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Chalker-Coddington network model Chalker-Coddington network model and its applications to various and its applications to various

quantum Hall systemsquantum Hall systems

V. KagalovskyV. Kagalovsky

Sami Shamoon College of Engineering Sami Shamoon College of Engineering

Beer-Sheva IsraelBeer-Sheva Israel

  

  

  

  

  

  

  

Delocalization Transitions and Multifractality November to 6 November 2008 2

Mathematics and Physics of Anderson localization: 50 Years After

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ContextContext

Integer quantum Hall effectInteger quantum Hall effectSemiclassical pictureSemiclassical pictureChalker-Coddington network modelChalker-Coddington network modelVarious applicationsVarious 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 thevicinity of the saddle point

potential

Transmission probability

11 exp(- )

T

Percolation + tunneling

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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

z2z3

z4

2

4

3

1

Z

Z

Z

ZM

4

3

2

1

0

0

coshsinh

sinhcosh

0

0

i

i

i

i

e

e

e

eM

Crit. value argument

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Fertig and Halperin, PRB 36, 7969 (1987)Exact transmission probability through the saddle-point potential

11 exp(- )

T

2 20

( )SPV U x y V

2 0 1( ( 1/2) )/E n E V E

2 cE 12cUE m for strong magnetic fields

For the network model

2

1cosh

T

2ln(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

M – system width Lyapunov exponents

1>2>…>M/2>0

Localization length for the system of width M M is related to the

smallest positive Lyapunov exponent:

M ~ 1/M/2

2

2

2

1

1/2

2

1

†( )M

M

N

N

N

N

N

N

e

e

ee

ee

T T

Loc. Length explanation

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Renormalized localization length as function of energy and system width

constM

M

One-parameter scaling fits data for different M on one curve

( )

( )M M

Mf

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

M=16 M=32 M=64 M=128

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

M/M

M=16 M=32 M=64 M=128

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( ) ~| |

Main result in agreement with experiment and other numerical simulations2.5 0.5

The thermodynamic localization length is then defined as function of energy and diverges as energy approaches zero

Is that it?

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Generalization: each link carries two channels.

Mixing on the links is unitary 2x2 matrix

cossin

sincosU

ii

iii

ee

eee

Lee and Chalker, PRL 72, 1510 (1994)

Main result – two different critical energies even for thespin degenerate case

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One of the results: Floating of extended states

(B)

Landau level

PRB 52, R17044 (1996) V.K., B. Horovitz and Y. Avishai

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General Classification :Altland, Zirnbauer, PRB 55 1142 (1997)

N

S

S

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Compact form of the Hamiltonian

††Tc1H c c

2 chh

The 4N states are arranged as (p,p,h,h)

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

11( , ),M f M M

M

with 1.12, 1.45

Four additional symmetry classes: combination of time-reversal and spin-rotational symmetries

Unidir. Motion argument

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Energies of extended states ( )c c

At the critical energy const,M f M MM

and is independent of M, meaning the ratio between two variables is constant!

Spin transport

0xy

1xy

2xy

PRL 82 3516 (1999) V.K., B. Horovitz, Y. Avishai, and J. T. Chalker

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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 broken spin-rotational symmetry

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θ θ

p-wave x

y

01 1 1cos( ) sin( )

k k ki

02 2 2

0 1 1

cos( ) sin( )

cos( ) sin( )k k k

k k

i

i

1 2k k only for 90

SNS with phase shift π

there is a bound state

x yk ik

1 2k k

Chiral edge states imply QHE (but neither charge nor spin) – heat transport with Hall coefficient 2 22

3B

xyk

Kh

/xyK TRatio is quantized

S N S

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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 )

0 with probability W with probability 1-Wl

The result: 0M !!

!M=2 exercise

cosh sinh 1 0 1 10 1sinh cosh

AeA A

After many iterations

[( ...) ] 1......

A AB ABCeABC

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cosh sinh 1 0 1 10 1sinh cosh

AeA A

After many iterations

[ ( ...) ] 1......

