The GW approximation

Size: px

Start display at page:

Download "The GW approximation"

Cynthia Rodgers
5 years ago
Views:

1 The GW approximation Matteo Gatti European Theoretical Spectroscopy Facility (ETSF) LSI - Ecole Polytechnique & Synchrotron SOLEIL - France matteo.gatti@polytechnique.fr TDDFT school - Benasque 2014

2 Outline 1 Photoemission 2 One-particle Green s function 3 GW approximation 4 In practice: G 0 W 0 and beyond 5 Beyond GW 6 Conclusions

3 References L. Hedin Phys. Rev. 139, A796 (1965). L. Hedin and S. Lundqvist Solid State Physics 23 (Academic, New York, 1969). G. Strinati Rivista del Nuovo Cimento 11, (12)1 (1988). G. Onida, L. Reining, and A. Rubio Rev. Mod. Phys. 74, 601 (2002). F. Bruneval and M. Gatti Topics in Current Chemistry, in press F. Bruneval PhD thesis, Ecole Polytechnique (2005) bruneval_these.pdf

4 Outline 1 Photoemission 2 One-particle Green s function 3 GW approximation 4 In practice: G 0 W 0 and beyond 5 Beyond GW 6 Conclusions

5 Photoemission Direct Photoemission Inverse Photoemission

6 Direct Photoemission photon in - electron out E(N) + hν = E(N 1, i) + E kin E i = E(N) E(N 1, i) = E kin hν...plus momentum conservation ARPES occupied states

7 Direct Photoemission Bulk silicon Photon energy 800 ev photon in - electron out E(N) + hν = E(N 1, i) + E kin Photoemission intensity [arb. units] ω 2ω ω E i = E(N) E(N 1, i) = E kin hν...plus momentum conservation ARPES Binding energy E-E F [ev] M. Guzzo et al., PRL 107 (2011).

8 Inverse Photoemission electron in - photon out E(N) + E kin = E(N + 1, i) + hν E i = E(N+1, i) E(N) = E kin hν aka Bremsstrahlung isochromat spectroscopy (BIS) empty states

9 Inverse Photoemission electron in - photon out Nickel oxide Sawatzky and Allen PRL 53 (1984)

10 Photoemission

11 Photoemission Not discussed here: matrix elements - cross sections (dependence on photon energy / photon polarization) sudden approximation vs. interaction photoelectron - system surface sensitivity... S. Hüfner, Photoelectron spectroscopy (1995) E. Papalazarou et al., PRB 80 (2009)

12 Photoemission: additional charge

13 Photoemission: additional charge

14 Photoemission: additional charge What happens?

15 Outline 1 Photoemission 2 One-particle Green s function 3 GW approximation 4 In practice: G 0 W 0 and beyond 5 Beyond GW 6 Conclusions

16 One-particle Green s function What is the one-particle Green s function G(1, 2) = G(x 1, x 2, t 1 t 2 )?

17 One-particle Green s function What is the one-particle Green s function G(1, 2) = G(x 1, x 2, t 1 t 2 )? The one-particle Green s function G 1 Propagation of one additional particle in the system ig(x 1, x 2, t 1 t 2 ) = N T [ ψ(x 1, t 1 )ψ (x 2, t 2 ) ] N How to calculate G?

18 One-particle Green s function What is the one-particle Green s function G(1, 2) = G(x 1, x 2, t 1 t 2 )? The one-particle Green s function G 1 Propagation of one additional particle in the system ig(x 1, x 2, t 1 t 2 ) = N T [ ψ(x 1, t 1 )ψ (x 2, t 2 ) ] N How to calculate G? 2 Resolvent of H(ω) = H 0 + Σ(ω) = h 0 + V H + Σ(ω): G 1 (ω) = (ω H 0 Σ(ω)) = (G 1 0 (ω) Σ(ω)) What is H(ω)? What is Σ(ω)?

19 One-particle Green s function The one-particle Green s function G Definition and meaning of G: ig(x 1, t 1 ; x 2, t 2 ) = N T [ ψ(x 1, t 1 )ψ (x 2, t 2 ) ] N for for t 1 > t 2 ig(x 1, t 1 ; x 2, t 2 ) = N ψ(x 1, t 1 )ψ (x 2, t 2 ) N t 1 < t 2 ig(x 1, t 1 ; x 2, t 2 ) = N ψ (x 2, t 2 )ψ(x 1, t 1 ) N

