Diagrammatic λ series for extremely correlated Fermi liquids
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Diagrammatic λ series for extremely correlated Fermi liquids Edward Perepelitsky
1, 2, 3 and B. Sriram Shastry
1, 2, 3 Physics Department, University of California, Santa Cruz, CA 95064, USA Centre de Physique Th´eorique, Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France Coll`ege de France, 11 place Marcelin Berthelot, 75005 Paris, France. (Dated: October 6, 2018)The recently developed theory of extremely correlated Fermi liquids (ECFL), applicable to modelsinvolving the physics of Gutzwiller projected electrons, shows considerable promise in understandingthe phenomena displayed by the t - J model. Its formal equations for the Greens function arereformulated by a new procedure that is intuitively close to that used in the usual Feynman-Dysontheory. We provide a systematic procedure by which one can draw diagrams for the λ -expansionof the ECFL introduced in Ref. (9), where the parameter λ ∈ (0 ,
1) counts the order of the terms.In contrast to the Schwinger method originally used for this problem, we are able to write downthe n th order diagrams ( O ( λ n )) directly with the appropriate coefficients, without enumerating all the previous order terms. This is a considerable advantage since it thereby enables the possibleimplementation of Monte Carlo methods to evaluate the λ series directly. The new procedure alsoprovides a useful and intuitive alternative to the earlier methods. Keywords: λ expansion; t - J model; Hubbard model; Extremely Correlated Fermi Liquid Model;Strongly correlated electrons. PACS numbers: 71.10.Fd
I. INTRODUCTIONA. Motivation
The t − J model is a model of fundamental importance in condensed matter physics, and is supposed to have thenecessary ingredients to explain the physics of the high-temperature cuprates . Its Hamiltonian can be written interms of the Hubbard X operators as H = − X ijσ t ij X σ i X σj − µ X iσ X σσi + 12 X ijσ J ij X σσi + 14 X ijσ σ J ij { X σ σ i X σ σ j − X σ σ i X σ σ j } . (1)The operator X abi = | a ih b | takes the electron at site i from the state | b i to the state | a i , where | a i and | b i areone of the two occupied states | ↑i , | ↓i , or the unoccupied state | i . In terms of electron operators C, C † , andthe Gutzwiller projection operator P G that eliminates double occupancy, we may explicitly write X σ i = P G C † iσ P G , X σi = P G C iσ P G and X σσ ′ i = C † iσ C iσ ′ P G . The key object of study for this model is the single-particle Green’sfunction, given by the expression G σ σ ( i, f ) = −h T τ X σ i ( τ i ) X σ f ( τ f ) i , (2)as well as higher order dynamical correlation functions. Several novel approaches for computing these objects havebeen tried in literature , but it has been found difficult to impose the Luttinger Ward volume theorem in a consistentway, while providing a realistic description of both quasiparticle peaks and background terms in the spectral function.The essential difficulties in computing these objects are (I) the non-canonical nature of the X operators, and hencethe absence of the standard Wick’s theorem, and (II) the lack of a convenient expansion parameter. In the recentlydeveloped extremely correlated Fermi liquid theory (ECFL) , Shastry proposed a formalism which successfullyresolves both difficulties. This formalism is based on Schwinger’s approach to field theory, which bypasses Wick’stheorem, and is more generally applicable than the Feynman approach that is fundamentally based upon Wick’stheorem. Building atop this powerful formalism, the ECFL theory consists of the following main ingredients: • (1) The product ansatz, in which the physical Green’s function G [ i, f ] is written as a product of the auxiliary(Fermi-liquid type) Green’s function g [ i, f ], and a caparison function e µ [ i, f ] (Eq. (9)). The former is a canon-ical, i.e. unprojected electron type Green’s function, while the latter is a dynamical correction, which arisesfundamentally from the removal of double occupancy from the Hilbert space. This addresses the difficulty (I)above. • (2) The introduction of an expansion parameter λ ∈ (0 , − J model withthe free Fermi gas, and enables the formulation of a systematic expansion. This parameter is related to theextent to which double occupancy is removed, and has a close parallel to the semiclassical expansion parameter S arising in the expansion of spin S (angular momentum) operators in terms of canonical Bosons .In addition the detailed calculations require certain crucial steps • (3) The introduction of a particle-number sum rule for the auxiliary Green’s function (Eq. (62)), fixing thenumber of auxiliary fermions to equal the number of physical fermions. This arises from requiring the charge ofthe particle to be unaffected by Gutzwiller projection, and is closely connected to the volume of the Fermi-surfaceof the physical fermions. In essence it ensures that the theory satisfies the Luttinger-Ward volume theorem . • (4) The introduction of the second chemical potential u , which ensures that g [ i, f ] and e µ [ i, f ] individually satisfythe shift invariance theorem , and together with the original chemical potential µ , facilitates the fulfilling ofthe two particle-number sum rules.In earlier work these ingredients are accomplished directly using the Schwinger equation of motion (EOM) for thet − J model. In particular, the fundamental objects g [ i, f ] and e µ [ i, f ] are defined through their respective equationsof motion, and the expansion parameter λ is inserted directly into the equation of motion. The practical issue ofcomputing objects to various orders in λ is also accomplished by iterating the EOM order by order. The technicaldetails are given in Ref. (9) and Ref. (10), and are summarized below in section II, facilitating a self containedpresentation.In recent papers, the O ( λ ) ECFL has been theoretically benchmarked using Dynamical Mean-Field Theory(DMFT) , Numerical Renormalization Group (NRG) calculations , and high-temperature series . In all cases,the low order ECFL calculation compares remarkably well with these well established techniques. On the experimen-tal side, a phenomenological version of ECFL which uses simple Fermi-liquid expressions for the self-energies Φ[ i, f ]and Ψ[ i, f ] (which are simply related g [ i, f ] and e µ [ i, f ] respectively) was successful in explaining the anomalous linesshapes of Angle-Resolved Photoemission Spectroscopy (ARPES) experiments . Encouraged by this, higher orderterms e.g. O ( λ ) are of considerable interest in order to probe densities closer to the Mott limit than possible withthe O ( λ ) theories, and in this context the present work is relevant. In this paper, we develop a diagrammatic λ expansion. This expansion allows one to calculate the Greens function and related objects to any order in λ bydrawing diagrams. These diagrams are reminiscent of those in the Feynman series , although more complicatedthan the former. This extra complication stems from the non-canonical nature of the X -operators and the absenceof Wick’s theorem. The diagrammatic formulation of the λ series has the following advantages: • It allows one to calculate the n th order contribution to any object by drawing diagrams directly for that order,without having to iterate the expressions from the previous orders. This not only allows for greater ease ofcomputation of analytical expressions, but is also essential for powerful numerical series summation techniques,such as diagrammatic Monte Carlo . Ultimately, it will allow the series to be evaluated to high orders in λ ,whereas presently, only a second order calculation has been possible . • It allows for the diagrammatic interpretation of the various objects in the theory such as the auxiliary Green’sfunction g [ i, f ] and the caparison factor e µ [ i, f ]. For example, one can see that the product ansatz (Eq. (9))is a natural consequence of the structure of the G [ i, f ] diagrams. In particular, it is necessitated by the extracomplexity introduced into the diagrams (over those of the Feynman series) by the projection of the doubleoccupancy. • It allows one to visualize the structure of the diagrams to all orders in λ , therefore facilitating diagrammaticre-summations based on some physical principle. B. Results
The main result of the paper is the formulation of diagrammatic rules to calculate the Green’s function to anyorder in λ . More precisely, the rules state how to generate numerical representations (see section IV B), whichare then converted into diagrams. A subset of these numerical representations (determined by a simple criterion)are in one-to-one correspondence with the standard Feynman diagrams. Therefore, the diagrams given here are anatural generalization of the Feynman diagrams. In this broader class of diagrams, we obtain a subset of numericalrepresentations which are not in one-to-one correspondence with the resulting non-Feynman diagrams. In particular,two different numerical representations can lead to the same (non-Feynman) diagram. This occurs since in thesenon-Feynman diagrams, an interaction vertex can have more than two pairs of Green’s function lines exiting andentering it (e.g. Fig. (40g)). However, the contributions of both numerical representations must be kept. We alsodiscuss below the relationship between ECFL and a formalism using the high-temperature expansion for the t − Jmodel due to Zaitsev and Izyumov in section VIII, and make some connections in the following.We find that a certain subset of the G [ i, f ] diagrams terminate with a self-energy insertion, rather than a singlepoint, as in the case of the Feynman diagrams. This expresses the diagrammatic necessity for the factorization of G into g and e µ . These are in turn expressed in terms of the two self-energies Φ and Ψ. It is interesting that withinthe Zaitsev-Izyumov formalism, a two self-energy structure for the Green’s function is necessary for the exact samereason. The fact that the two self-energy structure comes from three independent approaches, the λ expansion, thehigh-temperature expansion, and the factorization of the Schwinger EOM, shows that it is the correct representationof the Green’s function for this model. In addition, as already reported in Ref (11), the Dyson Maleev approachdeveloped by Harris, Kumar, Halperin and Hohenberg also leads to a similar two self energy scheme in quantumspin systems, where again the algebra of the basic variables is non-canonical.We derive diagrammatic rules for the constituent objects g , e µ , Φ, and Ψ from their definitions, starting from theSchwinger equations of motion. We avoid the use of dressed propagators (leading to skeleton terms), but ratherexpand various objects in powers of λ directly. The fact that these diagrammatic rules are consistent with thoseof G and the product ansatz serves as an independent proof of the rules given for G . We find that Φ consists oftwo independent pieces. The first can be obtained by adding a single interaction line to the terminal point of aΨ diagram, while the second one is completely independent of Ψ. We denote the second piece by the letter χ ,which leads to the relation Φ( ~k, iω k ) = ǫ ~k Ψ( ~k, iω k ) + χ ( ~k, iω k ) in momentum space. In a previous work by the sameauthors , we showed directly from the Schwinger equations of motion, that in the limit of infinite spatial dimensions,Φ( ~k, iω k ) = ǫ ~k Ψ( iω k ) + χ ( iω k ). Here, using the diagrammatic λ expansion, we show that this relationship continuesto make sense in finite dimensions. In going from finite to infinite dimensions, we lose momentum dependence sothat Ψ( ~k, iω k ) → Ψ( iω k ) and χ ( ~k, iω k ) → χ ( iω k ). We also derive the Schwinger EOM defining the object χ in finitedimensions.We derive diagrammatic rules for the three point vertices Λ and U , defined as functional derivatives of g − and e µ (Eq. (11)). Diagrammatically, their relationship to Φ and Ψ is seen to be consistent with the Schwinger equations ofmotion (Eq. (10)). We also derive a generalized Nozi`eres relation for these vertices, which differs from the standardone for the three-point vertices of the Feynman diagrams. We introduce the concept of a skeleton diagram into ourseries. This enables us to make the rather subtle connection between our diagrammatic approach for the λ expansion,and the iterative one used previously. Finally, we use our diagrammatic approach to derive analytical expressions forthe third order skeleton expansion of the objects g and e µ , whereas previously only the second order expressions hadbeen derived via iteration of the equations of motion. C. Outline of the paper
In section II, we begin by reviewing the ECFL formalism from Refs. (9) and (10) in the simplified case of J = 0.In section III, we introduce the λ expansion diagrams in a heuristic way, drawing an analogy with the standardFeynman diagrams. In section IV, we derive the rules for drawing and evaluating the bare diagrams for G to eachorder in λ . We also draw and evaluate the first and second order bare diagrams for G . In section V, we derive thediagrammatic rules for the constituent objects g , e µ , Φ, Ψ, χ , γ , Λ, and U . We also show how to evaluate diagramsin momentum space. We then introduce skeleton diagrams into the series, and complete the full circle by relatingour diagrammatic approach to the λ expansion to the original iterative one reviewed in section II. In section VI, wereview the ECFL formalism with J = 0, and introduce J into our diagrammatic series. In section VII, we computethe skeleton expansion to third order in λ for the objects g and e µ . We also discuss the high-frequency limit of G toeach order in the bare and skeleton expansions, as well as the “deviation” of the λ series from the Feynman series.Finally, in section VIII, we discuss the connection between the ECFL and the Zaitsev-Izyumov formalism for thehigh-temperature expansion of the t − J model.
