Introduction to many-body Green s functions

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1 Introduction to many-body Green s functions Matteo Gatti European Theoretical Spectroscopy Facility (ETSF) NanoBio Spectroscopy Group - UPV San Sebastián - Spain ELK school - CECAM 2011

2 Outline 1 Motivation 2 One-particle Green s functions: GW approximation 3 Two-particle Green s functions: Bethe-Salpeter equation 4 Micro-macro connection

3 References Francesco Sottile PhD thesis, Ecole Polytechnique (2003) francesco/tesi_dot.pdf Fabien Bruneval PhD thesis, Ecole Polytechnique (2005) bruneval_these.pdf Giovanni Onida, Lucia Reining, and Angel Rubio Rev. Mod. Phys. 74, 601 (2002). G. Strinati Rivista del Nuovo Cimento 11, (12)1 (1988).

4 Outline 1 Motivation 2 One-particle Green s functions: GW approximation 3 Two-particle Green s functions: Bethe-Salpeter equation 4 Micro-macro connection

5 Motivation Theoretical spectroscopy Calculate and reproduce Understand and explain Predict Exp. at 30 K from: P. Lautenschlager et al., Phys. Rev. B 36, 4821 (1987).

6 Theoretical Spectroscopy Which kind of spectra? Which kind of tools?

7 Why do we have to study more than DFT? Absorption spectrum of bulk silicon in DFT How can we understand this?

8 Why do we have to study more than DFT? Absorption spectrum of bulk silicon in DFT Spectroscopy is exciting!

9 MBPT vs. TDDFT: different worlds, same physics MBPT based on Green s functions one-particle G: electron addition and removal - GW two-particle L: electron-hole excitation - BSE moves (quasi)particles around is intuitive (easy) TDDFT based on the density response function χ: neutral excitations moves density around is efficient (simple)

10 Response functions External perturbation V ext applied on the sample V tot acting on the electronic system Potentials Dielectric function δv tot = δv ext + δv ind ɛ = δv ext δv tot δv ind = vδρ = 1 v δρ ɛ 1 = δv tot = 1 + v δv ext δv tot δρ δv ext

11 Response functions External perturbation V ext applied on the sample V tot acting on the electronic system Dielectric function P = ɛ = δv ext δv tot = 1 vp ɛ 1 = δv tot δv ext = 1 + vχ δρ δv tot χ = δρ δv ext χ = P + Pvχ P = χ 0 + χ 0 f xc P

12 Micro-macro connection Microscopic-Macroscopic connection: local fields χ G,G (q, ω) = P G,G (q, ω) + P G,G1 (q, ω)v G1 (q)χ G1,G (q, ω) ɛ 1 G,G (q, ω) = δ G,G + v G (q)χ G,G (q, ω) ɛ M (q, ω) = 1 ɛ 1 G=0,G =0 (q, ω) Adler, Phys. Rev. 126 (1962); Wiser, Phys. Rev. 129 (1963).

13 Micro-macro connection Microscopic-Macroscopic connection: local fields ɛ M (q, ω) = 1 v G=0 (q) χ G=0,G =0(q, ω) χ G,G (q, ω) = P G,G (q, ω) + P G,G1 (q, ω) v G1 (q) χ G1,G (q, ω) v G (q) = 0 for G = 0 v G (q) = v G (q) for G 0 Hanke, Adv. Phys. 27 (1978).

14 Absorption spectra Absorption spectra Abs(ω) = lim q 0 Imɛ M (q, ω) Abs(ω) = lim q 0 Im [v G=0 (q) χ G=0,G =0(q, ω)] Absorption response to V ext + V macro ind

15 Independent particles: Kohn-Sham Independent transitions: ɛ 2 (ω) = 8π2 ϕ Ωω 2 j e v ϕ i 2 δ(ε j ε i ω) ij

16 What is an electron?

17 Outline 1 Motivation 2 One-particle Green s functions: GW approximation 3 Two-particle Green s functions: Bethe-Salpeter equation 4 Micro-macro connection

