Luttinger Liquid at the Edge of a Graphene Vacuum H.A. Fertig, Indiana University Luis Brey, CSIC, Madrid I. Introduction: Graphene Edge States (Non-Interacting) II. III. Quantum Hall Ferromagnetism and a Domain Wall at the Edge Properties of the Domain Wall IV. Ecitations from Filled (Graphene) Landau Levels (with Drew Iyengar and Jianhui Wang) V. Summary Funding: NSF
I. Edge States for Graphene B Honeycomb lattice, two atoms per unit cell Lattice constant:.46å Nearest neighbor distance: 1.4Å A Simple tight-binding model for p z orbitals: H = t n 1 n1n = n. n. t.5-3 ev n
A and B sublattice sites in unit cell For each k there are eigenvalues at ± ε particle-hole symmetry Fermi energy at ε= E F
t a n n l 3 ), ( ± = τ ε ± = Ψ ) ( ) ( ), ( 1 l l n n k y k y ik e n K φ φ ± = Ψ ) ( ) ( ), ( 1 l l n n k y k y ik e n K φ φ = Ψ,) ( φ ik e K = Ψ,) ( φ ik e K Wavefunctions in a magnetic field: state = harmonic oscillator φ n Energies: ε Particle-hole conjugates k With valley and spin indices, each Landau level is 4-fold degenerate
Real samples in eperiments 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 DEG s (GaAs samples, Si MOSFET s) K K K y K y Tight-binding results, armchair edge
II. Quantum Hall Ferromagnetism and the Graphene Edge E F DOS Energy E F Interactions DOS Energy Echange tends to force electrons into the same level even when bare splitting between levels is small Renormalizes gap to much larger value than epected from non-interacting theory (even if bare gap is zero!)
This does happen in graphene (Zhang et al., 6). Plateaus at ν=?,±1. System may be a quantum Hall ferromagnet. cf. Alicea and Fisher, 6 Nomura and Macdonald, 6 Fuchs and Lederer, 6
Vacuum state (undoped graphene): n= n=1 K,K n= K,K EF n=-1 n=- Low-lying ecitations: (+) spin (+valley) waves EF Spin stiffness Analogy with Heisenberg ferromagnet.
Consequences for edge states: 4 n= states (E z =) Electron-like edge state. (X ) Include Zeeman coupling Hole-like edge state. (X ) Domain wall Spin polarized Spin unpolarized
Description of the domain wall: θ( X Ψ = ) + θ( X) iϕ + + cos C + sin e C C > +, X,, X,, X, X < L X θ = ; X L θ = π; ϕ = E Pseudospin stiffness = πl ρ s X < L dθ dx + X < L ( E X )) cosθ ( X ) z ( θ(x )/π.5.4.3..1. 4 6 8 1 1 Result of minimizing energy. Width of domain wall set by strength of confinement. X /l
III. Properties of the Domain Wall = +, y = -, 1. φ=: Broken U(1) symmetry linearly dispersing collective mode φ ~ in-plane angle of spins m ~ position of domain wall
. Spin-charge coupling gapless charged ecitations! Twist phase once X =k y l = weight in w/f X =k y l Fermion operator:
3. Tunneling from STM tip: power law IV not a Fermi liquid! Power law eponent a function of confinement potential tunneling t Domain wall STM tip y Graphene sheet I ~ t de G [ ( E) G ( E ev ) G ( E ev ) G ( E) ] adv ret ret adv tip DW tip DW G 1 τ κ = + 1/ / ; = 4π ~ ρ / ( τ ) ~ T ( ) ( ) τψ y = ; τ ψ + y = ; ~ κ ( ) Γ U(1) spin stiffness Γ ~ confinement potential Eponent sensitive to edge confinement!
4. Tunneling from a bulk lead: possibility of a quantum phase transition (into 3D metal). Lead Tunneling t y Model lead as non-interacting electrons in a magnetic field with Perturbative RG: dt dl = ( κ ) t Shrinking t DW a Luttinger liquid Growing t DW + lead = Fermi liquid?
IV. Inter-Landau Level Ecitations (Magnetoplasmons) Low-lying ecited states = particle-hole pairs Electron in empty band Standard DEG: ql q cf. Kallin and Halperin, 1984 Hole in filled 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. Two-Body Problem Electron Hole (ħ=ω c =l=1)
To diagonalize (A= -By): 1. Adopt center and relative coordinate R=(r 1 +r )/, r= r 1 -r. Apply unitary transformation H = U+ H U with r r ip ( zˆ P) iy P r = center of mass momentum U = e e with z=+iy Wavefunctions constructed from: with
Wavefunctions are 4-vectors n +,n - > constructed from with energies Electron Hole 3. Apply unitary transformation to interaction H 1 : 4. Compute eigenvalues of two-body eigenenergies with fied P
Results: γ =4 γ 1 =4 4,, 1, 1 3,, 5, 1 Energy ħ vf 1 Lowest filled LL =, 1, Energy ħ vf 1 Lowest filled LL = 1 4, 1 3, 1, 1 4 6 8 1 1 P 4 6 8 1 1 P Interaction scale: β=(e /εl)/(ħv F /l ) (c/v F ε)/137 =.73
γ <4 γ 1 <4,,, 1,, 4, 1, 1, 1, 1 3, 1 Energy ħ vf 1, Energy ħ vf, 1 1, 1 4 6 8 1 1 P 4 6 8 1 1 P Note negative energy
Comments: 1. Negative energies because we have not included loss of echange self-energy many-body approach needed. Landau level miing relatively small. Probability.1 4 6 8 1 1 P Note however for β 1, LL miing 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= n=1 n= 1 E F n=-1 n=- K,K Ecitations characterized by S z and τ z Also: Echange energy with infinite number of filled hole levels leads to (logarithmically) divergent self-energy. Fi this with an eplicit cutoff in number of filled Landau levels.
µ (1, 1) (1, 1) ( 1, 1) ( 1, 1) (, ) (, 1) (, 1) ( 1, ) (1, ) γ=4, =spin, double arrows=pseudospin (m,n)=(s z,t z )
Energy generically involves four terms: Noninteracting Direct (Ladders) Echange (Bubbles - RPA) Echange self-energy
Results: N= 1 E E e Ε γ <4 Intra-Landau level Two-body result (up to constant) E e Ε 4 6 8 1 1 P E 1 E 3 E 4 γ <4 E γ 4 n=1 Comments: 1. Change in kinetic energy and Zeeman energy must be added in. Gapless ecitations for ν=-1,1 3. Ecitation spectra identical for ν=-1,1: particle-hole symmetry 4 6 8 1 1 P
E 1 γ 1 =4 Dashed lines equivalent to two-body result. E e Ε 1 E Very large many-body correction! 4 6 8 1 1 P E e Ε 1 E 1 γ =4 E 4 6 8 1 1 P Minima/maima 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 ecitations, and gapless charged ecitations through pseudospin teture. 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 ecitations = ecitons Many-body corrections split and distort dispersions found in two-body problem