Lattice simulation of tight-binding theory of graphene with partially screened Coulomb interactions

Transcription

1 Lattice simulation of tight-binding theory of graphene with partially screened Coulomb interactions Dominik Smith Lorenz von Smekal 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 1

2 Outline Introduction ( Why am I in a QCD session? ) Setup and Results Conclusion and Outlook 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 2

3 Introduction Graphene: Carbon atoms arranged on 2D hexagonal ( honeycomb ) lattice. Two triangular sublattices: A-sites B-sites. In absence of two-body interactions: π-band electrons described by tight-binding theory. H TB = ( κ)(a x,s a y,s + a y,s a x,s), {a µ, a ν } = {a µ, a ν } = 0, {a µ, a ν} = δ µν x,y,s Semi-metallic behavior (no band gap). Linear dispersion at low energies! E( k) = ± vf k, vf c/300 Electrons are massless Dirac fermions! 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 3

4 Introduction Include EM interaction: H = H TB + H INT. v F c/300, α eff = e2 v F 2.2 Low-energy effective theory: (variant of) QED 2+1 But: Strongly coupled! Phenomenology much less clear! Expectation: Interactions generate mass-gap. Corresponds to transition to insulating phase! Sublattice symmetry-breaking Chiral symmetry-breaking in low-energy theory! Interaction strength can be controlled with substrate: α eff α eff /ǫ. Question: Does transition occur in physical parameter range? Use established non-perturbative methods: Dyson-Schwinger, renormalization group, Lattice simulations. 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 4

5 Introduction Low energy theory of graphene (variant of QED 2+1 ) simulated with staggered fermions: Drut, Lähde, Phys.Rev.Lett. 102, (2009) Drut, Lähde, Phys.Rev. B 79, (2009) Drut, Lähde, Phys.Rev. B 79, (2009) Thirring model in 2+1 dimensions simulated: Hands, Strouthos, Phys.Rev. B 78, (2008) Armour, Hands, Strouthos, Phys.Rev. B 81, (2010) Armour, Hands, Strouthos, Phys.Rev. B 84, (2011) Fully hexagonal system coupled to gauge-links simulated: Buividovich, Polikarpov, Phys.Rev. B 86, (2012) based on framework developed in: Brower,Rebbi, Schaich, PoS (Lattice 2011) 056 All find gapped phase for α eff > α c , well within physical region! 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 5

6 Introduction Low energy theory of graphene (variant of QED 2+1 ) simulated with staggered fermions: Drut, Lähde, Phys.Rev.Lett. 102, (2009) Drut, Lähde, Phys.Rev. B 79, (2009) Drut, Lähde, Phys.Rev. B 79, (2009) Thirring model in 2+1 dimensions simulated: Hands, Strouthos, Phys.Rev. B 78, (2008) Armour, Hands, Strouthos, Phys.Rev. B 81, (2010) Armour, Hands, Strouthos, Phys.Rev. B 84, (2011) Fully hexagonal system coupled to gauge-links simulated: Buividovich, Polikarpov, Phys.Rev. B 86, (2012) based on framework developed in: Brower,Rebbi, Schaich, PoS (Lattice 2011) 056 All find gapped phase for α eff > α c , well within physical region! But: Experiments suggest that suspended graphene is conductor! Why the difference? Elias et al. Nature Phys. 7, 701 (2011) Mayorov et al. Nano Lett. 12, 4629 (2012) 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 5

7 Introduction Interactions are screened by lower orbitals! Screening by (lower) σ-band electrons calculated in constrained random phase approximation (crpa): Wehling et al. Phys.Rev. Lett. 106, (2011) Results for on-site (V 00 ), nearest-neighbor (V 01 ), next-nearest-neighbor (V 02 ) and third-nearest-neighbor (V 03 ) potentials! Ulybyshev et al. Phys.Rev. Lett. 111, (2013) { V00, V V(r) = 01, V 02, V 03 : r 2a e 2 /(1.4 r) : r > 2a Simulations show: α c 3.14 ( 2.2)!, V(r), ev 10 But: Screening isn t constant at large r! Improvements are needed! 1 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 6 Non-compact gauge field Screened potentials Coulomb r, nm

