Electron-electron interactions and Dirac liquid behavior in graphene bilayers

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1 Electron-electron interactions and Dirac liquid behavior in graphene bilayers arxiv:85.35 S. Viola Kusminskiy, D. K. Campbell, A. H. Castro Neto Boston University Workshop on Correlations and Coherence in Quantum Matter Évora 28 S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 1 / 7

2 Non-Interacting System Bilayer Graphene S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 2 / 7

3 Non-Interacting System Bilayer Graphene Tight Binding Model H = 4 P p ψ p v F pe iφ(p) t v F pe iφ(p) t v F pe iφ(p) v F pe iφ(p) 1 C A ψp with ψ p = (c p,a 1, c p,b 1, c p,b 2, c p,a 2 ) S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 2 / 7

4 Non-Interacting System Bilayer Graphene Tight Binding Model H = 4 P p ψ p v F pe iφ(p) t v F pe iφ(p) t v F pe iφ(p) v F pe iφ(p) 1 C A ψp with ψ p = (c p,a 1, c p,b 1, c p,b 2, c p,a 2 ) via unitary transformation: H = 4v F Pp ψ p p σ vf mσ z v F mi p σ v F mσ z +v F mi «ψ p S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 2 / 7

5 Non-Interacting System Bilayer Graphene Tight Binding Model H = 4 P p ψ p v F pe iφ(p) t v F pe iφ(p) t v F pe iφ(p) v F pe iφ(p) 1 C A ψp with ψ p = (c p,a 1, c p,b 1, c p,b 2, c p,a 2 ) via unitary transformation: H = 4v F Pp ψ p p σ vf mσ z v F mi p σ v F mσ z +v F mi 2 massive Dirac equations v F = 3ta 2 velocity of light m = t 2v 2 F mass «ψ p S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 2 / 7

6 Non-Interacting System Bilayer Graphene Tight Binding Model H = 4 P p ψ p v F pe iφ(p) t v F pe iφ(p) t v F pe iφ(p) v F pe iφ(p) 1 C A ψp with ψ p = (c p,a 1, c p,b 1, c p,b 2, c p,a 2 ) Non-interacting dispersion via unitary transformation: H = 4v F Pp ψ p p σ vf mσ z v F mi p σ v F mσ z +v F mi 2 massive Dirac equations v F = 3ta 2 velocity of light m = t 2v 2 F mass «ψ p Relativistic dispersion: E 1 (k) = mv 2 F + q(mv 2 F )2 + (v F k) 2 j k 2 /2m k mvf v F k k mv F S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 2 / 7

7 Non-Interacting System Bilayer Graphene Tight Binding Model H = 4 P p ψ p v F pe iφ(p) t v F pe iφ(p) t v F pe iφ(p) v F pe iφ(p) 1 C A ψp with ψ p = (c p,a 1, c p,b 1, c p,b 2, c p,a 2 ) Non-interacting dispersion via unitary transformation: H = 4v F Pp ψ p p σ vf mσ z v F mi p σ v F mσ z +v F mi 2 massive Dirac equations v F = 3ta 2 velocity of light m = t 2v 2 F mass «ψ p Relativistic dispersion: E 1 (k) = mv 2 F + q(mv 2 F )2 + (v F k) 2 j k 2 /2m k mvf v F k k mv F Crossover energy scale: mv 2 F non-relativistic ultra-relativistic S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 2 / 7

8 Coulomb Interaction Compressibility Add Coulomb interactions: V ± (k) = 2πe2 ɛ (1±exp{ kd}) [2(k+βm)] β is the screening strength: β = Hartree - Fock approximation β > Thomas - Fermi approximation S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 3 / 7

9 Coulomb Interaction Compressibility Add Coulomb interactions: V ± (k) = 2πe2 ɛ (1±exp{ kd}) [2(k+βm)] β is the screening strength: β = Hartree - Fock approximation β > Thomas - Fermi approximation S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 3 / 7

10 Coulomb Interaction Compressibility Add Coulomb interactions: V ± (k) = 2πe2 ɛ (1±exp{ kd}) [2(k+βm)] β is the screening strength: β = Hartree - Fock approximation β > Thomas - Fermi approximation Take into account inter-band Vs. intra-band interactions S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 3 / 7

11 Coulomb Interaction Compressibility Add Coulomb interactions: V ± (k) = 2πe2 ɛ (1±exp{ kd}) [2(k+βm)] β is the screening strength: β = Hartree - Fock approximation β > Thomas - Fermi approximation Take into account inter-band Vs. intra-band interactions Inverse Compressibility: µ n e!"#$% &'()#'* &'()#'*+,+-(#.$ S. Viola Kusminskiy et al, Phys. Rev. Lett. 1, 1685 S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 3 / 7

Coulomb Interaction Compressibility Add Coulomb interactions: V ± (k) = 2πe2 ɛ (1±exp{ kd}) [2(k+βm)] β is the screening strength: Take into account inter-band Vs.

