Luttinger Liquid at the Edge of Liquid at the Edge of Luttinger a - - PowerPoint PPT Presentation

Luttinger Luttinger Liquid at the Edge of Liquid at the Edge of a a Graphene Graphene Vacuum Vacuum

H.A. Fertig, Indiana University Luis Brey, CSIC, Madrid

I. Introduction: Graphene Edge States (Non-Interacting) II. Quantum Hall Ferromagnetism and a Domain Wall at the Edge

• III. Properties of the Domain Wall
• IV. Excitations from Filled (Graphene) Landau Levels

(with Drew Iyengar and Jianhui Wang)

• V. Summary

Funding: NSF

• I. Edge States for Graphene

A B

Honeycomb lattice, two atoms per unit cell Lattice constant: 2.46Å Nearest neighbor distance: 1.42Å Simple tight-binding model for pz orbitals:

∑

=

− =

. . 2 1

2 1

n n n n

n n t H

t ≈ 2.5-3 eV

A and B sublattice sites in unit cell

• For each k there are eigenvalues at ±|ε| ⇒ particle-hole symmetry
• Fermi energy at ε=0

EF

t a n n l 3 ) , ( ± = τ ε

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ± = Ψ

−

) ( ) ( ) , (

2 2 1

l l

x n x n x

k y k y x ik e n K φ φ

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ± = ′ Ψ

−

) ( ) ( ) , (

2 1 2

l l

x n x n x

k y k y x ik e n K φ φ

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Ψ ) , ( φ x ik e K

x

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ′ Ψ ) , ( φ x ik e K

x

Wavefunctions in a magnetic field:

state

• scillator

harmonic =

n

φ

Energies: ε Particle-hole conjugates kx With valley and spin indices, each Landau level is 4-fold degenerate

• Real samples in experiments are very narrow (.1-1µm) ⇒ edges

can have a major impact on transport

• Can get a full description of QHE within Dirac equation
• Edge structure can be probed directly via STM at very small

length scales. Nothing comparable is possible in standard 2DEG’s (GaAs samples, Si MOSFET’s) Tight-binding results, armchair edge

K K´ K´y Ky

• II. Quantum Hall Ferromagnetism and the Graphene Edge

DOS

Energy

EF

Interactions

DOS

Energy

EF

• Exchange tends to force electrons into the same level even when

bare splitting between levels is small

• Renormalizes gap to much larger value than expected from

non-interacting theory (even if bare gap is zero!)

This does happen in graphene (Zhang et al., 2006).

• Plateaus at ν=0?,±1.
• System may be a quantum

Hall ferromagnet.

• cf. Alicea and Fisher, 2006

Nomura and Macdonald, 2006 Fuchs and Lederer, 2006

“Vacuum” state (undoped graphene):

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

EF

K,K’ K,K’

Low-lying excitations: 2 (+2) spin (+valley) waves EF

Spin stiffness

⇒ Analogy with Heisenberg ferromagnet.

Consequences for edge states:

4 n=0 states (Ez=0)

Electron-like edge state. Hole-like edge state. Spin polarized Spin unpolarized Include Zeeman coupling Domain wall

∆(X0) −∆(X0)

Description of the domain wall: > ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = Ψ

+ ↑ − < + ↓ − + ↑ +

∏

| 2 ) ( sin 2 ) ( cos

, , , , , , X L X X i X

C C e X C X

ϕ

θ θ

; ; = = → = −∞ → ϕ π θ θ L X X

( )

) ( cos ) (

2 2

X X E dX d E

L X z L X s

θ θ ρ π

∑ ∑

< <

∆ − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = l

Pseudospin stiffness

2 4 6 8 10 12 0.0 0.1 0.2 0.3 0.4 0.5

θ(X0)/2π X0/ℓ

Result of minimizing

• energy. Width of

domain wall set by strength of confinement.

y x

• III. Properties of the Domain Wall
• 1. φ=0: Broken U(1) symmetry ⇒ linearly dispersing collective mode

= +,↑ = -,↓

φ ~ in-plane angle of “spins” m ~ position of domain wall

• 2. Spin-charge coupling ⇒ gapless charged excitations!

X0=kyl2 = weight in w/f Twist phase once X0=kyl2

Fermion operator:

x y

Graphene sheet Domain wall

• 3. Tunneling from STM

tip: power law IV ⇒ not a Fermi liquid! Power law exponent a function

• f confinement potential

STM tip

( ) ( ) ( ) ( )

[ ]

∫

− − − E G eV E G eV E G E G dE t I

adv DW ret tip ret DW adv tip 2

~

( ) ( ) ( )

κ τ

τ ψ τ ψ τ 1 ~ ; ; ~ = =

+ y

y T G

( )

Γ = + = / ~ 4 ; 2 / / 1 ρ π κ x x x

⇒ Exponent sensitive to edge confinement!

tunneling t

U(1) spin stiffness Γ ~ confinement potential

Model lead as non-interacting electrons in a magnetic field ⇒

x y

Tunneling t Lead

• 4. Tunneling from a bulk lead: possibility of a quantum phase

transition (into 3D metal).

