GW Approximation in ωk space

Transcription

1 GW Approxmaton n ωk space Valero Olevano March 26, 208 Contents

2 GREEN FUNCTION G 0 2 Green functon G 0 G 0 ζ,ζ 2 ;ω = ζ ζ 2 +ηsgn ǫ 0 G 0 r,r 2 ;ω = r r 2 +ηsgn ǫ 0 G 0 r,r 2 ;ω = ξ ξ 2 G 0 ζ,ζ 2 ;ω = ξ ξ 2 = ξ = ξ = ζ ζ 2 +ηsgn ǫ 0 ζ +ηsgn ǫ 0 ζ +ηsgn ǫ 0 r 2 r,ξ = /2+ r,ξ = +/2 r 2,ξ 2 = /2+ r 2,ξ 2 = +/2 r 2 +ηsgn ǫ 0

3 2 SCREENED INTERACTION 3 2 Screened Interacton Wζ,ζ 2,ω = dζ 3 ε ζ,ζ 3,ωwζ 3,ζ 2 Wζ,ζ 2,ω = Wr,r 2,ω = dr 3 ε r,r 3,ωwr 3,r 2 spn ndependent but t s not the case of ε wr,r 2 = r r 2 coulomban many body nteracton wq,g = 2 fourer transform of the coulomban nteracton Wr,r 2,ω = Wr,r 2,ω = Wr,r 2,ω = dr 3 ε r,r 3,ωwr 3,r 2 dr 3 3 dq e q+gr ε q,g,g 3,ωe q+g3r3 BZ G,G 3 3 dq e q+g2r3 BZ q +G 2 2e q+g2r2 G 2 dr 3 e q+g3r3 e q+g2r3 = 3 δq q δg 3 G 2 3 dq e q+gr ε q,g,g 2,ω BZ q +G 2 2e q+g2r2 G,G 2 Wr,r 2,ω = e q+gr Wq,G,G 2,ωe q+g2r2 q,g,g 2 = e q+gr ε q,g,g 2,ω 2 2e q+g2r2 q,g,g 2 Wq,G,G 2,ω = ε q,g,g 2,ω 2 2 = ε q,g,g 2,ω 2 ε = symmetrzed epslon.

4 3 SINGLE PLASMON POLE MODEL 4 3 Sngle Plasmon Pole Model ε ω = δ + 2 ω 2 ω 2 ε G,G,q,ω = δg,g + 2 G,G,q ω 2 ω 2 G,G,q The poles are n ω = ± ω. AG,G,q = 2 G,G,q ω 2 G,G,q = ε G,G,q,0 δg,g [ ω 2 G,G AG,G ],q,q = ε G,G,q,0 ε G,G,q,E 0 E ε E 0 = δ + E0 2 ω2 ε E 0 [E0 2 + ω2 ] = δ[e0 2 + ω2 ] 2 ω 2 ω2 = δe0 2 +ε 0 ω 2 ω 2 ε E 0 δ = ε 0 ε E 0 E2 0 [ ε E 0 δ = ε 0 ε E 0 + ε 0 ε ] E 0 ε 0 ε E 0 E0 2 [ ω 2 ε ] [ ] 0 δ = ε 0 ε E 0 E0 2 A = ε 0 ε E 0 E 2 0

5 4 GW APPROXIMATION FOR THE SELF-ENERGY 5 4 GW approxmaton for the Self-Energy M x,x 2 = Gx,x 2 Wx +,x 2 M ζ,ζ 2,ω = dω e ω η Gζ,ζ 2,ω ω Wζ,ζ 2,ω W s spn-ndependent and G does not take factors 2 on spn sum: M r,r 2,ω = dω e ω η Gr,r 2,ω ω Wr,r 2,ω M r,r 2,ω = dω e ω η r r 2 ω ω ǫ 0 +ηsgn ǫ 0 e q+gr ε q,g,g 2,ω 2 e q+g2r2 q,g,g 2 5 Matrx elements of the Self-Energy operator j r M r,r 2,ω j r 2 = q,g,g 2 2 dr j r e q+gr r + dr 2 r 2 e q+g2r2 j r 2 dω e ω η ε q,g,g 2,ω ω ω ǫ 0 +ηsgn ǫ 0 = {n,k } ρ j G = dr re q+gr j r wth q = k j k G 0, q BZ j M ω j = δq k j k G 0 2 ρ jg ρ j G 2 G,G 2 + dω e ω η ε q,g,g 2,ω ω + +ηsgn ǫ 0 ε q,g,g,ω δg,g Σ x Sgma exchange ε q,g,g,ω 2 q,g,g ω 2 ω 2 q,g,g Σ c Sgma correlaton

6 6 SELF-ENERGY: EXCHANGE TERM 6 6 Self-Energy: exchange term j x ω j = δq k j k G 0 2 ρ jg 2 G dω e ω η ω + +ηsgn ǫ 0 Only the the poles of G n the upper half magnary ω-plane are ncluded n the closed, ant-clockwse path. They correspond to the partcle poles exctaton correspondng to occuped states. The ntegral yelds f = 0, occupaton number: dω... = f Res = f j x j = f δq k j k G 0 2 ρ jg 2 G t does not depend on ω because t consttutes only a shft term of the poles along the real axs whch doesn t change the ntegral.

