Download Anderson transition ???????? Critical Statistics

Universality in quantum chaos 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 PRL 94, 244102 (2005), PRE 2007 in press
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
Kramer, et al.
Anderson transition
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.
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  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
Altshuler, Introduction to mesoscopic
physics
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
1. Semiclassical origin

0
 quan  0
2. Induced by quantum effects clas
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.
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?
Random
g 
g 0
g  gc
QUANTUM
Delocalized
wavefunctions
Wigner-Dyson
Localized
wavefunctions
Poisson
Anderson
transition
Deterministic
Chaotic motion
Only?
Integrable motion
????????
Critical Statistics
Is it possible to define new universality class ?
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
N 
g 
Wigner-Dyson statistics applies
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
ET
 cla s
g ( L) 
L

Examples:
 clas
d 2

 0
de 
q
2
t
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

1D kicked rotor with singularities
H  p  V ( )  (t  nT )
2
k n 1  k n  V ' ( n )
 n 1   n  Tk n 1
n
Classical Motion
V ( )  K cos( )
Classical diffusion

V ( )   |  | V ( )   log |  | V ( ) 
Step
function
Classical Anomalous Diffusion
P( k , t )  1 / k

k  t
Quantum Evolution
2
2

T


T

Uˆ  exp( 
) exp( iV ( ) / ) exp(
)
2
2
4 
4 
Power-law localization
| (ri ) |
1
ri
 1
g ( L) 
ET

 cla s
L
 clas  
q
2
t
2
 1
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
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
P( k , t )  1 / k
Quantum Output

k  t
(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!
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.
V ( )  K cos( )

1 2
i ( , t )  
( , t )  V ( ) (t  n)( , t )
2
t
2 
n
Kicked rotor
Anderson model
Tm    Tm 2
T m um
r 0
W r um
r
E um
What happens if
it
  (0, t )  e u (0, t )

V ( )   |  | V ( )   log |  | V ( )  step
W ( )   tan(V ( ) / 2) Wm   W exp( im )d
Is there any relation between non-differentiability and Wr? Yes
Non-differentiability induces long range hopping
The associated Anderson model has long-range hopping
depending on the nature of the non-analyticity:
Wr 
1
r
 1
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
2
sin 2 (s)
Similar to Critical statistics . R2 ( s)   4 2 sinh 2 ( 2 s 2 )
2. Step like singularity Wr ~Aij/r withA  sin( (i  j ) / 2)
ij
Semi-Poisson statistics (Harper model, pseudo integrable
billiards) Exact treatment possible AGG PRE 2006
Experimental verification by using ultra-cold atoms techniques
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
H n  En n
1. Eigenvector statistics: P
q
Dq = d Metal
( q 1) d
L

Dq = 0 Insulator
2. Eigenvalue statistics:
2q
n
d
( q 1)( d  Dq )
(r ) d r ~ L
Dq = f(d,q) M-I transition
P( s)    s  i 1  i  /  
Level Spacing distribution:
i
 ( L  i   j / ) = n( L)  n( L)
2
Number variance:
Insulator  2 ( L)  L

( Poisson ) P( s)  e s
2
2
  2 ( L) ~ log L
Metal

β  As2
(Wigner  Dyson ) Ps  ~ s e
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
2
 ( n) ~
s  1
var
n
3. Eigenstates are multifractals.
  (r ) d r ~ L
2q
d
 Dq ( q 1)
var  s  s
2
2
s n   s n P(s)ds
n
Mobility edge
Anderson transition
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. Sub-Poisson Number variance
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 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
CONCLUSIONS
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
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.
ANDERSON TRANSITION
Main:Non trivial interplay between tunneling and interference leads to the
metal insulator transition (MIT)
Spectral correlations
Wavefunctions
Scale invariance
2
Multifractals
L
L
L 1
1
P s s s 1
P( s)  e  As s  1
Skolovski, Shapiro, Altshuler
  (r ) d r ~ L
2q
d
 Dq
n
Quantum Anomalous
diffusion P(k,t)~ t-D2
CRITICAL STATISTICS
Kravtsov, Muttalib
97
Density of Probability
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
Classical-Quantum diffusion
ANDERSON-MOTT TRANSITION
Main:Non trivial interplay between tunneling and interference leads to the
metal insulator transition (MIT)
Spectral correlations
Wavefunctions
Scale invariance
Multifractals
 (n) ~ n

P( s) ~ s
s  1
P( s) ~ e
s  1
2
  (r ) d r ~ L
2q
d
 Dq ( q 1 )
n
 As
Skolovski, Shapiro, Altshuler
CRITICAL STATISTICS
Kravtsov, Muttalib
97
"Spectral correlations are universal, they depend only on the
dimensionality of the space."
Mobility edge Mott
Anderson transition
Multifractality
Intuitive: Points in which the modulus of the wave function
is bigger than a (small) cutoff M. If the fractal dimension
depends on the cutoff M, the wave function is multifractal.
Formal: Anomalous scaling of the density moments.
I p =  ψ n r 
2p
r
I p n r 2p L
r
Dp p 1
Dp
L
Kravtsov, Chalker 1996
POINCARE SECTION
P
X
Is it possible a MIT in 1D ?
Yes, if long range hopping is permitted
H i   i i    ij F (i  j ) j
 i ,  ij [1,1]
j
Eigenstates power-law localized
Thermodynamics limit:

h
F (i  j ) 
| i  j |
|  (ri ) |
Eigenstates
1
ri

i  j  1
ri  1
Spectral
1
1
Multifractal
Critical statistics
Localized
Poisson statistics
1
Delocalized
Random Matrix
Analytical treatment by using the supersymmetry method (Mirlin
&Fyodorov)
Related to classical diffusion operator.
Eigenfunction characterization
1. Eigenfunctions moments:
1 Insulator
IPR    (r ) d r ~ 
Metal
V
4
d
1
n
2. Decay of the eigenfunctions:
e

 (r ) ~ 1 / V
 1/ r 

 r /
n
Insulator
Metal  d

  d
?
 d

Insulator
Critical
Metal
Looking for the metal-insulator
transition in deterministic Hamiltonians
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
Return Probability
A A
 W  =
A A
RMT
P
RMT
A =  s Ps ds

2
0
V(q) = log (q)
t = 50
CLASSICAL
V(q)= 10 log (q)