18 Photoemission Direct Photoemission Inverse Photoemission

19 Why do we have to study more than DFT? adapted from M. van Schilfgaarde et al., PRL 96 (2006).

20 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

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

22 One-particle Green s function What is G? Definition and meaning of G: [ ] G(x 1, t 1 ; x 2, 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. This yields G(x 1, t 1 ; x 2, t 2 ) = i j f j (x 1 )f j (x 2 )e iε j (t 1 t 2 ) [θ(t 1 t 2 )θ(ε j µ) θ(t 2 t 1 )Θ(µ ε j )]; where: ε j = E(N + 1, j) E(N), ε j > µ E(N) E(N 1, j), ε j < µ f j (x 1 ) = N ψ (x 1) N + 1, j, ε j > µ N 1, j ψ (x 1 ) N, ε j < µ

23 One-particle Green s function What is G? - Fourier transform G(x, x, ω) = j Fourier Transform: f j (x)f j (x ) ω ε j + iηsgn(ε j µ). Spectral function: A(x, x ; ω) = 1 π ImG(x, x ; ω) = j f j (x)f j (x )δ(ω ε j ).

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

25 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

26 One-particle Green s function Straightforward? G(x, t; x, t ) = i < N T [ ψ(x, t)ψ (x, t ) ] N > N > =??? Interacting ground state! Perturbation Theory? Time-independent perturbation theories: messy. Textbooks: adiabatically switched on interaction, Gell-Mann-Low theorem, Wick s theorem, expansion (diagrams). Lots of diagrams...

27 One-particle Green s function Straightforward? G(x, t; x, t ) = i < N T [ ψ(x, t)ψ (x, t ) ] N > N > =??? Interacting ground state! Perturbation Theory? Time-independent perturbation theories: messy. Textbooks: adiabatically switched on interaction, Gell-Mann-Low theorem, Wick s theorem, expansion (diagrams). Lots of diagrams...

28 One-particle Green s function Straightforward? G(x, t; x, t ) = i < N T [ ψ(x, t)ψ (x, t ) ] N > N > =??? Interacting ground state! Perturbation Theory? Time-independent perturbation theories: messy. Textbooks: adiabatically switched on interaction, Gell-Mann-Low theorem, Wick s theorem, expansion (diagrams). Lots of diagrams...

29 Functional approach to the MB problem Equation of motion To determine the 1-particle Green s function: [ ] i h 0 (1) G(1, 2) = δ(1, 2) i t 1 d3v(1, 3)G 2 (1, 3, 2, 3 + ) Do the Fourier transform in frequency space: [ω h 0 ]G(ω) + i vg 2 (ω) = 1 where h 0 = v ext is the independent particle Hamiltonian. The 2-particle Green s function describes the motion of 2 particles.

30 Unfortunately, hierarchy of equations G 1 (1, 2) G 2 (1, 2; 3, 4) G 2 (1, 2; 3, 4) G 3 (1, 2, 3; 4, 5, 6)...

31 Self-energy Perturbation theory starts from what is known to evaluate what is not known, hoping that the difference is small... Let s say we know G 0 (ω) that corresponds to the Hamiltonian h 0 Everything that is unknown is put in Σ(ω) = G 1 0 (ω) G 1 (ω) This is the definition of the self-energy Thus, [ω h 0 ]G(ω) Σ(ω)G(ω) = 1 to be compared with [ω h 0 ]G(ω) + i vg 2 (ω) = 1

32 Self-energy Perturbation theory starts from what is known to evaluate what is not known, hoping that the difference is small... Let s say we know G 0 (ω) that corresponds to the Hamiltonian h 0 Everything that is unknown is put in Σ(ω) = G 1 0 (ω) G 1 (ω) This is the definition of the self-energy Thus, [ω h 0 ]G(ω) Σ(ω)G(ω) = 1 to be compared with [ω h 0 ]G(ω) + i vg 2 (ω) = 1

33 One-particle Green s function Trick due to Schwinger (1951): introduce a small external potential U(3), that will be made equal to zero at the end, and calculate the variations of G with respect to U δg(1, 2) δu(3) = G 2 (1, 3; 2, 3) + G(1, 2)G(3, 3).

