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Universality in quantum chaos, Anderson Universality in quantum chaos, Anderson localization and the one parameter scaling localization and the one parameter scaling
theorytheory
Antonio M. García-García
[email protected] 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 In collaboration with Wang Jiao, Wang Jiao, NUS, Singapore, NUS, Singapore, PRL PRL 94, 244102 (2005) 94, 244102 (2005), , PRE, 73, 374167 PRE, 73, 374167 (2006).(2006).
OutlinOutline:e:
0. What is this talk about?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 theory1. Introduction to random matrix theory
2. Introduction to the theory of disordered systems2. Introduction to the theory of disordered systems
2.1 Localization and universality in disordered systems2.1 Localization and universality in disordered systems
2.2 The one parameter scaling theory2.2 The one parameter scaling theory
3. Introduction to quantum chaos3. Introduction to quantum chaos
3.1 Universality in QC and the BGS conjecture3.1 Universality in QC and the BGS conjecture
4. My research: One parameter scaling theory in 4. My research: One parameter scaling theory in QCQC
4.1 Limits of applicability of the BGS conjecture4.1 Limits of applicability of the BGS conjecture
4.2 Metal-Insulator transition in quantum chaos4.2 Metal-Insulator transition in quantum chaos
Relevant for:Relevant for:
1. Quantum classical transition.1. Quantum classical transition.
2. Nano-Meso physics. Quantum engineering.2. Nano-Meso physics. Quantum engineering.
3. Systems with interactions for which the exact 3. Systems with interactions for which the exact Schrödinger equation cannot be solved.Schrödinger equation cannot be solved.
Quantum Quantum ChaosChaos
Disordered Disordered systemssystems
(Simple) Quantum (Simple) Quantum mechanics mechanics
beyond textbooksbeyond textbooksImpact of classical Impact of classical chaos in quantum chaos in quantum
mechanicsmechanics
Quantum Quantum mechanics in a mechanics in a
random potentialrandom potential
1. Powerful analytical 1. Powerful analytical techniques.techniques.
2. Ensemble average.2. Ensemble average.
3. Anderson localization.3. Anderson localization.
??
1. Semiclassical techniques.1. Semiclassical techniques.
2. BGS conjecture.2. BGS conjecture.
Schrödinger equation + generic V(r) Quantum coherence
What information (if any) can I get What information (if any) can I get from a “bunch” of energy levels?from a “bunch” of energy levels?
This question was first raised in the context of This question was first raised in the context of nuclear physics in the 50‘snuclear physics in the 50‘s
-Shell model does not work-Shell model does not work
-Excitations seem to have -Excitations seem to have no patternno pattern
High energy nuclear High energy nuclear excitations excitations
i
ii EEssP /)( 1
P(s)
s
-Wigner carried out a statistical -Wigner carried out a statistical analysis of these excitations. analysis of these excitations.
- Surprisingly, P(s) and other spectral - Surprisingly, P(s) and other spectral correlator are correlator are universaluniversal and well and well described by random matrix theory described by random matrix theory (GOE).(GOE).
Random Matrix Random Matrix Theory:Theory:
Signatures of a RM spectrum (Wigner-Dyson):Signatures of a RM spectrum (Wigner-Dyson):
1. Level Repulsion 1. Level Repulsion
2. Spectral Rigidity2. Spectral Rigidity
= 1,2,4 for real,complex, quaternions= 1,2,4 for real,complex, quaternions
Signatures of an uncorrelated Signatures of an uncorrelated spectrum (Poisson) :spectrum (Poisson) :
In both cases spectral correlations are UNIVERSAL, namely, In both cases spectral correlations are UNIVERSAL, namely, independent of the chosen distribution. The only scale is independent of the chosen distribution. The only scale is the mean level spacing the mean level spacing . .
ii EEAsβ ses~sP 1
2
1log)()(22
2
if EELL~LnLn=LΣ
)exp()()(2 ssPLL
Random matrix theory Random matrix theory describes the eigenvalue describes the eigenvalue correlations of a matrix correlations of a matrix whose entries are random whose entries are random real/complex/quaternions real/complex/quaternions numbers with a (Gaussian) numbers with a (Gaussian) distribution.distribution.
s
P(s)
Two natural questions arise:
1. Why are the high energy excitations of nuclei well described 1. Why are the high energy excitations of nuclei well described by random matrix theory (RMT)?by random matrix theory (RMT)?
