Universality in quantum chaos and the one parameter scaling theory Antonio M. García-García [email protected] Princeton University Spectral correlations.

Universality in quantum chaos and the Universality in quantum chaos and the one parameter scaling theoryone parameter scaling theory

Antonio M. García-Garcí[email protected]

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 In collaboration with Wang JiaoWang Jiao PRL PRL 94, 244102 (2005) 94, 244102 (2005), PRE, 73, 374167 (2006) , PRE, 73, 374167 (2006)

Understanding localization and universalityUnderstanding localization and universality

1. Anderson’s paper (1958).1. Anderson’s paper (1958). Locator expansion. Locator expansion.

2. Abu Chacra, Anderson (1973).2. Abu Chacra, Anderson (1973). Self consistent theory. Self consistent theory. Exact only for d going to infinity.Exact only for d going to infinity.

3. Vollhardt and Wolffle (1980-1982).3. Vollhardt and Wolffle (1980-1982). Self consistent Self consistent theory for the 2 point function. Only valid for d =2+theory for the 2 point function. Only valid for d =2+ or or far from the transition. Consistent with previous far from the transition. Consistent with previous renormalization group arguments renormalization group arguments (Wegner)(Wegner)..

4. One parameter scaling theory (1980),4. One parameter scaling theory (1980), gang of four. gang of four.

Insulator:Insulator: For d < 3 or, in d > 3 for strong disorder. Diffusion eventually stops. For d < 3 or, in d > 3 for strong disorder. Diffusion eventually stops. Discrete spectrumDiscrete spectrum

Metal:Metal:d > 2 and weak disorder. Quantum effects do not alter d > 2 and weak disorder. Quantum effects do not alter

significantly the classical diffusion (weak localization). Continous spectrumsignificantly the classical diffusion (weak localization). Continous spectrum

Anderson transition:Anderson transition: For d > 2 a metal-insulator transition takes place.For d > 2 a metal-insulator transition takes place.

Can we apply this to deterministic systems?Can we apply this to deterministic systems?

Energy scales in a disordered systemEnergy scales in a disordered system

1. Mean level spacing:1. Mean level spacing:

2. Thouless energy: 2. Thouless energy:

ttTT(L) (L) is the typical (classical) travel time is the typical (classical) travel time through a system of size L through a system of size L

1

TE

g Dimensionless Dimensionless

Thouless conductanceThouless conductance22 dd

T LgLLE Diffusive motion Diffusive motion without quantum without quantum

correctionscorrections

1

1

gE

gE

T

T

Metal Wigner-Dyson

Insulator Poisson

TT thE /

Scaling theory of localizationScaling theory of localization

The change in the conductance with the system The change in the conductance with the system size only depends on the conductance itselfsize only depends on the conductance itself

)(ln

logg

Ld

gd

Beta function is universal but it depends on the global Beta function is universal but it depends on the global symmetries of the systemsymmetries of the system

0log)(1

/)2()(1/

2

ggegg

gdgLggL

d

Quantum

Weak localization

In 1D and 2D localization for any disorderIn 1D and 2D localization for any disorder

In 3D a metal insulator transition at gIn 3D a metal insulator transition at gcc , , (g(gcc) = 0) = 0

Scaling theory and anomalous diffusionScaling theory and anomalous diffusion

dde e is related to the fractal dimension of the spectrum. is related to the fractal dimension of the spectrum. The average is over initial The average is over initial

conditions and/or ensembleconditions and/or ensemble

UniversalityUniversality

Two routes to the Anderson transition Two routes to the Anderson transition

1. Semiclassical origin 1. Semiclassical origin

2. Induced by quantum effects 2. Induced by quantum effects

2

)( e

clasT

d

dL

ELg clas

clasquanclas 0

00 quanclas

0)( cg

)()( gfg clas

weak weak localization?localization?

