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Topological Aspects of the n=0 Landau Level in graphene Chiral - - PowerPoint PPT Presentation

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Graphene Week09 ESF-FWF Obergurgl, March 5, 2009 Topological Aspects of the n=0 Landau Level in graphene Chiral Symmetry & Hall Plateau Transition Y. Hatsugai (Univ. Tsukuba) T. Kawarabayashi (Toho Univ.) H. Aoki (Univ. Tokyo)

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SLIDE 1

Topological Aspects of the n=0 Landau Level in graphene Chiral Symmetry & Hall Plateau Transition

  • Y. Hatsugai (Univ. Tsukuba)
  • T. Kawarabayashi (Toho Univ.)
  • H. Aoki (Univ. Tokyo)

T.Kawarabayashi, Y. Hatsugai and H. Aoki, preprint

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

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SLIDE 2

Today’s Talk

Quantum Hall effect of graphene: stability and topology

Need to fill many Landau Levels with van Hove singularity Numerical technique : Non Abelian formulation & lattice gauge

Chiral Symmetry of graphene : speciality of the n=0 L.L.

Sublattice symmetry as chiral symmetry Physical outcomes of chiral symmetry

Quantum Hall plateau transition of graphene

Spatially correlated randomness for hopping as ripples Role of the correlation Quantum Hall critical states

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

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SLIDE 3

Observation of Anomalous QHE in Graphene

Anomalous QHE of gapless Dirac Fermions

Novoselov et al. Nature 2005 Zhang et al. Nature 2005

σxy = e2 h (2n + 1), n = 0, ±1, ±2, · · · = 2e2 h (n + 1 2)

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SLIDE 4

ǫn : No zero point energy shift

QHE of Graphene (Gapless Semiconductor)

Landau Level of Doubled Dirac Fermions

µF

E

  • k

E(B = 0) = ±c| k|

D(E) =

  • n

δ(E − ǫn)

ǫn = ±C √ Bn C = c

  • (2)e

McClure, 1956

1 √ 2 √ 3 2 √ 5

σxy [e2/h]

1 3 5 7 9

σxy(µF ) = e2 (2n + 1) n = 0, 1, 2, 3, · · · ǫn−1 < µF < ǫn

Zheng-Ando 2002 Gusynin-Sharapov, 2005 Peres-Guinea-Neto 2006 ...

σxy : e2 h × odd integer

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SLIDE 5

Hall conductance as a topological invariant

By Chern numbers of Bloch electrons

Thouless-Kohmoto-Nightingale-den Nijs 1982

ǫℓ(k) < µF , ℓ = 1, · · · , j

σj

xy = e2

h

j

  • ℓ=1

Cℓ, Cℓ = 1 2πi

  • BZ

dAℓ, Aℓ = ψℓ|dψℓ

Counting vortices in the band

Sum over the filled bands Need to sum many bands until E=0

E=0

Numerical difficulty for the weak field (experimental situation) Need to fill negative energy Dirac sea

Need to sum over them

{

graphene

E=0

{

with randomness Aoki-Ando 1986 van Hove singularity

Rammal 1985

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SLIDE 6

Bulk of the Filled Fermi sea & Dirac Sea

Integration of the NonAbelian Berry Connection of the “Fermi Sea” & “Dirac Sea”

σxy

Hatsugai 2004 Numerical advantage for graphene

Hj(k)|ψj(k) = ǫj(k)|ψj(k)

Ψ =( |ψ1, · · · , |ψM)

Collect M states below the Fermi level

AF S ≡ Ψ†dΨ =    ψ†

1|dψ1

· · · ψ†

1|dψM

. . . ... . . . ψ†

M|dψ1

· · · ψ†

M|dψM

  

Matrix vector potential of the Fermi ( Dirac ) Sea Non Abelian extension for the Chern numbers

σxy = e2 h 1 2πi

  • T 2 TrMdAF S
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SLIDE 7

Bulk of the Filled Fermi sea & Dirac Sea

Integration of the NonAbelian Berry Connection of the “Fermi Sea” & “Dirac Sea”

