Competing order in the fermionic Hubbard model on the hexagonal graphene lattice Southampton, 27 July 2016 Pavel Buividovich, Maksim Ulybyshev (Regensburg) Dominik Smith, Lorenz von Smekal (Giessen) Fachbereich 7 | Institut für Theoretische Physik | Lorenz von Smekal | 01
Introduction [courtesy L. Holicki] Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 2
Honeycomb Lattice - hexagonal Brillouin zone • triangular lattice (2 atoms per unit cell) graphene • single-particle energy bands E ± ( k ) = ± | Φ ( k ) | M structure factor: K' X e i k · δ i Φ (k) = t K i • massless dispersion around Dirac points K ± [Wallace, 1947] v f = 3 ta/ 2 ' 1 ⇥ 10 6 m / s ' c/ 300 E ( p ) = ± ~ v f | p | , Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 3
Honeycomb Lattice • mass terms (gaps) Graphene Gets a Good Gap on SiC Nevis et al ., PRL 115 (2015) 136802 1 X a † k , σ a k , σ − b † � � H m = m σ k , σ b k , σ N 2 k , σ (pseudo-spin) staggered on-site potential Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 4
Honeycomb Lattice • mass terms (gaps) Graphene Gets a Good Gap on SiC Nevis et al ., PRL 115 (2015) 136802 1 X a † k , σ a k , σ − b † � � H m = m σ k , σ b k , σ N 2 k , σ (pseudo-spin) staggered on-site potential � • spin (flavor) dependence m cdw = 1 2 ( m u + m d ) m sdw = 1 2 ( m u � m d ) Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 4
Honeycomb Lattice • mass terms (gaps) Graphene Gets a Good Gap on SiC Nevis et al ., PRL 115 (2015) 136802 1 X a † k , σ a k , σ − b † � � H m = m σ k , σ b k , σ N 2 k , σ (pseudo-spin) staggered on-site potential � • spin (flavor) dependence m cdw = 1 2 ( m u + m d ) m sdw = 1 2 ( m u � m d ) • Coulomb interaction e 2 α g = 4 πε ~ v f effective coupling Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 4
Honeycomb Lattice • mass terms (gaps) Graphene Gets a Good Gap on SiC Nevis et al ., PRL 115 (2015) 136802 1 X a † k , σ a k , σ − b † � � H m = m σ k , σ b k , σ N 2 k , σ (pseudo-spin) staggered on-site potential � with strong interactions: • spin (flavor) dependence Mott-insulator transition m → 0 m cdw = 1 2 ( m u + m d ) − → charge-density wave (CDW) m sdw = 1 2 ( m u � m d ) − → AF spin-density wave (SDW) • Coulomb interaction e 2 α g = 4 πε ~ v f effective coupling Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 4
Honeycomb Lattice • mass terms (gaps) Graphene Gets a Good Gap on SiC Nevis et al ., PRL 115 (2015) 136802 1 X a † k , σ a k , σ − b † � � H m = m σ k , σ b k , σ N 2 k , σ (pseudo-spin) staggered on-site potential � with strong interactions: • spin (flavor) dependence Mott-insulator transition m → 0 m cdw = 1 2 ( m u + m d ) − → charge-density wave (CDW) m sdw = 1 2 ( m u � m d ) − → AF spin-density wave (SDW) • Coulomb interaction 4 QSH QSH e 2 α g = V 2 2 CDW SDW 4 πε ~ v f SM 0 effective coupling SDW 0 CDW 0 1 2 4 2 U V 1 Raghu et al ., PRL 100 (2008) 156401 Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 4
Honeycomb Lattice • mass terms (gaps) Graphene Gets a Good Gap on SiC Nevis et al ., PRL 115 (2015) 136802 1 X a † k , σ a k , σ − b † � � H m = m σ k , σ b k , σ N 2 k , σ (pseudo-spin) staggered on-site potential � with strong interactions: • spin (flavor) dependence Mott-insulator transition m → 0 m cdw = 1 2 ( m u + m d ) − → charge-density wave (CDW) m sdw = 1 2 ( m u � m d ) − → AF spin-density wave (SDW) • Coulomb interaction 4 QSH QSH e 2 α g = V 2 2 CDW SDW 4 πε ~ v f SM 0 effective coupling SDW 0 CDW 0 1 2 4 2 U V 1 • sign-problem in HMC with m cdw > 0 Raghu et al ., PRL 100 (2008) 156401 Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 4
Potentially Strong Interactions • suspended graphene remains conducting, semimetal Elias et al ., Nature Phys. 2049 (2011) e 2 ≈ 300 ε → 1 α g = 137 ≈ 2 . 19 4 π ~ v f • puzzle α crit ∼ 1 predictions at the time Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 5
Potentially Strong Interactions • suspended graphene remains conducting, semimetal Elias et al ., Nature Phys. 2049 (2011) e 2 ≈ 300 ε → 1 α g = 137 ≈ 2 . 19 4 π ~ v f • puzzle α crit ∼ 1 predictions at the time Wehling et al ., PRL 106 (2011) 236805 • screening at short distances from σ - band electrons and localised higher energy states part. screened CRPA : • interpolate at intermediate distances ITEP screened std. Coulomb with dielectric thin-film model 10 U V(r) [eV] � 1 + 1 + ( � 1 − 1) e − kd k ) = 1 � − 1 ( ⃗ V 1 � 1 + 1 − ( � 1 − 1) e − kd � 1 V 2 ( � 1 = 2.4 and d = 2.8 ) 1 0 0.2 0.4 0.6 0.8 1 r [nm] Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 5