LWigner-Dyson Wigner-Dyson clasclas > 0> 0

Poisson Poisson clasclas < 0< 0

eddLtq /2

Universality in quantum chaosUniversality in quantum chaos Bohigas-Giannoni-Schmit conjectureBohigas-Giannoni-Schmit conjecture Classical chaos Wigner-Dyson Classical chaos Wigner-Dyson Exceptions:Exceptions: Kicked systems and arithmetic billiardsKicked systems and arithmetic billiards Berry-Tabor conjectureBerry-Tabor conjecture Classical integrability Poisson statisticsClassical integrability Poisson statisticsExceptions: Exceptions: Harmonic oscillatorsHarmonic oscillators Systems with a degenerate spectrumSystems with a degenerate spectrumQuestions:Questions:1. Are these exceptions relevant?1. Are these exceptions relevant?2. Are there systems not classically chaotic but still 2. Are there systems not classically chaotic but still

described by the Wigner-Dyson?described by the Wigner-Dyson?3. Are there other universality class in quantum 3. Are there other universality class in quantum

chaos? How many?chaos? How many?

RandomRandom QUANTUM QUANTUM DeterministicDeterministic

Delocalized Delocalized wavefunctions wavefunctions Chaotic motion Chaotic motion Wigner-DysonWigner-Dyson Only?Only? LocalizedLocalized wavefunctionswavefunctions Integrable motionIntegrable motion Poisson Poisson

Anderson Anderson transition ???????? transition ????????

Critical Statistics

Is it possible to define new universality class ?Is it possible to define new universality class ?

g

0g

cgg

Wigner-Dyson statistics in non-Wigner-Dyson statistics in non-random systemsrandom systems

1. Typical time needed to reach the “boundary” (in real or 1. Typical time needed to reach the “boundary” (in real or momentum space) of the system. Symmetries importantmomentum space) of the system. Symmetries important. Not . Not for mixed systems. for mixed systems.

In billiards it is just the ballistic travel time.In billiards it is just the ballistic travel time.

In kicked rotors and quantum maps it is the time needed to explore a fixed In kicked rotors and quantum maps it is the time needed to explore a fixed basis.basis.

In billiards with some (Coulomb) potential inside one can obtain this time by In billiards with some (Coulomb) potential inside one can obtain this time by mapping the billiard onto an Anderson model (Levitov, Altshuler, 97).mapping the billiard onto an Anderson model (Levitov, Altshuler, 97).

2. Use the Heisenberg relation to estimate the Thouless energy and 2. Use the Heisenberg relation to estimate the Thouless energy and the dimensionless conductance g(N) as a function of the system the dimensionless conductance g(N) as a function of the system

size N (in momentum or position). size N (in momentum or position). ConditionCondition::

Wigner-Dyson statistics appliesWigner-Dyson statistics applies

02

)(

eclas

T

d

dL

ELg clas

Anderson transition in non-random systemsAnderson transition in non-random systems

Conditions:Conditions: 11. . Classical phase space must be homogeneous. Classical phase space must be homogeneous. 2. Quantum power- 2. Quantum power-law localization. 3.law localization. 3.

Examples:Examples:

tq

d

dL

ELg

eclas

T clas 202

)(

1D:1D:=1, d=1, dee=1/2, Harper model, interval exchange maps =1/2, Harper model, interval exchange maps (Bogomolny)(Bogomolny)

=2, d=2, dee=1, Kicked rotor with classical singularities =1, Kicked rotor with classical singularities (AGG, WangJiao). (AGG, WangJiao).

2D: 2D: =1, d=1, dee=1, =1,

Coulomb billiard Coulomb billiard (Altshuler, Levitov).(Altshuler, Levitov).

3D: 3D: =2/3, d=2/3, dee=1, Kicked rotor at critical coupling, kicked rotor 3 =1, Kicked rotor at critical coupling, kicked rotor 3

incommensurate frequencies (incommensurate frequencies (Casati,Shepelansky).Casati,Shepelansky).

