## 50 Years of Anderson LocalizationIn his groundbreaking paper Absence of diffusion in certain random lattices (1958), Philip W. Anderson originated, described and developed the physical principles underlying the phenomenon of the localization of quantum objects due to disorder. Anderson''s 1977 Nobel Prize citation featured that paper, which was fundamental for many subsequent developments in condensed matter theory and technical applications. After more than a half century, the subject continues to be of fundamental importance. In particular, in the last 25 years, the phenomenon of localization has proved to be crucial for the understanding of the Quantum Hall Effect, mesoscopic fluctuations in small conductors, some aspects of quantum chaotic behavior, and most recently the localization and collective modes of electromagnetic and matter waves. This unique and invaluable volume celebrates the five decades of the impact of Anderson Localization on modern physics.In addition to the historical perspective on its origin, the volume provides a comprehensive description of the experimental and theoretical aspects of Anderson localization, together with its application in various areas, which include disordered metals and the metal-insulator transition, mesoscopic physics, classical systems and light, strongly-correlated systems, and mathematical models. The volume is edited by E Abrahams, who, as a major contributor in the field of localization, has published several papers on the celebrated scaling theory of localization. A distinguished group of experts, each of whom has left his mark on the developments of this fascinating theory, contribute their personal insights in this volume.They are: P W Anderson (Nobel Laureate, 1977), G Bergmann (University of Southern California), M Buttiker (University of Geneva), J Cardy (Oxford University), S Chakravarty (University of California, Los Angeles), V Dobrosavljevic (Florida State University), R C Dynes (University of California, San Diego), K B Efetov (Ruhr University Bochum), A M Finkel''stein (Texas A&M University), A Genack (Queens College, New York), N Giordano (Purdue University), Y Imry (Weizmann Institute), B Kramer(University of Hamburg), S V Kravchenko (Northeastern University), A Mirlin(University of Karlsruhe), A M M Pruisken (University of Amsterdam), T V Ramakrishnan (Indian Institute of Science), T Spencer (Institute of Advanced Study, Princeton), D J Thouless (University of Washington), D Vollhardt (University of Augsburg) and F J Wegner (University of Heidelberg). |

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### Contents

1 | |

7 | |

27 | |

43 | |

Chapter 5 Anderson Localization and Supersymmetry K B Efetov | 73 |

Criticality Symmetries and Topologies A D Mirlin F Evers I V Gornyi and P M Ostrovsky | 107 |

Chapter 7 Scaling of von Neumann Entropy at the Anderson Transition S Chakravarty | 151 |

Chapter 8 From Anderson Localization to Mesoscopic Physics M B uttiker and M Moskalets | 169 |

Chapter 13 Inhomogeneous Fixed Point Ensembles Revisited F J Wegner | 289 |

Chapter 14 Quantum Network Models and Classical Localization Problems J Cardy | 301 |

Chapter 15 Mathematical Aspects of Anderson Localization T Spencer | 327 |

Chapter 16 Finite Size Scaling Analysis of the Anderson Transition B Kramer A MacKinnon T Ohtsuki and K Slevin | 347 |

Established Facts and Open Questions S V Kravchenko and M P Sarachik | 361 |

Chapter 18 Disordered Electron Liquid with Interactions A M Finkelstein | 385 |

Chapter 19 TypicalMedium Theory of MottAnderson Localization V Dobrosavljevi c | 425 |

Chapter 20 Anderson Localization vs MottHubbard MetalInsulator Transition in Disordered Interacting Lattice Fermion Systems K Byczuk W Hofste... | 473 |

Electric and Thermal Transport Y Imry and A Amir | 191 |

Chapter 10 Localization and the MetalInsulator Transition Experimental Observations R C Dynes | 213 |

Chapter 11 Weak Localization and its Applications as an Experimental Tool G Bergmann | 231 |

Chapter 12 Weak Localization and ElectronElectron Interaction Effects in thin Metal Wires and Films N Giordano | 269 |

Chapter 21 Topological Principles in the Theory of Anderson Localization A M M Pruisken | 503 |

Chapter 22 Speckle Statistics in the Photon Localization Transition A Z Genack and J Wang | 559 |

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### Common terms and phrases

A. M. M. Pruisken American Physical Society amplitude Anderson localization Anderson transition average conductivity correlation function corresponding Coulomb interaction critical behavior critical exponent critical point density dephasing described diagram dimensional dimensions disordered systems DMFT Dobrosavljevic electron–electron interactions ensemble equation experimental exponentially Fermi energy Fermi liquid Fermi-liquid fermions films finite fixed point fluctuations frequency Green’s function Hamiltonian impurities integral JETP lattice Lett limit localization length magnetic field magnetoresistance matrix measurements mesoscopic metal–insulator transition metallic mobility edge modes MOSFETs Mott multifractal obtained P. W. Anderson particles phase transition physical potential quantum Hall effect random regime renormalization Reprinted Fig resistance sample scaling theory self-consistent shown in Fig speckle pattern spin supermatrix supersymmetry symmetry class T. V. Ramakrishnan temperature dependence thermopower Thouless tion transmission two-dimensional wave functions weak localization zero