20 One-particle Green s function t 1 > t 2 N ψ(r 1, t 1 )ψ (r 2, t 2 ) N t 1 < t 2 N ψ (r 2, t 2 )ψ(r 1, t 1 ) N

21 One-particle Green s function What is G? Definition and meaning of G: [ ] G(x 1, x 2, t 1 t 2 ) = i < N T ψ(x 1, t 1 )ψ (x 2, t 2 ) N > Insert a complete set of N + 1 or N 1-particle states and Fourier transform. This yields: G(x 1, x 2, ω) = j f j (x 1 )f j (x 2 ) ω E j + iηsgn(e j µ). where: { E(N + 1, j) E(N), Ej > µ E j = E(N) E(N 1, j), E j < µ { N ψ (x1 ) N + 1, j, E j > µ f j (x 1 ) = N 1, j ψ (x 1 ) N, E j < µ

22 Photoemission Direct Photoemission Inverse Photoemission One-particle excitations poles of one-particle Green s function G

23 One-particle Green s function Spectral function A useful definition: the spectral function A(x, x ; ω) = 1 π ImG(x, x ; ω) = j f j (x)f j (x )δ(ω E j ). One particle excitations peaks of spectral function A

24 One-particle Green s function One-particle Green s function From one-particle G we can obtain: one-particle excitation spectra ground-state expectation value of any one-particle operator: e.g. density ρ or density matrix γ: ρ(r, t) = ig(r, r, t, t + ) γ(r, r, t) = ig(r, r, t, t + ) ground-state total energy (e.g. Galitskii-Migdal)

25 One-particle Green s function: absorption?

26 One-particle Green s function: absorption?

27 One-particle Green s function: absorption?

28 How to calculate G? Straightforward? ig(x 1, t 1 ; x 2, t 2 ) = N T [ ψ(x 1, t 1 )ψ (x 2, t 2 ) ] N

29 How to calculate G? Straightforward? ig(x 1, t 1 ; x 2, t 2 ) = N T [ ψ(x 1, t 1 )ψ (x 2, t 2 ) ] N N > =??? Interacting ground state! See e.g. A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems

30 How to calculate G? G as the solution of a differential equation (with boundary conditions) Equation of motion [ ] i h 0 (1) G(1, 2) = δ(1, 2) i t 1 d3v(1, 3)G 2 (1, 3, 2, 3 + ) where h 0 = v ext Interaction two-particle excitations

31 How to calculate G? G as the solution of a differential equation (with boundary conditions) Equation of motion [ ] i h 0 (1) G(1, 2) = δ(1, 2) i t 1 d3v(1, 3)G 2 (1, 3, 2, 3 + ) where h 0 = v ext Interaction two-particle excitations Unfortunately, hierarchy of equations G(1, 2) G 2 (1, 2; 3, 4) G 2 (1, 2; 3, 4) G 3 (1, 2, 3; 4, 5, 6)...

32 Self-energy [ ] i h 0 (1) G(1, 2) = δ(1, 2) i d3v(1, 3)G 2 (1, 3, 2, 3 + ) t 1 Self-energy [ ] i h 0 (1) V H (1) G(1, 2) = δ(1, 2) + t 1 d3σ(1, 3)G(3, 2) Σ = non-local effective potential (exchange and correlation) Σ = Σ[G]: self-consistency

33 Self-energy [ ] i h 0 (1) V H (1) G(1, 2) = δ(1, 2) + d3σ(1, 3)G(3, 2) t 1 Fourier transform: G as resolvent [ ] ω h 0 (r 1 ) V H (r 1 ) G(r 1, r 2, ω) = dr 3 Σ(r 1, r 3, ω)g(r 3, r 2, ω) (ω H 0 )G(ω) = Σ(ω)G(ω) G 1 (ω) = (ω H 0 Σ(ω)) Σ = Σ(ω): folding of higher-order excitations

34 Dyson equation [ ] i h 0 (1) V H (1) G(1, 2) = δ(1, 2) + d3σ(1, 3)G(3, 2) t 1 [ ] i h 0 (1) V H (1) G 0 (1, 2) = δ(1, 2) t 1 Dyson equation G(1, 2) = G 0 (1, 2) + d34g 0 (1, 3)Σ(3.4)G(4.2)

35 Dyson equation Dyson equation G(1, 2) = G 0 (1, 2) + d34g 0 (1, 3)Σ(3.4)G(4.2) Two general properties: 1 G = G 0 + G 0 ΣG = G 0 + G 0 ΣG 0 + G 0 ΣG 0 ΣG = (G 1 0 Σ) 1