1D kicked rotor with singularities 1D kicked rotor with singularities

||log)(||)( VV

)4

exp()/)(exp()4

exp(ˆ2

2

2

2

T

iVT

U

11

1 )('

nnn

nnn

Tk

Vkk

n

nTtVpH )()(2

)cos()( KV Classical Motion

Quantum Evolution

Normal diffusion

Anomalous Diffusion

122/1),( tkktkP

'2'/1),( tkktkP

1. Quantum anomalous diffusion

2. No dynamical localization for <0

1. 1. > 0 Localization Poisson > 0 Localization Poisson

2. 2. < 0 Delocalization Wigner-Dyson < 0 Delocalization Wigner-Dyson

3. 3. = 0 MIT transition Critical statistics = 0 MIT transition Critical statistics

Anderson transitionAnderson transition

1. log and step singularities 1. log and step singularities

2. Multifractality and Critical statistics.2. Multifractality and Critical statistics.

Results are Results are stablestable under perturbations and under perturbations and sensitive to the removal of the singularitysensitive to the removal of the singularity

122)(

tqL

ELg clas

T clas

Analytical approach: From the kicked rotor to the 1D Anderson Analytical approach: From the kicked rotor to the 1D Anderson model with long-range hopping model with long-range hopping

Fishman,Grempel and Prange method:Fishman,Grempel and Prange method:

Dynamical localization in the kicked rotor is Dynamical localization in the kicked rotor is 'demonstrated''demonstrated' by mapping it onto by mapping it onto a 1D Anderson model with short-range interaction.a 1D Anderson model with short-range interaction.

Kicked rotorKicked rotor ),()()(),(2

1),(

2

2

tntVttt

in

),0(),0( tuet ti

1

1 r

Wr

The associated Anderson model has The associated Anderson model has long-range hoppinglong-range hopping depending depending on the nature of the non-analyticity:on the nature of the non-analyticity:

TTmm pseudo pseudo randomrandom

Explicit analytical results are possible, Fyodorov and Mirlin

Anderson Model

0r

mrmrmm EuuWuT

Signatures of a metal-insulator transitionSignatures 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.

)1(2

~)( qDdq

n

qLrdr

Skolovski, Shapiro, Altshuler

1~)(

1~)(

sesP

sssPAs

Mobility edge Anderson transition

nn ~)(3varvar

dssPssss nn )(var22

V(x)= log|x| Spectral Spectral MultifractalMultifractal =15 =15 χχ =0.026 D =0.026 D

22= 0.95= 0.95

=8 =8 χχ =0.057 D =0.057 D22= 0.89 D= 0.89 D22 ~ 1 – 1/ ~ 1 – 1/

=4 =4 χχ=0.13 D=0.13 D

22= 0.72= 0.72

=2 =2 χχ=0.30 D=0.30 D22= 0.5= 0.5

Summary of properties Summary of properties 1. Scale Invariant Spectrum1. Scale Invariant Spectrum2. Level repulsion2. Level repulsion3. Linear (slope < 1), 3. Linear (slope < 1), 33 ~ ~/15 /15 4. Multifractal wavefunctions4. Multifractal wavefunctions5. Quantum anomalous diffusion 5. Quantum anomalous diffusion

ANDERSON TRANSITON IN QUANTUM CHAOS

2~)( DttP

Ketzmerick, Geisel, Huckestein

3D kicked rotator3D kicked rotator

Finite size scaling analysis Finite size scaling analysis shows there is a transition shows there is a transition a MIT at ka MIT at kc c ~ 3.3~ 3.3

)cos()cos()cos(),,( 221321 kV

3/22 ~)( ttpquan

ttpclas

~)(2

In 3D, for =2/3

cgg

For a KR with 3 incommensurable frequencies see Casati, Shepelansky, 1997

Experiments and 3D Anderson transitionExperiments 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.

CONCLUSIONS1. One parameter scaling theory is a valuable 1. One parameter scaling theory is a valuable tool for the understading of universal features tool for the understading of universal features of the quantum motion.of the quantum motion.