34 Hedin s equation 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)

35 GW bandstructure: photoemission additional charge

36 GW bandstructure: photoemission 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

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

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

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

40 GW results M. van Schilfgaarde et al., PRL 96 (2006).

41 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

42 What is wrong? What is missing?

43 Absorption Two-particle excitations poles of two-particle Green s function L Excitonic effects = electron - hole interaction

44 Absorption Two-particle excitations poles of two-particle Green s function L Excitonic effects = electron - hole interaction

45 Absorption Two-particle excitations poles of two-particle Green s function L Excitonic effects = electron - hole interaction

46 Outline 1 Motivation 2 One-particle Green s functions: GW approximation 3 Two-particle Green s functions: Bethe-Salpeter equation 4 Micro-macro connection

47 Beyond RPA P(12) = ig(12)g(21) = P 0 (12) Independent particles (RPA)

48 Beyond RPA P(12) = ig(13)g(42)γ(342) Interacting particles (excitonic effects)

49 From Hedin s equations to BSE From Hedin... P = iggγ Γ = 1 + δσ δg GGΓ

50 From Hedin s equations to BSE From Hedin......to Bethe-Salpeter P = iggγ Γ = 1 + δσ δg GGΓ ( δσ ) L = L 0 + L 0 v + i L δg

51 The Bethe-Salpeter equation Exercise Formal derivation δg(12) L(1234) = i δv = +ig(15) δg 1 (56) ext(34) δv G(62) ext(34) = + ig(15) δ[g 1 0 (56) Vext(56) Σ(56)] G(62) δv ext(34) [ δvh (5)δ(56) = ig(13)g(42) + ig(15)g(62) δσ(56) ] δv ext(34) δv ext(34) [ δvh (5)δ(56) = ig(13)g(42) + ig(15)g(62) δσ(56) ] δg(78) δg(78) δg(78) δv ext(34) [ L(1234) =L 0 (1234) + L 0 (1256) v(57)δ(56)δ(78) + i δσ(56) δg(78) ] L(7834)

52 The Bethe-Salpeter equation [ L(1234) = L 0 (1234) + L 0 (1256) v(57)δ(56)δ(78) + i δσ(56) δg(78) ] L(7834) Polarizabilities δg(12) L(1234) = i δv ext (34) χ(12) = δρ(1) δv ext (2) L(1122) = χ(12)

53 The Bethe-Salpeter equation Approximations ( δσ ) L = L 0 + L 0 v + i L δg

54 The Bethe-Salpeter equation Approximations Σ igw ( δσ ) L = L 0 + L 0 v + i L δg Approximation:

55 The Bethe-Salpeter equation Approximations Σ igw ( δ(gw )) L = L 0 + L 0 v L δg Approximation: δ(gw ) δg = W + GδW δg W

56 The Bethe-Salpeter equation Approximations Final result: L = L 0 + L 0 (v W )L

57 The Bethe-Salpeter equation Bethe-Salpeter equation L(1234) = L 0 (1234)+ L 0 (1256)[v(57)δ(56)δ(78) W (56)δ(57)δ(68)]L(7834)

58 Absorption spectra in BSE Bulk silicon G. Onida, L. Reining, and A. Rubio, RMP 74 (2002).

59 Solving BSE L(1234) = L 0 (1234)+ L 0 (1256)[v(57)δ(56)δ(78) W (56)δ(57)δ(68)]L(7834) Static W Simplification: W (r 1, r 2, t 1 t 2 ) W (r 1, r 2 )δ(t 1 t 2 ) L(1234) L(r 1, r 2, r 3, r 4, t t ) L(r 1, r 2, r 3, r 4, ω)

60 Solving BSE L(1234) = L 0 (1234)+ L 0 (1256)[ v(57)δ(56)δ(78) W (56)δ(57)δ(68)] L(7834) Static W Simplification: W (r 1, r 2, t 1 t 2 ) W (r 1, r 2 )δ(t 1 t 2 ) L(1234) L(r 1, r 2, r 3, r 4, t t ) L(r 1, r 2, r 3, r 4, ω)

