1. Boris Altshuler Physics Department, Columbia University
Disorder and chaos in
quantum systems II.
Lecture 2.
Boris Altshuler Physics Department, Columbia University

2. Previous Lecture: Anderson Localization as Metal-Insulator Transition
Anderson model.
Localized and extended states. Mobility edges.
2. Spectral Statistics and Localization Poisson versus Wigner-Dyson.
Anderson transition as a transition between different types of spectra.
Thouless conductance
P(s)
Conductance g s

3. Lecture2.
1. Quantum Chaos,
Integrability and
Localization

4. 166Er s P(s) P(s) Particular nucleus
Spectra of several nuclei combined (after spacing)
rescaling by the mean level
N. Bohr, Nature 137 (1936) 344.

5. ? Q: Original answer: Later it became clear that
Why the random matrix theory (RMT) works so well for nuclear spectra
These are systems with a large number of degrees of freedom, and therefore the “complexity” is high
Original
answer:
Later it
became
clear that
there exist very “simple” systems with as many as 2 degrees of freedom (d=2), which demonstrate RMT - like spectral statistics

6. d integrals of motion Integrable Systems Examples
Classical (h =0) Dynamical Systems with d degrees of freedom
d integrals of motion
The variables can be separated and the problem reduces to d one-dimensional problems
Integrable Systems
Examples
1. A ball inside rectangular billiard; d=2
Vertical motion can be separated from the horizontal one
Vertical and horizontal components of the momentum, are both integrals of motion
2. Circular billiard; d=2
Radial motion can be separated from the angular one
Angular momentum and energy are the integrals of motion

7. B Classical Dynamical Systems with d degrees of freedom
Integrable Systems
The variables can be separated [ d one-dimensional problems [d integrals of motion
Rectangular and circular billiard, Kepler problem, , 1d Hubbard model and other exactly solvable models, . .
Chaotic Systems
The variables can not be separated [ there is only one integral of motion - energy
Examples B
Kepler problem in magnetic field
Stadium
Sinai billiard

8. Q: What does it mean Quantum Chaos ?
Nonlinearities
Exponential dependence on the original conditions (Lyapunov exponents)
Ergodicity
Classical Chaos h =0
Quantum description of any System with a finite number of the degrees of freedom is a linear problem – Shrodinger equation
Q: What does it mean Quantum Chaos ?

9. Bohigas – Giannoni – Schmit conjecture
Chaotic classical analog
Wigner- Dyson spectral statistics
No quantum numbers except energy

10. Classical
Quantum ?
Integrable
Poisson ?
Wigner-Dyson
Chaotic

11. Lecture1.
2. Localization beyond real space

12. Kolmogorov – Arnold – Moser (KAM) theory
A.N. Kolmogorov,
Dokl. Akad. Nauk SSSR, 1954.
Proc Int. Congress of Mathematics, North-Holland, 1957
Integrable classical Hamiltonian , d>1:
Separation of variables: d sets of action-angle variables
Andrey Kolmogorov
Vladimir Arnold
Jurgen
Moser

13. 1D classical motion – action-angle variables

14. Kolmogorov – Arnold – Moser (KAM) theory
A.N. Kolmogorov,
Dokl. Akad. Nauk SSSR, 1954.
Proc Int. Congress of Mathematics, North-Holland, 1957
Integrable classical Hamiltonian , d>1:
Separation of variables: d sets of action-angle variables
Quasiperiodic motion:
set of the frequencies, which are in general incommensurate. Actions are integrals of motion
Andrey Kolmogorov
…=>
Vladimir Arnold
Jurgen
Moser
tori

15. For d>1 each torus has measure zero on the energy shell !
Integrable dynamics:
Each classical trajectory is quasiperiodic and confined to a particular torus, which is determined by a set of the integrals of motion
space
Number of dimensions
real space d
phase space: (x,p) 2d
energy shell
2d-1
tori
For d>1 each torus has measure zero on the energy shell !

