Advanced Condensed Matter Theory

Transcription

1 Prof. Dr. Andreas Melke Insttut für Theoretsche Physk Unverstät Hedelberg Phlosophenweg Hedelberg Germany e-mal: melke@tphys.un-hedelberg.de Advanced Condensed Matter Theory Andreas Melke Wnter term 206/7 Tme: Tuesday, 9- Phlosophenweg 9 SR Ths scrpt s stll n development.

2 Abstract Strong correlatons are mportant for the understandng of many phenomena n modern condensed matter physcs. Examples are hgh temperature superconductvty (Nobel prze laureates/987/ndex.html), the fractonal quantum Hall effect (Nobel prze se/physcs/laureates/998/ndex.html) or qute recently topologcal phase transtons n two dmensons (Nobel prze The theory for strong correlatons needs new concepts and methods to descrbe such phenomena and to solve the correspondng models. The am of ths course s to provde these methods, to ntroduce the models, and to help understandng the phenomena n strongly correlated systems. Not the only one but the most mportant model to descrbe strongly correlated Fermons s the Hubbard model. In the modern theory of strong correlatons t plays the same role as the Isng model n statstcal physcs: It serves as a standard model to descrbe and understand most of the phenomena n strongly correlated systems. The Hubbard model was ntally ntroduced to descrbe the physcal behavour of transton metals, to understand magnetc phenomena n tnerant electron systems (ferromagnetsm, ant-ferromagnetsm, ferrmagnetsm), to descrbe the Mott transton, and to descrbe π-electron systems n quantum chemstry. In one space dmenson t descrbes a Luttnger lqud. Although the model has a very smple structure, the behavour depends strongly on the parameters, the nteracton strength, the densty, and the underlyng lattce. In ths course we try to gve an overvew over the physcs of strongly correlated electrons and the Hubbard model. The students are expected to know quantum mechancs and statstcal physcs. Some knowledge n condensed matter physcs s helpful as well. Acknowledgement I am grateful to Johannes Hölck. He carefully read the manuscrpt, found lots of errors, and had several deas for mprovements. About ths text The manuscrpt s avalable as pdf. Lnks for ctatons are coloured n dark red, lnks wthn the text n dark blue, lnks to external materal n dark green. 2

3 Contents Interactng Fermons and Bosons 5. The Hubbard model Quantum Systems wth many Partcles Creaton and Annhlaton Operators for Fermons Creaton and Annhlaton Operators for Bosons Sngle partcle operators Interactons Coherent States Coherent States for Bosons Grassmann Algebra Coherent states for fermons Gaussan Integrals Functonal-Integral Representaon The non-nteractng system Perturbaton Theory Frequency and momentum representaton Calculatng Greens Functons Ferm Lquds Quas Partcles Equlbrum Propertes of the normal Ferm Lqud Mcroscopc Dervaton Renormalsaton Man dea Effectve acton Renormalsaton group equatons for G eff The Hubbard model I, Renormalzaton The Hubbard model 5 4. Symmetres of the Hubbard model Some rgorous results Leb s Theorem The Mermn-Wagner Theorem Nagaoka s Theorem Flat-band systems Unform densty theorem Further rgorous results Hgh-Temperature Superconductvty: Doped Mott-Insulators Prelmnary Remarks Models of Hgh-Temperature Superconductors Sngle-partcle Hamltonan Interactons

4 Contents An effectve sngle-band model t J model Frustrated spn systems Some general deas RVB-states The Néel state Short-range correlatons The two-dmensonal Hesenberg model Feld theoretc descrpton Fermons and Schwnger bosons Feld theoretc formulaton The doped ant-ferromagnet: The t J modell The fractonal quantum Hall effect Introducton The nteger quantum Hall effect Dsorder Laughlns gauge argument The fractonal quantum Hall effect Wave functons Elementary exctatons Perodc boundary condtons Unversalty Classcal electrodynamcs n quantum Hall systems Quantsaton

5 Interactng Fermons and Bosons. The Hubbard model As mentoned n the abstract, the Hubbard model serves as a standard model for strongly correlated electrons. The Hamltonan of the Hubbard model s gven by H = H kn + H WW = t x,y c x,σc y,σ +U c x c x c x c x (.) x,y,σ x The model descrbes electrons n a tght bndng approxmaton on a lattce. x,y denote the lattce stes. σ {, } denotes the spn of the electron. c xσ s a creaton operator for an electron wth spn σ on lattce ste x, c xσ s the correspondng annhlaton operator. The frst part of the Hamltonan descrbes the hoppng of electrons on the lattce, t xy s the hoppng ampltude. The second part descrbes the nteracton between the electrons. Usually we assume the nteracton to be repulsve,.e. U > 0. The model was proposed ndependently by J. Hubbard [25] for the descrpton of transton metals, by J. Kanamor [28] for the descrpton of tnerant ferromagnetsm, and by M.C. Gutzwller [9] for the descrpton of the metal-nsulator transton. In Quantum Chemstry, the model s popular as well, and was ntroduced ten years earler [58,?, 60]. Under the name Parser-Parr-Pople model t has been used to descrbe extended π-electron systems. Recently, t has been used to descrbe hgh temperature superconductors. Today, the bosonc Hubbard model s of some nterest as well. Its Hamltonan s H = H kn + H WW = t x,y c xc y + U x,y 2 x n x (n x ) (.2) where now c x, c x are creaton and annhlaton operators for bosons. For students who are not famlar wth the notaton used n (.,.2), we ntroduce creaton and annhlaton operators n the next secton. Students who are famlar wth ths notaton may skp that and may contnue readng from secton.3..2 Quantum Systems wth many Partcles In realty, almost any system contans many partcles nteractng wth each other. A sngle partcle can be descrbed by a wave functon, whch s an element of the sngle partcle Hlbert space. The Hlbert space of two or more dstngushable partcles s a product of the Hlbert spaces for sngle partcles. For dentcal partcles, the stuaton s dfferent. Here, the type of the partcles s essental, they may be ether fermons or bosons. The mult partcle wave functon for bosons has to be symmetrc, for fermons t has to be ant-symmetrc aganst permutatons of partcles. The goal of the subsequent sectons s to obtan a compact notaton for such mult partcle states usng creaton or annhlaton operators for partcles. We ntroduce them frst separately for bosons and fermons. Many of the concepts n ths chapter may be found n standard text books lke [56]..2. Creaton and Annhlaton Operators for Fermons Let {φ ( r,σ)} be an orthonormal bass of sngle partcle states. We denote the coordnate and the spn by q = ( r,σ). In the case of fermons, a bass of N-partcle states can be buld out of Slater determnants of the 5