II. ECFL EQUATIONS OF MOTION AND THE λ EXPANSION
The Greens function is the fundamental object in this theory and is defined as usual by G σ i ,σ f [ i, f ] ≡ −hh X σ i i X σ f f ii = − Z [ V ] tr e − βH T τ (cid:16) e −A X σ i i ( τ i ) X σ f f ( τ f ) (cid:17) , (3)where A = P j R β X σσ ′ j ( τ ′ ) V σσ ′ j ( τ ′ ) dτ ′ , is the additional term in the action due to the Bosonic source V i ≡ V i ( τ i ),included in the partition functional Z [ V ] = tr e − βH T τ e −A . The angular brackets represent averages over the distri-bution in Eq. (3). The function G satisfies the Schwinger equation of motion for the t - J model as derived in Refs.(9), (10), and (2). { [( ∂ τ i − µ ) δ [ i, j ] − t [ i, j ]] δ σ ,σ j + V σ ,σ j i δ [ i, j ] } G σ j ,σ [ j , f ] = − δ [ i, f ] δ σ ,σ + λδ [ i, f ] γ σ ,σ [ i ] − λt [ i, j ] γ σ ,σ a [ i ] G σ a ,σ [ j , f ]+ λt [ i, j ] σ σ a δδ V ¯ σ , ¯ σ a i G σ a ,σ [ j , f ] + λ J [ i, j ] γ σ ,σ a [ j ] G σ a ,σ [ i, f ] − σ σ a δδ V ¯ σ , ¯ σ a j G σ a ,σ [ i, f ] ! , (4)where the bold repeated indices are summed over. The functional derivative takes place at time ( τ + i ), and we haveused the notation δ [ i, m ] = δ i,m δ ( τ i − τ m ), t [ i, m ] = t i,m δ ( τ i − τ m ), and γ σ ,σ [ i ] = σ σ G ¯ σ , ¯ σ [ i, i + ] .We next outline how one obtains the above equation of motion from the definition Eq. (3). We take a time derivative ∂ τ i of Eq. (3), yielding several terms. We start with a simple contribution, namely the time derivative of the θ ( τ i − τ f ),which involves the anticommutator { X σ i i , X σ j j } = δ ij (cid:16) δ σ i σ j − λ σ i σ j X ¯ σ i ¯ σ j i (cid:17) , (5)strictly speaking with λ = 1. We use the anticommutator, generalized as above by introducing the param-eter λ ∈ [0 , λ = 0 and the fullyGutzwiller projected value λ = 1. This process is fundamental to obtaining the λ expansion. From this, weget δ if δ ( τ i − τ f ) (cid:16) δ σ i σ f − λσ i σ f h X ¯ σ i ¯ σ f i ( τ i ) i (cid:17) . This is expressed back in terms of the Greens function by writing h X ¯ σ i ¯ σ f i i → G ¯ σ f ¯ σ i [ iτ i , iτ + i ] = σ i σ f γ σ i σ f [ i ], and thus to the first two terms on the right hand side of Eq. (4).Another contribution arises from the τ i dependence in the lower and upper limits of the time integrals in theexpression T τ e −A S X σ i i ( τ i ) = e − P j R βτi X σσ ′ j ( τ ′ ) V σσ ′ j ( τ ′ ) dτ ′ X σ i i ( τ i ) e − P j R τi X σσ ′ j ( τ ′ ) V σσ ′ j ( τ ′ ) dτ ′ , (6)involving the equal time commutator P j V σσ ′ j ( τ i )[ X σσ ′ j ( τ i ) , X σ i i ( τ i )] = V σ i σ ′ i ( τ i ) X σ ′ i ( τ i ). This leads to the thirdterm in the left hand side of Eq. (4).The non trivial term is obtained when the ∂ τ i X σi ( τ i ) is evaluated from the Heisenberg equation of motion [ H, X σi ]and the fundamental anticommutator Eq. (5) yielding[ X σi , H ] = − µX σi − t ij X σ i j + λ X jσ j t ij ( σ i σ j ) X ¯ σ i ¯ σ j i X σ j j − λ X j = i J ij ( σ i σ j ) X ¯ σ i ¯ σ j j X σ j i . (7)Note that the J term has an almost identical structure to the t term, with i ↔ j . The term involving J actually doesnot come with the external λ , we introduce it so that the λ = 0 limit is the Fermi gas. (This is permissible since weare finally intersted in the limit λ = 1.) A higher order Greens function hh X ¯ σ i ¯ σ j i ( τ i ) X σ j j ( τ i ) , X σ f f ( τ f ) ii is generatedby the third term and a similar one by the fourth term. These are re-expressed in terms of the Greens function byusing the identity due to Schwinger hh X ¯ σ i ¯ σ j i ( τ i ) X σ j j ( τ i ) , X σ f f ( τ f ) ii = hh X ¯ σ i ¯ σ j i ( τ i ) iihh X σ j j ( τ i ) , X σ f f ( τ f ) ii − δδ V ¯ σ i ¯ σ j i ( τ i ) hh X σ j j ( τ i ) , X σ f f ( τ f ) ii . (8)Using again hh X ¯ σ i ¯ σ j i ( τ i ) ii = G ¯ σ j ¯ σ i [ iτ i , iτ + i ] = σ i σ j γ σ i σ j [ i ], we obtain the last four terms on the right hand side ofequation Eq. (4). For ease of presentation we will initially set J → G [ i, f ] can be factored via the following product ansatz: G [ i, f ] = g [ i, j ] . e µ [ j , f ] , (9)where g [ i, f ] is the auxiliary Green’s function, e µ [ i, f ] is the caparison factor, all objects have been represented as 2 × g [ i, f ] and e µ [ i, f ] are defined by thetheir respective Schwinger equations of motion. g − [ i, m ] = ( µ − ∂ τ i − V i ) δ [ i, m ] + t [ i, m ] (1 − λγ [ i ]) − λ Φ[ i, m ] . e µ [ i, m ] = (1 − λγ [ i ]) δ [ i, m ] + λ Ψ[ i, m ]Φ[ i, m ] = − t [ i, j ] ξ ∗ . g [ j , n ] . Λ ∗ [ n , m ; i ]; Ψ[ i, m ] = − t [ i, j ] ξ ∗ . g [ j , n ] . U ∗ [ n , m ; i ] . (10)These exact relations give the required objects g and e µ in terms of the vertex functions. Here we also note that thelocal (in space and time) Green’s function γ [ i ], and the vertices Λ[ n, m ; i ] and U [ n, m ; i ], are defined as γ [ i ] = e µ ( k ) [ n , i + ] . g ( k ) [ i, n ]; Λ[ n, m ; i ] = − δδ V i g − [ n, m ]; U [ n, m ; i ] = δδ V i e µ [ n, m ] , (11)where we have used the notation M ( k ) σ ,σ = σ σ M ¯ σ , ¯ σ to denote the time reversed matrix M ( k ) of an arbitrary matrix M . These exact relations give the vertex functions in terms of the objects g and e µ . The vertices defined above (Λ and U ) have four spin indices, those of the object being differentiated and those of the source. For example, U σ σ σ a σ b [ n, m ; i ] = δδ V σaσbi e µ σ σ [ n, m ]. In Eq. (10), ξ σ a σ b = σ a σ b , and the ∗ indicates that these spin indices should also be carried over(after being flipped) to the bottom indices of the vertex, which is also marked with a ∗ . The top indices of the vertexare given by the usual matrix multiplication. An illustrative example is useful here: ( ξ ∗ . g [ j, n ] . U ∗ [ n , m ; i ]) σ σ = σ σ a g σ a ,σ b [ j, n ] δδ V ¯ σ σ a i e µ σ b ,σ [ n , m ].The λ expansion is obtained by expanding Eq. (10) and Eq. (11) iteratively in the continuity parameter λ . The λ = 0 limit of these equations is the free Fermi gas. Therefore, a direct expansion in λ will lead to a series in λ inwhich each term is made up of the hopping t ij and the free Fermi gas Green’s function g (0) [ i, f ]. As is the case inthe Feynman series, this can be reorganized into a skeleton expansion in which only the skeleton graphs are kept and g (0) [ i, f ] → g [ i, f ]. However, one can also obtain the skeleton expansion directly by expanding Eq. (10) and Eq. (11)in λ , but treating g [ i, f ] as a zeroth order (i.e. unexpanded) object in the expansion. This expansion is carried out tosecond order in Ref. (9). In doing this expansion, one must evaluate the functional derivative δ g δ V . This is done withthe help of the following useful formula which stems from the product rule for functional derivatives. δ g [ i, m ] δ V r = g [ i, x ] . Λ[ x , y , r ] . g [ y , m ] . (12)Within the λ expansion, the LHS is evaluated to a certain order in λ by taking the vertex Λ on the RHS to be of thatorder in λ . III. HEURISTIC DISCUSSION OF λ EXPANSION DIAGRAMSA. Numerical representations of Feynman diagrams
Before deriving the precise rules for the λ expansion diagrams, it is useful to have a heuristic discussion in whichwe compare them to the more familiar Feynman diagrams . To this end, we introduce numerical representationsfor the standard Feynman diagrams. These numerical representations will then be generalized to generate the λ expansion diagrams.Consider any Feynman diagram for the Green’s function G [ i, f ] such as those displayed in Fig. (1). There is aunique path which runs between i and f which uses only Green’s function lines, not counting the interaction lines.We denote this as the zeroth Fermi loop. It is drawn in red in Fig. (1). We number the interaction lines which connectto the zeroth Fermi loop in the order in which they appear in this loop. This list of numbers (along with f ) is placedin the top row of our numerical representation. In the case of both Fig. (1a) and Fig. (1b), it is1 2 f. If the zeroth Fermi loop does not exhaust all of the Green’s function lines in the diagram, such as in Fig. (1a),we proceed to the first Fermi loop. To identify the first Fermi loop, we find the interaction vertex with the highestnumber which connects to the zeroth Fermi loop with only one of its two sides. In this case, this is the interactionvertex labeled 2. The other side has one incoming line and one outgoing line. There is a unique path in the diagramwhich connects these two lines using only Green’s function lines, not interaction lines. This defines the first Fermiloop. It is drawn in blue in Fig. (1a).Since the interaction vertex 2 spawned the first Fermi loop, it is starred in the top row of the representation. Wealso include a lower row for the first Fermi loop. Therefore, the numerical representation of Fig. (1a) now reads.1 2 ∗ f ∗ : 0 f. The second row, which represents the first Fermi loop, is labeled by 2 ∗ , since it was spawned by the second interactionvertex in the zeroth Fermi loop. The fact that only 0 and f are present in the second row tells us that there wereno interaction vertices introduced in the first Fermi loop. That is to say there are no interaction vertices whichconnect to the first Fermi loop, but not to the previous ones (in this case the zeroth Fermi loop). Finally, after all ofthe Fermi loops have been recorded, all nonzero integers which are not starred indicate the position of one side of aninteraction vertex in a Fermi loop. We record the position of the other side as a subscript. Therefore, the completenumerical representation of Fig. (1a) is 1 2 ∗ f ∗ : 0 f. We can represent this in short as 1 2 ∗ f ; 2 ∗ : 0 f , where the semi colon indicates the next line. The completenumerical representation of Fig. (1b) is 1 2 f. Note that the order of appearance of the 1 and 2 as subscripts is important. Reversing them would yield the diagramin Fig. (2), which has the following numerical representation.1 2 f. a) b)i f i f1 21 2* FIG. 1: Second order Feynman diagrams for G [ i, f ]. The zeroth Fermi loop, which is the chain running from i to f is coloredin red. In panel a), the first Fermi loop is colored in blue. The numerical representation of the diagram in panel a) is1 2 ∗ f ; 2 ∗ : 0 f , while that of the diagram in panel b) is 1 2 f . i f1 2 FIG. 2: This Feynman diagram results from reversing the order of the subscripts in the numerical representation of the Feynmandiagram in Fig. (1b). Therefore, the numerical representation of this diagram is 1 2 f . We now consider the slightly more complicated diagram in Fig. (3) to illustrate the scope of this approach. We willnow show how the numerical representation of this diagram is derived. We first identify the zeroth Fermi loop, whichis drawn in red in Fig. (3). The top row now reads 1 2 3 4 f. In this case, the vertex with the highest number which connects to this loop with only one side is 4. Hence, 4 spawnsanother Fermi loop, and gets a star in the top row. 1 2 3 4 ∗ f. We identify this as the first Fermi loop. It is drawn in blue in Fig. (3). The numerical representation is modified toread 1 2 3 4 ∗ f ∗ : 0 1 2 f. Considering only the interaction vertices introduced in the first Fermi loop, we now search for the one with the highestnumber which connects to the first Fermi loop with one side, but whose other side is free , that is to say that it doesnot connect to any of the Fermi loops introduced thus far (zeroth and first). This is the interaction vertex 2. Hence,it gets a star, and spawns the second Fermi loop, which is drawn in green. The numerical representation now reads1 2 3 4 ∗ f ∗ : 0 1 2 ∗ f (4 , ∗ : 0 f. Here, the ordered pair (4 ,
2) is used to distinguish the 2 in the first Fermi loop from the 2 in the zeroth Fermi loop,the latter being denoted simply as 2. The first number in the pair is 4 since the fourth interaction vertex in the zerothFermi loop spawned the first Fermi loop. Also note that no interaction vertices are introduced in the second Fermiloop, hence its row only has a 0 and an f . Therefore, we have arrived at the end of our first sequence of nested Fermiloops. We now take a step back in this sequence and return to the first Fermi loop. Considering only the interactionvertices introduced in the first loop with number less than 2, we search for the one with the highest number whichconnects with one side to the first Fermi loop, but whose other side is free (i.e. does not connect to the zeroth, first,or second Fermi loops). There is no such interaction vertex. Therefore, we take another step back in the sequence,and return to the zeroth Fermi loop. We find that the interaction vertex 2 connects to this loop with one side, butthat the other side is free. Hence, 2 gets a star and spawns the fourth Fermi loop, which is drawn in turquoise. Thenumerical representation now reads 1 2 ∗ ∗ f ∗ : 0 1 2 ∗ f (4 , ∗ : 0 f ∗ : 0 f. Since there are no interaction vertices introduced in the fourth Fermi loop, we have arrived at the end of our secondsequence of nested Fermi loops. We take a step back to the zeroth Fermi loop and find that there are no moreinteraction vertices introduced in this loop which have one side free. Since all of the Fermi loops have been identified,as the final step, we must take the integers which are not starred, and place them in their final locations as subscripts.The complete numerical representation now reads 1 2 ∗ ∗ f ∗ : 0 ∗ f (4 , ∗ : 0 (4 ,
1) 1 f ∗ : 0 f. f3i 1 4*2* 2*1 FIG. 3: Sixth order Feynman diagram. The zeroth, first, second, and third Fermi loops are drawn in red, blue, green, andturquoise respectively. Interaction vertices introduced in a particular Fermi loop are numbered in the same color as thatloop. An interaction vertex is starred if it spawns a new Fermi loop. The numerical representation for this diagram is1 2 ∗ ∗ f ; 4 ∗ : 0 ∗ f ; (4 , ∗ : 0 (4 ,
1) 1 f ; 2 ∗ : 0 f . If we now wanted to formulate a set of rules for generating the numerical representations obtained from the Feynmandiagrams, they would be the following. • (1) Write a row of integers 1 . . . m f where m ≥
1, e.g.1 2 3 4 f. • (2) Assign a star to any of the integers in the row ( f does not count as an integer), e.g.1 2 ∗ ∗ f. • (3) Every starred integer gives rise to a lower row. The i th lower row also consists of integers 0 . . . m i f , where m i ≥
0, e.g. 1 2 ∗ ∗ f ∗ : 0 1 2 f ∗ : 0 f. • (4) In the lower rows, assign a star to any of the integers excluding 0, e.g.1 2 ∗ ∗ f ∗ : 0 1 2 ∗ f ∗ : 0 f. • (5) The integers starred in step 4 once again give rise to lower rows, etc. Continue this process until the lastrows which you create have no starred integers, e.g. 1 2 ∗ ∗ f ∗ : 0 1 2 ∗ f (4 , ∗ : 0 f ∗ : 0 f. • (6) Label each integer with a tuple (an ordered list of numbers) which traces that integer back to the first rowthrough the starred integers. For example, the 0 in the third row would be labeled (4 , , • (7) Between any 2 consecutive integers of a row (including 0’s and f ’s), one can place as subscripts an orderedlist of tuples from the following set: all those corresponding to non-starred integers except 0 whose tuple can beobtained from the tuple of the smaller of the 2 consecutive integers in question, by taking the first k ≤ l entriesof this tuple (where l is the length of the tuple), and subtracting a non-negative integer from the last entry.For example, suppose that the two consecutive integers in question are the 2 and f of the second row. Then alltuples (corresponding to non-starred integers) eligible to be used as subscripts between them are: (4 , ∗ ∗ f ∗ : 0 ∗ f (4 , ∗ : 0 (4 ,
1) 1 f ∗ : 0 f. If we think back to the order in which we generated Fermi loops (and hence the numerical representation) from agiven Feynman diagram, we can see that it complies exactly with rule (7) stated above. Doing things in this wayensures that the mapping between Feynman diagrams and numerical representations is one-to-one.
