article is published in Physics Reports.The article was published on 2000-03-01 and is currently open access. It has received 557 citations till now. The article focuses on the topics: Eigenfunction & Mesoscopic physics.
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TL;DR: In this paper, it was shown that in the absence of coupling of the electrons to any external bath dc electrical conductivity exactly vanishes as long as the temperature T does not exceed some finite value Tc.
1,699 citations
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TL;DR: In this paper, the physics of Anderson transition between localized and metallic phases in disordered systems is reviewed, including both metal-insulator transitions and quantum-Hall-type transitions between phases with localized states.
Abstract: The physics of Anderson transitions between localized and metallic phases in disordered systems is reviewed The term ``Anderson transition'' is understood in a broad sense, including both metal-insulator transitions and quantum-Hall-type transitions between phases with localized states The emphasis is put on recent developments, which include: multifractality of critical wave functions, criticality in the power-law random banded matrix model, symmetry classification of disordered electronic systems, mechanisms of criticality in quasi-one-dimensional and two-dimensional systems and survey of corresponding critical theories, network models, and random Dirac Hamiltonians Analytical approaches are complemented by advanced numerical simulations
1,505 citations
Cites background or methods or result from "Statistics of energy levels and eig..."
...A high-precision evaluation of the multifractal spectrum at the IQH transition was carried out in Evers et al. (2001) for the CCN model of a size L × L with L ranging from 16 to 1280....
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...The quasi-metallic regime b ≫ 1 can be studied (Evers and Mirlin, 2000; Mirlin, 2000b; Mirlin and Evers, 2000; Mirlin et al., 1996) via mapping onto the supermatrix σ-model, cf. Sec....
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...(2.62) For the case of the unitary symmetry class (A), a relation linking the exponents Xq to the wave function anomalous dimensions ∆q [and based on a result of Klesse and Zirnbauer (2001)] was obtained (Evers et al., 2001), Xq = { ∆q + ∆1−q , q < 1/2 2∆1/2 , q > 1/2....
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...III, and the 2D Anderson transition of the symplectic class (Mildenberger and Evers, 2007; Obuse et al., 2007b), Sec....
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...It is worth mentioning that the physics of the intermediate, quasiclassical regime, where the physics is dominated by the vicinity of the percolation fixed point (Evers and Brenig, 1994, 1998; Gammel and Brenig, 1996; Klesse and Metzler, 1995; Kratzer and Brenig, 1994; Mil’nikov and Sokolov, 1988) is interesting in its own right....
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TL;DR: The experimental observation of Anderson localization in a perturbed periodic potential is reported: the transverse localization of light caused by random fluctuations on a two-dimensional photonic lattice, demonstrating how ballistic transport becomes diffusive in the presence of disorder, and that crossover to Anderson localization occurs at a higher level of disorder.
Abstract: One of the most interesting phenomena in solid-state physics is Anderson localization, which predicts that an electron may become immobile when placed in a disordered lattice. The origin of localization is interference between multiple scatterings of the electron by random defects in the potential, altering the eigenmodes from being extended (Bloch waves) to exponentially localized. As a result, the material is transformed from a conductor to an insulator. Anderson's work dates back to 1958, yet strong localization has never been observed in atomic crystals, because localization occurs only if the potential (the periodic lattice and the fluctuations superimposed on it) is time-independent. However, in atomic crystals important deviations from the Anderson model always occur, because of thermally excited phonons and electron-electron interactions. Realizing that Anderson localization is a wave phenomenon relying on interference, these concepts were extended to optics. Indeed, both weak and strong localization effects were experimentally demonstrated, traditionally by studying the transmission properties of randomly distributed optical scatterers (typically suspensions or powders of dielectric materials). However, in these studies the potential was fully random, rather than being 'frozen' fluctuations on a periodic potential, as the Anderson model assumes. Here we report the experimental observation of Anderson localization in a perturbed periodic potential: the transverse localization of light caused by random fluctuations on a two-dimensional photonic lattice. We demonstrate how ballistic transport becomes diffusive in the presence of disorder, and that crossover to Anderson localization occurs at a higher level of disorder. Finally, we study how nonlinearities affect Anderson localization. As Anderson localization is a universal phenomenon, the ideas presented here could also be implemented in other systems (for example, matter waves), thereby making it feasible to explore experimentally long-sought fundamental concepts, and bringing up a variety of intriguing questions related to the interplay between disorder and nonlinearity.