36 Dyson equation Dyson equation G(1, 2) = G 0 (1, 2) + d34g 0 (1, 3)Σ(3.4)G(4.2) Two general properties: 1 G = G 0 + G 0 ΣG = G 0 + G 0 ΣG 0 + G 0 ΣG 0 ΣG = (G 1 0 Σ) 1 2 G = G 0 + G 0 ΣG { G1 = G 0 + G 0 Σ 1 G 1 G = G 1 + G 1 Σ 2 G

37 Outline 1 Photoemission 2 One-particle Green s function 3 GW approximation 4 In practice: G 0 W 0 and beyond 5 Beyond GW 6 Conclusions

39 What happens?

46 Screening W (r 1, r 2, ω) = dr 3 ɛ 1 (r 1, r 3, ω)v(r 3, r 2 )

47 Screening: quasiparticles

49 GW approximation additional charge reaction: polarization, screening GW approximation 1 polarization made of noninteracting electron-hole pairs (RPA) 2 classical (Hartree) interaction between additional charge and polarization charge

50 GW approximation

51 Hedin s equation and GW Hedin s equations Σ =igw Γ G =G 0 + G 0 ΣG Γ =1 + δσ δg GGΓ P = iggγ W =v + vpw L. Hedin, Phys. Rev. 139 (1965)

52 Hedin s equation and GW GW approximation: Γ = 1 Σ =igw G =G 0 + G 0 ΣG Γ =1 P = igg W =v + vpw L. Hedin, Phys. Rev. 139 (1965)

53 Self-energy and GW The self-energy Self-energy = exchange + induced Hartree + induced exchange-correlation Σ(12) = ig(12)v(12) + ig(12)w p (12) + ig(1 4) δσ( 42) δρ( 5) χ( 5 3)v( 31) W p (12) = W (12) v(12) = v(1 3)χ( 3 4)v( 42) GW = exchange + induced Hartree (with RPA χ) Remember: v = δv H /δρ

54 Correlation = coupling of excitations Dynamical screening Convolution in frequency domain Σ c (r 1, r 2, ω) = i dω e iηω G(r 1, r 2, ω + ω )W p (r 1, r 2, ω ) 2π

55 Correlation = coupling of excitations Dynamical screening Convolution in frequency domain Σ c (r 1, r 2, ω) = i dω e iηω G(r 1, r 2, ω + ω )W p (r 1, r 2, ω ) 2π W p (r 1, r 2, ω) = 2 s ω s W s (r 1, r 2 ) ω 2 (ω s iη) 2

56 Correlation = coupling of excitations Dynamical screening Convolution in frequency domain Σ c (r 1, r 2, ω) = i dω e iηω G(r 1, r 2, ω + ω )W p (r 1, r 2, ω ) 2π W p (r 1, r 2, ω) = 2 s Σ c (r 1, r 2, ω) = j,s 0 ω s W s (r 1, r 2 ) ω 2 (ω s iη) 2 f j (r 1 )f j (r 2 )W s (r 1, r 2 ) ω E j + ω s sgn(µ E j ) Coupling of addition/removal E i and neutral excitations ω s

57 Outline 1 Photoemission 2 One-particle Green s function 3 GW approximation 4 In practice: G 0 W 0 and beyond 5 Beyond GW 6 Conclusions

58 GW: Self-consistent solution G = G 0 + G 0 Σ[G] G Self-consistent GW bad for spectral properties in solids OK for atoms, small molecules necessary for total energy (conserving approximation) computationally very heavy!

59 G 0 W 0 : Quasiparticle corrections Standard perturbative G 0 W 0 H 0 (r)φ i (r) + dr Σ(r, r, ω = E i ) φ i (r ) = E i φ i (r)

60 G 0 W 0 : Quasiparticle corrections Standard perturbative G 0 W 0 H 0 (r)φ i (r) + dr Σ(r, r, ω = E i ) φ i (r ) = E i φ i (r) H 0 (r)ϕ i (r) + V xc (r)ϕ i (r) = ɛ i ϕ i (r)

61 G 0 W 0 : Quasiparticle corrections Standard perturbative G 0 W 0 H 0 (r)φ i (r) + dr Σ(r, r, ω = E i ) φ i (r ) = E i φ i (r) H 0 (r)ϕ i (r) + V xc (r)ϕ i (r) = ɛ i ϕ i (r) First-order perturbative corrections with Σ = igw : E i ɛ i = ϕ i Σ(E i ) V xc ϕ i

