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  1. Universality in quantum chaos and the one parameter scaling theory Antonio M. García-García Princeton University ICTP, TriesteIn the semiclassical limit the spectral properties 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. In collaboration with Wang JiaoPRL 94, 244102 (2005), PRE 2007 in press

  2. Universality in disordered systems Insulator For d < 3 or, in d > 3 for strong disorder all eigenstates are localized in space. Classical diffusion eventually stops Transition to localization is caused by destructuve Interference. Metal d > 2 and weak disorder eigenstates delocalized. Quantum effects do not alter significantly the classical diffusion. Anderson transition For d > 2 there is a critical density of impurities such that a metal-insulator Transition occurs. Kramer, et al. Anderson transition Sridhar, Insulator Metal

  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-DysonInsulator 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. Altshuler, Introduction to mesoscopic physics

  6. Scaling theory and anomalous diffusion Lapidus, fractal billiards 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 (g) > 0 Poisson (g) < 0 weak localization?

  7. How to apply this to quantum chaos? 1. Only for classical systems with an homogeneous phase space. Not mixed systems. 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 each system one has to map the quantum chaos problem onto an appropiate basis. For billiards, kicked rotors and quantum maps this is straightforward.

  8. 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 Exceptions: Harmonic oscillator 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?

  9. RandomQUANTUM Deterministic Delocalized • wavefunctions Chaotic motion • Wigner-DysonOnly? Localized wavefunctionsIntegrable motion Poisson Anderson transition ???????? Critical Statistics Is it possible to define new universality class ?

  10. Wigner-Dyson statistics in non-random systems 1. Evaluate the typical time needed to reach the boundary of the system. Take into account symmetries. 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) a potential inside one can obtain this time by mapping the billiard onto an Anderson model. 2. Use the Heisenberg relation to estimate the Thouless energy and the dimesionless conductance g(N) as a function of the system size N (in momentum or position). IF Wigner-Dyson statistics applies

  11. Anderson transition in non-random systems Conditions:1. Between chaotic and integrable but not a superposition. 2. Classical anomalous diffusion 3. Quantum power-law localization Examples: 1D: =1, de=1/2, Harper model =2, de=1, Kicked rotor with classical singularities, interval exchange maps. 2D: =1, de=1, Coulomb billiard 3D: =2/3, de=1, Kicked rotor at critical coupling

  12. 1D kicked rotor with singularities Classical Motion Classical diffusion Step function Classical Anomalous Diffusion Quantum Evolution Power-law localization

  13. 1.  > 0 Localization Poisson 2.  < 0 Delocalization Wigner-Dyson 3.  = 0 L-D transition Critical statistics Anderson transition 1. log (1/f noise) and step singularities 2. Multifractality and Critical statistics. Results are stable under perturbations and sensitive to the removal of the singularity

  14. 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 PRE69 066216) 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!

  15. 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 What happens if step Is there any relation between non-differentiability and Wr? Yes

  16. Non-differentiability induces long range hopping The associated Anderson model has long-range hopping depending on the nature of the non-analyticity: Already solved (Fyodorov, Mirlin,Seligman 1996, Levitov 1990) but long range hopping is now NOT random. Critical Cases 1. Log singularity Wr ~Aij/r with Aij pseudo-random Similar to Critical statistics . 2. Step like singularity Wr ~Aij/r with Semi-Poisson statistics (Harper model, pseudo integrable billiards) Exact treatment possible AGG PRE 2006 Experimental verification by using ultra-cold atoms techniques

  17. 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:

  18. 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

  19. V(x)= log|x| Spectral Multifractal =15 χ =0.026 D2= 0.95 =8 χ =0.057 D2= 0.89 D2 ~ 1 – 1/ =4 χ=0.13 D2= 0.72 =2 χ=0.30 D2= 0.5 Summary of properties 1. Scale Invariant Spectrum 2. Level repulsion 3. Sub-Poisson Number variance 4. Multifractal wavefunctions 5. Quantum anomalous diffusion ANDERSON TRANSITON IN QUANTUM CHAOS Ketzmerick, Geisel, Huckestein

  20. 3D kicked rotator In 3D, for =2/3 Finite size scaling analysis shows there is a transition a MIT at kc ~ 3.3

  21. Experiments and 3D Anderson transition Our findings for the 3D kicked rotor at kc and 1D with log singularities may be used to test experimentally the Anderson transition by using ultracold atoms techniques (Raizen). 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 2006) an approximate singularity will still show typical features of a metal insulator transition

  22. 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. CONCLUSIONS

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

  24. Density of Probability

  25. 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

  26. Classical-Quantum diffusion

  27. ANDERSON-MOTT TRANSITIONMain:Non trivial interplay between tunneling and interference leads to the metal insulator transition (MIT) Spectral correlations Wavefunctions Scale invarianceMultifractals 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

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


  30. 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.

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

  32. 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 Looking for the metal-insulator transition in deterministic Hamiltonians

  33. Return Probability

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

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