Keldysh Formalism: Non-equilibrium Green s Function

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1 Keldysh Formalism: Non-equilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Non-equilibrium Green s Funcion (NEGF), also named as Keldysh Formalism, is presened here. Perurbaion heory is also builded up sep by sep on he basis of non-ineracing NEGF, and hen leads o he Dyson s Equaion. Finally, applicaion of NEGF on igh-binding model wih random impuriy is explicily calculaed as an example. I. DEFINITION, FREE FIELD AND PERTURBATION When only he properies of ground sae is concerned, Zero-emperaure (single- and many-paricle) Green s funcion ogeher wih is perurbaion heory principally gives all he informaion. Also if he equilibrium hermal disribuion is concerned, we can urn o Masubara Green s funcion and is perurbaion heory[2]. However, we may need o consider some more general saes, such as a saionary non-equilibrium sae for example a sae wih nonzero curren, or even an arbirary sae ρ, how o ge correlaion funcion for such sysem and hen how o do he corresponding perurbaion expansion? In experimens of conduciviy, here are such non-equilibrium sysem, which has non-zero curren and includes ineracion beween paricles. So in his noe I will shorly review he idea and echnics of Non-equilibrium Green s funcion, also known as Keldysh Formalism. All he formulas defined in his noe is only for fermions, alhough i is no hard o make hem o also cover bosons. Generally we need o calculae he Green s funcion G (x, ; x, ) = i } ) (T T r ψ H (x, ) ψ H (x, ) ρ H, (1) where ψ H (x, ) is he annihilaion operaor in Heisenberg picure, and ρ is he densiy marix also in Heisenberg picure. We will se = 1 for a while. We also assume ρ is diagonal under paricle number basis, [N, ρ] = 0. More general densiy marix could be possible, bu no used anywhere ye. For free field, his can be calculaed quie srai forward, ha ψh (x, ) = k e iω(k)+ikx c k ψ H (x, ) = k eiω(k) ikx c, (2) k hen G 0 (x, ; x, ) = iθ ( ) k 1 n k e iω(k)( )+ik(x x ) iθ ( ) k n k e iω(k)( )+ik(x x ) (3) Following he roadmap of zero-emperaure Green s funcion, nex sep would be o urn ψ H (x, ) ino ineracion picure ψ I (x, ) and o somehow replace he real sae ρ H wih sae of non-ineracing sysem ρ 0 H. Le s firs organize he above mapping for zero-emperaure Green s funcion and hen see if he same procedure can be applied ono NEGF. Zero-emperaure Green s funcion is defined as a special case of eq(1), ρ H = Ω Ω, } G (x, ; x, ) = i Ω T ψ H (x, ) ψ H (x, ) Ω, (4) Firs, express T A H ( 1 ) B H ( 2 )} in ineracion picure. and H = H 0 + e ɛ 0 H 1, (5) A H () = e ih( 0) e ih0( 0) A I () e ih0( 0) e ih( 0). (6)

