[PDF] Anderson Localization In High Dimensional Lattices eBook

Author : Elena Tarquini
Publisher :
Page : 0 pages
File Size : 47,35 MB
Release : 2016
Category :
ISBN :

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In this thesis, we investigate the behavior of Anderson Localization in high dimension. In the first part we study Lévy Matrices (LMs), a Random Matrix model with long-range hopping presenting strong analogy with the problem of Anderson Localization on tree-like structure, representative of the limit of infinite dimensionality. We establish the equation determining the localization transition and obtain the phase diagram. We investigate then the unusual behavior of the delocalized phase. Using arguments based on supersymmetric field theory and Dyson Brownian motion we show that the eigenvalue statistics is the same one as of the Gaussian orthogonal ensemble in the whole delocalized phase and is Poisson-like in the localized phase. Our numerical analysis confirms this result, valid in the limit of infinitely large LMs, and provides information on the behavior of other observables like the wave-functions statistics. At the same time, numerical results also reveal that the characteristic scale governing finite size effects diverges much faster than a power law approaching the transition and is already very large far from it. This leads to a very wide crossover region in which the system looks as if it were in a mixed phase. In the second part we study numerically the behavior of the Anderson Model in dimension from 3 to 6 through exact diagonalization, Transfer Matrix method and an approximate Strong Disorder Renormalization Group technique. The results we find suggest that the upper critical dimension of Anderson Localization is infinite. Finally, we discuss the possible implications of this scenario on the anomalous behavior of the delocalized phase of models representative of the limit of infinite dimension, like Lévy Matrices and the Anderson model on tree-like structures.

Author : Elihu Abrahams
Publisher : World Scientific
Page : 610 pages
File Size : 41,16 MB
Release : 2010
Category : Science
ISBN : 9814299065

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This unique volume celebrates the five decades of the impact of Anderson localization on modern physics. In addition to the historical perspective on its origin, it provides a comprehensive description of the experimental and theoretical aspects of Anderson localization.

Author : Elihu Abrahams
Publisher : World Scientific
Page : 610 pages
File Size : 29,81 MB
Release : 2010
Category : Science
ISBN : 9814299073

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This unique volume celebrates the five decades of the impact of Anderson localization on modern physics. In addition to the historical perspective on its origin, it provides a comprehensive description of the experimental and theoretical aspects of Anderson localization.

Author : Hongyi Xie
Publisher : LAP Lambert Academic Publishing
Page : 208 pages
File Size : 43,54 MB
Release : 2012
Category :
ISBN : 9783659312892

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The problem of localization of interacting particles leads me to consider the question of competition between alternative propagation channels, which can be studied in great detail in the form of a single particle problem with two parallel, parametrically different, coupled channels. One could ask what happens to the localization properties when a less localized lattice is coupled to a more localized one? Will the less localized one dominate the localization of the system or the more localized? The qualitative answer to this question depends on the dimensionality of the system. Correspondingly, I exactly solved the Anderson models on a two-leg ladder and on a two-layer Bethe lattice. In one dimension, in the weak disorder limit the localization lengths of two coupled chains are of the order of the localization length of the more localized chain under resonance conditions. On the Bethe lattices, the less disordered lattice is not affected much by the more disordered lattice in the presence of coupling. These trends are believed to be persistent in high dimensions.

Author : Y. Nagaoka
Publisher : Springer Science & Business Media
Page : 236 pages
File Size : 28,46 MB
Release : 2012-12-06
Category : Science
ISBN : 3642818412

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This volume contains the proceedings of the Fourth Taniguchi International Symposium on the Theory of Condensed Matter, which was held at Senkari Semi nar House of Kwansei Gakuin Universi~y in Sanda-shi, Japan, during the period of 3-8 November 1981. The topic of the symposium was "Anderson rocalization," one of the most fundamental problems in condensed-matter physics. Since Anderson's classic paper was published in 1958, much theoretical and experimental effort has been performed to study the problem of electron localization in a random potential. Quite recently, Abrahams, Anderson, Licciardello, and Ramakrishnan proposed a scaling theory of the Anderson lo calization which made it possible to perform microscopic investigations. Rapid progress has followed and we are now getting a coherent picture of the behavior of electrons in disordered systems. When we organized the symposium, we asked Dr. Anderson to participate in it and to give a review talk on theoretical aspects of the problem. Though he kindly accepted our invitation, he could not come due to a sudden illness. A review talk was given by Professor Thouless who kindly accepted our request to take the place of Dr. Anderson. Fortunately, Dr. Anderson has since re covered from his illness.