σxy

Hatsugai 2004 Numerical advantage for graphene

Hj(k)|ψj(k) = ǫj(k)|ψj(k)

Ψ =( |ψ1, · · · , |ψM)

Collect M states below the Fermi level

AF S ≡ Ψ†dΨ =    ψ†

1|dψ1

· · · ψ†

1|dψM

. . . ... . . . ψ†

M|dψ1

· · · ψ†

M|dψM

  

Matrix vector potential of the Fermi ( Dirac ) Sea Non Abelian extension for the Chern numbers

σxy = e2 h 1 2πi

  • T 2 TrMdAF S

Technology 1

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SLIDE 8

Numerical Technique from the Lattice gauge theory

Topological Invariant on Discretized Lattice

σxy = e2 h 1 2πi

  • F1234

F1234 = Im log U12U23U34U41 Umn = det

j Ψ† mΨn,

Ψn = (ψ1(kn), · · · , ψj(kn))

Fukui-Hatsugai-Suzuki 2005

Lattice in k space ( discretization for the integral )

Fermi Sea of j filled bands

Technical Advantage for large Chern Numbers k

Uµ(k) ≡ n(k)|n(k + ˆ µ)/Nµ(k)

Uµ(k)

Nµ(k) = |n(k)|n(k + ˆ µ)|

˜ F12(k) ≡ ln U1(k)U2(k + ˆ 1)U1(k + ˆ 2)−1U2(k)−1

˜ F12(k)

−π < ˜ F12(k)/i ≤ π

(principal value) 1 2 NB

Brillouin Zone gauge invariant

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SLIDE 9

Numerical Technique from the Lattice gauge theory

Topological Invariant on Discretized Lattice

σxy = e2 h 1 2πi

  • F1234

F1234 = Im log U12U23U34U41 Umn = det

j Ψ† mΨn,

Ψn = (ψ1(kn), · · · , ψj(kn))

Fukui-Hatsugai-Suzuki 2005

Lattice in k space ( discretization for the integral )

Fermi Sea of j filled bands

Technical Advantage for large Chern Numbers k

Uµ(k) ≡ n(k)|n(k + ˆ µ)/Nµ(k)

Uµ(k)

Nµ(k) = |n(k)|n(k + ˆ µ)|

˜ F12(k) ≡ ln U1(k)U2(k + ˆ 1)U1(k + ˆ 2)−1U2(k)−1

˜ F12(k)

−π < ˜ F12(k)/i ≤ π

(principal value) 1 2 NB

Brillouin Zone gauge invariant

Technology 2

Chern number extension of the KSV formula for polarization

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SLIDE 10

Hall Conductace vs chemical potential

Accurate Hall conductance over whole spectrum

  • 3
  • 2
  • 1

1 2 3

  • 20
  • 10

10 20 30

  • 2

2

φ = 1/31 D(E) σxy [e2/h]

µ/t, t ≈ 1[eV] for graphene

Electron Like in this region Hole Like in this region Dirac Like in this region

Hatsugai-Fukui-Aoki ’06

single band model

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SLIDE 11

Chern numbers ( ) based on Realistic Band Calc.

σxy

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SLIDE 12

Chern numbers ( ) based on Realistic Band Calc.

quantized everywhere

σxy

M.Arai and Y.Hatsugai, Phys.Rev. B79, 075429 (2009)

σxy

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SLIDE 13

Chern numbers ( ) based on Realistic Band Calc.

quantized everywhere

σxy

M.Arai and Y.Hatsugai, Phys.Rev. B79, 075429 (2009)

σxy

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SLIDE 14

Chern numbers ( ) based on Realistic Band Calc.

quantized everywhere

σxy σxy

Fermi surface

EF

M.Arai and Y.Hatsugai, Phys.Rev. B79, 075429 (2009)

σxy

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SLIDE 15

Chern numbers ( ) based on Realistic Band Calc.

