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Anderson localization: from single particle to many body problems. Igor Aleiner (4 lectures) Windsor Summer School, 14-26 August 2012 ( Columbia University.

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Presentation on theme: "Anderson localization: from single particle to many body problems. Igor Aleiner (4 lectures) Windsor Summer School, 14-26 August 2012 ( Columbia University."— Presentation transcript:

1 Anderson localization: from single particle to many body problems. Igor Aleiner (4 lectures) Windsor Summer School, August 2012 ( Columbia University in the City of New York, USA )

2 Lecture # 1-2 Single particle localization Lecture # 2-3 Many-body localization

3 I V Conductivity: Conductance: Insulator Metal Transport in solids Superconductor

4 I V Conductivity: Conductance: Insulator Metal Transport in solids Focus of The course

5 Lecture # 1 Metals and insulators – importance of disorder Drude theory of metals First glimpse into Anderson localization Anderson metal-insulator transition (Bethe lattice argument; order parameter … )

6 Band metals and insulators MetalsInsulators Gapless spectrum Gapped spectrum

7 Metals Gapless spectrum Insulators Gapped spectrum But clean systems are in fact perfect conductors: Current Electric field

8 Metals Gapless spectrum Insulators Gapped spectrum But clean systems are in fact perfect conductors: (quasi-momentum is conserved, translational invariance)

9 Finite conductivity by impurity scattering One impurity Incoming flux Probability density Scattering cross-section

10 Finite conductivity by impurity scattering Finite impurity density Elastic mean free path Elastic relaxation time

11 Finite conductivity by impurity scattering Finite impurity density Drude conductivity CLASSICAL Quantum (band structure) Quantum (single impurity)

12 Conductivity and Diffusion Finite impurity density Einstein relation Diffusion coefficient

13 Conductivity, Diffusion, Density of States (DoS) Einstein relation Density of States (DoS)

14 Clean systems

15 Density of States (DoS) Clean systems Metals, gapless Insulators, gapped Phase transition!!!

16 But only disorder makes conductivity finite!!! Disordered systems Clean Disorder included Disordered

17 Spectrum always gapless!!! No phase transition??? Only crossovers??? Lifshitz tail

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

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

20 DoS extended Anderson Transition - mobility edges (one particle) Coexistence of the localized and extended states is not possible!!! Rules out first order phase transition

21 Temperature dependence of the conductivity (no interactions) DoS MetalInsulatorPerfect one particle Insulator No singularities in any thermodynamic properties!!!

22 To take home so far: Conductivity is finite only due to broken translational invariance (disorder) Spectrum (averaged) in disordered system is gapless Metal-Insulator transition (Anderson) is encoded into properties of the wave-functions

23 Anderson Model Lattice - tight binding model Onsite energies i - random Hopping matrix elements I ij j i I ij -W < i

24 One could think that diffusion occurs even for : Golden rule: Random walk on the lattice Pronounce words: Self-consistency Mean-field Self-averaging Effective medium ………….. ?

25 is F A L S E Probability for the level with given energy on NEIGHBORING sites Probability for the level with given energy in the whole system 2d attempts Infinite number of attempts

26 Perturbative Resonant pair

27 Bethe lattice: INFINITE RESONANT PATH ALWAYS EXISTS

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

29 Long hops? Resonant tunneling requires:

30 All states are localized means Probability to find an extended state: System size

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

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

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

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

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

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

37 But the Andersons argument is not complete:

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

39 Amplitude associated with the paths interfere with each other:

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

41 Summary of Lecture # 1 Conductivity is finite only due to broken translational invariance (disorder) Spectrum (averaged) in disordered system is gapless (Lifshitz tail) Metal-Insulator transition (Anderson) is encoded into properties of the wave-functions extended localized Metal Insulator

42 Distribution function of the local densities of states is the order parameter for Anderson transition metal insulator

43 Resonant pair Perturbation theory in (I/W) is convergent!

44 Perturbation theory in (I/W) is divergent!

45 To establish the metal insulator transition we have to show the convergence of (W/I) expansion!!!

46 Lecture # 2 Stability of metals and weak localization Inelastic e-e interactions in metals Phonon assisted hopping in insulators Statement of many-body localization and many- body metal insulator transition

47 Why does classical consideration of multiple scattering events work? 1 2 ClassicalInterference Vanish after averaging

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

49 Look for interference contributions that survive the averaging unitarity Correction to scattering crossection Phase coherence

50 Additional impurities do not break coherence!!! unitarity Correction to scattering crossection

51 Sum over all possible returning trajectories unitarity Return probability for classical random work

52 Sometimes you may see this… MISLEADING… DOES NOT EXIST FOR GAUSSIAN DISORDER AT ALL

53 Quantum corrections (weak localization) (Gorkov, Larkin, Khmelnitskii, 1979) 3D 2D 1D Finite but singular E. Abrahams, P. W. Anderson, D. C. Licciardello, and T.V. Ramakrishnan, (1979) Thouless scaling + ansatz:

54 2D 1D Metals are NOT stable in one- and two dimensions Localization length: Drude + corrections Anderson model,

55 Exact solutions for one-dimension x U(x) N ch Gertsenshtein, Vasilev (1959) N ch =1

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

57 Other way to analyze the stability of metal metal insulator Metal ??? Explicit calculation yields: Metal is unstable

58 To take home so far: Interference corrections due to closed loops are singular; For d=1,2 they diverges making the metalic phase of non-interacting particles unstable; Finite size system is described as a good metal, if, in other words For, the properties are well described by Anderson model with replacing lattice constant.

59 Regularization of the weak localization by inelastic scatterings (dephasing) e-h pair Does not interfere with

60 Regularization of the weak localization by inelastic scatterings (dephasing) e-h pair But interferes with e-h pair

61 Phase difference:

62 e-h pair Phase difference: - length of the longest trajectory;

63 Inelastic rates with energy transfer

64 Electron-electron interaction Altshuler, Aronov, Khmelnitskii (1982) Significantly exceeds clean Fermi-liquid result

65 Almost forward scattering: diffusive Ballistic

66 To take home so far: Interference corrections due to closed loops are singular; For d=1,2 they diverges making the metalic phase of non-interacting particles unstable; Interactions at finite T lead to finite System at finite temperature is described as a good metal, if, in other words For, the properties are well described by ??????

67 Transport in deeply localized regime

68 Inelastic processes: transitions between localized states (inelastic lifetime) –1 energy mismatch (any mechanism)

69 Phonon-induced hopping energy difference can be matched by a phonon Any bath with a continuous spectrum of delocalized excitations down to = 0 will give the same exponential Variable Range Hopping Sir N.F. Mott (1968) Without Coulomb gap A.L.Efros, B.I.Shklovskii (1975) Optimized phase volume Mechanism-dependent prefactor

70 insulator Drude metal Electron phonon Interaction does not enter

71 Q: Can we replace phonons with e-h pairs and obtain phonon-less VRH? insulator Drude metal Electron phonon Interaction does not enter

72 Metal-Insulator Transition and many-body Localization: insulator Drude metal [Basko, Aleiner, Altshuler (2005)] Interaction strength (Perfect Ins) and all one particle state are localized


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