Non-equilibrium Green functions I

Transcription

1 Non-equilibrium Green funcions I Joachim Keller Lieraure: H. Haug, A.-P. Jauho, Quanum Kineics in Transpor and Opics of Semiconducors J. Rammer, H. Smih, Quanum field-heoreical mehod in ranspor heory of meals, Rev. Mod. Phys. 58,33 (986 L.P. Kadanoff, G. Baym Quanum Saisical Mechanics (96) L.P. Keldysh, JETP 0, 08 (965) Basic definiions and equaions. Equilibrium Green s funcions We define G r (x, ; x, ) = iθ( ) {ψ(x, ), ψ (x, )} + G a (x, ; x, ) = +iθ( ) {ψ(x, ), ψ (x, )} + which are called rearded and advanced Green funcions (hey fulfill he usual equaions of moion of a Green funcion in he mahemaical sense). Here ψ (x, ) is a creaion operaor for a fermion a posiion x. The ime dependence of he operaors is deermined by he ime-independen Hamilonian H of he sysem: O() = O H () = e ih Oe ih The bracke denoes a hermal average wih he saisical operaor ρ = e βh /Z. In order o describe a grand-canonical ensemble we have o replace H by H µn. This can be accomplished by couning all single-paricle energies from he chemical poenial µ. We will do ha also for he Hamilonian describing he ime dependence.

2 In he following we also need he correlaion funcions G > (x, ; x, ) = i ψ(x, )ψ (x, ) G < (x, ; x, ) = +i ψ (x, )ψ(x, ) and he ime-ordered Green funcion G c (x, ;x ) = i T ψ(x, )ψ (x ) = iθ( ) ψ(x, )ψ (x, ) + iθ( ) ψ (x, )ψ(x, ) = Θ( )(G > (x, ; x, ) + Θ( )G < (x, ; x, ) Noe ha he rearded Green funcion can also be expressed by he wo correlaion funcions G r (x, ; x, ) = Θ( )(G > (x, ; x, ) G < (x, ; x, )) G a (x, ; x, ) = Θ( )(G > (x, ; x, ) G < (x, ; x, )) and G r G a = G > G <. In hermal equilibrium hese funcions depend only on he ime-difference =. Furhermore all hree are linked ogeher by he dissipaion flucuaion heorem. Inroducing a Fourier ransform (here we suppress he dependence on variables x, x for he momen) G(ω) = + e iω G()d one obains he imporan relaion beween he wo correlaion funcions G > (ω) = e βω G < (ω) Defining a specral funcion A(ω) = i[g r (ω) G a (ω)] = i[g > (ω) G < (ω)] we find G < (ω) = if(ω)a(ω), G > = i( f(ω))a(ω) where f(ω) = exp(βω) +

3 is he Fermi funcion. Noe ha he Green funcions and he specral funcion depend also on he oher variables x, x. In he case of free conducion elecrons wih ψ(x) = k c ke ikx / V ol we obain afer a Fourier ransformaion in space G r,a (kω) = ω ± iδ ɛ k + µ A(k, ω) = πδ(ω ɛ k + µ) Here we see ha A(kω) conains informaion abou he single-paricle exciaion specrum, while he disribuion funcion f(ω) which appears in he correlaion funcions describes he hermal occupaion of such saes. In a non-equilibrim heory he disribuion funcion will become an independen quaniy. Therefore we need he correlaion funcions in addiion o he rearded and advanced Green funcions.. Perurbaion heory for equilibrium Green funcions In order o calculae he Green funcion wih help of a perurbaion heory we spli he Hamilonian ino H = H 0 + H i where H 0 describes a non-ineracing elecron sysem. Going over o he ineracion represenaion he uniary operaor for he ime evoluion beween imes 0 and becomes e ih( 0) = e ih 0 V (, 0 )e ih 0 0, V (, 0 ) = T exp ( i 0 H i 0( )d ) where he ime dependence of he ineracion H i (and oher operaors wih subscrip 0) is deermined by he Hamilonian H 0 of he non-ineracing sysem: O 0 () = exp(ih 0 )O exp( ih 0 ). T denoes a ime ordering of operaors on he ime-axis from 0 o. Here he ime 0 is some arbirary reference ime. We can se 0 = 0. However, for laer use in he case of a ime dependen ineracion i has o be chosen earlier hen he swich-on ime of he ime-dependen ineracion. Laer we le 0. For fixed ime, we hen obain ψ( )ψ ( ) = e ih 0 0 S( 0, )ψ 0 ( )S(, )ψ 0( )S(, 0 )e ih 0 0 which can also be wrien as ψ( )ψ ( ) = e ih 0 0 T S ψ 0 ( )ψ 0( )e ih 0 0, S = exp ( i H0(τ)dτ ) i 3

