Spectral and Wavefunction Statistics (II)

V.E.Kravtsov,Abdus Salam ICTP

Wavefunction statistics Porter-Thomas distribution

Wavefunction statistics in one-dimensionalAnderson model

Weak multifractality in 2D disordered conductors

Wavefunction statistics and Coulomb peaks heights

RL

RLg

rdr

FERL

2

)( |)(|Contact

area

Bunching

Porter-Thomas distribution

-follows from random matrix theory

-describes local distribution of wavefunction intensity in chaotic systems

-fails to describe the “web” pattern

The Anderson model

arrrrr VH ,','

ˆ

i

)( if

2/W 2/W

1D: all states are localized with localization length W

3D: Anderson localization transition at W=16.5

2D: all states are localized with exponentially large localization radius ~exp(-a/W )

2

2

Distribution of eigenfunction amplitude for 1D Anderson model

yyL

yyP exp)||()( 2)1(

Porter-Thomas:

Take =2then the distribution for 1D Anderson model can be considered as limit of the Porter-Thomas distribution.

Poisson distribution as a of the Wigner surmise

2

2

16exp

]exp[ No surprise that the limit of Porter-Thomas

gives the distribution of localized functions in one

dimension.

Distribution of wavefunction amplitude in 2D conductors (L

)/(ln4

1exp)( 2 gxgxP

1)/ln(/2|| 0222 lLDgLx

)(ln xP

x

Porter-Thomas

Porter-Thomas with corrections

g g

Log-normal

Sample dimension from local measurement

2|)(| r

2DQuasi 1D

2|)(| rx

)(ln xP Porter-Thomas

Porter-Thomas with corrections

g g

Log-normal in 2D:

Stretch-exponential in quasi-1D

)/(ln4

1exp)( 22 gxLgxP

]2exp[ xA

Zero-dimensional

quantum dot

V2||

For g>>1 RMT behavior for a typical wavefunction

Dimension-specific behavior for large

amplitudes

Where the RMT works

)(ln xP Porter-Thomas

Porter-Thomas with corrections

g gV2||

1,/

2),/ln(/)/ln()/ln(/2 )(0

2

dL

dlLllLDg

o

g>>1For dynamic

phenomena g

D

LLgg ),()(Diffusion displacement for time 1/

qqdq

gq

qqq LLLx

2

0

12

22||

Weak multifractality in 2D conductors

)/(ln4

1exp)( 2 gxg

x

dxdxxP 22|| Lx

0

2g

qdq

12 0

20 Dg

2=dimensionality of space

q-dependent multifractal dimensionality

Magnetic field makes fractality

weaker

Fractal dimension of this map decreases with increasing the level=“multi”-fractality

Multifractality: qualitative picture

Map of the regions where exceeds the chosen

level

Arbitrary chosen level Weight of the

dark blue regions scales

like

HdLHd