1. Universality in quantum chaos and the one parameter scaling theory
Antonio M. García-García
Princeton University
Spectral correlations of classically chaotic Hamiltonian are universally described by random matrix theory. With the help of the one parameter scaling theory we propose an alternative characterization of this universality class. It is also identified the universality class associated to the metal-insulator transition. In low dimensions it is characterized by classical superdiffusion. In higher dimensions it has in general a quantum origin as in the case of disordered systems. Systems in this universality class include: kicked rotors with certain classical singularities, polygonal and Coulomb billiards and the Harper model. We hope that our results may be of interest for experimentalists interested in the Anderson transition.
In collaboration with Wang Jiao PRL 94, (2005), PRE, 73, (2006)

2. Understanding localization and universality
1. Anderson’s paper (1958). Locator expansion.
2. Abu Chacra, Anderson (1973). Self consistent theory. Exact only for d going to infinity.
3. Vollhardt and Wolffle ( ). Self consistent theory for the 2 point function. Only valid for d =2+ or far from the transition. Consistent with previous renormalization group arguments (Wegner).
4. One parameter scaling theory (1980), gang of four.
Insulator: For d < 3 or, in d > 3 for strong disorder. Diffusion eventually stops. Discrete spectrum
Metal:d > 2 and weak disorder. Quantum effects do not alter
significantly the classical diffusion (weak localization). Continous spectrum
Anderson transition: For d > 2 a metal-insulator transition takes place.
Can we apply this to deterministic systems?

3. Energy scales in a disordered system
1. Mean level spacing:
2. Thouless energy:
tT(L) is the typical (classical) travel time through a system of size L
Dimensionless Thouless conductance
Diffusive motion without quantum corrections
Metal Wigner-Dyson Insulator Poisson

4. Scaling theory of localization
The change in the conductance with the system size only depends on the conductance itself
Beta function is universal but it depends on the global symmetries of the system
Quantum
Weak localization
In 1D and 2D localization for any disorder
In 3D a metal insulator transition at gc , (gc) = 0

5. Scaling theory and anomalous diffusion
de is related to the fractal dimension of the spectrum The average is over initial conditions and/or ensemble
Universality
Two routes to the Anderson transition
1. Semiclassical origin
2. Induced by quantum effects
Wigner-Dyson clas > 0
Poisson clas < 0
weak localization?

6. Universality in quantum chaos
Bohigas-Giannoni-Schmit conjecture
Classical chaos Wigner-Dyson
Exceptions:
Kicked systems and arithmetic billiards
Berry-Tabor conjecture
Classical integrability Poisson statistics
Harmonic oscillators
Systems with a degenerate spectrum
Questions:
1. Are these exceptions relevant?
2. Are there systems not classically chaotic but still described by the Wigner-Dyson?
3. Are there other universality class in quantum chaos? How many?

7. Random QUANTUM Deterministic
Delocalized
wavefunctions Chaotic motion
Wigner-Dyson Only?
Localized
wavefunctions Integrable motion
Poisson
Anderson
transition ????????
Critical Statistics
Is it possible to define new universality class ?

8. Wigner-Dyson statistics in non-random systems
1. Typical time needed to reach the “boundary” (in real or momentum space) of the system. Symmetries important. Not for mixed systems.
In billiards it is just the ballistic travel time.
In kicked rotors and quantum maps it is the time needed to explore a fixed basis.
In billiards with some (Coulomb) potential inside one can obtain this time by mapping the billiard onto an Anderson model (Levitov, Altshuler, 97).
2. Use the Heisenberg relation to estimate the Thouless energy and the dimensionless conductance g(N) as a function of the system size N (in momentum or position). Condition:
Wigner-Dyson statistics applies

9. Examples: Anderson transition in non-random systems
Conditions: Classical phase space must be homogeneous Quantum power-law localization
Examples:
1D:=1, de=1/2, Harper model, interval exchange maps (Bogomolny)
=2, de=1, Kicked rotor with classical singularities (AGG, WangJiao) D: =1, de=1, Coulomb billiard (Altshuler, Levitov).
3D: =2/3, de=1, Kicked rotor at critical coupling, kicked rotor 3
incommensurate frequencies (Casati,Shepelansky).

10. Classical Motion Normal diffusion Anomalous Diffusion
1D kicked rotor with singularities
Classical Motion
Normal diffusion
Anomalous Diffusion
Quantum Evolution
1. Quantum anomalous diffusion
2. No dynamical localization for <0

11. 1.  > 0 Localization Poisson
2.  < 0 Delocalization Wigner-Dyson
3.  = 0 MIT transition Critical statistics
Anderson transition
1. log and step singularities
2. Multifractality and Critical statistics.
Results are stable under perturbations and sensitive to the removal of the singularity

12. Kicked rotor Anderson Model
Analytical approach: From the kicked rotor to the 1D Anderson model with long-range hopping
Fishman,Grempel and Prange method:
Dynamical localization in the kicked rotor is 'demonstrated' by mapping it onto a 1D Anderson model with short-range interaction.
Kicked rotor
Anderson Model
Tm pseudo random
The associated Anderson model has long-range hopping depending on the nature of the non-analyticity:
Explicit analytical results are possible, Fyodorov and Mirlin

13. Signatures of a metal-insulator transition
1. Scale invariance of the spectral correlations A finite size scaling analysis is then carried out to determine the transition point. 2.
3. Eigenstates are multifractals.
Skolovski, Shapiro, Altshuler
var
Mobility edge Anderson transition