8 Partially screened Coulomb potential Better: Use di-electric screening function at long distances! ǫ 1 ( k) = 1 ǫ 1 ǫ (ǫ 1 1)e kd ǫ (ǫ 1 1)e kd Obtain partially screened potential from Fourier back-transform of Ṽ0( k) = (2πe 2 )/k: V( r) = 1 d 2 k (2π) Ṽ0( k)ǫ 1 ( k) e i k r 2 K 2 =e 2 dk ǫ 1 ( k) J 0(kr). Wehling et al. Phys.Rev. Lett. 106, (2011) (ǫ 1 = 2.4 and d = 2.8Å) V(r) [ev] 10 part. screened CRPA ITEP screened std. Coulomb 0 Asymptotically approaches unscreened potential. Instantaneous approximation valid since v F c/300 c! r [nm] 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 7

9 Results Interacting tight-binding theory: H = ( κ)(a x,s a y,s +a y,s a x,s)+ 1 e 2 q x V xy q y, 2 x,y,s x,y q x = a x,1 a x,1 +a x, 1 a x, 1 1 Simulated on N x = N y = 18, N t = 20 lattice. Rectangular sheets with periodic boundary conditions. Add staggered mass to break sublattice ( chiral ) symmetry explicitly: H H + m S (a x,1 a y,1 a x, 1 a y, 1) (m S = ±m, x A, B) Like in lattice QCD: Take limit m 0! x Order parameter: Difference of spin-density ( chiral condensate ). { } 1 N = n A n B = (a x,1 N x N a y,1 a x, 1 a y, 1) (a x,1 a y,1 a x, 1 a y, 1). y x X A x X B 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 8

10 Results Two different setups: (a) Potential as ITEP (constant screening at large r, crpa at small r). Goal: Reproduction and consistency check. (b) Partially screened Coulomb potential. Goal: Improved results! Interaction strength controlled by re-scaling of α eff. di-electric constant : ǫ 2.2/α eff, α eff α eff /ǫ (V(r) V(r)/ǫ) 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 9

11 Results Two different setups: (a) Potential as ITEP (constant screening at large r, crpa at small r). Goal: Reproduction and consistency check. (b) Partially screened Coulomb potential. Goal: Improved results! Interaction strength controlled by re-scaling of α eff. di-electric constant : ǫ 2.2/α eff, α eff α eff /ǫ (V(r) V(r)/ǫ) Parameters: T K, (low temperature phase) α eff (ǫ = ) m = 0.1, 0.2, 0.3, 0.4, 0.5 ev Several hundreds of independent N measurements (binning!). 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 9

12 Results Chiral limit : m 0 extrapolation done with N = a 0 + a 1 m + a 2 m ITEP screened part. screened < n> α = 2.73 α = 3.36 α = 4.37 α = 4.86 < N > m [ev] α eff Phase transition sets in around α Far in unphysical regime! No significant difference between two setups! 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 10

13 Results Expressed in terms of di-electric screening ǫ 2.2/α eff = : < N > ITEP screened part. screened ε ) n ( n (m) =1.0 0 =0.7 = m, ev Figure on right: Ulybyshev et al. (ITEP), Phys.Rev. Lett. 111, (2013). Clearly consistent! 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 11

14 Conclusions and Outlook Conclusions: Transition from semi-metal to insulator is insensitive to long distance part of potential. Graphene in vacuum is indeed a conductor! Next project: Investigate effect of interactions on Van Hove singularity. Dietz,Smekal et al. arxiv: van Hove singularity van Hove singularity 2 k y Future: External magnetic fields, phonons, edge effects, curvature, Dirac frequency k x Publications: Smith, v.smekal, arxiv: PoS Lattice 2013 Smith, v.smekal, arxiv: (long version!) 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 12

15 Hardware Production runs: Nvidia Tesla GPUs on the Scout Cluster (C1060) of the Center of Scientific Computing of the University of Frankfurt and on the Lichtenberg Cluster (K20) of the Hochschulrechenzentrum TU Darmstadt. Development: Nvidia GTX 780 and GTX Titan cards. Thanks for coming! 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 13