4 0 > 0 < Abbildung : The conour used for he perurbaion heory in he imeevoluion i β Abbildung : The conour i used for he perurbaion heory including expansion of he saisical operaor where is he conour shown in Fig. and he ime ordering operaor T orders he ineracion operaors conained in S in he direcion of he conour a he righ places before, beween, and behind he fermion operaors a he fixed imes,. by Then Finally we have o expand also he saisical operaor, which can be expressed e βh = e i( 0 iβ)h 0 S( 0 iβ, 0 )e i 0H 0 T iψ( )ψ ( ) = T S i iψ 0 ( )ψ 0( ) 0 T is i 0 where now i is he conour shown in Fig. and 0 means a hermal average performed wih he saisical operaor ρ 0 = e βh 0 /Z 0. Due o he ime ordering conained in S requires ha a perurbaion heory 4

5 0 0 - i β Abbildung 3: onour i used for he perurbaion expansion of he ime-ordered Green funcion for he Green funcion can be formulaed easily only for a Green funcion wih ime-ordered operaors. Therefore we define a ime ordered Green funcion on he conour i : G(τ, τ ) = { i ψ(τ )ψ (τ ) for τ > c τ +i ψ (τ )ψ(τ ) for τ < c τ where we disinguish beween imes τ i on differen pars of he conour (see Fig. 3). Then for G he following perurbaion expansion holds: G(τ, τ ) = i T is iψ 0(τ )ψ 0(τ ) 0 T is i 0 I is relaed o he real-ime Green funcions inroduced above by G c (, ) for τ, τ G > (, ) for τ, τ G(τ, τ ) = G < (, ) for τ, τ G c (, ) for τ, τ The las beeing an aniime-ordered Green funcion. The rearded and advanced Green funcions can be obained from G > and G < in he usual way. The purpose of his discussion was o moivae he inroducion of a imeordered Green funcion on he special conour i. This will he basis for he following discussion of non-equilibrium Green funcions. In hermal equilibrium i is possible o simplify his conour considerably: As in hermal equilibrium all Green funcions depend only on he ime difference we can replace i by a 5

6 conour h which runs from 0 o 0 iβ, where we can pu 0 0 wihou loss of generaliy. One hen defines a ime-ordered Green funcion on he imaginary axis. The physical rearded and advanced Green funcions are hen obained by an analyical coninuaion in frequency space. This is he Masubara echnique..3 Green funcions wih ime dependen perurbaion A non-equilibrium siuaion is obained if we add o he Hamilonian H of he sysem a ime-dependen perurbaion H describing for insance he ineracion wih an exernal ime-dependen field: H = H + H wih H = H 0 +H i. We assume ha he ime dependen ineracion H vanishes for imes < 0. The ime dependen ineracion can be reaed by a perurbaion expansion in a similar way as he inernal ineracion. The final resul for he Green funcion can be wrien as a double expansion: G(τ, τ ) = i T is is ψ 0 (τ )ψ 0(τ ) 0 T is is 0 where ( ) S i = exp i dτh0(τ) i i ( ) S = exp i dτh τ,0(τ) and boh ineracion operaors are Heisenberg operaors consruced wih H 0. Noe ha H τ,0 conains an implicie ime dependence hrough he exernal field. Here he conour sars and ends a 0, while he conour i sars a 0 and ends a 0 iβ. The principal significance of his resul is ha for he ime-ordered Green funcion on he conour i he usual perurbaion expansion in form of Feynman graphs as in he case of equilibrium Green funcion can be applied. In paricular if he Hamilonian H 0 is bilinear in fermionic operaors Wick heorem can be applied. I saes ha in each order of he perurbaion heory he resul for he Green funcion can be expressed by producs of single paricle Green funcions of he non-ineracing sysem. The same holds for he saisical operaor. Therefore he denominaor cancels agains non-conneced graphs in he numeraor, leaving 6