B. Topologies of λ expansion diagrams The exact rules for drawing diagrams for the λ expansion, as defined in section II will be derived in section IV.There, it will be shown that the λ expansion diagrams are constructed from the 2 elements displayed in Fig. (4). Theone in panel a) is a generalization of the Feynman interaction vertex, in which one of the sides can have any number ofpairs of incoming and outgoing lines rather than just one pair. The one in panel b) is a generalization of the terminalpoint f in a Feynman diagram. In the case of the Feynman diagrams, it is a single point, while in the case of a λ expansion diagram, it is a single point along with any number of pairs of incoming and outgoing lines. These extralines come from the second term on the RHS of Eq. (17). This term, which itself comes from the anti-commutatorof the X -operators in Eq. (2), and which is absent in the EOM of canonical theories, allows a diagram to close in onitself in an iterative expansion of the EOM. ...a) ...fb) FIG. 4: The 2 elements used for construction the λ expansion diagrams. In Fig. (5), we have drawn two of the simplest non-Feynman diagrams which can be made from these elements.The one in panel a) has the following numerical representation. f ∗ f ∗ : 0 f. The zeroth Fermi loop runs from the site i to the site f and is drawn in red. The site f , which is the terminal pointof the zeroth Fermi loop spawns the first Fermi loop, drawn in blue. The one in panel b) has the following numericalrepresentation. 1 ∗∗ f ∗∗ : 0 f ∗ (1 , f ) ∗ : 0 f. f i fia) b) 1* ** FIG. 5: Two of the simplest non-Feynman diagrams in the λ expansion. A non-Feynman diagram occurs when the terminalpoint of a Fermi loop spawns another Fermi loop. The diagrams drawn in Fig. (5) are both valid λ expansion diagrams. However, as will be shown below, the allowedtopologies of λ expansion diagrams do not include all of the possible ways of combining the two elements in Fig. (4),but rather only a subset of these. To see which subset, consider the plausible diagram displayed in Fig. (6a), whichis not an allowed λ expansion diagram. This diagram is obtained from the Fock diagram in Fig. (6b) by adding aFermi loop to the latter. The numerical representation for the diagram in Fig. (6b) is1 f. We see that the point from which the first Fermi loop emanates in Fig. (6a) is represented by a subscript in Fig. (6b).Alternatively, using the terminology introduced in section III A, the first Fermi loop is spawned by the interactionvertex 1 of the zeroth Fermi loop. However, the other side of this interaction vertex is not free, but rather connects tothe zeroth Fermi loop itself. This is not allowed. In fact, we shall find below that when there are more than one pairof lines connected to a single point of an interaction vertex, each pair must both start and terminate a Fermi loop atthat point, as in Fig. (5b). Another diagram which is not an allowed λ expansion diagram is drawn in Fig. (7). a) b)1* 1*i if f FIG. 6: The diagram in panel a) is not allowed in the λ expansion. This is because first Fermi loop emanates from a pointwhich is represented by a subscript in the Fock diagram displayed in panel b). i f FIG. 7: A more elaborate version of the diagram in Fig. (6a), which is also not allowed in the λ expansion. λ expansion diagrams is that they are not in one-to-one correspondence with their numericalrepresentations. To see this, consider the diagrams drawn in Fig. (8). As usual, the zeroth, first, and second Fermiloops are drawn in red, blue, and green respectively. The diagram in (Fig. 8a) has the numerical representation1 2 ∗∗ f ∗∗ : 0 f ∗ (2 , f ) ∗ : 0 f. In words, this says that the interaction vertex 2 of the zeroth Fermi loop spawns the first Fermi loop. The interactionvertex 1 of the zeroth Fermi loop connects to the first Fermi loop. Finally, the terminal point of the first Fermi loopspawns the second Fermi loop. On the other hand, the diagram in (Fig. 8b) has the numerical representation1 2 ∗∗ f ∗∗ : 0 f ∗ (2 , f ) ∗ : 0 f. In this case, the interaction vertex 1 of the zeroth Fermi loop connects to the second Fermi loop rather than thefirst. We see that both of the above numerical representations lead to the same diagram, although they both have acontribution which must be accounted for. a)i f1 2* b)i f1 2** *
FIG. 8: A demonstration that unlike Feynman diagrams, λ expansion diagrams are not in one-to-one correspondence with theirnumerical representations. In the diagram in panel a), the interaction vertex 1 of the zeroth Fermi loop also connects to thefirst Fermi loop. In the diagram in panel b), it connects to the second Fermi loop. The topologies of both diagrams, however,are identical. A final point to mention in this discussion of the λ expansion diagrams is that when drawing the diagrams in realspace, the vertex appropriate for the t -interaction differs from the one appropriate for the J -interaction. While this isderived rigorously from the EOM below, one can understand it by examining the relevant terms in the Hamiltonian(Eq. (1)). First, we examine the t -term. Writing the X operators in terms of canonical creation and destructionoperators, we obtain − X ijσ t ij X σ i X σj = − X ijσ t ij c † iσ (1 − n i ¯ σ ) c jσ = − X ijσ t ij c † iσ c jσ + X ijσ t ij c † iσ n i ¯ σ c jσ . (13)Here, we have used the non-Hermitean mapping described in Ref. (11) X σ i → c † iσ (1 − n i ¯ σ ); X σj → c jσ . (14)As discussed in Ref. (11), it is permissible to drop the projection from the destruction operator, since if the systemstarts in the subspace of no double occupancy, the unprojected destruction operator cannot take it out of this subspace.The second term on the RHS of Eq. (13) can be represented with the interaction vertex drawn in Fig. (9a). Next, weexamine the J term. Since a spin flip operator or number operator cannot take the system out of the subspace of nodouble occupancy, the X operators in the J term can be replaced by their canonical counterparts. Therefore, writtenin terms of canonical operators, the J term looks like12 X ijσ J ij n iσ + 14 X ijσ σ { J ij c † iσ c iσ c † jσ c jσ − n iσ n jσ } . (15)2The first term amounts to a shift in the chemical potential µ , while the second one leads to the interaction vertexdrawn in Fig. (9b). The corresponding lines between Figs. (9a) and (9b) have been marked with corresponding letters.Throughout the text, we shall sometimes use the term “Feynman diagrams” to refer to the λ expansion diagramsformed solely from the interaction vertices in Fig. (9), and sometimes to refer to the usual Feynman diagrams .It should be clear what we mean from the context. To obtain the more general λ expansion diagrams, one must usethe vertices drawn in Fig. (10). Once again, the corresponding lines have been marked with corresponding lettersbetween the t -vertices in panel a) and the J -vertices in panel b). a) i ja bcd b) i jab cd FIG. 9: t -vertices in panel a) versus J -vertices in panel b) for the λ expansion diagrams which are also Feynman diagrams.The corresponding lines are marked with corresponding letters. a) ja b abi cd ef b) i j cd ef... ... FIG. 10: t -vertices in panel a) versus J -vertices in panel b) for the more general λ expansion diagrams. The correspondinglines are marked with corresponding letters. A λ expansion diagram drawn in real space will of course have a mix of t -vertices and J -vertices. Luckily, whendrawing the diagrams in momentum space, we can use only one type of vertex ( t or J ). The details of this procedureare discussed in section VI. To convert between “ t -diagrams” and “ J -diagrams”, we must rearrange every interactionvertex as indicated in Fig. (10). For example, the Hartree and Fock diagrams, when drawn using t -vertices, appearas in Fig. (12b) and Fig. (12c) respectively. In this introductory section, we have used the more familiar J -verticesto construct our diagrams, while in the rest of the paper, we shall take the point of view of using the t -vertices. Thecounterparts of the diagrams drawn in Figs. (1a), (1b), (2), (5b), and (8), are drawn in Figs. (17 g), (17 b), (17 c),(17 n), and (40g) respectively.To conclude this preliminary discussion, we point out that while it may be possible to define the λ diagrams as allinequivalent ways of combining the elements displayed in Fig. (4) with some topological constraints, this definitionwould not have much practical value. It also would not tell us how to evaluate the diagram once we had drawn it. Onthe other hand, the numerical representations of the λ expansion diagrams defined below are both easy to generatein a systematic manner, and easy to evaluate. In fact, one may argue that even for the standard Feynman diagrams,the definition in terms of the numerical representations presented in section III A is more useful than the usual one,since it gives a systematic way of generating, and a compact way of representing the diagrams.3 IV. BARE DIAGRAMMATIC λ EXPANSION FOR G [ i, f ] .A. Integral equation of motion and the first order λ expansion. As can be seen from Eq. (4), the parameter λ adiabatically connects the free Fermi gas at λ = 0 with the fullyprojected model at λ = 1. Therefore, in the bare λ series for G , to each order in λ , G [ i, f ] is expressed as a functionalof the free Fermi gas, g (0) [ i, f ] and the hopping t ij . In this section, we aim to derive a set of rules for drawing diagramsto compute the n th order contribution to the bare series for G [ i, f ]. We do this by rewriting Eq. (4) as an integralequation, and then iterating this equation in λ . An analogous expansion is done for the first couple of orders of theFeynman series in Kadanoff and Baym in Ref. (24). To this end, we rewrite Eq. (4) as − g − σ ,σ j [ i, j ] G σ j ,σ [ j , f ] = − δ [ i, f ] δ σ ,σ + λ × δ [ i, f ] γ σ ,σ [ i ] − λ × t [ i, j ] γ σ ,σ a [ i ] G σ a ,σ [ j , f ] + λ × t [ i, j ] σ σ a δδ V ¯ σ , ¯ σ a i G σ a ,σ [ j , f ] , (16)where g − [ i, f ], the inverse of the free Fermi gas Green’s function is obtained by setting λ = 0 in Eq. (10). RewritingEq. (16), we obtain the following integral equation for G [ i, f ]. G σ ,σ [ i, f ] = g (0) σ ,σ [ i, f ] − λ g (0) σ ,σ b [ i, f ] σ b σ G ¯ σ , ¯ σ b [ f, f + ] − λ × g (0) σ ,σ b [ i, k ] − t [ k , j ] σ b σ a G ¯ σ a , ¯ σ b [ k , k + ] G σ a ,σ [ j , f ] + t [ k , j ] σ b σ a δδ V ¯ σ b , ¯ σ a k G σ a ,σ [ j , f ] ! , (17)This expression has considerable parallels to a similar expression for the (canonical) Hubbard model, with one ex-ception, the second term on the RHS, (arising from the non-canonical nature of the X ’s) has no counterpart in thecanonical theory. If we drop this term, the series so generated is exactly the Feynman series.We now proceed to draw the diagrams for the zeroth and first order contributions to G . The zeroth order contributionto the Green’s function, which is given by the free Fermi gas g (0) [ i, f ], is represented by the diagram in Fig. (11). i fi i f f FIG. 11: The zeroth order contribution to the Green’s function: g (0) [ i, f ] To obtain the first order contribution to G [ i, f ], we plug g (0) [ i, f ] in for G [ i, f ] in the RHS of Eq. (17).This leads to the three diagrams displayed in Fig. (12). The diagrams a), b), and c) in Fig. (12) corre-spond to the three terms in the parenthesis on the RHS of Eq. (17) respectively. They correspond to the an-alytical expressions a): − λσ b σ g (0) ifσ σ b [ τ i , τ f ] g (0) ff ¯ σ ¯ σ b [ τ f , τ + f ]; b): λσ a σ b g (0) iaσ σ a [ τ i , τ a ] g (0) aa ¯ σ b ¯ σ a [ τ a , τ + a ] t ab g (0) bfσ b σ [ τ a , τ f ]; and c): − λσ a σ b g (0) iaσ σ a [ τ i , τ a ] t ab g (0) baσ b ¯ σ a [ τ a , τ + a ] g (0) af ¯ σ b σ [ τ a , τ f ]. In drawing the diagram in Fig. (12c), we have used the Schwingeridentity δ g (0) σ a ,σ b [ i, f ] δ V σ c σ d r = − g (0) σ a ,σ x [ i, x ] δ g − σ x ,σ y [ x , y ] δ V σ c σ d r g (0) σ y ,σ b [ y , f ] = g (0) σ a ,σ c [ i, r ] g (0) σ d ,σ b [ r, f ] . (18)In other words, the role of the functional derivative in the Eq. (17) is to pick a line in the diagram for G σ a σ [ j , f ], andto split it into two lines, one entering the point k , and the other one exiting it.4 b) − −aaba baa) −ii f1 bf bf −f 2 f 2 fii a1 aa b ii a1 bff 2ba − baaa aa −c)1:* 1 f*0 f 1 f1f :* f *0 f FIG. 12: The first order contribution to the Green’s function: G (1) [ i, f ]. The diagrams in panels a), b), and c) come from thefirst, second, and third terms on the RHS of Eq. (17), respectively. The reader would recognize that we bypassed the Wicks theorem, by utilizing instead the Schwinger identityEq. (18).
B. Rules for calculating the n th order contribution. By plugging in the first and zeroth order diagrams into the RHS of Eq. (17), we can obtain the second orderdiagrams. Using this iterative process, we can obtain diagrams for G to any order in λ . Moreover, by noticing thepattern in the iterative process, we can derive the rules for obtaining the n th order contribution to G directly withoutcalculating the lower order contributions. In the case of the Feynman diagrams, this is merely an alternate way ofderiving the rules obtained from using Wick’s theorem. However, in the present case, in which the standard Wick’stheorem is not available, this derivation is essential in going from the EOM definition of the λ expansion introduced inRef. (9) and the equivalent diagrammatic one developed here. We now present the diagrammatic rules for calculatingthe n th order contribution to G . • (1) Write a row of consecutive integers followed by the letter f , i.e. 1 . . . m f , where m ≥ m = 0, we simplywrite f ), e.g. 1 2 3 f. • (2) Give any number of stars (including no stars) to each these integers (including f ), e.g.1 ∗∗ f ∗ . • (3) Each integer (including f ) with p stars ( p ≥