2. Are there other physical systems whose spectral correlations 2. Are there other physical systems whose spectral correlations are well described by RMT? are well described by RMT?
Answers:Answers:
1. It was claimed that the reason is the many body “complex” 1. It was claimed that the reason is the many body “complex” nature of the problem. It is not yet fully understood!.nature of the problem. It is not yet fully understood!.
2.1 2.1 Quantum chaos (’84): Quantum chaos (’84): Bohigas-Giannoni-Schmit conjecture. Bohigas-Giannoni-Schmit conjecture. Classical chaos RMTClassical chaos RMT
2.2 Disordered systems(’84):2.2 Disordered systems(’84): RMT correlations for weak RMT correlations for weak disorder and d > 2. disorder and d > 2. Supersymmetry method. Microscopic Supersymmetry method. Microscopic justification. Efetov justification. Efetov
2.3 More recent applications: 2.3 More recent applications: Quantum Gravity (Amborjn), QCDQuantum Gravity (Amborjn), QCD, description of networks , description of networks (www).(www).
A few words about disordered systems:
c) A really quantitative theory of c) A really quantitative theory of strong localization is still missing strong localization is still missing but:but:
1. Self-consistent theory from the 1. Self-consistent theory from the insulator side, valid only for d insulator side, valid only for d >>1. No interference. >>1. No interference. Abu-Abu-Chakra, Anderson, 73Chakra, Anderson, 73
2. Self-consistent theory from the 2. Self-consistent theory from the metallic side, valid only for d ~ 2. metallic side, valid only for d ~ 2. No tunneling. No tunneling. Vollhardt and Wolffle,’82Vollhardt and Wolffle,’82
3 One parameter scaling 3 One parameter scaling theory(1980). Gang of four. theory(1980). Gang of four. Correct but qualitativeCorrect but qualitative.
The theory of disordered systems studies a quantum particle in a random potential.
1. How do quantum effects 1. How do quantum effects modify the transport properties modify the transport properties of a particle whose classical of a particle whose classical motion is diffusive?. motion is diffusive?.
a) Many of the main results of the a) Many of the main results of the field are already included in the field are already included in the original paper by Anderson 1957!!original paper by Anderson 1957!!
b) Weak localization corrections b) Weak localization corrections are well understood. Lee, are well understood. Lee, Altshuler.Altshuler.
Questions:
Answers:Answers:
<x2
>
t
Dquan
t
Dclast
Dquanta
a = ?
Dquan=f(d,W)?
Your intuition about localization
V(x)
X
Ea
Eb
Ec
Assume that V(x) is a truly disordered potential.Assume that V(x) is a truly disordered potential.
Question:Question: For any of the energies above, will the For any of the energies above, will the classical motion be strongly affected by quantum classical motion be strongly affected by quantum effects?effects?
0
Localisation according to Localisation according to the the one parameter scaling theoryone parameter scaling theoryInsulator (eigenstates localised)Insulator (eigenstates localised)When? When? For d < 3 or, (or d > 3 for strong disorder).For d < 3 or, (or d > 3 for strong disorder).
Why? Why? Caused by destructuve interference. Caused by destructuve interference.
How? How? Diffusion stops, Poisson statistics andDiffusion stops, Poisson statistics and
discrete spectrum.discrete spectrum.
Metal (eigenstates delocalised)Metal (eigenstates delocalised)When? When? d > 2 and weak disorder, eigenstates delocalized.d > 2 and weak disorder, eigenstates delocalized.
Why? Why? Interference effects are small.Interference effects are small.
How? How? Diffusion weakly slowed down, Wigner-Dyson Diffusion weakly slowed down, Wigner-Dyson statistics and continous spectrum. statistics and continous spectrum.
Anderson transitionAnderson transitionFor d > 2 there is a critical density For d > 2 there is a critical density
of impurities such that a metal-of impurities such that a metal-
insulator transition occursinsulator transition occurs..