2 2 FIG. 1: The ime-loop inegraion pah. Define uniary operaor U (, ) = e ih0( 0) e ih( 0) e ih0( 0), (7) which is he formal soluion of where H I () = e ih0( 0) H 1 e ih0( 0). Or he explici form is Then for 1 > 2 he ime-ordered operaors, i U (, ) = H I () U (, ), (8) U (, ) = T exp i dh I (). (9) T A H ( 1 ) B H ( 2 )} = A H ( 1 ) B H ( 2 ) = U ( 1, 0 ) A I ( 1 ) U ( 1, 0 ) U ( 2, 0 ) B I ( 2 ) U ( 2, 0 ) = U ( 0, 1 ) A I ( 1 ) U ( 1, 2 ) B I ( 2 ) U ( 2, 0 ). (10) = U ( 0, ) U (, 1 ) A I ( 1 ) U ( 1, 2 ) B I ( 2 ) U ( 2, ) U (, 0 ) Nex for rue ground sae Ω, we can relae i wih 0, he ground sae of non-ineracing field, U ( 0, ) 0 = lim T eih( T 0) e ih0( T 0) 0 = lim T e ih(t +0) 0. (11) According o Gell-Mann and Low Theorem, for adiabaic coupling, back in Schrödinger picure, ground sae is sable, φ 0 ( f ) U ( f, i ) φ 0 ( i ). In our case, Ω ( 0 ) U ( 0, T ) 0 ( T ), so up o some consan, U ( 0, ) 0 Ω U (, 0 ) Ω 0, (12) and similarly, U (, 0 ) Ω 0. Therefore, when 1 > 2 } Ω T ψ H (x, ) ψ H (x, ) Ω 0 U (, 1 ) A I () U ( 1, 2 ) B I () U ( 2, ) 0. (13) Finally, he consan can be eliminaed as 0 T G (x, ; x, ) = i } ψ I (x, ) ψ I (x, ) U (, ) 0 U (, ) 0 0, (14) where now all he quaniies are from free field, and herefore can be calculaed order by order by perurbaion. However, when we apply he same procedure o general ρ H, we require he Gell-Mann and Low heorem holds for excied saes for boh = ±. This condiion is oo srong. The idea o solve above problem is o remove some special poins from he hree special poins = (, 0, ), for example by seing 0. In his way we require ha only φ 0 () = 0 insead of boh φ 0 (± ) = 0. The pah is o sar along = o = 0 and hen come back along = 0 o =. And finally le 0, a picured in fig(1). As inhe case of usual Green s funcion, le firsly reorganize T A ( 1 ) B ( 2 )}, and hen he relaion beween ρ H and ρ 0 H. For 1 > 2, inser U (, ) ino eq(10), T A H ( 1 ) B H ( 2 )} = U ( 0, ) U (, ) U (, 1 ) A I ( 1 ) U ( 1, 2 ) B I ( 2 ) U ( 2, )} U (, 0 ). (15) Then boh sides relaes only wih U (, 0 ), which sill link he rue eigensaes n wih free-field eigensaes n 0, ha U (, 0 ) n n 0. (16)

3 3 (a) G + (, ) and G (, ) (b)g c (, ) (c) G c (, ) s (d) Impuriy Scaering FIG. 2: One possible Feynman rules for NEGF. Therefore, we arrive G (x, ; x, ) = i T r(u(, )Tψ I (x,)ψ I(x, )U(,)}ρ 0 H) T r(u(,)ρ 0 H) = i T r(tcψ I (x,)ψ I(x, )U(,)}ρ 0 H) T r(u(,)ρ 0 H), (17) where T c is he loop order as shown in fig(1) o order he ime along he loop from o. For example, a he above branch i s he same wih ime order T c = T, while a he lower branch i s ani-ime order T c = T. And U (, ) = T c exp i dhi (). (18) One hing need o be noiced ha in he final expression of Green s funcion, all he ime spos should be reaed as poins a above branch (also call posiive branch). However, laer on, we will see ha when we ry o calculae such Green s funcion, we will need o do inegral also over he lower branch (also call negaive branch). This means generally no all he ime spos are a he posiive branch, and he second line of eq(17) could no be separaed as U (, ) and U (, ) ouside and inside usual ime order T. Therefore, generally we need o define Green s funcion according o he branches of and. There are four differen configuraions of hem, as picured in fig(2). They are less and greaer Green s funcion[1], G + (x, ; x, ) = it r (ρ H T c ψ H (x, + ) ψ ( ) ( ) H x, = it r ρ H ψ H (x, ) ψ H (x, ) G (x, ; x, ) = it r (ρ H T c ψ H (x, ) ψ ( ) ( ), (19) H x, + = it r ρ H ψ H (x, ) ψ H (x, ) where ± means he ime is on posiive/negaive branch; ime ordered and ani-ime ordered Green s funcion, G c (x, ; x, ) = it r (ρ H T c ψ H (x, + ) ψ ( ) ( H x, + = it r ρ H T ψ H (x, ) ψ H (x, ) G c (x, ; x, ) = it r (ρ H T c ψ H (x, ) ψ ( ) ( ; (20) H x, = it r ρ H T ψ H (x, ) ψ H (x, )