61 Solving BSE L(1234) = L 0 (1234)+ L 0 (1256)[ v(57)δ(56)δ(78) W (56)δ(57)δ(68)] L(7834) Static W Simplification: W (r 1, r 2, t 1 t 2 ) W (r 1, r 2 )δ(t 1 t 2 ) L(1234) L(r 1, r 2, r 3, r 4, t t ) L(r 1, r 2, r 3, r 4, ω)

62 Solving BSE Dielectric function L(r 1 r 2 r 3 r 4 ω) = L 0 (r 1 r 2 r 3 r 4 ω) + dr 5 dr 6 dr 7 dr 8 L 0 (r 1 r 2 r 5 r 6 ω) [ v(r 5 r 7 )δ(r 5 r 6 )δ(r 7 r 8 ) W (r 5 r 6 )δ(r 5 r 7 )δ(r 6 r 8 )] L(r 7 r 8 r 3 r 4 ω) [ ɛ M (ω) = 1 lim v G=0 (q) q 0 ] drdr e iq(r r ) L(r, r, r, r, ω)

63 Solving BSE L(r 1 r 2 r 3 r 4 ω) = L 0 (r 1 r 2 r 3 r 4 ω) + dr 5 dr 6 dr 7 dr 8 L 0 (r 1 r 2 r 5 r 6 ω) [ v(r 5 r 7 )δ(r 5 r 6 )δ(r 7 r 8 ) W (r 5 r 6 )δ(r 5 r 7 )δ(r 6 r 8 )] L(r 7 r 8 r 3 r 4 ω) Transition space How to solve it? L (n1 n 2 )(n 3 n 4 )(ω) = φ n 1 (r 1 )φ n2 (r 2 ) L(r 1 r 2 r 3 r 4 ω) φ n 3 (r 3 )φ n4 (r 4 ) = L

64 Solving BSE L(r 1 r 2 r 3 r 4 ω) = L 0 (r 1 r 2 r 3 r 4 ω) + dr 5 dr 6 dr 7 dr 8 L 0 (r 1 r 2 r 5 r 6 ω) [ v(r 5 r 7 )δ(r 5 r 6 )δ(r 7 r 8 ) W (r 5 r 6 )δ(r 5 r 7 )δ(r 6 r 8 )] L(r 7 r 8 r 3 r 4 ω) Transition space How to solve it? L (n1 n 2 )(n 3 n 4 )(ω) = φ n 1 (r 1 )φ n2 (r 2 ) L(r 1 r 2 r 3 r 4 ω) φ n 3 (r 3 )φ n4 (r 4 ) = L

65 Exercise Calculate: L 0 (r 1, r 2, r 3, r 4, ω) = ij L 0 = (f j f i ) φ i (r 1)φ j (r 2 )φ i (r 3 )φ j (r 4) ω (E i E j ) f n1 f n2 ω (E n2 E n1 ) δ n 1 n 3 δ n2 n 4

66 Solving BSE BSE in transition space We consider only resonant optical transitions for a nonmetallic system: (n 1 n 2 ) = (vkck) (vc) L = L 0 + L 0 ( v W ) L L = [1 L 0 ( v W )] 1 L 0 L = [L 1 0 ( v W )] 1 L (vc)(v c )(ω) = [(E c E v ω)δ vv δ cc + (f v f c ) v W ] 1 (f c f v )

67 Solving BSE L (vc)(v c )(ω) = [(E c E v ω)δ vv δ cc + (f v f c ) v W ] 1 (f c f v ) Spectral representation of a hermitian operator [H exc ωi ] 1 = λ L (vc)(v c )(ω) = λ H exc A λ = E λ A λ A λ A λ E λ ω A (vc) λ A (v c ) λ E λ ω (f c f v )

68 Solving BSE L (vc)(v c )(ω) = [(E c E v ω)δ vv δ cc + (f v f c ) v W ] 1 (f c f v ) L [H exc ωi ] 1 Spectral representation of a hermitian operator [H exc ωi ] 1 = λ L (vc)(v c )(ω) = λ H exc A λ = E λ A λ A λ A λ E λ ω A (vc) λ A (v c ) λ E λ ω (f c f v )

