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A Short Introduction to Anderson Localization

Dirk Hundertmark

in Analysis and Stochastics of Growth Processes and Interface Models

Anderson localization is another physical problem that has spurred much mathematical research. The issue here is how disorder, such as random changes in the spacing of a crystal, influences the ... More

Anderson localization is another physical problem that has spurred much mathematical research. The issue here is how disorder, such as random changes in the spacing of a crystal, influences the movement of electrons and thus the crystal's conductivity. In 1977, Anderson was awarded the Nobel prize for his investigations on this subject. This chapter introduces the physical model, based on a random Schrodinger operator, and carefully reviews different notions of localization as well as rigorous proofs of localization. A very readable introduction to finite-volume criteria for localization via percolation arguments is followed by an elegant proof of localization for large disorder.Less

Disorder and interference: localization phenomena

Cord A. Müller and Dominique Delande

in Ultracold Gases and Quantum Information: Lecture Notes of the Les Houches Summer School in Singapore: Volume 91, July 2009

In the presence of disorder, classical transport is usually diffusive. This chapter deals with the effect of interference between multiply scattered waves on transport properties. Interference may ... More

In the presence of disorder, classical transport is usually diffusive. This chapter deals with the effect of interference between multiply scattered waves on transport properties. Interference may lead to reduced diffusive transport—this is known as weak localization—or to complete inhibition of transport—which is known as Anderson localization or strong localization. Key parameters are the dimension of the system and the strength of the disorder. After introductory sections on a transfer-matrix description of 1D transport, scaling theory of localization, and key numerical and experimental results, a general microscopic theory of transport in disordered systems is presented, with emphasis on experimental realizations with cold atomic gases. Simple examples are the propagation of light in a disordered medium, for which we show results from a live coherent backscattering expriment, and the propagation of atomic matter waves in an effective disordered potential created by an optical speckle. Finally, the dynamical localization transition of the kicked rotor, as observed with cold atoms, is discussed.Less

Disordered ultracold atomic gases

Maciej Lewenstein, Anna Sanpera, and Verònica Ahufinger

in Ultracold Atoms in Optical Lattices: Simulating quantum many-body systems

This chapter starts with an introduction to disordered systems, with an emphasis on disorder in condensed matter, and discusses such phenomena as the Anderson localization, and quantum phases such as ... More

This chapter starts with an introduction to disordered systems, with an emphasis on disorder in condensed matter, and discusses such phenomena as the Anderson localization, and quantum phases such as with Bose glass. It then turns to various experimental realizations of disorder in ultracold atomic gases. The chapter focuses on disordered Bose-Einstein condensates and the phenomenon of Anderson localization in speckle or quasi-periodic, quasi-random potential. It moves on to disordered ultracold fermionic systems, and disordered ultracold Bose–Fermi (B–F) and Bose–Bose (B–B) mixtures. Finally, the chapter examines spin glasses (providing a brief presentation of the mean field theory of Parisi, and the droplet model) and disorder induced order.Less

Introduction to Metal–Insulator Transitions

V. Dobrosavljević

in Conductor-Insulator Quantum Phase Transitions

This Overview provides a general introduction to metal-insulator transitions, with focus on specific mechanisms that can localize the electrons in absence of magnetic or charge ordering, and produce ... More

This Overview provides a general introduction to metal-insulator transitions, with focus on specific mechanisms that can localize the electrons in absence of magnetic or charge ordering, and produce well defined quantum critical behavior. The Overview contrasts the physical picture of Mott, who emphasized the role of electron-electron interactions, and that of Anderson, who stressed the possibility of impurity-induced bound state formation, as alternative routes to arrest the electronic motion. It also describes more complicated situations when both phenomena play coexist, leading to meta-stability, slow relaxation, and glassy behavior of electrons. A critical overview of the available theoretical approaches is then presented, contrasting the weak-coupling perspective, which emphasizes diffusion-mode corrections, and the strong-coupling viewpoint, which stresses inhomogeneous phases and local correlation effects. The Overview gives specific examples of experimental systems, providing clues on what should be the most profitable path forward in unraveling the mystery of metal-insulator transitions.Less

DISORDERED SYSTEMS

Thierry Giamarchi

in Quantum Physics in One Dimension

This chapter discusses the effects of disorder in fermionic systems, including Anderson localization. There are important differences for the disorder effects between the one-dimensional world, where ... More