4 and rearded and advanced Green s funcion, G r (x, ; x, ) = iθ ( ) T r (ρ H ψ H (x, ), ψ H (x, ). (21) G a (x, ; x, ) = iθ ( ) T r (ρ H ψ H (x, ), ψ H (x, ) Only hree of hem are independen, G r = G c G + = G G c, G a = G c G = G + G c. (22) This can be seen easily by separaing G ± as four pars, θ ( ) G ± and θ ( + ) G ±. Then G c = θ ( + ) G + + θ ( ) G G c = θ ( ) G + + θ ( + ) G (23) So G c + G c = G + + G. Noice for greaer/less Green s funcion, differen definiions and noaions are used in lieraure, for example, G > G and G < G + in [2]. Before we go furher o perurbaion calculaion of ineracing field, le s firs lis all he free-field Non-equilibrium Green s funcions. 4 G 0,+ (x, ; x, ) = i k n k e iω(k)( )+ik(x x ) G 0, (x, ; x, ) = i k 1 n k e iω(k)( )+ik(x x ) G 0,c (x, ; x, ) = i k θ ( ) n k e iω(k)( )+ik(x x ) G 0, c (x, ; x, ) = i k θ ( ) n k e iω(k)( )+ik(x x ) G 0,r (x, ; x, ) = iθ ( ) k e iω(k)( )+ik(x x ) G 0,a (x, ; x, ) = iθ ( ) k e iω(k)( )+ik(x x ) (24) Abou Eq(16), he expression similar wih of Gell-Mann and Low Theorem, i should be undersood in a kind inverse logic order. For an ineracing field H = H 0 + H 1, he non-equilibrium densiy marix ρ H is no clearly defined, while ρ 0 H of he corresponding free filed H 0 could be well defined. Then he meaning of eq(16) is ha we sar from ρ 0 H a = and arrive a sae a = 0, and hen jus use his new sae as ρ H. I s no guaraneed ha his reamen will correspond o he rue non-equilibrium disribuion ρ H. So for NEGF, even concepually insead of saring from he rue ρ H, we build up he heory from ρ 0 H. This could be a disadvanage of NEGF. Also several advanages need o be noiced. Firs, his NEGF can be used for sysem a hermal equilibrium sae, which is usually he subjec of Masubara Green s funcion. We will use as an example in III. Second, even H 1 explicily depends on, his NEFG is sill applicable, because we only se one end as H 1 () = 0, H 1 ( ) could be arbirary. A las, for non-equilibrium sysem as long as a unambiguous free-field corresponding non-equilibrium sae could be defined, we could always do he perurbaion calculaion order by order. II. SELF-ENERGY AND DYSON S EQUATION In order o arrive a he Dyson s equaion, we would firsly go o a lile pracice of firs and second order calculaion. Consider an example wih V = αβ M αβc αc β [2]. Then he firs order perurbaion gives, G + (µ, ; ν, ) = G 0,+ (µ, ; ν, ) dsm αβ T r ( T c Cµ () C ν ( ) C α (s) C β (s). (25) αβ The second erm has one disconneced erm and one non-zero conneced erm, which includes αβ dsmαβ Tc Cµ () C α (s) } T c C ν ( ) C β (s) } = [ αβ dsg0,c (µ, ; α, s) M αβ G 0,+ (β, s; ν, ) + ] dsg0,+ (µ, ; α, s) M αβ G 0, c (β, s; ν, ) (26) A he nex order, more erms will be coupled ino his expansion series. Similarly G also couples wih all oher Gs. We will work ou explicily a special case of his ineracion in he nex secion III. Skipping some deailed calculaion here, overall hese expansion series could be represened by a marix equaion, in he form of ( ) Ĝ = Ĝ0ΣĜ0 + Ĝ0ΣĜ0ΣĜ0 + = Ĝ0 1 + ΣĜ (27)

