1960-15 ICTP Conference Graphene Week 2008 25 - 29 August 2008 - - PowerPoint PPT Presentation

1960-15 ICTP Conference Graphene Week 2008

• Y. Hatsugai

25 - 29 August 2008 Institute of Physics, University of Tsukuba, Japan Bulk--edge correspondence in graphene with/without magnetic field Topological aspects of Dirac fermions in real materials

Bulk--edge correspondence in graphene with/without magnetic field

Topological aspects of Dirac fermions in real materials

• 3
• 2
• 1

1 2 3

• 30
• 20
• 10

10 20

Institute of Physics University of Tsukuba

JAPAN

• Y. Hatsugai

Graphene Week08 ICTP Trieste, Aug. 28, 2008

σxy = e2 h 1 2πi

• Tr dA

Edge States in Condensed Matter Physics

Bound states in quantum mechanics

Surface states in Semiconductors Solitons in polyacetylene Edge states in quantum Hall effects Local moments in integer spin chains near the impurities Zero bias conductance peaks of the d-wave superconductors Zero energy localized states of graphene Quantum Spin Hall Edge states

Hu, ‘94 Wang et al., PRL 100, 013905 ‘08 Scarola-Das Sarma., PRL 98, 210403 ‘07

more possibilities

Levinson’s theorem, Friedel’s sum rule Fujita et al.‘96 Kane-Mele‘05 Kennedy ‘90 Hagiwara-Katsumata-Affleck-Halperin-Renard ‘90 Halperin ‘82 Hatsugai ‘93 Su-Schriefer-Heeger ‘79 Ryu-Hatsugai‘02

Edge states in 2D cold atoms in optical lattice One-way edge modes in gyromagnetic photonic crystals

Bernevig-Hughes-Zhang ‘06

Accidental ? NO ! Inevitable reasons Universal Structures behind: Bulk determines the edges : Bulk-Edge Correspondence Protected by Topological constraints and also additional Symmetry

Why the Edge States are there??

Bulk-Edge Correspondence Revisit the past and discuss graphene Quantum Hall effects Several New results They Can Be detected by STM experiments !

Why the Edge States are there??

Quantum Hall Effects by edge states

Edge states and Hall conductance σxy Halperin ‘82

Landau gauge in y

L R

E

EF

Without boundaries

x ∼ 2

Bky ky = 2π Ly (ny + Φ Φ0 ) ny = 0, ±1, ±2, · · ·

EF

With boundaries E Left edge Right edge Edge potential Gapless excitations

Φ = 0

L R

H2D =

• ky

H1D(ky) ky

H1D(ky)

: harmonic osc. centered at = Laughlin’s undetermined : # of Landau Levels below

n

EF

Edge states are essential in the QHE ! Φ =Φ 0

Left edge Right edge 2 states are carried from L to R

Hall Conductance has a Topological meaning

Discussion by the Bloch electrons ( Peierls substitution ) preserve U(1) gauge symmetry without cutoff ambiguity recover continuum theory by scaling limit ( weak field limit)

Thouless-Kohmoto-Nightingale-den Nijs ‘82 Sum of the First Chern numbers below EF

(m,n) (m+1,n) (m+1,n+1) (m,n+1)

a a

• x

m,n+1

• x

m,n

• y

m+1,n

• y

m,n

σbulk

xy

= e2 h

• :(k)<EF

C σedge

xy

= e2 h I(αj, Cj)

When EF is in the j-th gap Winding number of the edge state in the complex energy surface Hatsugai ‘93a

σbulk

xy

= σedge

xy

Hatsugai ‘93b

H =

• ij

c†

ieiθijcj

2πφ =

• ij∈P

θij

P : plaquette

φ = Ba2 Φ0

Two topological quantities

Bulk ---- Edge Correspondence

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)

Graphene under magnetic field

In continuum, 2D = (1D harmonic oscillators with parameter ) Bloch electrons, 2D = (1D Harper problem with parameter )