This chapter discusses the effects of disorder in fermionic systems, including Anderson localization. There are important differences for the disorder effects between the one-dimensional world, where localization occurs because electrons bump back and forth between impurities, and the higher dimensional world, where Anderson's localization is a rather subtle interference mechanism. The discussion looks at one-dimensional electrons subject to weak and dense impurities, in which the disorder can be replaced by its Gaussian limit. The application of disordered systems to quantum wires, one of the ultimate weapons to study individual one-dimensional systems, is considered.Less

Anderson Localization

Keith Slevin and Tomi Ohtsuki

in Conductor-Insulator Quantum Phase Transitions

This chapter reviews briefly the theory of the Anderson localisation of electrons. In disordered materials at low temperatures, quantum interference may lead to the suppression of diffusion. If this ... More

This chapter reviews briefly the theory of the Anderson localisation of electrons. In disordered materials at low temperatures, quantum interference may lead to the suppression of diffusion. If this occurs, the material becomes an insulator at zero temperature and zero frequency even though the density of states at the Fermi level is finite. This transition from metal to insulator is called the Anderson transition. Anderson localisation occurs particularly easily in low dimensional systems. After describing very briefly some elements of the theory of Anderson localisation, the chapter focuses on numerical simulations of Anderson localisation using the transfer matrix method, and the analysis and interpretation of the results using finite size scaling. After mentioning other approaches such as diagonalisation, this chapter closes by describing some of the experimental signatures of Anderson localisation.Less

ELECTRON MOBILITY IN DENSE HE GAS

A.F. Borghesani

in Ions and electrons in liquid helium

Experiments on the mobility of electrons in dense helium gas elucidated how localized electron states develop when the gas density gas is increased. Up to 77 K, the density dependence of the mobility ... More

Experiments on the mobility of electrons in dense helium gas elucidated how localized electron states develop when the gas density gas is increased. Up to 77 K, the density dependence of the mobility clearly shows that the formation of electron bubbles is a continuous phenomenon. Localization of electrons in bubbles also appears at high temperatures if the density is so large that the free energy of the localized state is negative enough. Percolation and hydrodynamic models have been devised to explain the continuous transition from high-mobility states to low-mobility states. It is shown that density-dependent, quantum multiple scattering effects modify the energy of the nearly free electron in a way that can be accurately described by heuristically modifying the kinetic theory prediction.Less

Law in Disorder

Andrew Zangwill

in A Mind Over Matter: Philip Anderson and the Physics of the Very Many

A formal request by the theorists produces a stand-alone Solid-State Theory Group at Bell Labs. A summer visitor program leads several visiting theorists to conclude that localization occurred in ... More

A formal request by the theorists produces a stand-alone Solid-State Theory Group at Bell Labs. A summer visitor program leads several visiting theorists to conclude that localization occurred in Feher’s samples due to an electrostatic mechanism suggested by Nevill Mott. Anderson develops a theory for localization where the disorder in the positions of the dopants plays a crucial role. Mott champions Anderson’s theory and the Nobel Committee cites it when Anderson wins a share of the 1977 Nobel Prize with Mott and John Van Vleck. David Thouless re-ignites Anderson’s interest in localization and he leads the Gang of Four to develop a novel scaling theory of localization.Less

Optical lattices and quantum simulation

Massimo Inguscio and Leonardo Fallani

in Atomic Physics: Precise Measurements and Ultracold Matter

This chapter extends the investigation of ultracold atoms in optical lattices to the emerging field of ‘quantum simulation’, in which atoms are used to experimentally realize basic condensed-matter ... More

This chapter extends the investigation of ultracold atoms in optical lattices to the emerging field of ‘quantum simulation’, in which atoms are used to experimentally realize basic condensed-matter models to precisely investigate their properties and their quantum phase transitions in an ultimately clean setting, where decoherence or unwanted interactions with the environment can be avoided. The chapter discusses the superfluid/metal-to-insulator transition exhibited by strongly-interacting atoms realizing Hubbard models, as well as the Anderson localization determining the suppression of transport for atoms moving in a disordered potential. It concludes with an illustration of the most recent developments in the field, discussing novel experimental techniques and important frontiers in this research.Less

The renormalization group

Tom Lancaster and Stephen J. Blundell

in Quantum Field Theory for the Gifted Amateur

The renormalization group is introduced in this chapter. Examples treated include asymptotic freedom, Anderson localization, and the Kosterlitz–Thouless transition.