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

2. Lecture # 1 Metals and insulators – importance of disorder
Drude theory of metals
First glimpse into Anderson 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. Band metals and insulators
Gapped spectrum
Gapless spectrum

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

7. 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

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

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

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

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

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

13. Density of States (DoS)
Clean systems

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

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

16. No phase transition??? Only crossovers??? Spectrum always gapless!!!
Disordered
Spectrum always gapless!!!
No phase transition???
Only crossovers???

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

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

19. 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

20. Temperature dependence of the conductivity
DoS
DoS
DoS
Metal
Insulator
“Perfect” one particle
Insulator

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

22. Next time: Anderson transition in more detail
Non-existence of metals in two- and one-dimensional system (instability with respect to quantum corrections).