The Inter-band contribution is positive.!"#$% &'()#'*+,+-(#.$ &'()#'* This competition is responsible for non - monotonic behavior.

12 Coulomb Interaction Compressibility Add Coulomb interactions: V ± (k) = 2πe2 ɛ (1±exp{ kd}) [2(k+βm)] β is the screening strength: Take into account inter-band Vs. intra-band interactions β = Hartree - Fock approximation β > Thomas - Fermi approximation Inverse Compressibility: µ n e The Intra-band contribution is negative. The Inter-band contribution is positive.!"#$% &'()#'*+,+-(#.$ &'()#'* This competition is responsible for non - monotonic behavior. The inverse compressibility diverges negatively at very low densities. The bump signals a crossover from a 2DEG - like behavior to a graphene monolayer - like one. The effect is present only when the full, four bands model is considered. There exists a very small electron - hole asymmetry. S. Viola Kusminskiy et al, Phys. Rev. Lett. 1, 1685 S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 3 / 7

13 Coulomb Interaction Quasiparticle Energy Energy of a quasiparticle in the i th band: ɛ i (q) = δe/δn i (q) ni =n i E[δn i ] is the total energy E = E + E ex δn i (q) = n i (q) n i (q) n i (k) non-interacting occupation number ɛ i (q) = E i (q) 4 R P q α,j χα ij (q, p)χ α ji (p, q)nj (p)v α(q p) S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 4 / 7

14 Coulomb Interaction Quasiparticle Energy Energy of a quasiparticle in the i th band: ɛ i (q) = δe/δn i (q) ni =n i E[δn i ] is the total energy E = E + E ex δn i (q) = n i (q) n i (q) n i (k) non-interacting occupation number ɛ i (q) = E i (q) 4 R P q α,j χα ij (q, p)χ α ji (p, q)nj (p)v α(q p) S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 4 / 7

15 Coulomb Interaction Quasiparticle Energy Energy of a quasiparticle in the i th band: ɛ i (q) = δe/δn i (q) ni =n i Screening is necessary: diverging Fermi velocity as in 2DEG E[δn i ] is the total energy E = E + E ex δn i (q) = n i (q) n i (q) n i (k) non-interacting occupation number ɛ i (q) = E i (q) 4 R P q α,j χα ij (q, p)χ α ji (p, q)nj (p)v α(q p) S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 4 / 7

16 Coulomb Interaction Quasiparticle Energy Energy of a quasiparticle in the i th band: ɛ i (q) = δe/δn i (q) ni =n i Screening is necessary: diverging Fermi velocity as in 2DEG E[δn i ] is the total energy E = E + E ex δn i (q) = n i (q) n i (q) n i (k) non-interacting occupation number ɛ i (q) = E i (q) 4 R P q α,j χα ij (q, p)χ α ji (p, q)nj (p)v α(q p) Relativistic dispersion recovered! E 1 (k) = ɛ + (m vf 2)2 + (vf k)2 Both mass and light velocity get renormalized S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 4 / 7

17 Coulomb Interaction Quasiparticle Energy Energy of a quasiparticle in the i th band: ɛ i (q) = δe/δn i (q) ni =n i Screening is necessary: diverging Fermi velocity as in 2DEG E[δn i ] is the total energy E = E + E ex δn i (q) = n i (q) n i (q) n i (k) non-interacting occupation number ɛ i (q) = E i (q) 4 R P q α,j χα ij (q, p)χ α ji (p, q)nj (p)v α(q p) Relativistic dispersion recovered! E 1 (k) = ɛ + (m vf 2)2 + (vf k)2 Both mass and light velocity get renormalized due to inter-band interactions S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 4 / 7