16. V(x)= log|x| Spectral Multifractal  =15 χ =0.026 D2= 0.95
 = χ = D2= D2 ~ 1 – 1/
 = χ= D2= 0.72
 = χ= D2= 0.5
Summary of properties
1. Scale Invariant Spectrum
2. Level repulsion
3. Linear (slope < 1), 3 ~/15
4. Multifractal wavefunctions
5. Quantum anomalous diffusion
ANDERSON TRANSITON IN QUANTUM CHAOS
Ketzmerick, Geisel, Huckestein

17. 3D kicked rotator In 3D, for =2/3
Finite size scaling analysis shows there is a transition a MIT at kc ~ 3.3
For a KR with 3 incommensurable frequencies see Casati, Shepelansky, 1997

18. Experiments and 3D Anderson transition
Our findings may be used to test experimentally the Anderson transition by using ultracold atoms techniques.
One places a dilute sample of ultracold Na/Cs in a periodic step-like standing wave which is pulsed in time to approximate a delta function then the atom momentum distribution is measured.
The classical singularity cannot be reproduced in the lab. However (AGG, W Jiao, PRA 2006) an approximate singularity will still show typical features of a metal insulator transition.

19. CONCLUSIONS
1. One parameter scaling theory is a valuable tool for the understading of universal features of the quantum motion.
2. Wigner Dyson statistics is related to classical motion such that
3. The Anderson transition in quantum chaos is related to
4. Experimental verification of the Anderson transition is possible with ultracold atoms techniques.

20. ANDERSON TRANSITION Main:Non trivial interplay between tunneling and interference leads to the metal insulator transition (MIT)
Spectral correlations Wavefunctions
Scale invariance Multifractals
Quantum Anomalous
diffusion P(k,t)~ t-D2
CRITICAL STATISTICS
Kravtsov, Muttalib 97
Skolovski, Shapiro, Altshuler

21. Density of Probability

22. CLASSICAL
1. Stable under perturbation (green, black line log|(x)| +perturbation.
2. Normal diff. (pink) is obtained if the singularity (log(|x|+a)) is removed.
3. Red alpha=0.4, Blue alpha=-0.4

23. How to apply this to quantum chaos?
1. Only for classical systems with an homogeneous phase space. Not mixed phase space.
2. Express the Hamiltonian in a finite basis and see the dependence of observables with the basis size N.
3. The role of the system size in the scaling theory is played by N
4. For billiards, kicked rotors and quantum maps this is straightforward.

24. Classical-Quantum diffusion

25. Non-analytical potentials and the Anderson transition in deterministic systems
Classical Input (1+1D)
Non-analytical chaotic potential
1. Fractal and homongeneous phase space (cantori)
2. Anomalous Diffusion in momentum space
Quantum Output (AGG PRE )
Wavefunctions power-law localized
1. Spectral properties expressed in terms of P(k,t)
2. The case of step and log singularities (1/f noise) leads to:
Critical statistics and multifractal wavefunctions
Attention: KAM theorem does not hold and Mixed systems are excluded!

26. ANDERSON-MOTT TRANSITION Main:Non trivial interplay between tunneling and interference leads to the metal insulator transition (MIT)
Spectral correlations Wavefunctions Scale invariance Multifractals
CRITICAL STATISTICS
"Spectral correlations are universal, they depend only on the dimensionality of the space."
Kravtsov, Muttalib 97
Skolovski, Shapiro, Altshuler
Mobility edge Mott Anderson transition

27. Multifractality
Intuitive: Points in which the modulus of the wave function is bigger than a (small) cutoff M. If the fractal dimension depends on the cutoff M, the wave function is multifractal.
Formal: Anomalous scaling of the density moments.
Kravtsov, Chalker 1996

28. POINCARE SECTION P X

29. Is it possible a MIT in 1D ? Yes, if long range hopping is permitted
Eigenstates power-law localized
Thermodynamics limit: Eigenstates Spectral
Multifractal Critical statistics
Localized Poisson statistics
Delocalized Random Matrix
Analytical treatment by using the supersymmetry method (Mirlin &Fyodorov)
Related to classical diffusion operator.

30. Eigenfunction characterization
1. Eigenfunctions moments:
2. Decay of the eigenfunctions:

31. Looking for the metal-insulator transition in deterministic Hamiltonians
What are we looking for?
- Between chaotic and integrable but not a
superposition. NOT mixed systems.
1D and 2D : Classical anomalous diffusion and/or fractal spectrum
3D : Anomalous diffusion but also standard kicked rotor
Different possibilities
- Anisotropic Kepler problem. Wintgen, Marxer (1989) Billiard with a Coulomb scatterer. Levitov, Altshuler (1997)
- Generalized Kicked rotors, Harper model, Bogomolny maps

32. Other options: Look at eigenvalue and eigenvectors
How do we know that a metal is a metal? Texbook answer: Look at the conductivity or other transport properties
Other options: Look at eigenvalue and eigenvectors
1. Eigenvector statistics:
Dq = d Metal Dq = 0 Insulator Dq = f(d,q) M-I transition
2. Eigenvalue statistics:
Level Spacing distribution:
Number variance:

33. Return Probability

35. CLASSICAL
V(q) = log (q)
t = 50

36. V(q)= 10 log (q)

38. Altshuler, Introduction to mesoscopic
physics