Polarisation Propagator

1. Polarisation Propagator • Collective excitations (M227, F 171, F 558) • Poles of G are single-particle excitations (creation of particles or holes) • Poles of P are collective excitations • Density and Density Fluctuation operators

2. Polarisation Propagator • Dielectric Function • The dielectric function for a material relates the electrostatic potential due to all charges to an external electrostatic potential

3. P Po r’,t’ r,t r’,t’ r,t Polarisation Propagator • Space-time interpretation of Polarisation Propagator • Drawn as a pair of directed lines • Non-interacting propagatorPo represented by single directed lines • Interacting propagatorP represented by filled loop Create electron-hole pair Destroy electron-hole pair t > t’ time

4. Polarisation Propagator • Lehmann Representation(F 172 M 375) physical significance of P

5. Polarisation Propagator • Lehmann Representation(F 172 M 375) physical significance of P

6. Polarisation Propagator • Time-Ordered (Causal) Polarisation Propagator (i)2Go(r,r) Go(r’,r’) (i)2Go(r,r’) Go(r’,r) -(i)2Go(r,r) Go(r’,r’)

7. 1,t1 Po(1,2) 2,t2 Polarisation Propagator • Single-particle polarisation propagator Po in coordinate form

8. Polarisation Propagator • Single-particle polarisation propagator Po in coordinate form

9. Im(e) Advanced xxx xx xx xxx x x xxx Re(e) xxx xx xx xxx x x xxx Retarded Polarisation Propagator • Poles of time-ordered P or Po in the complex energy plane

10. r’ r’ r 1 2 r r’ 1 2 r Polarisation Propagator • Leading terms in expansion of Polarisation Propagator (i)2Go(r,r’) Go(r’,r)

11. Polarisation Propagator • Further Diagrams in the Polarisation Propagator • Zeroth Order • First Order • Second Order • Third Order

12. Polarisation Propagator • Classification of Diagrams in the Polarisation Propagator • Proper • Improper • Ladder • Ring (Bubble) Vertex Part

13. = + + + … + + … + + Polarisation Propagator • Effective Interparticle Interaction (M 189, F 111, 154) • Total inter-particle/hole interaction is sum of direct (instantaneous Coulomb) interaction plus (retarded) reaction from medium = + P V = v + v P v V effective (dressed) interaction v Coulomb (bare) interaction P Polarisation propagator (polarisation insertion)

14. Polarisation Propagator • Effective Interparticle Interaction • Summation in terms of proper polarisation insertion P* P* = + + + … V = v + v P*V = v + v P* (v + v P* V) = v + v P* v + v P* v P* (v + v P*)V P = P* + P* v P V = v + v P v V = v + v (P* + P* v P) v V = v + v P*v + v P* v (P* + P* v P) v

15. Polarisation Propagator • Effective Interparticle Interaction:Dielectric Function • V = v + v P* V • (1 - v P*) V = v • V = (1 - P*v)-1 v • e-1 = (1 - v P*)-1 • e = (1 - v P*) • P* = Po yields the Random Phase Approximation to e • eRPA = (1 - v Po)

16. Polarisation Propagator • Selective Summation of Ring Diagrams • Restricting P* to Po in e-1 sums ring diagrams to infinite order • e-1RPA v = (1 - v Po)-1v= (1 + v Po + v Po v Po + v Po v Po v Po + … ) v • = v + v Po v + v Po v Po v+ v Po v Po v Po v+ … = + + + …

17. EExt p1 p2 Polarisation Propagator • Model of the Density Response Function

18. Polarisation Propagator • Model of the Density Response Function

19. Eext Eext Eext Polarisation Propagator • Model of the Density Response Function -T12 -T12 2 2 1 1 1 2 -T21

20. Polarisation Propagator • Model of the Density Response Function Si Si Herrendörfer and Patterson J Phys Chem Solids 58, 207 (1997)

21. Polarisation Propagator Expt - - - - Model ____ • Model of the Density Response Function

22. Polarisation Propagator • Model of the Density Response Function • Reflectance Anisotropy of stepped silicon surfaces Hogan and Patterson, Phys. Rev. B 57, 14843 (1998)

23. Polarisation Propagator • Model of the Density Response Function • Expansion of the polarisability Po in s (occ) and p (unocc) ETB Bloch orbitals s Bloch state px Bloch state Nicastro, Galamic-Mulaomerovic and Patterson, J. Phys. Condens. Matt. 13, 1215 (2001)

24. Polarisation Propagator • Model of the Density Response Function • Polarisability and Coulomb expansion coefficients R1q R2q Tq

25. Polarisation Propagator • Model of the Density Response Function

26. 2 4 4 3 1 1 2 3 2 4 3 The GW Approximation • Self Energy • Expressed generally as S = G W G M 211 • G dressed Green’s function • W is the screened interaction W = v + v P v • G is the vertex part (vertex correction) S(1,2) = G(1,3) W(1,4) G(3,4,2) = + + + …

27. 2 W(q, e’) Go(k - q, e - e’) 1 The GW Approximation • Self Energy:GoWo COHSEX approximation • Wo= e-1RPA v Wo is the screened interaction in RPA • G = d(2 - 3) d(2 - 4) • S(1,2) = Go(1,3) Wo(1,4)d(2 - 3)d(2 - 4) = Go(1,2) Wo(1,2) Hybertsen and Louie Phys. Rev. B 34, 5390 (1986) (861 citations)

28. The GW Approximation • Self Energy:GoWo COHSEX approximation • Poles of Go: Screened Exchange (SEX) contribution to S • Poles of Wo: Coulomb hole (COH) contribution to S

29. The GW Approximation • Poles of time-ordered e-1xand Gox in the complex energy plane Im(e) Advanced m x xx xx xx x x xxx xxx xx xx xxx x x xxx Re(e) xxx xx xx xxx x x xxx xxx xx xx xxx x x xxx Retarded

30. The GW Approximation • Self Energy:GoWo COHSEX approximation • Poles of Go: Screened Exchange (SEX) contribution to S • Poles of Wo: Coulomb hole (COH) contribution to S

31. 2 1 The GW Approximation • Model of the Density Response Function • Self Energy:GoWo approximation Nicastro, Galamic-Mulaomerovic and Patterson, J. Phys. Condens. Matt. 13, 1215 (2001)

32. 2 1 The GW Approximation • Model of the Density Response Function • Self Energy:GoWo approximation