Tight-binding model of graphene with Coulomb interactions

Transcription

1 Tight-binding model of graphene with Coulomb interactions Dominik Smith Lorenz von Smekal Dedicated to the memory of Professor Mikhail Polikarpov 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 1

2 Introduction Many properties of graphene are well understood in etreme weak coupling limit. H = ( κ)(a,s a y,s + a y,s a,s), {a µ, a ν } = {a µ, a ν} = 0, {a µ, a ν} = δ µν,y,s Well described by tight-binding theory. Conical dispersion at low energies. Low energy effective Dirac theory. Van Hove singularity at saddle points. Semi-metallic behavior (no band gap).... Include electromagnetic interaction: Low energy theory becomes QED 2+1. Simulated with staggered Fermions. Drut, Lähde, Phys.Rev.Lett. 102, (2009) Armour, Hands, Strouthos, Phys.Rev.B81:125105, 2010 Presently: Go beyond low energies. Simulate heagons directly. 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 2 Buividovich et al. (ITEP), Phys. Rev. B 86 (2012),

3 Introduction First derivation of path-integral for heagonal lattice: Currently: Tight-binding with gauge-links. Tight-binding with instantaneous interactions. Our goal: Investigate effect of interactions on Van Hove singularity. Dietz,Smekal et al. arxiv: Status: CUDA code operational and producing. Plausibility checks (find α c ) and cross-checks (ITEP) van Hove singularity van Hove singularity Dirac frequency Brower,Rebbi,Schaich, PoS(Lattice 2011)056 Buividovich, Polikarpov, Phys. Rev. B 86 (2012) Ulybyshev et al. (ITEP), arxiv: k y k 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 3

4 Outline Interacting tight-binding model with normal-ordering terms (non-compact Hubbard field). Introducing the compact Hubbard field results. Improved Fermion discretization (ITEP) comparison. 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 4

5 Interacting tight-binding theory Since v F c/300 c, model interactions by non-local potential (Brower et al.): H = ( κ)(a,s a y,s + a y,s a,s) + e 2 q V y q y, q = a,1 a,1 + a, 1 a, 1 1,y,s Introduce hole operators b, b for spin 1 particles:,y b = a, 1, b = a, 1, a = a,+1, a = a,+1 q = a a b b. Apply normal ordering etra term from potential. Flip sign of b, b on one sub-lattice. H = ( κ)(a a y + b y b + h.c.) + e 2 :q V y q y : + e 2 V (a a + b b ),y,s Add staggered mass to break-sublattice symmetry:,y H H + m S (a a + b b ) (m S ± m, A, B) 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 5

6 Functional-integral for interacting theory Factor the eponential: e βh e δh e δh... e δh δ = β/n t. Epress partition function using coherent states ψ t,η t, ψ t,η t : Tr e βh = [ ] Nt 1 dψ,t dψ,t dη,t dη,t e (ψ,t+1 ψ,t+1+η,t+1 η,t+1) ψ t+1,η t+1 e δh ψ t,η t. Using ξ F (a λ, a λ) ξ = F (ξλ,ξ λ) e λ ξ λ ξ λ, obtain Nt [ ] 1 { [ Tr e βh = dψ,t dψ,t dη,t dη,t ep δ e 2 Q,t+1,t V yq y,t+1,t,y κ(ψ,t+1ψ y,t +ψy,t+1ψ,t +ηy,t+1η,t +η,t+1η y,t ) + m S (ψ,t+1ψ,t +η,t+1η,t ) ] + e 2 V (ψ,t+1ψ,t +η,t+1η,t ),y [ ψ,t+1(ψ,t+1 ψ,t ) +η,t+1(η,t+1 η,t ) ]}. where Q,t,t = ψ,tψ,t η,tη,t. Antiperiodic in time! Leading error iso(δ). 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 6

7 The Hubbard field Hubbard-Stratonovich transformation eliminates fourth powers: ( ) ep δe 2 Q,t+1,t V yq y,t+1,t = [det(...)] 1/2,y [ Nt 1 ] ( dφ,t ep δ 4 N t 1,y φ,t V 1 y φ y,t i eδ N t 1 Gaussian integral can be carried out to obtain Fermion determinant Tr e βh = [ N t 1 ] { dφ,t ep δ 4 N t 1,y φ,t Q,t+1,t ). } ( φ,t V 1 y φ y,t det M + ie β ) 2 φ,tδ yδ t 1,t N t M (,t)(y,t ) = δ y (δ tt δ t 1,t ) β N t κ n δ y,+ n δ t 1,t + β N t m S δ yδ t 1,t + β N t e 2 V δ yδ t 1,t Suitable for HMC simulations! No sign problem! 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 7

8 Hybrid Monte-Carlo for non-compact Hubbard field Force terms for Hubbard field φ and momentum p: d φ,t = p,t, dτ d dτ p,t = β (V 1 φ),t 2 β ) e Im (χ,t+1(b 1 χ),t 2N t N t Order parameter for sub-lattice symmetry breaking ( chiral condensate ) ] ( N = Tr [ Ne βh B = M + ie β ) φ,tδ yδ t 1,t N t = 1 DψDψ DηDη [ ( ) ψ,t+1ψ,t +η,t+1η,t ZN t X A,t X B,t = 1 [ ( Dφ βz m det BB )] e S[φ] = 2 ( Dφ det BB ) Re Tr βz = 2 N t N t 1 A B 1 (,t+1)(,t) B(,t+1)(,t) 1 B Doesn t work (no symmetry breaking)! Why?? ( ψ,t+1ψ,t +η,t+1η,t ) ] e βh ( ) B 1 db e S[φ] dm 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 8