quantized everywhere

σxy σxy

Fermi surface

EF

M.Arai and Y.Hatsugai, Phys.Rev. B79, 075429 (2009)

σxy

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SLIDE 16

Chern numbers ( ) based on Realistic Band Calc.

quantized everywhere

σxy σxy

Fermi surface

EF

σxy

Fermi surface

EF

M.Arai and Y.Hatsugai, Phys.Rev. B79, 075429 (2009)

σxy

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SLIDE 17

Chiral Symmetry of Graphene

Equivalence of Hopping only between

Chiral Symmetry

Chiral Symmetry of Graphene

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

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SLIDE 18

Chiral Symmetry of Graphene

Equivalence of Hopping only between

Chiral Symmetry

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-19
SLIDE 19

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry

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SLIDE 20

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Protect “Massless” Dirac cones

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SLIDE 21

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Protect “Massless” Dirac cones

t’ t’ t’ t t t

Perturbation preserving this chiral symmetry Example : t’ terms Energy dispersion Density of States

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SLIDE 22

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry

slide-23
SLIDE 23

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Protect n=0 Landau level at “E=0”

slide-24
SLIDE 24

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Protect n=0 Landau level at “E=0”

with chiral symmetry

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SLIDE 25

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Protect n=0 Landau level at “E=0”

with chiral symmetry

Splitting of n=0 L.L. by Chiral symmetry breaking

! "! #$ % & % " % ' ! ' " & (

  • H. Aoki, T.Fukui, Y. Hatsugai,
  • Int. J. Mod. Phys. 21 1133 (2007)

without chiral symmetry

slide-26
SLIDE 26

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry

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SLIDE 27

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Protect zero mode edge states at Zigzag boundaries (Fujita)

slide-28
SLIDE 28

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Protect zero mode edge states at Zigzag boundaries (Fujita)

with chiral symmetry

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SLIDE 29

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Protect zero mode edge states at Zigzag boundaries (Fujita)

  • S. Ryu & Y. Hatsugai,

Physica E22, 679 (2004)

with chiral symmetry chiral symmetry breaking

  • nly at edges

Splitting of the zero mode edge state Bulk is gapless still.

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SLIDE 30

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Protect zero mode edge states at Zigzag boundaries (Fujita)

  • S. Ryu & Y. Hatsugai,

Physica E22, 679 (2004)

with chiral symmetry chiral symmetry breaking

  • nly at edges

Splitting of the zero mode edge state Bulk is gapless still.

slide-31
SLIDE 31

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Protect zero mode edge states at Zigzag boundaries (Fujita)

  • S. Ryu & Y. Hatsugai,

Physica E22, 679 (2004)

with chiral symmetry chiral symmetry breaking

  • nly at edges

Splitting of the zero mode edge state Bulk is gapless still.

chiral symmetry breaking both bulk and edges

Full gap opens

slide-32
SLIDE 32

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Protect zero mode edge states at Zigzag boundaries (Fujita)

  • S. Ryu & Y. Hatsugai,

Physica E22, 679 (2004)

with chiral symmetry chiral symmetry breaking

  • nly at edges

Splitting of the zero mode edge state Bulk is gapless still.

chiral symmetry breaking both bulk and edges

Full gap opens

slide-33
SLIDE 33

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry

slide-34
SLIDE 34

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Hamiltonian anti-commutes with

{Γ, H} = ΓH + HΓ =0 , Γ2 = 1

∃Γ

Mathematically

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SLIDE 35

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Hamiltonian anti-commutes with

{Γ, H} = ΓH + HΓ =0 , Γ2 = 1

∃Γ

Γ =

  • I

O O† −I

  • H =

O D D† O

  • Mathematically
slide-36
SLIDE 36

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Hamiltonian anti-commutes with

{Γ, H} = ΓH + HΓ =0 , Γ2 = 1

∃Γ

Γ =

  • I

O O† −I

  • H =

O D D† O

  • Uniform site energy

Random hopping Antiferromagnetic order Staggered site energies Bond order Random potential

