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

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

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

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

5. { 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

6. 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
………….. ?

7. 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

8. Resonant pair
Perturbative

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

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

11. Long hops?
Resonant tunneling requires:

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

13. 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

14. 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

15. 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

16. 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

17. 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

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

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

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

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

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

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

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

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

26. Sum over all possible returning trajectories1 2
unitarity
Return probability for
classical random work

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

28. 2D 1D Metals are NOT stable in one- and two dimensions
Localization length:
Drude + corrections
Anderson model,

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

30. 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

31. 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

32. Next time:
Inelastic transport in insulators