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

Universality in quantum chaos and the one
parameter scaling theory
Antonio M. García-García
[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 Wang Jiao PRL 94, 244102 (2005), PRE, 73, 374167 (2006)
Understanding localization and universality
1. Anderson’s paper (1958). Locator expansion.
2. Abu Chacra, Anderson (1973). Self consistent theory.
Exact only for d going to infinity.
3. Vollhardt and Wolffle (1980-1982). Self consistent
theory for the 2 point function. Only valid for d =2+
or far from the transition. Consistent with previous
renormalization group arguments (Wegner).
4. One parameter scaling theory (1980), gang of four.
Insulator: For d < 3 or, in d > 3 for strong disorder. Diffusion eventually stops.
Discrete spectrum
Metal:d > 2 and
weak disorder. Quantum effects do not alter
significantly the classical diffusion (weak localization). Continous spectrum
Anderson transition: For d > 2 a metal-insulator transition takes place.
Can we apply this to deterministic systems?
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
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
d 2


de 
Universality
L 
Wigner-Dyson
clas > 0
Poisson
clas < 0
weak localization?
 ( g )   clas  f ( g )
Two routes to the Anderson transition
 ( gc )  0
 clas  0
 quan  clas
1. Semiclassical origin

0
 quan  0
2. Induced by quantum effects clas
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 oscillators
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. 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
Anderson transition in non-random systems
Conditions:
1. Classical phase space must be homogeneous.
2. Quantum power-law localization.
3.
E
d 2
g ( L) 
T

 clas
L
 clas 
de


0
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).
(AGG, WangJiao).
3D: =2/3, de=1, Kicked rotor at critical coupling, kicked rotor 3
incommensurate frequencies (Casati,Shepelansky).
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
MIT transition
Critical statistics
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
For a KR with 3 incommensurable frequencies
see Casati, Shepelansky, 1997
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 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
How to apply this to quantum chaos?
1. Only for classical systems with an
homogeneous phase space. Not mixed
phase space.
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 billiards, kicked rotors and quantum
maps this is straightforward.
Classical-Quantum diffusion
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!
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
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
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)
Altshuler, Introduction to mesoscopic
physics