Introduction to Green functions, GW, and BSE

Transcription

1 Introduction to Green functions, the GW approximation, and the Bethe-Salpeter equation Stefan Kurth 1. Universidad del País Vasco UPV/EHU, San Sebastián, Spain 2. IKERBASQUE, Basque Foundation for Science, Bilbao, Spain 3. European Theoretical Spectroscopy Facility (ETSF),

2 Outline One-particle Green functions The The Summary

3 Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation One-particle Green functions

4 Green functions in mathematics Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation consider inhomogeneous differential equation (1D for simplicity) ˆD x y(x) = f(x) where ˆD x is linear differential operator in x. Example: damped harmonic oscillator ˆD x = d2 dx 2 + γ d dx + ω2 general solution of inhomogeneous equation: y(x) = y hom (x) + y spec (x) where y hom is solution of the homogeneous eqn. ˆDx y hom (x) = 0 and y spec (x) is any special solution of the inhomogeneous equation.

5 Green functions in mathematics (cont.) Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation how to obtain a special solution of the inhomogeneous equation for any inhomogeneity f(x)? first find the solution of the following equation ˆD x G(x, x ) = δ(x x ) This defines the Green function G(x, x ) corresponding to the operator ˆD x. Once G(x, x ) is found, a special solution can be constructed by y spec (x) = dx G(x, x )f(x ) check: ˆDx dx G(x, x )f(x ) = dx δ(x x )f(x ) = f(x)

6 Hamiltonian of interacting electrons Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation consider system of interacting electrons in static external potential V ext (r) described by Hamiltonian Ĥ ) Ĥ = ˆT + ˆV ext + Ŵ = d 3 x ˆψ (x) ( V ext(r) ˆψ(x) d 3 x d 3 x ˆψ (x) ˆψ (x 1 ) r r ˆψ(x ) ˆψ(x) x = (r, σ): space-spin coordinate ˆψ (x), ˆψ(x): electron creation and annihilation operators

7 Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation One-particle Green functions at zero temperature Time-ordered 1-particle Green function at zero temperature ig(x, t; x, t ) = ΨN 0 ˆT [ ˆψ(x, t) H ˆψ (x, t ) H ] Ψ N 0 Ψ N 0 ΨN 0 Ψ N 0 : N-particle ground state of Ĥ: Ĥ ΨN 0 = EN 0 ΨN 0 ˆψ(x, t) H = exp(iĥt) ˆψ(x) exp( iĥt) : electron annihilation operator in Heisenberg picture ˆT : time-ordering operator ˆT [ ˆψ(x, t) H ˆψ(x, t ) H ] = θ(t t ) ˆψ(x, t) H ˆψ(x, t ) H θ(t t) ˆψ(x, t ) H ˆψ(x, t) H

8 Green functions as propagator Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation t 1 > t 2 (t > t ) create electron at time t 2 at position r 2 and propagate; then annihilate electron at time t 1 at position r 1 t 2 > t 1 (t > t) annihilate electron (create hole) at time t 1 at position r 1 ; then create electron (annihilate hole) at time t 2 at position r 2

9 Observables from Green functions Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Information which can be extracted from Green functions ground-state expectation values of any single-particle operator Ô = d 3 x ˆψ (x)o(x) ˆψ(x) e.g., density operator ˆn(r) = σ ˆψ (rσ) ˆψ(rσ) ground-state energy of the system Galitski-Migdal formula E0 N = i d 3 x lim (i 2 lim t t + r r ) 2 G(rσ, t; r t σ, t ) 2 spectrum of system: direct photoemission, inverse photoemission

10 Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Spectral (Lehmann) representation of Green function use completeness relation 1 = N,k ΨN k ΨN k and Fourier transform w.r.t. t t Lehmann representation = k G(x, x ; ω) g k (x)gk (x ) ω (E N+1 k E0 N) + iη + k ω + (E N 1 k f k (x)f k (x ) E N 0 ) iη with quasiparticle amplitudes f k (x) = Ψ N 1 k ˆψ(x) Ψ N 0 g k (x) = Ψ N N+1 0 ˆψ(x) Ψ k

