The two-body Green s function

Transcription

1 The two-body Green s function G ( x, x, x, x ) T ( x ) ( x ) ( x ) ( x ) (Heisenberg picture operators, average over interacting g.s.) Relevant to ground state energy and magnetism, screened interaction, dielectric function, Auger spectroscopy... G is related to G - Recall the EOM i h ( x) 0 G( x, z) ( x z) ( t t ) i dyv( x, y) G x, y, y, z. x z t x z i h ( x) 0 t y x z y The perturbation expansion leads to the same unlinked diagrams as before and the linked cluster theorem- The diagram rules hold- 1

2 Bethe-Salpeter equation for the two-body Green s function A Dyson-like analysis of series for the two-particle G ( x, x, x, x ) T ( x ) ( x ) ( x ) ( x ) leads to: only self-energy insertions G 1,,3,4 ig 3,1 ig 4, ig 3, ig 4,1 g d4d6d7d8 ig 3, 5 ig 7,1 ig 4,6 ig 8, 3 5,6,7,8 GF g Scattering amplitude=sum of linked diagrams with two legs in and out (legs are not included) (exchange for parallel spins only; for parallel spins every interaction consists of direct+exchange)

3 g is is the sum of all two-body diagrams, such that each ingoing line starts with an interaction each outgoing line leaves an interaction red lines not included in g while internal lines can have all sorts of self-energy insertions. How can we proceed with Dyson s philosophy? 3 3

4 In g one can separate out the irreducible interaction diagrams that cannot be split in two by cutting only two dressed lines let J denote their sum (irreducible scattering amplitude) J = Then we get the scattering amplitude by iteration g = J + J J

5 = J + J J +... The series can then be summed formally = J + J (boxes must have incoming and outgoing legs; propagators are fully dressed, of course) 5

6 From the expansion of g with J =irreducible scattering amplitude = J + J and the Dyson equation which defines g namely + one derives an equation for G + 6 6

7 + + Summing the series, () G (341) G(31) G(4) G(3) G(41) d d d d J ig ig G () (5678) (35) (46) (781) Bethe Salpeter equation J =irreducible scattering amplitude Prerequisites: one-body G and J (irreducible scattering amplitude) 7 7

8 Simplest: the Ladder Approximation J... exchange 7 8

9 Scattering amplitude g and self energy S Each diagram for g becomes a diagram for S: just add interaction ad close loop Diagram for g Diagram for S = = 9 9

10 Diagram for g Diagram for S Since there is a one-to-one correspondence between diagrams, S(1,1') i dd5d6d7 V g (6857) ig(5) ig(6) ig(71)

11 Simplest: the Ladder Approximation J... exchange S... exchange 11

12 Two-body Green s Function G ( x, x, x, x ) T ( x ) ( x ) ( x ) ( x ) (Heisenberg) in translationally invariant systems Consider the propagation of opposite spin particles G (r,r,r,r,t) T (r,0) (r,0) (r,t) (r,t) (more generally, G also describes particle-hole excitations) 1

13 Start from one-body propagator i kt ig ( t) e ( t)(1 n ) ( t) n Transform to, with 0 dt e k k k 0 i( x) t i i( x) t i, dt e, 0 x i x i i ( i) 1 nk nk igk ( ) (1 nk ) nk i[ ] i i i i k k k k build non-interacting two-body GF of total momentum Q with usual diagram rules: q ig t ig t e t n n t n n i( Qq q ) t Qq ( ) q ( ) ( )(1 Qq)(1 q) ( ) Qq q Q-q electrons holes 13

14 Product in time representation implies convolution in frequency space i(1 n )(1 n ) ( i) n n Qq q Qq q FT ig ( t) ig ( t) Qq q ( ) i ( ) i Qq q Qq q Rules require sum over internal momenta and energies q Q to build g ( Q, t), total momentum q,h Q-q, h ig ( Q, ) g (1 n )(1 n ) n n Qq q Qq q (Q, ) [ ] ( ) i ( ) i q Qq q Qq q 14 14

15 Consider the propagation of opposite spin particles G (r,r,r,r,t) T (r,0) (r,0) (r,t) (r,t) q,h Q-q, h ig interaction: iu Let's work out interacting G (0,0,0,0, ) for the Hubbard Model H=t h ci c j U ni ni ( on site interaction) in ladder approximation. i, j i The interaction has lattice periodicity and does not mix different Q. first-order interaction diagram: q q 1 ig (Q, )( iu )i g (Q, ) Q-q 1 Q-q 15 15

