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 BetheSalpeter Equation
3 Outline 1 Motivation 2 Green s functions 3 The GW Approximation 4 The BetheSalpeter 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 oneparticle G: electron addition and removal  GW twoparticle L: electronhole 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: nonlocal in space γ(r 1, r 2, t) Oneparticle Green s function: nonlocal 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: nonlocal in space γ(r 1, r 2, t) Oneparticle Green s function: nonlocal 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 BetheSalpeter Equation
10 Photoemission Direct Photoemission Inverse Photoemission
11 Oneparticle Green s function The oneparticle 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 Oneparticle Green s function The oneparticle 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 Oneparticle 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 Oneparticle 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 1particle 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 Oneparticle 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 Oneparticle excitations poles of oneparticle Green s function G
17 Oneparticle Green s function Oneparticle Green s function From oneparticle G we can obtain: oneparticle excitation spectra groundstate expectation value of any oneparticle 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 + ) groundstate total energy
18 Absorption Twoparticle excitations poles of twoparticle Green s function L Excitonic effects = electron  hole interaction
19 Absorption Twoparticle excitations poles of twoparticle Green s function L Excitonic effects = electron  hole interaction
20 Absorption Twoparticle excitations poles of twoparticle Green s function L Excitonic effects = electron  hole interaction
21 Table of characters G(1, 2): oneparticle Green s function (2 points) L(1, 2, 3, 4): twoparticle Green s function (4 points)
22 Outline 1 Motivation 2 Green s functions 3 The GW Approximation 4 The BetheSalpeter Equation
23 GW bandstructure: photoemission additional charge
24 GW bandstructure: photoemission additional charge reaction: polarization, screening GW approximation 1 polarization made of noninteracting electronhole pairs (RPA) 2 classical (Hartree) interaction between additional charge and polarization charge L. Hedin, Phys. Rev. 139 (1965)
25 GW and HartreeFock
26 GW and HartreeFock
27 GW and HartreeFock HartreeFock 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 HartreeFock from Brice Arnaud
29 GW and HartreeFock M. van Schilfgaarde et al., PRL 96 (2006).
30 Outline 1 Motivation 2 Green s functions 3 The GW Approximation 4 The BetheSalpeter 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 BetheSalpeter 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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