Self-consistent Green’s functions formalism with three-body interactions

Arianna Carbone,1 Andrea Cipollone,2 Carlo Barbieri,2 Arnau Rios,2 and Artur Polls1

1Departament d’Estructura i Constituents de la Materia and Institut de Ciencies del Cosmos,Universitat de Barcelona, E-08028 Barcelona, Spain

2Department of Physics, Faculty of Engineering and Physical Sciences,University of Surrey, Guildford, Surrey GU2 7XH, UK

(Dated: October 15, 2013)

We extend the self-consistent Green’s functions formalism to take into account three-body inter-actions. We analyze the perturbative expansion in terms of Feynman diagrams and define effectiveone- and two-body interactions, which allows for a substantial reduction of the number of diagrams.The procedure can be taken as a generalization of the normal ordering of the Hamiltonian to fullycorrelated density matrices. We give examples up to third order in perturbation theory. To de-fine nonperturbative approximations, we extend the equation of motion method in the presenceof three-body interactions. We propose schemes that can provide nonperturbative resummation ofthree-body interactions. We also discuss two different extensions of the Koltun sum rule to computethe ground state of a many-body system.

PACS numbers: 21.45.Ff,21.30.-x,21.60.De,24.10.Cn

I. INTRODUCTION

The description of quantum many-body systems,whether of nuclear, atomic or molecular nature, is aneverlasting challenge of theoretical physics [1, 2]. Even ifthe Hamiltonian is well known, the formalism needed todescribe such systems can be baffling. One of the issues isthat the actual inter-particle interactions in the mediumcan be very different from those in free space. Forstrongly correlated many-body systems, ordinary pertur-bation theory must be replaced by methods which per-form an all-order summation of Feynman diagrams. Theself-consistent Green’s functions (SCGF) formalism hasbeen precisely devised to treat the correlated behavior ofsuch systems [3]. The time-dependent many-body fieldcorrelation functions, also called Green’s functions (GFs)or propagators, contain information associated with theaddition or removal of a given number of particles fromthe correlated ground state. Consequently, they can beused to obtain invaluable microscopic information on themany-particle system. More importantly, knowledge ofthe N -body GFs translates into the ability of comput-ing all N -body operators and hence provides access toa wide range of observables. At the one-body level, thenonperturbative nature of the system is taken into ac-count through the self-consistent solution of the Dysonequation [4].

The original many-body Green’s functions formalismdates back to the 1960s [2, 5, 6]. In the past few decades,computational techniques have gradually improved to thepoint of allowing for fully ab-initio studies that takeinto account beyond-mean-field correlations. First prin-ciples calculations are now routinely performed in solidstate [7, 8], atomic and molecular physics [9–12] and nu-clear structure [13, 14]. In nuclear physics, finite nucleihave been studied using a variety of techniques, includ-ing the SCGF approach. The Faddeev random phaseapproximation (FRPA) has been used to describe closed-

shell isotopes [15, 16]. Medium-mass nuclei with openshells have been tackled within the Gorkov-Green’s func-tion method [17]. The behavior of correlations in infinitenuclear matter has been extensively studied using laddersummations [18–21].

Initially, the many-body Green’s functions frameworkwas developed with Hamiltonians containing up to two-body (2B) interactions in mind. In the specific case ofnuclear systems, however, three-body (3B) interactionsplay a substantial role. In infinite matter, three-bodyforces (3BFs) are long thought to be responsible for sat-uration [21–27]. Ab-initio calculations of light nuclei havepointed towards the essential role played by 3B interac-tions, particularly in reproducing the correct ground andexcited-state properties [28–31]. Recent breakthroughsin a wide variety of many-body techniques also indicatethat 3B interactions play a role in medium-mass nuclei[32–35]. In spite of all these advances, many-body the-ory with underlying 3B interactions has only been pushedforward in an intermittent fashion [31, 36–38].

Our aim here is to develop the formal tools needed toinclude 3B interactions in nonperturbative calculationswithin the SCGF formalism. While our main motivationare nuclear systems, the formalism can be easily appliedto other many-body systems. Such an approach is piv-otal both to provide theoretical foundations to approxi-mations made so far and to advance the many-body for-malisms for much-needed ab-initio nuclear structure. Inthe present paper, we present the extension of the SCGFformalism to include 3BFs by working out in full thefirst orders of the perturbation expansion and the self-consistent equations of motion. Our approach is mainlypractical. We want to put forward the basic rules that areneeded to extend present calculations to include 3B inter-actions. These include extensions in the perturbative ex-pansion, Feynman diagram rules and the equation of mo-tion (EOM) method. Moreover, we extend the Galitskii-Migdal-Koltun (GMK) sum-rule to compute the total en-

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ergy of the system including such interactions [39, 40]. Inprinciple, the approach is also able to incorporate 3B cor-relations in the many-body wave-function, but we willnot discuss these explicitly here [41]. We will also notcomment on the actual numerical implementation of theapproach, which can be found elsewhere in the literature[27, 35].

This study is made all the more timely in view ofthe notable recent efforts in improving the descriptionof the strong interaction acting between nucleons. Real-istic, phase-shift-equivalent 2B potentials have been tra-ditionally used both in finite and infinite matter calcu-lations [42–44]. Using such interactions as a startingpoint, however, one needs to evaluate the associated 3Bforces consistently. Traditionally, this has been done ei-ther phenomenologically [45] or through an extension ofthe meson-exchange picture [22]. In the last decade, how-ever, a more systematic approach has been devised, basedon the chiral effective field theory approach to low-energyQCD [46, 47]. A particularly appealing advantage of thisapproach is that it naturally gives rise to 2B, 3B andmany-body interactions as the order of the expansion in-creases. This avoids the somewhat ad hoc adjustment ofdifferent ingredients in 2B and 3B potentials.

Another important motivation to develop a formalismthat explicitly includes 3BFs comes from recent advancesin developing low-momentum interactions [48]. These ap-proaches reduce the computational effort on the many-body side by taming the strong force using renormaliza-tion group techniques. However, this results into the ap-pearance of induced 3BFs (and other many-body forces).A number of approximations have been proposed to in-clude both contributions in different many-body calcula-tions [25, 27, 31–33, 35, 37]. Most of these approachesinvolve, at some level, a normal ordering of the Hamilto-nian to average on the third, spectator particle. We willshow that such approach is justified within the GFs for-malism, provided that a class of interaction-irreduciblediagrams are discarded to avoid double countings. In-cidentally, the latter approach goes beyond the usualnormal ordering of perturbation theory by incorporatingfully correlated density matrices in the averaging proce-dure.

Induced 3BFs are not exclusive to nuclear physics, asthey arise from any truncation in the many-body modelspace. They play a role in a variety of other fields ofphysics [49, 50]. The bare and the induced 3B inter-actions, however, are treated on the same footing froma many-body perspective. Hence, techniques that dealwith 3BFs and 3B correlations within a many-body sys-tem are needed to describe such systems. The develop-ments proposed here are of relevance for applications onsystems where induced many-body forces play a role.

This paper is organized as follows. In Section II we ex-ploit effective one-body (1B) and 2B potentials to groupdiagrams in the self-energy in a more compact form. Webenchmark this approach by explicitly presenting the ex-pansion of the single-particle (SP) GF up to third order

in perturbation theory. We study, in Section III, thehierarchy of EOMs including 3BFs by means of the in-teracting vertex Γ functions. There, the truncation of Γis presented up to second order in the perturbative ex-pansion. We also show that this truncation leads to theirreducible self-energy diagrams obtained in Section II.Of relevance for practical applications, we present an im-proved ladder and ring summation that includes 3BFs.Section IV is devoted to the GMK sum rule and its use forthe calculation of the ground state energy of the many-body system. Two extensions of this rule are presentedwhich account for 3BFs. Appendix A provides a revisionof Feynman diagrams and rules with 3BFs. We pay par-ticular attention to additional symmetry factors due toequivalent groups of lines. The proof for the expressionsof the effective 1B and 2B interactions at arbitrary ordersin perturbation theory is presented in Appendix B.

II. PERTURBATIVE EXPANSION OF THEONE-BODY GREEN’S FUNCTIONS

We work with a system of N non-relativistic fermionsinteracting by means of 2B and 3B interactions. Wedivide the Hamiltonian into two parts, H = H0 + H1.H0 = T + U is an unperturbed, one-body contribution.It is the sum of the kinetic term and an auxiliary one-body operator, U , which defines the reference state forthe perturbative expansion, |ΦN0 〉, on top of which corre-lations will be added 1. The second term of the Hamil-tonian, H1 = −U + V + W , includes the interactions.V and W denote, respectively, the two- and three-bodyinteraction operators. In a second-quantized framework,the full Hamiltonian reads:

H =∑

α

ε0α a†αaα −

∑

αβ

Uα,β a†αaβ (1)

+1

4

∑

αγβδ

Vαγ,βδ a†αa†γaδaβ

+1

36

∑

αγεβδη

Wαγε,βδη a†αa†γa†εaηaδaβ .

The greek indices α,β,γ,. . . label a complete set of SPstates which diagonalize the unperturbed Hamiltonian,H0, with eigenvalues ε0

α. a†α and aα are creation and an-nihilation operators for a particle in state α. The matrixelements of the 1B operator U are given by Uα,β . Equiv-alently, the matrix elements of the 2B and 3B forces areVαγ,βδ and Wαγε,βδη. In the following, we work with an-tisymmetrized matrix elements in both the 2B and the3B sectors.

1 A typical choice in nuclear physics would be a Slater determinantof single-particle harmonic oscillator or a Woods-Saxon wavefunction.

3

The main ingredient of our formalism is the 1B GF,also called SP propagator or 2-point GF, which providesa complete description of one-particle and one-hole exci-tations of the many-body system. More specifically, theSP propagator is defined as the expectation value of thetime-ordered product of an annihilation and a creationoperators in the Heisenberg picture:

i~Gαβ(tα − tβ) = 〈ΨN0 |T [aα(tα)a†β(tβ)]|ΨN

0 〉 , (2)

where |ΨN0 〉 is the interacting N -body ground state of

the system. The time ordering operator brings operatorswith earlier times to the right, with the correspondingfermionic permutation sign. For tα − tβ > 0, this resultsin the addition of a particle to the state β at time tβand its removal from state α at time tα. Alternatively,for tβ − tα > 0, the removal of a particle from state αoccurs at time tα and its addition to state β at time tβ .These correspond, respectively, to the propagation of aparticle or a hole excitation through the system. We can

also introduce the 4-point and 6-point GFs:

i~G4−ptαγ,βδ(tα, tγ ; tβ , tδ) = (3)

〈ΨN0 |T [aγ(tγ)aα(tα)a†β(tβ)a†δ(tδ)]|ΨN

0 〉 ,

i~G6−ptαγε,βδη(tα, tγ , tε; tβ , tδ, tη) = (4)

〈ΨN0 |T [aε(tε)aγ(tγ)aα(tα)a†β(tβ)a†δ(tδ)a

†η(tη)]|ΨN

0 〉 .

Physically, the interpretation of Eq. (3) and Eq. (4) fol-lows that of the 2-point GF in Eq. (2). In these cases,more combinations of particle and hole excitations are en-countered depending on the ordering of the several timearguments. Note also that these propagators provide ac-cess to all 2B and 3B observables. The extension to for-mal expressions for higher many-body GFs is straightfor-ward.

In the following, we will consider propagators both intime representation, as defined above, or in energy repre-sentation. Note that, due to time-translation invariance,the m-point GF depends only on m− 1 time differencesor, equivalently, m − 1 independent frequencies. Hencethe Fourier transform to the energy representation is onlywell-defined when the total energy is conserved:

2πδ(ωα + ωγ + . . .− ωβ − ωδ − . . .)×Gm−ptαγ...,βδ...(ωα, ωγ , . . . ;ωβ , ωδ, . . .) = (5)

∫dtα

∫dtγ . . .

∫dtβ

∫dtδ . . . e

i(ωαtα+ωγtγ+...)Gm−ptαγ...,βδ...(tα, tγ , . . . ; tβ , tδ, . . .) e

−i(ωβtβ+ωδtδ+...) .

For the 1B GFs, one also considers,

Gαβ(ω) =

∫dτ eiωτGαβ(τ) = G2−pt

αβ (ω, ω) . (6)

Interactions in the many-body system can be treatedby means of a perturbative expansion. For the 1B prop-agator, G, this expansion reads [3, 4]:

Gαβ(tα − tβ) = − i

~

∞∑

n=0

(− i

~

)n1

n!

