Transcript Chalker-Coddington network model and its applications to various quantum Hall systems V. Kagalovsky Sami Shamoon College of Engineering Beer-Sheva Israel INI Mathematics and Physics of Anderson localization:

Chalker-Coddington network model
and its applications to various
quantum Hall systems
V. Kagalovsky
Sami Shamoon College of Engineering
Beer-Sheva Israel
INI
Mathematics and Physics of Anderson localization: 50 Years After
Delocalization Transitions and Multifractality
2 November to 6 November 2008
Context
Integer quantum Hall effect
Semiclassical picture
Chalker-Coddington network model
Various applications
Inter-plateaux transitions
Floating of extended states
New symmetry classes in dirty superconductors
Effect of nuclear magnetization on QHE
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Inter-plateaux transition is a critical phenomenon
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In the limit of strong magnetic field
electron moves along lines of constant potential
Scattering in the
vicinity of the
saddle point
potential
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Transmission
probability
T
1
1 exp(- )
Percolation + tunneling
The network model of Chalker and Coddington. Each node
represents a saddle point and each link an equipotential line of
the random potential (Chalker and Coddington; 1988)
z1 z4
z3
z2
 Z1 
 Z4 
   M  
 Z2 
 Z3 
 e i
M  
 0
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1
0  cosh

i 
e  sinh 
2
sinh   e i

cosh  0
3
0 

i 
e 
4
Crit. value argument
Fertig and Halperin, PRB 36, 7969 (1987)
Exact transmission probability through the
saddle-point potential VSP U (x2  y2)V0
T
1
1 exp(- )
  (E (n 1/2)E2 V0)/ E1
E2  c
U
E1  m2
c
for strong magnetic fields
For the network model
T
1
cosh2
 2 ln(sinh )
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Total transfer matrix T of the system is a result of N iterations.
Real parts of the eigenvalues are produced by diagonalization
of the product
 1N

e

1/2
(T †T )
M – system width

















2 N
e





M N

2


M N

2



2 N


1N 

e
e
e
e
Lyapunov exponents
1>2>…>M/2>0
Localization length for the system of width M M is related to the
smallest positive Lyapunov exponent:
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M ~ 1/M/2
Loc. Length explanation
Renormalized localization length as function of energy and
system width
0.9
0.9
0.8
0.8
M
M
 const
0.7
0.7
M=16
M=32
M=64
M=128
0.6
0.6
0.5
0.5
M/M

0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0
0
0.0
2
0.2
4
0.4
6
0.6


One-parameter scaling fits data
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8
0.8
 M ( )
M
10
1.0
 M 
 f

  ( ) 
The thermodynamic localization length is then defined as function
of energy and diverges as energy approaches zero
 ( ) ~|  |
Main result
  2.5  0.5
in agreement with experiment and
other numerical simulations
Is that it?
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Generalization: each link carries two channels.
Mixing on the links is unitary 2x2 matrix
 e cos
U  e 
 i
  e sin 
i
i
e sin  

 i
e cos 
i
Lee and Chalker, PRL 72, 1510 (1994)
Main result – two different critical energies even for the
spin degenerate case
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One of the results: Floating of extended states

Landau level
 (B)
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PRB 52, R17044 (1996)
V.K., B. Horovitz and Y. Avishai
General Classification:
Altland, Zirnbauer, PRB 55 1142 (1997)
S
N
S
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Compact form of the Hamiltonian
 


 




T  


c

h

1
†
ˆ c c
H

†


2
 h c 





The 4N states are arranged as (p,p,h,h)
Four additional symmetry classes: combination of time-reversal
and spin-rotational symmetries
Class C – TR is broken but SROT is preserved – corresponds to
SU(2) symmetry on the link in CC model (PRL 82 3516 (1999))
Renormalized localization length
M ( ,)
M
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with







 f  M 1 ,M

1 





 1.12,  1.45
Unidir. Motion argument
At the critical energy
M
M
 f   M , M











 const
and is independent of M, meaning the ratio between two
variables is constant!
Energies of extended states

c()c 
 xy  0
 xy 1
Spin transport
 xy  2
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PRL 82 3516 (1999) V.K., B. Horovitz, Y. Avishai, and J. T. Chalker
Class D – TR and SROT are broken
Can be realized in superconductors with a p-wave
spin-triplet pairing, e.g. Sr2RuO4 (Strontium Ruthenate)
The A state (mixing of two different representations) –
total angular momentum Jz=1

broken time-reversal symmetry
Triplet
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
broken spin-rotational symmetry
y
   kx  ik y 

k1 k 2

θ
p-wave
x
k 2 0  cos(k 2)  i sin(k 2) 
k1 0  cos(k1) isin(k1) 
k1  k 2
SNS with phase shift π
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θ
0  cos(k1)  i sin(k1) 
only for

