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Conserving approximations in nonequilibrium green function theoryStan, Adrian

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Chapter 7Total energies from variational functionals ofthe Green function and the renormalizedfour-point vertex

Robert van Leeuwen, Nils Erik Dahlen and Adrian Stan

1Rijksuniversiteit Groningen, Materials Science Centre, Theoretical Chemistry,Nijenborgh 4, 9747AG Groningen, The Netherlands.

2Department of Physics, Nanoscience Center, FIN 40014, University of Jyvaskyla, Jyvaskyla, Finland.3European Theoretical Spectroscopy Facility (ETSF).

Physical Review, B74, 195105 (2006)

Abstract

We derive variational expressions for the grand potential or action in terms of the many-body Green function G whichdescribes the propagation of particles and the renormalized fourvertex Γ which describes the scattering of two particles inmany-body systems. The main ingredient of the variational functionals is a term we denote as the Ξ-functional which playsa role analogously to the usual Φ-functional studied by Baym (G.Baym, Phys.Rev. 127, 1391 (1962)) in connection with theconservation laws in many-body systems. We show that any Ξ-derivable theory is also Φ-derivable and therefore respectsthe conservation laws. We further set up a computational scheme to obtain accurate total energies from our variationalfunctionals without having to solve computationally expensive sets of self-consistent equations. The input of the functionalis an approximate Green function G and an approximate fourvertex Γ obtained at a relatively low computational cost. Thevariational property of the functional guarantees that the error in the total energy is only of second order in deviationsof the input Green function and vertex from the self-consistent ones that make the functional stationary. The functionalsthat we will consider for practical applications correspond to infinite order summations of ladder and exchange diagramsand are therefore particularly suited for applications to highly correlated systems. Their practical evaluation is discussedin detail.

65

66 THE RENORMALIZED FOUR-POINT VERTEX

7.1 Introduction

Total energy calculations play an important role in con-densed matter physics and quantum chemistry. Forsolid state physicists they are essential in predictingstructural changes and bulk moduli in solids. In chem-istry molecular bonding curves and potential energysurfaces are essential to understand phenomena likemolecular dissociation and chemical reactions. How-ever, accurate total energy calculations are notoriouslydifficult and computationally demanding. In quantumchemistry there are advanced wavefunction methodslike configuration interaction and coupled cluster the-ory [1] to calculate energies but they can only be ap-plied to relatively small molecules. In solid state physicsmost total energy calculations for crystals or surfacesare based on density functional theory [2] where thedensity functionals are mostly based on the local den-sity approximation (LDA) and generalized gradientapproximations (GGA) [3]. These functionals have hadgreat success but there are many cases where the func-tionals fail, in which case there is no clear systematicroute to improvement. We have therefore recently ad-vanced a different scheme which involves variational en-ergy functionals of the many-body Green function andapplied it succesfully to atoms, molecules [4, 5, 6, 7, 8]and the electron gas [9]. A variety of such functionalscan be systematically contructed using diagrammaticperturbation theory in which the different functionalscorrespond to different levels of perturbation theory.For these functionals we use input Green functions thatare relatively easy to obtain at low computational cost,for instance from a local density or Hartree-Fock calcu-lation. The variational property of the functional thenassures that the errors in the energy are only of secondorder in the difference between our approximate Greenfunction and the actual Green function that makes thefunctional stationary. This is the essential feature thatallows one to obtain accurate total energies at a rela-tively low computational cost. The remaining questionis then how to select approximate variational function-als that yield good total energies.

In a diagrammatic expansion in many-body pertur-bation theory the building blocks are Green functionlines G which describe the propagation of particles andholes and interaction lines v which in electronic sys-tems is represented by the Coulomb repulsion betweenthe electrons. From this diagrammatic structure onecan proceed to construct variational functionals in sev-eral ways. First of all we can renormalize the Greenfunction lines. This leads to a functional that has beenintroduced by Luttinger and Ward [10] and leads to afunctional we will call the Φ-functional Φ[G, v], depend-ing on the dressed Green function and the bare two-