∫dt1 . . .

∫dtn〈ΦN0 |T [H1(t1) . . . H1(tn)aIα(tα)aIβ

†(tβ)]|ΦN0 〉conn , (7)

where |ΦN0 〉 is the unperturbed many-body ground state,

i.e. our reference N -body state. aIα and aIβ†

are nowoperators in the interaction picture with respect to H0.The subscript “conn” implies that only connected dia-grams have to be considered when performing the Wickcontractions of the time-ordered product.

H1 contains contributions from 1B, 2B and 3B interac-tions. Thus, the expansion involves terms with individ-ual contributions of each force, as well as combinations ofthese. Feynman diagrams are essential to keep track ofsuch a variety of different contributions. The set of Feyn-man diagrammatic rules that stems out of Eq. (7) in thepresence of 3B interactions is reviewed in detail in Ap-

pendix A. In general, these are unchanged with respectto the 2B case. However, we provide a few examples toillustrate the appearance of non-trivial symmetry factorswhen 3B are considered. This complicates the rules ofthe symmetry factors and illustrates some of the difficul-ties associated with many-particle interactions. In thefollowing, we will work mostly with unlabelled Feynmandiagrams. We also work with antisymmetrized matrixelements but, in contrast to Hugenholtz diagrams [51],we expand the interaction vertices and show them withdifferent types of lines for clarity.

A first reorganization of the contributions generatedby Eq. (7) is obtained by considering one-particle re-ducible diagrams, i.e. diagrams that can be disconnected

4

by cutting a single fermionic line. Reducible diagrams aregenerated by an all-orders summation through Dyson’sequation [3],

Gαβ(ω) = G(0)αβ(ω) +

∑

γδ

G(0)αγ (ω)Σ?γδ(ω)Gδβ(ω) . (8)

Thus, in practice, one only needs to include one-particleirreducible (1PI) contributions to the self-energy, Σ?.The uncorrelated SP propagator, G(0), is associated withthe system governed by the H0 Hamiltonian and repre-sents the n = 0 order in the expansion of Eq. (7). In theprevious equation, ω corresponds to the energy of thepropagating particle or hole excitation. The irreducibleself-energy Σ?, appearing in Eq. (8), describes the kernelof all 1PI diagrams. This operator plays a central rolein the GF formalism and can be interpreted as the non-local and energy-dependent interaction that each fermionfeels due to the interaction with the medium. At posi-tive energies, Σ?(ω) is also identified with the opticalpotential for scattering of a particle from the many-bodytarget [51–55].

A further level of simplification in the self-energy ex-pansion can be obtained if unperturbed propagators,G(0), in the internal fermionic lines are replaced bydressed GFs, G. This process is generically called propa-gator renormalization and further restricts the set of dia-grams to skeleton diagrams [3, 51]. These are defined as1PI diagrams that do not contain any portion that canbe disconnected by cutting a fermion line twice at anytwo different points. These portions would correspond toself-energy insertions, which are already re-summed intothe dressed propagator G by Eq. (8). The SCGF ap-proach is precisely based on diagrammatic expansions ofsuch skeleton diagrams with renormalized propagators.

In principle, this framework offers great advantages.First, it is intrinsically nonperturbative and completelyindependent from any choice of the reference state andauxiliary 1B potential, U , which automatically drops outof Eq. (8). Second, many-body correlations are expandeddirectly in terms of SP excitations which are closer tothe exact solution than those associated with the unper-turbed state, |ΦN0 〉. Third, the number of diagrams isvastly reduced to 1PI skeletons. Fourth, a full SCGFcalculation automatically satisfies the basic conservationlaws [3, 56, 57]. In practice, however, calculating dia-grams with dressed propagators is computationally moreexpensive than using the plain G(0) in perturbation the-ory. Moreover, self-consistency requires an iterative so-lution for Σ? and for G via the Dyson equation, Eq. (8).Therefore, the SCGF scheme is not always applied infull detail, but it is often employed to provide importantguidance in developing working approximations to theself-energy.

= + + 14

GII

FIG. 1. Diagrammatic representation of the effective 1B in-teraction of Eq. (10). This is given by the sum of the original1B potential (dotted line), the 2B interaction (dashed line)contracted with a dressed SP propagator, G (double line witharrow), and the 3B interaction (long-dashed line) contractedwith a dressed 2B propagator GII . The correct symmetryfactor of 1/4 in the last term is also shown explicitly.

= +

FIG. 2. Diagrammatic representation of the effective 2B inter-action of Eq. (11). This is given by the sum of the original 2Binteraction (dashed line) and the 3B interaction (long-dashedline) contracted with a dressed SP propagator, G.

A. Interaction-irreducible diagrams

It is possible to further restrict the set of relevant dia-grams by exploiting the concept of effective interactions.Let us consider an articulation vertex in a generic Feyn-man diagram. A 2B, 3B or higher interaction vertex is anarticulation vertex if, when cut, it gives rise to two dis-connected diagrams 2 Formally, a diagram is said to beinteraction-irreducible if it contains no articulation ver-tices. Equivalently, a diagram is interaction reducible ifthere exist a group of fermion lines (either interacting ornot) that leave one interaction vertex and eventually allreturn to it.

When an articulation vertex is cut, one is left with acycle of fermion lines that all connect to the same inter-action. If there were p lines connected to this interactionvertex, this set of closed lines would necessarily be part ofa 2p-point GF 3. If this GF is computed explicitly in thecalculation, one can use it to evaluate all these contribu-tions straight away. This eliminates the need for comput-ing all the diagrams looping in and out of the articulationvertex, at the expense of having to find the many-bodypropagator. An n-body interaction vertex with p fermionlines looping over it is an n−p effective interaction oper-ator. Infinite sets of interaction-reducible diagrams canbe sub-summed by means of effective interactions.

The two cases of interest when 2B and 3B forces arepresent in the Hamiltonian are shown in Figs. 1 and 2

2 1B vertices cannot be split and therefore cannot be articulations.3 More specifically, these fermion lines contain an instantaneous

contribution of the many-body GF that enters and exits the sameinteraction vertex, corresponding to a p−body reduced densitymatrix.

5

that give, respectively, the diagrammatic definition of the1B and 2B effective interactions. The 1B effective inter-action is obtained by adding up three contributions: theoriginal 1B interaction; a 1B average over the 2B interac-tion; and a 2B average over the 3B force. The 1B and 2Baverages are performed using fully dressed propagators.Similarly, an effective 2B force is obtained from the orig-inal 2B interaction plus a 1B average over the 3B force.Note that these go beyond usual normal-ordering “aver-ages” in that they are performed over fully-correlated,many-body propagators. Similar definitions would holdfor higher-order forces and effective interactions beyondthe 3B level.

Hence, for a system with up to 3BFs, we define an

effective Hamiltonian,

H1 = U + V + W (9)

where U and V represent effective interaction operators.The diagrammatic expansion arising from Eq. (7) with

the effective Hamiltonian H1 is formed only of (1PI,skeleton) interaction-irreducible diagrams to avoid anypossible double counting. Note that the 3B interaction,W , remains the same as in Eq. (1) but enters only theinteraction-irreducible diagrams with respect to 3B in-teractions. The explicit expressions for the 1B and 2Beffective interaction operators are:

U =∑

αβ

−Uα,β − i~

∑

γδ

Vαγ,βδ Gδγ(t− t+) +i~4

∑

γεδη

Wαγε,βδη GIIδη,γε(t− t+)

a†αaβ , (10)

V =1

4

∑

αγβδ

[Vαγ,βδ − i~

∑

εη

Wαγε,βδη Gηε(t− t+)

]a†αa

†γaδaβ . (11)

We have introduced a specific component of the 4-pointGFs,

GIIδη,γε(t− t′) = G4−ptδη,γε(t

+, t; t′, t′+) , (12)

which involves two-particle and two-hole propagation.This is the so-called two-particle and two-time Green’sfunction. Let us also note that the contracted propaga-tors in Eqs. (10) and (11) correspond to the full 1B and2B reduced density matrices of the many-body system:

ρ1Bδγ = 〈ΨN

0 | a†γaδ |ΨN0 〉 = −i~Gδγ(t− t+) , (13)

ρ2Bδη,γε = 〈ΨN

0 | a†γa†εaηaδ |ΨN0 〉 = i~GIIδη,γε(t− t+) . (14)

In a self-consistent calculation, effective interactionsshould be computed iteratively at each step, using corre-lated 1B and 2B propagators as input.

The effective Hamiltonian of Eq. (9) not only regroupsFeynman diagrams in a more efficient way, but also de-fines the effective 1B and 2B terms from higher orderinteractions. Averaging the 3BF over one and two spec-tator particles in the medium is expected to yield themost important contributions to the many-body dynam-ics in nuclei [31, 33]. We note that Eqs. (10) and (11)are exact and can be derived rigorously from the per-turbative expansion. Details of the proof are discussedin Appendix B. As long as interaction-irreducible dia-grams are used together with the effective Hamiltonian,

H1, this approach provides a systematic way to incorpo-rate many-body forces in the calculations and to gener-ate effective in-medium interactions. More importantly,the formalism is such that symmetry factors are properlyconsidered and no diagram is over-counted.

This approach can be seen as a generalization ofthe normal ordering of the Hamiltonian with respectto the reference state |ΦN0 〉, a procedure that has al-ready been used in nuclear physics applications with3BFs [31, 33, 48]. In both the traditional normal or-

dering and our approach, the U and V operators con-tain contributions from higher order forces, while W re-mains unchanged. The normal ordered interactions affectonly excited configurations with respect to |ΦN0 〉, but notthe reference state itself. Similarly, the effective oper-ators discussed above only enter interaction-irreduciblediagrams. As a matter of fact, if the unperturbed 1Band 2B propagators were used in Eqs. (10) and (11),

the effective operators U and V would trivially reduceto the contracted 1B and 2B terms of normal ordering.In the present case, however, the contraction goes beyondnormal ordering because it is performed with respect tothe exact correlated density matrices. To some extent,

one can think of the effective Hamiltonian, H, as be-ing reordered with respect to the interacting many-bodyground-state |ΨN

0 〉, rather than the non-interacting |ΦN0 〉.This effectively incorporates correlations that, in the nor-mal ordering procedure, must be instead calculated ex-plicitly by the many-body approach. Calculations indi-cate that such correlated averages are important in boththe saturation mechanism of nuclei and nuclear matter[27, 35].

Note that a normal ordered Hamiltonian also containsa 0B term equal to the expectation value of the origi-nal Hamiltonian, H, with respect to |ΦN0 〉. Likewise, the

full contraction of H, according to the procedure of Ap-

6

pendix B, will yield the exact ground state energy:

EN0 = −i~∑

αβ

Tαβ Gβα(t− t+)

+i~4

∑

αγβδ

Vαγ,βδ GIIβδ,αγ(t− t+)

− i~36

∑

αγεβδη

Wαγε,βδη GIIIβδη,αγε(t− t+)

= 〈ΨN0 |H |ΨN

0 〉 , (15)

in accordance with our analogy between the effective

Hamiltonian, H = H0 + H1, and the usual normal or-dering.

Before we move on, let us mention a subtlety arising inthe Hartree-Fock (or lowest-order) approximation to the

two-body propagator. If one were to insert V into thesecond term of the right hand side of Eq. (10), one wouldintroduce a double counting of the pure 3BF Hartree-Fock component. This is forbidden because the diagramin question would be interaction reducible. The correct3BF Hartree-Fock term is actually included as part ofthe last term of Eq. (10) (see also Fig. 1). Consequently,there is no Hartree-Fock term arising from the effectiveinteractions. Instead, this lowest-order contribution isfully taken into account within the 1B effective interac-tion.

B. Self-energy expansion up to third order

As a demonstration of the simplification introduced bythe effective interaction approach, in this subsection wewill derive all interaction-irreducible contributions to theproper self-energy up to third order in perturbation the-ory. We will discuss these contributions order by order,thus providing an overview of how the approach can beextended to higher-order perturbative and also to non-perturbative calculations. Among other things, this ex-ercise will illustrate the amount and variety of new dia-grams that need to be considered when 3BFs are used.

For a pure 2B Hamiltonian, the only possibleinteraction-reducible contribution to the self-energy isthe generalized Hartree-Fock diagram. This correspondsto the second term on the right hand side of Eq. (10)(see also Fig. 1). Note that this can go beyond theusual Hartree-Fock term in that the internal propaga-tor is dressed. This diagram appears at first order inany SCGF expansion and it is routinely included in mostGF calculations with 2B forces. Thus, regrouping dia-grams in terms of effective interactions, such as Eqs. (10)and (11), give no practical advantages unless 3BFs (orhigher-body forces) are present.

