Universality in quantum chaos, Anderson localization and the one parameter scaling theory Antonio M. García-García Princeton University ICTP, Trieste In 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 Jiao, NUS, Singapore, PRL 94, (2005), PRE, 73, (2006). PRE, 73, (2006).

Outline: 0. What is this talk about? 0.1 Why are these issues interesting/relevant? 0.1 Why are these issues interesting/relevant? 1. Introduction to random matrix theory 2. Introduction to the theory of disordered systems 2.1 Localization and universality in disordered systems 2.1 Localization and universality in disordered systems 2.2 The one parameter scaling theory 2.2 The one parameter scaling theory 3. Introduction to quantum chaos 3.1 Universality in QC and the BGS conjecture 3.1 Universality in QC and the BGS conjecture 4. My research: One parameter scaling theory in QC 4.1 Limits of applicability of the BGS conjecture 4.1 Limits of applicability of the BGS conjecture 4.2 Metal-Insulator transition in quantum chaos 4.2 Metal-Insulator transition in quantum chaos

Relevant for: 1. Quantum classical transition. 2. Nano-Meso physics. Quantum engineering. 3. Systems with interactions for which the exact Schrödinger equation cannot be solved. Quantum Chaos Disordered systems (Simple) Quantum mechanics beyond textbooks Impact of classical chaos in quantum mechanics Quantum mechanics in a random potential 1. Powerful analytical techniques. 2. Ensemble average. 3. Anderson localization. ? 1. Semiclassical techniques. 2. BGS conjecture. Schrödinger equation + generic V(r) Quantum coherence

What information (if any) can I get from a “bunch” of energy levels? This question was first raised in the context of nuclear physics in the 50‘s -Shell model does not work -Excitations seem to have no pattern High energy nuclear excitations P(s) s -Wigner carried out a statistical analysis of these excitations. - Surprisingly, P(s) and other spectral correlator are universal and well described by random matrix theory (GOE).

Random Matrix Theory: Signatures of a RM spectrum (Wigner-Dyson): 1. Level Repulsion 2. Spectral Rigidity  = 1,2,4 for real,complex, quaternions  = 1,2,4 for real,complex, quaternions Signatures of an uncorrelated spectrum (Poisson) : In both cases spectral correlations are UNIVERSAL, namely, independent of the chosen distribution. The only scale is the mean level spacing . Random matrix theory describes the eigenvalue correlations of a matrix whose entries are random real/complex/quaternions numbers with a (Gaussian) distribution. s P(s)

Two natural questions arise: 1. Why are the high energy excitations of nuclei well described by random matrix theory (RMT)? 2. Are there other physical systems whose spectral correlations are well described by RMT? Answers: 1. It was claimed that the reason is the many body “complex” nature of the problem. It is not yet fully understood!. 2.1 Quantum chaos (’84): Bohigas-Giannoni-Schmit conjecture. Classical chaos RMT 2.2 Disordered systems(’84): RMT correlations for weak disorder and d > 2. Supersymmetry method. Microscopic justification. Efetov 2.3 More recent applications: Quantum Gravity (Amborjn), QCD, description of networks (www).

A few words about disordered systems: c) A really quantitative theory of strong localization is still missing but: c) A really quantitative theory of strong localization is still missing but: 1. Self-consistent theory from the insulator side, valid only for d >>1. No interference. Abu-Chakra, Anderson, Self-consistent theory from the insulator side, valid only for d >>1. No interference. Abu-Chakra, Anderson, Self-consistent theory from the metallic side, valid only for d ~ 2. No tunneling. Vollhardt and Wolffle,’82 2. Self-consistent theory from the metallic side, valid only for d ~ 2. No tunneling. Vollhardt and Wolffle,’82 3 One parameter scaling theory(1980). Gang of four. Correct but qualitative 3 One parameter scaling theory(1980). Gang of four. Correct but qualitative. The theory of disordered systems studies a quantum particle in a random potential. 1. How do quantum effects modify the transport properties of a particle whose classical motion is diffusive?. a) Many of the main results of the field are already included in the original paper by Anderson 1957!! b) Weak localization corrections are well understood. Lee, Altshuler. Questions:Answers: t D quan t D clas t D quan t a a = ? D quan =f(d,W)?