62 G 0 W 0 : Quasiparticle corrections Standard perturbative G 0 W 0 H 0 (r)φ i (r) + dr Σ(r, r, ω = E i ) φ i (r ) = E i φ i (r) H 0 (r)ϕ i (r) + V xc (r)ϕ i (r) = ɛ i ϕ i (r) First-order perturbative corrections with Σ = igw : E i ɛ i = ϕ i Σ(E i ) V xc ϕ i Σ(E i ) = Σ(ɛ i ) + (E i ɛ i ) ω Σ(ω) ɛi Quasiparticle energies E i = ɛ i + Z i ϕ i Σ(ɛ i ) V xc ϕ i Z i = (1 ω Σ(ω) ɛi ) 1 Hybersten and Louie, PRB 34 (1986); Godby, Schlüter and Sham, PRB 37 (1988)

63 G 0 W 0 : QP results M. van Schilfgaarde et al., PRL 96 (2006)

64 G 0 W 0 results Great improvement over LDA. Drawback: dependency on the starting point G 0 W 0 results OK for sp electron systems questionable for df electron systems (and whenever LDA is bad)

65 Beyond G 0 W 0 : alternative starting points Looking for a better starting point Kohn-Sham with other functionals (EXX, LDA+U) - e.g. Rinke 2005, Miyake 2006, Jiang 2009,... Generalized Kohn-Sham: hybrid functionals (HSE06) - e.g. Fuchs 2006,... effective quasiparticle Hamiltonians - QSGW scheme - Faleev Hedin s COHSEX approximation - Bruneval 2005

66 Beyond G 0 W 0 : alternative starting points Looking for a better starting point Kohn-Sham with other functionals (EXX, LDA+U) - e.g. Rinke 2005, Miyake 2006, Jiang 2009,... Generalized Kohn-Sham: hybrid functionals (HSE06) - e.g. Fuchs 2006,... effective quasiparticle Hamiltonians - QSGW scheme - Faleev Hedin s COHSEX approximation - Bruneval 2005

67 Beyond G 0 W 0 : QSGW scheme Only retain hermitian part of GW Σ and iterate QP: φ i Σ φ j = 1 2 Re[ φ i Σ(E i ) φ j + φ i Σ(E j ) φ j ] S. V. Faleev, M. van Schilfgaarde, and T. Kotani, PRL 93 (2004) M. van Schilfgaarde, T. Kotani, and S. V. Faleev, PRL 96 (2006)

68 Outline 1 Photoemission 2 One-particle Green s function 3 GW approximation 4 In practice: G 0 W 0 and beyond 5 Beyond GW 6 Conclusions

69 Beyond GW: vertex corrections Beyond GW multiple plasmon satellites: cumulant expansion self-screening atomic limit additional interactions: T matrix

70 Multiple satellites in silicon: PES Bulk silicon Photon energy 800 ev Photoemission intensity [arb. units] ω 2ω ω Binding energy E-E F [ev] M. Guzzo et al., PRL 107 (2011).

71 Multiple satellites in silicon: GW GW spectral function: top valence at Γ A very weak plasmon satellite

72 Multiple satellites in silicon: GW GW spectral function: bottom valence at Γ A plasmaron satellite B. I. Lundqvist, Phys. Kondens. Mater. 6 (1967)

73 Multiple satellites in silicon: GW GW spectral function

74 Decoupling approximation: exponential solution Equation of motion of G : G = G 0 + G 0 V H G + G 0 UG + i G 0 v δg δu with G 0 = (ω h 0 ) 1 1 Linearize: V H = V 0 H + vχu +... G = G 0 + G 0 ŪG + ig 0 W δg δū with Ū = ɛ 1 U, G 0 = (ω h 0 VH) 0 1

75 Decoupling approximation: exponential solution Equation of motion of G : G = G 0 + G 0 V H G + G 0 UG + i G 0 v δg δu with G 0 = (ω h 0 ) 1 1 Linearize: V H = V 0 H + vχu +... G = G 0 + G 0 ŪG + ig 0 W δg δū with Ū = ɛ 1 U, G 0 = (ω h 0 V 0 H) 1 2 Optimize QP such that G and G QP are diagonal in the basis k : holes: G QP k (τ) = iθ( τ)e iɛqp k τ k : G = G QP + G QP (Ū QP )G + ig QP W δg δū

76 Decoupling approximation: exponential solution Equation of motion of G : G = G 0 + G 0 V H G + G 0 UG + i G 0 v δg δu with G 0 = (ω h 0 ) 1 1 Linearize: V H = V 0 H + vχu +... G = G 0 + G 0 ŪG + ig 0 W δg δū with Ū = ɛ 1 U, G 0 = (ω h 0 V 0 H) 1 2 Optimize QP such that G and G QP are diagonal in the basis k : holes: G QP k (τ) = iθ( τ)e iɛqp k τ Exact solution: k : G = G QP + G QP (Ū QP )G + ig QP W δg δū G(t 1, t 2 ) = G QP (t 1 t 2 )e i QP (t 1 t 2 ) e i t 2 t1 dt [ U(t ) t ] 2 t dt W (t,t )