If 3BFs are considered, the only first-order,interaction-irreducible contribution is precisely given by

(a) (b)

FIG. 3. 1PI, skeleton and interaction-irreducible self-energydiagrams appearing at second order in the perturbative ex-pansion of Eq. (7), using the effective Hamiltonian of Eq. (9).

the one-body effective interaction depicted in Fig. 1,

Σ?,(1)αβ = Uα,β . (16)

Since U is in itself a self-energy insertion, it will not ap-pear in any other, higher-order skeleton diagram. Eventhough it only contributes to Eq. (16), the effective1B potential is very important since it determines theenergy-independent part of the self-energy. It thereforerepresents the (static) mean-field seen by every parti-cle, due to both 2B and 3B interactions. As alreadymentioned, Eq. (10) shows that this potential incorpo-rates three separate terms, including the Hartree-Fockpotentials due to both 2B and 3BFs and higher-order,interaction-reducible contributions due to the dressed Gand GII propagators. Thus, even the calculation ofthis lowest-order term Σ?,(1) requires an iterative pro-cedure to evaluate the internal many-body propagatorsself-consistently.

Note that, if one were to stop at the Hartree-Fock level,the 4-point GF would reduce to the direct and exchangeproduct of two 1B propagators. In that case, the lastterm of Eq. (10) (or Fig. 1) would reduce to the pure 3BFHartree-Fock contribution with the correct 1/2 factor infront, due to the two equivalent fermionic lines. Thisapproximate treatment of the 2B propagator in the 1Beffective interaction has been employed in most nuclearphysics calculations up to date, including both finite nu-clei [32, 33, 35] and nuclear matter [25–27, 37, 58, 59] ap-plications. Numerical implementations of averages withfully correlated 2B propagators are underway.

At second order, there are only two interaction-irreducible diagrams, that we show in Fig. 3. Dia-gram 3a has the same structure as the well-known con-tribution due to 2BFs only, involving two-particle–one-hole (2p1h) and two-hole–one-particle (2h1p) intermedi-ate states. This diagram, however, is computed with the2B effective interaction (notice the wiggly line) instead ofthe original 2B force and hence it corresponds to furtherinteraction-reducible diagrams. By expanding the effec-tive 2B interaction according to Eq. (11), the contribu-tion of Fig. 3a splits into the four diagrams of Fig. 4 [seealso a similar example in Fig. 16].

The second interaction-irreducible diagram arises fromexplicit 3BFs and it is given in Fig. 3b. One may ex-pect this contribution to play a minor role due to phasespace arguments, as it involves 3p2h and 3h2p excita-tions at higher excitation energies. Moreover, 3BFs are

7

(a) (b)

(c) (d)

FIG. 4. These four diagrams are contained in diagram 3a.They correspond to one 2B interaction-irreducible diagram,(a), and three interaction-reducible diagrams, (b) to (d).

generally weaker than the corresponding 2BFs (typically,

< W >≈ 110 < V > for nuclear interactions [22, 46]).

Summarizing, at second order in standard self-consistentperturbation theory, one would find a total of 5 skeletondiagrams. Of these, only 2 are interaction irreducibleand need to be calculated when effective interactions areconsidered.

Fig. 5 shows all the 17 interaction-irreducible diagramsappearing at third order. Again, note that, expanding

the effective interaction V , would generate a much largernumber of diagrams (53 in total). Diagrams 5a and 5bare the only third order terms that would appear in the2BF case. Numerically, these two diagrams only requireevaluating Eq. (11) beforehand, but can otherwise bedealt with using existing 2BF codes. They have alreadybeen exploited to include 3BFs in nuclear structure stud-ies [21, 25, 27, 35, 37].

The remaining 15 diagrams, from 5c to 5q, appearwhen 3BFs are introduced. These third-order diagramsare ordered in Fig. 5 in terms of increasing numbers of3B interactions and, within these, in terms of increasingnumber of particle-hole excitations. Qualitatively, onewould expect that this should correspond to a decreasingimportance of their contributions. Diagrams 5a-5c, forinstance, only involve 2p1h and 2h1p intermediate con-figurations, normally needed to describe particle additionand removal energies to dominant quasiparticle peaks aswell as total ground state energies.

Diagram 5c includes one 3B irreducible interactionterm and still needs to be investigated within the SCGFmethod. Normal-ordered Hamiltonian studies [31, 33]clearly suggest that this brings in a small correction tothe total energy with respect to diagrams 5a and 5b.This is in line with the qualitative analysis of the num-

ber of V and W interactions entering these diagrams.Diagrams 5a-5c all represent the first order term in anall order summation needed to account for configurationmixing between 2p1h or 2h1p excitations. Nowadays,resummations of these configurations are performed rou-

tinely for the first two diagrams in third-order algebraicdiagrammatic construction, ADC(3), and FRPA calcula-tions [10, 11, 16].

The remaining diagrams of Fig. 5 all include 3p2hand 3h2p configurations. These become necessary to re-produce the fragmentation patterns of shakeup config-urations in particle removal and addition experiments,i.e. Dyson orbits beyond the main quasiparticle peaks.These contributions are computationally more demand-ing. Diagrams 5d to 5k all describe interaction between2p1h (2h1p) and 3p2h (3h2p) configurations. These aresplit into four contributions arising from two effective2BFs and four that contain two irreducible 3B interac-tions. Similarly, diagrams 5l to 5q are the first contri-butions to the configuration mixing among 3p2h or 3h2pstates.

Appendix A provides the Feynman diagram rules tocompute the contribution associated with these dia-grams. Specific expressions for some diagrams in Fig. 5are given. We note that the Feynman rules remain un-altered whether one uses the original, U and V , or the

effective, U or V , interactions. Hence, symmetry factorsdue to equivalent lines remain unchanged.

III. EQUATION OF MOTION METHOD

The perturbation theory expansion outlined in the pre-vious section is useful to identify new contributions aris-ing from the inclusion of 3B interactions. However, dia-grams up to third order alone do not necessarily incor-porate all the necessary information to describe stronglycorrelated quantum many-body systems. For example,the strong repulsive character of the nuclear force at shortdistances requires explicit all-orders summations of lad-der series. All-order summations of 2p1h and 2h1p arealso required in finite systems to achieve accuracy for thepredicted ground state and separation energies, as wellas to preserve the correct analytic properties of the self-energy beyond second order.

To investigate approximation schemes for all-ordersummations including 3BFs, we now develop the EOMmethod. This will provide special insight into possibleself-consistent expansions of the irreducible self-energy,Σ?. For 2B forces only, the EOM technique defines ahierarchy of equations that link each n-body GF to the(n−1)- and the (n+1)-body GFs. When extended to in-clude 3BFs, the hierarchy also involves the (n+ 2)-bodyGFs. A truncation of this Martin-Schwinger hierarchyis necessary to solve the system of equations [5] and canpotentially give rise to physically relevant resummationschemes. Here, we will follow the footprints of Ref. [60]and apply truncations to obtain explicit equations for the4-point (and 6-point, in the 3BF case) vertex functions.

8

(a) (b) (c)

(d) (e) (f) (g)

(h) (i) (j) (k)

(l) (m) (n)

(o) (p) (q)

FIG. 5. 1PI, skeleton and interaction-irreducible self-energy diagrams appearing at third order in the perturbative expansionof Eq. (7) using the effective Hamiltonian of Eq. (9).

A. Equation of motion for G and proper self-energy

The EOM for a given propagator is found by takingthe derivative of its time arguments. The time argumentsare linked to the creation and annihilation operators inEqs. (2) to (4) and hence the time dependence of theseoperators will drive that of the propagator [51]. Theunperturbed 1B propagator can be written as the n = 0

order term of Eq. (7),

i~G(0)αβ(tα − tβ) = 〈ΦN0 |T [aIα(tα)aIβ

†(tβ)]|ΦN0 〉 , (17)

Its time derivative will be given by the von Neumannequation for the operators in the interaction picture [6]:

i~∂

∂taIα(t) = [aIα(t), H0] = ε0

αaIα(t) . (18)

9

= + + +

G4−pt G6−pt

FIG. 6. Diagrammatic representation of the EOM, Eq. (22),for the dressed 1B propagator, G. The first term, given bya single line, defines the free 1B propagator, G(0). The sec-ond term denotes the interaction with a bare 1B potential,whereas the third and the fourth terms describe interactionsinvolving the intermediate propagation of two- and three-particle configurations.

Taking the derivative of G(0) with respect to time andusing Eq. (18), we find

i~

∂

∂tα− ε0

α

G

(0)αβ(tα − tβ) = δ(tα − tβ)δαβ . (19)

Note that the delta functions in time arise from thederivatives of the step-functions involved in the time-ordered product.

The same procedure applied to the exact 1B propaga-tor, G, requires the time-derivative of the operators inthe Heisenberg picture. For the original Hamiltonian ofEq. (1), the EOM for the annihilation operator reads:

i~∂

∂taα(t) = [aα(t), H] = ε0

αaα(t)−∑

δ

Uα,δaδ(t) +1

2

∑

εδµ

Vαε,δµa†ε(t)aµ(t)aδ(t) (20)

+1

12

∑

εθδµλ

Wαεθ,δµλa†ε(t)a

†θ(t)aλ(t)aµ(t)aδ(t) .

This can now be used to take the derivative of the full 1B propagator in Eq. (2):

i~∂

∂tα− ε0

α

Gαβ(tα − tβ) = δ(tα − tβ)δαβ −

∑

δ

Uα,δGδβ(tα − tβ) (21)

+1

2

∑

εδµ

Vαε,δµG4−ptδµ,εβ(tα, tα; t+α , tβ)

+1

12

∑

εθδµλ

Wαεθ,δµλG6−ptδµλ,εθβ(tα, tα, tα; t++

α , t+α , tβ) .

This equation links the 2-point GF to both the 4- andthe 6-point GFs. Note that the connection with the lat-ter is mediated by the 3BF and hence does not appearin the pure 2BF case. Regarding the time-argumentsin Eq. (21), the t+α and t++

α present in the 4- and 6-point GFs are necessary to keep the correct time-ordering

in the creation operators when going from Eq. (20) toEq. (21). An analogous equation can be obtained forthe derivative of the time argument tβ . After some ma-nipulation, involving the Fourier transforms of Eqs. (5)and (6), one obtains the equation of motion for the SPpropagator in frequency representation:

Gαβ(ω) = G(0)αβ(ω)−

∑

γδ

G(0)αγ (ω)Uγ,δGδβ(ω) (22)

−1

2

∑

γεδµ

G(0)αγ (ω)Vγε,δµ

∫dω1

2π

∫dω2

2πG4−ptδµ,βε(ω1, ω2;ω, ω1 + ω2 − ω)

+1

12

∑

γεθδµλ

G(0)αε (ω)Wγεθ,δµλ

∫dω1

2π

∫dω2

2π

∫dω3

2π

∫dω4

2πG6−ptδµλ,γβθ(ω1, ω2, ω3;ω4, ω, ω1 + ω2 + ω3 − ω4 − ω) .

Again, this involves both the 4- and the 6-point GFs, which appear due to the 2B and 3B interactions, respec-

10

tively. The equation now involves n − 2 frequency inte-grals of the n-point GFs. The diagrammatic representa-tion of this equation is given in Fig. 6.

The EOMs, Eqs. (21) and (22), connect the 1B propa-gator to GFs of different orders. In general, starting froman n-body GF, the derivative of the time-ordering oper-ator generates a delta function between an incoming andoutgoing particle, effectively separating a line and leav-ing an (n− 1)-body propagator. Conversely, the 2B partof the Hamiltonian introduces an extra pair of creationand annihilation operators that adds another particle andleads to an (n+ 1)-body GF. For a 3B Hamiltonian, the(n+2)-body GF enters the EOM due to the commutatorin Eq. (20). This implies that higher order GFs will beneeded, at the same level of approximation, in the EOMhierarchy with 3BFs.

Eq. (22) gives an exact equation for the SP propaga-tor G that, however, requires the knowledge of both the4-point and 6-point GFs. Full equations for the latter

require applying the EOMs to these propagators as well.Before that, however, it is possible to further simplifycontributions in Eq. (22) by splitting the n-point GFsinto two terms. The first one is relatively simple, involv-ing the properly antisymmetrized independent propaga-tion of n dressed particles. The second term will involvethe interaction vertices, Γ4−pt and Γ6−pt, 1PI vertexfunctions that include all interaction effects [51]. Thesecan be neatly connected to the irreducible self-energy.