5 5 Like he example abou disordered sysem in lecure noes[3], concep of self-energy Σ can be inroduced. Then he Dyson s equaion of NEGF could be organized as Ĝ (, ) = Ĝ0 (, ) + d 1 d 2 Ĝ 0 (, 1 ) ˆΣ ( 1, 2 ) Ĝ ( 2, ), (28) where ( Ĝc Ĝ Ĝ = Ĝ + Ĝ c ), (29) and Ĝ (, ) is he operaor form of G (x, ; x, ) = x Ĝ (, ) x ; Ĝ 0 = ˆD is he free-field NEGF; and ˆΣ is he self-energy erm coming from ineracion, ( ) ˆΣc ˆΣ ˆΣ = ˆΣ +. (30) ˆΣ c We will explain boh heoreically and by examples how o ge he self-energy erm from ineracion laer. Le s emporally suppose hey are known. Because only hree are independen of hese six NEGFs, he above formula can be simplified furher by choosing he righ hree independen ones. Le s use Ĝa, Ĝr and From eq(22) i is easy o see ha his ransformaion is uniary, ( Ĝc Ĝ U Ĝ + Ĝ c ˆF = Ĝc + Ĝ c. (31) ) ( ) U 0 Ĝ = a Ĝ r Ǧ, (32) ˆF where U = 1 ( ) 1 1. (33) Induced by his ransformaion, he self-energy erm will ransform as ( ) ˆΣc ˆΣ ˇΣ U ˆΣ + U, (34) ˆΣ c And finally, as we will see from he example in secion III, because generally Σ c + Σ c = (Σ + + Σ ), ( ) Ω Σ r ˇΣ Σ a, (35) 0 Therefore, eq(28) sill holds, bu in a new form, or simply denoed as, Ǧ = Ǧ0 + Ǧ0 ˇΣǦ. (36) Then he calculaion of Green s funcion becomes calculaion of self-energy, where differen approximaions, such as Born or Self-Consisen Born approximaion, could be used. III. EXAMPLE Le s apply he general procedure above ono he following problem, he random impuriy, a special case of he example in secion II, H = k ɛ (k) C k C k + k,q,x M (q) e iq X C k+q V C k, (37)

6 6 s s FIG. 3: Firs order expansion on loop pah, LHS for s on posiive branch, RHS for s on negaive branch. Boh of hem are zero if M (0) = 0. s 1 s 2 s 1 s 2 FIG. 4: Second order expansion of G + on loop pah, for s 1, s 2 boh on posiive branch. wih M (0) = 0 means average effec of impuriy is zero. X is uniformly disribued so ha k is sill a good quanum number, so G 0 (k, ; k, ) = G 0 (k, ; k, ) δ (k k ) and G (k, ; k, ) = G (k, ; k, ) δ (k k ). Le s firs work in ime domain, laer on we will do he Fourier ransformaion o work in he frequency domain. Firs order expansion of G +, eq(26) for our special case will be X 1 V [ dsg 0,c (k, ; k, s) M (0) G 0,+ (k, s; k, ) + ] dsg 0,+ (k, ; k, s) M (0) G 0, c (k, s; k, ) = 0. (38) Or equivalenly by Feynman Diagram, as in fig(3). The second order expansion of G + is, i 2V 2 ds 1 ds 2 e iq1 X1 iq2 X2 M (q 1 ) M (q 2 ) C k 1,q 1,X 1 k 2,q 2,X 2 T r (T, (39) c C k () C k ( ) C k 1+q 1 (s 1 ) C k1 (s 1 ) C k 2+q 2 (s 2 ) C k2 (s 2 ) where by Wick s Theorem, = T r +T r C k () C k 2+q 2 (s 2 ) T r T r C k () C k 1+q 1 (s 1 ) T r C k () C k ( ) C k 1+q 1 (s 1 ) C k1 (s 1 ) C k 2+q 2 (s 2 ) C k2 (s 2 ) C k1 (s 1 ) C k ( ) T r T r C k2 (s 2 ) C k ( ) C k2 (s 2 ) C k 1+q 1 (s 1 ). (40) C k1 (s 1 ) C k 2+q 2 (s 2 ) This has o be calculaed for he four configuraions of s 1, s 2. For example, here le s work ou for case of boh s 1, s 2 on posiive branch. i 2V 2 ds 1 ds 2 M (q 1 ) 2 e iq1 X1+iq1 X2 G 0,c (k, ; k, s 2 ) G 0,+ (k, s 1 ; k, ) G 0,c (k + q 1, s 2 ; k + q 1, s 1 ) q 1,X 1,X 2 + i 2V 2 ds 1 ds 2 M (q 2 ) 2 e iq2 X1 iq2 X2 G 0,c (k, ; k, s 1 ) G 0,+ (k, s 2 ; k, ) G 0,c (k + q 2, s 1 ; k + q 2, s 2 ). (41) q 2,X 1,X 2 = i V 2 ds 1 ds 2 M (q) 2 e iq X1 iq X2 G 0,c (k, ; k, s 1 ) G 0,+ (k, s 2 ; k, ) G 0,c (k + q, s 1 ; k + q, s 2 ) q,x 1,X 2 Those wo erm give he same conribuion because of he inegraion and summaion over s 1, s 2, q 1,2. Excep ha here G + couples wih G 0,c and ec, his expression has exacly he same form wih he second order of he usual

