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, August 2012

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

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

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

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
Gapped spectrum
Gapless spectrum

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

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

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

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

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

12. Conductivity and Diffusion
Finite impurity density
Diffusion coefficient
Einstein relation

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

14. Density of States (DoS)
Clean systems

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

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

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

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

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

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

21. Temperature dependence of the conductivity (no interactions)
DoS
DoS
DoS
Metal
Insulator
“Perfect” 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. { I i and j are nearest Iij = 0 otherwise
Anderson Model
Lattice - tight binding model
Onsite energies ei - random
Hopping matrix elements Iij j i
Iij
Iij =
I i and j are nearest
neighbors
0 otherwise {
Critical hopping:
-W < ei <W uniformly distributed

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

25. Infinite number of attempts
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. Resonant pair
Perturbative

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

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

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)
𝜈 𝑖 (𝜀)= 𝛼 𝜓 𝛼 𝑖 2 𝛿(𝜀− 𝜉 𝛼 )
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)
𝜈 𝑖 (𝜀)= 𝛼 𝜓 𝛼 𝑖 2 𝛿(𝜀− 𝜉 𝛼 )
Metal
Insulator

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)
Metal
Insulator

34. 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)
𝜈 𝑖 (𝜀)= 𝛼 𝜓 𝛼 𝑖 2 𝛿(𝜀− 𝜉 𝛼 )
Metal
Insulator

35. 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
insulator
h→0
metal
~ h
behavior for a
given realization
probability distribution
for a fixed energy

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

37. But the Anderson’s 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
insulator
metal

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
Vanish after averaging 2
Classical
Interference

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

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

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

51. Sum over all possible returning trajectories1 2
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)
Finite but singular 3D 2D 1D
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
U(x)
Nch
Gertsenshtein, Vasil’ev (1959)
Nch =1

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

57. Other way to analyze the stability of metal
insulator
Explicit calculation yields:
Metal ???
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)
Does not interfere with
e-h pair

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

61. Phase difference:
e-h pair
e-h pair

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

63. Inelastic rates with energy transfer

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

65. Almost forward scattering:
Ballistic
diffusive

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

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

70. 𝜆 𝑒−𝑝ℎ ⟶ 0 ????? “metal” Drude “insulator” Electron phonon
𝜆 𝑒−𝑝ℎ ⟶ 0 ?????
Drude
“metal”
Electron phonon
Interaction does not enter
“insulator”

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

72. 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