  90
S
N
S
there is a bound state
Chiral edge states imply QHE (but neither charge nor spin) –
heat transport with Hall coefficient
2 2kB2
K xy 
Ratio Kxy /T is quantized
3h
Class D – TR and SROT are broken – corresponds to O(1)
symmetry on the link – one-channel CC model with
phases on the links (the diagonal matrix element )







l  0 with probability W
 with probability 1-W
The result:
M 0 !!!
M=2 exercise






cosh sinh 1 0 1  e A 1
A
sinh cosh 0 A 1






























After many iterations
... e[( A AB ABC...) ]
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











1
ABC...






cosh sinh 1 0  1   e A  1 


sinh cosh   0 A 1
  A







After many iterations

... e[( A AB ABC...) ] 

1 
  ABC...




After many iterations there is a constant probability 
for ABC…=+1, and correspondingly 1-  for the value -1.
Then: W+(1- )(1-W)= 
=1/2 except for W=0,1
Both eigenvectors have EQUAL probability , and their
contributions therefore cancel each other leading to
 =0
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Change the model
Node matrix  cosh A sinh A






sinh A cosh A






Cho, M. Fisher PRB 55, 1025 (1997)
Random variable A=±1 with probabilities W and 1-W respectively






cosh A sinh A
sinh A cosh A
 
 

 
 
 
1 0 cosh sinh 1 0
0 A sinh cosh 0 A


















Disorder in the node is equivalent to correlated
disorder on the links – correlated O(1) model
M=2 exercise






cosh A sinh A 1  e A 1
1
sinh A cosh A 1
























=0 only for <A>=0, i.e. for W=1/2
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Sensitivity to the disorder realization!
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0.6
0.4
xy=1
METAL
0.2

Heat transport
0.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
W
-0.2
xy=0
-0.4
-0.6
PRB 65, 012506 (2001)
J. T. Chalker, N. Read, V. K., B. Horovitz, Y. Avishai, A. W. W. Ludwig
I. A. Gruzberg, N. Read, and A. W. W. Ludwig, Phys. Rev. B 63, 104422 (2001)
A. Mildenberger, F. Evers, A. D. Mirlin, and J. T. Chalker, Phys. Rev. B 75, 245321 (2007)
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Another approach to the same problem
M/M
M/M
M =16
M = 32
M = 64
M = 128
M = 256
2
M =16
M = 32
M = 64
M = 128
M = 256
2
(a)
(b)
1
1
0
0
0.0
0.2
0.4
0.6
0.8
1.0

W=0.1 is fixed
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0
5
10
15
20
25
1/
M
=1.4
30
35
M/M
M/M
2
2
M = 16
M = 32
M = 64
M = 128
M = 16
M = 32
M = 64
M = 128
(b)
(a)
1
0.00
1
0.05
0.10
0.15
W
=0.1 is fixed
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0
1
2
3
4
5
1/
W-0.2|M
=1.4
6
M/M
7
M=16
M=32
M=64
M=128
6
5
4
3
2
1
0
0
1
2
1/
W-0.19|M
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=1.4
3
PRL 101, 127001 (2008)
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V.K. & D. Nemirovsky
0.6
Pure Ising transition
=1 >1
0.4
xy=1
METAL
0.2

0.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
W
-0.2
xy=0
-0.4
-0.6
A. Mildenberger, F. Evers, A. D. Mirlin,
and J. T. Chalker,
Phys. Rev. B 75, 245321 (2007)
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W≡p
For W=0.1 keeping only higher M systems causes a
slight increase in the critical exponent  from 1.4 to
1.45 indicating clearly that the RG does not flow
towards pure Ising transition with =1, and
supporting (ii) scenario: W=0.1>WN
In collaboration with Ferdinand Evers
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W=0.02
M/M
3
M=16
M=32
M=64
M=128
2
1
0
0.00
0.05
0.10
0.15
0.20

