Benasque, August 31, 2012 Photo made during the conference can be found on the web page (see album...

Benasque, August 31, 2012

http://picasaweb.google.com/felix.izrailev

Photo made during the conference can be found on the web page (see album “Benasque”):


From closed to open 1D Anderson model:Transport versus spectral statistics

F.M.IzrailevInstituto de Física, BUAP, Puebla, México

andMichigan State University, USA

V. G. Zelevinsky (NSCL, E.Lansing, USA)

In collaboration with:

S. Sorathia (IFUAP, Puebla, Mexico) , G.Celardo (Univesita di Cattolica, Brescia, Italy),


Open 1D Anderson model

Ideal semi-infinite LEFT lead

Ideal semi-infinite RIGHT lead

• How scattering properties depend on the degree of internal chaos and coupling strength ?

Published in Phys. Rev. E (2012)


Level statistics

sssBAssP f

21

216exp1 2

2

nnn EEs 1~

n

nn s

ss ~

~

sP

Spacing between neighbouring eigenvalues

Spacing normalized to mean local spacing

Level spacing distribution - provides information about the degree of chaos

Ensemble average combined with average over eigenvalues that lie within a small energy window at the band centre

1 ssP

Regular

picket fence

4

exp2

2sssP

Chaotic

Wigner-Dyson

ssP exp

Localized

Poisson

011

16874.0

212

f

Phenomenological spacing distribution interpolates between all three regimes

2D Coulomb gas on a circle

repulsion parameter


2D Coulomb gas on a circle F.M.Izrailev, 1990-91

F.Dyson, 1962


Distribution of spacings between eigenvalues in the 1D Anderson model is the same as that for spacings between classical charged particles in the Coulomb gas model !

Chaotic

Poisson

Delta function

N

lx


N

l 3.2

N

lx

Repulsion parameter is the properly normalized localization length !


c

cn

cm

mnmn iEE

EAEAEdHE

0H

2

2

2

,1, 0

2/11

iEE

EEdH Nn

Rn

Lmn

Non-Hermitian Effective Hamiltonian

EEiEEiEE

1

0

1 22

2

2

)2/(120

2/11Ei

E

iEE

EEd

N,nR

,nL

mnmn )/E(iE

HE

1

2 22124

H

c

cn

cmmnmnmn AAWW

iH 002;

2mnH

using the identity

The exact non-Hermitian effective Hamiltonian

Near the centre of the energy band this reduces to:

c

ncnA 1,0


ND

2 RL - Coupling parameter for

- Transmission coefficient2LRST

Average transmission and variance

22 TTTVar - Variance of the transmission

N

lz

z

dzz

l

N

N

lT

2exp

cosh2exp

2 2

0

22/3

N

lz

z

dzzzz

l

N

N

lT

2exp

cosh4

2sinh2

2exp

2 2

0 3

22/32

Analytical expressions for 1D continuous (i.e. not discrete)

disordered media

Excellent agreement between discrete Anderson model and

formally exact analytical results for continuous system


General case

21

42

2

lnl

NT ln

for any coupling:

One can introduce an effective localization length:

111

llef

lwith

where

efl

NT ln

2

4

11 21 ln

Nl


N

l2

Benasque, August 31, 2012 Photo made during the conference can be found on the web page (see album...

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