particle interaction v. The Luttinger-Ward functionalhas been applied, with great success, to the calculationof total energies of the electron gas [11, 9], and atomsand molecules [4, 5, 6, 7, 8, 12]. This type of function-als has also received considerable attention for Hubbardlattice type systems [13, 14, 12, 15, 16, 17]. Apart fromrenormalization of the Green function lines, we can alsodecide to renormalize the interaction lines by replac-ing the bare interaction by a dynamically screened one,usually denoted by W . This leads to the functionalΨ[G,W ] first introduced in a paper by Hedin [18] andelaborated upon by Almbladh et al.[11, 9] which hasbeen applied with succes to calculations of the totalenergy of the electron gas [11, 9] and atoms [5]. Thistype of functionals has also received considerable at-tention in the Dynamical Mean Field Theory (DMFT)community [19, 20, 21]. The natural place to use thisfunctional is in extended systems in which screening ofthe long range Coulomb interaction is essential. Finallythere is also the possibility to renormalize the four-vertices and replace them by a renormalized fourver-tex Γ. In this work we will concentrate on this type offunctionals.

The natural place for variational functionals of theGreen function G and the fourvertex Γ is in systemswhere short range correlations play an important rolesuch as in highly correlated systems. Such a type oftheory was recently discussed in work of Janis [13, 14]on the Hubbard model in which it was demonstratedhow to derive the so-called parquet approximation froma functional of the Green function and the fourver-tex. Furthermore Katsnelson and Lichtenstein [22] haveconsidered the electronic structure of correlated met-als in which the building blocks of the theory are anapproximate T -matrix and a bare or noninteractingGreen function (or a bare Green function in an effectivecorrelated medium when using Dynamical Mean FieldTheory[23]). For describing the structural properties ofsuch materials it would therefore be of great importanceto be able to calculate the total energy from variationalenergy expressions in terms of the Green function andthe fourvertex where we use an approximate G and Γas an input. The variational property then guaranteesthat the errors in the energy are only of second orderin the deviations of the input Green function and ver-tex from the true quantities that make the functionalstationary.

The construction of energy functionals in terms ofG and Γ is most naturally done by the use of theHugenholtz diagram technique [24, 25, 26, 27] whichhas the bare four-vertex as a diagrammatic buildingblock. This procedure has been carried out, initially byDe Dominicis [28, 29] and later in more generality by DeDominicis and Martin [30, 31] and leads to a functional

7.2. Defining equations 67

we will call the Ξ-functional Ξ[G,Γ]. In the latter worksthe derivation has been carried out for a very generalmany-body system with not only one- and two-bodyinteractions but also with 1

2-body and 3

2-body interac-

tions that describe Bose-condensed and superconduct-ing phases. Unfortunately this leads to rather involvedequations and disguises the simpler case in which thereare only one and two-body interactions. For instance, inthe general Bose-condensed and super-conducting caseno particle-particle and particle-hole contributions tothe fourvertex can be distinguished. The work of DeDominicis and Martin was aimed at demonstrating thatone could express all thermodynamic quantities com-pletely in terms of distribution functions rather thanat a practical application of the formalism. In theirwork there is, therefore, no discussion of approximatefunctionals and of ways of evaluating them. However,nowadays the functionals can be subjected to numeri-cal computation and it is therefore timely to discuss theformalism from this point of view and to present com-putational schemes to evaluate the functionals. This isexactly the purpose of this work.

If we consider the first of the two papers of De Do-minicis and Martin [30] we see that they use a purelyalgebraic approach to construct their functional whichis not capable of displaying its full structure. Their sec-ond paper [31] uses a purely diagrammatic approach toderive in much more detail the structure of the func-tional but the derivation is quite difficult due to nu-merous intricate topological theorems that need to bediscussed in order to avoid double counting of the dia-grams. However, we found that a combination of bothmethods discussed in these two papers leads to a muchquicker derivation of the final results. Therefore, inthis work we derive, in a as simple as possible man-ner, a variational energy or action functional for normalsystems using a purely algebraic method in combina-tion with a diagrammatic analysis. We use, however,one generalization of the formalism of DeDominicis andMartin: since the Green functions are generated bydifferentiation of our functionals with respect to time-nonlocal potentials, the most natural framework to useis the Keldysh Green function technique [32, 33, 34, 35].We therefore consider generally time-dependent sys-tems that are initially in thermodynamic equilibrium.This has two other advantages. Firstly it allows foran elegant discussion of conservation laws which, aswas shown by Baym [36], are closely connected to Φ-derivability. Such conservation laws were earlier dis-cussed for variational energy and action functionalswithin the Φ- and Ψ-formalism in connection with time-dependent density-functional theory [37]. In particularwe will in this paper show that also Ξ-derivable theoriesare conserving. Secondly, the use of finite temperature

allows for an elegant treatment of the boundary condi-tions on the Green functions. These are, for instance,essential in going from the equations of motion for theGreen function to the Dyson equation which will playan important role in our derivations.