• ky
• ky

ky ky

Rammal 1985

Hofstadter diagram for the honeycomb

Energy

Landau gauge

One particle Energy vs flux/hexagon (in flux quantum)

φ φ

TKNN formula: as a topological invariant

Topological meaning of the Hall Conductance

Thouless-Kohmoto-Nightingale-den Nijs ‘82

σxy = e2 h

• :(k)<EF

C

:First Chern number of the -th Band

• Sum over the bands below EF

C = 1 2πi

• T 2:BZ

F F = dA = dψ|dψ A = ψ|dψ

intrinsically integer unless the energy gap collapses

∀k, ǫ(k) = ǫ±1(k)

regularity of the Berry connection Kubo formula

H(k)|ψ(k) = ǫ(k)|ψ(k) k ∈ T 2 = {k = (kx, ky)| 0 ≤ kx, ky ≤ 2π} d = dkµ ∂ ∂kµ

BZ

σxy

Bulk Hall conductance of graphene

Hall conductance by Chern number

Thouless-Kohmoto-Nightingale-den Nijs 1982

ǫ(k) < µF , = 1, · · · , j

σj

xy = e2

h

j

• =1

C, C = 1 2πi

• BZ

dA, A = ψ|dψ

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

Bulk of the Filled Fermi sea & Dirac Sea

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

σxy

Hatsugai 2004 Technical 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

Hall Conductace vs chemical potential

Hall conductance of graphene over the 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

E(kx, ky) = ±

• (1 + cos kx + cos ky)2 + (sin kx + sin ky)2
• 3
• 2
• 2
• Electron Like

in this region

1 2 3

2

µ/t, t

E(k k ) ±

• (1

v

1

E(k k ) ±

• (1 +

Hole Like in this region

YH, T. Fukui & H. Aoki, ‘06

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

Hall Conductace vs chemical potential

Hall conductance of graphene over the 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

g p

( ) ( )

1 1

• 20
• 10

10 xy

• 1

g p

2

30

2

b

30

b in n

20 20

n

2

n

E(kx, ky) = ±

• (1 + cos kx + cos ky)2 + (sin kx + sin ky)2

YH, T. Fukui & H. Aoki, ‘06

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

Hall Conductace vs chemical potential

Hall conductance of graphene over the 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

Quantum phase transition at the van Hove Energies Singularity breaks Topological Stability

g p

( ) ( )

1 1

• 20
• 10

10 xy

• 1

g p

2

30

2

b

30

b in n

20 20

n

2

n

E(kx, ky) = ±

• (1 + cos kx + cos ky)2 + (sin kx + sin ky)2

YH, T. Fukui & H. Aoki, ‘06

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

3 types of Edge states in Graphene

• I. QH edge states in graphene

II.Zero modes edge state without magnetic field III.Zero modes with magnetic field

• S. Ryu & YH, ‘02
• M. Arikawa, H. Aoki & YH

arXiv:0805.3240 & 0806.2429

YH, T. Fukui & H. Aoki, ‘06 See also, L. Brey, H. A. Fertig, ‘06 Edge transport : D. A. Abanin, P. A. Lee, L. S. Levitov ‘07

Fujita et al., ‘96

Laughlin’s Argument & Edge States

Adiabatic Charge Transfer

B V I y

x

• I y

Ly

Edge States determine the Hall Conductance

• R+
• R

B E

Complex Energy surface

• f Harper eq.