69 Absorption spectra in BSE Independent (quasi)particles Abs(ω) vc v D c 2 δ(e c E v ω) Excitonic effects [H el + H hole +H el hole ] A λ = E λ A λ Abs(ω) λ vc A (vc) λ v D c 2 δ(e λ ω) mixing of transitions: v D c 2 vc A(vc) λ v D c 2 modification of excitation energies: E c E v E λ

70 BSE calculations A three-step method 1 LDA calculation Kohn-Sham wavefunctions ϕ i 2 GW calculation GW energies E i and screened Coulomb interaction W 3 BSE calculation solution of H exc A λ = E λ A λ with: H (vc)(v c ) exc = (E c E v )δ vv δ cc + (f v f c ) vc v W v c excitonic eigenstates A λ, E λ spectra ɛ M (ω)

71 A bit of history derivation of the equation (bound state of deuteron) E. E. Salpeter and H. A. Bethe, PR 84, 1232 (1951). BSE for exciton calculations L.J. Sham and T.M. Rice, PR 144, 708 (1966). W. Hanke and L. J. Sham, PRL 43, 387 (1979). first ab initio calculation G. Onida, L. Reining, R. W. Godby, R. Del Sole, and W. Andreoni, PRL 75, 818 (1995). first ab initio calculations in extended systems S. Albrecht, L. Reining, R. Del Sole, and G. Onida, PRL 80, 4510 (1998). L. X. Benedict, E. L. Shirley, and R. B. Bohn, PRL 80, 4514 (1998). M. Rohlfing and S. G. Louie, PRL 81, 2312 (1998).

72 Continuum excitons Bulk silicon G. Onida, L. Reining, and A. Rubio, RMP 74 (2002).

73 Bound excitons Solid argon F. Sottile, M. Marsili, V. Olevano, and L. Reining, PRB 76 (2007).

74 The Wannier model Bethe-Salpeter equation H exca λ = E λ A λ H (vc)(v c ) exc = (E c E v)δ vv δ cc + v W Wannier model two parabolic bands E c E v = E g + k 2 2µ 2 2µ no local fields ( v = 0) and effective screened W W (r, r ) = 1 ɛ 0 r r solution = Rydberg series for effective H atom E n = E g R eff n 2 with R eff = R µ ɛ 2 0

75 Exciton analysis Exciton amplitude: Ψ λ (r h, r e ) = vc A (vc) λ φ v(r h )φ c (r e ) Graphene nanoribbon Manganese Oxide D. Prezzi, et al., PRB 77 (2008). C. Rödl, et al., PRB 77 (2008).

76 Outline 1 Motivation 2 One-particle Green s functions: GW approximation 3 Two-particle Green s functions: Bethe-Salpeter equation 4 Micro-macro connection

77 Micro-macro connection Observation At long wavelength, external fields are slowly varying over the unit cell: dimension of the unit cell for silicon: 0.5 nm visible radiation 400 nm < λ < 800 nm Total and induced fields are rapidly varying: they include the contribution from electrons in all regions of the cell. Large and irregular fluctuations over the atomic scale.

78 Micro-macro connection Observation One usually measures quantities that vary on a macroscopic scale. When we calculate microscopic quantities we need to average over distances that are large compared to the cell parameter small compared to the wavelength of the external perturbation. The differences between the microscopic fields and the averaged (macroscopic) fields are called the crystal local fields.

79 Suppose that we are able to calculate the microscopic dielectric function ɛ, how do we obtain the macroscopic dielectric function ɛ M that we measure in experiments?

80 Micro-macro connection Fourier transform In a periodic medium, every function V (r, ω) can be represented by the Fourier series V (r, ω) = qg V (q + G, ω)e i(q+g)r or: V (r, ω) = q e iqr G V (q + G, ω)e igr = q e iqr V (q, r, ω) where: V (q, r, ω) = G V (q + G, ω)e igr V (q, r, ω) is periodic with respect to the Bravais lattice and hence is the quantity that one has to average to get the corresponding macroscopic potential V M (q, ω).