1) gives rise to another row of integers which now starts with 0(as opposed to 1), and which ends with an f with p − f , each of which can have any number of stars, giving rise to further rows. 0 is notallowed to have any stars, e.g. 1 ∗∗ f ∗ ∗∗ : 0 1 2 ∗∗∗ f ∗ (1 , ∗∗∗ : 0 1 f ∗∗ (1 , , f ) ∗∗ : 0 f ∗ (1 , , f, f ) ∗ : 0 1 f (1 , f ) ∗ : 0 1 ff ∗ : 0 1 2 f. Note that each integer in the above diagram is uniquely specified by a tuple which traces it back to the first rowthrough the starred integers. For example, the number 1 in the fifth row corresponds to the tuple (1 , , f, f, • (4) Let ν be the total number of integers without stars excluding 0’s and f ’s. Let s f be the total number ofstars on the f in the top row, and let s be the total number of stars excluding those on f ’s. Then the order n must satisfy the relation n = ν + s f + s . In the above example, ν = 8, s f = 1, and s = 5. Therefore this a 14 th order diagram.5 • (5) Between any 2 consecutive integers of a row (including 0’s and f ’s), one can place as subscripts an orderedlist of tuples from the following set: All those corresponding to non-starred integers (except 0’s and f ’s) whosetuple can be obtained from the tuple of the smaller of the 2 consecutive integers in question, by taking the first k ≤ l entries of this tuple (where l is the length of the tuple), and subtracting a non-negative integer from thelast entry. We have taken f ’s to be integers greater than all other integers in their respective rows. For example,suppose that the two consecutive integers in question are 1 and f in the fifth row of the above diagram. Thenall integers eligible to be used as subscripts between them are: (1 , , f, f, , , , f ’s) must be used exactly once in this way. e.g.1 ∗∗ f ∗ ∗∗ : 0 1 2 ∗∗∗ f ∗ (1 , ∗∗∗ : 0 1 f ∗∗ (1 , , f ) ∗∗ : 0 (1 , , f ∗ (1 , , f, f ) ∗ : 0 1 (1 , ,f,f, f (1 , f ) ∗ : 0 1 (1 ,f,
1) (1 , ff ∗ : 0 1 2 ( f,
1) ( f, f. • (6) We use the numerical representation to draw the diagram in the following way. Each integer excluding 0’sand f ’s corresponds to an interaction vertex shown in Fig. (13). The interaction vertices displayed in panelsa), b), c), and d) correspond to 0, 1, 2, and 3 stars respectively on the integer in question. On the top rightof each panel, we indicate how the spins contribute to the sign of the diagram. Note that when two outgoingor two incoming lines share the same spin label, this spin contributes to the sign of the diagram, while whenan outgoing and an incoming line share the same spin label, this spin does not contribute to the sign of thediagram. For example, in panel d) σ a and σ d contribute to the sign while σ b and σ c do not.The f in the top row corresponds to a terminal point shown in Fig. (14). The terminal points displayed inpanels a), b), c), and d) correspond to 0, 1, 2, and 3 stars respectively on the f in the top row. On the top rightof each panel, we indicate how the spins contribute to the sign of the diagram. Note that the same general ruleholds as in the case of the interaction vertices, except that now for the case of 1 or more stars, the spin σ alsocontributes to the sign of the diagram. For the case of one or more stars, one can obtain the terminal pointsin Fig. (14) from the interaction vertices in Fig. (13) by removing the interaction line and the Green’s functionline to the right of it, and making the substitution σ a → σ . The interaction vertices displayed in Fig. (13) andthe terminal points displayed in Fig. (14) continue to follow the same pattern for greater than three stars.To actually draw the diagram, let us momentarily ignore the subscripts in our numerical representation, andcorrespondingly the Green’s function lines labeled by ¯ σ a and ¯ σ b in panel a) of Fig. (13), (the case of 0 stars).Then the top row of the numerical representation corresponds to a chain of interaction lines connected to eachother by Green’s function lines running from the point i to the point f . The lower rows also correspond to asimilar chain running from a single point back to itself. This is the point k (displayed in panels b), c), and d)in Fig. (13)) on the interaction vertex corresponding to the starred integer which gives rise to this lower row.Thus, the number of such chains beginning and ending at a point of a particular interaction vertex is equalto the number of stars on the starred integer which corresponds to this vertex. For the example given above,following this procedure yields the intermediate diagram displayed in Fig. (15).Finally, to put the subscripts back into the diagram, we break each chain at any place where there are subscriptsbetween two consecutive vertices of the chain, and pass the chain through the (non-starred) vertices indicatedby the subscripts in the order in which they are written, after which it resumes its original course. This isaccomplished with the help of the two Green’s function lines labeled by ¯ σ a and ¯ σ b on the non-starred vertices,(displayed in panel a) of Fig. (13)),which were ignored in drawing the intermediate diagram in Fig. (15). Thefinal diagram is displayed in Fig. (16).Note that when drawing a vertex (or terminal point) with multiple stars, such as that displayed in Fig. 13d),the lines ¯ σ a and σ c (incoming) correspond to the row with 2 stars on its f , the lines σ b (outgoing) and σ d correspond to the row with 1 star on its f , and the lines σ c (outgoing) and ¯ σ d correspond to the row with 0stars on its f . Therefore, in Fig. (16), on the point k corresponding to the vertex (1 , ∗∗∗ , the lines ¯ σ m and σ n (incoming) are part of the row (1 , ∗∗∗ (3 rd row) in the numerical representation, the lines σ l (outgoing) and σ o are part of the row (1 , , f ) ∗∗ (4 th row) in the numerical representation, and the lines σ n (outgoing) and ¯ σ o are part of the row (1 , , f, f ) ∗ (5 th row) in the numerical representation.6 • (7) Each solid line in the diagram contributes a non-interacting Green’s function, each wavy line contributes ahopping matrix element. An equal-time Green’s function is always taken to be g (0) ( τ, τ + ), i.e. the incoming(creation) line is given the greater time. • (8) The total sign of the diagram is given by ( − n ( − s ( − s f − × (sign from the spins), where in thecase of s f = 0, ( − s f − ≡
1, and the way in which the spins contribute to the sign is indicatedFigs. (13) and (14). Therefore, the diagram in Fig. (16) has sign ( − ( − ( − (sign from the spins) = − σ b σ h σ c σ d σ e σ g σ v σ σ w σ x σ y σ z σ t σ u σ j σ k σ m σ o σ p σ q σ r σ s . • (9) Sum over internal sites and spins, and integrate over internal times.According to the above rules, the contribution of the diagram drawn in Fig. (16) is − σ b σ h σ c σ d σ e σ g σ v σ σ w σ x σ y σ z σ t σ u σ j σ k σ m σ o σ p σ q σ r σ s g (0) iaσ σ a [ τ i , τ a ] t ab g (0) bcσ b σ c [ τ a , τ b ] t cd g (0) deσ d σ e [ τ b , τ c ] t eg g (0) geσ g ¯ σ e [ τ c , τ + c ] g (0) ec ¯ σ g ¯ σ c [ τ c , τ b ] g (0) cf ¯ σ d σ v [ τ b , τ f ] g (0) ah ¯ σ b ¯ σ j [ τ a , τ d ] t hj g (0) jkσ k σ l [ τ d , τ l ] t kl g (0) laσ m σ n [ τ l , τ a ] g (0) koσ e ¯ σ p [ τ e , τ g ] t op g (0) pkσ q σ n [ τ g , τ e ] g (0) ko ¯ σ m σ p [ τ e , τ g ] g (0) ok ¯ σ q σ o [ τ g , τ e ] g (0) kqσ n σ r [ τ e , τ r ] t qr g (0) rqσ s ¯ σ r [ τ r , τ + r ] g (0) qk ¯ σ s ¯ σ o [ τ r , τ e ] g (0) amσ a σ t [ τ a , τ s ] t mn g (0) nmσ u ¯ σ t [ τ s , τ + s ] g (0) mh ¯ σ u ¯ σ j [ τ s , τ d ] g (0) ha ¯ σ k ¯ σ h [ τ d , τ a ] g (0) fs ¯ σ σ w [ τ f , τ u ] t st g (0) tuσ x σ y [ τ u , τ v ] t uv g (0) vsσ z ¯ σ w [ τ v , τ u ] g (0) su ¯ σ x ¯ σ y [ τ u , τ v ] g (0) uf ¯ σ z ¯ σ v [ τ v , τ f ] . Upon turning off the sources, the Green’s functions become spin diagonal, i.e. g (0) σ σ [ i, f ] = δ σ σ g (0) ↑↑ [ i, f ] = δ σ σ g (0) ↓↓ [ i, f ] ≡ δ σ σ g (0) [ i, f ]. This allows one to evaluate the spin sum and the sign of the above expression. A goodway to evaluate the spin sum is to break the diagram into spin loops in the following manner. Recall that at eachinteraction vertex and at the terminal point, lines are paired according to spin. They share the same spin if one isincoming and the other is outgoing, and they have opposite spins if both lines of the pair are incoming or both areoutgoing. Starting with the line exiting i , follow the path of Green’s function lines created by the spin pairings untilyou reach the line labeled by σ (or ¯ σ if f has one or more stars). These spins are all set by the value of σ = σ , andtherefore this is the zeroth spin loop. If not all of the lines have been used up by the zeroth loop, find a random lineand follow the path created by the spin pairings to reach the line to which it is paired. This is the first spin loop, etc.Continue to do this until you have used up all of the lines in the diagram. Let F s denote the number of spin loopsin the diagram. Then, the spin sum is 2 F s . We emphasize that unlike the case of the standard Feynman diagrams,the spin loops of the λ expansion diagrams do not coincide with the Fermi loops (where each row of the numericalrepresentation can be thought of as a Fermi loop).To determine the sign of the diagram, assign values to the spins in a manner consistent with the spin loops (i.e.the value of any one spin in the spin loop determines the values of all of them). Then, plug these values into theanalytical expression for the diagram. It is important to note that the reason we can compute the spin sum and thesign independently, is that the choice we make for the values of the spins does not affect the sign of the diagram.To see this note that every spin loop consists of an even number of pairs that have either two incoming lines or twooutgoing lines (since it has an equal number of each kind), and an arbitrary number which have one incoming lineand one outgoing line. However, only the former contributes to the sign, while the latter does not (see Figs. 13 and14.) Moreover, each pair contributes a distinct spin and appears in exactly one spin loop. Therefore, by flipping allof the spins in a spin loop, we flip an even number of spins, and therefore do not change the sign of the diagram. Theonly exception to this line of reasoning is the zeroth spin loop, in the case when the terminal point f has 1 or morestars (see Fig. (14)). In this case, the zeroth spin loop must have one more pair where both lines are incoming thanit has pairs where both lines are outgoing. This is due to the fact that in this case both the spins σ and ¯ σ exit thesites i and f respectively. It is also consistent with the fact that the terminal point f now has one more pair withtwo incoming lines than two outgoing lines. Therefore, the spin pairs in the zeroth spin loop now contribute an oddnumber of spins. However, the spin σ from the zeroth spin loop now also appears explicitly in the sign. Therefore,flipping all of the spins in the zeroth loop once again does not change the sign of the diagram. In Fig. (16), we find( σ ) = ( σ a ) = σ t = ¯ σ u = ¯ σ j = σ b = σ c = σ g = ¯ σ e = ¯ σ d = σ v = σ z = ¯ σ w = σ ; σ x = σ y ; σ h = σ k = ( σ l ) = ¯ σ p = σ m ; σ q = ¯ σ o = ¯ σ s = σ r = ( σ n ) , F s = 3. Theloops contribute ( − = ( − = 1 to the sign. Therefore, the final contribution of the diagram in Fig. (16) is − × g (0) ia [ τ i , τ a ] t ab g (0) bc [ τ a , τ b ] t cd g (0) de [ τ b , τ c ] t eg g (0) ge [ τ c , τ + c ] g (0) ec [ τ c , τ b ] g (0) cf [ τ b , τ f ] g (0) ah [ τ a , τ d ] t hj g (0) jk [ τ d , τ l ] t kl g (0) la [ τ l , τ a ] g (0) ko [ τ e , τ g ] t op g (0) pk [ τ g , τ e ] g (0) ko [ τ e , τ g ] g (0) ok [ τ g , τ e ] g (0) kq [ τ e , τ r ] t qr g (0) rq [ τ r , τ + r ] g (0) qk [ τ r , τ e ] g (0) am [ τ a , τ s ] t mn g (0) nm [ τ s , τ + s ] g (0) mh [ τ s , τ d ] g (0) ha [ τ d , τ a ] g (0) fs [ τ f , τ u ] t st g (0) tu [ τ u , τ v ] t uv g (0) vs [ τ v , τ u ] g (0) su [ τ u , τ v ] g (0) uf [ τ v , τ f ] , where all sites and times other than i and f , and τ i and τ f are summed/integrated over. k jbk akak −bk − a b k j akbk ak − bk − a bk j kbk ak − ck −bk ck aa c k j akbk ak − ck bkck dk − dk a da) b)c) d) FIG. 13: Interaction vertices appearing in the diagrams. Panels a), b), c), and d) correspond to 0, 1, 2, and 3 stars on thenumber representing the interaction vertex, respectively. Note that the lines are broken into pairs based on spin. A pair of twoincoming or two outgoing lines share opposite spins, while a pair of one incoming and one outgoing line share the same spin.Moreover, in the case of the former, the spin contributes to the sign of the diagram, while in the case of the latter, it does not.The contribution to the sign is written in the top right of each panel. a) f2f b) fbf 2f − bf − b2c) f2f −bf cf c cf −bf cf df − bf df cf FIG. 14: Terminal point in the diagram corresponding to the f in the top row. Panels a), b), c), and d) correspond to 0, 1, 2,and 3 stars on the f in the top row, respectively. Same comments regarding spin apply as in Fig. (13). Note that in the caseof one or more stars on the f in the top row, the line labeled by ¯ σ is outgoing. This is compensated by the fact there are twomore lines entering the point f than exiting it. FIG. 15: Intermediate step in the process of drawing the diagram corresponding to the numerical representation in step 5 ofthe rules. All of the interaction vertices are drawn in. To complete the diagram, we must split some of the Green’s functionlines through the unused points in the interaction vertices in a manner indicated by the numerical representation. fi a b c d e ghj klo pq r st uvm n−s t−s u−dj dji 1 aaaas t s uab− −d kdk el emahen r reoem−el−gp gp −gq gqen−eo −r s r s−r rab bc −bc −bd bd c e−c g −c e c g −f v−f 2uw−uw v z−ux −v yux v y −v zah− f v
FIG. 16: Diagram corresponding to the numerical representation in step 5 of the rules.