The paper is divided as follows. We first discuss somedefinitions that form the basis of our subsequent analy-sis. We then derive self-consistent equations that relatethe Green function and the renormalized fourvertex.Then we provide a general construction of the varia-tional functional using purely algebraic methods andwe subsequently analyze the structure of the functionalusing diagrammatic methods. We then briefly discussthe conserving properties of the functional. Finally wediscuss approximate functionals with details for theirpractical evaluation and present our conclusions andoutlook.

7.2 Defining equations

In the following we will consider a many-body systeminitially in thermodynamic equilibrium. At an initialtime t0 the system is subjected to a time-dependentexternal field. The Hamiltonian of the system in antime-dependent external potential w(xt) is (in atomicunits) given by

H(t) = h0(t) + V (7.1)

where in the usual second quantization notation theone- and two-body parts of the Hamiltonian are givenby

h0(t) =

Zdxψ†(x)h0(xt)ψ(x) (7.2)

V =1

2

Zdxdx′v(r, r′)ψ†(x)ψ†(x′)ψ(x′)ψ(x). (7.3)

Here x = (r, σ) is a space-spin coordinate. The two-body interaction will usually be taken to be a Coulom-bic repulsion, i.e. v(r, r′) = 1/|r − r′|. The one-bodypart of the Hamiltonian has the explicit form

h0(xt) = −1

2∇2 + w(xt) − µ. (7.4)

We further introduced the chemical potential µ in theone-body part of the Hamiltonian of Eq.(7.4) in antic-ipation of a finite temperature treatment of the sys-tem. We first consider the expectation value of anoperator O for the case that the system is initiallyin an equilibrium state before a certain time t0. Fort < t0 the expectation value of operator O in theSchrodinger picture is then given by 〈O〉 = TrρOwhere ρ = e−βH0/Tre−βH0 is the density matrix andH0 is the time-independent Hamiltonian that describesthe system before the perturbation is switched on. We

68 THE RENORMALIZED FOUR-POINT VERTEX

(t0,−iβ)

t0t2

t1

-

6

Figure 7.1: The Keldysh contour drawn in the complextime plane

further defined β = 1/kBT , with kB the Boltzmannconstant, to be the inverse temperature, and the traceinvolves a summation over a complete set of states inthe Hilbert space. After we switch on the field theexpectation value becomes a time-dependent quantitygiven by

〈O〉(t) = TrnρOH(t)

o(7.5)

where OH(t) = U(t0, t)O(t)U(t, t0) is the operator inthe Heisenberg picture. The evolution operator U ofthe system is defined as the solution to the equations

i∂tU(t, t′) = H(t)U(t, t′) (7.6)

i∂t′ U(t, t′) = −U(t, t′)H(t′) (7.7)

with the boundary condition U(t, t) = 1 . The formalsolution of Eq. (7.6) can be obtained by integration toyield U(t, t′) = T exp (−i

R t

t′dτH(τ )) for t > t′ with a

similar expression with anti-chronological time-ordering

for t′ > t. The operator e−βH0 can now be regardedas an evolution operator in imaginary time, i.e. U(t0 −iβ, t0) = e−βH0 , if we define H(t) to be equal to H0

on the contour running straight from t0 to t0 − iβ inthe complex time plane. We can therefore rewrite ourexpression for the expectation value as

〈O〉(t) =Tr

nU(t0 − iβ, t0)U(t0, t)OU(t, t0)

o

TrnU(t0 − iβ, t0)

o (7.8)

If we read the time arguments of the evolution operatorsin the numerator of this expression from left to right wemay say that the system evolves from t0 along the realtime axis to t after which the operator O acts. Thenthe system evolves back along the real axis from time tto t0 and finally parallel to the imaginary axis from t0to t0 − iβ. This observation motivates us to define thefollowing action functional (compare with the actionfunctionals used in Refs.[36, 38])

Y = i lnTrnU(t0 − iβ, t0)

o, (7.9)

where we define the evolution operator on the contouras

U(t0 − iβ, t0) = TC exp(−iZdtH(t)). (7.10)