9 The compact Hubbard field No sub-lattice symmetry breaking observed. Possible reason: Non-compact Hubbard field φ in Fermion determinant (thanks Maksim Ulybyshev!) ( ) The determinant of M + ie β N t φ,t δ y δ t 1,t is φ V uncontrollable errors from floating point rounding. Solution: Use compact Hubbard field instead! Replacement: ( ) β N t e 2 V δ y δ t 1,t + ie β N t φ,t δ y δ t 1,t ep Changes Fermion force term: d dτ p,t = β 2N t (V 1 φ),t 2 β N t e Im Sub-lattice symmetry breaks! Brower,Rebbi,Schaich, PoS(Lattice 2011)056 (ie βnt φ,t ) δ y δ t 1,t ( ) ) χ,t+1 ep (ie βnt φ,t (B 1 χ),t 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 9

10 The potential Constructed piecewise: Constrained random phase approimation (crpa) at short distances (Wehling et al. PRL 106, (2011)), Coulomb otherwise. Corrected for periodic boundary (one image in each direction). Differs from ITEP. { V00, e 2 V 01, V 02, V 03 : r 2a V (r) = e 2 /r : r > 2a standard periodic (1st) Ulybyshev et al. (ITEP), arxiv: V(r) V(r), ev r [nm] 1 Non-compact gauge field Screened potentials Coulomb r, nm 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 10

11 Results Simulating at β = 2.0 on N = 6. Etrapolating N t from N t = 8, 10, 12, 14, 16. Several hundreds of independent measurements for each set (N t,α, m). < n> m = 0.1 m = 0.2 m = 0.3 m = 0.4 m = α < n> α = 4.36 α = 3.94 α = 3.61 α = 2.55 α = 1.87 α = m Limit m 0 from N = a 0 + a 1 m + a 2 m 2. Probably large finite-volume errors! (In progress: Improve Vy 1 computation larger N will be feasible.) 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 11

12 Improved discretization Error of standard Fermion operator is O(δ). Strategy for improvement: split Hamiltonian (ITEP). [ ] [ ] 2N t 1 Tr e βh Tr e βh TB e βh C = dψ,t dψ,t dη,t dη,t { N t 1 e (ψ,2t ψ,2t +η,2t η,2t +ψ,2t+1 ψ,2t+1+η,2t+1 η,2t+1) } ψ 2t,η 2t e δh TB ψ 2t+1,η 2t+1 ψ 2t+1,η 2t+1 e δh C ψ 2t+2,η 2t+2. Leads to 2nd-order Fermion action: [ N t 1 S F [φ] = ψ,2t (ψ,2t ψ,2t+1) δκ + ψ,2t+1 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 12 <,y> (ψ,2t+1 e iδeφ,t ψ,2t+2 ) + δ ( ψ,2tψ y,2t+1 +ψ y,2tψ,2t+1 ) ±mψ,2tψ,2t+1 ].

13 Improved discretization (II) Improved Fermion matri: δ y (δ tt δ t+1,t ) β N t κ δ y,+ n δ t+1,t + β N t m S δ yδ t+1,t M (,t)(y,t ) = n δ yδ tt δ yδ t+1,t ep( i β N t eφ,(t 1)/2 ) : t even : t odd Hubbard field only on odd timeslices! ( ) ) HMC force: d dτ p,k = β 2N t (V 1 φ),k 2β N t e Im χ,2k+1 ep (i βnt eφ,k (M 1 χ),2k+2 Order parameter: N DψDψ DηDη [ ( ) ( )] ψ,2tψ,2t+1 +η,2tη,2t+1 ψ,2tψ,2t+1 +η,2tη βh,2t+1 e = 1 βz X A,t [ ( Dφ m det MM )] e S[φ] = 2 N t 1 N t A Operator inserted only on even timeslices! Perhaps a problem... X B,t M 1 (,2t)(,2t+1) M(,2t)(,2t+1) 1 B 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 13

14 Comparison N t etrapolation for naive and improved Fermion operators. < n> β = 2. α = 1.87 m = 0.3 < n> = a*(1/n 0.22 t )+b : a = / b = / : a = / b = / < n> β = 2. α = 1.87 m = 0.5 < n> = a*(1/n t )+b : a = / b = / : a = / b = / /N t 1/N t Within errors no difference. O(δ) behavior in both cases. Reason currently unknown. Coulomb energy shows similar behavior. 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 14

15 Comparison (II) Similar results for different α. Self-consistency is confirmed, but no improvement... < n> β = 2. α = 2.18 m = 0.5 < n> = a*(1/n t )+b : a = / b = / : a = / b = / < n> β = 2. α = 2.55 m = 0.5 < n> = a*(1/n t )+b : a = / b = / : a = / b = / /N t 1/N t Conclusion: Much work ahead. 1. August 2013 IKP TUD / SFB 634-D4 D. Smith, L. Smekal 15