C. Second order contribution
Using the rules from section IV B, we draw the diagrams that contribute to G [ i, f ] in second order in Fig. (17), andcalculate their contributions below. fa) 11 22 f f22 fi f f ) 1 1: 0 f2* 2 f*b) 1 221 f fii f f1 122 fc)i f f d) 1*1*: 0 f fii e) 11 2 f*: 0 f2* i k ) 11 f *: 0 ff * fg) 11 2*: 0 f2*i l) 11 f *: 0 ff *i f f2*i: 0 f2*h) 1*1*: 0 f f *i: 0 ff *m )1*1*: 0 f fi fi) 1*1*: 0 ff1(1,1) o) f *: 0 f1(f,1)f *i fj) 1*1*: 0 ff1(1,1) :* 0 f * fi p) f *: 0 f f1(f,1) :* 0 f *if *n) * f*1 i f1* : 0 f* *:0 f(1,f)* q) * f*f:0 f(f,f)*: 0 f* *f * i F I G . : T h e s e c o nd o r d e r d i ag r a m s c o n t r i bu t i n g t o t h e G r ee n ’ s f un c t i o n : G ( ) [ i , f ], a nd t h e i r c o rr e s p o nd i n g nu m e r i c a l r e p r e - s e n t a t i o n s . N o t e t h a tt h e d i ag r a m s a )t h r o u g h j ) a r e t h e s t a nd a r d s e c o nd o r d e r F e y n m a nd i ag r a m s . T h e o t h e r d i ag r a m s a r e n o t . a ) g (0) ia [ τ i , τ a ] t ab g (0) ba [ τ a , τ + a ] g (0) ac [ τ a , τ b ] t cd g (0) dc [ τ b , τ + b ] g (0) cf [ τ b , τ f ] b ) − g (0) ia [ τ i , τ a ] t ab g (0) bc [ τ a , τ c ] t cd g (0) da [ τ c , τ a ] g (0) ac [ τ a , τ c ] g (0) cf [ τ c , τ f ] c ) g (0) ia [ τ i , τ a ] t ab g (0) bc [ τ a , τ b ] t cd g (0) dc [ τ b , τ + b ] g (0) ca [ τ b , τ a ] g (0) af [ τ a , τ f ] d ) g (0) ia [ τ i , τ a ] g (0) aa [ τ a , τ + a ] t ab g (0) bc [ τ a , τ b ] t cd g (0) dc [ τ b , τ + b ] g (0) cf [ τ b , τ f ] e ) g (0) ia [ τ i , τ a ] t ab g (0) ba [ τ a , τ + a ] g (0) ac [ τ a , τ b ] g (0) cc [ τ b , τ + b ] t cd g (0) df [ τ b , τ f ] f ) g (0) ia [ τ i , τ a ] t ab g (0) bc [ τ b , τ + b ] t cd g (0) da [ τ b , τ a ] g (0) af [ τ a , τ f ] g ) − g (0) ia [ τ i , τ a ] t ab g (0) bc [ τ a , τ b ] g (0) ca [ τ b , τ a ] g (0) ac [ τ a , τ b ] t cd g (0) df [ τ b , τ f ] h ) g (0) ia [ τ i , τ a ] g (0) aa [ τ a , τ + a ] t ab g (0) bc [ τ a , τ b ] g (0) cc [ τ b , τ + b ] t cd g (0) df [ τ b , τ f ] i ) g (0) ia [ τ i , τ a ] t ab g (0) bf [ τ a , τ f ] g (0) ac [ τ a , τ b ] t cd g (0) dc [ τ b , τ + b ] g (0) ca [ τ b , τ a ] j ) g (0) ia [ τ i , τ a ] t ab g (0) bf [ τ a , τ f ] g (0) ac [ τ a , τ b ] g (0) cc [ τ b , τ + b ] t cd g (0) da [ τ b , τ a ] k ) − g (0) ia [ τ i , τ a ] t ab g (0) ba [ τ a , τ + a ] g (0) af [ τ a , τ f ] g (0) ff [ τ f , τ + f ] l ) 2 g (0) ia [ τ i , τ a ] t ab g (0) bf [ τ a , τ f ] g (0) fa [ τ f , τ a ] g (0) af [ τ a , τ f ] m ) − g (0) ia [ τ i , τ a ] g (0) aa [ τ a , τ + a ] t ab g (0) bf [ τ a , τ f ] g (0) ff [ τ f , τ + f ] n ) − g (0) ia [ τ i , τ a ] g (0) aa [ τ a , τ + a ] g (0) aa [ τ a , τ + a ] t ab g (0) bf [ τ a , τ f ] o ) − g (0) if [ τ i , τ f ] g (0) fa [ τ f , τ a ] t ab g (0) ba [ τ a , τ + a ] g (0) af [ τ a , τ f ] p ) − g (0) if [ τ i , τ f ] g (0) fa [ τ f , τ a ] g (0) aa [ τ a , τ + a ] t ab g (0) bf [ τ a , τ f ] q ) g (0) if [ τ i , τ f ] g (0) ff [ τ f , τ + f ] g (0) ff [ τ f , τ + f ] . V. DIAGRAMMATIC λ EXPANSION FOR CONSTITUENT OBJECTSA. Introduction of the two self-energies
We next consider the auxiliary Green’s function g [ i, f ]. Using Eq. (10) for g − [ i, f ], we can write the analog ofEq. (17) for g [ i, f ]. g σ ,σ [ i, f ] = g (0) σ ,σ [ i, f ] − λ g (0) σ ,σ b [ i, k ] − t [ k , j ] σ b σ a G ¯ σ a , ¯ σ b [ k , k + ] g σ a ,σ [ j , f ] + t [ k , j ] σ b σ a δδ V ¯ σ b , ¯ σ a k g σ a ,σ [ j , f ] ! , (19)Comparing the iterative expansion of G [ i, f ] through Eq. (17) with that of g [ i, f ] through Eq. (19), we see that theterms in the parenthesis are identical in both expansions. However, the second term on the RHS of Eq. (17), missingin Eq. (19), allows a Green’s function diagram to close on itself in the iterative expansion, merging the initial point i and the terminal point f . Such a diagram must necessarily have more than one line connected to its terminal point,and therefore at least one star on the f in the top row. Therefore, the diagrams for g [ i, f ] are the subset of thediagrams for G [ i, f ] which have no stars on the f in the top row. In Fig. (17), these are diagrams a) through j), anddiagram n).We see that in the diagrams for g [ i, f ], the terminal point labeled by f is connected to the rest of the diagram onlyby a single line. Therefore, it will be possible to describe these diagrams in terms of a Dyson equation, with a Dysonself-energy. This is not the case for the other diagrams in Fig. (17) (those which do have a star on the f in the toprow), and these diagrams require the introduction of a second-self energy. We now proceed to define these two typesof self-energies.1We shall denote the Dyson self-energy for g [ i, f ] by Σ a . As is the case in the Feynman diagrams, it is obtainedfrom the diagrams for g [ i, f ] by removing the external line coming in from the point i , and the one going out to thepoint f . If a diagram for Σ a can be split into two pieces by cutting a single line, then it is reducible. Otherwise, it isirreducible. Denote the irreducible part of Σ a by Σ ∗ a .Now consider those diagrams which do have a star on the f in the top row. The second self-energy, Σ b , is obtainedfrom these diagrams by removing the external line coming from the point i . Once again, if a diagram for Σ b can besplit into two pieces by cutting a single line, then it is reducible. Otherwise, it is irreducible. Denote the irreduciblepart of Σ b by Σ ∗ b .From the diagrammatic structure of the series, it is clear that G [ i, f ] = g [ i, f ] + g [ i, j ] . Σ ∗ b [ j , f ]. Comparing withEq. (9), we see that e µ [ i, f ] = δ [ i, f ] + Σ ∗ b [ i, f ]. Also, from Dyson’s equation, we know that g − [ i, f ] = g − [ i, f ] − Σ ∗ a [ i, f ]. We shall give an independent proof of these formulae starting from the equations ofmotion for g − and e µ (Eq. (10)) in section V B. B. g − and e µ We shall now prove, starting with the equations of motion in Eq. (10), the observations already made in sectionV A, that e µ [ i, f ] = δ [ i, f ] + Σ ∗ b [ i, f ]; g − [ i, f ] = g − [ i, f ] − Σ ∗ a [ i, f ] . (20)This is equivalent to showing thatΣ ∗ b [ i, f ] = − λγ [ i ] δ [ i, f ] + λ Ψ[ i, f ]; Σ ∗ a [ i, f ] = λγ [ i ] t [ i, f ] + λ Φ[ i, f ] . (21)We rewrite the EOM for e µ [ i, f ] (Eq. (10)) in expanded form. e µ σ σ [ i, f ] = ( δ σ σ − λσ σ G ¯ σ ¯ σ [ i, i + ]) δ [ i, f ] + λ Ψ σ σ [ i, f ]Ψ σ σ [ i, f ] = − t [ i, j ] σ σ a g σ a ,σ b [ j , n ] δδ V ¯ σ ¯ σ a i e µ σ b σ [ n , f ] . (22)We now proceed to prove the first of Eqs. (20) using induction in λ . The lowest order contribution to Σ ∗ b [ i, f ] comesfrom diagram a) in Fig. (12). Removing the incoming external line, we obtain Σ ∗ (1) bσ σ [ i, f ] = − λσ σ δ [ i, f ] g (0) ff ¯ σ ¯ σ [ τ f , τ + f ].Using Eq. (22) to obtain the first order contribution to e µ [ i, f ], we get e µ (1) σ σ [ i, f ] = − λσ σ g (0)¯ σ , ¯ σ [ i, i + ] δ [ i, f ]. Clearly,these two are equal, and we have that Σ ∗ (1) bσ σ [ i, f ] = e µ (1) σ σ [ i, f ].Now consider the m th order contribution Σ ∗ ( m ) bσ σ [ i, f ]. This will be obtained from the corresponding m th order G [ i, f ]diagram upon dropping the incoming external line. If in the numerical representation for this G [ i, f ] diagram, thereare no numbers other than f ∗ ... ∗ in the top row (e.g. panels o), p), and q) in Fig. (17)), then the contribution of thisdiagram to G [ i, f ] is G ( m ) σ σ [ i, f ] = − λ g (0) σ σ b [ i, f ] σ b σ G ( m − σ ¯ σ b [ f, f + ] (see Fig. (18)). The resulting contribution to Σ ∗ b ,which we shall denote by Σ ∗ b , is Σ ∗ ( m ) b σ σ [ i, f ] = − λ σ σ G ( m − σ ¯ σ [ f, f + ] δ [ i, f ] . (23) fbfi 1i 2f − bf −G FIG. 18: Schematic representation for a Green’s function diagram with only the number f ∗ ... ∗ in the top row. Upon removingthe incoming external line, it contributes to Σ ∗ b [ i, f ]. f ∗ ... ∗ in the top row of the numerical representation of the corresponding m th order G [ i, f ] diagram. Then, the top row reads 1 . . . f ∗ ... ∗ (e.g. panels k) through m) of Fig. (17)). In this case,we know that for the resulting Σ b [ i, f ] diagram to be irreducible, i.e. for it to contribute to Σ ∗ b [ i, f ], the number 1in the top row should not be starred. Therefore, we can represent the diagram schematically as in Fig. (19). Thisrepresentation is obtained as follows. If we consider just the part of the diagram between the points j and f , we knowthat a line in this part of the diagram (denoted in Fig. (19) by the letter s ) is split by the point k . If we restore s by removing the lines labeled by ¯ σ a and ¯ σ b from the point k , then the part of the diagram running from j to f is aGreen’s function diagram which contributes to g [ j , n ] . Σ ∗ b [ n , f ] (since the f has at least one star on it). However, theline s can’t be contained in the g [ j , n ] part of the diagram (represented in Fig. (19) by a double line), since then theresulting Σ b (of the overall diagram) would be reducible. Therefore it must be contained in the Σ ∗ b [ n , f ] part of thediagram. The analytical expression for the diagram in Fig. (19) is − λ g (0) σ σ b [ i, k ] t [ k , j ] σ b σ a g σ a σ c [ j , n ] δδ V ¯ σ b ¯ σ a k Σ ∗ bσ c σ [ n , f ] . (24)Removing the incoming external line, and using the inductive hypothesis, we obtain the contribution of these typesof diagrams to Σ ∗ b [ i, f ], which we shall denote as Σ ∗ b [ i, f ].Σ ∗ ( m ) b σ σ [ i, f ] = − λt [ i, j ] σ σ a g ( m ) σ a σ b [ j , n ] δδ V ¯ σ ¯ σ a i e µ ( m ) σ b σ [ n , f ] , (25)where m = m + m + 1. Comparing Eq. (25) with Eq. (22), we see thatΣ ∗ b σ σ [ i, f ] = λ Ψ σ σ [ i, f ] . (26)Combining Eq. (23) and Eq. (26), we find thatΣ ∗ bσ σ [ i, f ] = Σ ∗ b σ σ [ i, f ] + Σ ∗ b σ σ [ i, f ] = − λ σ σ G ¯ σ ¯ σ [ f, f + ] δ [ i, f ] + λ Ψ σ σ [ i, f ] . (27)Therefore, comparing Eq. (27) with Eq. (22), we have shown the first of Eqs. (20) to be true. ak −bk −i 1 skbki j ak ncn b* f ...2−f FIG. 19: Schematic representation for a Green’s function diagram whose top row is 1 . . . f ∗ ... ∗ . Upon removing the incomingexternal line, it contributes to Σ ∗ b [ i, f ]. Now consider the EOM for g − [ i, f ] (Eq. (10)) in expanded form. g − σ σ [ i, f ] = g − σ σ [ i, f ] − λ t [ i, f ] σ σ G ¯ σ ¯ σ [ i, i + ] − λ Φ σ σ [ i, f ]Φ σ σ [ i, f ] = t [ i, j ] σ σ a g σ a ,σ b [ j , n ] δδ V ¯ σ ¯ σ a i g − σ b σ [ n , f ] . (28)Our goal is to prove the second of Eqs. (20) using Eq. (28). To this end, we note that diagrams for Σ ∗ a [ i, f ] can besplit into four groups. Recall that a diagram for Σ a [ i, f ] is obtained from a g [ i, f ] diagram (or equivalently from a G [ i, f ] diagram with no stars on the f in the top row) by removing the incoming and outgoing external lines. Considera g [ i, f ] diagrams whose numerical representation has the following property. There are no subscripts between thenumber immediately to the left of f in the top row (which we shall denote by c ) and f (e.g. panels e), g) through3j), and n) of Fig. (17)). This implies that c has at least one star, as otherwise c must be a subscript between c and f . Therefore, the top row looks like 1 . . . c ∗ ... ∗ f . In the case that c = 1 (e.g. panels i), j), and n) of Fig. (17)), thesediagrams can be represented schematically as in Fig. (20). We denote the corresponding contribution to Σ ∗ a by Σ ∗ a . If c > ∗ a by Σ ∗ a . Comparing Fig. (18) with Fig. (20) and Fig. (19) with Fig. (21),and removing the external lines, we find thatΣ ∗ a σ σ [ i, f ] = − Σ ∗ b σ σ [ i, j ] t [ j , f ]; Σ ∗ a σ σ [ i, f ] = − Σ ∗ b σ σ [ i, j ] t [ j , f ] . (29)Here, the minus comes from rule (8) of section IV B, where there is a minus sign discrepancy between the factors( − s f − (applicable to f ∗ ... ∗ in Σ ∗ b ) and ( − s (applicable to c ∗ ... ∗ in Σ ∗ a ). Using Eq. (23) and Eq. (26), we find that − Σ ∗ a σ σ [ i, f ] = − λ σ σ G ¯ σ ¯ σ [ i, i + ] t [ i, f ]; − Σ ∗ a σ σ [ i, f ] = λ Ψ σ σ [ i, j ] t [ j , f ] . (30) k1i G−bk bk −i j fak 2fak FIG. 20: Schematic representation for Green’s function diagram whose top row is 1 ∗ ... ∗ f . Upon removing the incomingand outgoing external lines, it contributes to Σ ∗ a [ i, f ]. One can obtain Σ ∗ b [ i, f ] displayed in Fig. (18) by also removing theinteraction line exiting the point k . ski j ncn li 1 bk d−lakak −bk − a1* + a2* l d 2fm f... FIG. 21: Schematic representation for Green’s function diagram whose top row is 1 . . . c ∗ ... ∗ f . Upon removing the outgoingand incoming external lines, it contributes to Σ ∗ a [ i, f ]. One can obtain Σ ∗ b [ i, f ] displayed in Fig. (19) by also removing theinteraction line exiting the point l . Motivated by this observation, we define a new object χ σ σ [ i, f ] defined by the formulaΦ σ σ [ i, f ] = − Ψ σ σ [ i, j ] t [ j , f ] + χ σ σ [ i, f ] (31)Plugging this formula into Eq. (28), we obtain g − σ σ [ i, f ] = g − σ σ [ i, f ] − λ t [ i, f ] σ σ G ¯ σ ¯ σ [ i, i + ] + λ Ψ σ σ [ i, j ] t [ j , f ] − λχ σ σ [ i, f ] (32)Plugging Eq. (32) into the equation for Φ (Eq. (28)), we obtainΦ σ σ [ i, f ] = − t [ i, j ] σ ¯ σ g ¯ σ ¯ σ [ j , i ] δ [ i, f ] − Ψ σ σ [ i, j ] t [ j , f ] − λt [ i, j ] σ σ a g σ a σ b [ j , n ] δδ V ¯ σ ¯ σ a i χ σ b σ [ n , f ] , (33)where we have used Eq. (22) to handle the second and third terms on the RHS of Eq. (32). Comparing Eq. (33) withEq. (31), we obtain the following EOM for χ σ σ [ i, f ]. χ σ σ [ i, f ] = − t [ i, j ] σ ¯ σ g ¯ σ ¯ σ [ j , i ] δ [ i, f ] − λt [ i, j ] σ σ a g σ a σ b [ j , n ] δδ V ¯ σ ¯ σ a i χ σ b σ [ n , f ] . (34)4Comparing Eq. (32) and Eq. (30), we see that − Σ ∗ a σ σ [ i, f ] and − Σ ∗ a σ σ [ i, f ] account for the second and third termson the RHS of Eq. (32) respectively. Therefore, we now show that the remainder of the − Σ ∗ a [ i, f ] diagrams accountfor the fourth term. To this end, we consider all g [ i, f ] diagrams which do have a subscript between the number c (the number immediately to the left of f in the top row) and f in the top row. The top row now looks like 1 . . . c ... f (e.g. panels a) through d) and f) of Fig. (17)). Note that for the resulting Σ a [ i, f ] diagram to be irreducible, thenumber 1 cannot have any stars. We further subdivide this group of g [ i, f ] diagrams into 2 groups. In the first group,whose contribution to Σ ∗ a [ i, f ] shall be denoted by Σ ∗ a [ i, f ], the subscript immediately preceding f in the top row is1. The top row for these diagrams looks like 1 . . . c ... f (e.g. panels c) and f) of Fig. (17)). In the second group, whosecontribution to Σ ∗ a [ i, f ] shall be denoted by Σ ∗ a [ i, f ], the subscript immediately preceding f in the top row is not 1.The top row for these diagrams looks like 1 . . . c ...d f , where d = 1 (e.g. panels a), b), and d) of Fig. (17)). Our goalis to show that λχ σ σ [ i, f ] = Σ ∗ a σ σ [ i, f ] + Σ ∗ a σ σ [ i, f ].We do this by induction. The g [ i, f ] diagrams contributing to Σ ∗ a [ i, f ] are shown in Fig. (22). The contribution ofthis diagram becomes − λσ a σ b g (0) i a σ σ a [ τ i , τ a ] t ab g ba σ b ¯ σ a [ τ a , τ + a ] g (0) a f ¯ σ b σ [ τ a , τ f ] . (35)After removing the two external lines, we find thatΣ ∗ a σ σ [ i, f ] = − λσ ¯ σ t i b g b i ¯ σ ¯ σ [ τ i , τ + i ] δ [ i, f ] . (36) −ii a1 bff 2ba baaa aa − FIG. 22: Schematic representation for Green’s function diagram whose top row is 1 . . . c ... f . Upon removing the incoming andoutgoing external lines, it contributes to Σ ∗ a [ i, f ]. This is the analog of the Fock diagram in the standard Feynman skeletonexpansion. Thus, Σ ∗ a σ σ [ i, f ] is equal to the first term on the RHS of Eq. (34). Note that since this term contains the lowest ordercontribution to χ [ i, f ], this covers the base case of the induction. We now want to show that Σ ∗ a σ σ [ i, f ] equals thesecond term on the RHS of Eq. (34). The g [ i, f ] diagrams contributing to Σ ∗ a σ σ [ i, f ] can be represented schematicallyas in Fig. (23). Here, the reasoning is similar to that which led to Fig. (19). If the line s were contained in g jn σ a σ c [ τ k , τ n ],then the resulting Σ a (of the overall diagram) would be reducible, while if s was the bare line g (0) d f ¯ σ d σ [ τ d , τ f ], the diagramwould contribute to Σ ∗ a (see Fig. (22)). The box can be either a Σ ∗ a insertion or a Σ ∗ a insertion, but can’t be a Σ ∗ a insertion or a Σ ∗ a insertion, since in this case the diagram would contribute to Σ ∗ a (see Fig. (21)). The analyticalcontribution of Fig. (23) is − λ g (0) σ σ b [ i, k ] t [ k , j ] σ b σ a g σ a σ c [ j , n ] δδ V ¯ σ b ¯ σ a k (cid:18) Σ ∗ a σ c ¯ σ d [ n , d ] + Σ ∗ a σ c ¯ σ d [ n , d ] (cid:19) g (0)¯ σ d σ [ d , f ] . (37)Dropping the external lines, and using the inductive hypothesis, we obtainΣ ∗ a σ σ [ i, f ] = − λ t [ i, j ] σ σ a g σ a σ b [ j , n ] δδ V ¯ σ ¯ σ a i χ σ b σ [ n , f ] . (38)Combining this with Eq. (36) and comparing with Eq. (34), we find thatΣ ∗ a σ σ [ i, f ] + Σ ∗ a σ σ [ i, f ] = λχ σ σ [ i, f ] . (39)Using Eq. (39), Eq. (30), and Eq. (32), we have proven the second of Eqs. (20).5 ak −bk −i 1 skbki j ak ncn d fdd − 2f+ a4*a3* FIG. 23: Schematic representation for Green’s function diagram whose top row is 1 . . . c ...d f , where d = 1. Upon removing theincoming and outgoing external lines, it contributes to Σ ∗ a [ i, f ]. C. Diagrams in momentum space
