Anderson localization: from single particle to many body problems.

1. Anderson localization:from single particle to many body problems. (4 lectures) Igor Aleiner ( Columbia University in the City of New York, USA ) Windsor Summer School, 14-26 August 2012

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

3. extended localized 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 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

5. 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 ??????

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  energy mismatch  (inelastic lifetime)–1 (any mechanism)

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

10. ⟶ 0 ????? (All one-particle states are localized) Drude Electron phonon Interaction does not enter “metal” “insulator”

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

29. Matrix elements: ???? 2

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

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

33. j1 l1 l2 j2 Interaction only within the same cell;

34. Statistics of matrix elements?

35. random signs Parameters:

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

37. Probability Distribution Note: metal insulator Look for:

38. Iterations: Cayley tree structure

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

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

41. Probability Distributions “Non-ergodic” metal

42. Drude metal

43. Kinetic Coefficients in Metallic Phase

44. Kinetic Coefficients in Metallic PhaseWiedemann-Frantz law ?

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

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

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

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

49. Estimate for the transition temperature for general case 1) Start with T=0; 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: Interaction matrix element # of possible decay processes of an excitations allowed by interaction Hamiltonian; Energy mismatch

50. 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”