Here the integral is taken on the contour and TC de-notes time-ordering along the contour [33, 35]. Whenwe evaluate this quantity for the equilibrium system wesee that

iY = − ln Trne−βH0

o= βΩ (7.11)

where Ω is the grand potential. Therefore the totalenergy E of the system is obtained from the zero-temperature limit

limT→0

iY

β= lim

T→0Ω = E − µN (7.12)

where N denotes the number of particles in the system.Let us now see how this functional can be used as agenerating functional by making variations with respectto parameters in the Hamiltonian. To do this one needsto consider changes in the evolution operator U whichare readily evaluated using Eqs.(7.6) and (7.7). Forinstance, when we make a perturbation δV (t) in theHamiltonian we have using Eq.(7.6)

i∂t δU(t, t′) = δV (t)U(t, t′) + H(t)δU(t, t′) (7.13)

with a similar differential equation with respect to t′

and boundary condition δU(t, t) = 0. The solution tothis equation is given by

δU(t, t′) = −iZ t

t′dτU(t, τ )δV (τ )U(τ, t′) (7.14)

from which variations in the action can be calculated.For instance, if we choose the perturbation to be a time-dependent and spatially nonlocal potential of the form

δV (t) =

Zdx1dx2 δu(x1,x2, t)ψ

†(x1)ψ(x2) (7.15)

we obtain the time-dependent one-particle density ma-trix as a functional derivative with respect to Y

〈ψ†(x1)ψ(x2)〉(t) =δY

δu(x1,x2, t). (7.16)

Similarly, when we consider a time-dependent two-bodypotential of the form

δV (t) =

Zd(x1x2x3x4) δV (x1x2x3x4, t)

×ψ†(x1)ψ†(x2)ψ(x3)ψ(x4) (7.17)

we obtain the time-dependent two-particle density ma-trix as a derivative

〈ψ†(x1)ψ†(x2)ψ(x3)ψ(x4)〉(t)

=δY

δV (x1x2x3x4, t)(7.18)

7.3. Hedin’s equations 69

Note that in order to derive Eqs.(7.16) and (7.18) wehad to make variations δu and δV for time-variableson the contour. After the variation is made all observ-ables are, of course, evaluated for physical quantitiesthat are the same on the upper and lower branch ofthe contour. In the remainder of the paper we willheavily use the action functional as a generating func-tional for the many-body Green functions. To do thiswe have to generalize the time-local potentials u andV to time-nonlocal ones, such that the derivatives of Ywith respect to these potentials become time-orderedexpectation values that we can identify with the one-and two-particle Green functions G and G2. By a sub-sequent Legendre transform we then can construct avariational functional in terms of G and G2. Let usstart out by defining the n-body Green function as

Gn(1 . . . n, 1′ . . . n′) = (7.19)

(−i)n〈TC [ψH(1) . . . ψH(n)ψ†H(1′) . . . ψ†

H(n′)]〉

where we introduced the short notation 1 = (x1t1) andwhere we defined the expectation value of a Heisenbergoperator as

〈O〉 =Tr

nU(t0 − iβ, t0)OH(t)

o

TrnU(t0 − iβ, t0)

o . (7.20)

The many-body Green functions satisfy the followinghierarchy equations [39, 40] which connect the n-bodyGreen function to the n + 1 and n − 1 body Greenfunction:

(i∂t1 − h0(1))Gn(1 . . . n, 1′ . . . n′) =nX

j=1

δ(1j′)(−1)n−jGn−1(2 . . . n, 1′ . . . j′ − 1, j′ + 1 . . . n′)

−iZdxv(x1,x)Gn+1(1 . . . n,xt1,xt

+1 , 1

′ . . . n′). (7.21)

These equations follow directly from the definition ofthe Green functions, the anti-commutation relations ofthe field operators and the equations of motion of theevolution operators in Eq.(7.6) and (7.7). The Greenfunctions are defined for time-arguments on the timecontour. Such contour Green functions were first in-troduced by Keldysh [32] and are often denoted asKeldysh Green functions [33, 34, 35] and play an impor-tant role in nonequilibrium systems. The one-particleGreen function G1 = G obeys the boundary conditionG(x1t0, 2) = −G(x1t0 − iβ, t2) as is readily derivedusing the cyclic property of the trace. The propertyG(1,x2t0) = −G(1,x2t0 − iβ) for the other argumentis likewise easily verified as well as similar relations for

V0(1234) = 4 3

1 2

= 4 3

1 2

− 4 3

1 2

Figure 7.2: Vertex corresponding to the Hugenholtzdiagram technique.

the n-body Green functions. These boundary condi-tions are sometimes referred to as the Kubo-Martin-Schwinger conditions [41, 39] and are essential in solv-ing the equations of motion for the Green function [35].After these preliminaries we are now ready to derive theequations that connect the one- and two-body Greenfunctions which we will use to construct the variationalfunctional Y .

