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