Download What is quantum chaos?

Universality in quantum chaos, Anderson
localization and the one parameter scaling theory
Antonio M. García-García
[email protected]
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, 244102 (2005),
PRE, 73, 374167 (2006).
Outline:
0. What is this talk about?
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
2.2 The one parameter scaling theory
3. Introduction to quantum chaos
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
4.2 Metal-Insulator transition in quantum chaos
(Simple) Quantum mechanics
Schrödinger equation + generic V(r)
Quantum coherence
beyond textbooks
Impact of classical chaos
in quantum mechanics
?
Quantum mechanics in
a random potential
Quantum Chaos
Disordered systems
1. Semiclassical techniques.
1. Powerful analytical techniques.
2. BGS conjecture.
2. Ensemble average.
Relevant for:
3. Anderson localization.
1. Quantum classical transition.
2. Nano-Meso physics. Quantum engineering.
3. Systems with interactions for which the exact
Schrödinger equation cannot be solved.
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
High energy nuclear excitations
-Shell model does not work
-Excitations seem to have no
pattern
P( s)    s  Ei 1  Ei  /  
P(s)
i
-Wigner carried out a statistical analysis of
these excitations.
s
- Surprisingly, P(s) and other spectral
correlator are universal and well described
by random matrix theory (GOE).
Random Matrix Theory:
Random matrix theory describes the
eigenvalue correlations of a matrix
whose entries are random
real/complex/quaternions numbers
with a (Gaussian) distribution.
Signatures of a RM spectrum (Wigner-Dyson):
1. Level Repulsion
Ps  ~ s e
β  As 2
2. Spectral Rigidity
s
Ei1  Ei

Σ2 L = n( L)  n( L) ~ log L L 
2
2
E f  Ei

 1
 = 1,2,4 for real,complex, quaternions
Signatures of an uncorrelated
spectrum (Poisson) :
2 (L)  L
P(s)  exp( s)
P(s)
s
In both cases spectral correlations are UNIVERSAL,
namely, independent of the chosen distribution. The
only scale is the mean level spacing .
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:
Questions:
Answers:
1. How do quantum effects modify
the transport properties of a particle
whose classical motion is diffusive?.
Dclast
Dquant
<x2>
Dquanta
t
The theory of disordered systems studies
a quantum particle in a random potential.
a=?
Dquan=f(d,W)?
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.
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, 73
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.
Your intuition about localization
Ea
V(x)
Eb
0
Ec
X
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?
Localisation according to
the one parameter scaling theory
Insulator (eigenstates localised)
Anderson transition
When? For d < 3 or, (or d > 3 for strong disorder).
Why? Caused by destructuve interference.
How? Diffusion stops, Poisson statistics and
discrete spectrum.
Metal (eigenstates delocalised)
When? d > 2 and weak disorder, eigenstates delocalized.
Why? Interference effects are small.
Kramer, et al.
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 metalinsulator transition occurs.
Sridhar,et.al
Insulator
Metal
Energy scales in a disordered system
1. Mean level spacing:
2. Thouless energy:
  1
ET  h / tT
tT(L) is the typical (classical) travel time
through a system of size L
Dimensionless
Thouless conductance
Diffusive motion
without quantum corrections
ET  
ET  
g  1
g  1
ET
g

2
ET  DL
Metal
Insulator
 L
d
d 2
gL
Wigner-Dyson
Poisson
Scaling theory of localization
The change in the conductance with the system
size only depends on the conductance itself
d log g
  (g)
d ln L
Beta function is universal but it depends on the global
symmetries of the system
Quantum
d 2
g  1 g  L
g  1 g  e  L / 
 ( g )  (d  2)   / g
 ( g )  log g  0
In 1D and 2D localization for any disorder
In 3D a metal insulator transition at gc , (gc) = 0
Weak
localization
0
Altshuler, Introduction to mesoscopic
physics
What is quantum
chaos?
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
Gutzwiller-Berry-Tabor conjecture
Integrable classical
motion
Integrability in d dimensions
Poisson
statistics
(Insulator)
P(s)
d canonical momenta are conserved
s
Momentum is a good quantum number
System is localized in momentum space
Poisson statistics is also related to localisation but in momentum space
Universality and its exceptions
Bohigas-Giannoni-Schmit conjecture
Exceptions:
1. Kicked systems
H  p  V ( )  (t  nT )
2
n
Classical
<p2>
Quantum
t
2. Harper model
3. Arithmetic billiard
V ( )  K cos( )
Dynamical localization
in momentum space
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
g 
Delocalized
wavefunctions
Wigner-Dyson
g 0
Localized
wavefunctions
Poisson
g  gc
Anderson
transition
Critical Statistics
Deterministic
Chaotic motion
Only?
Integrable motion
????????
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
q t
2