7.3 Hedin’s equations

In order to derive a variational energy functional interms of the Green function G and the renormalizedfour-vertex Γ we start out by deriving coupled equa-tions between these quantities, similar to the familiarHedin equations [18]. However, instead of the usualcoupled equations in terms of the Green function Gand the screened interaction W we have equations interms of the Green function G and the fourvertex Γ.Since our aim is to derive equations in terms of therenormalized fourvertex it is advantageous to write ourequations in terms of the bare fourvertex first. This ismost conveniently done within the Hugenholtz diagramtechnique [24, 25, 26, 27]. We will therefore first rewritethe two-particle interaction as a fourpoint function as

V =1

2

Zdxdx′v(r, r′)ψ†(x)ψ†(x′)ψ(x′)ψ(x)

=1

4

Zd(x1x2x3x4)V0(x1x2x3x4)

×ψ†(x1)ψ†(x2)ψ(x3)ψ(x4) (7.22)

where we defined

V0(x1x2x3x4) = v(r1, r2)[δ(x2 − x3)δ(x1 − x4)

−δ(x1 − x3)δ(x2 − x4)] (7.23)

This term is used as a basic entity in the Hugenholtzdiagram technique and is displayed pictorially in fig.7.2.We now make use of the fact that the Green functioncan be obtained as a derivative of the functional

iY [u] = − ln Tr U [u](t0 − iβ, t0) (7.24)

70 THE RENORMALIZED FOUR-POINT VERTEX

with respect to a nonlocal (in space and time) potentialu(12), where

U [u](t0 − iβ, t0) = TC exp(−iZdtH(t)−

i

Zd1

Zd2 ψ†(x1)u(12)ψ(x2)) . (7.25)

Since this expression contains a double time-integralone has to define precisely how the time-ordering in thisequation is defined. The details of this are presented inAppendix A where we further show that

G(12) = iδY [u]

δu(21). (7.26)

By a subsequent differentiation (see Appendix A ) wecan obtain the two-particle Green function as

G2(1234) = − δG(14)

δu(32)+G(14)G(23). (7.27)

If the derivatives are taken at u = 0 we obtain theGreen functions as defined in Eq.(7.20). If the deriva-tive is taken at nonzero u then there is no direct relationbetween the Green function and expectation values oftime-ordered field operators. However, as shown in theAppendix A the Green functions in the presence of anonlocal potential u still satisfy a set of hierarchy equa-tions. The first ones are

(i∂t1 − h(1))G(11′) = δ(11′)

+

Zd2u(12)G(21′)

− i

2

Zd(234)V (1234)G2(4321

′) (7.28)

and its adjoint

(−i∂t′1− h(1′))G(11′) = δ(11′)

+

Zd2G(12)u(21′)

− i

2

Zd(234)G2(1234)V (4321′) (7.29)

Here we defined

V (1234) = v(r1, r2)δ(t1, t2)[δ(23)δ(14)

−δ(13)δ(24)]θ1234 (7.30)

where δ(ij) = δ(ti, tj)δ(xi − xj) and θ1234 = 1 ift1 > t2 > t3 > t4 (on the contour) and zero other-wise. The function θ1234 therefore ensures that the op-erators have the proper time-ordering before the equaltime limits, described by the delta functions, are taken.In the next section we will also allow for more generalforms of V (1234) in order to obtain the two-particle

Green function as a functional derivative with respectto V . The higher order hierarchy equations relate thetwo-particle Green function to the one- three-particleGreen function and so on. To cut this hierarchy chainit is customary to introduce the self-energy operator Σand its adjoint Σ by the equations

Zd2Σ(12)G(21′) =

− i

2

Zd(234) V (1234)G2(4321

′) (7.31)

Zd2G(12)Σ(21′) =

− i

2

Zd(234)G2(1234)V (4321′) (7.32)

such that we have the equations of motion

(i∂t1 − h(1))G(11′) = δ(11′)

+

Zd2(u(12) + Σ(12))G(21′) (7.33)

(−i∂t′1− h(1′))G(11′) = δ(11′)

+

Zd2G(12)(u(21′) + Σ(21′)). (7.34)

