1. Anderson localization: from single particle to many body problems. Igor Aleiner (4 lectures) Windsor Summer School, 14-26 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. Summary of Lectures # 1,2 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

4. Distribution function of the local densities of states is the order parameter for Anderson transition Interference corrections due to closed loops are singular; For d=1,2 they diverges making the metalic phase of non-interacting particles unstable; Finite T alone does not lift localization; metal insulator

6. Lecture # 3 Inelastic transport in deep insulating regime Statement of many-body localization and many-body metal insulator transition Definition of the many-body localized state and the many-body mobility threshold Many-body localization for fermions (stability of many-body insulator and metal)

7. Transport in deeply localized regime

8. Inelastic processes: transitions between localized states (inelastic lifetime) –1   energy mismatch (any mechanism)

9. 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

10. “insulator” Drude “metal” Electron phonon Interaction does not enter

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

12. Q: Can we replace phonons with e-h pairs and obtain phonon-less VRH? A#1: Sure 3) Use the electric noise instead of phonons. 1) Recall phonon-less AC conductivity: Sir N.F. Mott (1970) 2) Calculate the Nyquist noise (fluctuation dissipation Theorem). 4) Do self-consistency (whatever it means). Easy steps: Person from the street (2005)(2005-2011)

13. Q: Can we replace phonons with e-h pairs and obtain phonon-less VRH? A#1: Sure [Person from the street (2005)] A#2: No way (for Coulomb interaction in 3D – may be) [L. Fleishman. P.W. Anderson (1980)] is contributed by rare resonances     R Thus, the matrix element vanishes !!! 0 *

14. 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

15. Many-body mobility threshold insulator metal [Basko, Aleiner, Altshuler (2005)] Many body DoS All STATES LOCALIZED All STATES EXTENDED -many-body mobility threshold

16. “All states are localized “ means Probability to find an extended state: System volume

17. Many body localization means any excitation is localized: Extended Localized

18. Entropy Many body DoS All STATES LOCALIZED All STATES EXTENDED States always thermalized!!! States never thermalized!!!

19. Many body DoS Is it similar to Anderson transition? One-body DoS Why no activation?

20. Physics: Many-body excitations turn out to be localized in the Fock space

21. Fock space localization in quantum dots (AGKL, 1997) - one-particle level spacing; e    e e e e   ´ No spatial structure ( “0-dimensional” )

22. Fock space localization in quantum dots (AGKL, 1997) 1-particle excitation 3-particle excitation 5-particle excitation Cayley tree mapping

23. Fock space localization in quantum dots (AGKL, 1997) 1-particle excitation 3-particle excitation 5-particle excitation - one-particle level spacing; 1.Coupling between states: 2.Maximal energy mismatch: 3.Connectivity:

24. insulator metal Metal-Insulator Transition in the bulk systems Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] Interaction strength Metal-Insulator “Transition” in zero dimensions [Altshuler, Gefen, Kamenev,Levitov (1997)] In the paper:

25. Metal-Insulator Transition in the bulk systems Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] Metal-Insulator “Transition” in zero dimensions [Altshuler, Gefen, Kamenev,Levitov (1997)] - one-particle level spacing; 1-particle level spacing in localization volume; 1) Localization in Fock space = Localization in the coordinate space. 2) Interaction is local;

26. Metal-Insulator Transition in the bulk systems Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] Metal-Insulator “Transition” in zero dimensions [Altshuler, Gefen, Kamenev,Levitov (1997)] - one-particle level spacing; 1-particle level spacing in localization volume; 1,2) Locality: 3) Interaction matrix elements strongly depend on the energy transfer,  :

27. Matrix elements: ???? In the metallic regime:

28. Matrix elements: ???? In the metallic regime: 2

29. Matrix elements: ???? 2

30. Metal-Insulator Transition in the bulk systems Vs. finite T Metal-Insulator Transition in the bulk systems [Basko, Aleiner, Altshuler (2005)] Metal-Insulator “Transition” in zero dimensions [Altshuler, Gefen, Kamenev,Levitov (1997)] - one-particle level spacing; 1-particle level spacing in localization volume; 1,2) Locality: 3) Interaction matrix elements strongly depend on the energy transfer,  :

31. MIT Effective Hamiltonian for MIT. We would like to describe the low-temperature regime only. Spatial scales of interest >> 1-particle localization length Otherwise, conventional perturbation theory for disordered metals works. Altshuler, Aronov, Lee (1979); Finkelshtein (1983) – T-dependent SC potential Altshuler, Aronov, Khmelnitskii (1982) – inelastic processes

32. No spins Reproduces correct behavior of the tails of one particle wavefunctions

33. j1j1 j2j2 l1l1 l2l2 Interaction only within the same cell;

34. Statistics of matrix elements?

35. Parameters: random signs

36. What to calculate? – random quantity Idea for one particle localization Anderson, (1958); MIT for Cayley tree: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) No interaction: Metal Insulator

37. Probability Distribution metal insulator Note: Look for:

38. How to calculate? Parameters: non-equilibrium (arbitrary occupations) → Keldysh SCBA allow to select the most relevant series Find the distribution function of each diagram

39. Iterations: Cayley tree structure

40. after standard simple tricks: + kinetic equation for occupation function Nonlinear integral equation with random coefficients Decay due to tunneling Decay due to e-h pair creation

