Matsubara-Gree s Fuctios Time Orderig : Cosider the followig operator If H = H the we ca trivially factorise this as, E(s = e s(h+ E(s = e sh e s I geeral this is ot true. However for practical applicatio we eed such a factorizatio. Thus we write, E(s = e sh U(s U(s = e sh e s(h+ We wat to express U(s i some simply way i terms of. To this ed let us examie some of its properties. I particular we compute, du(s ds = H e sh e s(h + + e sh (H + e s(h + = H e sh e s(h + + e sh H e s(h + + e sh e s(h + = e sh e s(h + = e sh e sh e sh e s(h + Defie, Thus we have, ˆ (s = e sh e sh du(s ds If H, ] = the (s = ad the solutio would simply be, = ˆ (s U(s ( U(s = e s I geeral this is ot the case hece the solutio is symbolically deoted as, U(s = T e ] s ˆ (t The meaig of the symbol T...] will be made clear soo. It may be regarded as a symbolic way of writig the iterative solutio to Eq.(. The solutio to Eq.( ca be geerated as follows Homewor : Prove this by actual substitutio ]. U(s = + ˆ (t + t ˆ (t ˆ (t + t t To uderstad the meaig of the symbol T...] we mae the followig defiitios : T (A(tB(t = A(tB(t ; t > t ˆ (t ˆ (t ˆ (t +... T (A(tB(t = B(t A(t ; t > t that is, the largest times come to the extreme left ad the smallest go to the extreme right. We may also write this as, U(s = + ˆ (t + θ(t t ˆ (t ˆ (t + θ(t t θ(t t ˆ (t ˆ (t ˆ (t +...
2 Examie the followig idetity obtaied by simply iterchagig t ad t. θ(t t ˆ (t ˆ (t = θ(t t ˆ (t ˆ (t If A = B the A = 2 (A + B, hece we may add these two equal quatities ad the tae half the resultat. θ(t t ˆ (t ˆ (t = 2 θ(t t ˆ (t ˆ (t + θ(t t ˆ (t ˆ (t] By a similar method we may coclude, = 2 T ˆ (t ˆ (t ] Therefore we may write, U(s = + θ(t t θ(t t ˆ (t ˆ (t ˆ (t = 3! ˆ (t + 2! T ˆ (t ˆ (t ] + 3! T ˆ (t ˆ (t ˆ (t ] T ˆ (t ˆ (t ˆ (t ] +... = T e ˆ (t ] Now we wish to defie a certai quatity ow as the Gree fuctio. Imagie that a N-particle system at time t = is i a eigestate of the hamiltoia : H = d 3 r c ( r 2 2 2m c( r + 2 d 3 r d 3 r ( r r c ( rc ( r c( r c( r Let us deote this eigestate as I >. The subscript I deotes the I-th excited state, I = correspods to the groud state. We ow cosider the time evolutio of operators i imagiary time. c( r, t = e i th c( r, e i th This is i the absece of a exteral potetial. I the presece of the exteral potetial we have to defie the time evolutio differetly. We formally itroduce a time depedet exteral potetial that is also defied i imagiary time. c full ( r, t = Ê (t c( r, Ê(t i (H Ê(t = + d 3 r W ( r, tc ( rc( r Ê(t i Ê ( (t = Ê (t H + d 3 r W ( r, tc ( rc( r Ê( = ˆ. Usig the time orderig decompositio we may write, Û(t = T Ê(t = e i th Û(t e i t ] d 3 r W ( r,sc ( r,sc( r,sds = T S(t]