Author : Margherita Disertori
Publisher : SMF
Page : 244 pages
File Size : 29,24 MB
Release : 2008
Category : Mathematics
ISBN :

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During the last thirty years, random Schrodinger operators, which originated in condensed matter physics, have been studied intensively and very productively. The theory is at the crossroads of a number of mathematical fields: the theory of operators, partial differential equations, the theory of probabilities, in particular the study of stochastic processes and that of random walks and Brownian motion in a random environment. This monograph aims to give the reader a panorama of the subject, from the now-classic foundations to very recent developments.

Author : Yu.V. Kopaev
Publisher : Elsevier
Page : 353 pages
File Size : 13,15 MB
Release : 2012-12-02
Category : Science
ISBN : 0444600396

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Electronic Phase Transitions deals with topics, which are presently at the forefront of scientific research in modern solid-state theory. Anderson localization, which has fundamental implications in many areas of solid-state physics as well as spin glasses, with its influence on quite different research activities such as neural networks, are two examples that are reviewed in this book. The ab initio statistical mechanics of structural phase transitions is another prime example, where the interplay and connection of two unrelated disciplines of solid-state theory - first principle electronic structure calculations and critical phenomena - has given rise to impressive new insights. Clearly, there is more and more need for accurate, stable numerical simulations of models of interacting electrons, presently discussed with great vigor in connection with high-Tc superconductors where the superconducting transition is close to a magnetic transition, i.e. an antiferromagnetic spin structure. These topics and others are discussed and reviewed by leading experts in the field.

Author : Shi-Ju Ran
Publisher : Springer Nature
Page : 160 pages
File Size : 13,50 MB
Release : 2020-01-27
Category : Science
ISBN : 3030344894

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Tensor network is a fundamental mathematical tool with a huge range of applications in physics, such as condensed matter physics, statistic physics, high energy physics, and quantum information sciences. This open access book aims to explain the tensor network contraction approaches in a systematic way, from the basic definitions to the important applications. This book is also useful to those who apply tensor networks in areas beyond physics, such as machine learning and the big-data analysis. Tensor network originates from the numerical renormalization group approach proposed by K. G. Wilson in 1975. Through a rapid development in the last two decades, tensor network has become a powerful numerical tool that can efficiently simulate a wide range of scientific problems, with particular success in quantum many-body physics. Varieties of tensor network algorithms have been proposed for different problems. However, the connections among different algorithms are not well discussed or reviewed. To fill this gap, this book explains the fundamental concepts and basic ideas that connect and/or unify different strategies of the tensor network contraction algorithms. In addition, some of the recent progresses in dealing with tensor decomposition techniques and quantum simulations are also represented in this book to help the readers to better understand tensor network. This open access book is intended for graduated students, but can also be used as a professional book for researchers in the related fields. To understand most of the contents in the book, only basic knowledge of quantum mechanics and linear algebra is required. In order to fully understand some advanced parts, the reader will need to be familiar with notion of condensed matter physics and quantum information, that however are not necessary to understand the main parts of the book. This book is a good source for non-specialists on quantum physics to understand tensor network algorithms and the related mathematics.

Author : Tobias Brandes
Publisher : Springer Science & Business Media
Page : 446 pages
File Size : 46,6 MB
Release : 2003-09-11
Category : Science
ISBN : 9783540407850

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The phenomenon of localization of the electronic wave function in a random medium can be regarded as the key manifestation of quantum coherence in a condensed matter system. As one of the most remarkable phenomena in condensed matter physics discovered in the 20th century, the localization problem is an indispensable part of the theory of the quantum Hall effects and rivals superconductivity in its significance as a manifestation of quantum coherence at a macroscopic scale. The present volume, written by some of the leading experts in the field, is intended to highlight some of the recent progress in the field of localization, with particular emphasis on the effect of interactions on quantum coherence. The chapters are written in textbook style and should serve as a reliable and thorough introduction for advanced students or researchers already working in the field of mesoscopic physics.