6 Interactng Fermons and Bosons sngle partcle states:, 2,..., N = N! φ (q ) φ (q 2 )... φ (q N ) φ 2 (q ) φ 2 (q 2 )... φ 2 (q N )... φ N (q ) φ N (q 2 )... φ N (q N ) (.3) Ths state can be wrtten n the form, 2,..., N = N! P S N ( ) P N j= φ j (q P( j) ) (.4) Because of the constructon as a determnant, the state s ant-symmetrc f one permutes a par of ndces, as t should be for fermons.,..., α,..., β,..., N =,..., β,..., α,..., N (.5) Normalsaton: Orthogonalty, 2,..., N, 2,..., N = (.6) { P ( ) P j, j 2,..., j N, 2,..., M = k δ jk, P(k) fn = M 0 fn M. We now defne the creaton operator of a partcle n the state by (.7) c, 2,..., N =,, 2,..., N (.8) c maps states wth N onto states wth N + partcles. The rght-hand sde may vansh, ths happens f and only f one of the ndces k =. One has Ths holds for all states, so we have c c j, 2,..., N =, j,, 2,..., N Therefore we have c c = 0. Furthermore one may wrte = j,,, 2,..., N For each operator c we ntroduce the hermtan conjugate operator c : j,..., j M c,..., N =,..., N c j,..., j M = c j c, 2,..., N (.9) c c j = c j c (.0), 2,..., N = c c 2...c N vak. (.) =,..., N, j,..., j M { δ =,δ 2, j... δ, j δ 2, ±... alltogether N! permutatons, f N = M + 0 f N M +. = δ, 2,..., N j,..., j M δ 2,, 3,..., N j,..., j M +...(N terms). (.2) and therefore and c,..., N = δ, 2,..., N δ 2,, 3,..., N +...(N terms). (.3) c vak. = 0 (.4) 6

7 Interactng Fermons and Bosons The commutaton relatons for the creaton operators c can be carred over onto the annhlaton operators c : c c j = c j c (.5) We now ntroduce the ant-commutator [A,B] + = AB + BA for arbtrary operators A and B. Then we may wrte Because of we obtan [c,c j ] + = 0, [c,c j ] + = 0 (.6) c c j,..., N = δ, j, 2,..., N δ 2, j,, 3,..., N +... (.7) c j c,..., N = δ, j,..., N δ, j, 2,..., N + δ 2, j,, 3,..., N +... (.8) and snce ths holds for any state, we have.2.2 Creaton and Annhlaton Operators for Bosons (c j c + c c j),..., N = δ, j,..., N (.9) [c,c j] + = δ, j (.20) Bosonc wave functons are symmetrc. Therefore we could make the ansatz, n analogy to the fermons, 2,..., N = N! φ j (q P( j) ) (.2) j But ths state s not normalsed. Snce these wave functons are symmetrc aganst permutatons of two ndces, they do not vansh f two ndces are dentcal. Let n be the number of partcles n the state. Then we have Therefore, the correct normalsaton s P, 2,..., N, 2,..., N = n! (.22) {,..., N }, 2,..., N = N! {,..., N } n! P j φ j (q P( j) ) (.23) and therefore j, j 2,..., j N, 2,..., M = For bosons, the creaton operators can be defned as { {,..., M } n! P k δ jk, P(k) f N = M 0 f N M. (.24) c, 2,..., N = n +,, 2,..., N (.25) Here, n s the number of partcles n the sngle partcle state contaned n, 2,..., N. Ths s n complete analogy to the operators one ntroduces n the typcal text-book treatment of the harmonc oscllator. The hermtan conjugate operators to c are c and we obtan: j,..., j M c,..., N =,..., N c j,..., j M = n +,..., N, j,..., j M { n + δ =,δ 2, j... + δ, j δ 2, +... alltogether N! permutatons, f N = M + {,..., N } n! 0 f N M +. = n + (δ, 2,..., N j,..., j M + δ 2,, 3,..., N j,..., j M +...) (N terms). (.26) 7

8 Interactng Fermons and Bosons Here agan n s the number of partcles n the sngle partcle state contaned n j,..., j M. Ths means c,..., N = n (δ, 2,..., N + δ 2,, 3,..., N +...) (N terms) (.27) where now n s the number of partcles n the sngle partcle state contaned n,..., N (one partcle more than n j,..., j M ). In complete analogy to the fermonc case treated before, we obtan [c,c j ] = 0 (.28) [c,c j ] = 0 (.29) [c,c j ] = δ, j (.30) where now [.,.] s the usual commutator. The creaton operators can be used to form the mult partcle states:, 2,..., N = {,..., N } n! c c 2...c N vak. (.3) Summary: Creaton and Annhlaton Operators We ntroduce the varable ζ, whch s for fermons, + for bosons. Wth the help of ths varable, we may wrte the formula for both types of partcles n the compact form, 2,..., N = j, j 2,..., j N, 2,..., M = N! {,..., N } n! P { ζ P φ j (q P( j) ) (.32) j {,..., M } n! P ζ P k δ jk, P(k) f N = M 0 f N M. (.33) c, 2,..., N = n +,, 2,..., N (.34) c,..., N = n (δ, 2,..., N + ζ δ 2,, 3,..., N +...) (N terms) (.35) [c,c j ] ζ = 0 (.36) [c,c j ] ζ = 0 (.37), 2,..., N = [c,c j ] ζ = δ, j (.38) {,..., N } n! c c 2...c N vak. (.39) 8

9 Interactng Fermons and Bosons.2.3 Sngle partcle operators Let us now dscuss the operator One has and therefore ˆN = N = c c (.40) c c,..., N = (δ, + δ 2, δ N,),..., N (.4) ˆN,..., N = N,..., N (.42) ˆN s the partcle number operator. It s a sngle partcle operator, snce t may operate on sngle partcle states. Any sngle partcle operator T (for nstance the knetc energy or a potental) operates on the sngle partcle bass. One has T = We consder frst operators whch are dagonal n the chosen bass For N-partcle states we have smlarly N = t, (.43) T = t (.44) T,..., N = t j,..., N (.45) j The operator acts on each partcle ndependently. We now want to show that T can be wrtten as Wth ths form of T we calculate T, 2,..., N = T n + c 2,..., N = T = t c c (.46) n + [T,c ] 2,..., N + n + c T 2,..., N (.47) and further on [T,c ] = t [c c,c ] = t c (.48) T, 2,..., N = t, 2,..., N + c T 2,..., N = t j,..., N (.49) j whch shows that the representaton (.46) of T s correct. We wll deal wth sngle partcle operators whch are non dagonal. Snce any hermtan operator can be dagonalsed wth the help of a untary transformaton, we have to know how a untary transformaton acts on the creaton and annhlaton operators. Let us ntroduce a new bass α = u α where U = (u α ) wth u α = α s a untary matrx. Let c α be the new creaton operators. We have c α vak. = α = u α = = u α c vak. α c vak. (.50) 9