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0.25
0.30
W=0.02
M/M
=1.29
3
M=16
M=32
M=64
M=128
2
1
5
10
1/1.29
M
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W=0.02
M/M
1.8
=1.09
1.6
1.4
1.2
M=32
M=64
M=128
1.0
0.8
0.6
0.4
0.2
0.0
0
5
10
15
20
1/1.09
M
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25
30
W=0.02
RG flows towards the pure
Ising transition with =1!
M/M
1.0
=1
0.8
0.6
W=0.02<WN
M=64
M=128
0.4
0.2
0.0
0
5
10
15
20
1/11
M
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25
30
35
40
W=0.04
M=16, 32, 64, 128 =1.34
M=32, 64, 128 =1.11
M=64, 128 =0.97
RG flows towards the pure
Ising transition with =1!
W=0.04<WN
We probably can determine the exact position of the
repulsive fixed point WN and tricritical point WT?
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Back to the original network model
Height of the barriers fluctuate - percolation
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Random hyperfine fields
Hint  n Ii He
Nuclear spin
Magnetic filed produced by electrons


8

H e   g   se  re  Ri 


e 3
Additional potential

Vhf  B Bhf
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Nuclear spin relaxation
Spin-flip in the vicinity of long-range impurity
S.V. Iordanskii et. al., Phys. Rev. B 44, 6554 (1991)
Yu.A. Bychkov et. al., Sov. Phys-JETP Lett. 33, 143 (1981) ,
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First approximation – infinite barrier with probability p
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If p=1 then 2d system is broken into M 1d chains
All states are extended independent on energy
Lyapunov exponent =0 for any system size as
in D-class superconductor
Naive argument – a fraction p of nodes is missing,
therefore a particle should travel a larger distance
(times 1/(1-p)) to experience the same number of
scattering events, then the effective system width
is M(1-p)-1 and the scaling is
M
M
 1 p 
1
f
 M 


  ( ) 
But “missing” node does not allow particle to
propagate in the transverse direction. Usually M~M,
we, therefore, can expect power >1
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M/M
20
M=16
M=32
M=64
18
16
14
12
10
8
6
4
2
0
0.0
0.2
0.6
0.4
0.8
1.0
p
Renormalized localization length at critical energy
=0 as function of the fraction of missing nodes p for
different system widths. Solid line is the best fit
1.24(1-p)-1.3. Dashed line is the fit with "naive" exponent
=1
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(1-p)1.3/M
M=16 p=0.5
M=32 p=0.5
M=64 p=0.5
M=16 p=0
M=32 p=0
M=64 p=0
M=16 =0.3
M=32 =0.3
M=64 =0.3
M=16 =0.5
M=32 =0.5
M=64 =0.5
M=16 =0
M=32 =0
M=64 =0
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
10
12
14
16
18
20
22
24
26
M2.5
Data collapse for all energies , system widths
M and all fractions p≠1 of missing nodes
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The effect of directed percolation can be responsible
for the appearance of the value ≈1.3.
By making a horizontal direction preferential, we have
introduced an anisotropy into the system.
Our result practically coincides with the value of
critical exponent for the divergent temporal correlation
length in 2d critical nonequilibrium systems, described
by directed percolation models
H. Hinrichsen, Adv. Phys. 49, 815 (2000)
G. Odor, Rev. Mod. Phys. 76, 663 (2004)
S. Luebeck, Int. J. Mod. Phys. B 18, 3977 (2004)
It probably should not come as a surprise if we
recollect that each link in the network model can be
associated with a unit of time
C. M. Ho and J. T. Chalker, Phys. Rev. B 54, 8708 (1996).
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Thanks to Ferdinand Evers
Scaling
M
M
 cl  4 / 3
 1 p






 cl
f  M  q 


 q  2.5
The fraction of polarized nuclei p is a relevant parameter
PRB 75, 113304 (2007)
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V.K. and Israel Vagner
Summary
Applications of CC network model
QHE – one level – critical exponents
QHE – two levels – two critical energies – floating
QHE – current calculations
QHE – generalization to 3d
QHE - level statistics
SC – spin and thermal QHE – novel symmetry classes
SC – level statistics
SC – 3d model for layered SC
Chiral ensembles
RG
QHE and QSHE in graphene
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