Upon turning off the sources, all objects become translationally-invariant in both space and time. We define theFourier transform of all objects with two external points (e.g. G [ i, f ]), denoted below by the generic symbol Q [ i, f ],as Q [ i, f ] = 1 N s β X k e ik ( i − f ) Q ( k ) , (40)where N s is the number of sites on the lattice, β is the inverse temperature, k ≡ ( ~k, iω k ), and k ( i − f ) ≡ ~k · ( ~R i − ~R f ) − ω k ( τ i − τ f ). For the rest of the paper, we shall not write the explicit factor N s β that goesalong with each momentum sum. To obtain the momentum space contribution of a given g ( k ) diagram, we assignmomentum k to the outgoing and incoming external lines, and sum over the momenta of the internal lines, in sucha way that momentum is conserved at each point in the diagram. We also associate with each Green’s function linethe factor g (0) ( q ), where q is the momentum label of that line, and with each interaction line the factor − ǫ q , where q is the momentum label of that interaction line, and t [ i, f ] ≡ − P q e iq ( i − f ) ǫ q . The other rules are the same as inthe coordinate space evaluation. For example, consider the diagram in panel b) of Fig. (17), whose momentum spacelabels are displayed in Fig. (24). The momentum space contribution of this diagram is − g (0) ( k ) ǫ p g (0) ( p ) ǫ q g (0) ( q ) g (0) ( k + q − p ) g (0) ( k ) (41)where a sum over the internal momenta p and q is implied. Upon removing the external lines, we obtain the followingcontribution to Σ ∗ a ( k ), or equivalently to χ ( k ): − ǫ p g (0) ( p ) ǫ q g (0) ( q ) g (0) ( k + q − p ) . (42) k kpqk+q−pp q FIG. 24: Momentum space representation of diagram for g ( k ) from Fig. (17b). Upon removing the incoming and outgoingexternal lines, it contributes to χ ( k ). Additionally, consider the diagram for G ( k ) displayed in panel l) of Fig. (17), whose momentum space labels aredisplayed in Fig. (25). The incoming external line carries momentum k into the diagram, while the terminal pointabsorbs this momentum without transferring it to an outgoing external line. The momentum space contribution ofthis diagram is − g (0) ( k ) ǫ p g (0) ( p ) g (0) ( q ) g (0) ( k + q − p ) (43)6Upon removing the incoming external line, we obtain the following contribution to Σ ∗ b ( k ), or equivalently to Ψ( k ): − ǫ p g (0) ( p ) g (0) ( q ) g (0) ( k + q − p ) (44) k p pqk + q − p FIG. 25: Momentum space representation of diagram for G ( k ) from Fig. (17l). Upon removing the incoming external line, itcontributes to Ψ( k ). D. The vertices Λ and U In section V B, we showed that our diagrammatic series is consistent with the ECFL EOM, Eq. (10) and Eq. (11).We rewrite them here for convenience. g − [ i, m ] = ( µ − ∂ τ i − V i ) δ [ i, m ] + t [ i, m ] (1 − λγ [ i ]) − λ Φ[ i, m ] . e µ [ i, m ] = (1 − λγ [ i ]) δ [ i, m ] + λ Ψ[ i, m ]Φ[ i, m ] = − t [ i, j ] ξ ∗ . g [ j , n ] . Λ ∗ [ n , m ; i ]; Ψ[ i, m ] = − t [ i, j ] ξ ∗ . g [ j , n ] . U ∗ [ n , m ; i ] . (45) γ [ i ] = e µ ( k ) [ n , i + ] . g ( k ) [ i, n ]; Λ[ n, m ; i ] = − δδ V i g − [ n, m ]; U [ n, m ; i ] = δδ V i e µ [ n, m ] . (46)We now examine the vertices Λ σ a σ b σ c σ d [ n, m ; i ] ≡ − δδ V σcσdi g − σ a σ b [ n, m ] and U σ a σ b σ c σ d [ n, m ; i ] ≡ δδ V σcσdi e µ σ a σ b [ n, m ] in moredetail. The zeroth order vertices, also called the bare vertices, are given byΛ (0) σ a σ b σ c σ d [ n, m ; i ] = δ [ n, m ] δ [ n, i ] δ σ a σ c δ σ b σ d ; U (0) σ a σ b σ c σ d [ n, m ; i ] = 0 . (47)The higher order terms contributing to Λ σ a σ b σ c σ d [ n, m ; i ] arise from splitting a line in Σ ∗ aσ a σ b [ n, m ] through the point i .The higher order terms contributing to U σ a σ b σ c σ d [ n, m ; i ] arise from splitting a line in Σ ∗ bσ a σ b [ n, m ] through the point i .These terms can be represented schematically as in Fig. (26). * sb[n,m]ni cdii m ...b−ma) b)s[n,m]n mi cdii ba *a ba FIG. 26: Schematic diagram for the vertices. Λ σ a σ b σ c σ d [ n, m ; i ] is displayed in panel a) while U σ a σ b σ c σ d [ n, m ; i ] is displayed in panel b). From Fig. (26a), we see that in Λ σ a σ b σ c σ d [ n, m ; i ], the external points n and m accommodate an incoming and outgoingexternal Green’s function line, respectively, while the external point i accommodates an incoming external Green’sfunction line and an external interaction line (Compare with Fig. (13a)). In Fig. (26b), we see that in U σ a σ b σ c σ d [ n, m ; i ],7the external point n accommodates an incoming external Green’s function line, while the external point i accommo-dates an incoming external Green’s function line and an external interaction line. However, the external point m is theterminal point and does not accommodate any external lines. Therefore, the vertices are represented schematicallyas in Fig. (27). In the case of the bare vertex Λ (0) [ n, m, i ], the diagram in Fig. (27a) collapses onto a single point,which corresponds to the point k in Fig. (13a). a) n mi[n,m,i] ba dc ba n miU [n,m,i]b) dc an bm an FIG. 27: Schematic diagram for the vertices. Λ σ a σ b σ c σ d [ n, m ; i ] is displayed in panel a) while U σ a σ b σ c σ d [ n, m ; i ] is displayed in panel b). In Eq. (45), the self-energies Φ and Ψ are expressed in terms of the vertices Λ and U respectively. These relationshipscan be expressed diagrammatically as in Fig. (28). a) n mi n mib) ai 1ibnai 1i 2mbn [n,m,i] U [n,m,i] − −−− a1 FIG. 28: Schematic diagram for the self-energies in terms of the vertices. In panel a) Φ σ σ [ i, m ] is expressed in terms ofΛ σ a σ b σ c σ d [ n, m ; i ] and in panel b) Ψ σ σ [ i, m ] is expressed in terms of U σ a σ b σ c σ d [ n, m ; i ]. We now turn the sources off, so that we can represent the vertices in momentum space, as in Fig. (29). In the caseof Λ( p, k ), the external lines carry a total of zero momentum out of the vertex. In the case of U ( p, k ), the terminalpoint (the one with no external lines coming in or out) absorbs momentum k , and therefore the remainder of theexternal lines have to bring momentum k into the vertex. Therefore, comparing Fig. (29) with Fig. (27), the Fouriertransform of the three point vertices, denoted below by the generic symbol Q [ n, m, i ], is: Q [ n, m, i ] = X kp e ipn e − ikm e i ( k − p ) i Q ( p, k ) = X kp e ip ( n − i ) e ik ( i − m ) Q ( p, k ) . (48)Furthermore, there are only four non-zero spin configurations contributing to the vertex. These are Q (1) ≡ Q σσσσ , Q (2) ≡ Q σσ ¯ σ ¯ σ , Q (3) ≡ Q σ ¯ σσ ¯ σ , and Q (4) ≡ Q σ ¯ σ ¯ σσ . These four spin configurations are related by the equation Q (1) − Q (2) = Q (3) + Q (4) . (49)We shall now state the rules for computing the Q ( i ) and derive Eq. (49). Recall that to obtain a diagram for Λ ( U ),we must split a line in the self-energy Σ ∗ a (Σ ∗ b ). This will give us an extra Green’s function line in the diagram, andwe must assign momenta to the external lines as indicated in Fig. (29), at the same time summing over the momentaof the internal lines in such a way as to conserve momentum at each point of the diagram. Also recall from sectionIV B that the Green’s function lines in the diagrams for Σ ∗ a (Σ ∗ b ) are partitioned into anywhere between 0 and F s spinloops, where the zeroth loop contains the lines with the labels σ and σ . The spins carried by the Green’s functionlines in a single loop are allowed to alternate. However, the spin carried by each Green’s function line in the loop isdetermined by that of any one of them (in the case of the zeroth loop it is the fixed spin σ ).Now, in the case that the line split in going from Σ ∗ → Q is from a loop which is not the zeroth loop, the resultingvertex diagram contributes only to Q (1) and Q (2) with a factor of relative to the contribution of the original diagramto Σ ∗ . In the case that the line split in going from Σ ∗ → Q is from the zeroth loop, the line split could either carryspin σ in the original Σ ∗ diagram or spin ¯ σ . In the case of the former, the resulting vertex diagram contributes toboth Q (1) and Q (3) with a factor of 1 relative to the contribution of the original diagram to Σ ∗ . In the case of the8latter, the resulting vertex diagram contributes to Q (2) with a factor of 1, and to Q (4) with a factor of ( − ∗ . Eq. (49) immediately follows. Note that in the Feynman diagrams,we have the simpler situation in which all of the Green’s function lines in a single spin loop (also referred to as Fermiloop), carry the same spin . Then, the very last case described above becomes impossible, Q (4) →
0, and Eq. (49)reduces to the standard Nozi`eres relation Q (1) − Q (2) = Q (3)19 .Following Ref. (9), we define Q ( a ) ≡ Q (2) − Q (3) . Fourier transforming Eq. (45), we obtain:Φ( k ) = X p ǫ p g ( p )Λ ( a ) ( p, k ); Ψ( k ) = X p ǫ p g ( p ) U ( a ) ( p, k ) (50)These relations are represented diagrammatically in Fig. (30) a) b)bk aap (p,k) dc ba U (p,k) dc ba pqk+q−p qk+q−p
FIG. 29: Schematic diagram for the vertices in momentum space. Λ σ a σ b σ c σ d ( p, k ) is displayed in panel a) while U σ a σ b σ c σ d ( p, k ) isdisplayed in panel b). a) b) Uk kpp (p,k) (p,k)kp p (a) (a) FIG. 30: Schematic diagram for the self-energies in terms of the vertices. In panel a) Φ( k ) is expressed in terms of Λ ( a ) ( p, k )and in panel b) Ψ( k ) is expressed in terms of U ( a ) ( p, k ). E. Skeleton diagrams
Consider the diagrammatic expansion for the irreducible self-energies that we have been using thus far, in whicheach diagram is composed of bare Green’s function lines g (0) [ i, f ], and hopping matrix elements t if . We aim toreorganize this expansion in such a way that we only keep a subset of these diagrams, in which we replace each bareGreen’s function line g (0) [ i, f ], by the full auxiliary Green’s function g [ i, f ], thereby accounting for the diagrams whichwe discarded. We shall now define this subset of diagrams, which is referred to as the skeleton diagrams.The skeleton diagrams are those diagrams in which one can’t separate a self-energy insertion Σ a from the rest ofthe diagram by cutting two Green’s function lines. For example, consider the Σ ∗ a diagrams in Fig. (31) (the sameconsiderations will apply to Σ ∗ b diagrams). From left to right, these are the irreducible self-energies correspondingto the g diagrams in Fig. (17b), Fig. (17c), and Fig. (12c). We see that the Σ ∗ a diagram in panel b) of Fig. (31)is a non-skeleton diagram, since by cutting the two Green’s function lines labeled by the letter c , we isolate the Σ a self-energy insertion enclosed in the box. In contrast, the Σ ∗ a diagram in panel a) of Fig. (31) is a skeleton diagram,since it is impossible to isolate a Σ a insertion by cutting two Green’s function lines. Finally, the diagram in panelc) of Fig. (31) is also a skeleton diagram. Furthermore, we see that by placing the self-energy insertion enclosed inthe box into the Green’s function line of the diagram in Fig. (31c), we reproduce the diagram in Fig. (31b). Since afull auxiliary Green’s function line consists of an arbitrary self-energy insertion Σ a surrounded by two bare Green’sfunction lines g (0) , we see that the whole series is reproduced by keeping only the skeleton diagrams and making thesubstitution g (0) [ i, f ] → g [ i, f ].9 cca) b) c)s FIG. 31: Examples of skeleton and non-skeleton diagrams for the irreducible self-energy Σ ∗ a . The diagram in panel a) is askeleton diagram. The diagram in panel b) is not, since we can isolate a self-energy insertion by cutting the two lines labeledby c . The non-skeleton diagram in panel b) can be obtained from the skeleton diagram in panel c) by inserting the self-energyinsertion enclosed by the box into the Green’s function line. However, by splitting the line labeled by s in the diagram in panelb), through another external point, we obtain a skeleton diagram for the vertex Λ. Now, consider the vertices Λ[ n, m ; i ] and U [ n, m ; i ]. Recall from Fig. (26), that these correspond to splitting aGreen’s function line through the point i in Σ ∗ a [ n, m ] and Σ ∗ b [ n, m ], respectively. How do we obtain the skeletondiagrams for the vertices? A naive guess would be that we do so by splitting a Green’s function line in the skeletondiagrams for the irreducible self-energies. However, this is only partially correct. To see this, consider again thenon-skeleton Σ ∗ a diagram in panel b) of Fig. (31). If we choose to split either of the two lines labeled by c , thenwe leave the self-energy insertion surrounded by the box intact, and the resulting diagram for Λ is a non-skeletondiagram. However, if we split the Green’s function line labeled by s , this breaks up this self-energy insertion, andleads to a skeleton diagram for Λ.Taking this reasoning a step further, consider the diagram for Σ ∗ a in panel a) of Fig. (32). This diagram can beobtained from the diagram in Fig. (31b) by inserting