7 7 G + G G c G c Non-Zero Impuriy Line = Σ 1,c Σ 1, Σ 1, Σ 1, c FIG. 5: Anoher se of Feynman rules for NEGF, second order expansion is shown as example o express in hese new rules. Furher perurbaion could be done in his diagram form. Green s funcion wih impuriy. Similarly, G + also coupled wih G 0,, G 0, c and obviously G 0,+ iself. So par of he firs order of self-energy erm could be defined as Σ c (k; s 2, s 1 ) i 2V 2 M (q 1 ) 2 e iq1 X1+iq1 X2 G 0,c (k + q, s 2 ; k + q, s 1 ). (42) q 1,X 1,X 2 So we ge a erm of G + a he second order in he form of If we check he erm abou G + in eq(28), we will ge G 2,+ = G 0,+ Σ 1,c G 0,c +... (43) G 2,+ = G 0,+ Σ 1,c G 0,c + G 0, c Σ 1,+ G 0,c + G 0,+ Σ 1, G 0,+ + G 0, c Σ 1, c G 0,+, (44) which does include he above erm we already calculaed. The diagram expression of all hose four erms are shown in he follow fig(5). A deail calculaion include all he configuraion of s 1, s 2 will give us he lef hree erms. Similarly we can do he second order expansion of G,c c. Then he hird order expansion will give us more self-energy erms. Finally, we can ge he Dyson s equaion, which afer Fourier ransformaion will lead us o frequency domain Dyson s equaion. In he Feynman diagrams in fig(2), where he ime loop is explicily down o be an indicaor of he ype of Green s funcion. Besides his, considering he characer of our specific case, M (0) = 0, we may jus use differen lines o represen differen Green s funcions, so ha he diagrams could be simpler, Here we skip he boring derivaion of eq(36), he marix form Dyson s equaion, hopefully his will no sop one undersanding he general picure of NEGF. IV. SUMMARY Keldysh NEGF Formula is a generalizaion of he usual zero and non-zero emperaure Green s funcion. I can be applied ono he usual equilibrium cases and non-equilibrium case. The way we derive such formula here is o consruc a loop from o and back o, and inroducing Green s funcions G (, ) corresponding o he differen configuraion of (, ) over branches. Formally he srucure of Green s funcion and heir Dyson s Equaion

8 looks equivalen wih he one for usual Green s Funcion. So all he perurbaion calculaion is quie sraighforward. However, in order o do he perurbaion, a well-defined non-ineracing sae and Green s funcion is a necessary saring poin. Wheher such saring poin leads o he real non-equilibrium sae is quesionable, and relies on he comparison beween calculaion and experimens. There is anoher way o ge his formula in [4], where he auhors use conour pah over complex plan o replace above loop pah. The derivaion is more elegan in a sense, bu he problem of picking up he righ saring poin sill exiss, alhough he auhors claim heir reamen is exac more or less. 8 [1] C. Caroli and R. Combresco and P. Nozieres and D. Sain-James, Direc calculaion of he unneling curren, J. Phys. C: Solid S. Phys, 4(1971), [2] G. D. Manhan, Many-Paricle Physics(1990 New York), Plenum Press. [3] M. Berciu, Lecure noes for Phys503, 2005, hp://www.physics.ubc.ca/ berciu/. [4] H. Haug and A.P. Jauho, Quanum Kineics in Transpor and Opics of Semiconducors(1996 Berlin Heidelberg), Springer- Verlag.

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