For the 4-point GF, this separation is shown diagram-matically in Fig. 7. The first two terms involve twodressed fermion lines propagating independently, andtheir exchange as required by the Pauli principle. Theremaining part, stripped of its external legs, can containonly 1PI diagrams which are collected in a vertex func-tion, Γ4−pt. This is associated with interactions and, atlowest level, it would correspond to a 2BF. As we willsee in the following, however, 3B interactions also pro-vide contributions to Γ4−pt. The 4-point vertex functionis defined by the following equation:

G4−ptαγ,βδ(ωα, ωγ ;ωβ , ωδ) = i~

[2πδ(ωα − ωβ)Gαβ(ωα)Gγδ(ωγ)− 2πδ(ωγ − ωβ)Gαδ(ωα)Gγβ(ωγ)

](23)

+(i~)2∑

θµνλ

Gαθ(ωα)Gγµ(ωγ)Γ4−ptθµ,νλ(ωα, ωγ ;ωβ , ωδ)Gνβ(ωβ)Gλδ(ωδ) .

G4−pt = − + Γ4−pt

FIG. 7. Exact separation of the 4-point Green’s function,G4−pt, in terms of non-interacting lines and a vertex function,as given in Eq. (23). The first two terms are the direct andexchange propagation of two non-interacting and fully dressedparticles. The last term defines the 4-point vertex function,Γ4−pt, involving the sum of all 1PI diagrams.

Eq. (23) is exact and is an implicit definition of Γ4−pt.Different many-body approximations arise when approx-imations are performed on this vertex function [3, 14].

A similar expression holds for the 6-point GF. In thiscase, the diagrams that involve non interacting lines cancontain either all 3 dressed propagators moving indepen-dently from each other or groups of two lines interactingthrough a 4-point vertex function. The remaining termsare collected in a 6-point vertex function, Γ6−pt, whichcontains terms where all 3 lines are interacting. Thisseparation is demonstrated diagrammatically in Fig. 8.The Pauli principle requires a complete antisymmetriza-tion of these diagrams. For the “free propagating” term,

G6−pt = + +

6

Γ4 Γ6−pt

3

3

FIG. 8. Exact separation of the 6-point Green’s function,G6−pt, in terms of non-interacting dressed fermion lines andvertex functions, as given in Eq. (24). The first two termsgather non-interacting dressed lines and subgroups of inter-acting particles that are fully connected to each other. Roundbrackets with numbers above (below) these diagrams indi-cate the numbers of permutations of outgoing (incoming) legsneeded to generate all possible diagrams. The last term de-fines the 6-point 1PI vertex function Γ6−pt.

this implies all 3! = 6 permutations of the 3 lines. Thesecond term, involving Γ4−pt, requires 32 = 9 cyclic per-mutations within both incoming and outgoing legs. The6-point vertex function is already antisymmetrized andhence no permutations are needed.

The equation corresponding to Fig. 8 is exact and pro-

11

vides an implicit definition of the Γ6−pt vertex function:

G6−ptαγε,βδη(ωα, ωγ , ωε;ωβ , ωδ, ωη) = (24)

(2π)2(i~)2A[αωα,γωγ,εωε][δ(ωα − ωβ) δ(ωγ − ωδ)Gαβ(ωα)Gγδ(ωγ)Gεη(ωε)

]

+2π(i~)3 Pcycl.[αωα,γωγ,εωε]P

cycl.[βωβ,δωδ,ηωη]

[δ(ωα − ωβ)Gαβ(ωα)×

∑

θµνλ

Gγθ(ωγ)Gεµ(ωε)Γ4−ptθµ,νλ(ωγ , ωε;ωδ, ωη)Gνδ(ωδ)Gλη(ωη)

]

+(i~)4∑

θµχνλξ

Gαθ(ωα)Gγµ(ωγ)Gεχ(ωε)Γ6−ptθµχ,νλξ(ωα, ωγ , ωε;ωβ , ωδ, ωη)Gνβ(ωβ)Gλδ(ωδ)Gξη(ωη) .

= + +

Γ6−pt

Σ∗

Γ4−pt

FIG. 9. Diagrammatic representation of the irreducible self-energy Σ? by means of effective 1B and 2B potentials and1PI vertex functions, as given in Eq. (25). The first term isthe energy independent part of Σ? and contains all diagramsdepicted in Fig.1. The second and third terms are dynamicalterms consisting of excited configurations generated through2B and 3BFs. This is an exact equation for Hamiltoniansincluding 3BFs and it is not derived from perturbation theory.

Here, we have introduced the antisymmetrization opera-

tor, A, which sums all possible permutations of pairs ofindices and frequencies, αωα, with their correspond-ing sign. Likewise, Pcycl. sums all possible cyclic per-mutations of the index-frequency pairs. Again, let usstress that both Γ4−pt and Γ6−pt are formed of 1PI dia-grams only, since they are defined by removing all exter-nal dressed legs from the G4−pt and G6−pt propagators.However, they are still two-particle reducible, since theyinclude diagrams that can be split by cutting two lines.In general, Γ4−pt and Γ6−pt are solution of all-orderssummations analogous to the Bethe-Salpeter equation, inwhich the kernels are 2PI and 3PI vertices [see Eqs. (27)to (29) below].

Inserting Eqs. (23) and (24) into Eq. (22), and exploit-ing the effective 1B and 2B operators defined in Eqs. (10)and (11), one recovers the Dyson equation, Eq. (8). Onecan therefore identify the exact expression of the irre-ducible self-energy Σ? in terms of 1PI vertex functions:

Σ?γδ(ω) = Uγ,δ (25)

− (i~)2

2

∑

µνλ

∑

ξθε

Vγµ,νλ

∫dω1

2π

∫dω2

2πGνξ(ω1)Gλθ(ω2)Γ4−pt

ξθ,δε(ω1, ω2;ω, ω1 + ω2 − ω)Gεµ(ω1 + ω2 − ω)

+(i~)4

12

∑

µφλνχ

∑

θξηεσ

Wµγφ,λνχ

∫dω1

2π

∫dω2

2π

∫dω3

2π

∫dω4

2πGλθ(ω1)Gνξ(ω2)Gχη(ω3)×

Γ6−ptθξη,εδσ(ω1, ω2, ω3;ω4, ω, ω1 + ω2 + ω3 − ω4 − ω)Gεµ(ω4)Gσφ(ω1 + ω2 + ω3 − ω4 − ω) .

The diagrammatic representation of Eq. (25) is shownin Fig. 9. We note that, as an irreducible self-energy, thisshould include all the connected, 1PI diagrams. Thesecan be regrouped in terms of skeleton and interaction-irreducible contributions, as long as Γ4−pt and Γ6−pt areexpressed that way. Note that effective interactions are

used here. The interaction-reducible components of U ,

V and W are actually generated by contributions involv-

ing partially non-interacting propagators contributionsinside G4−pt and G6−pt. The first two terms in bothEqs. (23) and (24) only contribute to generate effectiveinteractions. Note, however, that the 2B effective inter-action does receive contributions from both Γ4−pt andΓ6−pt in the self-consistent procedure.

The first term entering Eq. (25) is the energy-independent contribution to the irreducible self-energy,

12

G4−pt = − +

G4−pt

+ +

G6−pt G8−pt

FIG. 10. Diagrammatic representation of the EOM for thefour-point propagator, G4−pt, given in Eq. (26). The lastterm, involving an 8-point GF, arises due to the presence of3B interactions.

already found in Eq. (16). This includes the subtraction

of the auxiliary field, U , as well as the 1B interaction-irreducible contributions due to the 2B and 3BFs. Onceagain, we note that the definition of this term, shown inFig. 1, involves fully correlated density matrices. Conse-quently, even though this is a static contribution, it goesbeyond the Hartree-Fock approximation. The dispersivepart of the self-energy is described by the second andthird terms on the right-side of Eq. (25). These accountfor all higher-order contributions and incorporate corre-lations on a 2B and 3B level associated with the vertexfunctions Γ4−pt and Γ6−pt, respectively. In Sec. III Cbelow, we will expand these vertices up to second orderand show that Eq. (25) actually generates all diagramsderived in Sec. II B.

B. Equation of motion for G4−pt and Γ4−pt

We now apply the EOM method to the 4-point GF.This will provide insight into approximation schemes thatinvolve correlations at or beyond the 2B-level. Let usstress that our final aim is to obtain generic nonpertur-bative approximation schemes in the many-body sector.Taking the time derivative of the first argument in Eq. (3)and following the same procedure as in Sec. III A, we find:

G4−ptαγ,βδ(ωα, ωγ ;ωβ , ωδ) = i~ [2πδ(ωα − ωβ)G

(0)αβ(ωα)Gγδ(ωγ)− 2πδ(ωγ − ωβ)G

(0)αδ (ωα)Gγβ(ωγ)] (26)

+∑

µλ

G(0)αµ(ωα)Uµ,λG

4−ptλγ,βδ(ωα, ωγ ;ωβ , ωδ)

−1

2

∑

µελθ

G(0)αµ(ωα)Vµε,λθ

∫dω1

2π

∫dω2

2πG6−ptλθγ,βεδ(ω1, ω2, ωγ ;ωβ , ω1 + ω2 − ωα, ωδ)

+1

12

∑

µεχλθη

G(0)αµ(ωα)Wµεχ,λθη

∫dω1

2π

∫dω2

2π

∫dω3

2π

∫dω4

2π

G8−ptλθηγ,βεχδ(ω1, ω2, ω3, ωγ ;ωβ , ω4, ω1 + ω2 + ω3 − ωα − ω4, ωδ) .

which is the analogous of Eq. (22) for the SP propaga-tor. As expected, the EOM connects the 2-body (4-point)GFs to other propagators. The 1B propagator term justprovides the non-interacting dynamics, with the properantisymmetrization. The interactions bring in admix-tures with the 4-point GFs itself, via the one-body po-tential, but also with the 6- and 8-point GFs, via thethe 2B and the 3B interactions, respectively. Similarlyto what we observed in Eq. (22), the dynamics involven − 4 frequency integrals of the n-point GFs. The dia-grammatic representation of this equation is given in Fig.10.

To proceed further, we follow the steps of the previoussection and of Ref. [60] and split the 8-point GF intofree dressed propagators and 1PI vertex functions. Thisdecomposition is shown in Fig. 11. In addition to the

already-defined vertex functions, one needs 1PI objectswith 4 incoming and outgoing indices. To this end, weintroduce the 8-point vertex Γ8−pt in the last term. Notethat due care has to be taken of all antisymmetrizationpossibilities when groups of fermion lines that are notconnected by Γ8−pt are considered. The first term, forinstance, involves 4 non-interacting but dressed fermionlines, and there are 4! = 24 possible combinations. Thereare

(42

)(42

)12 = 72 equivalent terms involving two non-

interacting lines and a single Γ4−pt, as in the second termof Fig. 11. The double Γ4−pt contribution (third term)can be obtained in 6× 3 = 18 equivalent ways.

With this decomposition at hand, one can now pro-ceed and find an equation for the 4-point vertex function,Γ4−pt. Inserting the exact decompositions of the 4-, 6-and 8-point GFs, given respectively by Figs. 7, 8 and 11,

13

G8−pt = +

24

Γ4

6

+ +Γ4

3

+Γ6

4

Γ8−ptΓ4

12

6 4

FIG. 11. Exact separation of the 8-point Green’s function,G8−pt, in terms of non-interacting lines and vertex functions.The first four terms gather non-interacting dressed lines andsubgroups of interacting particles that are fully connected toeach other. Round brackets with numbers above (below) thesediagrams indicate the numbers of permutations of outgoing(incoming) legs needed to generate all possible diagrams. Thelast term defines the 8-point 1PI vertex function Γ8−pt.

into the EOM, Eq. (26), one obtains an equation withΓ4−pt on both sides. The diagrammatic representationof this self-consistent equation is shown in Fig. 12.

A few comments are in order at this point. The lefthand side of Eq. (26) in principle contains two dressedand non interacting propagators, as shown in the firsttwo terms of Fig. 7. In the right hand side, however, oneof the 1B propagators is not dressed. When expandingthe GFs in Eq. (26) in terms of the Γ2n−pt vertex func-tions, the remaining contributions to the Dyson equationarise automatically (Fig. 6). The free unperturbed line,therefore, becomes dressed. As a consequence, the pairof dressed non-interacting propagators cancel out exactlyon both sides of Eq. (26). This dressing procedure of theG(0) propagator happens only partially in the last three

terms of the equation and has been disregarded in ourderivation. In this sense, Fig. 12 should be taken as anapproximation to the exact EOM for G4−pt.