16 first-order interaction diagram: q q 1 ig (Q, )( iu )i g (Q, ) Q-q 1 Q-q The only irreducible diagram=j scattering amplitude g g(q, ) 1... iu Ug UgUg (external lines excluded) iu 1 Ug (Q, ) 16 16

17 no exchange if spins are opposite: then, -particle GF G scattering amplitude g ig ig ( ig ) g ( ig ) In frequency representation, recalling g (Q, iu ) 1 Ug one finds: iu ig ig ( ig ) ( ig ) i 1Ug 1Ug (Kanamori 1963) g 17 17

18 Other equivalent method: Irreducible interaction is bare interaction J = =-iu The scattering amplitude can be obtained from the irreducible interaction by solving g = J J g g(q, ) iu ( iu ) ig g g(q, ) iu 1 Ug (Q, ) 18 18

19 The scattering amplitude allows to compute G Alternatively, from the irreducible interaction one can solve the Bethe-Salpeter Bethe-Salpeter G G J ig ig ig ig J ig ig ( ) igig g iu G 1 Ug (Kanamori) 19 19

20 Screened Interaction = Coulomb interaction. In any diagram for G or G where it appears, we can make replacements that produce new diagrams, for instance we may replace the Coulomb dotted line by: One can sum (formally) all such diagram in one strike by replacing the Coulomb interaction by an effective or dressed interaction, which depends on frequency; but this must be done in such a way that all the diagrams are reproduced exactly once. 0 0

21 Here we need the Irreducible polarization part p: cannot be split by dividing a bare interaction line... lowest order: p ( q, ) 0 if [1 f ] if [1 f ] p n m m n Recall: i ( ) ( ) 0 i0 i0 mn, n m n m A Dyson equation applies to the dressed interaction lines : 1 1

22 Mathematical expression for the Dyson equation W (1,) V (1,) V (1,3) p (3,4) W (4,) d(3,4), V (1,) 1 r 1 In Jellium, translational invariance simplifies matters. W ( x x ) V ( x x ) V ( x x ) p ( x x ) W ( x x ) dx dx Fourier transforming, W ( q, ) V ( q) V ( q) p ( q, ) W ( q, ) W ( q, )[1 V ( q) p ( q, )] V ( q) Vq W( q, ), with ( q, ) ( q, ) 1 V p ( q, ) dielectric function q p( q, ) irreducible polarization part.

23 High-density Electron Gas-Jellium For Jellium, due to translational invariance, we impose momentum conservation at each interaction, and re-label the second-order diagram accordingly with: k k m q k q m That is: q k q (, ) k k bubble : 3 d m d ip ( q, ) ig m, ig ( m q, ) p p p ( q, ) 3 dm 0 3 p fm(1 fmq ) fmq (1 fm) m mq ih m mq ih 3 3

24 From p ( q, ) 3 dm 0 3 p fm(1 fmq ) fmq (1 fm) m mq ih m mq ih Re p ( q, ) 0 Im p ( q, ) p f ( 1 f ) ( ) 0 m mq m m mq m f f m mq m mq 3 dq d 0 ( )= ig (k-q, )( ) ( ) 3 q p 0(, ) is iv i q ( p) p V q 4p e S( ) diverges as q 0; the long-distance divergence q is not physical as screening restores the situation. 4 4

25 High-density Electron Gas Random Phase Approximation p( q, ) irreducible polarization part. p ( q, ) 0 Random Phase Approximation replaces ( q, ) 1 V p ( q, ) RPA This turns out to be exact for r 0, 0 since any insertion 0 for r 0 faster. Let us see why. q s s p by p 0 5 5

26 g Suppose we modify To get g The extra interaction line introduces a factor. dg k E F 3 p m kf q dg p 1 G E q V GG q 6 6

27 g g therefore the extra interaction line introduces a factor dg p q F VqGG EkF k F E E E 1 where E kf and this goes like kf rs 0 at high density. We have a sound criterion for choosing the diagrams! K For metals r s is between and 6 7 7

28 The explicit calculation ( Lindhard 1954) 3 dm fm(1 fmq ) fmq (1 f ) m p0( q, ) 3 p m mq ih m mq ih fmq (1 fm) In set: m q n, f f, f f f ih m Call m the occupied states in both integrals mq mq n m nq nq 3 dm fm(1 fmq ) fn (1 f ) nq p0( q, ) 3 p m mq ih nq n ih and with n m in the second fraction, 3 dm 1 1 p0( q, ) f ( 1 ). 3 m fmq p m mq ih mq m ih 1 1 Using P i p (x) the imaginary part is easy because of the. x i0 x 8 8

