# Introduction to Green s functions

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1 Introduction to Green s functions Matteo Gatti ETSF Users Meeting and Training Day Ecole Polytechnique - 22 October 2010

2 Outline 1 Motivation 2 Green s functions 3 The GW Approximation 4 The Bethe-Salpeter Equation

3 Outline 1 Motivation 2 Green s functions 3 The GW Approximation 4 The Bethe-Salpeter Equation

4 Electronic Spectroscopy Here only two categories: 1 charged excitations: photoemission and inverse photoemission 2 neutral excitations: absorption and electron energy loss

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

6 MBPT vs. TDDFT: different worlds, same physics (TD)DFT based on the density response function χ: neutral excitations moves density around is efficient (simple) 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)

7 Table of characters Density: local in space and time ρ(r 1, t 1 ) Density matrix: non-local in space γ(r 1, r 2, t) One-particle Green s function: non-local in space and time: G(r 1, r 2, t 1, t 2 ) G(1, 2) G(r 1, r 2, t 1, t 2 ) G(r 1, r 2, t 1, t 2 ) = G(r 1, r 2, t 1 t 2 ) G(r 1, r 2, ω) ρ(r 1, t 1 ) = ig(r 1, r 1, t 1, t + 1 )

8 Table of characters Density: local in space and time ρ(r 1, t 1 ) Density matrix: non-local in space γ(r 1, r 2, t) One-particle Green s function: non-local in space and time: G(r 1, r 2, t 1, t 2 ) G(1, 2) G(r 1, r 2, t 1, t 2 ) G(r 1, r 2, t 1, t 2 ) = G(r 1, r 2, t 1 t 2 ) G(r 1, r 2, ω) ρ(r 1, t 1 ) = ig(r 1, r 1, t 1, t + 1 )

9 Outline 1 Motivation 2 Green s functions 3 The GW Approximation 4 The Bethe-Salpeter Equation

10 Photoemission Direct Photoemission Inverse Photoemission

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

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

13 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

14 One-particle Green s function What is G? Definition and meaning of G: h i G(r 1, t 1 ; r 2, t 2 ) = i < N T ψ(r 1, t 1 )ψ (r 2, t 2 ) N > Insert a complete set of N + 1 or N 1-particle states. This yields G(r 1, t 1 ; r 2, t 2 ) = i X j f j (r 1 )f j (r 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 (r 1 ) = N ψ (r 1) N + 1, j, ε j > µ N 1, j ψ (r 1 ) N, ε j < µ

15 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 ).

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

17 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

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

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

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

21 Table of characters G(1, 2): one-particle Green s function (2 points) L(1, 2, 3, 4): two-particle Green s function (4 points)

22 Outline 1 Motivation 2 Green s functions 3 The GW Approximation 4 The Bethe-Salpeter Equation

23 GW bandstructure: photoemission additional charge

24 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 L. Hedin, Phys. Rev. 139 (1965)

25 GW and Hartree-Fock

26 GW and Hartree-Fock

27 GW and Hartree-Fock Hartree-Fock GW Σ(12) = ig(12)v(1 + 2) Σ(12) = ig(12)w (1 + 2) v infinite range in space v is static Σ is nonlocal, hermitian, static W is short ranged W is dynamical Σ is nonlocal, complex, dynamical

28 GW and Hartree-Fock from Brice Arnaud

29 GW and Hartree-Fock M. van Schilfgaarde et al., PRL 96 (2006).

30 Outline 1 Motivation 2 Green s functions 3 The GW Approximation 4 The Bethe-Salpeter Equation

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

32 What is wrong? What is missing?

33 Beyond RPA Independent particles (RPA)

34 Beyond RPA Interacting particles (excitonic effects)

35 Absorption spectra in BSE Independent (quasi)particles Abs(ω) vc v D c 2 δ(e c E v ω) Excitonic effects: the Bethe-Salpeter equation [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

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

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

38 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).

39 Electronic Spectroscopy Here only two categories: 1 charged excitations: photoemission and inverse photoemission = GW 2 neutral excitations: absorption and electron energy loss = BSE

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