10 Interactng Fermons and Bosons and therefore we let Then we have c α = α c (.5) c = α c α (.52) c α = α c (.53) c = α c α (.54) Ths s the general form of a sngle partcle operator. Examples: T = t c c = t α β c αc β,α,β = t α,β c αc β (.55) α,β Potental: V ( r) t, j = d 3 r φ ( r)v ( r)φ j ( r) (.56) Knetc energy: t, j = In the orthonormal bass φ k,σ = V exp( k r)χ σ one obtans.2.4 Interactons ) d 3 rφ ( r) ( h2 φ j ( r) (.57) 2m h T = 2 k 2 2m c k,σ c k,σ (.58) k,σ The am of ths course s to treat nteractng systems. Almost any nteracton s an nteracton between two partcles. Such nteractons can be descrbed as two partcle operators. Genercally, we cannot expect that nteractons are dagonal n the mult partcle states constructed out of a gven sngle partcle bass. But, for smplcty, we wll start wth ths case. Let V be the operator of the nteractons, then, n ths bass, we have V, j = V j, j. (.59), j s a two partcle state. For matrx elements of states wth N partcles, we obtan j... j N V... N = ζ P P 2 j Pk, j Pk V k, k j Pl l k k l k,k ( ) = 2 V k, k j... j N... N (.60) k k 0

11 Interactng Fermons and Bosons Here 2 k k s the sum over all pars of partcles n the states... N. For, the number of pars of partcles n the states and s n n. For =, t s n (n ). The number of pars therefore s n n j δ, j n = c c c j c j δ, j c c = ζ c c j c c j = c c j c jc (.6) Therefore we have V = 2 V, j c c j c jc =, j 2, j V, j c c j c jc (.62), j Transformng ths nto a general bass, we obtan V = 2, j V k,l c c j c lc k (.63), j,k,l Ths s the general form of any two partcle operator. Each two partcle nteracton can be wrtten n that form. The representaton of any operator, e.g. the Hamltonan of a gven model, wth the help of creaton and annhlaton operators s thus a smple short form of wrtng down the matrx elements of that operator n a mult partcle bass that has been constructed from a sngle partcle bass by ether formng completely antsymmetrc states, Slater determnants, n the case of fermons or completely symmetrc states n the case of bosons. The advantage of ths representaton s the smple algebrac relaton shp between the creaton and annhlaton operators n the form of commutaton relatons (bosons) or ant-commutaton relatons (fermons). Many calculatons are much easer n ths representaton..3 Coherent States Creaton and annhlaton operators map states of one Hlbert space onto states of another Hlbert space. A state wth N partcles s mapped to a state wth N ± partcles. Successve applcaton of creaton and annhlaton operators yelds states out of Hlbert spaces wth an arbtrary number of partcles. The drect sum of all Hlbert spaces wth N partcles, N = 0,..., s called Fock space. The entre Fock space can be spanned by applyng creaton operators onto the vacuum. In many cases t s useful to work n the Fock space nstead of a Hlbert space wth a fxed number of partcles. Ths s esp. the case, f the number of partcles s not a good quantum number, as e.g. n the case of phonons, or f one treats a problem n a grand canoncal ensemble. Up to now we used the N-partcle states bult out of sngle partcle states as a bass of the Fock space. There s another, actually over-complete set of states whch proved to be useful, coherent states. These are egenstates of the annhlaton operators. Frst of all, t s easy to see that a creaton operator cannot have an egenstate. Suppose that such an egenstate exsts. It would be a sum of states of a dfferent number of partcles. Wthn ths sum, there would be necessarly a state wth the lowest number of partcles. The creaton operator actng on ths state would ncrease the lowest number of partcles by one. Therefore ths sum of states cannot be an egenstate. A correspondng argument does not exst for the annhlaton operator, snce n the Fock space, there s no state wth a maxmal number of partcles. Suppose that we have got an egenstate of the annhlaton operator,.e. c φ = φ φ. (.64) For bosons, annhlaton operators commute. Therefore ther egenvalues are usual complex numbers. We shall see later that for fermons, ths s not the case.

12 Interactng Fermons and Bosons.3. Coherent States for Bosons A state wth n bosons n the sngle partcle states, =,...,N can be wrtten n the form One has n,n 2,...,n N = For the coherent state φ, we make the ansatz The condton c φ = φ φ yelds and therefore Fnally we obtan N = ( ) n! (c )n vak. (.65) c n,n 2,...,n N = n n,n 2,...,n,...,n N (.66) φ = φ n,n 2,...,n N n,n 2,...,n N (.67) n,n 2,...,n N φ φ n,...,n,...,n N = n + φ n,...,n +,...,n N (.68) φ = φ n,n 2,...,n N = N n,n 2,...,n N = = exp( A creaton operator actng on ths state yelds N = N = φ n n! ( φ n ) n! (c )n vak. (.69) φ c ) vak. (.70) Now we calculate the scalar product of two coherent states: ψ φ = c φ = c exp( φ c ) vak. = φ φ (.7) n,...,n N m...m N = n,...,n N ( ψ n n! ( ψ m φ n m!n! ) φ n ) m,...,m N n,...,n N = exp(ψ φ ) (.72) One can show that coherent states are overcomplete. Ths property s very mportant. We now show that D[φ]exp( φ φ ) φ φ = (.73) where we use the notaton d(rφ )d(iφ ) D[φ] = π (.74) 2

13 Interactng Fermons and Bosons To show ths, we calculate the commutator One obtans [c, D[φ]exp( φ φ ) φ φ ] = [c, φ φ ] = (φ φ ) φ φ (.75) D[φ]exp( φ φ )(φ φ ) φ φ (.76) Partell ntegraton of the second term on the rght hand sde shows, that the commutator vanshes. Therefore, the operator D[φ]exp( φ φ ) φ φ commutes wth all annhlaton operators c. Therefore, t must be a number. To calculate that number, we take t s expectaton value n the vacuum. We obtan D[φ]exp( φ φ ) vac. φ φ vac. = D[φ]exp( φ φ ) = (.77) Ths shows the completeness. A trace of an arbtrary operator A can be expressed as TrA = D[φ]exp( φ φ ) φ A φ (.78) Smlarly, we can expand an arbtrary state f n terms of coherent states f = D[φ]exp( φ φ ) φ f φ (.79) The expresson f (φ ) := φ f (.80) s the representaton of f n terms of coherent states, lke f (x) = x f s called coordnate representaton of f. In ths representaton, the annhlaton operator s smply the dervatve ( ) φ c f = f c φ = f φ = f (φ ) (.8) φ and the creaton operator s the multplcaton operator so that we wrte φ φ c f = f c φ = (φ f φ ) = φ f (φ ) (.82) c = φ, c = φ (.83) Ths representaton fulfls the usual commutaton relatons. As a consequence, the Hamltonan, whch s usually expressed by creaton and annhlaton operators n some form H({c,c }) can be wrtten as H({φ, φ }) n the coherent state representaton and the egenvalue equaton s H({φ Matrx elements of an operator A({c,c }) are, φ }) f (φ ) = E f (φ ) (.84) φ A({c,c }) ψ = A({φ,ψ })exp(φ ψ ) (.85) where one has to take nto account that all creaton operators must be placed left of all annhlaton operators (normal orderng). The expectaton value of the number operator n = c c s therefore φ φ and the expectaton value of the number of partcles s N = φ φ. Furthermore φ ( ˆN N) 2 φ φ φ = φ φ = N (.86) so that the relatve devaton of the partcle number from t s expectaton value s N /2. 3