29 3 dm p 0( q, ) p f (1 ) ( ) ( ). 3 m f mq m mq mq m p q q measuring q in units of k F,set, a, a. E q q F q q [1 a ], q, q q q q Im( p ) q, q, q 0 0 otherwise. mkf [1 a ], q, q q 0 4p q 9

30 q q measuring q in units of k F,set, a, a. E q q F The real part is the Hilbert transform. mkf 1 1a 1a Re( p 0) [ 1 {[1 a ]log [ 1 a ]log } ] p q a 1a 1 Static Screening me kf me q kf q For 0, ( q,0) 1 ( k )ln 3 F p q p q 4 k q F For q 0, K ( q,0) ~ 1 q Perfect screening at long distances TF 4kF, where KTF is the Thomas-Fermi wave vector. p a B 30 30

31 KTF ( q,0) ~ 1 is the Thomas-Fermi approximation leading to a screened q ktf r 4p e e potential V s(q)= V s(r)=. q k r ( TF ) The actual behavior of the screening charge at long distances in given by cos( kr F ) d ( q,0) the Friedel oscillations going like, due to at q k. 3 F r dq Dynamical Screening Behavior for q 0 P 3 kf P ( q, ) ~ 1 ( ) q 4 ( ih ) 5 m ( ih ) Plasmons : arise when ( q, ) 0 (self-sustained oscillations). p Drude qualitatively right : ( ) 1 i ( ) 31 31

32 Plasmon dispersion ( q) 1 3 ( k ) q F P 10 m P (particle-like dispersion) In semiconductors and insulators plasmons are seen in a similar way, since the gap is often small compared to p. With increasing q, eventually the plasmon branch enters the electronhole continuum and becomes unstable against converting into a pair. Thus, the plasmons dominate the high-frequency, long wavelength screening, while electron-hole pairs in metals are slower and act as shorter distance (higher q) screening modes. 3 3

33 Actually they are quite sharp only at q=0 since they can decay into multiple pairs Plasmon satellites are typically observed in electron spectroscopies q of metals in the 10 to 5 ev range Ultraviolet transparency metals become transparent at > p ( the electrons cannot follow the field)

34 Low-density Fermi gas Galitzkii (1958) Diagrams that dominate self-energy if n<<1 S and g are linked: S g Galitzkii (1958) Low-density approximation scattering amplitude g as in closed-band system S =0 in closed-band system 34

35 Reason: if band is almost empty, n<<1 ig 0 i ( k, ) diagram involves k summations, amounting to a i sign( ) k k F 0 localized propagator g ( k, ) g ( ) g ( ) k e h electron electron hole d ( p ) d ( p ) g g e h ( ) 1 electronic quantum state ( ) n 1 hole quantum state There is a lot of phase space to create an electron, little to create a hole. hole Extra interaction -> new electron line in bubble, 1-n states available electron Extra interaction -> new hole line, n<<1 states available because there are few electrons to remove 35

36 The Low Density Electron Gas Low-density electron gas: interaction >> kinetic energy Collective modes and Many-body interactions unimportant compared to two-body ones Almost empty and almost filled bands can be studied few electrons in empty band canonical electron- hole transformation few holes in full band E F E F Ti (3 d) (4 s) V (3 d) (4 s) 3 Ni (3 d) (4 s) 8 Pd d d s metal (4 ) (4 ) (5 ) ( ) 36 36

37 Auger CVV spectra of open-band systems (M. Cini, Surface Science 1979) the band can be polarized by the primary hole, so a two-hole Green's function cannot tell the whole story. the existence of electron-hole excitations offers an intra-band decay channel for the two-hole resonances. 37

38 Valence on-site repulsion U Fermi level Almost full valence band n h <<1 Core-valence interaction W core H c c Un n W ( n n ) ij ij i j coreb b b b (3) Auger Spectrum G ( ) G ( t) Tbt ( ) c ( t) c ( t) c (0) c ( 0) b ( 0) (3) g t Tb b t core( ) (0) ( ) one can show that n 1 (3) G ( t) Tbt () c ( t) c ( t) c (0) c (0) b g core ( t) Tc ( t) c ( t) c (0) c ( 0) h W (0) 38

39 Core propagator by linked cluster theorem g t t (0) i coret ( ) ( ) e can be factored out core summing over time orderings of disconnected parts and putting back core g ( t) ( t) e e core i t Ct () g (0) core =core-valence W =valence U = valence GF C(t)= shake-up effects in core photoemission 39

40 Valence propagator: Lehmann representation for U=0 T=top B=bottom T A() T B E F B E F B B() E F 0 0 T B A B ig( ) d d i0 i0 40