2. Wigner Dyson statistics is related to classical 2. Wigner Dyson statistics is related to classical motion such that motion such that

3. The Anderson transition in quantum chaos is 3. The Anderson transition in quantum chaos is related to related to

4. Experimental verification of the Anderson 4. Experimental verification of the Anderson transition is possible with ultracold atoms transition is possible with ultracold atoms techniques.techniques.

gN

cggN

ANDERSON TRANSITIONANDERSON TRANSITIONMain:Main:Non trivial interplay between tunneling and interference leads to the Non trivial interplay between tunneling and interference leads to the

metal insulator transition (MIT) metal insulator transition (MIT) Spectral correlations Wavefunctions

Scale invarianceScale invariance MultifractalsMultifractals

Quantum Anomalous

diffusion P(k,t)~ t-D2

CRITICAL STATISTICSCRITICAL STATISTICS

2 L LL 1 1

P s s s 1

Kravtsov, Muttalib 97

1)( sesP As

qDdq

nLrdr ~)(

2

Skolovski, Shapiro, Altshuler

Density of Probability

1. Stable under perturbation (green, black line log|(x)| +perturbation. 1. Stable under perturbation (green, black line log|(x)| +perturbation.

2. Normal diff. (pink) is obtained if the singularity (log(|x|+a)) is removed. 2. Normal diff. (pink) is obtained if the singularity (log(|x|+a)) is removed.

3. Red alpha=0.4, Blue alpha=-0.43. Red alpha=0.4, Blue alpha=-0.4

CLASSICAL

How to apply this to quantum chaos?How to apply this to quantum chaos?

1. Only for classical systems with an 1. Only for classical systems with an homogeneous phase space. Not mixed homogeneous phase space. Not mixed phase space.phase space.

2. Express the Hamiltonian in a finite 2. Express the Hamiltonian in a finite basis and see the dependence of basis and see the dependence of observables with the basis size N.observables with the basis size N.

3. The role of the system size in the 3. The role of the system size in the scaling theory is played by Nscaling theory is played by N

4. For billiards, kicked rotors and 4. For billiards, kicked rotors and quantum maps this is straightforward.quantum maps this is straightforward.

Classical-Quantum diffusion

Non-analytical potentials and the Anderson Non-analytical potentials and the Anderson transition in deterministic systemstransition in deterministic systems

Classical Input (1+1D)Classical Input (1+1D) Non-analytical chaotic potentialNon-analytical chaotic potential 1. Fractal and homongeneous phase space (cantori)1. Fractal and homongeneous phase space (cantori) 2. Anomalous Diffusion in momentum space 2. Anomalous Diffusion in momentum space Quantum OutputQuantum Output (AGG PRE69 066216)(AGG PRE69 066216)

Wavefunctions power-law localized Wavefunctions power-law localized 1. Spectral properties expressed in terms of P(k,t)1. Spectral properties expressed in terms of P(k,t) 2. The case of step and log singularities (1/f noise) leads to: 2. The case of step and log singularities (1/f noise) leads to: Critical statistics and multifractal wavefunctionsCritical statistics and multifractal wavefunctions Attention:Attention: KAM theorem does not hold and KAM theorem does not hold and Mixed systems are excluded!Mixed systems are excluded!

tkktkP /1),(

ANDERSON-MOTT TRANSITIONANDERSON-MOTT TRANSITIONMain:Main:Non trivial interplay between tunneling and interference leads to the Non trivial interplay between tunneling and interference leads to the

metal insulator transition (MIT) metal insulator transition (MIT)

Spectral correlations Wavefunctions Scale invarianceScale invariance MultifractalsMultifractals

CRITICAL STATISTICSCRITICAL STATISTICS

""Spectral correlations are universal, they depend only on the dimensionality of the space."