14 Interactng Fermons and Bosons.3.2 Grassmann Algebra We saw that coherent states for bosons may be useful. In the followng we wll often make use of coherent states. The goal of the present secton s to ntroduce coherent states for fermons. To do ths we need to deal wth objects whch ant-commute. Such objects are called Grassmann varables. One can defne usual operatons for them, they form a so called Grassmann algebra. The goal of ths subsecton s to ntroduce the man deas behnd ths concept. An excellent ntroducton (n German) on Grassmann varables s the scrpt by Franz Wegner [76]. A Grassman algebra s bult up from a set of generatng elements {ξ, =,...,N}. It contans all polynomals of these generatng elements wth complex coeffcents. The fundamental rule s and therefore ξ ξ j + ξ j ξ = 0 (.87) ξ 2 = 0 (.88) In the followng, we use an enlarged Grassmann algebra, that s formed by the generators {ξ,ξ, =,...,N}. The symbolc operaton obeys the followng rules (ξ ) = ξ (.89) (ξ ) = ξ (.90) (ξ ξ j ) = ξ j ξ (.9) We now consder functons of the varables ξ and ξ. These are polynomals of ξ and ξ, where n each monom, a factor ξ occurs only once, snce ts square vanshes. Known functons are defned by ther Taylor expanson, whch termnates automatcally. E.g. one has exp(ξ ) = + ξ. Havng defned functons, the next pont s to defne dervatves. One has ξ ξ = (.92) For a product of varables, one has frst to nterchange the varables so that the varable and the dervatve are close to each other. For example, one has ξ ξ j ξ k ξ l ξ = ξ ( ξ ξ j ξ k ξ l ) = ξ j ξ k ξ l (.93) The dervatve of an expresson wth respect to a varable vanshes, f the expresson does not contan the varable. Smlarly, all other known rules lke the product rule hold, except for addtonal sgns whch occur due to the exchangng of varables. Lke varables, dervatves ant-commute: ξ = (.94) ξ j ξ j ξ Next we defne ntegrals. Clearly, there s no ntegral whch s analogue to the usual Remann ntegral for real numbers. The dea s to defne the ntegral as a lnear mappng of functons to real numbers, whch behaves smlar to ntegrals of usual ntegrable functons whch vansh at nfnty. Furthermore we need that the ntegral of a total dfferental vanshes. There are varous possbltes to defne the ntegral, a typcal conventon s dξ = 0 (.95) dξ ξ = (.96) 4

15 Interactng Fermons and Bosons The same must be true for the ξ. But one has to pay attenton concernng ant-commutaton and sgns. For nstance, we have for f (ξ,ξ 2 ) = a 0 + a ξ + a 2 ξ 2 + a 2 ξ ξ 2 (.97) and dξ 2 f (ξ,ξ 2 ) = a 2 a 2 ξ (.98) dξ dξ 2 f (ξ,ξ 2 ) = a 2 (.99) but dξ 2 dξ f (ξ,ξ 2 ) = a 2 (.00) The expressons dξ ant-commute lke the varables ξ tself. Wth these defntons, arbtrary ntegrals can be calculated, for nstance one obtans dξ dξ exp( ξ ξ ) = dξ dξ ( ξ ξ ) = (.0) We now defne a scalar product for functons of Grassmann varables. Let f and g be functons of ξ,...,ξ N, then we defne f g = D[ξ ]exp( ξ ξ ) f (ξ,...,ξ N )g(ξ,...,ξn) (.02) where D[ξ ] = (dξ dξ ) (.03) Usng Grassmann varables we can now construct coherent states for fermons..3.3 Coherent states for fermons Frst, we need a conventon about the behavour of a product of Grassmann varables and creaton and annhlaton operators for fermons. It s natural to choose for all, j. As an ansatz for the coherent state we take [c,ξ j ] + = [c,ξ j] + = [c,ξ j ] + = [c,ξ j ] + = 0 (.04) ξ = exp( ξ c ) vak. = ( ξ c ) vak. (.05) The second expresson can be obtaned from the frst by takng nto account that all the expressons ξ c commute wth each other so that the exponental of the sum s a product of exponentals. Furthermore each annhlaton operator c j commutes wth ξ c f j. For = j we have Ths now yelds whch s ndeed the defnng property for coherent states. c ξ c = ξ c c = ξ ( c c ) (.06) c ξ = ξ ξ (.07) 5

16 Interactng Fermons and Bosons Let us now calculate, how a creaton operator acts on a coherent state. For the scalar product of two coherent states we obtan c ξ = c ( ξ c ) ( ξ j c j ) vak. j = c ( ξ j c j ) vak. j = ( ξ c ξ ) ( ξ j c j ) vak. j = ξ ξ (.08) ξ χ = vak. ( c ξ )( χ c ) vak. = ( + ξ χ ) = exp( ξ χ ) (.09) Next we want to show the completeness relaton D[ξ ]exp( ξ ξ ) ξ ξ = (.0) To do ths, we defne and show that holds. Frst, one obtans and therefore Furthermore, we have E = D[ξ ]exp( ξ ξ ) ξ ξ (.),..., n E j,..., j m =,..., n j,..., j m (.2),..., n ξ = vak. c n...c ξ = ξ n...ξ (.3),..., n E j,..., j m = D[ξ ] ( ξ ξ )ξ n...ξ ξ j...ξ j m (.4) dξ dξ ( ξ ξ )ξ ξ = (.5) dξ dξ ( ξ ξ )ξ = 0 (.6) dξ dξ ( ξ ξ )ξ = 0 (.7) dξ dξ ( ξ ξ ) = (.8) Ths frst of all means that one gets a non-vanshng contrbuton only f n = m and {,..., n } = { j,..., j m }. If ths condton s fulflled, the only thng that remans s to permute the ξ k so that they occur n the same order as the ξ j k. The result s smply the sgn, one gets, f one calculates the scalar product n (.2). Ths shows the completeness relaton. Usng t, we drectly obtan TrA = D[ξ ]exp( ξ ξ ) ξ A ξ (.9) where the addtonal mnus sgn n ξ occurs because we have to exchange ξ wth ξ, so that each Grassmann varable gets an addtonal mnus sgn. Due to the completeness relaton one has also f = D[ξ ]exp( ξ ξ ) ξ f ξ (.20) 6