Unifi fied Unifi d

j=1,···

n C n C

j

plex ex

• ergy

rgy su su face

Chern # = winding # Difference between the neighboring gaps

σxy

bulk = σxy edge Bulk-Edge Correspondence

• f the topological numbers

Cj = Ij − Ij−1

genus g=q -- 1: number of the gaps

φ = p/q

Analytic Continuation of the Bloch State to the complex energy (Riemann surface)

YH, T. Fukui & H. Aoki, ‘06

Another type of Edge states in Graphene

Quantum Hall edge states

Topologically protected edge states

Symmetry protected edge states: zero modes ( topological origin & stability )

• S. Ryu & YH ‘02

without magnetic field with magnetic field

• M. Arikawa, H. Aoki & YH

arXiv:0805.3240 & 0806.2429

New feature : topological compensation It can be observed by STM experiments under magnetic field

Fujita et al. ‘96 : discovery : topological reason

Without magnetic field

Graphene on a Cylinder

Localized Boundary State in Carbon Sheet (1)

Tight-binding Model Calculation

“ Peculiar Localized State at Zigzag Graphite Edge “ M. Fujita, K. Wakabayashi, K. Nakada and K. Kusakabe, JPSJ 65, 1920 (1996)

cnt-fujita – p.

now called as Graphene

Localized Boundary State in Carbon Sheet (2)

Local Spin Density Functional Appr. Calculation

“Magnetic Ordering in Hexagonally Bonded Sheets with First-Row Elements”, Okada, Oshiyama, Phys. Rev. Lett. 87, 146803 (2001)

cnt-oshiyama – p.

Zero Bias Conductance Peak

.

in Anisotropic Superconductivity Zero Energy Boundary States

.

• f Anisotropic Superconductivity

.

Tanaka-Kashiwaya

• L. J. Buchholtz,G. Zwicknagl, Phys. Rev. B 23, 5788 (1981) (p wave )

C.-R. Hu, Phys. Rev. Lett. 72, 1526 (1994) (d wave )

• S. Kashiwaya, Y. Tanaka, Phys. Rev. Lett. 72, 1526 (1994)
• M. Matsumoto and H. Shiba, JPSJ, 1703 (1995)

(fig.) M. Aprili, E. Badica, and L. H. Greene,Phys. Rev. Lett. 83, 4630 (1999)

zbcp – p.

d-wave superconductivity

When the zero modes exist (1D) ?

• A =ψ(k)|

∇kψ(k)

Determined by the Berry phase of the bulk (without boundaries)

S.Ryu & Y.Hatsugai, Phys. Rev. Lett. 89, 077002 (2002) Y.Hatsugai., J. Phys. Soc. Jpn. 75 123601 (2006) Kuge, Maruyama, Y. Hatsugai, arXiv:0802.2425

Edge states with boundaries

γ =

• A =
• d

k · A

Zak

{Γ, H} = ΓH + HΓ =0

Require Local Chiral Symmetry (ex. bipartite )

Quantized

γ =

• A =

π

: There exists odd number of zero modes

Zero energy localized states EXIST

γ = π

Bulk-edge correspondence: “Bulk determines the edges”

Lattice analogue of Witten’s SUSY QM

With magnetic field

Close look at E=0

n=0 Landau Level φ = 1/4 φ = 1/7 φ = 1/21

Bulk Landau Level and the zero mode edge states coexist

Charge density around E=0

Integrated local density of state over the n=0 LL

I(x) = 1 2π EC

−EC

dE π

−π

dky |Ψ(x, ky, E)|2

x

E

ky

EC ∼ 0.05 t

This can be observed by the STM image

n=0 Landau Level

• M. Arikawa, H. Aoki & YH arXiv:0805.3240 & 0806.2429

LDOS around E=0 with Landau Level

Armchair

I(x) = 1 2π EC

−EC

dE π

−π

dky |Ψ(x, ky, E)|2

I(x)/I0

x/lB

0.5 1 1.5 2 2.5 3 0.5 1 1.5

=1/41 =1/31 =1/21

Suppression near the edge

Standard behavior due to edge potential

x

Boundary

Zigzag

I(x) = 1 2π EC

−EC

dE π

−π

dky |Ψ(x, ky, E)|2

=1/41 =1/31 =1/21

Strong enhancement near the edge

Characteristic feature of the Graphene Zigzag edges!