 L
 d / de
de is related to the fractal dimension of the spectrum.
The average is over initial conditions and/or ensemble
g ( L) 
ET

 clas
L
 clas
Lapidus,
fractal
billiards
d 2


de 
Universality
L 
Wigner-Dyson
(g) > 0
Poisson
(g) < 0
weak localization?
 ( g )   clas  f ( g )
Two routes to the Anderson transition
 (g)  0
 clas  0
 quan  clas
 quan  0
2. Induced by quantum effects  clas  0
1. Semiclassical origin
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
E or position). Condition
d: 2
g ( L) 
T

 cla s
L
 clas 
de

Wigner-Dyson statistics applies

0
q2  t
Anderson transition in non-random systems
Conditions:
1. Classical phase space must be homogeneous.
2. Quantum power-law localization.
3.
ET
 clas
g ( L) 
L

d 2
 clas    0
de 
q t
2

Examples:
1D:=1, de=1/2, Harper model, interval exchange maps (Bogomolny)
=2, de=1, Kicked rotor with classical singularities
2D: =1, de=1, Coulomb billiard (Altshuler, Levitov).
3D: =2/3, de=1, 3D Kicked rotor at critical coupling.
(AGG, WangJiao).
1D kicked rotor with singularities
H  p  V ( )  (t  nT )
2
n
Classical Motion
V ( )  K cos( )
k n 1  k n  V ' ( n )
 n 1   n  Tk n 1
Normal diffusion

V ( )   |  |
V ( )   log |  |
P(k , t )  1 / k

k
2
t
Anomalous Diffusion
2
 1
Quantum Evolution
2
2

T


T

Uˆ  exp(
) exp( iV ( ) / ) exp(
)
2
2
4 
4 
P(k , t )  1 / k
'
k
2
t
 '
1. Quantum
anomalous
diffusion
2. No dynamical
localization for
<0
g ( L) 
ET

 cla s
L
 clas  
q
2
t
2
 1
1.
>0
Localization
Poisson
2.
<0
Delocalization
Wigner-Dyson
3.
=0
Anderson tran. Critical statistics
AGG, Wang Jjiao, PRL 2005
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
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
  (0, t )  eit u (0, t )
Anderson
Model

1 2
i ( , t )  
( , t )  V ( ) (t  n)( , t )
2
t
2 
n
Tmum  Wr um r  Eum
Tm pseudo random
r 0
Wr 
1
r
 1
The associated Anderson model has long-range hopping depending
on the nature of the non-analyticity:
Explicit analytical results are possible, Fyodorov and Mirlin
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.
Skolovski, Shapiro, Altshuler
2.

P( s) ~ s
 As
P( s) ~ e
s  1
 3 ( n) ~
s  1
var
n
3. Eigenstates are multifractals.
  (r ) d r ~ L
2q
n
d
 Dq ( q 1)
Mobility edge
var  s 2  s
Anderson transition
2
s n   s n P(s)ds
V(x)= log|x|
 =15
 =8
 =4
 =2
Spectral
χ =0.026
χ =0.057
χ=0.13
χ=0.30
Multifractal
D2= 0.95
D2= 0.89
D2 ~ 1 – 1/
D2= 0.72
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
P(t ) ~ t  D
ANDERSON
TRANSITON IN
QUANTUM CHAOS
2
Ketzmerick, Geisel, Huckestein
3D kicked rotator
In 3D,
g  gc
for =2/3
V (1 , 2 ,3 )  k cos(1 ) cos( 2 ) cos( 2 )
Finite size scaling analysis
shows there is a transition
a MIT at kc ~ 3.3
2
p (t )
quan
2
p (t )
clas
~t
2/3
~t
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
N 
g 
3. The Anderson transition in quantum chaos is
related to N  
g  gc  
4. Experimental verification of the Anderson
transition is possible with ultracold atoms
techniques.