the self-energy insertion enclosed by the box into the line labeledby s in Fig. (31b). Once again, if we split any line other than the one labeled by s in Fig. (32a), the resulting diagramfor Λ will be a non-skeleton diagram, while if we split the line labeled by s , the resulting diagram for Λ will be askeleton diagram. Meanwhile, for the Σ ∗ a diagram in Fig. (32b), obtained from the diagram in Fig. (31b) by puttinga reducible self-energy insertion into the line labeled by s in Fig. (31b), it is not possible to split any line in such away that the resulting diagram for Λ will be a skeleton diagram. sa) b) FIG. 32: In this case both Σ ∗ a diagrams displayed in panels a) and b) are non-skeleton diagrams. However, the diagram inpanel a) contains only irreducible self-energy insertions, while the one in panel b) contains a reducible self-energy insertion.One can obtain a skeleton diagram for the vertex Λ only by splitting the line labelled by s in the diagram in panel a). It isimpossible to obtain a skeleton diagram for the vertex Λ from the diagram in panel b) regardless of which line we split. Therefore, we see that to construct the skeleton diagrams for Λ[ n, m ; i ] ( U [ n, m ; i ]), we have to use the followingprocedure. Take a skeleton diagram for Σ ∗ a [ n, m ] (Σ ∗ b [ n, m ]), and insert into at most one line of this diagram, a skeletondiagram for Σ ∗ a . Then, insert into at most line of that diagram, a skeleton diagram for Σ ∗ a , and so on. This producesa sequence of skeleton diagrams for the irreducible self-energies. Then, in the last skeleton diagram of the sequence,split a single Green’s function line through the point i . This procedure is represented schematically (for the case ofΛ[ n, m ; i ]) in Fig. (33a).Now consider the part of Fig. (33a) enclosed by the second box (counting from the very outer box). This is itself askeleton diagram for the vertex Λ[ w, v ; i ], where w and v are internal variables. Therefore, we see that one can obtainthe skeleton expansion for Λ[ n, m ; i ] ( U [ n, m ; i ]) from the skeleton expansion for Σ ∗ a [ n, m ] (Σ ∗ b [ n, m ]) by replacing ineach skeleton diagram for Σ ∗ a [ n, m ] (Σ ∗ b [ n, m ]), a single Green’s function line g [ x, y ], with g [ x, w ] . Λ[ w , v ; i ] . g [ v , y ],where Λ[ w , v ; i ] is the full vertex. This is represented schematically in Fig. (33b). The case in which there is onlyone box in Fig. (33a) corresponds to plugging in the bare vertex into Fig. (33b).0 *a,s *a,s. . . *a,s *a,sa) b)*a,s i in m n m FIG. 33: Panel a) demonstrates the general procedure for obtaining a skeleton diagram for the vertex Λ from a Σ ∗ a diagramconsisting of a sequence of Σ ∗ a skeleton diagrams. The original Σ ∗ a diagram is itself a skeleton diagram only if there is only oneskeleton diagram in the sequence, i.e. the Σ ∗ a diagram in question. If we remove the outermost box in panel a), we are stillleft with a general skeleton diagram for the vertex Λ. Therefore, to obtain a skeleton diagram for Λ, one must insert the fullvertex Λ into a green’s function line of a skeleton diagram for Σ ∗ a . This is displayed in panel b). In the case that the originalΣ ∗ a diagram in panel a) is itself a skeleton diagram (i.e. there is only one skeleton diagram in the sequence), the Λ vertex inpanel b) is a bare vertex. We now have three skeleton expansions. The first is the original skeleton expansion of the self-energies in terms ofthe auxiliary Green’s function. g − ≡ g − [ g ]; e µ ≡ e µ [ g ] , (51)The second is the skeleton expansion for the vertices in terms of the auxiliary Green’s function. This is the skeletonexpansion represented in Fig. (33a). Λ ≡ Λ[ g ]; U ≡ U [ g ] . (52)The third is the skeleton expansion for the vertices in terms of the auxiliary Green’s function and the full vertex Λ.This is the skeleton expansion represented in Fig. (33b).Λ ≡ Λ[ g , Λ];
U ≡ U [ g , Λ] . (53)Using the diagrammatic rules developed here, we have access to all three of these skeleton expansions at any order.However, in the absence of these rules, we could derive the terms in these skeleton expansions by using Eqs. (51), (52),(53), and (45) in the following manner. Suppose that we have the skeleton expansions in Eqs. (51) - (53) through m th order in λ . Then, plugging the m th order term of the skeleton expansion from Eq. (52) into Eq. (45) yields the m + 1 st order term of the skeleton expansion in Eq. (51). Then, applying the rule g → g Λ g to the m + 1 st order term of theskeleton expansion in Eq. (51), yields the m + 1 st order contribution to the skeleton expansion in Eq. (53). Finally,plugging the k th order term of the skeleton expansion from Eq. (52) (0 ≤ k ≤ m ) into the m + 1 − k th term of theskeleton expansion from Eq. (53) yields the m + 1 st order term of the skeleton expansion from Eq. (52), after whichwe can iterate the process again. This process starts at zeroth order by plugging the bare vertex into Eq. (45) andcalculating the first order contribution to the skeleton expansion in Eq. (51), and so on. This is the approach usedin the original ECFL papers , and reviewed in section II. It reveals the power of the Schwinger approach in thatit enables one to bypass the bare series and work directly with the skeleton expansion. However, the utility of thediagrams developed here is that they enable one to obtain the contribution of a given order directly, without iteration,and also to visualize all the higher order terms diagrammatically, therefore facilitating diagrammatic re-summations. VI. PUTTING J BACK INTO THE EQUATIONS
Let us rewrite Eq. (4) in the form of an integral equation as in Eq. (17), but this time keeping J . G σ ,σ [ i, f ] = g (0) σ ,σ [ i, f ] − λ g (0) σ ,σ b [ i, f ] σ b σ G ¯ σ , ¯ σ b [ f, f + ] − λ g (0) σ ,σ b [ i, k ] (cid:18) − t [ k , j ] σ b σ a G ¯ σ a , ¯ σ b [ k , k + ] G σ a ,σ [ j , f ] + t [ k , j ] σ b σ a δδ V ¯ σ b , ¯ σ a k G σ a ,σ [ j , f ] (cid:19) , − λ g (0) σ ,σ b [ i, k ] J [ k , j ] σ b σ a G ¯ σ a , ¯ σ b [ j , j + ] G σ a ,σ [ k , f ] − J [ k , j ] σ b σ a δδ V ¯ σ b , ¯ σ a j G σ a ,σ [ k , f ] ! . (54)The λ -expansion of Eq. (54) is given by the same set of rules as in section IV B, with the only difference being thatnow each vertex can be either a t -vertex or a J -vertex. Comparing the second and third lines on the RHS of Eq. (54),1we see that the J -vertices can be obtained from the t -vertices in Fig. (13) by moving the line labeled by σ a from thepoint j to the point k , and moving all lines but the one labeled by σ b from the point k to the point j . The J -verticesare displayed in Fig. (34). They are more reminiscent of the standard Feynman diagram vertices. a) b)c) d)k jbk a bk jbk k jbk a bk jbk − ak − ak − bk −ak ak − ck −bk ck akak a c a dakbk ak − ck bkck dk − dk FIG. 34: The J -vertices in the diagrams of the λ expansion. They are more reminiscent of the Feynman diagram vertices thanthe t -vertices displayed in Fig. (13). The two types of vertices can be obtained from each other by interchanging lines betweenthe two points of the vertex. Now, let us compare an arbitrary t -vertex and an arbitrary J -vertex in momentum space. The t -vertex is shownin panel a) of Fig. (35), while the J -vertex is shown in panel b). Conserving momentum at each point of the t and J vertices yields the relation p a + n X m =1 p m = p b + n X m =1 p m − . (55)In Fig. (35a), the interaction line contributes a factor of ǫ p b , while in Fig. (35b), the interaction line contributes afactor of J p a − p b . At first, it seems as though for each diagram with i interaction vertices, we must now draw 2 i separate diagrams, since for each vertex we must decide whether it will be a t -vertex or a J -vertex. For example,consider the diagram in Fig. (25), also displayed in Fig. (36a), in which the interaction vertex is a t -vertex. In Fig.(36b), it is drawn with a J -vertex. However, we see that the Green’s function lines in both diagrams have the samemomentum labels, and the only difference is the momentum label of the interaction line. This is because the twothings which determine the momentum labels of the Green’s function lines are • (1) the interconnections (via Green’s function lines) between the interaction vertices (irrespective of where onthese vertices these lines appear), • (2) Eq. (55).Since the J -vertex simply reshuffles the lines on the t -vertex, and Eq. (55) applies equally well to both types ofvertices, both (1) and (2) are unaffected by the choice of t vertex vs. J vertex. Therefore, we can choose to use eitherthe diagram in Fig. (36a) or the diagram in Fig. (36b) if we associate with the interaction line in each diagram thefactor ǫ p + J k − p . In general, we can construct diagrams either from the vertices in Fig. (35a) (as we have alreadybeen doing), or from the vertices in Fig. (35b) (which would be more reminiscent of the Feynman diagrams), as longas we associate with each interaction vertex the factor ǫ p b + J p a − p b . a) b)pa ... pa ...−pap2n−1 pb pbp1 p2p2n pb p1 p2p2np2n−1pb FIG. 35: The t and J vertices are displayed in momentum space in panels a) and b) respectively. For each interaction vertexof a diagram, we can choose to use either one as long as we associate with it the factor ǫ p b + J p a − p b . p pqk + q − pa) k k p k + q − pqk−pb) FIG. 36: The G ( k ) diagrams drawn in panel a) and panel b) correspond to the same diagram. The one in panel a) is drawnusing a t -vertex, while the one in panel b) is drawn using a J -vertex. In both cases, we associate the factor ǫ p + J k − p withthe interaction vertex. The ECFL equations with J included are given as follows . g − [ i, m ] = ( µ − ∂ τ i − V i ) δ [ i, m ] + t [ i, m ] (1 − λγ [ i ]) + λ J [ i, j ] γ [ j ] δ [ i, m ] − λ Φ[ i, m ] , e µ [ i, m ] = (1 − λγ [ i ]) δ [ i, m ] + λ Ψ[ i, m ] , Φ[ i, m ] = L [ i, n ] . g − [ n , m ]; Ψ[ i, m ] = − L [ i, n ] . e µ [ n , m ] , (56)where the operator L is given by: L σ σ [ i, m ] = t [ i, j ] σ σ a g σ a σ [ j , m ] δδ V ¯ σ ¯ σ a i − J [ i, j ] σ σ a g σ a σ [ i, m ] δδ V ¯ σ ¯ σ a j . (57)Using the same decomposition as in Eq. (31), i.e.Φ[ i, m ] = − Ψ[ i, j ] t [ j , m ] + χ [ i, m ] , (58)we find that χ [ i, m ] = L [ i, n ] . g − [ n , m ] + λ J [ m, k ] L [ i, m ] γ [ k ] − λ L [ i, n ] χ [ n , m ] , (59)where ( L [ i, n ] . g − [ n , m ]) σ σ = t [ i, j ] σ σ g ¯ σ ¯ σ [ j , i ] δ [ i, m ] − J [ i, m ] σ σ g ¯ σ ¯ σ [ i, m ] . (60)Finally, we note that the equations associated with Fig. (30) now becomeΦ( k ) = X p ( ǫ p + 12 J k − p ) g ( p )Λ ( a ) ( p, k ); Ψ( k ) = X p ( ǫ p + 12 J k − p ) g ( p ) U ( a ) ( p, k ) . (61) VII. FINITE ORDER CALCULATIONSA. Zeroth through third order calculation
In this section, we compute the skeleton expansion for the objects γ , Ψ, and χ through second order in λ inmomentum space. As can be seen from Eq. (63) below, this yields the skeleton expansion for g − and e µ through thirdorder in λ . Before proceeding with this computation, we follow Ref. (10) in introducing a second chemical potential u into the theory. As explained in Ref. (10), there is a so-called shift identity of the t − J model, which states thatadding an onsite term to the hopping affects G only through a shift of the chemical potential µ . However, the sameis not true of the constituent factors g and e µ , which will be affected by such a shift. To remedy this, in Ref. (10), thesecond chemical potential u is introduced directly into the definitions of g − and e µ (Eq. (56)) through the formula t [ i, j ] → t [ i, j ] + u δ [ i, j ] in every term but the t [ i, f ] term in the equation for g − [ i, f ]. Now, an onsite shift in thehopping affects g and e µ only through a shift in the second chemical potential u . Moreover, the fact that G will not3be affected for any value of u (other than through a shift of the original chemical potential µ ) is a consequence ofthe shift identity. Furthermore, the two chemical potentials µ and u can now be used to satisfy the two sum rules X k G ( k ) = n X k g ( k ) = n . (62)The first of these ensures the correct particle sum-rule for the physical electrons. The second one states that theauxiliary fermions must satisfy the same particle sum-rule as the physical ones. We can think of the Hubbardoperator X σi = c iσ (1 − n i ¯ σ ) as representing the physical fermions, and the canonical operator c iσ as representing theauxiliary fermions. Since, the number operator is a charge neutral object, charge conservation implies that the physicaland auxiliary fermions must satisfy the same particle sum-rule. As a consequence of this, the physical electrons havea Fermi-surface which complies with the Luttinger-Ward volume theorem (see Ref. (9) where these sum rules wereoriginally introduced and their implications discussed).We now proceed to present the diagrams and analytical expressions for g − and e µ through third order in λ . Takingthe Fourier transform of Eq. (56) and Eq. (9), and using Eq. (58), we obtain g − ( k ) = iω k + µ ′ − ( ǫ k − u e µ ( k ) − λχ ( k ) ,µ ′ = µ − u λ γJ e µ ( k ) = (1 − λγ ) + λ Ψ( k ) , G ( k ) = g ( k ) e µ ( k ) , (63)where J is the zero-momentum component of the Fourier transform of J ij . Our strategy is to compute the skeletonexpansion for γ , Ψ, and χ through second order in λ (i.e. γ = γ (0) + γ (1) + γ (2) , etc.) After plugging in the expressionsfrom this skeleton expansion into Eq. (63), we must set λ = 1, and solve the resulting integral equations. The twoLagrange multipliers µ and u are then determined by the sum rules in Eq. (62).In Fig. (37), we have drawn the skeleton diagrams for γ (which is just a constant when the sources are off) throughsecond order in λ . Therefore, γ is the sum of the following terms a ) n b ) − λ (cid:16) n (cid:17) c ) λ (cid:16) n (cid:17) d ) − λ X plq g ( p ) g ( l ) g ( q ) g ( p + l − q )( ǫ q − u J p − q ) . (64) a) b) c) d) FIG. 37: Second order skeleton expansion for γ . Only the diagram in panel a) is a standard Feynman diagram. γ (0) is givenby the diagram in panel a), γ (1) is given by the diagram in panel b), and γ (2) is given by diagrams in panels c) and d). Weconserve momentum at each interaction vertex as indicated in Figs. (35) and (36). In Fig. (38), we have done the same for Ψ( k ). Therefore, Ψ( k ) is the sum of the following terms4 a ) − λ X pq g ( p ) g ( q ) g ( k + q − p )( ǫ p − u J k − p ) b ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( k + q − l )( ǫ p − u J k − p )( ǫ l − u J p − l ) c ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( q + l − p )( ǫ p − u J k − p )( ǫ l − u J p − l ) d ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( p + l − q )( ǫ p − u J k − p )( ǫ q − u J p − q ) e ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( k + l − p )( ǫ p − u J k − p )( ǫ q − u J l − q ) f ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( l + p − k )( ǫ p − u J k − p )( ǫ l − u J p − k ) g ) λ n X pq g ( p ) g ( q ) g ( k + q − p )( ǫ p − u J k − p ) (65) a) b) c)d) e)f) g) FIG. 38: Second order skeleton expansion for Ψ( k ). All diagrams but the one in panel g) are standard Feynman diagrams(with one interaction line set to unity). Ψ (1) is given by the diagram in panel a), and Ψ (2) is given by the diagrams in panelsb) through g). We conserve momentum at each interaction vertex as indicated in Figs. (35) and (36). The skeleton diagrams for χ ( k ) have been split into two groups. Those drawn in Fig. (40), whose contribution willbe denoted by χ B ( k ), can be obtained from the Ψ( k ) diagrams in Fig. (38) by attaching an interaction line to theterminal point of those diagrams. Due to the decomposition Eq. (58), this interaction line will contribute only a J term, but no ǫ term, to the expression for χ B ( k ). The rest of the χ ( k ) diagrams, whose contribution will be denotedby χ A ( k ), are drawn in Fig. (39). Then, χ ( k ) = χ A ( k ) + χ B ( k ), where χ A ( k ) is the sum of the terms in Eq. (66) and5 χ B ( k ) is the sum of the terms in Eq. (67). a ) − X p g ( p )( ǫ p − u J k − p ) b ) − λ X pq g ( p ) g ( q ) g ( k + q − p )( ǫ p − u J k − p )( ǫ q − u J p − q ) c ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( l + q − p )( ǫ p − u J k − p )( ǫ l − u J p − l )( ǫ l + q − p − u J p − q ) d ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( k + q − l )( ǫ p − u J k − p )( ǫ l − u J p − l )( ǫ q − u J l − q ) e ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( k + l − p )( ǫ p − u J k − p )( ǫ l − u J p − l )( ǫ q − u J l − q ) f ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( p + l − q )( ǫ p − u J k − p )( ǫ q − u J p − q )( ǫ l − u J k + q − p − l ) g ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( k + l − p )( ǫ p − u J k − p )( ǫ l − u J p − l )( ǫ k + l − p − u J p − k )(66) a) b) c)d) e)f) g) FIG. 39: Second order skeleton expansion for χ A ( k ). These diagrams are independent of those for Ψ( k ). All diagrams arestandard Feynman diagrams. The diagram in panel a) contributes to χ (0) , the diagram in panel b) contributes to χ (1) , and thediagrams in panels c) through g) contribute to χ (2) . We conserve momentum at each interaction vertex as indicated in Figs.