Multifractality of random
wavefunctions: recent progress
V.E.Kravtsov
Abdus Salam ICTP
Anderson transition
W /V
disorder
L
Extended states
Critical states
Localized
states
Multifractal wave functions
Map of the regions
with amplitude
larger than the
chosen level
L
L
Multifractal metal and insulator
Multifractal
insulator
Multifractal
metal
Quantitative description: fractal
dimensions and spectrum of
multifractality
Weight of the map
where wavefunction
amplitude |y|2 ~ L-a
is by definition Lf(a)
1
r | i (r ) |  L (q)
2q
L
L
f (a ) - qa
|
y
(
r
)
|
~
d
a
L


2q
r
df (a q )
da
 q;  (q)  qa q - f (a q )
Saddle-point approximation
-> Legendre transform
Weak and strong fractality
4D
3D
2+e
4D
3D
Dq = d – g q
metal
2+e
Weak
fractality
PDF of wave function amplitude
2



1
ln
|
y
|
2

P(| y | ) ~ d
exp ln L f  2
L |y |
ln L 


For weak multifractality
Log-normal distribution with
the variance ~ ln L
Altshuler, Kravtsov, Lerner,
1986
Symmetry relationship
 q   (q) - d (q - 1)
1
r | i (r) |  L (q)
2q
 q  1- q
Mirlin, Fyodorov, 2006
Gruzberg,Ludwig,Zirnbauer, 2011
Statistics of large and small
amplitudes are connected!
Unexpected consequence
2
2
|
y
(
r
)
|

1

|
y
|

1

|
y
|

|
y
|
(q  1)

2
2q
r
 | y (r ) |
 (q)  0 for q  1
2q
 L- ( q )  1, (q  1)
r
Small moments
exaggerate
small
amplitudes
L |y |
d
2q
min
For infinitely
sparse fractal
d
-2 d q
~ L (L
)

-d
L
, metal


| y |2min ~  e - L /  , insulator
 L- 2 d , sparse fractal


Supplement
(d q - d )( q - 1)  (d1- q - d )( -q)
d q 1/ 2  0
Dominated by
large
amplitudes
(d q - d )( q - 1)  -d (-q),
q  1/ 2
Dominated by small
amplitudes
 q  2d (q - 1 / 2)
Critical RMT: large- and small- bandwidth cases
Mirlin & Fyodorov, 1996
Kravtsov & Muttalib, 1997
Kravtsov & Tsvelik 2000
criticality
1
g ij  | H ij | 
| i - j |2
1
b2
2
d_2/d
fractality
Eigenstates are multifractal
at all values of b
2+e 1
3D
Anderson,
O class
0.6
d  ?
Weak
fractality
1/b
Strong
fractality
pbb =1.64
pbb=1.39
pbb=1.26
The nonlinear sigma-model and the dual representation
~
i
p

-1
Sigma-model: F  -(pg ) 2
g
StrLQi 
ij Str Qi Q j 

ij
i
2
Valid for b>>1

Q=ULU is a geometrically constrained
supermatrix:
gij  | H ij |2 
1
| i - j |2
1
b2
- functional:

Q 1
2
StrQ  0
Duality!
1
~
F  ij g ij Str Qi Q j - i i StrLQi  - iE i StrQi 
2
2


Convenient to expand in
small b for strong
multifractality
Q  y y
Virial expansion in the number of
resonant states
Gas of low density ρ
Almost diagonal RM
bΔ
ρ1
2-particle collision
2-level interaction
b1
Δ
ρ2
3-particle collision
3-level interaction
b2
Virial
expansion
as
re-summation
O.M.Yevtushenko,
A.Ossipov, V.E.Kravtsov
2003-2011
 1
 exp H nm
m n
 2
F2
F3

StrQn Qm   1   Vn(,2m)   Vn(,3m) ,l  ...

nm
n  m l
2
( 2)
n,m
V
e
- H nm
2
StrQnQm 
-1
Vn(,3m),l  Vn(,2m)Vn(,2l )Vm( 2,l)  Vn(,2m)Vn(,2l )  Vn(,2m)Vm( 2,l)  Vn(,2l )Vm( 2,l)
Term containing m+1 different matrices Q
gives the m-th term of the virial expansion
Virial expansion of correlation
functions

C (r ,  )   [C0  bC1 (r / b)  b C2 (r / b)  ...]
2
At the Anderson transition in d –dimensional space
rr
m
d
Each term proportional to b gives a result of
interaction of m+1 resonant states
Parameter b enters both as a parameter of
expansion and as an energy scale -> Virial
expansion is more than the locator expansion
Two wavefunction correlation:
ideal metal and insulator
 Vd
Metal:
Insulator:
d
r n (r ) m (r )
2
1
V V
V
V d
1
d
2
1
1
V
d
 
d

V
1

  1

Small amplitude
100% overlap
Large amplitude
but rare overlap
Critical enhancement of
wavefunction correlations
| E - E ' |-1 d 2 / d
Amplitude higher than in
a metal but almost full
overlap
States rather remote (d\E-E’|<E0) in energy
are strongly correlated
Another difference between sparse
multifractal and insulator wave functions
C (  En - Em )   Vd r n (r ) m (r )
d
2
2
 1
|  |1- d 2 / d , d 2  0 sparse fractal
C ( )  
d ( ), hard insulator


Wavefunction correlations in a
normal and a multifractal metal
E0 ~
D
D
~
Wc 16.5
C(E - E' ) 
V  d
d
r n (r ) m (r ) d ( En - E )d ( Em - E ' )
n ,m
 d (E
Wc
Wc - W
2
n
- E )d ( Em - E ' )
n,m
New length
scale l0,
new energy
scale
E0=1/ l0 3

2
Multifractal metal:  l 0
1- d 2 / d
 E0 


 E - E' 


Critical power law
persists
Normal metal:  l 0

d
E0
Density-density correlation function
D( r , t ) 

y n ( R)y n* ( R  r )y m ( R  r )y m* ( R) d ( En - E ) exp i( En - Em )t 
R ,n,m
D(r,t)
???
Return probability for multifractal
wave functions
P (t )  D ( r  1, t )
Kravtsov,
Cuevas,
2011
Numerical
result
t
Analytical
result
-d2 / d
Quantum diffusion at criticality and classical
random walk on fractal manifolds
Quantum critical case
D( r , t )  t
-d2 / d
d
f (r / t )
Random walks on fractals
D( r , t )  t
-dh / dw
f (r / t ), d w  2d h / d s
dw
Similarity of description!
Oscillations in return probability
Akkermans et al. EPL,2009
P(t )  D(r  0, t )
Classical random walk
on regular fractals
Analytical
result
Multifractal
wavefunctions
Kravtsov, Cuevas, 2011
Real experiments
 ( p, t )  P ( r , t )