Eq. (26) links 1B, 2B, 3B and 4B propagators. Corre-spondingly, Fig. 12 involves higher order vertex functions,such as Γ6−pt and Γ8−pt, which are in principle coupled,through their own EOMs, to more complex GFs. Thehierarchy of these equations has to be necessarily trun-cated. In Ref. [60], truncation schemes were exploredby neglecting the Γ6−pt vertex function at the level ofFig. 12 (Γ8−pt did not appear in the 2BF-only case).This level of truncation is already sufficient to retainphysically-relevant subsets of diagrams, such as laddersand rings. Let us note, in particular, that the summationof these infinite series leads to nonperturbative many-body schemes. For completeness, we show in Fig. 12 allcontributions coming also from the Γ6−pt and Γ8−pt ver-tices, many of them arising from 3BFs.

We have ordered the diagrams in Fig. 12 in terms ofincreasing contributions from 3BFs and in the order ofperturbation theory at which they start contributing toΓ4−pt. Intuitively, we expect that this should order themin decreasing importance. Diagrams 12a, 12b, 12c and12f are those that are also present in the 2BF-only case.Diagram 12f, however, is of a mixed nature: it can con-tribute only at third order with effective 2BFs, but doescontain interaction-irreducible 3BF contributions at sec-ond order that are similar to diagrams 12d and 12e.Diagrams 12d-h all contribute to Γ4−pt at second order,

although the first three require a combination of a Vand a W term. The remaining diagrams in this group,12g and 12h, require two 3B interactions at second or-der and are expected to be subleading. Note that 12d isantisymmetric in α and γ, but it must also be antisym-metrized with respect to β and δ. Its conjugate contri-bution, 12e, should not be further antisymmetrized in αand γ, because such exchange term is already included in12f. All the remaining terms, 12i-k, only contribute fromthe third order on.

The simplest truncation schemes to Γ4−pt come fromconsidering the first three terms of Fig. 12, which involveeffective 2BFs only. In the pure 2B case, these have al-ready been discussed in the literature [60]. Retainingdiagrams 12a and 12b leads to the ladder resummationused in recent studies of infinite nucleonic matter [21, 27]:

Γ4ladd

αγ,βδ(ωα, ωγ ;ωβ , ωα + ωγ − ωβ) = Vαγ,βδ (27)

+i~2

∫dω1

2π

∑

εµθλ

Vαγ,εµGεθ(ω1)Gµλ(ωα + ωγ − ω1)Γ4ladd

θλ,βδ(ω1, ωα + ωγ − ω1;ωβ , ωα + ωγ − ωβ) ,

where we have explicitly used the fact that Γ2n−pt is only defined when incoming and outgoing energies are conserved.Likewise, diagrams 12a and the direct contribution of 12c generate a series of ring diagram which correspond to the

14

=Γ4−pt + +

Γ4−pt Γ4−pt

α γ

β δ

γ

β δ

α γ

β δ

α

γβ

δ

α

(a)

(b)

(c)

+ ++

Γ4−pt

+ +Γ4−pt

Γ6−pt Γ6−pt Γ6−pt

α γ

β

δ

α

γ

β δ

α

γ

β δ

α γ

β δ

α

γβ

δ

γ(d)

(e)(f)

(g)

(h)

+

Γ4−pt Γ4−pt

α

γ

β δ

(i)

+

Γ4−pt

Γ4−pt

α

β δ

(j)

+

Γ8−pt

α

γ

β δ(k)

ex.

ex. ex.

ex.

FIG. 12. Self-consistent expression for the Γ4−pt vertex function derived from the EOM for G4−pt. The round bracketsunderneath some of the diagrams indicate that the term obtained by exchanging the βωβ and δωδ arguments must also beincluded. Diagrams (a), (b), (c) and (f) are the only ones present for 2B Hamiltonians, although (f) also contains some intrinsic3BF contributions such as the αωα ↔ γωγ exchange of (e). All other diagrams arise from the inclusion of 3B interactions.Diagram (b) is responsible for generating the ladder summation, the direct part of (c) generates the series of antisymmetrizedrings, and the three sets together [(b), (c) and the exchange of (c)] would give rise to a Parquet-type resummation.

antisymmetrized version of the random phase approximation (RPA):

Γ4ring

αγ,βδ(ωα, ωγ ;ωβ , ωα + ωγ − ωβ) = Vαγ,βδ (28)

−i~∫

dω1

2π

∑

εµθλ

Vαε,βµGµλ(ω1)Gθε(ω1 − ωα + ωβ)Γ4ring

λγ,θδ(ω1, ωγ ;ω1 − ωα + ωβ , ωα + ωγ − ωβ) .

Adding up the first three contributions together, 12a-c, and including the exchange, will generate a Parquet-type ofresummation, with ladders and rings embedded into each other:

Γ4Parquet

αγ,βδ (ωα, ωγ ;ωβ , ωα + ωγ − ωβ) = Vαγ,βδ (29)

+i~∫

dω1

2π

∑

εµθλ

[1

2Vαγ,εµGεθ(ω1)Gµλ(ωα + ωγ − ω1)Γ

4Parquet

θλ,βδ (ω1, ωα + ωγ − ω1;ωβ , ωα + ωγ − ωβ)

−Vαε,βµGµλ(ω1)Gθε(ω1 − ωα + ωβ)Γ4Parquet

λγ,θδ (ω1, ωγ ;ω1 − ωα + ωβ , ωα + ωγ − ωβ)

+Vαε,δµGµλ(ω1)Gθε(ω1 + ωγ − ωβ)Γ4Parquet

λγ,θβ (ω1, ωγ ;ω1 + ωγ − ωβ , ωβ)].

Eqs. (27) and (28) can be solved in a more or less sim-ple fashion because the corresponding vertex functionseffectively depend on only one frequency (Ω = ωα + ωγand Ω = ωα−ωβ , respectively). Hence these two resum-mation schemes have been traditionally used to study ex-tended systems [7, 8, 14]. The simultaneous resummation

of both rings and ladders within the self-energy is possiblefor finite systems, and it has been routinely used in bothquantum chemistry and nuclear physics [11, 35, 61, 62].The Parquet summation, as shown in Eq. (29), does re-quire all three independent frequencies and it is difficultto implement numerically. Specific approximations to

15

rewrite these in terms of two-time vertex functions havebeen recently attempted [63], but further developmentsare still required.

The next approximation to Γ4−pt would involve dia-grams 12d, 12e, and the exchange part included in 12f.All these should be added together to preserve the anti-symmetry and conjugate properties of the vertex func-

tion. The resulting contributions still depend on allthree frequencies and cannot be simply embedded in all-order summations such as the ladder, Eq. (27), or thering, Eq. (28), approximations. However, these diagramscould be used to obtain corrections, at first order in theinteraction-irreducible W , to the previously calculated4-point vertices. The explicit expression for these termsis:

∆Γ4d+e+e′

αγ,βδ (ωα, ωγ ;ωβ , ωα + ωγ − ωβ) =(i~)2

2

∫dω1

2π

∫dω2

2π

∑

εµξθλν

(30)

[−Wανγ,εµδ Gεθ(ω1)Gµλ(ω2) Γθλ,βξ(ω1, ω2;ωβ , ω1 + ω2 − ωβ)Gξν(ω1 + ω2 − ωβ)

+Wανγ,εµβ Gεθ(ω1)Gµλ(ω2) Γθλ,δξ(ω1, ω2;ωα + ωγ − ωβ , ω1 + ω2 − ωα − ωγ + ωβ)Gξν(ω1 + ω2 − ωα − ωγ + ωβ)

−Γγν,εµ(ωγ , ω1 + ω2 − ωγ ;ω1, ω2)Gεθ(ω1)Gµλ(ω2)Wαθλ,βδξ Gξν(ω1 + ω2 − ωγ)

+Γαν,εµ(ωα, ω1 + ω2 − ωα;ω1, ω2)Gεθ(ω1)Gµλ(ω2)Wγθλ,βδξ Gξν(ω1 + ω2 − ωα)] .

Eq. (30) has some very attractive features. First, itshould provide the dominant contribution beyond those

associated with the effective 2B interaction, V . Perhapsmore importantly, this contribution can be easily calcu-lated in terms of one of the two-time vertex functions,Γ4ladd and Γ4ring . This could then be inserted in Eq. (23)to generate corrections of expectation values of 2B oper-ators stemming from purely irreducible 3B contributions.A similar correction for the irreducible self-energy is alsodiscussed in the next section.

Once a truncation scheme is chosen at the level of thevertex functions, one can immediately derive a diagram-matic approximation for the self-energy [3]. Conservingapproximations can plausibly be derived from some ofthese truncation schemes [56]. A general derivation ofΦ-derivability with 3BF should be possible, but goes be-yond the scope of this work.

C. Contributions to the irreducible self-energy

In this subsection, we demonstrate the correspondencebetween the techniques derived in Section II and theEOM method. In particular, we want to show how theperturbative expansion of Eq. (7) leads to the self-energyobtained with the EOM expression, Eq. (25). We will dothis by expanding the self-energy up to third order andshowing the equivalence of both approaches at this order.To this end, we need to expand the vertex functions in

terms of the effective Hamiltonian, H1. The lowest or-der terms entering Γ4−pt can be easily read from Fig. 12.We show these second-order, skeleton and interaction-irreducible diagrams in Fig. 13. Only the first three termswould contribute for a 2BF. There are two terms involv-ing mixed 2BFs and 3BFs, whereas the final two contri-butions are due to two independent 3BFs. Note that, to

+= +γ

δ

α

βΓ4−2nd

α γ

β δ

α

γβ

δ

α γ

β δ

ex.

(a)(b)

(c)

ex.

(d)

++α γ

β

δ

α

γ

β δ

ex.

(e + f)

α γ α

++

β δ

γβ

δ

(g)

(h)

ex.

FIG. 13. Skeleton and interaction-irreducible diagrams con-tributing to the Γ4−pt vertex function up to second order.The round brackets above (below) some diagrams indicatethat the exchange diagram between the αωα and γωγ(βωβ and δωδ) arguments must also be included.

get the third order expressions of the self-energy, we ex-pand the vertex functions to second order, i.e. one orderless.

Analogously, we display the expansion up to secondorder of Γ6−pt in Fig. 14. Most contributions to thisvertex function contain 3BFs. The lowest order term, forinstance, is given by the 3B interaction itself. Note, how-ever, that second order terms formed of 2B effective inter-actions are possible, such as the second term on the righthand side of Fig. 14. These will eventually be connected

16

= +Γ6−2nd

α γ ǫ

β δ η

α γ

δ

α

δ(a)

cycl.

(b)

+ +

α

δ

(c)

δ

α

(d)cycl.

cycl.

cycl.

cycl.

+

cycl.

α

δ

(e)cycl.

+ +

α

δ

α

δ

(f)

(g)

cycl.

γ

γ γ

γ

γ γ

ββ

β β

β

β

β

ǫ

ǫ

ǫ

ǫ

ǫ

ǫ

ǫ

η η

η

η

η

η η

FIG. 14. The same as Fig. 13 for the Γ6−pt vertex func-tion. The round brackets above (below) some diagrams indi-cate that cyclic permutations of the αωα, γωγ and εωε(βωβ, δωδ and ηωη) arguments must also be included.

with a 3BF to give a mixed self-energy contribution [seeEq. (25) and Fig. 9].

If one includes the diagrams in Figs. 13 and 14 intothe irreducible self-energy Σ? of Fig. 9, all the diagramsdiscussed in Eq. 16, Fig. 3 and Fig. 5 of Sec. II B arerecovered. This does prove, at least up to third order,the correspondence between the perturbative expansionapproach and the EOM method for the GFs. Proceed-ing in this manner to higher orders, one should obtainequivalent diagrams all the way through.

It is important to note that diagrams representing con-jugate contributions to Σ? are generated by different, notnecessarily conjugate, terms of Γ4−pt and Γ6−pt. For in-stance, diagram Fig. 5d is the result of the term 13(e+f)and its exchange, on the right hand side of Fig. 13. Itsconjugate self-energy diagram, 5f, however, is generatedby the second contribution to Γ6−pt, Fig. 14b. Thisterm is also related to diagram 5g. More specifically, theterm 14b has 9 cyclic permutations of its indices, of which6 contribute to diagram 5f and 3 to diagram 5g. On theother hand, the conjugate of 5g is diagram 5e, which isentirely due to the exchange contribution of the 13d termin Γ4−pt. The direct contribution of this same term leadsto diagram 5c, which is already self-conjugate.