Mathematically Perturbations

slide-37
SLIDE 37

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Hamiltonian anti-commutes with

{Γ, H} = ΓH + HΓ =0 , Γ2 = 1

∃Γ

Γ =

  • I

O O† −I

  • H =

O D D† O

  • Uniform site energy

Random hopping Antiferromagnetic order Staggered site energies Bond order Random potential

Mathematically Perturbations

slide-38
SLIDE 38

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Hamiltonian anti-commutes with

{Γ, H} = ΓH + HΓ =0 , Γ2 = 1

∃Γ

Γ =

  • I

O O† −I

  • H =

O D D† O

  • Uniform site energy

Random hopping Antiferromagnetic order Staggered site energies Bond order Random potential

Mathematically Perturbations

slide-39
SLIDE 39

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Hamiltonian anti-commutes with

{Γ, H} = ΓH + HΓ =0 , Γ2 = 1

∃Γ

Γ =

  • I

O O† −I

  • H =

O D D† O

  • Uniform site energy

Random hopping Antiferromagnetic order Staggered site energies Bond order Random potential

Mathematically Perturbations

slide-40
SLIDE 40

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Hamiltonian anti-commutes with

{Γ, H} = ΓH + HΓ =0 , Γ2 = 1

∃Γ

Γ =

  • I

O O† −I

  • H =

O D D† O

  • Uniform site energy

Random hopping Antiferromagnetic order Staggered site energies Bond order Random potential

Mathematically Perturbations

slide-41
SLIDE 41

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Hamiltonian anti-commutes with

{Γ, H} = ΓH + HΓ =0 , Γ2 = 1

∃Γ

Γ =

  • I

O O† −I

  • H =

O D D† O

  • Uniform site energy

Random hopping Antiferromagnetic order Staggered site energies Bond order Random potential

Mathematically Perturbations

slide-42
SLIDE 42

Chiral Symmetry of Graphene

Equivalence of Hopping between

Chiral Symmetry Hamiltonian anti-commutes with

{Γ, H} = ΓH + HΓ =0 , Γ2 = 1

∃Γ

Γ =

  • I

O O† −I

  • H =

O D D† O

  • Uniform site energy

Random hopping Antiferromagnetic order Staggered site energies Bond order Random potential

Mathematically

Ripples ?

Perturbations

slide-43
SLIDE 43

Quantum Hall plateau transition

Khmelnitskii ’84 Laughlin ’84 Kivelson-Lee-Zhang ’92 Ludwig-Fisher-Shankar-Grinstein ’94

Quantum phase transition driven by disorder among states with different topological numbers

Sheng-Wen ’97 Hatsugai-Ishibashi-Morita ’99 Focus: Delocalized states within Landau Levels

Hall plateau transition of graphene & effects of randomness in graphene

Suzuura-Ando ’02 McCann et al. ’06 Aleiner-Efetov ’06 Altland ’06 Many... Sheng-Sheng-Wen ’06 Schweitzer-Markos ’08 Nomura-Ryu-Koshino-Mudry-Furusaki ’08

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

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SLIDE 44

Quantum Hall plateau transition

Khmelnitskii ’84 Laughlin ’84 Kivelson-Lee-Zhang ’92 Ludwig-Fisher-Shankar-Grinstein ’94

Quantum phase transition driven by disorder among states with different topological numbers

Sheng-Wen ’97 Hatsugai-Ishibashi-Morita ’99 Focus: Delocalized states within Landau Levels

Hall plateau transition of graphene & effects of randomness in graphene

Suzuura-Ando ’02 McCann et al. ’06 Aleiner-Efetov ’06 Altland ’06 Many... Sheng-Sheng-Wen ’06 Schweitzer-Markos ’08 Nomura-Ryu-Koshino-Mudry-Furusaki ’08