11 Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Spectral information contained in Green function Green function contains spectral information on single-particle excitations changing the number of particles by one! The poles of the GF give the corresponding excitation energies. direct photoemission inverse photoemission

12 Spectral function Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Spectral function A(x, x ; ω) = 1 π Im GR (x, x ; ω) = k g k (x)gk (x )δ(ω+e0 N E N+1 k )+f k (x)fk (x )δ(ω+e N 1 k E0 N ) A(x, x ; ω): local density of states

13 Perturbation theory for Green function Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Green function G(x, t; x, t ) = i Ψ N 0 ˆT [ ˆψ(x, t) H ˆψ(x, t ) H ] ΨN 0 is a complicated object, it involves many-body ground state Ψ N 0 perturbation theory to calculate Green function: split Hamitonian in two parts Ĥ = Ĥ0 + Ŵ = ˆT + ˆV ext + Ŵ treat interaction Ŵ as perturbation machinery of many-body perturbation theory: Wick s theorem, Gell-Mann-Low theorem, and, most importantly, Feynman diagrams

14 Feynman diagrams Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Feynman diagrams: graphical representation of perturbation series elements of diagrams: Green function G 0 of noninteracting system (Ĥ0) Green function G of interacting system Coulomb interaction v Clb (x, t; x, t ) = δ(t t ) r r

15 Diagrammatic series for Green function Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Perturbation series for G: sum of all connected diagrams = Lots of diagrams!

16 Self energy: reducible and irreducible Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Self energy insertion and reducible self energy Self energy insertion: any part of a diagram which is connected to the rest of the diagram by two G 0 -lines, one incoming and one outgoing Reducible self energy Σ: sum of all self-energy insertions =

17 Self energy: reducible and irreducible Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Proper self energy insertion and irreducible (proper) self energy Proper self energy insertion: any self energy insertion which cannot be separated in two pieces by cutting a single G 0 -line Irreducible self energy Σ: sum of all proper self-energy insertions =

18 Dyson equation Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation = + = = + Dyson equation G(x, x ; ω) = G 0 (x, x ; ω) + d 3 y d 3 y G 0 (x, y; ω)σ(y, y ; ω)g(y, x ; ω)

19 Skeletons and dressed skeletons Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Skeleton diagram: self-energy diagram which does contain no other self-energy insertions except itself Dressed skeleton: replace all G 0 -lines in a skeleton by G-lines irreduzible self energy: sum of all dressed skeleton diagrams Σ becomes functional of G: Σ = Σ[G]

20 Equation of motion for Green function Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Lehmann representation for G 0 G 0 (x, x ; ω) = k θ(ε k ε F )ϕ k (x)ϕ k (x ) ω ε k + iη + k θ(ε F ε k )ϕ k (x)ϕ k (x ) ω ε k iη act with operator ω ĥ0(x) = ω ( 2 x 2 + v ext (x)) on G 0 Equation of motion for non-interacting Green function G 0 (ω ĥ0(x))g 0 (x, x ; ω) = k ϕ k (x)ϕ k (x ) = δ(x x ) G 0 is a mathematical Green function!

21 Green functions in mathematics One-particle Green functions Perturbation theory and Feynman diagrams Self energy and Dyson equation Equation of motion for Green function (cont.) act with ω ĥ0(x) on Dyson equation for G Equation of motion for interacting Green function G (ω ĥ0(x))g(x, x ; ω) = δ(x x ) + d 3 y Σ(x, y ; ω)g(y, x ; ω) or with time arguments ( i ) t ĥ0(x) G(x, t; x, t ) = δ(x x )δ(t t ) + d 3 y dt Σ(x, t; y, t )G(y, t, x ; t )

22 Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Polarization propagator and two-particle Green functions

23 Linear density response Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Suppose we expose our interacting many-electron system to an external, time-dependent perturbation ˆV (t) = d 3 x δv (x, t)ˆn(x) we are interested in the change of the density δn(x, t) = Ψ N (t) ˆn(x) Ψ N (t) Ψ N 0 ˆn(x) Ψ N 0 to linear order in δv (x, t) time-dependent Schrödinger equation i ) t ΨN (t) = (Ĥ + ˆV (t) Ψ N (t) in Heisenberg picture Ψ N (t) H = exp(iĥt) ΨN (t) i t ΨN (t) H = ˆV (t) H Ψ N (t) H