17 Interactng Fermons and Bosons and we can defne the expresson f (ξ ) := ξ f (.2) lke n the bosonc case as the coherent representaton of a state f. One obtans drectly and smlarly ξ c f = ( f c ξ ) = ξ f ξ = ξ f (ξ ) (.22) ξ c f = ξ f (ξ ) (.23) The creaton operators c and the annhlaton operators c for fermons n the coherent representaton are therefore smply ξ and. The antcommuaton relatons are clearly fulflled. Lke n the bosonc case, we ξ obtan for a normal ordered (creaton operators left of all annhlaton operators) operator A({c,c }) ξ A({c,c }) χ = A({ξ, χ }) ξ χ (.24) An mportant pont s that these are not real numbers. For nstance, the expectaton value of the number operator s ξ ξ and ths s not a real number. In contrast to the bosonc case, fermonc coherent states are not elements of the usual Fock space. Nevertheless they are useful. Summary: Coherent States Let φ be a coherent state for bosons or fermons. Lke above, we ntroduce ζ = for bosons and ζ = for fermons. Then we have c φ = φ φ. (.25) TrA = f = c φ = ζ φ φ (.26) ψ φ = exp(ψ φ ) (.27) D[φ]exp( φ φ ) φ φ = (.28) D[φ]exp( φ φ ) ζ φ A φ (.29) D[φ]exp( φ φ ) ζ φ f φ (.30) c = f (φ ) := ζ φ f (.3) φ The last expresson s true for normal ordered operators only., c = ζ φ (.32) φ A({c,c }) ψ = A({φ,ψ })exp(φ ψ ) (.33) 7

18 Interactng Fermons and Bosons.4 Gaussan Integrals In the followng we wll often need to compute Gaussan ntegrals for complex varables or for Grassmann varables. For complex varables, one has ) = [deth] exp(z (H ) j z j ) (.34) D[φ]exp( φ, j h j φ j + z φ +z φ where H = (h j ) s a matrx wth a postve hermtan part. For Grassmann varables we have smlarly ) = [deth]exp( D[ξ ]exp( ξ, j h j ξ j + χ ξ + χ ξ, j, j χ (H ) j χ j ) (.35) Let us derve these two relatons. Frst for complex varables: If H has a postve hermtan part, the nverse H s defned. Completng the square as usual, we can ntroduce a new ntegraton varable ψ = φ j (H ) j z j. The left hand sde of (.34) becomes D[ψ]exp(, j, j ψ h j ψ j +z (H ) j z j ) (.36), j Next we dagonalse H usng a untary transformaton. Let U be ths untary transformaton. The we ntroduce ϕ = j (U ) j ψ j. We obtan ψ h j ψ j = h ϕ ϕ (.37) where h are the egenvalues of H. Furthermore, we have drφ diφ exp( h φ φ ) = (.38) π h whch fnally yelds the desred result. For Grassmann varables, one can do the same. The problem s, that up to now we have not ntroduced substtutons of varables n ntegrals. Let us therefore treat a general ntegral D[ξ ] f (ξ...ξ N,ξ...ξ N) (.39) and let us ntroduce a transformaton of the form ξ = j u j η j, ξ = j u j η j. The only non-vanshng contrbuton n the ntegral over ξ comes from the term n f whch contans each varable exactly once as a factor. Let us denote ths last term f 0 ξ ξ. Then, f 0 s the ntegral. Let us now calculate D[η] f (ξ (η)...ξ N (η),ξ (η )...ξ N (η )) = f 0 D[η] The polynomal expanson of the rght hand sde can be calculated. Ths yelds f 0 D[η] u P η P u P η P P,P where the sum runs over all permutatons. Now we can calculate the ntegral. We obtan f 0 P ( ) P u P P ( ) P ( j u j η ju kηk ) (.40) k (.4) u P = f 0 detu detu = f 0 (.42) Ths means that substtutons n ntegrals over Grassmann varables can be done as usual. For the Gaussan ntegral, one obtans n the end dη dη exp( h η η ) = h (.43) Ths fnally yelds (.35). 8

19 Interactng Fermons and Bosons.5 Functonal-Integral Representaon In the followng we want to deal wth a system of nteractng partcles. We consder frst systems where the partcle number s a good quantum number. Let us assume that we have a usual two-partcle nteracton. Then, the Hamltonan s of the form Ĥ = ε c c + V, j,k,l c c j c lc k (.44), j,k,l The sngle partcle contrbuton contans typcally the knetc energy and some sngle partcle potental. We assume, that ths contrbuton can be dagonalsed, and we chose the bass for the representaton of the Hamltonan such that t s dagonal. Furthermore, we assume that we have a fnte system where the sngle partcle energes ε are dscrete. Eventually, we may take the thermodynamc lmt n the end. Havng such a system, one typcally wants to calculate expectaton values of some operators A = A({c,c }). At fnte temperatures, they are A({c,c }) = Z Tr[A({c,c })exp( β(ĥ µ ˆN))] (.45) where β = /T s the nverse temperature, µ s the chemcal potental and Z = Trexp( β(ĥ µ ˆN)) (.46) s the grand canoncal partton functon of the system. The traces are calculated over the entre Fock space. We shall calculate the traces usng coherent states. We calculate frst the partton functon, whch yelds drectly some of the mportant thermodynamcal quanttes. We have Z = D[φ]exp( φ φ ) ζ φ exp( β(ĥ µ ˆN)) φ (.47) The frst problem now s that Ĥ and ˆN are normal ordered, but exp( β(ĥ µ ˆN)) s not. For small values of ε we have exp( ε(ĥ µ ˆN)) =: exp( ε(ĥ µ ˆN)) : +O(ε 2 ) (.48) where :. : means that the expresson s normal ordered. To see ths, smply expand the exponental on both sdes. Next we wrte [ exp( β(ĥ µ ˆN)) = exp( β M ˆN))] M (Ĥ µ (.49) We wll choose M suffcently large so that β/m s small. Then, the term n the parentheses on rght hand sde s normal ordered up to a small error. We now obtan Z = D[φ]exp( φ φ ) ζ φ exp( β M (Ĥ µ ˆN))...exp( β M (Ĥ µ ˆN)) φ (.50) where the expectaton value contans M factors. Between each par of factors we put a whch we wrte as D[φ]exp( φ φ ) φ φ = (.5) Dong ths, we have to change our notaton a lttle bt. Snce there are M factors,we need M coherent states. The states are denoted by φ k, k =,...,M. The varables φ k depends on are denoted by φ,k. The frst ndex denotes the sngle partcle states, the second the coherent states. D[φ] s now the ntegral over all varables φ,k and φ,k. Furthermore, I ntroduce φ 0 = ζ φ M. Then we obtan Z = D[φ]exp( φ,k M,kφ,k ) k= φ k exp( β M (Ĥ µ ˆN)) φ k (.52) 9

