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Photo made during the conference can be found on the web page (see album “Benasque”):. http://picasaweb.google.com/felix.izrailev. From closed to open 1D Anderson model:Transport versus spectral statistics. F.M.Izrailev Instituto de Física, BUAP, Puebla, México and - PowerPoint PPT Presentation
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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”):
Benasque, August 31, 2012
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),
Benasque, August 31, 2012
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)
Benasque, August 31, 2012
Benasque, August 31, 2012
Benasque, August 31, 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
Benasque, August 31, 2012
2D Coulomb gas on a circle F.M.Izrailev, 1990-91
F.Dyson, 1962
Benasque, August 31, 2012
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
Benasque, August 31, 2012
N
l 3.2
N
lx
Repulsion parameter is the properly normalized localization length !
Benasque, August 31, 2012
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
Benasque, August 31, 2012
Benasque, August 31, 2012
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
Benasque, August 31, 2012
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
Benasque, August 31, 2012
N
l2
Benasque, August 31, 2012