1,368 citations
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01 Jan 2001
TL;DR: Mostafa Adimy as mentioned in this paper Directeur de Recherches à l’INRIA Dir. de thèse Ionel S. CIUPERCA Mâıtre de Conférence à l'Université Lyon 1 Examinateur Michael C. MACKEY Directeur of Recherche et al.
Abstract: Mostafa ADIMY Directeur de Recherches à l’INRIA Dir. de thèse Ionel S. CIUPERCA Mâıtre de Conférence à l’Université Lyon 1 Examinateur Michael C. MACKEY Directeur de Recherche à l’Université Mc GIll Dir. de thèse Sylvie MÉLÉARD Professeur à l’Ecole Polytechnique Examinatrice Sophie MERCIER Professeur à l’Université de Pau et des Pays de l’Adour Rapportrice Laurent PUJO-MENJOUET Mâıtre de Conférence à l’Université Lyon 1 Dir. de thèse Marta TYRAN-KAMIŃSKA Professeur à l’University of Silesia Examinatrice Bernard YCART Professeur à l’Université de Grenoble Rapporteur
713 citations
Book•
13 Jul 2011TL;DR: Random matrix theory is a wide and growing field with a variety of concepts, results, and techniques and a vast range of applications in mathematics and related sciences as mentioned in this paper, which is a good reference for researchers in various areas of mathematics and mathematical physics.
Abstract: Random matrix theory is a wide and growing field with a variety of concepts, results, and techniques and a vast range of applications in mathematics and the related sciences. The book, written by well-known experts, offers beginners a fairly balanced collection of basic facts and methods (Part 1 on classical ensembles) and presents experts with an exposition of recent advances in the subject (Parts 2 and 3 on invariant ensembles and ensembles with independent entries). The text includes many of the authors' results and methods on several main aspects of the theory, thus allowing them to present a unique and personal perspective on the subject and to cover many topics using a unified approach essentially based on the Stieltjes transform and orthogonal polynomials. The exposition is supplemented by numerous comments, remarks, and problems. This results in a book that presents a detailed and self-contained treatment of the basic random matrix ensembles and asymptotic regimes. This book will be an important reference for researchers in a variety of areas of mathematics and mathematical physics. Various chapters of the book can be used for graduate courses; the main prerequisite is a basic knowledge of calculus, linear algebra, and probability theory.
443 citations
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"Statistics of energy levels and eig..." refers background in this paper
...The corresponding boundary condition [268,269] di!ers from Eq....
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Book•
01 Jan 1978
TL;DR: This IEEE Classic Reissue presents a unified introduction to the fundamental theories and applications of wave propagation and scattering in random media and is expressly designed for engineers and scientists who have an interest in optical, microwave, or acoustic wave propagate and scattering.
Abstract: A volume in the IEEE/OUP Series on Electromagnetic Wave Theory Donald G. Dudley, Series Editor This IEEE Classic Reissue presents a unified introduction to the fundamental theories and applications of wave propagation and scattering in random media. Now for the first time, the two volumes of Wave Propagation and Scattering in Random Media previously published by Academic Press in 1978 are combined into one comprehensive volume. This book presents a clear picture of how waves interact with the atmosphere, terrain, ocean, turbulence, aerosols, rain, snow, biological tissues, composite material, and other media. The theories presented will enable you to solve a variety of problems relating to clutter, interference, imaging, object detection, and communication theory for various media. This book is expressly designed for engineers and scientists who have an interest in optical, microwave, or acoustic wave propagation and scattering. Topics covered include:
5,877 citations
TL;DR: In this paper, it was shown that the conductance of disordered electronic systems depends on their length scale in a universal manner, and asymptotic forms for the scaling function were obtained for both two-dimensional and three-dimensional systems.
Abstract: Arguments are presented that the $T=0$ conductance $G$ of a disordered electronic system depends on its length scale $L$ in a universal manner. Asymptotic forms are obtained for the scaling function $\ensuremath{\beta}(G)=\frac{d\mathrm{ln}G}{d\mathrm{ln}L}$, valid for both $G\ensuremath{\ll}{G}_{c}\ensuremath{\simeq}\frac{{e}^{2}}{\ensuremath{\hbar}}$ and $G\ensuremath{\gg}{G}_{c}$. In three dimensions, ${G}_{c}$ is an unstable fixed point. In two dimensions, there is no true metallic behavior; the conductance crosses over smoothly from logarithmic or slower to exponential decrease with $L$.
4,466 citations