From the honeycomb lattice to the square lattice: a new look at graphene. Timo A. Lähde

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1 From the honeycomb lattice to the square lattice: a new look at graphene Timo A. Lähde Helsinki Institute of Physics / Department of Applied Physics, Aalto University (ex HUT), FI Espoo, Finland Simulations and Symmetries: Cold Atoms, QCD and Few-Hadron systems, Institute for Nuclear Theory, University of Washington, Seattle, USA, March 15 - May 21, 2010 in collaboration with: Joaquín E. Drut (Ohio State U.) Lauri Suoranta (Aalto U.)

2 Graphene phenomenology Origin of the semimetallic band structure... Graphene at low energies Does a gap form due to Coulomb interactions? Lattice Monte Carlo simulation From the honeycomb to the square lattice... Results Is suspended graphene an insulator? Lattice artifacts Are the simulation results realistic? Future projects Where to go from here?

3 Graphene phenomenology I: what is graphene? Carbon atoms form strong covalent bonds... graphene single graphite layer graphite stacked graphene nanotube rolled graphene fullerene wrapped graphene

4 Graphene phenomenology II: a promising material for applied physics... Graphene-based gate-controlled current switch... K. Sääskilahti, A. Harju, P. Pasanen, Appl. Phys. Lett. 95, (2009). Suspended graphene devices... K.I. Bolotin et al., Phys. Rev. Lett. 101, (2008), V. Crespi, Physics 1, 15 (2008).

5 Graphene phenomenology III:... and for fundamental physics as well! Ultrarelativistic (Dirac) dispersion relation in a non-relativistic system... Pictures: Lanzara research group, UC Berkeley Direct study of the electronic dispersion relation in graphene and related materials via ARPES...

6 Graphene phenomenology IV: not surprisingly, a very active field of study... Quarterly # of publications on graphene (yellow) and bilayer graphene (grey) on arxiv... (1) - discovery of graphene (2) - discovery of the QHE in graphene

7 Graphene phenomenology V: a closer look at the physics... Hybridized electron orbitals form a hexagonal honeycomb lattice... 1s 2 2s 2 2p 2 3 sp 2 + p lattice free electron Figure: A.H. Castro Neto, Materials Today 13, 1 (2010). The hexagonal symmetry leads to linear dispersion at low energies... P. R. Wallace, Phys. Rev. 71, 622 (1947).

8 Graphene phenomenology VI: theory of the electronic band structure... S. Reich et al, Phys. Rev. B66, (2002). Tight-binding description of the electron-ion interactions in graphene... Creation and annihilation operators for electrons on sublattices (A,B) Conical dispersion around two valleys, centered around the Dirac points (K,K )... A B k y b 1 " " a 3 1 a 2 " 1 2! K M K k x Theory of non-interacting electrons b 2

9 Graphene at low energies I: massless Dirac quasiparticles... The non-interacting theory describes a gapless semimetal... However: the quasiparticle velocity is a fraction of the speed of light in vacuum! In the vicinity of a Dirac point : Fermions (in 2+1 dimensions) Dirac flavors, N f = 2 describes a graphene monolayer... Velocity of quasiparticles

10 Graphene at low energies II: quasiparticles with instantaneous Coulomb interactions... Gauge field (in 3+1 dimensions) Electrostatic Coulomb interaction... Gauge invariance Whether the Coulomb interaction is significant depends on the dielectric constant of the environment... α g e2 4πɛ 0 v 300α 1 Fine-structure constant of graphene

11 Graphene at low energies III: spontaneous chiral symmetry breaking... G.W. Semenoff, Phys. Rev. Lett. 54, 2449 (1984), C. Burden, A.N. Burkitt, Eur. Phys. Lett. 3, 545 (1987). The massless quasiparticles possess chiral symmetry, which can be spontaneously broken... U(2Nf) U(Nf) x U(Nf) Order parameter, chiral condensate A conjectured electronic phase diagram, as a function of inverse coupling and fermion flavors... chiral symmetry unbroken, quasiparticles remain massless. chiral symmetry spontaneously broken, quasiparticles massive.