41. Stability of metallic phase Assume is Gaussian: >> () 2

42. Probability Distributions “Non-ergodic” metal

43. Drude metal

44. Kinetic Coefficients in Metallic Phase

45. Kinetic Coefficients in Metallic Phase Wiedemann-Frantz law ?

46. Non-ergodic+Drude metal So far, we have learned: Trouble !!! Insulator ???

47. Nonlinear integral equation with random coefficients Stability of the insulator Notice: for is a solution Linearization:

48. metal Recall: insulator  probability distribution for a fixed energy # of interactions # of hops in space unstable STABLE

49. So, we have just learned: Insulator Metal Non-ergodic+Drude metal

50. Estimate for the transition temperature for general case Energy mismatch Interaction matrix element # of possible decay p rocesses of an excitations allowed by interaction Hamiltonian; 2) Identify elementary (one particle) excitations and prove that they are localized. 3) Consider a one particle excitation at finite T and the possible paths of its decays: 1) Start with T=0;

51. Summary of Lecture # 3: Existence of the many-body mobility threshold is established. The many body metal-insulator transition is not a thermodynamic phase transition. It is associated with the vanishing of the Langevine forces rather the divergences in energy landscape (like in classical glass) Only phase transition possible in one dimension (for local Hamiltonians) Detailed paper: Basko, I.A.,Altshuler, Annals of Physics 321 (2006) 1126-1205 Shorter version:…., cond-mat/0602510; chapter in “Problems of CMP”

52. I.A, Altshuler, Shlyapnikov, NATURE PHYSICS 6 (2010) 900-904 Lecture #4. Many body localization and phase diagram of weakly interacting 1D bosons TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAA

53. Outline: Remind: Many body localization and estimate for the transition temperature; Remind: Single particle localization in 1D; Remind: “Superconductor”-insulator transition at T=0; Many-body metal-insulator transiton at finite T;

54. 1. Localization of single-electron wave-functions: extended localized d=1; All states are localized M.E. Gertsenshtein, V.B. Vasil’ev, (1959) Exact solution for one channel: “Conjecture” for one channel: Sir N.F. Mott and W.D. Twose (1961) Exact solution for   for one channel: V.L. Berezinskii, (1973)

55. Many-body localization; – random quantity Idea for one particle localization Anderson, (1958); MIT for Cayley tree: Abou-Chakra, Anderson, Thouless (1973); Critical behavior: Efetov (1987) No interaction: !0!0 metal insulator behavior for a given realization metal insulator  probability distribution for a fixed energy

56. metal insulator  probability distribution for a fixed energy unstable STABLE Perturbation theory for the fermionic systems: + stability of the metallic phase at

57. Estimate for the transition temperature for general case Energy mismatch Interaction matrix element # of possible decay p rocesses of an excitations allowed by interaction Hamiltonian; 1) Identify elementary (one particle) excitations and prove that they are localized. 2) Consider a one particle excitation at finite T and the possible paths of its decays:

58. Fermionic system: # of electron-hole pairs

59. Weakly interacting bosons in one dimension

60. Phase diagram 1 Crossover???? No finite T phase transition in 1D See e.g. Altman, Kafri, Polkovnikov, G.Refael, PRL, 100, 170402 (2008); 93,150402 (2004).

61. Finite temperature phase transition in 1D I.A., Altshuler, Shlyapnikov arXiv:0910.434; Nature Physics (2010)

62. I.M. Lifshitz (1965); Halperin, Lax (1966); Langer, Zittartz (1966)

64. I.A., Altshuler, Shlyapnikov arXiv:0910.434; Nature Physics (2010) High Temperature region

65. Bose-gas is not degenerate: occupation numbers either 0 or 1 # of bosons to interact with

66. Bose-gas is not degenerate: occupation numbers either 0 or 1

67. I.A., Altshuler, Shlyapnikov arXiv:0910.434; Nature Physics (2010) Intermediate Temperature region

68. Intermediate temperatures: Bose-gas is degenerate; typical energies ~ |  | > 1 matrix elements are enhanced

69. # of levels with different energies Intermediate temperatures:

70. Delocalization occurs in all energy strips

71. I.A., Altshuler, Shlyapnikov arXiv:0910.434; Nature Physics (2010) Low Temperature region

72. Low temperatures: Occupation Start with T=0 Spectrum is determined by the interaction but only Lifhsitz tale is important; Gross-Pitaevskii mean-field on strongly localized states: Optimal occupation # Random, non-integer:

73. Low temperatures: Occupation Start with T=0 Tunneling between Lifshitz states See Altman, Kafri, Polkovnikov, G.Refael, PRL, 100, 170402 (2008); 93,150402 (2004).

74. Low temperatures:Start with T=0 Everything is determined by the weakest links: T=0 transition: L Interaction relevant: Interaction irrelevant: Insulator “Superfluid”

75. Low temperatures:Start with T=0 Insulator: “Superfluid” All excitations are localized; many-body Localization transition temperature finite; Localization length of the low-energy excitations (phonons) diverges As their energy goes to zero; The system is delocalized at any finite Temperature;

76. I.A., Altshuler, Shlyapnikov arXiv:0910.434; Nature Physics (2010) Transition line terminate is QPT point

77. Disordered interacting bosons in two dimensions (conjecture)

78. Summary of Lectures # 3,4: Existence of the many-body mobility threshold is established. The many body metal-insulator transition is not a thermodynamic phase transition. It is associated with the vanishing of the Langevine forces rather the divergences in energy landscape (like in classical glass) Only finite T phase transition possible in one dimension (for local Hamiltonians)