7 7 SELF-ENERGY: CORRELATION TERM 7 7 Self-Energy: correlaton term j c ω j = δq k j k G 0 2 ρ jg ρ j G 2 G,G 2 dω e ω η 2 q,g,g 2 ω + +ηsgn ǫ 0 ω 2 ωq,g,g 2 η 2 The plasmon pole model presents a pole n the upper half plane for ω negatve at ω +η and a pole n the bottom half plane for ω postve at ω η dω e ω η ω + +ηsgn ǫ 0 ω ω +ηω + ω η The ncluded poles are n ω = ω +η and f f = n ω = ǫ 0 ω +η. dω f... = ǫ 0 ω 2 ω + 2 ω 2 ω = π ωω 2f 2 ω 2 = π = π + ω ω + ω ω ω2f ω + ω ω ω2f ω + ω ω + ω2f + ω2f ω ǫ0 2 ω 2 2f 2 = π 2 ω 2 ω + ω2f = π ω + ω2f Here we mean that we should replace ω ω η and ǫ 0 ǫ 0 ηsgnǫ 0 As 2f 2 s always. j c ω j = δq k j k G 0 ρ j G ρ j G 2 2 G,G 2 2 q,g,g 2 ωq,g,g 2 + ωq,g,g 2 2f

8 8 SELF-ENERGY: CORRELATION TERM WITHOUT PLASMON POLE MODEL 8 8 Self-Energy: correlaton term wthout plasmon pole model j c ω j = δq k j k G 0 2 ρ jg ρ j G 2 q,g,g 2 + where the correlaton part of the nverse delectrc matrx s ε c q,g,g 2,ω = ε q,g,g 2,ω δg,g 2 the delectrc functon s symmetrc n ω ε q,g,g 2, ω = ε q,g,g 2,ω dω e ω η ε ω + +ηsgn ǫ 0 c q,g,g 2,ω j c ω j = δq k j k G 0 2 ρ jg ρ j G 2 q,g,g 2 { dω ε c q,g,g 2,ω π 0 2 +ω + 2 } + ε c q,g,g 2,ǫ 0 ω θ θµ ǫ 0 The second term comes nto paly only n two cases: when s conducton wth + sgn many cases at hgh energes: µ < ǫ 0 < ω and when s valence wth sgn few cases: ω < ǫ 0 < µ For the delectrc functon we must use the tetraedral method to ntegrate or otherwse use a small magnary part n ω to account for fnte k-pont samplng.

9 9 IMAGINARY PART OF THE SELF-ENERGY 9 9 Imagnary part of the Self-Energy c r,r 2,ω = dω e ω η Gr,r 2,ω ω W c r,r 2,ω Gr,r 2,ω = dω Aω ω ω +ηsgnω µ W c r,r 2,ω = Wr,r 2,ω vr,r 2 W c r,r 2,ω = dω Dω ω ω +ηsgnω W c r,r 2,ω = vp dω Dω ω ω πsgnω δω ω Dω = vp dω Dω ω ω πsgnωdω Dr,r 2,ω = π IW cr,r 2,ωsgnω Dr,r 2, ω = Dr,r 2,ω W c r,r 2,ω = W c r,r 2, ω RW c ω = vp dω Dω ω ω = vp dω D ω ω +ω = RW c ω c r,r 2,ω = dω e ω η Gr,r 2,ω +ω W c r,r 2,ω c r,r 2,ω = dω e ω η µ Aω dω ω +ω ω η + 0 Dω 2 dω 2 ω ω 2 η + o µ Aω dω ω +ω ω +η dω 2 Dω 2 ω ω 2 +η The terms contrbutng to the ntegral n ω are only those who present poles both up and down the real axs, as for those presentng poles only up or only down, you can close the contour ntegraton path n the upper or n the bottom half plane resultng n zero. c r,r 2,ω = dω e ω η + µ µ dω Aω ω +ω ω +η 0 Aω dω ω +ω ω η 0 Dω 2 dω 2 ω ω 2 η Dω 2 dω 2 ω ω 2 +η Then, closng the contour ntegraton up e ω η and consderng the resdual of the poles whch are n ω = ω 2 +η for the frst term and n ω = ω ω+η for the second term, n the upper plane Res c r,r 2,ω = { 0 Aω Dω 2 µ } dω dω 2 µ ω +ω 2 ω +η + Aω Dω 2 dω dω 2 0 ω +ω ω 2 +η { Aω Dω 2 µ } = dω dω 2 θ ω 2 ω +ω 2 ω +η + Aω Dω 2 dω dω 2 θω 2 ω +ω ω 2 +η µ