(35) and (36). a ) − λ X pq g ( p ) g ( q ) g ( k + q − p )( ǫ p − u J k − p ) J p − k b ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( k + q − l )( ǫ p − u J k − p )( ǫ l − u J p − l ) J l − k c ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( q + l − p )( ǫ p − u J k − p )( ǫ l − u J p − l ) J l − k d ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( p + l − q )( ǫ p − u J k − p )( ǫ q − u J p − q ) J q − p e ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( k + l − p )( ǫ p − u J k − p )( ǫ q − u J l − q ) J p − k f ) − λ X pql g ( p ) g ( l ) g ( q ) g ( k + q − p ) g ( l + p − k )( ǫ p − u J k − p )( ǫ l − u J p − k ) J p − k g ) λ n X pq g ( p ) g ( q ) g ( k + q − p )( ǫ p − u J k − p ) J p − k (67) a)d)f) c)b) e)g) FIG. 40: Second order skeleton expansion for χ B ( k ), which vanishes when J = 0. These diagrams can be obtained from thosefor Ψ( k ) in Fig. (38) by adding an interaction line to the terminal point of those diagrams. However, this interaction linecontributes only a factor of J , and not a factor of ǫ . All diagrams but the one in panel g) are standard Feynman diagrams. Thediagram in panel a) contributes to χ (1) , and the diagrams in panels b) through g) contribute to χ (2) . We conserve momentumat each interaction vertex as indicated in Figs. (35) and (36). B. High frequency limit
We know from the anti-commutation relations for the Hubbard X operators, that the high frequency limit of theGreen’s function is lim iω k →∞ G ( k ) = − n iω k . From Eq. (63), we see that the high frequency limit of the Green’s functioncan also be expressed as lim iω k →∞ G ( k ) = − λγiω k . Since γ = P k G ( k ) = n in the exact theory, after setting λ = 1 the twoexpressions for the high frequency limit are equivalent.From Eq. (63), we see that to obtain g − ( k ) and e µ ( k ) to m th order in λ , we must calculate γ , Ψ( k ), and χ ( k ) toorder m −
1. If we are doing this using the bare expansion, then in order to satisfy the sum rules in Eq. (62) order byorder, we must also expand the two chemical potentials µ and u in λ . µ = µ (0) + µ (1) + . . . u = u (0)0 + u (1)0 + . . . , (68)where µ (0) is zeroth order in λ , µ (1) is first order in λ , etc. Denoting g , e µ , γ , Ψ, and χ by the generic symbol Q , andplugging the expansions from Eq. (68) into the bare expansion for Q ( m ) = Q ( m ) ( µ, u ), the latter is rearranged withthe various orders being mixed due to the expansion of the chemical potentials. Then, we can solve for the various7quantities µ (0) , µ (1) , etc such that in the rearranged series for γ ( m ) = γ ( m ) ( n ) and g ( m ) = g ( m ) ( n ), γ ( m ) = δ m, n X k g ( m ) ( k ) = δ m, n . (69)Then, substituting the expression for γ ( m ) back into Eq. (63), we see that only G (0) ( k ) and G (1) ( k ) contribute to thehigh-frequency limit of the Green’s function, and that lim iω k →∞ G ( k ) = − λγiω k = − λ n iω k .In the skeleton expansion, the situation is different. In this case, after we set λ = 1, the diagrams from all ordersin the skeleton expansion are mixed together on equal footing to generate one integral equation which together withthe sum rules in Eq. (62) determines g , µ , and u . The other objects are then obtained from these. In this case,if the skeleton expansions for γ , Ψ( k ), and χ ( k ) have been carried out to m − st order before being plugged intoEq. (63), then the sum rule Eq. (62) implies that (after setting λ = 1) P ml =0 γ ( l ) = n . However, from Eq. (63), thehigh frequency limit is given by lim iω k →∞ G ( k ) = − P m − l =0 γ ( l ) iω k . Therefore, the error in the high frequency limit is equal to γ ( m ) , and we have that lim iω k →∞ G ( k ) = 1 − n + γ ( m ) iω k . (70)This error vanishes as m → ∞ . C. Analysis of the λ expansion: Feynman type diagrams and non-Feynman diagrams. The λ series for G differs from the Feynman series for G in two fundamental ways. The first is the presence of theterm − λγ + λ Ψ( k ) in the numerator of G ( k ). In the Feynman series, this term is absent. To discuss the second one,let us identify λγ with the Hartree term in the Feynman diagrams, and λ Φ with all self-energy diagrams other thanthe Hartree term. Ψ forms a subset of Φ (except for a missing interaction line which is not important for the presentdiscussion), and hence all considerations which apply to Φ will apply equally well to Ψ. Hence, the second importantdifference is that there are diagrams which contribute to λγ which do not contribute the Hartree term of the Feynmanseries, and there are diagrams that contribute to λ Φ which do not contribute to the other self-energy diagrams of theFeynman series.From Fig. (37), we can see that the first order λγ diagram is exactly the Hartree term of the Feynman series,while the others are all diagrams which do not contribute to the Hartree term of the Feynman series. However, fromFig. (39) and Fig. (40), we can see that the only diagram in the 3rd order skeleton expansion for λ Φ which is not aFeynman diagram, is diagram g) in Fig. (40) (Feynman diagrams are the same order in λ as they are in the interaction,while non-Feynman diagrams are not). Therefore, the deviation of λ Φ and λ Ψ from the Feynman series grows ratherslowly as compared with the growth of the series itself. Moreover, if we consider the fact that the infinite series for γ must sum to n , we see that to “leading order”, the only difference between the λ series and the Feynman seriesis the presence of the term − λγ + λ Ψ( k ) in the numerator of G ( k ). This leads us to the point of view taken in thephenomenological ECFL , in which γ → n , and the self-energies Ψ( k ) and Φ( k ) are given simple Fermi-liquidforms. Then, the main correction to Fermi-liquid behavior is not seen as coming from the self-energies themselves,but from the interplay between the numerator and denominator of the single-particle Green’s function. VIII. CONNECTION WITH ZAITSEV-IZYUMOV FORMALISM
The Zaitsev-Izyumov formalism is a technique for doing an expansion in t and J around the atomic limit of thet − J model (given by t → J → t and J must necessarily appear with a factor of β . The diagrams of this series give rise to the sametwo self-energy structure for the single-particle Green’s function as found in ECFL. In particular, Eq. (3.6) of Ref. (8)reads G σ = h F σ i + ∆ σ ( G σ ) − − Σ σ . (71)We can make the identifications h F σ i → − γ ; ∆ σ → Ψ( k ); ( G σ ) − → g − ; Σ σ → − ǫ k γ + Φ( k ) . (72)8As is the case in the λ series, the fundamental object in the Zaitsev-Izyumov high-temperature series is the auxiliaryGreen’s function g .The main difference between the two series is the dimensionless expansion parameter. In the case of ECFL, it isthe continuity parameter λ . In the case of the high-temperature series, it is βt and βJ . To see this more explicitly,consider the simplest diagram in both series, which is the zeroth order diagram for γ . In ECFL, this is the diagramin Fig. (37a). In Ref. (8), it is represented by a dot. The relationship between the two is shown in Fig. (41). Inthis figure, the dashed line indicates an atomic limit auxiliary Green’s function g t → ,J → ( iω k ) = iω k + µ . The big dotindicates the atomic limit value of γ , i.e. γ t → ,J → = ρ , where ρ = e βµ e βµ is the atomic limit density. The wigglyline indicates a hoping ǫ k . Finally, the solid line indicates the bare auxiliary Green’s function g (0) ( k ) = iω k + µ − ǫ k .In panel a), the zeroth order γ from the high-temperature series is expanded as an infinite series in λ . Here, eachloop corresponds to P iω k g t → ,J → ( iω k ) = ρ − ρ , and there is a minus sign between the successive terms of the series.Summing the geometric series, we find that ρ − ρ · ρ − ρ = ρ . In panel b), the zeroth order γ from the λ series isexpanded as an infinite series in the hopping ǫ k . This gives the geometric series P k g (0) ( k ) = P k P ∞ n =0 ǫ nk ( iω k + µ ) n +1 .We see that to get from the high-temperature series to the λ series, one would have to break up all atomic limitobjects into an infinite series in terms of λ , and replace every atomic limit auxiliary Green’s function with a barepropagating one.We can summarize the fundamental difference between the two approaches as follows. In the case of zero magneticfield, the high-temperature series is an expansion around the atomic limit, i.e. an exponentially degenerate manifoldof states, without giving preference to any one of them. In doing so, it is difficult to recover the adiabatic continuityaspect of physics relating to the Fermi-surface and the Luttinger-Ward volume theorem . In contrast, ECFL buildsthe Fermi-surface into the λ expansion at zeroth order, by expanding around the free Fermi gas and by maintainingcontinuity in λ . Finally, by enforcing that the number of auxiliary fermions equals the number of physical onesthrough the second chemical potential u , ECFL is able to satisfy the Luttinger-Ward volume theorem. = + + + . . .a) b) = + + + . . . FIG. 41: In panel a), the zeroth order γ diagram from the high-temperature series (the big dot) is expanded as an infiniteseries in λ . The dashed lines indicate auxiliary Green’s functions in the atomic limit g t → ,J → ( iω k ). In panel b), the zerothorder γ diagram from the λ series is expanded as an infinite series in the hopping. The solid line indicates a bare propagatingauxiliary Green’s function g (0) ( k ), while the wavy line indicates the hopping ǫ k . IX. CONCLUSION
In conclusion, starting with the λ expansion as defined through iteration of the Schwinger EOM around the freeFermi gas , we derived a set of diagrammatic rules to calculate the n th order contribution to the physical Green’sfunction G in the t - J model. The resulting diagrams suggested the need for two self-energies, which we denotedby Σ a and Σ b . Using the Schwinger equations of motion defining the ECFL objects, g , e µ , γ , Φ, and Ψ, we deriveddiagrammatic rules for calculating these objects and found that they could be related simply to Σ ∗ a and Σ ∗ b , theirreducible parts of Σ a and Σ b . We also discovered diagrammatically that Ψ diagrams are simply a subset of theΦ diagrams, with an interaction line missing. Denoting the remainder of the Φ diagrams by the symbol χ , thisimplied the expression Φ( k ) = ǫ k Ψ( k ) + χ ( k ). We had already found this to be the case in the limit of infinite spatialdimensions with χ ( k ) → χ ( iω k ) and Ψ( k ) → Ψ( iω k ) in Ref. (23), and here we generalized it to finite dimensions.We also derived the Schwinger EOM for the object χ . We derived diagrammatic rules for the three point verticesΛ and U , defined as the functional derivatives of g − and e µ respectively, with respect to the source. We deriveda generalized Nozi`eres relation for these vertices, which differs from the standard one for the Feynman diagrams.We then introduced skeleton diagrams into our series, thereby allowing us to make the connection with the iterativeexpansion of the Schwinger equations of motion (as done in Refs. (9) and (10)), which deals exclusively with skeletondiagrams.9We then derived the third order skeleton expansion for g and e µ . Previously, this had been done only up to secondorder. We then discussed the error in the high-frequency limit incurred in the skeleton expansion carried out toany order in λ . We also discussed the “deviation” of the λ series from the Feynman series, thereby justifying on aqualitative level, the phenomenological ECFL , which has already been successful in explaining lines shapes foundboth from ARPES experiments , and from DMFT calculations . Finally, we discussed the connection between ECFLand the Zaitsev-Izyumov high-temperature series. We found that while both formalisms dealt with the projectionof double occupancy by introducing two self-energies, they had fundamentally different approaches to dealing withthe problem of the Fermi-surface. While the high-temperature series is an expansion around a completely degeneratemanifold of states, ECFL makes an adiabatic connection with the Fermi-surface and preserves the Luttinger-Wardvolume theorem.Our main motivation in deriving these diagrammatic rules is that they will allow the λ expansion to be evaluatedto high orders using powerful numerical techniques such as diagrammatic Monte Carlo, and also that the intuitiongained from the diagrams themselves could facilitate infinite re-summations guided by some physical principles. X. ACKNOWLEDGEMENTS
This work was supported by DOE under Grant No. FG02-06ER46319. The authors thank Professor AntoineGeorges, CPHT-Ecole Polytechnique and Coll`ege de France, supported in part by the DARP/MURI-OLE program,for their warm hospitality, where this work was completed. P. W. Anderson, Science , 1196 (1987); The Theory of Superconductivity , Princeton University Press, Princeton,NJ,1997; arXiv:0709.0656 (unpublished). B. S. Shastry, Phys. Rev.
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