16. Kolmogorov – Arnold – Moser (KAM) theory
Integrable classical Hamiltonian , d>1:
A.N. Kolmogorov,
Dokl. Akad. Nauk SSSR, 1954.
Proc Int. Congress of Mathematics, North-Holland, 1957
Separation of variables: d sets of action-angle variables
Quasiperiodic motion: set of the frequencies,
which are in general incommensurate
Actions are integrals of motion
…=> ? Q:
Will an arbitrary weak perturbation
of the integrable Hamiltonian destroy the tori and make the motion ergodic (when each point at the energy shell will be reached sooner or later)
Andrey Kolmogorov A:
Most of the tori survive weak and smooth enough perturbations
KAM theorem
Vladimir Arnold
Jurgen
Moser

17. KAM theorem:
Most of the tori survive weak and smooth enough perturbations ?
Each point in the space of the integrals of motion corresponds to a torus and vice versa
Finite motion.
Localization in the space of the integrals of motion

18. KAM theorem:
Most of the tori survive weak and smooth enough perturbations
Energy shell

19. Consider an integrable system.
Each state is characterized by a set of quantum numbers.
It can be viewed as a point in the space of quantum numbers. The whole set of the states forms a lattice in this space.
A perturbation that violates the integrability provides matrix elements of the hopping between different sites (Anderson model !?)
Weak enough hopping:
Localization - Poisson
Strong hopping:
transition to Wigner-Dyson

20. Localized momentum space
Sinai
billiard
Square
billiard
Disordered
localized
Disordered
extended
Localized real space
Localized momentum space
extended

21. Glossary Classical Quantum Integrable KAM Localized
Ergodic – distributed all over the energy shell
Chaotic
Extended ?

22. Invariant (basis independent) definition
Extended states:
Level repulsion, anticrossings, Wigner-Dyson spectral statistics
Localized states:
Poisson spectral statistics
Invariant
(basis independent)
definition

23. Integrable Chaotic All chaotic systems resemble each other.
Sinai
billiard
Square
billiard
All integrable systems are integrable in their own way
Disordered
extended
Disordered
localized

24. ? Q: What is the statistics of the many-body spectra?
Consider a finite system of quantum particles, e.g., fermions. Let the one-particle spectra be chaotic (Wigner-Dyson).
What is the statistics of the many-body spectra? ? Q:
The particles do not interact with each other.
Poisson:
individual energies are conserving quantum numbers.
b. The particles do interact. ????

25. Lecture 2.
3. Many-Body excitation in finite systems

26. Decay of a quasiparticle with an energy e in Landau Fermi liquid
Fermi Sea

27. Quasiparticle decay rate at T = 0 in a clean Fermi Liquid.

 
Fermi Sea
Reasons:
At small  the energy transfer, w , is small and the integration over  and w gives the factor 2. …………………………………………………………………
The momentum transfer, q , is large and thus the scattering probability at given  and w does not depend on  , w or 

28. Quasiparticle decay rate at T = 0 in a clean Fermi Liquid.
II. Low dimensions e
Small moments transfer, q , become important at low dimensions because the scattering probability is proportional to the squared time of the interaction, (qvF. )-2 vF
1/q

29. Quasiparticle decay rate at T = 0 in a clean Fermi Liquid.

 
Fermi Sea
Conclusions:
1. For d=3,2 from it follows that , i.e., that the qusiparticles are well determined and the Fermi-liquid approach is applicable.
2. For d= is of the order of , i.e., that the Fermi-liquid approach is not valid for 1d systems of interacting fermions.
Luttinger liquids

30. Quantum dot – zero-dimensional case ?
Decay of a quasiparticle with an energy e in Landau Fermi liquid
Quantum dot – zero-dimensional case ? e
e-w
e1+w
Fermi Sea e1

31. Decay of a quasiparticle with an energy e in Landau Fermi liquid
Quantum dot – zero-dimensional case ? e
Decay rate of a quasiparticle with energy e
e-w
( U.Sivan, Y.Imry & A.Aronov,1994 )
Fermi Golden rule:
e1+w
Fermi Sea e1
Mean level spacing
Thouless energy