24 Linear density response (cont.) to linear order in δv (x, t) we have Ψ N (t) = exp( iĥt) (1 i Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations t 0 dt ˆV (t ) H ) Ψ N 0 and for δn(x, t) = d 3 x 0 dt χ(x, t; x, t )δv (x, t ) with linear density response function iχ(x, t; x, t ) = iπ R (x, t; x, t ) = θ(t t ) ΨN 0 [ˆñ(x, t) H, ˆñ(x, t ) H ] Ψ N 0 Ψ N 0 ΨN 0 with ˆñ(x, t) H = ˆn(x, t) H Ψ N 0 ˆn(x) ΨN 0

25 Linear density response (cont.) Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Lehmann representation of linear density response function χ(x, x ; ω) = Π R (x, x ; ω) = k k Ψ N 0 ˆñ(x) Ψ N k ΨN k ˆñ(x ) Ψ N 0 ω (Ek N EN 0 ) + iη Ψ N 0 ˆñ(x ) Ψ N k ΨN k ˆñ(x) Ψ N 0 ω + (Ek N EN 0 ) + iη note: the poles of χ are at the optical excitation energies of the system, i.e., excitations for which the number of particles does not change!

26 Macroscopic response in solids Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Optical absorption from macroscopic dielectric function ɛ M (ω) δv ext (r, t) = V ext (q)e i(ωt qr), q G In a periodic system, induced potential δv ind contains all components with k = q + G δv ind (r, t) = e iωt G VG ind (q)e i(q+g)r Fourier component of the total potential in a solid: V tot G (q) = δ G,0 V ext (q)+vg ind (q) = [δ G,0 + v G (q)χ G,0 (q, ω)] V ext (q) with response function in momentum representation χ G,G (q, ω) = dr 1 dr 2 e i(q+g)r 1 χ(r 1, r 2, ω)e i(q+g )r 2

27 Macroscopic response in solids Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Macroscopic field and macroscopic dielectric function Macroscopic (averaged) potential: Macroscopic dielectric function: V ext (q) = ɛ M (q, ω)v tot M (q) ɛ M (q, ω) = optical absorption rate VM tot (q) = V tot G=0 (q) v G=0 (q)χ 0,0 (q, ω) Abs(ω) = lim q 0 Im ɛ M (q, ω)

28 Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Two-particle Green function and polarization propagator Two-particle Green function i 2 G (2) (x 1, t 1 ; x 2, t 2 ; x 3, t 3 ; x 4, t 4 ) = 1 Ψ N 0 ΨN 0 Ψ N 0 ˆT [ ˆψ(x 1, t 1 ) H ˆψ(x2, t 2 ) H ˆψ (x 3, t 3 ) H ˆψ (x 4, t 4 ) H ] Ψ N 0 Polarization propagator iπ(x, t; x, t ) = ΨN 0 ˆT [ˆñ(x, t) H ˆñ(x, t ) H ] Ψ N 0 Ψ N 0 ΨN 0 relation between the two: i 2 G (2) (x 1, t 1 ; x 2, t 2 ; x 1, t + 1 ; x 2, t + 2 ) = iπ(x 1, t 1 ; x 2, t 2 )+n(x 1 )n(x 2 )

29 Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Lehmann representation of polarization propagator Π(x, x ; ω) = k Ψ N 0 ˆñ(x) Ψ N k ΨN k ˆñ(x ) Ψ N 0 ω (Ek N EN 0 ) + iη k Ψ N 0 ˆñ(x ) Ψ N k ΨN k ˆñ(x) Ψ N 0 ω + (Ek N EN 0 ) iη compare with Lehmann representation of linear density response χ(x, x ; ω) = Π R (x, x ; ω) = k k Ψ N 0 ˆñ(x) Ψ N k ΨN k ˆñ(x ) Ψ N 0 ω (Ek N EN 0 ) + iη Ψ N 0 ˆñ(x ) Ψ N k ΨN k ˆñ(x) Ψ N 0 ω + (Ek N EN 0 ) + iη