18 Coulomb Interaction Renormalized quantities e density dependence: m and v F Coulomb interactions: decrease the quasiparticle mass: m < m increase the light velocity v F > v F S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 5 / 7

19 Coulomb Interaction Renormalized quantities e density dependence: m and v F e density dependence: n c and E c Coulomb interactions: decrease the quasiparticle mass: m < m increase the light velocity v F > v F Effect on the NR UR crossover: increase the crossover energy E c = mv 2 F : E c > E c decrease the crossover density n c = (mv F ) 2 /π: n c < n c S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 5 / 7

20 Fit to experimental data Non-interacting BLG in B-field: Landau Levels E ± n ωc =± j n r n positive integer h i ff 1+16r 4 +16r 2 n+ 1 1/2 1/2 2 ω c = v F p 2eB/c cyclotron frequency r = mv 2 F /ωc S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 6 / 7

21 Fit to experimental data Non-interacting BLG in B-field: Landau Levels E ± n ωc =± j n r n positive integer h i ff 1+16r 4 +16r 2 n+ 1 1/2 1/2 2 ω c = v F p 2eB/c cyclotron frequency r = mv 2 F /ωc Interacting BLG in a B-field: Quasiparticle dispersion with renormalized parameters m, v F Replace in LL m m v F v F S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 6 / 7

22 Fit to experimental data Non-interacting BLG in B-field: Landau Levels E ± n ωc =± j n r n positive integer h i ff 1+16r 4 +16r 2 n+ 1 1/2 1/2 2 ω c = v F p 2eB/c cyclotron frequency r = mv 2 F /ωc Free parameters Bare light velocity v F = 3ta 2 Bare mass m = t 2v 2 F Dimensionless coupling α = e 2 / v F ɛ α.5 for SiO 2 Screening strength: β RPA = 4α (1 + E F /m) β 1 5 for typical densities Interacting BLG in a B-field: Quasiparticle dispersion with renormalized parameters m, v F Replace in LL m m v F v F S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 6 / 7

23 Fit to experimental data Non-interacting BLG in B-field: Landau Levels E ± n ωc =± j n r n positive integer h i ff 1+16r 4 +16r 2 n+ 1 1/2 1/2 2 ω c = v F p 2eB/c cyclotron frequency r = mv 2 F /ωc Interacting BLG in a B-field: Free parameters Bare light velocity v F = 3ta 2 Bare mass m = t 2v 2 F Dimensionless coupling α = e 2 / v F ɛ α.5 for SiO 2 Screening strength: β RPA = 4α (1 + E F /m) β 1 5 for typical densities Cyclotron Resonance Data Quasiparticle dispersion with renormalized parameters m, v F Replace in LL m m v F v F!E LL [mev] "= "=4 2 "= "= B [T] B [T] B [T] Data from E. A. Henriksen et al, Phys. Rev. Lett. 1, S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 6 / 7

24 Fit to experimental data Non-interacting BLG in B-field: Landau Levels E ± n ωc =± j n r n positive integer h i ff 1+16r 4 +16r 2 n+ 1 1/2 1/2 2 ω c = v F p 2eB/c cyclotron frequency r = mv 2 F /ωc Interacting BLG in a B-field: Free parameters Bare light velocity v F = 3ta 2 Bare mass m = t 2v 2 F Dimensionless coupling α = e 2 / v F ɛ α.5 for SiO 2 Screening strength: β RPA = 4α (1 + E F /m) β 1 5 for typical densities Cyclotron Resonance Data Quasiparticle dispersion with renormalized parameters m, v F Replace in LL m m v F v F Results (solid line)!e LL [mev] "=4 2 "= "= α =.5 β = 4 v F = m/s vf = m/s t =.33 ev t = ev B [T] B [T] 2 1 "= B [T] Data from E. A. Henriksen et al, Phys. Rev. Lett. 1, S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 6 / 7

25 Conclusions and Open Questions Within a Hartree-Fock-Thomas-Fermi approximation, after the inclusion of Coulomb interactions, the quasiparticle dispersion can still be described by a Lorentz invariant dispersion but with renormalized parameters. This is a consequence of inter-band interactions which are dominant. The interaction renormalizes up both in-plane (t) and out of plane (t ) hopping parameters. Further corrections? S. Viola Kusminskiy (BU) e-e interactions in BLG Évora8 7 / 7

Luttinger Liquid at the Edge of a Graphene Vacuum

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