81 Micro-macro connection Averages V M (q, ω) = 1 Ω c drv (q, r, ω) V (q, r, ω) = G V (q + G, ω)e igr Therefore: V M (q, ω) = G V (q + G, ω) 1 Ω c dre igr = V (q + 0, ω) The macroscopic average V M corresponds to the G = 0 component of the microscopic V. Example V ext (q, ω) = ɛ M (q, ω)v tot,m (q, ω)

82 Micro-macro connection Averages V M (q, ω) = 1 Ω c drv (q, r, ω) V (q, r, ω) = G V (q + G, ω)e igr Therefore: V M (q, ω) = G V (q + G, ω) 1 Ω c dre igr = V (q + 0, ω) The macroscopic average V M corresponds to the G = 0 component of the microscopic V. Example V ext (q, ω) = ɛ M (q, ω)v tot,m (q, ω)

83 Micro-macro connection Fourier transforms Fourier transform of a function f (r, r, ω): f (q + G, q + G, ω) = drdr e i(q+g)r f (r, r, ω)e +i(q+g )r f G,G (q, ω) Therefore the relation V tot (r 1, ω) = dr 2 ɛ 1 (r 1, r 2, ω)v ext (r 2, ω) in the Fourier space becomes: V tot (q + G, ω) = G ɛ 1 G,G (q, ω)v ext (q + G, ω)

84 Micro-macro connection Fourier transforms Fourier transform of a function f (r, r, ω): f (q + G, q + G, ω) = drdr e i(q+g)r f (r, r, ω)e +i(q+g )r f G,G (q, ω) Therefore the relation V tot (r 1, ω) = dr 2 ɛ 1 (r 1, r 2, ω)v ext (r 2, ω) in the Fourier space becomes: V tot (q + G, ω) = G ɛ 1 G,G (q, ω)v ext (q + G, ω)

85 Micro-macro connection Example Macroscopic dielectric function V tot,m (q, ω) = ɛ 1 M (q, ω)v ext(q, ω) V tot(q + G, ω) = G ɛ 1 G,G (q, ω)vext(q + G, ω) V ext is a macroscopic quantity: V M,tot (q, ω) = V tot(q + 0, ω) V tot,m (q, ω) = ɛ 1 G=0,G =0 (q, ω)vext(q, ω) ɛ 1 M (q, ω) = ɛ 1 ɛ M (q, ω) = G=0,G =0 1 (q, ω) ɛ 1 G=0,G =0 (q, ω)

86 Micro-macro connection Example Macroscopic dielectric function V tot,m (q, ω) = ɛ 1 M (q, ω)v ext(q, ω) V tot(q + G, ω) = G ɛ 1 G,G (q, ω)vext(q + G, ω) V ext is a macroscopic quantity: V M,tot (q, ω) = V tot(q + 0, ω) V tot,m (q, ω) = ɛ 1 G=0,G =0 (q, ω)vext(q, ω) ɛ 1 M (q, ω) = ɛ 1 ɛ M (q, ω) = G=0,G =0 1 (q, ω) ɛ 1 G=0,G =0 (q, ω)

87 Micro-macro connection Example Macroscopic dielectric function V tot,m (q, ω) = ɛ 1 M (q, ω)v ext(q, ω) V tot(q + G, ω) = G ɛ 1 G,G (q, ω)vext(q + G, ω) V ext is a macroscopic quantity: V M,tot (q, ω) = V tot(q + 0, ω) V tot,m (q, ω) = ɛ 1 G=0,G =0 (q, ω)vext(q, ω) ɛ 1 M (q, ω) = ɛ 1 ɛ M (q, ω) = G=0,G =0 1 (q, ω) ɛ 1 G=0,G =0 (q, ω)

88 Micro-macro connection Example Macroscopic dielectric function V tot,m (q, ω) = ɛ 1 M (q, ω)v ext(q, ω) V tot(q + G, ω) = G ɛ 1 G,G (q, ω)vext(q + G, ω) V ext is a macroscopic quantity: V M,tot (q, ω) = V tot(q + 0, ω) V tot,m (q, ω) = ɛ 1 G=0,G =0 (q, ω)vext(q, ω) ɛ 1 M (q, ω) = ɛ 1 ɛ M (q, ω) = G=0,G =0 1 (q, ω) ɛ 1 G=0,G =0 (q, ω)