20 Interactng Fermons and Bosons Up to an error of the order O(M 2 ) we can use the formula (.33). Ths yelds Z = lm Z M = lm D[φ]exp( S M [φ]) (.53) M M S M [φ] = ε M k= [ ( φ,k φ,k φ,k ε ) ] µφ,k + H({φ,k,φ,k }) (.54) where ε = β/m. Typcally one ntroduces n the lmt M the functon φ (τ), where φ,k = φ (εk). Then we may wrte and φ,k φ,k ε = φ (εk) φ (εk ε) ε ε M k= φ (τ 0 + ) τ (.55) β dτ (.56) 0 ( ) β S[φ] = dτ φ (τ)( 0 τ µ)φ (τ 0 + ) + H({φ (τ),φ (τ 0 + )}) (.57) Z = D[φ] exp( S[φ]) (.58) φ (β)=ζ φ (0) These expressons are called functonal ntegrals. We obtaned a functonal ntegral representaton of the orgnal model Hamltonan. Often, t s easy to perform calculatons usng these expressons. But one should always keep n mnd that (.58), (.57) are meant as lmts of the expressons (.53), (.54). The problem s now to solve these ntegrals. We wll do t frst for the non-nteractng system..6 The non-nteractng system In a system wthout nteracton one has Therefore S M [φ] = ε M k= Ĥ = ε c c (.59) ( ) φ,k φ,k φ,k + (ε µ)φ,k ε and the ntegral n (.53) s a Gaussan ntegral. We obtan ( M ( ) ) Z = lm D[φ]exp ε φ,k φ,k φ,k + (ε µ)φ,k M k= ε ( = lm M ( ) ) D[φ ]exp ε φ,k φ,k φ,k + (ε µ)φ,k M ε = lm M k= (.60) [dets () ] ζ (.6) where S () s the M M-matrx S () = 0 0 ζ a a a a a (.62) 20

21 Interactng Fermons and Bosons wth a = β M (ε µ) (.63) The entry ζ a n the upper rght corner stems from the condton φ,0 = ζ φ,m. We expand the determnant and perform the lmt M. Ths yelds The partton functon therefore s lm M dets() = lm [ + M ( )M ζ ( a) M ] = lm M [ ζ ( β(ε µ) M ) M ] = ζ exp( β(ε µ)) (.64) Z = [ ζ exp( β(ε µ))] ζ (.65) Ths s the well known result for free fermons or bosons. We obtan the correspondng expressons for the occupaton numbers as c c = n = lnz = (.66) β ε exp(β(ε µ)) ζ and any further well known results for the free system can be obtaned as well (see e.g. L.D. Landau, E.M. Lfschtz, Theoretcal Physcs, Volume V, Statstcal Physcs, Chapter V). For later purposes t s nterestng to take a closer look on the last result. We obtan the dentty c c = β lnz ε (.67) by takng a dervatve of the partton functon Tr exp( β(ĥ µ ˆN)), expressed by creaton and annhlaton operators. The dervatve can as well be wrtten as calculated from ( M ( ) ) Z = lm D[φ]exp ε φ,k φ,k φ,k + (ε µ)φ,k (.68) M ε Ths yelds Z = lm ε M D[φ] [ ε M k= k= φ,kφ,k ]exp β lnz ε ( = β ε β 0 M k= ( ) ) φ,k φ,k φ,k + (ε µ)φ,k (.69) ε dτ φ (τ)φ (τ 0 + ) (.70) Here, 0 + n the argument of φ denotes that the argument of φ has to be nfntesmally smaller than that of φ. Ths wll be mportant later. We shall see that the expectaton value φ (τ)φ (τ ) has a dscontnuty at τ = τ. There s another mportant pont to menton here: The expectaton value of the partcle number for fermons n a coherent state s φ φ, whch s no real number. Nevertheless, we can calculate t usng the above expresson and coherent states. Later, we need expectaton values of the form φ (τ)φ j (τ ) (.7) and other expressons contanng several felds as well. To calculate such an expresson, we defne a generatng functon for them: ( ) Z M (J,J) = D[φ]exp ε + (ε µ)φ,k + (.72) M k= ( φ,k φ,k φ,k ε M k= ) (J,kφ,k + J,k φ,k) 2

22 Interactng Fermons and Bosons Usng the generatng functon one gets φ,kφ j,l = 2 Z M (J,J) Z M J j,l J,k and n the lmt M φ (τ)φ j (τ ) = Z δ 2 Z(J,J) δj j (τ )δj (τ) J=J =0 J=J =0 = δ 2 lnz(j,j) δj j (τ )δj (τ) J=J =0 (.73) (.74) Z(J,J) = lm M Z M(J,J) (.75) It s clear that the generatng functon can as well be used f one wants to calculate expectaton values of expressons contanng more than two felds. For the free system, t s not dffcult to calculate Z M (J,J). It s a smple Gaussan ntegral. Frst of all we have for the free system Z(J,J) = Z () (J,J ) (.76) where Z () (J,J ) = lm M D[φ]exp ( ε M k= ( φ,k φ,k φ,k ε + (ε µ)φ,k ) + M k= (J,kφ,k + J,k φ,k) ) (.77) Ths ntegral s of the form D[φ]exp ( M φ,ks () k,l= k,l φ,l + M k= (J,kφ,k + J,k φ,k) ) (.78) and yelds Therefore we obtan and furthermore [dets () ] ζ exp( Z M (J,J) = Z M M k,l= exp( J,k(S () ) k,l J,l ) (.79) M k,l= φ,kφ j,l = 2 Z M (J,J) Z M J j,l J,k J,k(S () ) k,l J,l ) (.80) J=J =0 = δ, j (S () ) l,k (.8) The nverse s S () = ζ a M ζ a M ζ a M 2 ζ a a ζ a M ζ a 2 a 2 a ζ a 3. a 2 a.. a 2... a M ζ a M 2 a M 2 a M 3 ζ a M a M a M 2 a M 3 a (.82) 22

23 Interactng Fermons and Bosons where a = β M (ε µ). Therefore we obtan for l k and a l k φ,kφ j,l = δ, j ζ a M (.83) φ,kφ j,l = δ, j a M+l k ζ a M (.84) for l < k. Let l = τm/β, k = τ M/β and use as before that n the lmt M ( x/m) M exp( x). The we obtan n that lmt The above expresson φ (τ)φ j (τ ) = δ, j exp( (ε µ)(τ τ ))(θ(τ τ) + ζ n ) (.85) n = β can be calculated as before. The expresson β 0 dτ φ (τ)φ (τ 0 + ) (.86) g (τ τ ) = exp( (ε µ)(τ τ ))(θ(τ τ) + ζ n ) (.87) s called the sngle partcle propagator. Expectaton values wth more than two felds can be calculated smlarly. We need such expressons below. An expresson of the form φ (τ )φ 2 (τ 2 )...φ n (τ n )φ jn (τ n)φ jn (τ n )...φ j (τ ) (.88) can be calculated usng the dervatve δ 2n lnz(j,j) δj (τ )...δj n (τ n )δj jn (τ n)...δj j (τ ) J=J =0 (.89) As before, the calculaton s done for fnte M and at the end we perform the lmt M. Takng frst the dervatves wth respect to all J j,l (τ ), we get for each dervatve a factor k J j,k (S( j) ) k,l. The dervatves wth respect to J,k act on these factors, snce all other contrbutons vansh at the end when we let J = J = 0. Ths means that n all these expresson we may connect always a varable J j,l wth a varable J,k and replace them by a contrbuton δ, j (S () ) k,l. Ths holds as well for the varables φ,k and φ j,l. Therefore, we obtan φ (τ )φ 2 (τ 2 )...φ n (τ n )φ jn (τ n)φ jn (τ n )...φ j (τ ) = P ζ P φ k (τ k )φ jp(k) (τ P(k) ) (.90) k Ths result s often called Wck s Theorem. In the present calculaton, t s a drect consequence of the fact that Z(J,J) s a Gaussan ntegral..7 Perturbaton Theory The results for the non-nteractng system can be used to obtan structured perturbatve expressons for the nteractng system. We frst dscuss the perturbaton expanson for the partton functon. Perturbatve expressons 23