41 Photoemission spectrum of Ni rectangular band model - n h =0.1 Im g U 0 Fermi 41

42 S (0) d ' g ( ') S( ) iu( n, g g g (0) p 1 Ug ( ') (0) (0) (0) computed photoemission spectrum rectangular band model- n h =0.1 Im g U 0 broadened twohole resonance U 1.5width of occupied band two-hole resonance in one-body spectrum due to interaction with background holes Fermi level 4

43 Experimental Ni spectrum broadened twohole resonance 6 0 Binding energy (ev) David R. Penn,J.Applied Phys. (1979) (M. Cini, Surface Science 1979) 43

44 Auger CVV spectrum-rectangular band model non-interacting-filled band

45 Auger CVV spectrum-rectangular band model U 1.5width of band, filled-band theory two-hole resonance non-interacting-filled band

46 Auger CVV spectrum-rectangular band model U 1.5width of occupied band, low-density theory broadened and shifted two-hole resonance new structure as broad as band Red: the two-hole resonance in closed band case Blue: 90% occupied band in the Low Density Approximation non-interacting-filled band -4-0 This theory gave good agreement with experiment for Ni, Pd Next step: Self-consistent LDA 46 0

47 LDA =Skeleton diagrams The following scheme is a natural consequence of the above discussion: Compute one-electron propagator by the self-consistent low-density approximation. Use dressed one-electron propagator to work out the ladder approximation for G Compute the Auger spectrum. G = = + S S =

48 We found however that the approximation for G is poor when compared to the exact G (which we could calculate for small cluster models) 48

49 Self-consistent LDA worsenes the approximation for Auger Spectrum. Bare-Ladder Approximation is superior! G = = + S S = Reason vertex corrections counteract dressing of lines! 49

50 M. Cini and C. Verdozzi

51 G.Seibold and J. Lorenzana PRL 86, 605 (001) Have proposed a new approach (time-dependent Gutzwiller approximation) which is superior to the BLA, but is much more complicated. See also 51

52 The full Screened Interaction polarization part = any graph that can be inserted into an interaction line Irreducible polarization part : cannot be split by cutting a single V line. 5

53 53 Irreducible polarization part : cannot be split by cutting a single V line. p=sum of all Irreducible polarization parts= filled ellypse = By a Dyson equation, one can determine the screened interaction W in terms of the bare (Coulomb) interaction v. W(1,) v(1,) v(1,3) p 3,4 W(4,) d(3,4) or also W(1,) v(1,) v(4,) W(1,3) p 3,4 d(3,4)

54 p( q, ) irreducible polarization part.... W (1,) V (1,) V (1,3) p (3,4) W (4,) d(3,4) Recall: for Jellium Fourier transforming W ( q, ) V V p ( q, ) W ( q, ) V V p ( q, ) V V p ( q, ) p ( q, ) V... q q q q q q q W q V V V (, ) q[1 p q ( p q)...] Vq Vq W ( q, ) screened interaction with ( q, ) 1 Vqp ( q, ) 1 V p ( q, ) ( q, ) q Exact!! But we need p( q, ) 54 54

55 THE VERTEX FUNCTION Actually, we can write an equation for the irreducible polarization. = Terms like are included by dressing the propagators: Terms like are not, so we could consider bubbles with dressed propagators like

56 Formal summation a la Dyson: we need a vertex to be used in = Vertex: the sum of all linked diagrams that fit between an incoming propagator, an outgoing propagator and an interaction line To avoid double counting we need to distinguish reducible and irreducible vertex parts. Is a reducible vertex part (can by split by cutting an interaction) Summing all the irreducible vertex parts we get the irreducible vertex

57 Irreducible vertex (1,;3) lowest order : (1,; 3) (1,) (1,3) ( so cal led 'Migdal theorem') The full p can be obtained in terms of the irreducible vertex. p(1,) i G(,3) G(4, ) (3,4,1) d(34) 57

58 p(1,) i G(,3) G(4, ) (3,4,1) d(34) + because the line arrives from 3, picks the photon and then goes to 4. Knowing the vertex one finds the polarization and the screened interaction. A similar analysis is possible for the proper self-energy = 3 Vertex: the sum of all linked diagrams that fit between an incoming propagator, an outgoing propagator and an interaction line 58

59 Jellium: Formal summations of the series for S One can start by the prototype diagram for self-energy d p 0 i V ig k, kabk k k m q k q m k 3 dq d 0 ( )= ig (k-q, )( ) ( ) 3 q p 0(, ) is iv i q ( p) p V q 4p e q 59

60 Iterating the bubble one avoids the divergence: + + d i ig k, [ V VV VVV...] k p d 0 V d 0 i ig k, i p ig k, V 1 V p 0 k k RPA Vq VRPA( q, ) screened interaction in RPA 1 V p ( q, ) q 0 60