12 Lattice Monte Carlo simulation I: evaluating the condensate... J.E. Drut, T.A. Lähde, Phys. Rev. B79, (2009). Integrate out the fermion fields... Positive definite probability measure for MC calculation Evaluate observables stochastically by generating snapshots of the gauge field... Compute the condensate at different (inverse) interaction strengths... β = ɛ 0v e 2

13 Lattice Monte Carlo simulation II: finding the transition... M. Göckeler et al, Nucl. Phys. B487, 313 (1997). Discretize on a square lattice with finite lattice spacing (acts as an UV cutoff)... Perform calculations at finite fermion mass (acts as an IR cutoff)... m! # $ % & ( ) +,-./ :;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_ 0 abcdefghijklmnopqrstuvwxyz{ }~ 1st order Tricritical point β c 2nd order β We want to study the critical region of the theory, however: Continuum physics is recovered in the vicinity of the second-order transition 1) Dirac operator develops small eigenvalues 2) Large finite-volume effects (correlation length diverges) In practice, extrapolate using results obtained at different masses and couplings...

14 Lattice Monte Carlo simulation III: discretized theory... H. Rothe, Lattice gauge theories, World Scientific (2005). Non-compact gauge action... Doubling problem for chiral lattice fermions: 1 staggered fermion flavor gives 2 continuum flavors! H.B. Nielsen, M. Ninomiya, Nucl. Phys. B185, 20 (1981); Nucl. Phys. B193, 173 (1981). C. Burden, A.N. Burkitt, Eur. Phys. Lett. 3, 545 (1987). Gauge invariance on the lattice: gauge links in the fermion action...

15 Lattice Monte Carlo simulation IV: generation of gauge configurations, Hybrid Monte Carlo... S. Duane et al., Phys. Lett. B 195, 216 (1987). Add to the Euclidean action a random Gaussian noise component... Introduce pseudofermions to provide efficient updating of the fermion action... Enables global updates of the lattice gauge potential Evolve the gauge field by numerically integrating the EOM (Molecular Dynamics)... Exact method: MD evolution error corrected by Metropolis step...

16 Lattice Monte Carlo simulation V: extrapolation to the critical point... A. Kocić, J.B. Kogut, K.C. Wang, Nucl. Phys. B398, 405 (1993). M. Göckeler et al, Nucl. Phys. B487, 313 (1997). Equation of state analysis... Trial function (also for QED4)... Simultaneous fit to condensate and susceptibility Dependence on σ: Information on critical exponents δ, βm! Dependence on β: Information on critical coupling βc! f 1 (σ) =σ δ 1/β m f 3 (σ) =σ δ

17 Results I: quantum phase transition into a gapped phase... J.E. Drut, T.A. Lähde, Phys. Rev. Lett. 102 (2009) J.E. Drut, T.A. Lähde, Phys. Rev. B79 (2009) βc ~ ± 0.002! 0.10 m 0 = 0, extrapolation 8 x 28 3, fitted excluded 12 x 32 3, fitted excluded # l 8 N f = 2, m 0 = Critical coupling from EOS analysis! " "

18 Results II: is the semimetal-insulator transition observable? Graphene on a SiO 2 substrate β ~ 0.10 Our critical coupling βc ~ Suspended graphene β ~ Figure: A.H. Castro Neto, Physics 2, 30 (2009) Under ideal circumstances: should be observable for suspended graphene samples!!

19 Results III: critical exponents, critical # of flavors... The EOS extrapolation indicates a secondorder transition, critical exponents: δ = 2.2 ± 0.1 β = 0.83 ± 0.05 γ = 1.0 ± 0.04 Consistent with J.E. Drut, T.A. Lähde, L.J. Suoranta, arxiv: Simulations for N f = 4 show a transition at stronger coupling, for N f = 6 nothing is observed: 4 < Ncrit < 6 J.E. Drut, T.A. Lähde, Phys. Rev. Lett. 102 (2009)