More importantly, however, nonperturbative self-

Γ4−pt

Γ4−pt

FIG. 15. Diagrammatic representation of the self-energy cor-rection ∆Σ?ΓWΓ given in Eq. (31).

energy expansions can be obtained by means of otherhierarchy truncations at the level of Γ4−pt and Γ6−pt.Translating these into self-energy expansions is then justan issue of introducing them in Eq. (25). According tothe approximation chosen for the vertex functions ap-pearing in Fig. 9, we will be summing specific sets of di-agrams when solving the Dyson equation, Eq. (8). How-ever, from the above discussion it should be clear thatextra care must be taken to guarantee that the trunca-tions lead to physically coherent results. In particular, itis not always possible to naively neglect Γ6−pt. The lasttwo terms of the self-energy equation, Eq. (25), generateconjugate contributions. Hence, neglecting one term orthe other will spoil the analytic properties of the self-energy which require a Hermitian real part and an anti-Hermitian imaginary part. In the examples discussedabove, diagrams 5f and 5g would be missing if Γ6−pt hadnot been considered.

When no irreducible 3B interaction terms are presentin the hierarchy truncation, only the Γ4−pt term con-tributes to Eq. (25). The ladder and the ring trunca-tions, shown in Eqs. (27) and (28) generate their ownconjugate diagrams and can be used on their own to ob-tain physical approximations to the self-energy. However,this need not be true in general. A counterexample is ac-tually provided by the truncation of Eq. (30) which, ifinserted in Eq. (25) without the corresponding contri-butions to Γ6−pt, cannot generate a correct self-energy.Because of its diagrammatic content, Eq. (30) can onlybe used as a correction to Γ4−pt.

As far as 3B interaction-irreducible diagrams are con-cerned, the most important contribution should be thatassociated with Fig. 5c as discussed in Sec. II B. Furthercontributions with similar structures are also expected tocontribute to the correlation dynamics. To include suchterms, one can go beyond third order by replacing theeffective 2BFs at the upper and lower ends by ladder orring summations. Note that this is precisely the struc-ture that arises from the hierarchy truncation associatedto Eq. (30). This would lead to a generalized contri-bution, whose diagrammatic content is summarized byFig. 15. The corresponding expression for the self-energy

17

would read:

∆Σ?ΓWΓ

αβ (ω) =− (i~)4

4

∫dω1

2π· · ·∫

dω4

2π

∑

γδνστχ

∑

µελξηθ

Γ4−ptαγ,δν(ω, ω1 + ω2 − ω;ω1, ω2)Gδµ(ω1)Gνε(ω2) × (31)

Gθγ(ω1 + ω2 − ω)Wµελ,ξηθ Gξσ(ω3)Gητ (ω4)Gχλ(ω3 + ω4 − ω) Γ4−ptστ,βχ(ω3, ω4;ω, ω3 + ω4 − ω) .

To quantify the importance of these terms, they wouldneed to be included in the self-consistent procedure.Moreover, these corrections should also be consideredwhen computing the total energy, as we will see next.

To conclude this Section, we would like to stress thefact that extensions to include 3BFs beyond effective 2B

interactions, like V , are a completely virgin territory. Toour knowledge, these have not been evaluated for nuclearsystems (or any other system, for that matter) with di-agrammatic formalisms. Truncation schemes, like thoseproposed here, should provide insight on in-medium 3Bcorrelations. The advantage that the SCGF formalismprovides is the access to nonperturbative, conserving ap-proximations that contain pure 3B dynamics without theneed for ad hoc assumptions.

IV. GROUND STATE ENERGY

SCGF calculations aim at providing reliable calcula-tions for the SP propagator of correlated systems via di-agrammatic techniques. Traditionally, there have beentwo motivations to do this. On the one hand, the SPpropagator provides access to all 1B operators and henceis a useful tool to characterize a wide range of the sys-tem’s properties. On the other hand, the ground stateenergy is a critically important 2B observable that canbe obtained from the 1B GF itself. This is a crucial re-sult, that arises from the GMK sum rule [39, 40]. Thesum rule is valid both at zero and at finite temperature,where the 1B propagator also provides access to the en-ergy and, at least approximately, to all other thermody-namical properties [64]. In this Section, we investigatethe modifications of the GMK sum rule when 3BF areincluded in the Hamiltonian.

Not all the information content from the propagator isneeded to obtain the ground state energy. The hole part,which includes details about the transition amplitude forthe removal of a particle from the many-body system,is enough for these purposes. One therefore defines thediagonal part of the hole spectral function:

Shα(ω) =1

πImGαα(ω) (32)

=∑

n

∣∣⟨ΨN−1n |aα|ΨN

0

⟩∣∣2 δ[~ω −

(EN0 − EN−1

n

)],

for energies below the Fermi energy, ~ω < ε−F = EN0 −

EN−10 . The nth excited state of the N−1 particle system

is described by the many-body wave function |ΨN−1n 〉 and

has a total energy EN−1n . The transition amplitude be-

tween the N and the N−1 body systems is closely relatedto the definition of the theoretical spectroscopic factor[14]. It also provides information on removal strengthdistributions as well as energy centroids [65]. Hence, thehole spectral function represents a direct link betweentheory and experiment.

To obtain the GMK sum rule, one starts by consideringthe first moment of the hole spectral function:

Iα =

∫ ε−F

−∞dω ω Shα(ω) . (33)

From the spectral representation above, it is easy to seethat this sum-rule is also the expectation value over themany-body state of the commutator:

Iα = 〈ΨN0 |a†α[aα,H]|ψN0 〉 . (34)

Using the Hamiltonian in Eq. (1), one can evaluate thecommutator to find:

Iα = 〈ΨN0 |∑

β

Tαβ a†αa†β +

1

2

∑

γβδ

Vαγ,βδ a†αa†γaδaβ (35)

+1

12

∑

γεβδη

Wαγε,βδη a†αa†γa†εaηaδaβ |ΨN

0 〉 .

Note that, in general, T represents the 1B part of theHamiltonian which, in addition to the kinetic energy,might also contain the 1B potential. Summing over allthe external SP states, α, one finds,

∑

α

Iα = 〈ΨN0 |T + 2V + 3W |ΨN

0 〉 . (36)

In other words, the sum over SP states of the first mo-ment of the spectral function yields a particular linearcombination of the contributions of the 1B, 2B and 3Bpotentials to the ground state energy,

EN0 = 〈ΨN0 |H|ΨN

0 〉 = 〈ΨN0 |T + V + W |ΨN

0 〉 . (37)

Since T is a 1B operator, one can actually compute itsexpectation value from the SP propagator itself:

〈ΨN0 |T |ΨN

0 〉 =1

π

∫ ε−F

−∞dω∑

αβ

TαβImGβα(ω) . (38)

18

The energy integral on the right hand side yields the 1Bdensity matrix element, Eq. (13):

ρ1Bβα =

1

π

∫ ε−F

−∞dω ImGβα(ω) , (39)

which can be used to simplify the previous expression.For the 2B case, this is enough to provide an indepen-dent constraint and hence allows for the calculation ofthe total energy. The ground state energy can then becomputed from the 1B propagator alone.

When 3BF are present, however, one needs a thirdindependent linear combination of 〈T 〉, 〈V 〉 and 〈W 〉.Knowledge of the 1B propagator is therefore not enoughto compute the total energy, since either the 2B or the 3Bpropagators are needed to compute 〈V 〉 or 〈W 〉 exactly.Depending on which of the two operators is chosen, oneis left with different expressions for the energy of theground state. This freedom in choice could in principle beexploited to test the validity of different approximations.In practical applications, however, one should choose thecombination that provides minimum uncertainty.

Let us start by considering the case where the 3B op-erator is eliminated. Adding 2〈T 〉 and 〈V 〉 to the sumrule, Eq. (36), one finds the following exact expressionfor the total ground state energy:

EN0 =1

3π

∫ ε−F

−∞dω

∑

αβ

(2Tαβ + ωδαβ)ImGβα(ω)

+1

3〈ΨN

0 |V |ΨN0 〉 . (40)

The calculation of this expression requires the hole part ofthe 1B propagator and the two-hole part of the 2B prop-agator, which would appear in the second term. We notethat this expression is somewhat equivalent to the orig-inal GMK, in that the ground state energy is computedfrom 1B and 2B operators, even though the Hamiltonianitself is a 3B operator. This might prove advantageousin calculations where the 2B propagator is computed ex-plicitly.

Alternatively, one can eliminate the 2B contributionfrom the GMK sum rule by adding 〈T 〉 and subtracting

〈W 〉 to the sum rule, Eq. (36). This leads to the expres-sion:

EN0 =1

2π

∫ ε−F

−∞dω

∑

αβ

(Tαβ + ωδαβ)ImGβα(ω)

−1

2〈ΨN

0 |W |ΨN0 〉 (41)

The first term in this expression is formally the same asthat obtained in the case where only 2BFs are present inthe Hamiltonian. In that sense, the second term can bethought of as a correction to the total energy associatedwith the 3BF. Note, however, that the 3BF does influ-ence the 1B propagator on the first term and hence thecorrection should only be applied at the very end of theself-consistent procedure.

Eqs. (40) and (41) are both exact. Which of the twois employed in actual calculations will mostly depend onthe accuracy associated with the evaluation of the ex-pectation values, 〈V 〉 and 〈W 〉. If the 2B interactionis dominant with respect to the 3BF, for instance, theformer will be a large contribution. Small errors in thecalculation of the 2B propagator could eventually yieldartificially large corrections in the ground state energy.In nuclear physics, the 3BF expectation value is expectedto provide a smaller contribution than the 2BF [22, 46].Consequently, approximations in Eq. (41) should lead tosmaller absolute errors. This has been the approach thatwe have recently followed in both finite nuclei and infi-nite nuclear matter [27, 35]. In finite nuclei, evaluating

〈W 〉 at first order in terms of dressed propagators leadsto satisfactory results. However, accuracy is lost if freepropagators, G(0) are used instead. Eq. (40) may eventu-ally be useful in calculations of infinite matter, in whichthe Γ4−pt is calculated nonperturbatively.

This first-order approximation with undressed propa-gators has been traditionally used in nuclear structure.In this context, three-body forces have been often dis-cussed in the Hartree-Fock approximation with Skyrmeor Gogny functionals [1, 66]. Zero-range forces have alsobeen employed in ab-initio-type calculations [67]. It isperhaps instructive to point out at this stage that theprevious formulae apply to this case as well. In partic-ular, the Hartree-Fock approximation with 3BF can bealternatively derived from the variational principle, un-der the assumption that the many-body state is describedby a Slater determinant, |ΦN0 〉. Diagonalizing an effective1B hamiltonian leads to a series of Hartree-Fock orbitalswith single-particle energies, εα. The total energy un-der a 2B Hamiltonian is not the sum of these energies,but rather requires a correction to avoid double-counting[1]. Similarly, in the 3B case, the energy is computed asfollows:

EHF0 =∑

α

εα − 〈V 〉HF − 2〈W 〉HF . (42)

This result is straightforwardly derived by noticingthat, in the Hartree-Fock approximation, the sum rule,Eq. (36), reduces to the first term. Within this approxi-mation, the expectation values can be directly computedfrom the uncorrelated 1B density matrix:

〈V 〉HF =1

2

∑

αγβδ

Vαγ,βδρHFβα ρ

HFδγ , (43)

〈W 〉HF =1

6

∑

αγεβδη

Wαγε,βδηρHFβα ρ

HFδγ ρHFηε . (44)

If the 3B contribution is perturbative, one would expectEq. (44) to provide a good starting point to evaluatethe full 3BF contribution of Eq. (41). This seems tobe the case in finite nuclei where, however, the internaldensity matrices should be appropriately dressed [35] tofind accurate results.

19

To close this section, let us comment on the use of ef-fective interactions in the calculation of the ground stateenergy itself. Two errors can arise in this context. Thefirst would involve incorrectly accounting for the differentpre-factor in the 1B and 2B effective interactions. Thisdouble counting of the HF potential has already beendiscussed at the end of Sec. II. The second issue wouldarise if a 3B correction to the energy was neglected. TheHartree-Fock case provides a good example of the latter.Replacing the bare 2B interaction in Eq. (42) by the effec-tive 2B force would immediately lead to errors. The 3Bcorrection in Eq. (42) would necessarily have to changeto provide the same result. Consequently, performing acalculation with an effective 2B force and simply comput-ing the energy with the usual 2B formulae is not enough.The 3B correction is needed anyway and is different ifone uses the bare or the effective interaction.