30 Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Particle-hole propagator: diagrammatic representation Definition of particle-hole propagator The particle-hole propagator is the two-particle Green function with a time-ordering such that both the two latest and the two earliest times correspond to one creation and one annihilation operator Diagrammatic representation: x 4, t 4 x 1, t 1 L(x 1,t 1 ;x 2,t 2 ;x 3,t 3 ;x 4,t 4 ) = + x 4, t 4 x 1, t 1 x 2, t 2 x 3, t 3 x 2, t 2 x 3, t 3 Diagrammatic representation of polarization propagator:

31 Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Polarization propagator and irreducible polarization insertions Irreducible polarization insertion A diagram for the polarization propagator which cannot be reduced to lower-order diagrams for Π by cutting a single interaction line Def: Dyson-like eqn. for full polarization propagator

32 Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Effective interaction and dielectric function Effective interaction W =: ε 1 v Clb = v Clb + v Clb P W Dielectric function ε = 1 v Clb P Inverse dielectric function ε 1 = 1 + v Clb Π

33 Vertex insertions Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Vertex insertion (part of a) diagram with one external in- and one outgoing G 0 -line and one external interaction line Irreducible vertex insertion A vertex insertion which has no self-energy insertions on the inand outgoing G 0 -lines and no polarization insertion on the external interaction line

34 Irreducible vertex and Hedin s equations Irreducible vertex Linear density response Two-particle Green function and polarization propagator Particle-hole propagator: diagrammatic representation Hedin s equations Hedin s equations (exact!) Hedin s equations Σ = v Hart + igw Γ ip = GGΓ G = G 0 + G 0 ΣG L. Hedin, Phys. Rev. 139 (1965) W = v Clb + v Clb P W Γ = 1 + δσ δg GGΓ

35 Band gaps and quasiparticle bands in GW Optical absorption in GW The

36 Band gaps and quasiparticle bands in GW Optical absorption in GW In the the vertex is approximated as: Γ 1 Hedin s equations (exact)

37 Band gaps and quasiparticle bands in GW Optical absorption in GW Perturbative GW corrections ĥ 0 (r)φ i (r) + ĥ 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 = GW : E i ɛ i = ϕ i Σ V xc ϕ i Hybertsen and Louie, PRB 34, 5390 (1986); Godby, Schlüter and Sham, PRB 37, (1988)

38 Band gaps and quasiparticle bands in GW Optical absorption in GW GW propaganda slide: improvement of gaps over LDA Schilfgaarde, PRL 96, (2006)

39 GW quasiparticle bands in copper Band gaps and quasiparticle bands in GW Optical absorption in GW dashed: LDA, solid: GW, dots: expt. Marini, Onida, Del Sole, PRL 88, (2001)

40 Band gaps and quasiparticle bands in GW Optical absorption in GW Absorption by independent Kohn-Sham particles Independent transitions: Im[ɛ(ω)] = 8π 2 ω 2 ij ϕ j e ˆv ϕ i 2 δ(ε j ε i ω)

41 Band gaps and quasiparticle bands in GW Optical absorption in GW Absorption by independent Kohn-Sham particles Independent transitions: Im[ɛ(ω)] = 8π 2 ω 2 ij ϕ j e ˆv ϕ i 2 δ(ε j ε i ω) Particles are interacting!

42 Band gaps and quasiparticle bands in GW Optical absorption in GW Optical absorption in GW: Independent quasiparticles Independent transitions: Im[ɛ(ω)] = 8π 2 ω 2 ij ϕ j e ˆv ϕ i 2 δ(e j E i ω)

43 Band gaps and quasiparticle bands in GW Optical absorption in GW Optical absorption in GW: Independent quasiparticles Independent transitions: Im[ɛ(ω)] = 8π 2 ω 2 ij ϕ j e ˆv ϕ i 2 δ(e j E i ω) Something still missing: the vertex, i.e., particle-hole interactions!

