## Fluctuations and Localization in Mesoscopic Electron SystemsThe quantum phenomena of tunneling and interference show up not only in the microscopic world of atoms and molecules, but also in cold materials of the real world, such as metals and semiconductors. Though not fully macroscopic, such mesoscopic systems contain a huge number of particles, and the holistic nature of quantum mechanics becomes evident already in simple electronic measurements. The measured quantity fluctuates as a function of applied fields in an unpredictable, yet reproducible way. Despite this fingerprint character of fluctuations, their statistical properties are universal, i.e. they are the same for a large class of different mesoscopic systems, having only very few parameters in common. Localization of electrons is a dramatic effect of destructive interference. As a consequence a metal can become an insulator while reaching mesoscopic scales. Based on elementary quantum and statistical physics, this text introduces the theory of mesoscopic electron systems. It focuses on universal characteristics of fluctuations and on the localization mechanism. General concepts and methods are stressed, such as scaling laws for distribution functions. Tools from condensed matter theory are used flexibly. Involved technical details are skipped so as to present a broad overview of the field, including topics like quantum dots, the quantum Hall effect and a number of the most recent developments. Contents: Experimental Facts; Basic Theoretical Models and Tools; Idealized Systems; Towards Realistic Systems; The Localization-Delocalization Transition. Readership: Condensed matter and theoretical physicists. |

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### Contents

Experimental Facts | 11 |

Basic Theoretical Models and Tools | 39 |

Idealized Systems | 103 |

Towards Realistic Systems | 117 |

The LocalizationDelocalization Transition | 153 |

191 | |

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### Common terms and phrases

Altshuler amplitudes asymptotic average value behavior box probability calculations Chalker-Coddington model characterized classical conductance distribution conductance g conformal mapping constant correlation function correlation length corresponding critical exponent defined delocalized phase denoted density dependence described diffusion disorder distribution function Efetov eigenstates eigenvalues eigenvectors electron systems ensemble exponential Fermi energy field theory filling factors finite fixed point flux Fokker-Planck equation Gaussian Hall conductance Hamiltonian idealized independent inverse Janssen Landau band LD transition leads length scale level spacing linear localization length localization-delocalization localized phase log-normal log-normal distribution magnetic field mean free path mesoscopic conductors mesoscopic systems microscopic Mirlin multifractal network model one-parameter scaling order parameter physical quantities point-contact conductances potential power law quantum dots quantum Hall effect quasi-lD systems random matrix theory regime relevant renormalization S-matrix scaling theory scaling variable scattering strengths tails thermodynamic Thouless energy transport typical value unitary vanishes wave function weak localization zero