HMC on Hexagonal Lattice • chiral extrapolation m sdw → 0 0.4 N x = 36 1 N x = 18 0.3 0.8 0.6 < Δ N > < Δ n> 0.2 0.4 α = 2.73 0.1 0.2 α = 3.36 α = 4.37 0 0 α = 4.86 0 0.1 0.2 0.3 0.4 0.5 0.6 2 2.5 3 3.5 4 4.5 5 m [eV] α eff • semimetal-insulator transition in unphysical regime Ulybyshev, Buividovich, Katsnelson, Polikarpov, α crit ≈ 3 > 2 . 19 PRL 111 (2013) 056801 Smith, LvS, PRB 89 (2014) 195429 Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 6
Dyson-Schwinger Equations • hexagonal lattice, screened Coulomb Manon Bischoff, MSc, TU Da (2015) Katja Kleeberg, MSc, JLU Gi (2015) Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 7
Dyson-Schwinger Equations • hexagonal lattice, screened Coulomb Manon Bischoff, MSc, TU Da (2015) Katja Kleeberg, MSc, JLU Gi (2015) Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 8
Dyson-Schwinger Equations • hexagonal lattice, screened Coulomb Manon Bischoff, MSc, TU Da (2015) Katja Kleeberg, MSc, JLU Gi (2015) Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 9
Dyson-Schwinger Equations • hexagonal lattice, screened Coulomb Manon Bischoff, MSc, TU Da (2015) Katja Kleeberg, MSc, JLU Gi (2015) Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 10
Dyson-Schwinger Equations • hexagonal lattice, screened Coulomb graphene’s single-particle band structure α crit ≈ 1 . 5 • no Lindhard screening Π ( ! , ~ q ) = + + from π - band electrons Manon Bischoff, MSc, TU Da (2015) Katja Kleeberg, MSc, JLU Gi (2015) • what about CDW and the other insulating phases? Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 11
Dyson-Schwinger Equations • hexagonal Hubbard model, Hartree-Fock − 1 − − 1 i Σ ( ~ p ) = = − fermion self-energy Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 12
Dyson-Schwinger Equations • hexagonal Hubbard model, Hartree-Fock − 1 − − 1 i Σ ( ~ p ) = with on-site U and nearest-neighbor V = − first order fermion self-energy Katja Kleeberg et al. , in preparation Araki and Semenoff, PRB 86 (2012) 121402(R) Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 12
HMC on Hexagonal Lattice m sdw → 0 • chiral extrapolation, SDW 12 only on-site U first N x = 6 N x = 12 11 10 max m = 0.0259259 κ 9 χ con 8 N x = 6 N t = 80 8 N x = 12 6 N x = 18 7 < χ dis > 4 6 6 N x = 6 0 0.01 0.02 0.03 0.04 N x = 12 2 m / κ 5 U peak / κ 0 4 4.4 4.8 5.2 5.6 4 U 00 / κ 3 2 0 0.01 0.02 0.03 0.04 m / κ Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 13
HMC on Hexagonal Lattice m sdw → 0 • chiral extrapolation, SDW 12 only on-site U first N x = 6 N x = 12 11 10 max m = 0.0259259 κ 9 χ con 8 N x = 6 N t = 80 8 N x = 12 6 N x = 18 7 < χ dis > 4 6 6 N x = 6 0 0.01 0.02 0.03 0.04 N x = 12 2 m / κ 5 U peak / κ 0 4 4.4 4.8 5.2 5.6 4 U 00 / κ 3 Sorella, Tosatti, EPL 19 (1992) 699: U c ≈ 4 . 5 κ 2 0 0.01 0.02 0.03 0.04 Assaad, Herbut, PRX 3 (2013) 031010: U c ≈ 3 . 8 κ m / κ Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 13
HMC with Geometric Mass • hexagonal Brillouin zone 8 × 8 lattice 12 × 12 lattice • removes Dirac points • preserves symmetries • improves invertibility Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 14
Suitable Order Parameters for zero(geometric)-mass simulations, use s⌦� X 1 � 2 ↵ � 2 ↵ ⌦� X O = O i O i + L 2 i ∈ A i ∈ B with Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 15
Suitable Order Parameters for zero(geometric)-mass simulations, use s⌦� X 1 � 2 ↵ � 2 ↵ ⌦� X O = O i O i + L 2 i ∈ A i ∈ B with • spin-density wave: ⇢ a i , ~ � σσ 0 i ∈ A O i → ~ X c † S i = c i, σ 0 c i = i, σ 2 b i , i ∈ B σ , σ 0 Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 15
Suitable Order Parameters for zero(geometric)-mass simulations, use s⌦� X 1 � 2 ↵ � 2 ↵ ⌦� X O = O i O i + L 2 i ∈ A i ∈ B with • spin-density wave: ⇢ a i , ~ � σσ 0 i ∈ A O i → ~ X c † S i = c i, σ 0 c i = i, σ 2 b i , i ∈ B σ , σ 0 • charge-density wave: X c † � � O i → Q i = i, σ c i, σ − 1 σ Lattice 2016 27 July 2016 | Lorenz von Smekal | p. 15
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