Multifractality of random wavefunctions: recent progress
V.E.Kravtsov
Abdus Salam ICTP
Anderson transition
L
Critical states Localized states
Extended states
disorderVW /
Multifractal wave functions
Map of the regions with amplitude larger than the chosen level
LL
Multifractal metal and insulator
Multifractal metal
Multifractal insulator
)(2 1|)(|
qr
qi Lr
Quantitative description: fractal dimensions and spectrum of
multifractality
LL
Weight of the map where wavefunction amplitude L is by definition L
2
f
qfq
r
Ldr )(2~|)(|
)()(;)(
qqq fqqq
d
df
Saddle-point approximation -> Legendre transform
Weak and strong fractality
3D
4D
2+
4D
3D Dq = d – q
Weak fractality
2+metal
PDF of wave function amplitude
LfL
LP
d ln
||lnlnexp
||
1~)|(|
2
22
For weak multifractality
Log-normal distribution with the variance ~ ln L
Altshuler, Kravtsov, Lerner, 1986
Symmetry relationship
)1()( qdqq
qq 1
Mirlin, Fyodorov, 2006
Gruzberg,Ludwig,Zirnbauer, 2011
Statistics of large and small amplitudes are connected!
)(2 1|)(|
qr
qi Lr
Unexpected consequence
10)( qforq
)1(||||1||1|)(| 2222 qr q
r
)1(,1|)(| )(2 qLr qq
r
qddqd LLL )(~|| 22min
fractalsparseL
insulatore
metalL
d
L
d
,
,
,
~||2
/2min
Small q
shows that t
he sparse fr
actal is
differe
nt fro
m lo
calizatio
n by sta
tistic
ally
signifi
cant min
imal a
mplit
ude
Small moments exaggerate
small amplitudes
For infinitely sparse fractal
Supplement
))(()1)(( 1 qddqdd qq
02/1 qdDominated by
large amplitudes
)2/1(2
2/1
),()1)((
qd
q
qdqdd
q
q
Dominated by small amplitudes
Critical RMT: large- and small- bandwidth cases
2
22
||1
1||
bji
Hg ijij
criticality
fractality
d_2/d
1/b
1
Weak fractality
Strong fractality
0.63D
Anderson, O class
2+
Eigenstates are multifractal at all values of b
Mirlin & Fyodorov, 1996 Kravtsov & Muttalib, 1997
Kravtsov & Tsvelik 2000
?d
b =1.64
b=1.39
b=1.26
The nonlinear sigma-model and the dual representation
Q=UU is a geometrically constrained supermatrix: 12 Q 0StrQ
Convenient to expand in small b for strong
multifractality
Duality!
i ijiij
ij QStri
QQStrggF2
~)( 12 Sigma-model:
Valid for b>>1
functional: i ii ijiij ij QStriEQStriQQStrgF
2
~
2
1
Q
2
22
||1
1||
bji
Hg ijij
Virial expansion in the number of resonant states
2-particle collision
Gas of low density ρ
3-particle collision
ρ1
ρ2
Almost diagonal RM
b12-level interaction
Δ
bΔ
b23-level interaction
Virial expansion as re-summation
...12
1exp )3(
,,)2(
,
2
lmnlmn
mnmnmnnm
nmVVQQStrH
1
2
)2(,
mnnm QQStrH
mn eV
)2(,
)2(,
)2(,
)2(,
)2(,
)2(,
)2(,
)2(,
)2(,
)3(,, lmlnlmmnlnmnlmlnmnlmn VVVVVVVVVV
F2 F3
Term containing m+1 different matrices Q gives the m-th term of the virial expansion
O.M.Yevtushenko, A.Ossipov, V.E.Kravtsov
2003-2011
Virial expansion of correlation functions
...])/()/([),( 22
10 brCbbrbCCrC
Each term proportional to gives a result of interaction of m+1 resonant states
mb
Parameter b enters both as a parameter of expansion and as an energy scale -> Virial
expansion is more than the locator expansion
At the Anderson transition in d –dimensional space drr
Two wavefunction correlation: ideal metal and insulator
22)()( rrrVd mn
d
Metal: 111
VVVV Small
amplitude 100% overlap
Insulator:
111
VV
d
ddd
Large
amplitude but rare overlap
Critical enhancement of wavefunction correlations
ddEE /1 2|'|
Amplitude higher than in a metal but almost
full overlap
States rather remote \E-E’|<E0) in energy are strongly correlated
Another difference between sparse multifractal and insulator wave functions
22)()()( rrrVdEEC mn
dmn
insulatorhard
fractalsparsedC
dd
),(
0,||
1
)(2/1 2
Wavefunction correlations in a normal and a multifractal metal
Normal metal: l
0
0E
New length
scale l0, new
energy scale
E0=1/l0
3dd
EE
E/1
0
2
'
Multifractal metal: l
0
Critical power law persists
WW
W
c
c
mnmn
mnmnmn
d
EEEE
EEEErrrdV
EEC
,
22
,
)'()(
)'()()()(
)'(
5.16~~0
D
W
DE
c
???
D(r,t)
mnR
mnnmmnn tEEiEERrRrRRtrD,,
** )(exp)()()()()(),(
Density-density correlation function
Return probability for multifractal wave functions
),1()( trDtP
ddt /2
Kravtsov, Cuevas,
2011
Analytical result
Numerical result
Quantum diffusion at criticality and classical random walk on fractal manifolds
)/(),( /2 trfttrD ddd
Quantum critical case
Random walks on fractals
shwddd dddtrfttrD wwh /2),/(),( /
Similarity of description!
Oscillations in return probabilityAkkermans et al. EPL,2009
),0()( trDtP
Classical random walk on regular fractals
Multifractal wavefunctions
Kravtsov, Cuevas, 2011
Analytical result
Real experiments
),(),( trPtp