24 Interactng Fermons and Bosons for correlaton functons can be obtaned n a smlar way. The partton functon of the nteractng system can be wrtten as ( M ( ) ) Z = lm D[φ]exp ε φ,k M φ,k φ,k + (ε µ)φ,k ε M k= ε V ({φ,k,φ,k }) k= ( ) = lm M Z 0,M = Z 0 exp exp ε M k= V ({φ,k,φ,k }) 0,M ( β ) dτ V ({φ (τ),φ (τ 0 + )}) 0 0 (.9) where the ndex 0 at Z 0 and. 0 denotes the non-nteractng system. V ({φ (τ),φ (τ 0 + )}) s a general nteracton. A meanngful physcal ansatz for the nteracton s the Coulomb nteracton or smlar two-partcle nteractons. The exponental functon n the last term can be wrtten as a seres; ths yelds the perturbaton expanson: ( β ) exp dτ V ({φ (τ),φ (τ 0 + )}) 0 0 = n=0 ( ) n β β dτ... dτ n n! 0 0 V ({φ (τ ),φ (τ 0 + )})...V ({φ (τ n ),φ (τ n 0 (.92) + )}) 0 For a concrete form of the nteracton the rght hand sde becomes a sum of expectaton values for products of varables, whch can be calculated usng Wck s Theorem. In prncple, the perturbaton expanson can be calculated for any many-body nteracton. In the followng, we restrct ourselves to two-partcle nteractons, whch have the general form V ({φ (τ),φ (τ 0 + )}) = 2 V, j,k,l φ (τ)φ j (τ)φ l (τ 0 + )φ k (τ 0 + ) (.93), j,k,l I assume that V, j,k,l s ether symmetrc (for bosons) or antsymmetrc (for fermons) n the frst two and n the last two ndces. The n-th term n the expanson s then of the form ( ) n 2 n n!... j k l β V m j m k m l m n j n k n l n m 0 β dτ... dτ n φ (τ )φ j (τ 0 + )φ l (τ )φ k (τ 0 + )... 0 φ n (τ n )φ j n (τ n )φ ln (τ n 0 + )φ kn (τ n 0 + ) (.94) The expectaton value can be wrtten as a sum of products of the form φ φ. Exactly one φ and one φ belong to a par. The expectaton value of the par s called a contracton. The sum runs over all possble combnatons of contractons. Each contracton φ (τ)φ j(τ ) yelds a factor δ, j g (τ τ ). The total sum can be represented n a graphcal form. Ths representaton goes back to Feynman, the dagrams are called Feynman dagrams. There are varous ways to ntroduce Feynman dagrams. We choose one to show the man prncples, for others we refer to the text books. A matrx element V, j,k,l s represented as a pont wth four lnes, two ncomng and two outgong lnes. Each lne gets an ndex, j, k, or l. The lnes belongng to and j belong to creaton operators, these are the outgong lnes. The ncomng lnes have ndces k or l, they belong to annhlaton operators. A term of n-th order contans n such ponts. The lnes of these ponts are now connected n such a way, that each outgong lne meets an ncomng lne. Each pont gets an ndex τ. Each lne now corresponds to a factor φ (τ)φ (τ ) = g (τ τ ), where τ and τ belong to the ndces of the two ponts connected by that lne. The lne has one ndex, the orgnal two ndces of the ncomng and outgong lnes are dentcal due to the factor δ j n the expectaton value above. Now, an mportant pont s that dfferent combnatons of contractons can belong to the same dagram. These combnatons of contractons dffer only n the order of the ndces τ at the vertces. But, n the above expresson, we perform the ntegral over all τ. Therefore all the combnatons of 24

25 Interactng Fermons and Bosons contractons belongng to the same dagram yeld the same contrbuton. The number of dfferent combnatons s called the symmetry factor of the dagram. The rules for the dagrams and ther contrbutons to the sum are thus:. Draw all dagrams wth m ponts and lnes between them so that each lne s orented from one pont to another. A pont can be connected wth tself. At each pont, we must have two ncomng and two outgong lnes. Two dagrams are dfferent f they cannot be obtaned from each other by permutng the ponts. Mathematcally, these dagrams are drected graphs. 2. Calculate the symmetry factor S of the dagram. To do that, each pont gets an ndex τ. S s the number of permutatons of the ndces whch map the dagram to tself. 3. Each lne gets an ndex. For each lne wrte down a factor g (τ τ ), where τ s the ndex of the end pont of the lne and τ s the ndex of the startng pont. 4. For each pont wrte done a factor V, j,k,l where and j are the ndces of the outgong and k and l are the ndces of the ncomng lnes. 5. Sum over all ndces of all lnes and ntegrate over all ndces τ of the vertces from 0 to β. 6. Multply the result wth a factor. ( ) m ζ n L 2 n (.95) e S Here, S s the symmetry factor. n e s the number of equvalent lnes. Two lnes are equvalent, f they have the same startng pont and the same end pont. n L s the number of loops n the dagram. 7. Addng up all these contrbutons yelds the m-th order of Z/Z 0. It s nstructve to calculate several examples usng these rules. Dong that, one notces that many calculatons are easer f one calculates ln(z/z 0 ). The perturbaton seres for ths expresson contans only connected dagrams. Ths fact s often called Lnked Cluster Theorem. There are several possbltes to prove t. The drect proof s to show that exp(sum of all connected dagrams) yelds the sum of all dagrams, thus Z/Z 0..8 Frequency and momentum representaton In most cases t s possble to calculate the above at least partally n frequency and momentum space. If the system s translatonally nvarant, the sngle partcle Hamltonan s dagonal n momentum space and the egenfunctons are plane waves V /2 exp(kx). Here V s the volume of the system. For smplcty we wll use perodc boundary condtons. The sngle partcle energes are denoted as ε k. In a system where the sngle partcle Hamltonan contans only the knetc energy, one has ε k = k 2 /(2m). In a sold, we can start wth a lattce model, n that case ε k s gven by the dsperson relaton of the lattce. For the nteracton, one has due to momentum conservaton V k k 2 k 3 k 4 V δ k +k 2,k 3 +k 4 (.96) The functons φ and φ are ether perodc (bosons) or ant-perodc (fermons) functons of τ. Therefore we may wrte g k (τ τ ) = β ω n exp( ω n (τ τ )) g k (ω n ) (.97) where g k (ω n ) = = β 0 dτ exp((ω n (ε k µ))τ)[θ(τ)( + ζ n k ) + ζ θ( τ)n k ] (ε k µ) ω n (.98) 25