0.5 1 1.5 2 2.5 3 0.5 1 1.5 2

x/lB

I(x)/I0

scaled by the magnetic length

STM observable

x

LDOS around E=0 with Landau Level

• M. Arikawa, H. Aoki & YH

arXiv:0805.3240 & 0806.2429

Boundary

Topological Compensation by Bulk

Bulk does give substantial contribution

Characteristic feature of the Graphene Zigzag edges!

I(x)/I0

x/lB

E=0 contribution n=0 LL (|E| < 0.05)

2 4 6 8 10 12 1 2

total bulk E=0 I(x)/I0

1 2 3 0.1 0.2 0.3 0.4

Only the Zero mode

power law : No Length scale

pn ∼ n−2/(π √ 3) scaled by

• B

I(x) = 1 2π EC

−EC

dE π

−π

dky |Ψ(x, ky, E)|2

B

Possible quantum liquid in Graphene

many body effects edge states as basic objects to condense edge states as 2D analogue of the solitons in 1D

• Y. Hatsugai, T. Fukui, H. Aoki,

“Topological low-energy modes in N=,0 Landau levels of graphene: a possibility of a quantum-liquid ground state”, arXiv:0804.4762, Physica E 40, 1530 (2008).

bond-order parameter! A possible candidate = Bond-ordering

Origin ? ( with Magnetic field )

Electron-Electron interaction

(Jahn-Teller?)

Preserve Chiral Symmetry Three Fold Degeneracy Local Kekule pattern Enlarged unit cell ( 3times )

Breaking the equivalence among 3 kinds of bonds in a Kekule pattern

Affleck-Marston ’88 Voit ’92 Nakamura ‘99

Can the E=0 Landau level be split in graphene??

c†

acb

Energy Dispersion (without Magnetic Field)

Enlarged Unit cell Chiral Symmetry is preserved Dirac Fermions get masses

The energy gap opens up

E(k) ≈ ±| det D(k)|

D =   tRe−i(2k1−k2)) tG tB tB tRe−i(−k1+k2)) tG tG tB tReik1  

(small gap & near E=0) Particle-Hole Symmetric

With bond-ordering

Landau level Structures with bond-orders

Characteristic E=0 Landau Level

• f the Dirac Fermion splits !

2 4 6 8 10 12

• 3
• 2
• 1

1 2 3 'dos_1o18.dat' 1 2 3 4 5 6

• 3
• 2
• 1

1 2 3 'dos_1o18_wd.dat'

One particle Density of states

without bond-order with bond-order

split ! E=0 LL Peierls Instability of the Flat band as of the E=0 Landau Level Bond-Order Formaion ( in a mean field level )

Bond-Ordering with Domains (Strip domain)

Straight line domain boundaries

A possible snap shot configuration

• f the dynamical bond-order

Charge Density of the in-gap states

Domain shape ( strip )

Bond-Ordering with Domains (circular domain)

Closed loop domain boundaries

A possible snap shot configuration

• f the dynamical bond-order

Charge Density of the in-gap states

Domain shape ( closed loop )

True ground state of graphene with interaction

edge states as basic objects to condense edge states as 2D analogue of the solitons in 1D

• Y. Hatsugai, T. Fukui, H. Aoki, arXiv:0804.4762, Physica E 40, 1530 (2008).

Quantum liquid by the condensation of the edge states?

Snapshots ?

2 4 6 8 10 12

• 3
• 2
• 1

1 2 3

0.5 1 1.5 2 2.5 3

• 0.4
• 0.2

0.2 0.4

One particle Density of states

without boundaries with boundaries

}

In-gap states

energy

In-gap states between the split E=0 Landau Level

(circular domain)

Summary

Topological Aspects of Graphene

QHE by the Bulk QHE by the edge Bulk --- Edge Correspondence

Another Edge states of Graphene

without magnetic field with magnetic field : Coexistence of the Bulk and edge states at E=0 (STM observable)

Possible quantum liquids with bond order

As a condensate of the loops by the edge states