Keldysh Formalism: Nonequilibrium Green s Function


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1 Keldysh Formalism: Nonequilibrium 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 Nonequilibrium 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 nonineracing NEGF, and hen leads o he Dyson s Equaion. Finally, applicaion of NEGF on ighbinding 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, Zeroemperaure (single and manyparicle) 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 nonequilibrium 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 nonequilibrium sysem, which has nonzero curren and includes ineracion beween paricles. So in his noe I will shorly review he idea and echnics of Nonequilibrium 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 zeroemperaure 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 nonineracing sysem ρ 0 H. Le s firs organize he above mapping for zeroemperaure Green s funcion and hen see if he same procedure can be applied ono NEGF. Zeroemperaure 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 imeloop 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 imeordered 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 nonineracing field, U ( 0, ) 0 = lim T eih( T 0) e ih0( T 0) 0 = lim T e ih(t +0) 0. (11) According o GellMann 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 GellMann 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 freefield 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 aniime 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 aniime 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 freefield Nonequilibrium 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 GellMann and Low Theorem, i should be undersood in a kind inverse logic order. For an ineracing field H = H 0 + H 1, he nonequilibrium 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 nonequilibrium 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 nonequilibrium sysem as long as a unambiguous freefield corresponding nonequilibrium sae could be defined, we could always do he perurbaion calculaion order by order. II. SELFENERGY 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 nonzero 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 selfenergy 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 freefield NEGF; and ˆΣ is he selfenergy erm coming from ineracion, ( ) ˆΣc ˆΣ ˆΣ = ˆΣ +. (30) ˆΣ c We will explain boh heoreically and by examples how o ge he selfenergy 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 selfenergy 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 selfenergy, where differen approximaions, such as Born or SelfConsisen 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 NonZero 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 selfenergy 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 selfenergy 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 nonzero emperaure Green s funcion. I can be applied ono he usual equilibrium cases and nonequilibrium 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 welldefined nonineracing sae and Green s funcion is a necessary saring poin. Wheher such saring poin leads o he real nonequilibrium 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. SainJames, Direc calculaion of he unneling curren, J. Phys. C: Solid S. Phys, 4(1971), [2] G. D. Manhan, ManyParicle 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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