7 us wih an expansion conaing only conneced graphs. For hese, finally, a Dyson equaion can be formulaed, which will be he basis of he following discussion..4 Dyson equaion In order o be specific le us assume a ime-dependen ineracion of he form H = dxψ (x)u(x, )ψ(x) hen he ime-ordered Green funcion on he conour fulfills he following Dyson equaion G(, ) = G 0 (, ) + d3g 0 (, )U(3)G(3, ) d3 d4g 0 (, 3)Σ(3, 4)G(4, ) where (i) is a shor-hand noaion for (x i, τ i ) and di = dx dτ i i. We did no specify he inernal ineracion furher. I is conained in he (irreducible) self-energy Σ. The self-energy can be calculaed and expressed by single-paricle Green funcions in any specific case. In principle he ime-inegraions have o be performed on he conour i. A simplificaion occurs if we pu 0. Then he conribuion from he pah 0 o 0 iβ can be negleced, if he inernal ineracions wash ou he memory of iniial condiions. Then for he inernal inegraions we may use he conour (Fig. 4) exending from o + (par ) and back (par ). This will be called he Keldysh conour in he following. There sill remains he problem of expressing he inegraions of producs of differen funcions by inegraions over he usual ime axis < < +. This will be he ask of he following secion..5 Langreh rules, Keldysh formalism In he Dyson equaions appear inegraions over he conour of producs of wo and hree funcions. Le us sar wih he invesigaion of a folding of wo funcions of he form (here we suppress he dependence on he spaial variables) (τ, τ ) = dτ 3 A(τ, τ 3 )B(τ 3, τ ) Our aim is o express his inegral on by inegrals over he simple ime-axis < < +. The resul depends on he choice of variables on he conour. If 7

8 τ τ Abbildung 4: Relaion beween real imes and imes on he differen branches of he Keldish conour we are ineresed, for insance, in < (, ) we place as in Fig. 4 τ on and τ on and evaluae he inegrals on he conour. Then we obain: < (, ) = + d [ A r (, )B < (, ) + A < (, )B a (, ) ] For a derivaion of his resul we spli he inegrals: < (, ) = dτa(τ, τ)b(τ, τ ) + dτa(τ, τ)b(τ, τ ) On he funcion B is a funcion B < while A = A c. On he funcion A is a funcion A > while B = B c. Then for he firs inegral we obain + dτa(τ, τ)b(τ, τ ) = da c (, )B < (, ) = da > (, )B < (. ) + For he second inegral we find dτa(τ, τ)b(τ, τ ) = da > (, )B c (, ) + = da > (, )B > (, ) + da < (, )B > (, ) + da > (, )B < (, ) Using he relaions A r (, ) = Θ( )(G > (, ) G < (, )) and A a (, ) = Θ( )(G > (, ) G < (, )) we can show he equivalence of he wo resuls. In a similar way we can show > (, ) = + d [ A r (, )B > (, ) + A > (, )B a (, ) ] 8

9 and r (, ) = + da r (, )B r (, ), a (, ) = + In he Dyson equaion we also encouner a folding of he form (τ, τ ) = dτa(τ, τ)u(τ)b(τ, τ ) his is ransformed ino < (, ) = + da a (, )B a (, ) d [ A r (, )U()B < (, ) + A < (, )U()B r (, ) ] For a produc of hree funcions we obain similar rules, for insance, le D(τ, τ ) = dτ 3 dτ 4 A(τ, τ 3 )B(τ 3, τ 4 )(τ 4, τ ) hen D < = d [ A r B r < + A r B < a + A < B a a] The rules for foldings of producs can also be cas ino marix muliplicaion form (Keldish marices) by wriing: ( ) A r A Â = <,> 0 A a hen in he above example ˆD = Â ˆBĈ A able of rules for all he oher combinaions necessary o calculae he selfenergy can be found in Haug/Jauho. As an example le us sudy he linear response of he densiy of free conducion elecrons on a scalar poenial. Here he self-energy is absen and we have he equaion: δg(x, τ ; x, τ ) = dτ 3 dx 3 G 0 (x, τ ; x 3, τ 3 )U(x 3, τ 3 )G 0 (x 3, τ 3 ; x, τ ) The densiy is direcly described by he funcion n(x, ) = ig < (x, ; x, ). For his quaniy we obain + δn(x, ) = dx 3 d 3 [ G r 0 (x x 3, 3 )U(x 3, 3 )G < 0 (x 3 x, 3 ) G < 0 (x x 3, 3 )U(x 3, 3 )G a 0(x 3 x, 3 ) ] 9

10 where we already have used ha he equilibrium Green funcions depend only on he difference of argumens. For a ime-dependence of he scalar poenial of he form U(x, ) = δu exp(iqx iω) and going over o a Fourier represenaion we find G 0 (x, ) = V ol δn(x, ) = iδu exp(iqx iω) V ol Now we use he equilibrium relaion: k dω π G 0(k, ω) exp(ikx iω) dω [ G r π 0 (k + q, ω + Ω)G < 0 (k, ω) G < 0 (k + q, ω + Ω)G a 0(k, ω) ] G < (k, ω) = if(ω)a(k, ω) = πif(ω)δ(ω ɛ k + µ), G r,a (kω) = ω ɛ k + µ ± iδ and obain δn(x, ) = V ol k f(ɛ k µ) f(ɛ k+q µ) U(x, ) Ω + ɛ k ɛ k+q + iδ Here we recover he well-known Lindhard funcion. 0