44 Absorption Band gaps and quasiparticle bands in GW Optical absorption in GW Neutral excitations poles of two-particle Green s function L in GW: excitonic effects, i.e., electron-hole interaction missing

45 Absorption Band gaps and quasiparticle bands in GW Optical absorption in GW Neutral excitations poles of two-particle Green s function L in GW: excitonic effects, i.e., electron-hole interaction missing

46 Absorption Band gaps and quasiparticle bands in GW Optical absorption in GW Neutral excitations poles of two-particle Green s function L in GW: excitonic effects, i.e., electron-hole interaction missing

47 Derivation of BSE Standard approximations to BSE The

48 Derivation of BSE Standard approximations to BSE Derivation of the Bethe-Salpeter equation (1) Propagator of e-h pair in a many-body system: 1st step: Dressing one-particle propagators (Dyson equation) Propagation of dressed interacting electron and hole:

49 Derivation of BSE Standard approximations to BSE Derivation of the Bethe-Salpeter equation (2) 2nd step: Classification of scattering processes Identify two-particle irreducible blocks where γ(1234) is the electron-hole stattering amplitude

50 Derivation of BSE Standard approximations to BSE Derivation of the Bethe-Salpeter equation (3) Final step: Summation of a geometric series Bethe-Salpeter equation

51 Derivation of BSE Standard approximations to BSE Derivation of the Bethe-Salpeter equation (3) Final step: Summation of a geometric series Bethe-Salpeter equation Analytic form of the Bethe-Salpeter equation (j = {x j, t j }) L(1234) = L 0 (1234)+ L 0 (1256)[v(57)δ(56)δ(78) γ(5678)]l(7834)d5d6d7d8

52 Derivation of BSE Standard approximations to BSE The Bethe-Salpeter equation: Approximation BSE determines electron-hole propagator L(1234), provided that self-energy Σ(12) and e-h scattering amplitude γ(1234) are given. Standard approximation: Approximate Σ by GW diagram: Σ(12) = G(12)W (12) = = + Approximate γ by W : γ(1234) = W (12)δ(13)δ(24)

53 Derivation of BSE Standard approximations to BSE The Bethe-Salpeter equation: Approximation Approximate Bethe-Salpeter equation Analytic form of the approximate 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)d5d6d7d8 L 0 (1234) = G(12)G(43) and W (12) from GW calculation

54 BSE calculations Derivation of BSE Standard approximations to BSE 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 L = L 0 + L 0 (v γ)l e-h propagator L(r 1 r 2 r 3 r 4 ω) spectra ɛ M (ω)

55 Results: Continuum excitons (Si) Derivation of BSE Standard approximations to BSE Bulk silicon G. Onida, L. Reining, and A. Rubio, RMP 74, 601 (2002)

56 Results: Bound excitons (solid Ar) Derivation of BSE Standard approximations to BSE Solid argon F. Sottile, M. Marsili, V. Olevano, and L. Reining, PRB 76, (R) (2007)

57 Summary Derivation of BSE Standard approximations to BSE Green functions: important concept in many-particle physics Diagrammatic analysis of Green functions (deceptively) simple, actual calculation of specific diagrams much harder for self energy Bethe-Salpeter equation: captures excitons in optical absorption

58 Literature Derivation of BSE Standard approximations to BSE endless number of textbooks on Green functions My favorites E.K.U. Gross, E. Runge, O. Heinonen, Many-Particle Theory (Hilger, Bristol, 1991) A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill, New York, 1971) and later edition by Dover press G. Stefanucci, R. van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction (Cambridge, 2003)

59 Literature Derivation of BSE Standard approximations to BSE on GW and BSE G. Onida, L. Reining, A. Rubio, Rev. Mod. Phys. 74, 601 (2002) S. Botti, A. Schindlmayr, R. Del Sole, and L. Reining, Rep. Progr. Phys. 70, 357 (2007)

60 Thanks Derivation of BSE Standard approximations to BSE Ilya Tokatly and Matteo Gatti for some figures YOU for your patience!

61 Thanks Derivation of BSE Standard approximations to BSE Ilya Tokatly and Matteo Gatti for some figures YOU for your patience!