Quantum Hall plateau transition Chiral symmetry & n=0 L.L. of graphene Our focus

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

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SLIDE 45

Ripples of graphene

Ripples as an imaginary random gauge field

Neto-Guinea-Peres-Novoselov-Geim ’09 (Meyer, Geim et al, Nature 2007)

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

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SLIDE 46

Ripples of graphene

Ripples as an imaginary random gauge field

Neto-Guinea-Peres-Novoselov-Geim ’09

Gauge out by non-unitary gauge transformation

(Meyer, Geim et al, Nature 2007)

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-47
SLIDE 47

Ripples of graphene

Ripples as an imaginary random gauge field

Neto-Guinea-Peres-Novoselov-Geim ’09

Gauge out by non-unitary gauge transformation

(Meyer, Geim et al, Nature 2007)

hopping amplitude is modified ( not a phase )

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-48
SLIDE 48

Ripples of graphene

Ripples as an imaginary random gauge field

Neto-Guinea-Peres-Novoselov-Geim ’09

Gauge out by non-unitary gauge transformation

(Meyer, Geim et al, Nature 2007)

hopping amplitude is modified ( not a phase ) Random hopping model on a honeycomb lattice with spatial correlation

zero modes are protected by the index theorem (conceptually)

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

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SLIDE 49

2D Honeycomb Lattice in disordered hopping amplitude Disordered components δ t

effective width ηt : correlation of random hopping

Density of states

The Green function method Schweitzer, Kramer, MacKinnon (1984)

a b

P.B.C.

φ =φuni

Model

Y.Hatsugai, X.G.Wen, M.Kohmoto, Phys.Rev. B56, 1061 (1997)

c.f. Dirac fermions on square lattice

Correlated Random Hopping

B.D.Simons, private communication

slide-50
SLIDE 50

Broadening of Landau levels uncorrelated disordered hopping

φuni/φ0=1/41 (l =2.4|a|) Wδt /t =0, 0.2, 0.4, 0.5, 0.6

DOS E/t

n=0 n=1 n=2 n= ---1 n= ---2 n=3 n= ---3

finite width for any n

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-51
SLIDE 51

Correlated Random Hopping

Wδt /t = 2.0

Landscape of hopping amplitude (distribution of gauge field ) for calculation of density of states

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-52
SLIDE 52

Correlated Random Hopping

Wδt /t = 2.0

Landscape of hopping amplitude (distribution of gauge field ) for calculation of density of states

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-53
SLIDE 53

Effect of spatial correlation

φuni/φ0=1/41 (l =2.4|a|) Wδt /t = 0.4 0.0, 1.0, 2.0, 3.0, 5.0

Broadening is smaller than the numerical resolution. No significant difference from Wδt=0 case

DOS E/t

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-54
SLIDE 54

φuni/φ0=1/41 (l =2.4|a|) Wδt /t = 0.6

γ = 0.000625t 0.0 1.0 1.5 2.0 2.5 3.0 Effect of spatial correlation almost no broadening at n=0 LL

n=0 n=1 n=2 n= ---1 n= ---2 n=3 n= ---3

EF /t

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-55
SLIDE 55

Bulk by topological numbers

Integration of the NonAbelian Berry Connection of the “Fermi Sea” & “Dirac Sea”

σxy

Hatsugai 2004

Hj(k)|ψj(k) = ǫj(k)|ψj(k) Ψ =( |ψ1, · · · , |ψM)

σxy = e2 h 1 2πi

  • T 2 TrMdAF S

σxy = e2 h 1 2πi

  • F1234

F1234 = Im log U12U23U34U41 Umn = det

j Ψ† mΨn,

Ψn = (ψ1(kn), · · · , ψj(kn))

Uµ(k) ≡ n(k)|n(k + ˆ µ)/Nµ(k)

Uµ(k)

Nµ(k) = |n(k)|n(k + ˆ µ)|

˜ F12(k) ≡ ln U1(k)U2(k + ˆ 1)U1(k + ˆ 2)−1U2(k)−1

˜ F12(k)

