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Disorder and chaos in quantum system: Anderson localization and its generalization

Boris Altshuler (Columbia ). Disorder and chaos in quantum system: Anderson localization and its generalization. (6 lectures). Igor Aleiner (Columbia ). Lecture # 2. Stability of insulators and Anderson transition Stability of metals and weak localization. extended. localized.

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Disorder and chaos in quantum system: Anderson localization and its generalization

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  1. Boris Altshuler (Columbia) Disorder and chaos in quantum system:Anderson localization and its generalization (6 lectures) Igor Aleiner (Columbia)

  2. Lecture # 2 • Stability of insulators and Anderson transition • Stability of metals and weak localization

  3. extended localized Anderson localization (1957) Only phase transition possible!!!

  4. extended localized Anderson localization (1957) Strong disorder d=3 Any disorder, d=1,2 Anderson insulator Localized Extended Weaker disorder d=3 Localized Extended Localized

  5. Anderson Model • Lattice - tight binding model • Onsite energies ei- random • Hopping matrix elements Iij I i andjare nearest neighbors 0otherwise Iij= { i j Iij Critical hopping: -W < ei <Wuniformly distributed

  6. Resonant pair Bethe lattice: INFINITE RESONANT PATH ALWAYS EXISTS

  7. Resonant pair Bethe lattice: Decoupled resonant pairs INFINITE RESONANT PATH ALWAYS EXISTS

  8. Long hops? Resonant tunneling requires:

  9. “All states are localized“ means Probability to find an extended state: System size

  10. Order parameter for Anderson transition? Idea for one particle localizationAnderson, (1958); MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) Insulator Metal

  11. Order parameter for Anderson transition? Idea for one particle localizationAnderson, (1958); MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) Insulator Metal

  12. Order parameter for Anderson transition? Idea for one particle localizationAnderson, (1958); MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) Insulator Metal

  13. Order parameter for Anderson transition? Idea for one particle localizationAnderson, (1958); MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) Insulator Metal

  14. Order parameter for Anderson transition? Idea for one particle localizationAnderson, (1958); MIT for Bethe lattice: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) metal insulator insulator h!0 metal ~ h behavior for a given realization probability distribution for a fixed energy

  15. Probability Distribution Note: metal insulator Can not be crossover, thus, transition!!!

  16. On the real lattice, there are multiple paths connecting two points:

  17. Amplitude associated with the paths interfere with each other:

  18. To complete proof of metal insulator transition one has to show the stability of the metal

  19. Back to Drude formula Finite impurity density CLASSICAL Quantum (single impurity) Drude conductivity Quantum (band structure)

  20. Why does classical consideration of multiple scattering events work? 1 Vanish after averaging 2 Classical Interference

  21. Look for interference contributions that survive the averaging Phase coherence 2 1 2 Correction to scattering crossection 1 unitarity

  22. Additional impurities do not break coherence!!! 2 1 2 Correction to scattering crossection 1 unitarity

  23. Sum over all possible returning trajectories 2 1 Return probability for classical random work 2 1 unitarity

  24. (Gorkov, Larkin, Khmelnitskii, 1979) Quantum corrections (weak localization) Finite but singular 3D 2D 1D

  25. 2D 1D Metals are NOT stable in one- and two dimensions Localization length: Drude + corrections Anderson model,

  26. Exact solutions for one-dimension x U(x) Nch Nch=1 Gertsenshtein, Vasil’ev (1959)

  27. Exact solutions for one-dimension x U(x) Nch Efetov, Larkin (1983) Dorokhov (1983) Nch>>1 Universal conductance fluctuations Altshuler (1985); Stone; Lee, Stone (1985) Weak localization Strong localization

  28. We learned today: • How to investigate stability of insulators (locator expansion). • How to investigate stability of metals (quantum corrections) • For d=3 stability of both phases implies metal insulator transition; The order parameter for the transition is the distribution function • For d=1,2 metal is unstable and all states are localized

  29. Next time: • Inelastic transport in insulators

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