32. ( U.Sivan, Y.Imry & A.Aronov,1994 )
Decay rate of a quasiparticle with energy e in 0d.
( U.Sivan, Y.Imry & A.Aronov,1994 )
Fermi Golden rule:
Recall:
Thouless conductance
Mean level spacing
Thouless energy
Zero dimensional system
Def:
One particle states are extended all over the system

33. e1+e2 = e’1 + e’2 Decay rate of a quasiparticle with energy e in 0d.
Problem: e
zero-dimensional case
e-w
one-particle spectrum is discrete
e1+w
equation
e1+e2 = e’1 + e’2
can not be satisfied exactly
Fermi Sea e1
Recall: in the Anderson model the site-to-site hopping does not conserve the energy

34. e e-w e’+ w e’ Offdiagonal matrix element
Decay rate of a quasiparticle with energy e in 0d. e
e-w
e’+ w e’
Offdiagonal
matrix
element

35. . . . . Chaos in Nuclei – Delocalization? Delocalization in Fock space1 2 3 4 5 6
Delocalization in Fock space e
generations e’
Can be mapped (approximately) to the problem of localization on Cayley tree
e1’ 1 2 3 4 5
Fermi Sea e1

36. Conventional Anderson Model
one particle,
one level per site,
onsite disorder
nearest neighbor hoping
labels sites
Basis:
Hamiltonian:

37. 0d system; no interactionsea
many (N ) particles no interaction:
Individual energies and thus occupation numbers are conserved eb eg
N conservation laws
“integrable system” ed
integrable system

38. 0d system with interactionsea
Basis: eb
occupation numbers
labels levels eg
Hamiltonian: ed

39. Conventional Anderson Model Many body Anderson-like Model
Basis:
Basis:
occupation numbers
labels levels
labels sites
“nearest neighbors”:

40. Q: ? Isolated quantum dot – 0d system of fermions
Exact many-body states:
Ground state, excited states
Exact means that the imaginary part of the energy is zero!
Quasiparticle excitations:
Finite decay rate Q: ?
What is the connection

41. S QD D No e-e interactions – resonance tunneling gate source drain
current
No e-e interactions – resonance tunneling

42. S QD D g No e-e interactions – resonance tunneling VSD gate source
drain D
current g
No e-e interactions – resonance tunneling
Mean level spacing d1
VSD

43. S
gate QD
source
drain D
current g
The interaction leads to additional peaks – many body excitations
No e-e interactions – resonance tunneling
VSD

44. Inelastic cotunnelingD D
Resonance tunneling
Peaks
Inelastic cotunneling
Additional peak

45. S
gate QD
source
drain S
current g
The interaction leads to additional peaks – many body excitations
VSD

46. Landau quasiparticle with the width gSIA
gate QD
source
drain D
current g
Landau quasiparticle with the width gSIA NE
Ergodic - WD
loc
VSD

47. Landau quasiparticle with the width gSIANE
Ergodic - WD
loc
VSD
extended
Localized - finite # of the satelites
Extended - infinite # of the satelites
(for finite e the number of the satelites is always finite)
Ergodic – nonergodic crossover!

48. Anderson Model on a Cayley tree

49. I, W K – branching number Anderson Model on a Cayley tree
Resonance at every generation
Sparse resonances

50. Definition:
We will call a quantum state ergodic if it occupies the number of sites on the Anderson lattice, which is proportional to the total number of sites :
ergodic
nonergodic
Localized states are obviously not ergodic: Q:
Is each of the extended state ergodic ? A:
In 3D probably yes
For d>4 most likely no

51. nonergodic states transition ergodicity crossover
Such a state occupies infinitely many sites of the Anderson model but still negligible fraction of the total number of sites
nonergodic states
Example of nonergodicity: Anderson Model Cayley tree:
transition
– branching number
ergodicity
crossover

52. Resonance is typically far
localized
Typically there is no resonance at the next step
nonergodic
Typically there is a resonance at every step
nonergodic
Typically each pair of nearest neighbors is at resonance
ergodic