2 2

) 2 ( t dl dt − − = κ

with Perturbative RG:

Shrinking t ⇒ DW a Luttinger liquid Growing t ⇒ DW + lead = Fermi liquid?

• IV. Inter-Landau Level Excitations (Magnetoplasmons)

Low-lying excited states = particle-hole pairs Standard 2DEG:

ql2 q Hole in filled band Electron in empty band

• Measurable in cyclotron resonance, inelastic light scattering.
• This picture is largely the same for graphene, just need

to be careful about spinor structure of particle and hole states.

Electron Hole

Two-Body Problem

(ħ=ωc=l=1)

• cf. Kallin and

Halperin, 1984

To diagonalize (A= -Byx):

• 1. Adopt center and relative coordinate R=(r1+r2)/2, r= r1-r2
• 2. Apply unitary transformation H´0 = U+H0U with

ixY P z p i

e e U

− × ⋅

=

) ˆ ( r r

with

Wavefunctions constructed from: with

= center of mass momentum

P r

z=x+iy

Wavefunctions are 4-vectors |n+,n-> constructed from with energies

Electron Hole

• 3. Apply unitary transformation to interaction H1:
• 4. Compute eigenvalues of

⇒ two-body eigenenergies with fixed P

2 4 6 8 10 12 P 1 2 EnergyvF 2, 1 3, 1 4, 1 5, 1 2, 0

2 4 6 8 10 12 P 1 2 EnergyvF 1, 0 2, 0 3, 0 4, 0, 1,1

Lowest filled LL = 0 Lowest filled LL = 1

Interaction scale: β=(e2/εl)/(ħvF/l ) ≈ (c/vFε)/137 = 0.73 Results:

γ0=4 γ1=4

2 4 6 8 10 12 P 1 2 EnergyvF 0, 0 0, 1, 1, 0 0, 2, 2, 0

2 4 6 8 10 12 P 1 EnergyvF 1, 1 2, 1 3, 1 1, 0, 4, 1

γ0<4 γ1<4 Note negative energy

Comments:

• 1. Negative energies because we have not included loss of

exchange self-energy ⇒ many-body approach needed

• 2. Landau level mixing relatively small

2 4 6 8 10 12 P 0.1 0.2 Probability

Note however for β ≈ 1, LL mixing becomes much more pronounced ⇒ system on cusp between weakly and strongly interacting

Many-Body Particle-Hole Approach

• A generalization of spin-wave calculation

Almost, but not quite.

Must watch out for degeneracies

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

EF

K,K’

1 2

• Excitations characterized

by ∆Sz and ∆τz

Also: Exchange energy with “infinite” number of filled hole levels leads to (logarithmically) divergent self-energy. Fix this with an explicit cutoff in number of filled Landau levels.

(−1,−1) (0 , 0) (0 ,−1) (0 , 1) (−1 , 0) (1 , 0) (1, 1) (−1, 1) (1 ,−1)

µ

γ=4 ↑,↓=spin, double arrows=pseudospin (m,n)=(sz,tz)

Energy generically involves four terms:

Noninteracting Direct (Ladders) Exchange (Bubbles - RPA) Exchange self-energy

Results: N=0

Comments: 1. Change in kinetic energy and Zeeman energy must be added in 2. Gapless excitations for ν=-1,1 3. Excitation spectra identical for ν=-1,1: particle-hole symmetry Two-body result (up to constant)

2 4 6 8 10 12 P 1 E2 Ε E0

γ0<4 Intra-Landau level

2 4 6 8 10 12 P 2 E2 Ε E1 E2 E3 E4

γ0<4 γ0≤4 ∆n=1

2 4 6 8 10 12 P 1 E2 Ε E1 E2 2 4 6 8 10 12 P 1 2 E2 Ε E1 E2

γ1=4 γ2=4 Very large many-body correction! Dashed lines equivalent to two-body result.

• Minima/maxima may be visible in inelastic light scattering or

microwave absorption.

Summary

• Graphene: a new and interesting material both for fundamental

and applications reasons.

• Clean system is likely a quantum Hall ferromagnet.
• Armchair edges: oppositely dispersing spin up and down bands

⇒ domain wall

• Domain wall supports gapless collective excitations, and

gapless charged excitations through pseudospin texture.

• Domain wall supports power law IV (Luttinger liquid).
• Domain wall may undergo quantum phase transition when

coupled to a bulk lead.

• Collective inter-Landau level excitations = excitons
• Many-body corrections split and distort dispersions found in

two-body problem