−π < ˜ F12(k)/i ≤ π

Fukui-Hatsugai-Suzuki 2005 This non-Abelian formulation is crucial Otherwise: need to sum up Chern numbers for many Landau sub-bands split by randomness

slide-56
SLIDE 56

Bulk by topological numbers

Integration of the NonAbelian Berry Connection of the “Fermi Sea” & “Dirac Sea”

σxy

Hatsugai 2004

Hj(k)|ψj(k) = ǫj(k)|ψj(k) Ψ =( |ψ1, · · · , |ψM)

σxy = e2 h 1 2πi

  • T 2 TrMdAF S

σxy = e2 h 1 2πi

  • F1234

F1234 = Im log U12U23U34U41 Umn = det

j Ψ† mΨn,

Ψn = (ψ1(kn), · · · , ψj(kn))

Uµ(k) ≡ n(k)|n(k + ˆ µ)/Nµ(k)

Uµ(k)

Nµ(k) = |n(k)|n(k + ˆ µ)|

˜ F12(k) ≡ ln U1(k)U2(k + ˆ 1)U1(k + ˆ 2)−1U2(k)−1

˜ F12(k)

−π < ˜ F12(k)/i ≤ π

Fukui-Hatsugai-Suzuki 2005 This non-Abelian formulation is crucial Otherwise: need to sum up Chern numbers for many Landau sub-bands split by randomness

slide-57
SLIDE 57

Wδt /t = 0.4 φuni/φ0=1/50

System size 20×20 Chern Number 100 samples

σxy = e2 h 1 2πi

  • T 2 TrMdAF S

EF /t

n=0 n=1 n=2 n= ---1 n= ---2

with spatial correlation without spatial correlation

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-58
SLIDE 58

Wδt /t = 0.4 φuni/φ0=1/50

System size 20×20 Chern Number 100 samples

σxy = e2 h 1 2πi

  • T 2 TrMdAF S

EF /t

Spatial correlation makes n=0 LL special No critical region only for n=0 n=0 n=1 n=2 n= ---1 n= ---2

with spatial correlation without spatial correlation

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-59
SLIDE 59

Wδt /t = 0.4 φuni/φ0=1/50

System size 20×20 Chern Number 100 samples

σxy = e2 h 1 2πi

  • T 2 TrMdAF S

EF /t

Spatial correlation makes n=0 LL special No critical region only for n=0 n=0 n=1 n=2 n= ---1 n= ---2

with spatial correlation without spatial correlation

Nomura-Ryu-Koshino-Mudry-Furusaki, Phys.

  • Rev. Lett. 100 246806 (2008)

without chiral symmetry c.f.

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-60
SLIDE 60

Size-dependence

Wδt /t = 0.4 φuni/φ0=1/50

100 samples

σxy = e2 h 1 2πi

  • T 2 TrMdAF S

EF /t

n=0 n=1 n=2 n=3 n= ---3 n= ---2 n= ---1

with spatial correlation

c.f. Critical zero mode of Random Dirac fermion: Y.Morita & Y.Hatsugai, Phys.Rev. Lett. 79, 3728,(1997) Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-61
SLIDE 61

Size-dependence

Wδt /t = 0.4 φuni/φ0=1/50

100 samples

σxy = e2 h 1 2πi

  • T 2 TrMdAF S

EF /t

Exact fixed point even with finite system size n=0 n=1 n=2 n=3 n= ---3 n= ---2 n= ---1

with spatial correlation

c.f. Critical zero mode of Random Dirac fermion: Y.Morita & Y.Hatsugai, Phys.Rev. Lett. 79, 3728,(1997) Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト

slide-62
SLIDE 62

Summary

Ripples of graphene as Spatially correlated random hopping model

  • n honeycomb lattice

Chiral Symmetry of graphene

Realization of an Exact Fixed point Even at finite system sizes Quantum Hall plateau critical point Critical states of random Dirac fermions

Speciality of the n=0 Landau Level of Dirac fermions

Graphene Week09 ESF-FWF Obergurgl, March 5, 2009ト