61 Iteration of bubble leads to S ( k, ) dqdhg ( k q, h) V ( q, ), RPA 0 0 Vq VRPA( q, ) screened interaction in RPA 1 V p ( q, ) q RPA Since p ( q, ) is an approximation to p ( q, ) it is natural to generalize 0 the RPA as follows: S ( k, ) dqdhg ( k q, h) V ( q, ), V RPA RPA 0 RPA Vq ( q, ) fully screened interaction 1 V p( q, ) q k m q k q m k 61

62 S p = Exact? No! a This is the GW approximation, which is often used in computations (often within the RPA). However even the full GW approximation is not exact. Consider the next diagram: k n m Note that b->k->m->n->a. This can be drawn in other ways b 6 6

63 a can also been drawn this way can also been drawn this way n m b k m n a b k m n a k can also been drawn this way, and a vertex part is clearly involved b m n Exchange second-order selfenergy b k a Recall the definition of the Vertex: the sum of all linked diagrams that fit between an incoming propagator, an outgoing propagator and an interaction line: interaction vertex propagator in m n propagator out k a 63 63

64 S S(1,) i G(1,4) (4,; 3) W(1,3) d(34) in terms of the dressed interaction W, the dressed propagator G and of the vertex Γ. The fact that the same function Γ indeed appears in the expressions for π and Σ will be more apparent shortly. lowest order : (1, ; 3) (1, ) (1, 3) (so called 'Migdal theorem') At Weak coupling, the vertex is just a dot GW approximation works fine for weak correlation However a Dyson-like argument does not allow to find the vertex. One needs a new ingredient. 64

65 In homogeneous systems the irreducible vertex function is q, dressed interaction 3 interaction vertex (1,,3, kq,, ) propagator in q k 1 a propagator out q k 65

66 Summary of Equations: can we find G? G( x, x') G ( x, x') dx dx G ( x, x ) S( x, x ) G( x, x') Needs S S (1,) i G(1,4) (4,; 3) W(1,3) d(34) W(1,) v(1,) v(1,3) 3,4 W(4,) d(3,4) p Needs W Needs p p(1,) i G(,3) G(4, ) (3,4,1) d(34) Closed! 4 equations for 5 unknowns 1 equation for and physics is over (a purely computational problem) 66 66

67 Functional derivatives t The action S dtl( q, q, t) is a familiar functional S S[ q( t)] Principle of least action: t 1 vary q with with h( t ) 0 h( t ), t S[ q h, q h] h h dtl( q, q, t) 0. t1 t t 1 L dt h q L h0 q t t t L L t d L dt h h h dt q q t dt q S t L d L S L d L 0 dth 0 0 t1 q dt q q q dt q S One defines functional derivative q S 0 Euler-Lagrange equation. q 67 t 1

68 n Given a functional F[ ] d xf[ ( x),, x] of functions in a n-dimensional space with gradient F f f ( x) Extension to several times If F does not depend on the derivatives n F[ ] d xf[ ( x)] F ( x) f Then, the functional derivative of F [ ] is just the derivative of f

69 Functional differentiation of diagrams diagram F[ g], g( a, b) propagator from a to b F g dadbdx g a b For instance, n [ ] (, )... g 4,5 F[ ] just erases g 4,

70 Heuristic derivation of last of Hedin equations: S(1,) (1,,3) (1, ) (1,3) (4, 6) (7,5) (6, 7,3) (4567) G(4,5) G G d (1,,3) 1 3 Contribution to S(1, ) 4 G(4,5) Contribution to S(1, ) G(4,5)

71 (1,,3) 1 3 G(4,6) Contributo a (,1,3) 5 G(7,5) 7 1 last of Hedin equations: S(1,) (1,,3) (1, ) (1,3) (4, 6) (7,5) (6, 7,3) (4567) G(4,5) G G d 71 71

72 Hedin s equations: G( x, x') G ( x, x') dx dx G ( x, x ) S( x, x ) G( x, x') S (1,) i G(1,4) (4,; 3) W(1,3) d(34) W(1,) v(1,) v(1,3) 3,4 W(4,) d(3,4) p p(1,) i G(,3) G(4, ) (3,4,1) d(34) S(1,) (1,,3) (1, ) (1,3) (4,6) (7,5) (6,7,3) (4567) G(4,5) G G d Hedin obtained the last of these exact equations that formally determine selfenergy, polarization, vertex function and Green s function. Although they were not solved exactly, they lie at the heart of powerful approximate methods for firstprinciple calculations. 7 7

73 Recent re-derivation 73