26 For bosons one has ω n = 2πn β Interactng Fermons and Bosons, for fermons ω n = (2n+)π β. They obey exp(βω n ) = ζ. Due to the (ant-) perodcty, each vertex contans a factor δ ωn +ω n2,ω n3 +ω n4, where ω n and ω n2 belong to the two φ and ω n3 and ω n4 belong to the two φ. The rules above may now be rewrtten. One obtans:. as above. 2. as above. 3. To each lne assocate an ndex k. Momentum conservaton at each vertex restrcts the possble values. In a dagram wth m vertces, we can choose m + values k for the lnes ndependently, the others are fxed due to momentum conservaton. To each lne we assocate a frequency. Here as well, n dagrams wth m vertces we may choose m+ values for ω, the rest s fxed due to the factors δ ωn +ω n2,ω n3 +ω n4 for each vertex. Each lne yelds a factor g k (ω n ). For lnes whch connect one vertex wth tself, we need an addtonal factor exp(ω n η). At the end, we take the lmt η For each vertex, we add a factor V k k 2 k 3 k 4. Snce momentum conservaton has already been taken nto account, the factor δ k +k 2,k 3 +k 4 can be dropped. 5. Take the sum over all k (or V (2π) d d d k) and the sum over all ω n. 6. Multply the result wth an addtonal factor β m, where m s the number of vertces. Snce each vertex contans a factor /V and each sum over k yelds a factor V, the fnal result contans a factor V n c, where n c the number of connected components of the dagram. Due to the Lnked Cluster Theorem, ln(z/z 0 ) s the sum of all connected dagrams and therefore V. It s an extensve quantty as t should be snce the logarthm of the grand canoncal partton functon s up to a factor /β the grand canoncal potental..9 Calculatng Greens Functons Greens functons can be calculated usng a generatng functon. Ths s smlar to the non-nteractng system. The generatng functon can be defned n a smlar way. It s ( [ ]) G(J,J) = β D[φ]exp dτ Z φ (τ)( 0 τ µ)φ (τ 0 + ) + H({φ (τ),φ (τ 0 + )}) ( ) β exp dτ [J (τ)φ (τ 0 + ) + φ (τ)j (τ 0 + )] 0 ( ) β = exp dτ [J (τ)φ (τ 0 + ) + φ (τ)j (τ 0 + )] (.99) 0 Here A s the expectaton value of A n the nteractng system. The Greens functons are dervatves of G(J,J). The calculaton can be done agan usng perturbaton theory. Sngle partcle propagators Let us now calculate expressons lke φ (τ)φ j (τ ) (.200) The correspondng perturbatonal seres contans dagrams wth two outer lnes, one ncomng and one outgong correspondng to the two φ ( ) n the propagator. Snce ths expresson contans a dvson by Z, only connected dagrams occur. The rules are smlar to the ones above: 26

27 Interactng Fermons and Bosons. Draw all dfferent connected dagrams wth 2 outer lnes and m vertces. One outer lne corresponds to φ, t ends at a vertex. The other one corresponds to φ and starts at a vertex. Two dagrams are dfferent f they cannot be mapped onto each other by a permutaton of nner lnes and vertces. For each dagram we do the followng calculatons: 2. Each vertex gets an nner ndex τ k. Each lne gets an ndex l. For each nner lne we wrte a factor g l (τ k τ k ), f t runs from τ k to τ k. The ncomng lne yelds a factor g (τ τ k ), the outgong lne yelds a factor g j (τ k τ ), where τ k s the correspondng nner vertex. For m = 0 one has only a factor g j (τ τ ). 3. For each vertex wrte down a factor V l l 2 l 3 l 4 where l and l 2 are ndces of the outgong, l 3 and l 4 are ndces of the ncomng lnes. 4. Now take the sum over all l, the ntegral over all τ. 5. Multply the result wth a factor ( ) m ζ n L where n L s the number of loops. Snce the outer lnes are fxed, there s no symmetry factor S. Generatng functon for connected Greens functons Next, one may want to calculate hgher order Greens functons lke In 0-th order Wck s Theorem yelds φ (τ )φ 2 (τ 2 )φ j2 (τ 2)φ j (τ ) (.20) φ (τ )φ 2 (τ 2 )φ j2 (τ 2)φ j (τ ) 0 = φ (τ )φ j (τ ) 0 φ 2 (τ 2 )φ j2 (τ 2) 0 φ (τ )φ j2 (τ 2) 0 φ 2 (τ 2 )φ j (τ ) 0 (.202) The perturbatve expanson contans contrbutons where the nteracton occurs only n one of the factors φ φ. Clearly, there are also terms where the nteracton s between two factors. Ths can easly be seen f one draws the correspondng dagrams. One fnally obtans φ (τ )φ 2 (τ 2 )φ j2 (τ 2)φ j (τ ) = φ (τ )φ j (τ ) φ 2 (τ 2 )φ j2 (τ 2) φ (τ )φ j2 (τ 2) φ 2 (τ 2 )φ j (τ ) + φ (τ )φ 2 (τ 2 )φ j2 (τ 2)φ j (τ ) c (.203) Here, the contrbuton φ (τ )φ 2 (τ 2 )φ j2 (τ 2 )φ j (τ ) c s the sum of all connected dagrams. The other contrbutons have already been calculated. Smlarly, all hgher Greens functons wth more than 4 felds can be decomposed nto a connected part and unconnected parts that are n turn composed of lower ordered connected dagrams. It s therefore sutable to calculate a generatng functon for the connected parts only. Smlarly to the seres for ln(z/z 0 ) whch contans only connected dagrams, we can show that W(J,J) = lng(j,j) (.204) contans only connected dagrams. An easy way (but not rgorous) to show that s the replca trck. Here, one frst looks at the dagrams contrbutng to G(J,J) n for natural numbers n. To do that, one ntroduces n copes φ α and φα of the orgnal felds φ and φ. A connected component contans only felds wth the same ndex α snce there s no nteracton of felds wth dfferent ndces α. Therefore, each dagram contans smply a factor n n C, where n c s the number of connected components of that dagram. Let us now perform the contnuaton to real n and calculate W(J,J) = lm n 0 n G(J,J) n (.205) Because of the lmt n 0 all contrbutons vansh where the power of n s larger than. Therefore W(J,J) contans only dagrams wth n C =,.e. only connected dagrams. 27