3 First we show that Û(t is uitary. This is the same as provig that Ê(t is uitary. Ê (tê(t = Û (te i th e i th Û(t = Û (tû(t First we show that Ê (tê(t is a costat. The we show that that this costat is uity. i i Ê (Ê (tê(t = (t Ê(t + Ê (t i Ê(t = Ê (t ( H + d 3 r W ( r, tc ( rc( r Ê(t + Ê (t ( H + d 3 r W ( r, tc ( rc( r Ê(t = But E (E( = ˆ this meas Ê (tê(t = ˆ ad Ê(t ad therefore Û(t is uitary. Cotiuig o we have, c full ( r, t = Û (te i th c( r, e i th Û(t = Û (t c( r, t Û(t We defie the followig quatities that are ow as the Gree fuctios of the system. This maes use of the grad caoical esemble of statistical mechaics. ] T r e β(h µn T S( c( r, tc ( r, t ] G( r, t; r, t = i T r e β(h µn T S(] ] (2 Here T r...] = I=,N= < I, N...] I, N >. ] G( r, t; r, t I=,N= < I, N e β(ɛ µn T S( c( r, tc ( r, t ] I, N > = i I=,N= e β(ɛ µn < I, N T S(] I, N > Here ɛ is the eergy of the I-th eigestate of a N-particle system. That is, H I, N >= ɛ I, N > ad N is the total umber of particles. Here we have to properly defie the time orderig. Cosider the iterval, ]. We say that >. That is, the times t, t e.t.c. go from the smallest possible value to the largest possible value alog the vertical imagiary axis. For particles with statistics = s, we may defie the followig time orderig prescriptios : ad, T c( r, tc( r, t ] = c( r, tc( r, t ; if t > t T c( r, tc( r, t ] = s c( r, t c( r, t ; if t > t Here t > t meas t is closer to tha t ad both lie o the imagiary axis betwee the poits ad. Some Simple Cases : Before studyig the difficult problem of mutually iteractig particles i a exteral field, which is what the Gree fuctio i Eq.(2 tries to do, we first focus o the simple case of o-iteractig particles ( = with o exteral field (W =. I this case the Gree fuctio is simply, ] T r e β(h µn T c( r, tc ( r, t ] G ( r, t; r, t = i T r e ] (3 β(h µn Here, c( r, t = e i Ht c( re i Ht (4 H = d 3 r c ( r 2 2 c( r (5 2m
4 I order to facilitate progress we use the mometum state represetatio. c( r = e i r c (6 I three dimesios (IMPORTANT :...] (2π 3 d 3...] where is the volume of the system. Firstly, sice c( r, c( r ] s = ad c( r, c ( r ] s = δ( r r, we must have, c, c ] s = ad c, c ] s = δ,. If we substitute Eq.(6 ito Eq.(5 we get, H = ɛ c c (7 where ɛ = 2 2 2m. c( r, t = e i r e i Ht c e i Ht = e i r c e i ɛ t (8 Let us first assume that i Eq.(3 t > t, this meas t is closer to tha t ad both lie betwee ad o the egative imagiary axis. Further we set = T re β(h µn ]. I other words, G < ( r, t; r, t = s i G < ( r, t; r, t = s i ] e T r β(h µn c ( r, t c( r, t e i( r r e i (ɛ t ɛ t (9 < I, N e β ] p (ɛ p µc p c p c c I, N > ( The states of a o-iteractig system are obtaied by creatig particles with well-defied mometa. For bosos we ca have a arbitrary umber of particles i each mometum state but for fermios we ca have oly oe. < I, N e β ] p (ɛ p µc p c p c c I, N >= e β p (ɛ p µ p < I, N c c I, N > The states c I, N > ad c I, N > are orthogoal for ad p I, N >= N p I, N >. If s = (fermios the above expressio becomes, e β p (ɛ p µ p < I, N c c {N I, N >= δ p}=, e β p (ɛp µnp N =, e β(ɛ µn N, {N p}=, e β p (ɛp µnp N =, e β(ɛ µn If s = (bosos the above expressio becomes, e β p (ɛp µp < I, N c c I, N >= δ, {N p }=,,2,3,.. e β p (ɛ p µn p N =,,2,3,... e β(ɛ µn N {N p }=,,2,3,... e β p (ɛ p µn p N =,,2,3,... e β(ɛ µn Defie the followig quatities, ad f B (λ = N=,,2,3,... e λn = e λ f F (λ = e λn = + e λ N=,