Your intuition about localization V(x) X EaEa EbEb EcEc Assume that V(x) is a truly disordered potential. Question: For any of the energies above, will the classical motion be strongly affected by quantum effects? 0

Localisation according to the one parameter scaling theory Insulator (eigenstates localised) When? For d 3 for strong disorder). Why? Caused by destructuve interference. How? Diffusion stops, Poisson statistics and discrete spectrum. discrete spectrum. Metal (eigenstates delocalised) When? d > 2 and weak disorder, eigenstates delocalized. Why? Interference effects are small. How? Diffusion weakly slowed down, Wigner-Dyson statistics and continous spectrum. Anderson transition For d > 2 there is a critical density of impurities such that a metal- insulator transition occurs. Metal Insulator Anderson transition Sridhar,et.al Kramer, et al.

Energy scales in a disordered system 1. Mean level spacing: 2. Thouless energy: t T (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

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 g c,  (g c ) = 0

Altshuler, Introduction to mesoscopic physics 0

1. Quantum chaos studies the quantum properties of systems whose classical motion is chaotic. 2. More generally it studies the impact on the quantum dynamics of the underlying deterministic classical motion, chaotic or not. Bohigas-Giannoni-Schmit conjecture Classical chaos Wigner-Dyson Energy is the only integral of motion Momentum is not a good quantum number Eigenfunctions delocalized in momentum space in momentum space What is quantum chaos?

Gutzwiller-Berry-Tabor conjecture Integrable classical motion motion Poisson statistics (Insulator) (Insulator) Integrability in d dimensions d canonical momenta are conserved d canonical momenta are conserved Momentum is a good quantum number System is localized in momentum space Poisson statistics is also related to localisation but in momentum space Poisson statistics is also related to localisation but in momentum space s P(s)

Universality and its exceptions Bohigas-Giannoni-Schmit conjecture Exceptions: 1. Kicked systems 1. Kicked systems Dynamical localization in momentum space 2. Harper model 3. Arithmetic billiard t Classical Quantum

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? 4. Is localization relevant in quantum chaos?

Random QUANTUM Deterministic Random QUANTUM Deterministic Delocalized Delocalized wavefunctions Chaotic motion wavefunctions Chaotic motion Wigner-Dyson Only? Wigner-Dyson Only? Localized Localized wavefunctions Integrable motion wavefunctions Integrable motion Poisson Poisson Anderson Anderson transition ???????? transition ???????? Critical Statistics

Main point of this talk Adapt the one parameter scaling theory in quantum chaos in order to: 1. Determine the universality class in quantum chaos related to the metal-insulator transition. 2. Determine the class of systems in which Wigner-Dyson statistics applies. 3. Determine whether there are more universality class in quantum chaos.

How to apply scaling theory to quantum chaos? 1. Only for classical systems with an homogeneous phase space. Not mixed systems. 2. Express the Hamiltonian in a finite momentum basis and study the dependence of observables with the basis size N. 3. For each system one has to map the quantum chaos problem onto an appropriate basis. For billiards, kicked rotors and quantum maps this is straightforward.

Scaling theory and anomalous diffusion d e 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 weak localization? Wigner-Dyson  (g) > 0 Poisson  (g) < 0 Lapidus, fractal billiards

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

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

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

1.  > 0 Localization Poisson 1.  > 0 Localization Poisson 2.  < 0 Delocalization Wigner-Dyson 2.  < 0 Delocalization Wigner-Dyson 3.  = 0 Anderson tran. Critical statistics 3.  = 0 Anderson tran. Critical statistics Anderson transition Anderson transition 1. log and step singularities 1. log and step singularities 2. Multifractality and Critical statistics. 2. Multifractality and Critical statistics. Results are stable under perturbations and sensitive to the removal of the singularity AGG, Wang Jjiao, PRL 2005

Analytical approach: From the kicked rotor to the 1D Anderson model with long-range hopping Analytical approach: From the kicked rotor to the 1D Anderson model with long-range hopping Fishman,Grempel and Prange method: 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 The associated Anderson model has long-range hopping depending on the nature of the non-analyticity: T m pseudo random Explicit analytical results are possible, Fyodorov and Mirlin Anderson Model

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 Eigenstates are multifractals. Skolovski, Shapiro, Altshuler Mobility edge Anderson transition var

V(x)=  log|x| Spectral Multifractal =15 χ =0.026 D 2 = 0.95  =15 χ =0.026 D 2 = 0.95 =8 χ =0.057 D 2 = 0.89 D 2 ~ 1 – 1/   =8 χ =0.057 D 2 = 0.89 D 2 ~ 1 – 1/  =4 χ=0.13 D 2 = 0.72  =4 χ=0.13 D 2 = 0.72 =2 χ=0.30 D 2 = 0.5  =2 χ=0.30 D 2 = 0.5 Summary of properties 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

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

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.

CONCLUSIONS 1. One parameter scaling theory is a valuable tool in the understanding 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.