89 Micro-macro connection Example Macroscopic dielectric function V tot,m (q, ω) = ɛ 1 M (q, ω)v ext(q, ω) V tot(q + G, ω) = G ɛ 1 G,G (q, ω)vext(q + G, ω) V ext is a macroscopic quantity: V M,tot (q, ω) = V tot(q + 0, ω) V tot,m (q, ω) = ɛ 1 G=0,G =0 (q, ω)vext(q, ω) ɛ 1 M (q, ω) = ɛ 1 ɛ M (q, ω) = G=0,G =0 1 (q, ω) ɛ 1 G=0,G =0 (q, ω)

90 Micro-macro connection Macroscopic dielectric function V ext (q + G, ω) = G ɛ G,G (q, ω)v tot (q + G, ω) Remember: V ext is a macroscopic quantity: V ext (q, ω) = G ɛ G=0,G (q, ω)v tot (q + G, ω) V ext (q, ω) = ɛ G=0,G =0(q, ω)v tot,m (q, ω)+ G 0 ɛ G=0,G (q, ω)v tot (q+g, ω) V ext (q, ω) = ɛ M (q, ω)v tot,m (q, ω) ɛ M (q, ω) ɛ G=0,G =0(q, ω)

91 Micro-macro connection Macroscopic dielectric function V ext (q + G, ω) = G ɛ G,G (q, ω)v tot (q + G, ω) Remember: V ext is a macroscopic quantity: V ext (q, ω) = G ɛ G=0,G (q, ω)v tot (q + G, ω) V ext (q, ω) = ɛ G=0,G =0(q, ω)v tot,m (q, ω)+ G 0 ɛ G=0,G (q, ω)v tot (q+g, ω) V ext (q, ω) = ɛ M (q, ω)v tot,m (q, ω) ɛ M (q, ω) ɛ G=0,G =0(q, ω)

92 Micro-macro connection Macroscopic dielectric function V ext (q + G, ω) = G ɛ G,G (q, ω)v tot (q + G, ω) Remember: V ext is a macroscopic quantity: V ext (q, ω) = G ɛ G=0,G (q, ω)v tot (q + G, ω) V ext (q, ω) = ɛ G=0,G =0(q, ω)v tot,m (q, ω)+ G 0 ɛ G=0,G (q, ω)v tot (q+g, ω) V ext (q, ω) = ɛ M (q, ω)v tot,m (q, ω) ɛ M (q, ω) ɛ G=0,G =0(q, ω)

93 Micro-macro connection Spectra ɛ M (q, ω) = 1 ɛ 1 G=0,G =0 (q, ω) Abs(ω) = lim q 0 Imɛ M (ω) = lim EELS(ω) = lim Imɛ 1 M q 0 q 0 Im (ω) = lim q 0 Imɛ 1 1 ɛ 1 G=0,G =0 (q, ω) G=0,G =0 (q, ω)

94 Micro-macro connection Spectra ɛ M (q, ω) = 1 ɛ 1 G=0,G =0 (q, ω) Abs(ω) = lim q 0 Imɛ M (ω) = lim EELS(ω) = lim Imɛ 1 M q 0 q 0 Im (ω) = lim q 0 Imɛ 1 1 ɛ 1 G=0,G =0 (q, ω) G=0,G =0 (q, ω)

95 BSE vs. TDDFT: what in common? BSE L = L 0 + L 0 (v + Ξ)L TDDFT χ = χ 0 + χ 0 (v + f xc )χ

96 The Coulomb term v The Coulomb term v = v 0 + v

97 Local fields reloaded Microscopic-Macroscopic connection: local fields χ G,G (q, ω) = P G,G (q, ω) + P G,G1 (q, ω)v G1 (q)χ G1,G (q, ω) G 1 ɛ 1 G,G (q, ω) = δ G,G + v G (q)χ G,G (q, ω) ɛ M (q, ω) = 1 ɛ 1 G=0,G =0 (q, ω) Adler, Phys. Rev. 126 (1962); Wiser, Phys. Rev. 129 (1963).