20 Results IV: supporting results by other groups (some examples)... Analytical Dyson-Schwinger calculations: consistent with our results, however infinite-order transition... D.V Khveschenko, Phys. Rev. Lett. 87, (2001) E.V. Gorbar et al., Phys. Rev. B66, (2002) Lattice Monte Carlo simulations: strong-coupling limit, closely related to Thirring model in (2+1) dimensions... S.J. Hands, C. Strouthos, Phys. Rev. B78, (2008) S.J. Hands, C. Strouthos, W. Armour, arxiv: Large N f treatment: second-order transition verified, critical exponents disagree... I. Herbut, V. Juričić, O. Vafek, Phys. Rev. B80, (2009) Strong-coupling expansion of the lattice theory: chiral EFT technique, transition to insulating phase confirmed... Y. Araki, T. Hatsuda, arxiv:

21 Lattice artifacts I: photon self-interactions on the lattice... We chose arbitrarily to simulate the non-compact theory of graphene... Equally well, we could simulate the compact theory instead, identical continuum limit... Gauge links introduce higher-order vertices (self-interactions, tadpoles)... How do these affect the simulation?

22 Lattice artifacts II: effects of photon self-interactions in the compact theory... J.E. Drut, T.A. Lähde, L.J. Suoranta, arxiv: First order transition, no approach to the continuum limit (!) βc ~ 0.40! , , # l , , " As for (3+1) dimensional QED, the compact theory of graphene bears little resemblance to continuum physics... "

23 Lattice artifacts III: tadpole improvement... G.P. Lepage, P.B. Mackenzie, Phys. Rev. D48, 2250 (1993). However, UV divergent tadpole contributions do not vanish as a power of the lattice spacing... Integrate out the tadpole contributions by renormalizing the gauge links... Renormalized gauge links give results closer to the continuum limit, estimate the correction a posteriori...

24 Lattice artifacts IV: tadpole improvement of the non-compact theory... J. Giedt, A. Skinner, S. Nayak, arxiv: J.E. Drut, T.A. Lähde, L.J. Suoranta, arxiv: Overall: effectively a shift in the parameters of the theory! Additionally, for the compact theory: 1 P x28 3, , P Average plaquettes in the noncompact and compact theories ! , , !

25 Lattice artifacts V: unimproved results for the non-compact theory... J.E. Drut, T.A. Lähde, L.J. Suoranta, arxiv: βc ~ ± Data at strong coupling cannot be fitted, scaling violations...! m = 0, extrapolation 12 x 32 3, 32 4, fitted excluded 8 x 28 3, fitted excluded # l N f = 2, m = " "

26 Lattice artifacts VI: tadpole-improved non-compact theory... J.E. Drut, T.A. Lähde, L.J. Suoranta, arxiv: Significant effect, insulating phase more likely! Fit range and stability much improved... βc ~ ± 0.001! m = 0, extrapolation # l N f = 2, m = " "

27 Lattice artifacts VII: tadpole-improved compact theory? J.E. Drut, T.A. Lähde, L.J. Suoranta, arxiv: TI βc ~ 0.40 βc ~ 0.25 Transition becomes less likely - effect of TI very different from the non-compact case!! , , # l , , " "

28 Lattice artifacts VIII: staggered fermions and chiral symmetry... Our objective is to study the following chiral symmetry breaking pattern... U(4) U(2) x U(2) However, using one flavor of staggered fermions at finite lattice spacing, we have only a smaller symmetry... U(1) x U(1) U(1) C. Burden, A.N. Burkitt, Eur. Phys. Lett. 3, 545 (1987). The full symmetry is restored in the continuum limit, but is the extrapolation reliable? Ultimately: simulate graphene with overlap fermions!!

29 Future projects I: fermion velocity renormalization... Collaboration: Lauri Suoranta (Aalto U.), master s thesis project Joaquín Drut (Ohio State U.) S.J. Hands, C. Strouthos, W. Armour, arxiv:

30 Future projects II: conductivity of graphene... Collaboration: Eero Tölö, Lauri Suoranta (Aalto U.) Joaquín Drut (Ohio State U.) K.I. Bolotin et al., Phys. Rev. Lett. 101, (2008), V. Crespi, Physics 1, 15 (2008).

31 Future projects III: exciton condensation in bilayer graphene... Collaboration: Joaquín Drut (Ohio State U.) Allan MacDonald (UT Austin)... M.Yu. Kharitonov, K. Efetov, arxiv: