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

Luttinger Liquid at the Edge of Liquid at the Edge of Luttinger a Graphene Graphene Vacuum Vacuum a 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 Honeycomb lattice, two atoms per unit cell Lattice constant: 2.46Å B Nearest neighbor distance: 1.42Å A Simple tight-binding model for p z orbitals: ∑ = − H t n n 1 2 = n n n . n . 1 2 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 E F

Wavefunctions in a magnetic field: ⎛ ⎞ ⎛ ⎞ ± φ − 2 ± φ − 2 ( y k ) ( y k ) ik x l ik x l ⎜ ⎟ ⎜ ⎟ ′ Ψ = − Ψ = n 1 x n x ( K , n ) e ( K , n ) e x x ⎜ ⎟ ⎜ ⎟ φ − φ − 2 2 ( y k ) ( y k ) l l ⎝ ⎠ ⎝ ⎠ − n 1 x n x ⎛ ⎞ φ 0 ⎛ ⎞ ik x ik x ⎜ ⎟ ′ Ψ = ⎜ ⎟ Ψ = 0 ( K , 0 ) e ( K , 0 ) e x x ⎜ ⎟ ⎜ ⎟ φ 0 ⎝ ⎠ ⎝ ⎠ 0 φ = harmonic oscillator state n Energies: ε a ε τ = ± ( , n ) 3 n t Particle-hole 0 l conjugates With valley and spin indices, each Landau level is 4-fold degenerate k x

• 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) 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 • 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=2 n=1 K,K’ E F n=0 K,K’ n=-1 n=-2 Low-lying excitations: 2 (+2) spin (+valley) waves E F Spin stiffness ⇒ Analogy with Heisenberg ferromagnet.

Consequences for edge states: Electron-like ∆ (X 0 ) 4 n=0 states edge state . (E z =0) Hole-like −∆ (X 0 ) edge state . Include Zeeman coupling Domain wall Spin polarized Spin unpolarized

Description of the domain wall: θ θ ⎡ ⎤ ( X ) ( X ) ∏ + ϕ + + Ψ = + > i 0 0 cos C sin e C C | 0 ⎢ ⎥ + ↑ − ↓ − ↑ , X , , X , , X , ⎣ 2 2 ⎦ 0 0 0 < X L 0 → −∞ θ = → θ = π ϕ = X 0 ; X L ; 0 0 0 2 ⎛ ⎞ θ d ( ) ∑ ∑ ⎜ ⎟ = l π 2 ρ + − ∆ θ E E ( X ) cos ( X ) ⎜ ⎟ s z 0 0 dX ⎝ ⎠ < < X L X L 0 0 0 Pseudospin stiffness 0.5 Result of minimizing 0.4 θ (X 0 )/2 π energy. Width of 0.3 domain wall set by 0.2 strength of confinement. 0.1 0.0 0 2 4 6 8 10 12 X 0 / ℓ

III. Properties of the Domain Wall = +, ↑ y = -, ↓ x 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! Twist phase once X 0 =k y l 2 X 0 =k y l 2 = weight in w/f Fermion operator:

STM tip 3. Tunneling from STM tip: power law IV ⇒ not a Fermi liquid! tunneling y t Power law exponent a function x of confinement potential Graphene sheet Domain wall [ ] ( ) ( ) ( ) ( ) ∫ − − − 2 adv ret ret adv I ~ t dE G E G E eV G E eV G E tip DW tip DW 1 ( ) ( ) + y ( ) τ ψ = τ ψ = G ~ T y 0 ; 0 ; 0 ~ τ κ τ ( ) ~ κ = + = π ρ Γ x 1 / x / 2 ; x 4 / U(1) spin stiffness Γ ~ confinement potential ⇒ Exponent sensitive to edge confinement!

4. Tunneling from a bulk lead: possibility of a quantum phase transition (into 3D metal). Lead Model lead as non-interacting electrons in a magnetic field y ⇒ Tunneling t x with 2 dt Perturbative Shrinking t ⇒ DW a Luttinger liquid = − κ − 2 ( 2 ) t Growing t ⇒ DW + lead = Fermi liquid? RG: dl

IV. Inter-Landau Level Excitations (Magnetoplasmons) Low-lying excited states = particle-hole pairs Electron in empty band cf. Kallin and Standard 2DEG: q l 2 Halperin, 1984 q 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 ( ħ = ω c = l =1) Hole Electron

To diagonalize (A = - By x ) : 1. Adopt center and relative coordinate R =( r 1 + r 2 )/2, r = r 1 - r 2 2. Apply unitary transformation H´ 0 = U + H 0 U with r r r ⋅ × − = ˆ P i p ( z P ) ixY = center of mass momentum U e e with z=x+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 fixed P

Results: γ 0 =4 γ 1 =4 � 4, 0 � , 2 � 2, 0 � � 1, � 1 � � 5, 1 � � 3, 0 � � 4, 1 � Energy � � � � v F � � � Energy � � � � v F � � � 2 � 2, 0 � � 3, 1 � 1 � 1, 0 � � 2, 1 � 1 Lowest filled LL = 0 Lowest filled LL = 1 2 4 6 8 10 12 2 4 6 8 10 12 P � � P � � Interaction scale: β =(e 2 / ε l )/( ħ v F / l ) ≈ (c/v F ε )/137 = 0.73

γ 0 <4 γ 1 <4 2 � 0, � 2 � , � 1, 0 � , � 2, 0 � � 4, 1 � � 0, � 1 � , � 3, 1 � 1 � 1, 0 � Energy � � � � v F � � � 1 Energy � � � � v F � � � � 2, 1 � 0 � 0, 0 � 0 � 1, 1 � 2 4 6 8 10 12 2 4 6 8 10 12 P � � P � � 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 0.2 Probability 0.1 2 4 6 8 10 12 P � � 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=2 n=1 1 E F n=0 2 n=-1 n=-2 • Excitations characterized by ∆ S z and ∆ τ z K,K’ 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.

µ (1 ,−1) (1, 1) (−1,−1) (−1, 1) (0 , 0) (0 ,−1) (0 , 1) (−1 , 0) (1 , 0) γ =4 ↑ , ↓ =spin, double arrows=pseudospin (m,n)=(s z ,t z )

Energy generically involves four terms: Noninteracting Direct (Ladders) Exchange (Bubbles - RPA) Exchange self-energy

Results: N=0 1 � E 0 � E � � � � 2 � Ε � � � Two-body result (up to constant) γ 0 <4 Intra-Landau level 0 0 2 4 6 8 10 12 Comments: P � � 1. Change in kinetic energy and γ 0 <4 Zeeman energy must be added in � E 3 � E 4 2. Gapless excitations for ν =-1,1 2 γ 0 ≤ 4 3. Excitation spectra identical for � E � � � � 2 � Ε � � � � E 1 ν =-1,1: particle-hole symmetry ∆ n=1 � E 2 0 2 4 6 8 10 12 P � �

Dashed lines equivalent to two-body result. γ 1 =4 � E 1 � E � � � � 2 � Ε � � � 1 Very large many-body correction! � E 2 0 2 4 6 8 10 12 P � � 2 � E 1 γ 2 =4 � E � � � � 2 � Ε � � � 1 � E 2 0 2 4 6 8 10 12 P � � • 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