V. CONCLUSIONS

We have presented an extension of the SCGF approachto include 3B interactions. The method allows to incor-porate consistently 2B and 3B forces on an equal foot-ing and should be interesting for nuclear physics applica-tions. Other many-body systems in which induced 3BFare generated by cuts in the model space could poten-tially benefit from this treatment as well.

The 3BF has been introduced in two different butequivalent ways in the formalism. On the one hand,we have studied the diagrammatic perturbative expan-sion of the propagator including 1B, 2B and 3B forces.The expansion is analogous to cases previously studiedin the literature, but the 3BF requires some careful han-dling. We present in Appendix A the Feynman rules as-sociated with this expansion. Within a SCGF approach,where propagators are dressed and the Dyson equationis used iteratively, only 1PI skeleton diagrams enter theexpansion. The number of diagrams can be further re-duced by introducing effective interactions, which sumup sub-series of interaction-reducible diagrams. Theseeffective interactions can be interpreted as a generaliza-tion of the normal ordering of the many-body hamilto-nian. Instead of ordering with respect to an uncorrelatedstate, however, the effective interactions include the effectof many-body correlations by construction. The properself-energy can be defined from 1B, 2B and 3B forces andstill be computed within the Dyson’s equation. We haveshown how this effective grouping of operators reducesthe number of diagrams by considering the perturbationexpansion up to third order. The equivalence betweenthe original diagrammatic expansion and that obtainedfrom the effective interaction at any arbitrary order isproven in Appendix B.

On the other hand, the propagator method can be ex-pressed using the EOM. We have re-derived the basicequations of this method, consistently including 3BFs.The Martin-Schwinger hierarchy now connects the (n)-

body propagator to the (n−1)-, the (n+1)- and, via the3BF, the (n+ 2)-body GF. In turn, this requires the in-troduction of vertex functions beyond the 4-point level.Through the hierarchy of the EOM, we have found anexpression for the vertex function Γ4-pt, which embod-ies the higher order interacting contributions beyond themean-field. Truncation to second order of this function,together with complete second order expression for the6-point Γ function, provides the third order approxima-tion for the irreducible self-energy. The correspondenceto the diagrams obtained in the perturbative expansionindicates that these two different approaches are equiva-lent.

Moreover, we have shown that, using the 2B effectiveinteraction in truncation schemes based on Γ4-pt, leadsto either ladder, ring or parquet approximations that ef-fectively include some 3B terms. Within these approxi-mations, the general structure of the formalism, based on2BFs, remains unaltered [14]. Results obtained recentlyboth in calculations for infinite nuclear matter [27] aswell as nuclei [35] exploit this expanded self-consistentGreen’s functions approach to include 3B nuclear forces.

More importantly, however, this approach is able toprovide a general mechanism to devise nonperturbativeresummation schemes. In particular, we have proposeda potentially relevant correction to the self-energy thatincludes interaction-irreducible three-body effects explic-itly. Extensions to include 3BFs beyond the effective 2BFapproach are lacking in the literature and could prove tobe potentially relevant in some instances, particularly innuclear physics.

Finally, we have presented a general method to com-pute the energy of the many-body ground state by meansof the GMK sum rule. The sum rule still allows for thecalculation of the ground state energy from the 2(n− 1)-point GFs in spite of the fact that the energy itself isan n-body operator. Two possible approaches have beenproposed, which require the calculation of either a 2B ora 3B expectation value. Depending on the relative im-portance of the 2B and the 3B forces in the system, oneor the other might be preferable.

Calculations performed using this extended SCGF for-malism have already been presented in the literature[27, 35]. This expanded approach provides a firm basis forfurther studies of nuclear systems from a Green’s func-tions point of view. The formalism can be extended tofinite temperature and off-equilibrium settings. More im-portantly, it provides a framework to compute many rel-evant quantities for the description of a quantum many-body system, from binding energies, to thermodynamicalproperties or even pairing.

We believe that this extended approach is an inter-esting tool to study quantum many-body systems froman ab-initio microscopic point of view. In principle, theframework provides a coherent description of the corre-lated, nonperturbative dynamics of systems with many-body interactions. The generalization to Hamiltoniansincluding N -body forces can be performed along similar

20

lines. In addition to its academic interest, advances ininteraction-tunable ultra cold gases might require thesedevelopments. In nuclear physics, the importance of4BFs, either bare or induced, could also be assessed usinganalogous techniques. Ultimately, the methods presentedhere should be a good starting point to foster new initia-tives that systematically address the issue of many-bodyforces in quantum many-body systems.

ACKNOWLEDGMENTS

This work is supported by the Consolider Inge-nio 2010 Programme CPAN CSD2007-00042, GrantNo. FIS2011-24154 from MICINN (Spain) and GrantNo. 2009SGR-1289 from Generalitat de Catalunya(Spain); and by STFC, through Grants ST/I005528/1and ST/J000051/1.

Appendix A: Feynman diagram rules for 2- and3-body interactions

We present in this appendix the Feynman rules as-sociated with the diagrams arising in the perturbativeexpansion of Eq. (7). The rules are given both in timeand energy formulation, and some specific examples willbe considered at the end. We pay particular attentionto non-trivial symmetry factors arising in diagrams thatinclude many-body interactions. We work with antisym-metrized matrix elements, but for practical purposes rep-resent them by extended lines.

We provide the Feynman diagram rules for a given p-body propagator, such as Eqs. (3) and (4). These arisefrom a trivial generalization of the perturbative expan-sion of the 1B propagator in Eq. (7). At k-th orderin perturbation theory, any contribution from the time-ordered product in Eq. (7), or its generalization, is rep-resented by a diagram with 2p external lines and k inter-action lines (from here on called vertices), all connectedby means of oriented fermion lines. These fermion linesarise from contractions between annihilation and creationoperators,

aIδ(t)aI †γ (t′) ≡ 〈ΦN0 |T

[aIδ(t)a

I †γ (t′)

]|ΦN0 〉 = i~G(0)

δγ (t−t′).

Applying the Wick theorem to any such arbitrary dia-gram, results in the following Feynman rules.

Rule 1: Draw all, topologically distinct and connecteddiagrams with k vertices, and p incoming and p out-going external lines, using directed arrows. For in-teraction vertices the external lines are not present.

Rule 2: Each oriented fermion line represents a Wickcontraction, leading to the unperturbed propagator

i~G(0)αβ(t − t′) [or i~G(0)

αβ(ωi)]. In time formulation,

the t and t′ label the times of the vertices at theend and at the beginning of the line. In energy

formulation, ωi denotes the energy carried by thepropagator.

Rule 3: Each fermion line starting from and endingat the same vertex is an equal-time propagator,

−i~G(0)αβ(0−) = ρ

(0)αβ .

Rule 4: For each 1B, 2B or 3B vertex, write down a fac-tor i

~Uαβ , − i~Vαγ,βδ or − i

~Wαγξ,βδθ, respectively.

For effective interactions, the factors are − i~ Uαβ ,

− i~ Vαγ,βδ.

When propagator renormalization is considered, onlyskeleton diagrams are used in the expansion. In that case,

the previous rules apply with the substitution i~G(0)αβ →

i~Gαβ . Furthermore, note that Rule 3 applies to di-agrams embedded in the one-body effective interaction(see Fig. 1) and therefore they should not be consideredexplicitly in an interaction-irreducible expansion. In cal-culating U , however, one should use the correlated ραβinstead of the unperturbed one.

Rule 5: Include a factor (−1)L where L is the number ofclosed fermion loops. This sign comes from the oddpermutation of operators needed to create a loopand does not include loops of a single propagator,already accounted for by Rule 3.

Rule 6: For a diagram representing a 2p-point GF, adda factor (−i/~), whereas for a 2p-point interac-tion vertex without external lines (such as Σ? andΓ2p−pt) add a factor i~.

The next two rules require a distinction between the timeand the energy representation. In the time representa-tion:

Rule 7: Assign a time to each interaction vertex. Allthe fermion lines connected to the same vertex ishare the same time, ti.

Rule 8: Sum over all the internal quantum numbers andintegrate over all internal times from −∞ to +∞.

Alternatively, in energy representation:

Rule 7’: Label each fermion line with an energy ωi, un-der the constraint that the total incoming energyequals the total outgoing energy at each interactionvertex,

∑i ω

ini =

∑i ω

outi .

Rule 8’: Sum over all the internal quantum numbersand integrate over each independent internal en-

ergy, with an extra factor 12π , i.e.

∫ +∞−∞

dωi2π .

Each diagram is then multiplied by a combinatorialfactor S that originates from the number of equivalentWick contractions that lead to it. This symmetry fac-tor represents the order of the symmetry group for onespecific diagram or, in other words, the order of the per-mutation group of both open and closed lines, once the

21

same int.

· · ·

16

112

1 12

112

(a)

same int.

· · ·

16

112

1 12

112

(b)

same int.

· · ·

16

112

1 12

112

(c)

same int.

· · ·

16

112

1 12

112

(d)

FIG. 16. Examples of diagrams containing symmetric andinteracting lines, with explicit symmetry factors. Diagrams(b) to (d) are obtained by expanding the effective interactionof diagram (a) according to Eq (11). Swapping the 3B and2B internal vertices in (c) gives a distinct, but topologicallyequivalent, contribution.

vertices are fixed. Its structure, assuming only 2BFs and3BFs, is the following :

S =1

k!

1

[(2!)2]q[(3!)2]k−q

(k

q

)C =

∏

i

Si . (A1)

Here, k represents the order of expansion. q (k − q) de-notes the number of 2B (3B) vertices in the diagram.The binomial factor counts the number of terms in theexpansion (V + W )k that have q factors of V and k − qfactors of W . Finally, C is the overall number of distinctcontractions and reflects the symmetries of the diagram.Stating general rules to find C is not simple. For ex-ample, explicit simple rules valid for the well-known λφ4

scalar theory are still an object of debate [68]. An ex-plicit calculation for C has to be carried out diagram bydiagram [68]. Eq. (A1) can normally be factorized in aproduct factors Si, each due to a particular symmetry ofthe diagram. In the following, we list a series of specificexamples which is, by all means, not exhaustive.

Rule 9: For each group of n symmetric lines, or symmet-ric groups-of-lines as defined below, multiply by asymmetry factor Si=

1n! . The overall symmetry fac-

tor of the diagram will be S =∏i Si. Possible cases

include:

(i) Equivalent lines. n equally-oriented fermion linesare said to be equivalent if they start from the sameinitial vertex and end on the same final vertex.

(ii) Symmetric and interacting lines. n equally-orientedfermion lines that start from the same initial vertexand end on the same final vertex, but are linked viaan interaction vertex to one or more close fermionline blocks. The factor arises as long as the dia-gram is invariant under the permutation of the twoblocks.

!U +12 +1

8 + . . .

1

. . .

12

12

12

!12

(a)

!U +12

+18

+ . . .

1

. . .

12

12

12

!12

(b)

same int.

· · ·

16

112

1 12

112

(c)

FIG. 17. Examples of diagrams entering the static part of theself-energy. Applying rule 9-ii, diagrams (a) and (b) take afactor Ssi = 1

2from the symmetry between the two bubbles

attached to the upper three body vertex. The symmetry isbroken in diagram (c), where the overall factor is Ssi = 1

(iii) Equivalent groups of lines. These are blocks of in-teracting lines (e.g. series of bubbles) that are equalto each other: they all start from the same initialvertex and end on the same final vertex.

Rule 9-i is the most well-known case and applies, forinstance, to the two second order diagrams of Fig. 3. Dia-gram 3a has 2 upward-going equivalent lines and requiresa symmetry factor Se=

12! . In contrast,diagram 3b has

3 upward-going equivalent lines and 2 downward-goingequivalent lines, that give a factor Se=

12! 3!=

112 .

Figs. 16 and 17 give specific examples of the applica-tion of rule 9-ii. Diagram 16a has 3 upward-going equiv-alent, non-interacting lines, which yield a symmetry fac-tor Se=

13! due to rule 9-i. However, there are also two

downward-going symmetric and equivalent lines, that in-teract through the exchange of a bubble and thus giverise to a factor Ssi=

12! . The total factor is therefore

S=Se×Ssi= 112 . Let us now expand the two 2B effective

interactions that are connected to the intermediate bub-ble according to Eq. (11). Diagram 16a is now seen tocontain three contributions, diagrams 16b to 16d, withthe symmetry factors shown in the figure. Note thatdrawing the contracted 3B vertex above or below thebubble in 16c leads to two topologically equivalent dia-grams that must only be drawn once, i.e. diagram 16c.However, since the diagram is no longer symmetric un-der the exchange of the two downward-going equivalentlines, rule 9-ii does not apply anymore and the Ssi factoris no longer needed.