MetalInsulator
Anderson transition
Sridhar,et.al
Kramer, et al.
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 LgLLDE 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
Altshuler, Introduction to mesoscopic
physics
0
1. Quantum chaos studies the 1. Quantum chaos studies the quantum properties of systems quantum properties of systems whose classical motion is chaotic. whose classical motion is chaotic. 2. More generally it studies the 2. More generally it studies the impact on the quantum dynamics of impact on the quantum dynamics of the underlying deterministic classical the underlying deterministic classical motion, chaotic or not.motion, chaotic or not.
Bohigas-Giannoni-Schmit Bohigas-Giannoni-Schmit conjectureconjecture
Classical chaos Wigner-Classical chaos Wigner-DysonDyson
Energy is the only integral of motionEnergy is the only integral of motion
Momentum is not a good quantum number Eigenfunctions Momentum is not a good quantum number Eigenfunctions delocalized delocalized
in momentum in momentum spacespace
What is quantum chaos?
Gutzwiller-Berry-Tabor Gutzwiller-Berry-Tabor conjectureconjecture
Integrable classical Integrable classical
motion motion Poisson Poisson statisticsstatistics
(Insulator)(Insulator)Integrability in d dimensions Integrability in d dimensions
d canonical momenta are conservedd canonical momenta are conserved
Momentum is a good quantum numberMomentum is a good quantum number
System is localized in momentum System is localized in momentum spacespace
Poisson statistics is also related to localisation but in momentum Poisson statistics is also related to localisation but in momentum spacespace
s
P(s)
Universality and its exceptionsUniversality and its exceptions
Bohigas-Giannoni-Schmit conjectureBohigas-Giannoni-Schmit conjecture
Exceptions:Exceptions: 1. Kicked systems1. Kicked systems
n
nTtVpH )()(2 )cos()( KV
Dynamical localization Dynamical localization in momentum spacein momentum space
2. Harper model2. Harper model3. Arithmetic 3. Arithmetic billiardbilliard
<p2
>
t
Classical
Quantum
Questions:Questions:
1. Are these exceptions relevant?1. Are these exceptions relevant?2. Are there systems not classically 2. Are there systems not classically
chaotic but still described by the chaotic but still described by the Wigner-Dyson?Wigner-Dyson?
3. Are there other universality class in 3. Are there other universality class in quantum chaos? How many?quantum chaos? How many?
4. Is localization relevant in quantum 4. Is localization relevant in quantum chaos?chaos?
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
g
0g
cgg
Main point of this talkMain point of this talk
Adapt the one parameter scaling theory in quantum chaos in order to:
1. Determine the universality class in 1. Determine the universality class in quantum chaos related to the metal-quantum chaos related to the metal-insulator transition.insulator transition.
2. Determine the class of systems in which 2. Determine the class of systems in which Wigner-Dyson statistics applies.Wigner-Dyson statistics applies.
3. Determine whether there are more 3. Determine whether there are more universality class in quantum chaos. universality class in quantum chaos.
How to apply scaling theory to How to apply scaling theory to quantum chaos?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 systems.systems.
2. Express the Hamiltonian in a finite 2. Express the Hamiltonian in a finite momentum basis and study the momentum basis and study the dependence of observables with the dependence of observables with the basis size N.basis size N.
3. For each system one has to map the 3. For each system one has to map the quantum chaos problem onto an quantum chaos problem onto an appropriate basis. For billiards, kicked appropriate basis. For billiards, kicked rotors and quantum maps this is rotors and quantum maps this is straightforward.straightforward.
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)( g
)()( gfg clas
weak weak localization?localization?
LWigner-Dyson Wigner-Dyson (g) (g) > 0> 0
Poisson Poisson (g) (g) < 0< 0
Lapidus, fractal billiards
eddLtq /2
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
tq 2
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, 3D Kicked rotor at critical coupling.=1, 3D Kicked rotor at critical coupling.
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 Anderson tran. Critical statistics = 0 Anderson tran. 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
AGG, Wang Jjiao, PRL 2005
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
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 in the understanding of universal features tool in the understanding 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