98 Local fields reloaded Microscopic-Macroscopic connection: local fields ɛ M (q, ω) = 1 v G=0 (q) χ G=0,G =0(q, ω) χ G,G (q, ω) = P G,G (q, ω) + P G,G1 (q, ω) v G1 (q) χ G1,G (q, ω) G 1 v G (q) = 0 for G = 0 v G (q) = v G (q) for G 0 Hanke, Adv. Phys. 27 (1978).

99 Absorption Abs(ω) = lim q 0 Imɛ M (q, ω) Abs(ω) = lim q 0 Im [v G=0 (q) χ G=0,G =0(q, ω)] χ = P + P v χ EELS Absorption response to V ext + V macro ind Eels(ω) = lim q 0 Im[1/ɛ M (q, ω)] Eels(ω) = lim q 0 Im [v G=0 (q)χ G=0,G =0(q, ω)] χ = P + P(v 0 + v)χ Eels response to V ext

100 The Coulomb term v The Coulomb term v = v 0 + v long-range v 0 difference between Abs and Eels

101 Coulomb term v 0 : Abs vs. Eels F. Sottile, PhD thesis (2003) - Bulk silicon: absorption vs. EELS.

102 The Coulomb term v The Coulomb term v = v 0 + v long-range v 0 difference between Abs and Eels what about v?

103 The Coulomb term v The Coulomb term v = v 0 + v long-range v 0 difference between Abs and Eels what about v? v is responsible for crystal local-field effects

104 Coulomb term v: local fields v: local fields ɛ M = 1 v G=0 χ G=0,G =0 Set v = 0 in: χ G,G = χ 0 G,G + G 1 χ 0 G,G 1 v G1 χ G1,G χ G,G = χ 0 G,G Result: ɛ M = 1 v G=0 χ 0 G=0,G =0 that is: no local-field effects! (equivalent to Fermi s golden rule)

105 Coulomb term v: local fields v: local fields ɛ M = 1 v G=0 χ G=0,G =0 Set v = 0 in: χ G,G = χ 0 G,G + G 1 χ 0 G,G 1 v G1 χ G1,G χ G,G = χ 0 G,G Result: ɛ M = 1 v G=0 χ 0 G=0,G =0 that is: no local-field effects! (equivalent to Fermi s golden rule)

106 Coulomb term v: local fields v: local fields ɛ M = 1 v G=0 χ G=0,G =0 Set v = 0 in: χ G,G = χ 0 G,G + G 1 χ 0 G,G 1 v G1 χ G1,G χ G,G = χ 0 G,G Result: ɛ M = 1 v G=0 χ 0 G=0,G =0 that is: no local-field effects! (equivalent to Fermi s golden rule)

107 Coulomb term v: local fields Bulk silicon: absorption

108 Coulomb term v: local fields A. G. Marinopoulos et al., PRL 89 (2002) - Graphite EELS

109 What are local fields? Effective medium theory Uniform field E 0 applied to a dielectric sphere with dielectric constant ɛ in vacuum. From continuity conditions at the interface: P = 3 ɛ 1 4π ɛ + 2 E 0 Jackson, Classical electrodynamics, Sec. 4.4.

110 What are local fields? Effective medium theory Regular lattice of objects dimensionality d of material ɛ 1 in vacuum Maxwell-Garnett formulas dot (O D system) wire (1D system) Imɛ 1 (ω) Imɛ M (ω) 9 [Reɛ 1 (ω) + 2] 2 + [Imɛ 1 (ω)] 2 Imɛ M (ω) Imɛ 1(ω) Imɛ Imɛ 1 (ω) M (ω) 4 [Reɛ 1 (ω) + 1] 2 + [Imɛ 1 (ω)] 2

111 What are local fields? F. Bruneval et al., PRL 94 (2005) - Si nanowires S. Botti et al., PRB 79 (2009) - SiGe nanodots

112 MBPT & TDDFT MBPT helps improving DFT & TDDFT DFT & TDDFT help improving MBPT

113 Conclusion (TD)DFT & MBPT... try to learn both!

114 Many thanks!

115 Acknowledgements Silvana Botti Fabien Bruneval Valerio Olevano Lucia Reining Francesco Sottile Valérie Véniard

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