A similar situation occurs when the two interactingfermion lines start and end on the same vertex, as inFig. 17. Consider the left-most and right-most externalfermion bubbles. In all three diagrams, they are con-nected to each other by a 3B interaction vertex aboveand by a series of interactions and medium polarizationsbelow. The intermediate bubble interactions in diagrams17a and 17b are symmetric under exchange. There are

22

!U +12

+18

+ . . .

1

. . .

12

12

12

!12

FIG. 18. Diagrams entering the effective one-body interaction, Eq. (10), obtained by substituting the right hand side of Fig. 13into Eq. (23). The two bubble terms correctly reproduce the symmetric factor inferred by applying rules 9-i and 9-ii.

!U +12 +1

8 + . . .

1

. . .

12

12

12

!12

(a)

!U +12

+18

+ . . .

1

. . .

12

12

12

!12

(b)

FIG. 19. Examples of a diagram where equivalent group oflines are present and one where rule 9-iii does not apply.Swapping the two chains of bubbles in (a), one finds an iden-tical diagram. This is precisely the case of rule 9-iii, whichbrings in a factor Segl=

12. Performing the same exchange in

diagram (b) generates a graph where the direction of the in-ternal loop is reversed. No symmetry rule applies here andSegl=1

therefore two sets of symmetric interacting lines (the twoup-going and two down-going fermion lines) and henceboth diagrams take a factor Ssi = 1/2. In contrast, thetwo external loops in 17c are not symmetric under ex-change due to the lower 3B vertex. Rule 9-ii does notapply anymore and Ssi = 1. If all the vertices betweenthe external loops where equal (e.g. effective 2B terms

V ), a factor Ssi=1/2 would still apply.The case of Fig. 17 is of particular importance be-

cause these diagrams directly contribute to the energy-independent 1B effective interaction. In the EOM ap-proach, these contributions arise from the first threeterms on the right hand side of Fig. 13. Note that the lad-

der diagram has a symmetry factor Se=1/2 and that theexchange contribution in the bubble term has to be con-sidered. Using these diagrams to the define the 2B prop-agator in Eq. (23) and inserting these in the last term

of Eq. (10), one finds the contributions to U shown inFig. 18. The two bubble terms have summed up to formdiagram 17a, each of them contributing a factor 1/4 fromEq. (10). Consequently, the approach leads to the cor-rect overall Ssi=1/2 symmetry factor. In our approach,there is no need to explicitly compute these diagrams,since they are automatically included by Eq. (10).

Finally, rule 9-iii applies to the diagram in Fig. 19a. Inthis case, the two chains of bubble diagrams are equal andstart and end at the same 3BF vertices. Hence, they areequivalent groups of lines and the diagram takes a factorSegl = 1

2 . Diagram 19b is different because the exchangeof all the bubbles generates a diagram in which the direc-tion of the internal fermion loop is reversed. Thereforeno symmetry rule applies and the symmetry factor is justSegl = 1. This is, however, topologically equivalent to theinitial diagram and hence must be counted only once.

As an example of the application of the above Feynmanrules, we give here the formulae for some of the diagramsin Fig. 5. Let us start by a contribution that has beendiscussed in Section III, diagram 5c. There are two setsof upward-going equivalent lines, which contribute to asymmetry factor Se = 1

22 . Considering the overall factorof Eq. (A1) and the presence of one closed fermion loop,one finds:

Σ(5c)αβ (ω) = − (i~)4

4

∫dω1

2π· · ·∫

dω4

2π

∑

γδνµελξηθστχ

Vαγ,δνG(0)δµ (ω1)G(0)

νε (ω2)Wµελ,ξηθG(0)ξσ (ω3)G(0)

ητ (ω4)× (A2)

G(0)θγ (ω1 + ω2 − ω)Vστ,βχG

(0)χλ(ω3 + ω4 − ω) .

Diagrams 5h and 5i differ only for the orientation of aloop. Hence, there are two pairs of equivalent lines in

the first case and one pair and one triplet of equivalentlines in the second, which is reflected in their differentsymmetry factors:

23

Σ(5h)αβ (ω) =

(i~)5

4

∫dω1

2π· · ·∫

dω5

2π

∑

γδεξθσµνλητφχζ

Vαγ,δεG(0)δξ (ω1)G(0)

νγ (ω2)Wξθσ,µνλG(0)µη (ω3)G

(0)χθ (ω4)× (A3)

G(0)ετ (ω − ω1 + ω2)Wητφ,βχζ G

(0)λφ (ω5)G

(0)ζσ (ω2 + ω3 + ω5 − ω1 − ω4) ,

Σ(5i)αβ (ω) =

(i~)5

12

∫dω1

2π· · ·∫

dω5

2π

∑

γδεξθσµνλητφχζ

Vαγ,δεG(0)δξ (ω1)G

(0)εθ (ω2)Wξθσ,µνλG

(0)µη (ω3)G(0)

ντ (ω4)× (A4)

G(0)χγ (ω1 + ω2 − ω)Wητφ,βχζ G

(0)λφ (ω5)G

(0)ζσ (ω3 + ω4 + ω5 − ω1 − ω2) .

Appendix B: Interaction-irreducible diagrams witheffective 1B and 2B interactions at all orders

Interaction-irreducible diagrams can be used to dis-tinguish between two different many-body effects in theSCGF approach. On the one hand, effective interac-tions sum all the instantaneous contributions associatedwith “averaging out” subgroups of particles that lead tointeraction-reducible diagrams. This has the advantageof reducing drastically the number of diagrams at eachorder in the perturbative expansion. It also gives riseto well-defined in-medium interactions. On the otherhand, the remaining diagrams will now include higher-order terms summed via the effective interaction itself.

In this Appendix, we proof that the perturbative

expansion can be recast into a set containing onlyinteraction-irreducible diagrams at any given order, aslong as properly defined effective interactions are used.The argument we propose has been often used to demon-strate how disconnected diagrams cancel out in the per-turbative expansion. We now apply it to a slightly differ-ent case that requires extra care. We focus on the caseof a diagram that includes only 2B and 3BFs. The ex-tension to the general case of many-body forces shouldbe straightforward.

Eq. (7) gives the perturbative expansion of the 1B GFin terms of the Hamiltonian, H(t), in the interaction pic-ture. The k-th order term of the perturbative expansionreads:

G(k−th)αβ (t− t′) =

(−i

~

)k+11

k!

∫· · ·∫

dtk

k terms

〈ΦN0 |T[aIα(t)aI†β (t′)H(t1) · · ·H(tk)

]|ΦN0 〉conn . (B1)

Only connected contributions are allowed and we takethe interaction picture external creation and destruc-tion operators to the left for convenience. Let us as-sume, without loss of generality, that the diagram is com-posed of q 2B and k − q 3B interaction operators. Thisgives rise to

(kq

)identical contributions when expanding

H(t) = T (t) + V (t) +W (t) in the time-ordered product,as discussed right after Eq. (A1) above.

Let us denote with O(t) a generic operator, represent-ing either a 2B, V (t), or a 3B, W (t), potential. Supposenow that there is a sub-set of m operators that are arbi-trary connected to each other, but that share the externallinks with a unique operator, O(tn), outside the subset.In other words, O(tn) is the only way to enter and exitthe subset of m- operators O(tn+1), · · · , O(tn+m) as

drawn below:

O(t1) · · ·O(tn−1) · O∣∣∣∣∣ (tn) ·

O(tn+1) · · ·O(tn+m)

(B2)O(tn) is also necessarily connected to the other interac-tions and, hence, this is an articulation vertex. In gen-eral, there can be an arbitrary number of articulationvertices, such as O(tn), at any given order. Each one ofthese vertices would isolate a particular subset of oper-ators. The following arguments can be applied to eachsubset separately.

For simplicity, let us restrict the argument to the sim-plest case of one articulation vertex only. Suppose that,among m terms, there are a 2B and b 3B interactions,with a + b = m. The number of time-ordered productsV (t) and W (t) in Eq. (B1) that is consistent with the

24

above decomposition is

(k

q

)(q

a

)(k − qb

)=

(k

m

)(m

a

)(n

q − a

)(B3)

where m+ n = k.Let us consider the case in which O(tn) is a 3B opera-

tor, with matrix elements Wµγδ,θσξ connected with fourlegs to the internal subset of m vertices and with twolegs to the rest of the diagram. We can factorize the am-plitude in Eq. (B1) by adding an intermediate identityoperator as follows:

1

n!

(n

q − a

)∫· · ·∫

dtn

n terms

〈ΦN0 |T[aIα(t)aI†β (t′)O(t1) · · · O(tn−1) aI†µ (t+n )aIθ(tn)

]|ΦN0 〉 Wµγδ,θσξ

1

(3!)2

(3

1

)2

× 1

(m)!

(m

a

)∫· · ·∫

dtk

m terms

〈ΦN0 |T[aI†γ (t+n )aI†δ (t+n )aIξ(tn)aIσ(tn)O(tn+1) · · ·O(tk)

]|ΦN0 〉δk,n+m. (B4)

Note that the factorization of the time ordered product,by inserting a |ΦN0 〉〈ΦN0 |, is possible because the Wicktheorem normal-orders these products with respect to thereference state, |ΦN0 〉. In other words, both Eqs. (B1)and (B4) lead to exactly the same results after all Wickcontractions have been carried out.

All possible orders in which a general O(t) entersEq. (B4) are equivalent and are accounted for by thebinomial factors. The factor

(31

)accounts for all the pos-

sible ways, eventually decided by contractions, in whichthe six creation/annihilation operators in W (tn) can beseparated in the two factors [see also Eq. (B7) below].

We also include an additional factor(

31

)coming from all

the possible ways to choose one creation/annihilation op-erator among the three possible pairs. The correct timeordering for creation and annihilation operators associ-ated with W (tn) is preserved using a†(t+n ).

With this decomposition, we can identify the secondline of Eq. (B4) as an m-th order contribution (with a2B and m − a 3B operators) to the perturbative expan-

sion of G4−ptσξ,γδ(tn, tn; t+n , t

+n ) = GIIσξ,γδ(tn − t+n ). Collect-

ing all possible contributions of form (B2) and (B4) inwhich the first n operators are unchanged, the k-th or-der interaction-reducible contribution to G becomes:

G(k−th)αβ (t− t′)→

(−i~)n+1 1

n!

(nq−a)∫· · ·∫

dtn〈ΦN0 |T[aIα(t)aI†β (t′)O(t1) · · ·O(tn−1)aI†µ (t+n )aIθ(tn)

]|ΦN0 〉int-irr

×Wµγδ,θσξi ~

(2!)2GII (m−th,a)σξ,γδ (tn − t+n )

U effµ,θ

, (B5)

where GII (m−th,a) sums all the diagrams at m-th orderwith a two-body operators. Note that the last term nolonger depends on time and can be seen as an energy-independent correction to the 1B potential. We can au-tomatically take into account these interaction-reducibleterms by reformulating the initial hamiltonian to includethe effective 1B vertex:

Uµ,θ → Uµ,θ +Wµγδ,θσξi~

(2!)2GIIσξ,γδ (t− t+)

− i~ ρ

2Bσξ,γδ

(B6)

where now we use an exact GII . The perturbative expan-sion obtained with this effective interaction should onlycontain interaction-irreducible diagrams to avoid doublecounting.

Note that in Eq. (B5) we automatically obtain the cor-rect symmetry factor 1/(2!)2 associated with the contrac-tion of W with the two pairs of incoming and outgoinglines of GII . In the general case, a c-body vertex can bereduced to a d-body one (with d < c) by using a (c− d)-body GF. The overall combinatorial factor in that casewill be:

1

(c!)2

(c!

d!(c− d)!

)2

=1

(d!)2

︸ ︷︷ ︸new vertex

1

((c− d)!)2

︸ ︷︷ ︸c−d equal lines

. (B7)

This yields both the correct combinatorial factors enter-ing the new effective d-body vertex and the symmetryfactor associated with the contraction with the (c − d)-body GF. The above arguments can be generalized to any

25

starting n-body Hamiltonian. Applying these derivation to all possible cases for a 3B Hamiltonians leads to theeffective interactions discussed in Eqs. (10) and (11).

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