77 Multiple satellites in silicon: exponential solution Plasmon-pole approximation to W : W (τ) = iλ k [ θ(τ)e i ω k τ + θ( τ)e i ω k τ ] Exponential solution - cumulant expansion A k (ω) = e a k π n=0 a n k n! (ω Reɛ QP k Imɛ QP k + n ω k ) 2 + (Imɛ QP k ), 2

78 Multiple satellites in silicon: exponential solution Plus contributions from: extrinsic effects, interference effects, cross sections, background M. Guzzo et al., PRL 107 (2011)

79 Satellites in strongly correlated materials: SrVO 3 (a) A(ω) [arb. units] US 1 QP only GW GW+E USP US 2 LS Energy E-E F [ev] M. Gatti and M. Guzzo, PRB 87 (2013)

80 GW approximation

81 Self-screening Particle in a box: add or remove ( 2 /2 + V box )φ = εφ ε = (E N=0 E N=1 ) = E N=1 E N=0

82 Self-screening Particle in a box: add or remove ( 2 /2 + V box )φ = εφ ε = (E N=0 E N=1 ) = E N=1 E N=0 Kohn-Sham ( 2 /2 + V box + ρv ρv)φ = εφ Hartree-Fock ( 2 /2 + V box + ρv)φ φ φvφ = εφ GW ( 2 /2 + V box + ρv)φ φ φvφ + Σ c φ = εφ W = v + W p = v + vχ RPA v W p should be zero!

83 Self-screening Corrections to GW W test-charge use exact χ instead of χ RPA χ = χ 0 W p = vχ 0 v 0 W test-electron local vertex: W p = (v + f xc )χ 0 v = 0 (f xc = v) W. Nelson, P. Bokes, P. Rinke, and R. W. Godby, Phys. Rev. A 75 (2007) P. Romaniello, S. Guyot, and L. Reining, JCP 131 (2009) F. Aryasetiawan, R. Sakuma, and K. Karlsson, arxiv:

84 Atomic limit One electron in two-site Hubbard model P. Romaniello, S. Guyot, and L. Reining, JCP 131 (2009)

86 GW and T matrix GW Add one primary electron e, spin Disexcitation (e, ) (e, ) Creation of electron-hole pairs e 2 -h in both spin channels. Note primary electron: final spin = initial spin (no spin flips) no interaction between primary electron and secondary particles analogously for additional hole

GW and T matrix T matrix Add one primary electron e, spin Disexcitation (e, ) (e 2, ) Creation of electron-hole pairs e 1 -h in both spin channels Interaction between primary electron and hole of

87 GW and T matrix T matrix Add one primary electron e, spin Disexcitation (e, ) (e 2, ) Creation of electron-hole pairs e 1 -h in both spin channels Interaction between primary electron and hole of electron-hole pair (A,B) Interaction between primary electron and electron of electron-hole pair (C) Note (B) spin flips: coupling with spin-waves, magnons, paramagnons analogously for additional hole V. P. Zhukov, E. V. Chulkov, and P. M. Echenique, PRB 72 (2005)

88 T matrix T matrix: hole-hole interaction 6 ev satellite in Nickel M. Springer, F. Aryasetiawan, and K. Karlsson, PRL 80 (1998)

89 Outline 1 Photoemission 2 One-particle Green s function 3 GW approximation 4 In practice: G 0 W 0 and beyond 5 Beyond GW 6 Conclusions

90 Independent (quasi)particles: GW Independent transitions: ɛ 2 (ω) = 8π2 ϕ Ωω 2 j e v ϕ i 2 δ(e j E i ω) ij

91 What is wrong? What is missing?

92 What is wrong? What is missing? We need the BSE...

93 What is wrong? What is missing? We need the BSE... and Ilya.

94 Many thanks!

95 Acknowledgements Fabien Bruneval Rex Godby Valerio Olevano Lucia Reining Pina Romaniello Francesco Sottile

Many-Body Perturbation Theory. Lucia Reining, Fabien Bruneval

Many-Body Perturbation Theory. Lucia Reining, Fabien Bruneval , Fabien Bruneval Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France European Theoretical Spectroscopy Facility (ETSF) Belfast, 27.6.2007 Outline 1 Reminder 2 Perturbation Theory