10 9 IMAGINARY PART OF THE SELF-ENERGY 0 dω ω ±η = vp dω ω πδω { µ I c r,r 2,ω = dω πaω D ω +ω θ ω +ω+ dω πaω D ω +ω θ ω +ω µ I c r,r 2,ω = = = = In the G 0 W RPA approxmaton dω πaω D ω +ω θω ωθµ ω θω ω θω µ dω πaω Dω ω θω ωθµ ω +θω ω θω µ dω πaω π IW cω ω sgnω ω θω ωθµ ω +θω ω θω µ dω Aω IW c ω ω θω ωθµ ω θω ω θω µ Ar,r 2,ω = r r 2 δ I c r,r 2,ω = r r 2 IW c r,r 2, θǫ 0 ωθµ ǫ 0 θ θǫ 0 I c r,r 2,ω = j I c ω j = { occ r r 2 IW c r,r 2, θǫ 0 ω ω µ + unocc r r 2 IW c r,r 2, θ ω > µ dr dr 2 j r j r 2 r r 2 θǫ 0 ωθµ ǫ 0 θ θǫ 0 q,g,g 2 e q+gr I ε c q,g,g 2, e q+g2r2 2 So that the fnal result s j I c ω j = 0 θǫ ωθµ ǫ 0 θ θǫ 0 δq k j k G 0 ρ jg ρ j G 2 2 I ε c q,g,g 2, q,g,g 2 θǫ 0 ωθµ ǫ 0 θω ǫ 0 θǫ 0 = θµ ǫ 0 θω ǫ 0 = θǫ 0 ωf θω ǫ 0 f Iε c ω = Iε c ω even functon

11 A DEFINITIONS, NOTATIONS A Defntons, notatons = N k cell Crystal Volume BZ = 3 cell 3 = BZdk BZ BZ BZ k dk = BZ N k k B Fourer transform defnton fω = dte ωt ft ft = dωfωe ωt drect fourer transform nverse fourer transform

12 C FOURIER TRANSFORM OF A TWO LATTICE INDICES QUANTITY 2 C Fourer transform of a two lattce ndces quantty fq,g,g 2 = 3 dr dr 2 e q+gr fr.r 2 e q+g2r2 fr,r 2 = 3 dq e q+gr fq,g.g 2 e q+g2r2 BZ G,G 2 fr,r 2 = e q+gr fq,g.g 2 e q+g2r2 q,g,g 2 Demonstraton: fq,q 2 = 3 dr dr 2 e Qr fr.r 2 e Q2r2 fr,r 2 = 3 dq dq 2 e Qr fq.q 2 e Q2r2 defnton fourer transform defnton nverse fourer transform fr +R,r 2 +R = fr,r 2 lattce perodcty 3 dq dq 2 e Qr+R fq.q 2 e Q2r2+R = 3 dq dq 2 e Qr fq.q 2 e Q2r2 3 dq dq 2 e q+gr+r fq.q 2,G,G 2 e q2+g2r2+r G,G 2 = 3 dq dq 2 e GR = e G2R = G,G 2 e q+gr fq.q 2,G,G 2 e q2+g2r2 e q q2r = q = q 2 fq.q 2,G,G 2 = fq,g,g 2 δq q 2

13 D CASE Q 0,G = 0 FOR ρ 2 Q,G = 0/Q 2 3 D Case q 0,G = 0 for ρ 2 q,g = 0/q 2 F = BZ q fq q 2 = fq = 0I SZ + BZ q 0 fq q 2 I SZ = 3 dq BZ/N k q 2 = N k dq BZ BZ/N k q 2 If we assume a spherc Brlloun zone of volume V and radus 3V/ /3 : V V I SZ = 7.79 dq q 2 = V 3V/ /3 0 2/3 BZ N k dq = 3 /3 2/3 V 2/3 In the case of a Brlloun Zone shape such for an fcc materal: I SZ = /3 BZ N k For other materals: sc: 6.88 fcc: 7.43 bcc: wz: hcp for deal u=3/8 wurzte