Dirk Hundertmark
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0009
- Subject:
- Mathematics, Probability / Statistics, Analysis
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 ...
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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
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.
Cord A. Müller and Dominique Delande
- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199603657
- eISBN:
- 9780191729515
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199603657.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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 ...
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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
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.
Maciej Lewenstein, Anna Sanpera, and Verònica Ahufinger
- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199573127
- eISBN:
- 9780191775048
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573127.003.0009
- Subject:
- Physics, Atomic, Laser, and Optical Physics
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 ...
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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
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.
V. Dobrosavljević
- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199592593
- eISBN:
- 9780191741050
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199592593.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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 ...
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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
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.
Thierry Giamarchi
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198525004
- eISBN:
- 9780191711909
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525004.003.0009
- Subject:
- Physics, Condensed Matter Physics / Materials
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 ...
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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
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.
Keith Slevin and Tomi Ohtsuki
- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199592593
- eISBN:
- 9780191741050
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199592593.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
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 ...
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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
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.
A.F. Borghesani
- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780199213603
- eISBN:
- 9780191707421
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213603.003.0027
- Subject:
- Physics, Condensed Matter Physics / Materials
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 ...
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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
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.
Massimo Inguscio and Leonardo Fallani
- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780198525844
- eISBN:
- 9780191780059
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525844.003.0007
- Subject:
- Physics, Atomic, Laser, and Optical Physics
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 ...
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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
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.
Vladimir Dobrosavljevic, Nandini Trivedi, and James M. Valles, Jr. (eds)
- Published in print:
- 2012
- Published Online:
- September 2012
- ISBN:
- 9780199592593
- eISBN:
- 9780191741050
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199592593.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Quantum phase transitions describe the violent rearrangement of electrons or atoms as they evolve from well defined excitations in one phase to a completely different set of excitations in another. ...
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Quantum phase transitions describe the violent rearrangement of electrons or atoms as they evolve from well defined excitations in one phase to a completely different set of excitations in another. The book chapters give insights into how a coherent metallic or superconducting state can be driven into an incoherent insulating state by increasing disorder, magnetic field, carrier concentration and inter-electron interactions. They illustrate the primary methods employed to develop a multi-faceted theory of many interacting particle systems. They describe how recent experiments probing the microscopic structure, transport, charge and spin dynamics have yielded guiding insights. What sets this book apart is this strong dialog between experiment and theory, which reveals the recent progress and emergent opportunities to solve some major problems in many body physics. The pedagogical style of the chapters has been set for graduate students starting in this dynamic field.Less
Quantum phase transitions describe the violent rearrangement of electrons or atoms as they evolve from well defined excitations in one phase to a completely different set of excitations in another. The book chapters give insights into how a coherent metallic or superconducting state can be driven into an incoherent insulating state by increasing disorder, magnetic field, carrier concentration and inter-electron interactions. They illustrate the primary methods employed to develop a multi-faceted theory of many interacting particle systems. They describe how recent experiments probing the microscopic structure, transport, charge and spin dynamics have yielded guiding insights. What sets this book apart is this strong dialog between experiment and theory, which reveals the recent progress and emergent opportunities to solve some major problems in many body physics. The pedagogical style of the chapters has been set for graduate students starting in this dynamic field.
Tom Lancaster and Stephen J. Blundell
- Published in print:
- 2014
- Published Online:
- June 2014
- ISBN:
- 9780199699322
- eISBN:
- 9780191779435
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199699322.003.0035
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
The renormalization group is introduced in this chapter. Examples treated include asymptotic freedom, Anderson localization, and the Kosterlitz–Thouless transition.
The renormalization group is introduced in this chapter. Examples treated include asymptotic freedom, Anderson localization, and the Kosterlitz–Thouless transition.