5 The, d N=,,2,3,... dλ Lf B(λ] = N e λn = N=,,2,3,... e λn e λ ad d N=, dλ Lf F (λ] = N e λn = N=, e λn e λ + If we set λ = β(ɛ µ the we ca say, G < ( r, t; r, t = si e i ( r r e i ɛ (t t F/B ( ( F/B ( = Similarly we ca show that whe t > t (Homewor, G > ( r, t; r, t = i e β(ɛ µ s We may ow relate these two quatities usig a clever observatio. Now cosider, G > ( r, t = ; r, t = i e βɛ ( + s F/B ( = e βµ e β(ɛ µ ( + s F/B ( = e βµ e β(ɛ µ ( + Therefore, Therefore we have, G > ( r, t = ; r, t = i e i ( r r e i ɛ (t t ( + s F/B ( (2 e i ( r r e i ɛ t e βɛ ( + s F/B ( (3 s e β(ɛ µ s = e βµ F/B ( e i ( r r e i ɛ t e βµ F/B ( = se βµ G < ( r, t = ; r, t (4 G ( r, t = ; r, t = se βµ G ( r, t = ; r, t (5 Here G is the time-ordered Gree fuctio. The above equatio is ow as the Kubo-Marti-Schwiger (KMS boudary coditio. We may ow rewrite the Gree fuctio of the oiteractig system as G ( r r, t t sice we ow that it depeds oly o the differeces betwee the time ad positio coordiates. A quatity that obeys such a periodicity property may be discrete Fourier trasformed. These discrete frequecies are ow as Matsubara frequecies. We may write, G ( r r, t t = If this has to obey the KMS boudary coditio we must have, Or, e i.( r r e i µ(t t e z (t t G (, z (6 e z ( t = s e z ( t e iz β = s
6 This meas that z β = odd umber π if s = ad z β = eve umber π if s =. Thus z = (2+π β s = ad z = 2π β if s =. We mae some observatios about these Matsubara frequecies, e z(t t = δ P (t t if Here δ P (t t is the periodic delta-fuctio. It has the property δ P ( t = s δ P ( t. For ay fuctio f(t defied i the iterval, ] ad obeyig the property, f( = sf(, we have, We ca similarly defie a periodic step fuctio. Or, From Eq.( ad Eq.(2 we fid, δ P (t t f(t = f(t θ P (t t = δ P (t t θ P (t t = e z (t t z + iδ But we ow that, i G>/< ( r, t; r, t = 2 2 2m G>/< ( r, t; r, t G ( r, t; r, t = θ P (t t G > ( r, t; r, t + θ P (t t G < ( r, t; r, t Therefore, i G ( r, t; r, t = i θ P (t t G > ( r, t; r, t + i θ P (t t G < ( r, t; r, t From Eq.( ad Eq.(2 we have, +i θ P (t t G> ( r, t; r, t + i θ P (t t G< ( r, t; r, t = i δ P (t t G > ( r, t; r, t G < ( r, t; r, t] + 2 2 2m G ( r, t; r, t G > ( r, t; r, t G < ( r, t; r, t] = i e i ( r r ( + s F/B ( + si e i ( r r F/B ( I other words, = i (i + 2 2 2m e i ( r r = iδ( r r G( r r ; t t = δ P (t t δ( r r (7 If we substitute Eq.(6 ito Eq.(7, we get a formula for G(, z. e i.( r r e i µ(t t e z (t t (i z ɛ + µg (, z = δ P (t t δ( r r (8
7 If we choose (i z ɛ + µg (, z =, the the left had side becomes, LHS = e i.( r r e i µ(t t e z(t t = d 3 ei ( r r (2π 3 e z(t t e i µ(t t Therefore the Gree fuctio i Fourier space has a particularly simple form, = δ( r r δ P (t t = RHS (9 G (, z = (i z ɛ + µ (2 I the literature it is customary to wor i atural uits where = which maes the above Gree fuctio tae the familiar form, G (, z = (iz ɛ + µ (2 where z = (2+π β for fermios ad z = 2π β for bosos.