A AB ABCeABC

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

cosh sinh 1 0 cosh sinh 1 00 0sinh cosh sinh cosh

A AA AA A

Node matrix cosh sinhsinh cosh

A AA A

Cho, M. Fisher PRB 55, 1025 (1997)

Random variable A=±1 with probabilities W and 1-W respectively

Disorder in the node is equivalent to correlated disorder on the links – correlated O(1) model

cosh sinh 1 11 1sinh cosh

AA A eA A

M=2 exercise

=0 only for <A>=0, i.e. for W=1/2Sensitivity to the disorder realization!

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0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

ME

TA

L

xy=0

xy

=1

W

PRB 65, 012506 (2001)

Heat transport

Another approach to the same problem

I. A. Gruzberg, N. Read, and A. W. W. Ludwig, Phys. Rev. B 63, 104422 (2001)

J. T. Chalker, N. Read, V. K., B. Horovitz, Y. Avishai, A. W. W. Ludwig

A. Mildenberger, F. Evers, A. D. Mirlin, and J. T. Chalker, Phys. Rev. B 75, 245321 (2007)

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0.0 0.2 0.4 0.6 0.8 1.00

1

2

M =16 M = 32 M = 64 M = 128 M = 256

M/M

(a)

0 5 10 15 20 25 30 350

1

2

M =16 M = 32 M = 64 M = 128 M = 256

(b)

M1/

M/M

=1.4W=0.1 is fixed

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0.00 0.05 0.10 0.15

1

2 M = 16 M = 32 M = 64 M = 128

(a)

M/M

M/M

W

0 1 2 3 4 5 6

1

2 M = 16 M = 32 M = 64 M = 128

(b)

W-0.2|M1/

=0.1 is fixed =1.4

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0 1 2 3

0

1

2

3

4

5

6

7

W-0.19|M1/

M/M

M=16 M=32 M=64 M=128

=1.4

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PRL 101, 127001 (2008) V.K. & D. Nemirovsky

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0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

ME

TA

L

xy=0

xy

=1

W

=1 >1

A. Mildenberger, F. Evers, A. D. Mirlin ,and J. T. Chalker ,

Phys. Rev. B 75, 245321 (2007)

Pure Ising transition

W≡p

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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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0.00 0.05 0.10 0.15 0.20 0.25 0.300

1

2

3

M/M

M=16 M=32 M=64 M=128

W=0.02

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5 10

1

2

3=1.29

M1/1.29

M=16 M=32 M=64 M=128

M/M

W=0.02

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0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

M/M

M=32 M=64 M=128

M1/1.09

=1.09

W=0.02

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0 5 10 15 20 25 30 35 40

0.0

0.2

0.4

0.6

0.8

1.0

M/M

M1/11

=1

M=64 M=128

W=0.02

RG flows towards the pure Ising transition with =1!

W=0.02<WN

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We probably can determine the exact position of therepulsive fixed point WN and tricritical point WT?

W=0.04

RG flows towards the pure Ising transition with =1!

W=0.04<WN

M=16, 32, 64, 128 =1.34M=32, 64, 128 =1.11

M=64, 128 =0.97

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Back to the original network model

Height of the barriers fluctuate - percolation

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Random hyperfine fields

int n eiH I H

83e e e i

eH g s r R

Nuclear spin

Magnetic filed produced by electrons

Bhf hfV B

Additional potential

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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

If p=1 then 2d system is broken into M 1d chainsAll states are extended independent on energyLyapunov exponent =0 for any system size as

in D-class superconductor

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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

11

( )M Mp fM

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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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

0.0 0.2 0.4 0.6 0.8 1.0

0

2

4

6

8

10

12

14

16

18

20

M=16 M=32M=64

M/M

p

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0 2 4 6 8 10 12 14 16 18 20 22 24 260.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

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

M2.5

(1-p)1.3/M

Data collapse for all energies , system widthsM 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 modelsH. 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).

Thanks to Ferdinand Evers

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1 cl qM p f MM

4 / 3cl

Scaling

2.5q

The fraction of polarized nuclei p is a relevant parameter

PRB 75, 113304 (2007) V.K. and Israel Vagner

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SummaryApplications 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