Kravtsov, Muttalib 97

)1(2

~)( qDdq

n

qLrdr

Skolovski, Shapiro, Altshuler

1~)(

1~)(

~)(2

sesP

sssP

nn

As

Mobility edge Mott Anderson transition

MultifractalityMultifractality

Intuitive: Intuitive: Points in which the modulus of the wave function Points in which the modulus of the wave function is bigger than a (small) is bigger than a (small) cutoffcutoff MM.. If the fractal dimension If the fractal dimension depends on thedepends on the cutoff M,cutoff M, the wave function is the wave function is multifractal.multifractal.

Formal:Formal: Anomalous scaling of the density moments. Anomalous scaling of the density moments. Kravtsov, Chalker 1996

I pr

n r 2 p L Dp p 1 pD

r

pnp Lrψ=I

2

POINCARE SECTION

P

X

Is it possible a MIT in 1D ?Yes, if long range hopping is permittedYes, if long range hopping is permitted

Eigenstates power-law localizedEigenstates power-law localized

ThermodynamicsThermodynamics limit: limit: Eigenstates Eigenstates SpectralSpectral

Multifractal Critical statistics Multifractal Critical statistics

Localized Poisson statisticsLocalized Poisson statistics

Delocalized Random MatrixDelocalized Random Matrix

Analytical treatment by using the supersymmetry method (Analytical treatment by using the supersymmetry method (Mirlin &Fyodorov)Mirlin &Fyodorov)

Related to classical diffusion operator.Related to classical diffusion operator.

11

1

j

ijijijiii jiji

hjiFjiFH 1

||)(]1,1[,)(

11

|)(| i

i

i rr

r

Eigenfunction characterization

1. Eigenfunctions moments:

2. Decay of the eigenfunctions:

MetalV

InsulatorrdrIPR d

n 1

4 1~)(

?/1

/1~)(

/

r

MetalV

Insulatore

r

r

n

Metald

Criticald

Insulatord

Looking for the metal-insulator transition in deterministic Hamiltonians

What are we looking for?What are we looking for? - Between chaotic and integrable but not a - Between chaotic and integrable but not a superposition. NOT mixed systems.superposition. NOT mixed systems.

1D and 2D : 1D and 2D : Classical anomalous Classical anomalous diffusion and/or fractal spectrumdiffusion and/or fractal spectrum 3D : Anomalous diffusion but also 3D : Anomalous diffusion but also standard kicked rotor standard kicked rotor Different possibilitiesDifferent possibilities- Anisotropic Kepler problem. - Anisotropic Kepler problem. Wintgen, Marxer (1989)Wintgen, Marxer (1989) - Billiard with a Coulomb scatterer. - Billiard with a Coulomb scatterer. Levitov, Levitov, Altshuler (1997) Altshuler (1997)

- Generalized Kicked rotors, Harper model, Bogomolny maps- Generalized Kicked rotors, Harper model, Bogomolny maps

How do we know that a metal is a metal?How do we know that a metal is a metal?Texbook answer:Texbook answer: Look at the conductivity or other transport propertiesLook at the conductivity or other transport properties

1. Eigenvector statistics:1. Eigenvector statistics:

DDq q = d = d Metal Metal D Dq q = 0 = 0 InsulatorInsulator D Dqq = f(d,q) = f(d,q) M-I M-I transitiontransition

2. Eigenvalue statistics:2. Eigenvalue statistics:

Level Spacing distribution: Level Spacing distribution:

Number variance: Number variance:

))(1(2)1( ~)( qDdqdq

ndq

q LrdrLP

Other options: Look at eigenvalue and eigenvectorsOther options: Look at eigenvalue and eigenvectors

2

2 log~)(

)( Asβes~sP

LL

DysonWigner

Metal

sesP

LL

Poisson

Insulator

)(

)(

)(

2

i

iissP /)( 1

222 )()()/( LnLn=L ji

nnn EH

Return Probability

dssPs=A

AA

AA=W

RMTP

RMT

0

2

V(q) = log (q)

t = 50

CLASSICAL

V(q)= 10 log (q)

Altshuler, Introduction to mesoscopic

physics