In order to derive a self-consistent set of equations wehave to give a relation between the two-particle Greenfunction and the self-energy. We first note that Σ = Σ.This can derived by applying to Eq.(7.32) the operator(i∂t1 − h(1)) and to Eq.(7.31) the operator (−i∂t′

1−

h(1′)). With the use of the equations of motion of theone- and two-particle Green functions from Eq.(7.21)the result then follows. As a remark we note that formore general initial conditions Σ is not longer equal toΣ [42]. From the equality of Σ and Σ it follows thatthe Green function has a unique inverse given by theDyson equation

G−1(12) = (i∂t1 − h(1))δ(12) − u(12) − Σ(12)

= G−10 (12) − u(12) − Σ(12) (7.35)

which satisfies

Zd2G−1(12)G(21) =

Zd2G(12)G−1(21′) = δ(11′)

(7.36)For later reference we also defined the inverse G−1

0 ofthe noninteracting Green function in Eq.(7.35). We arenow ready to express the two-particle Green functionin terms of the self-energy. If we differentiate Eq.(7.36)with respect to u we obtain

δG(14)

δu(32)= −

Zd(56)G(15)

δG−1(56)

δu(32)G(64) (7.37)

7.4. Construction of a variational functional 71

Σ = + Γ

Γ = − δΣδG+ δΣδG Γ

Figure 7.3: Graphical display of the Hedin equationsthat relate the selfenergy Σ to the vertex Γ.

If we subsequently differentiate Eq.(7.35) with respectto u we have

δG−1(56)

δu(32)= −δ(35)δ(26) − δΣ(56)

δu(32)(7.38)

and we see that the combination of Eqs.(7.37), (7.38)and (7.27) gives an expression for the two-particleGreen function in terms of the self-energy

G2(1234) = G(14)G(23) −G(13)G(24)

−Zd(56)G(15)G(64)

δΣ(56)

δu(32)(7.39)

The first line in Eq.(7.39) is simply the Hartree-Fock ap-proximation to the two-particle Green function whereasthe second line describes the higher order terms. If wedefine the renormalized fourvertex Γ by the equation

δΣ(56)

δu(32)= −

Zd(78) Γ(5786)G(27)G(83) (7.40)

then the two-particle Green function has the form

G2(1234) = G(14)G(23) −G(13)G(24)

+

Zd(5678)G(15)G(27)Γ(5786)G(83)G(64) (7.41)

This expression is displayed pictorially in Fig.7.4. Thefourvertex Γ has the interpretation of a renormalizedinteraction that describes the scattering of two particlesand will play an important role in our energy functionallater. From Eqs.(7.31) and (7.41) we see that we canwrite the self-energy in terms of Γ as

Σ(18) = −iZd(23)V (1238)G(32)

− i

2

Zd(234567)V (1234)G(36)

× G(45)Γ(5678)G(72) (7.42)

where in the derivation we used that V (1234) =−V (1243). To close the set of equations we finally note

that

δΣ(12)

δu(34)=

Zd(56)

δΣ(12)

δG(56)

δG(56)

δu(34)

= −Zd(5678)

δΣ(12)

δG(56)G(57)

δG−1(78)

δu(34)G(86)

=

Zd(56)

δΣ(12)

δG(56)G(53)G(46) +

Zd(5678)

δΣ(12)

δG(56)G(57)G(86)

δΣ(78)

δu(34)(7.43)

Therefore from Eq.(7.43) and (7.40) we obtain

Γ(1234) = − δΣ(14)

δG(32)+

Zd(5678)

δΣ(14)

δG(65)G(67)G(85)Γ(7238) (7.44)

The Eqns.(7.42) and (7.44), which are pictorially dis-played in Fig.7.3, represent a self-consistent set of equa-tions, equivalent to the so-called Hedin equations [18],that generate the perturbation series for the self-energyΣ[G, V ] in terms of the Green function and the inter-action V . For instance, if one starts by taking Γ = 0 inEq.(7.42) then from Eq.(7.44) one obtains an improvedfourvertex Γ which inserted in Eq.(7.42) leads to an im-proved self-energy. In the next section we will show howthe equations derived here can be used to construct theaction or grand potential in terms of G and Γ.

7.4 Construction of a varia-

tional functional

In this section we will construct a variational energy oraction functional of the dressed Green function G andthe renormalized four-vertex Γ. The main reason for in-vestigating such a functional is to obtain in a simple waycontributions to the total energy that correspond to theinfinite summation of ladder-type diagrams. Such dia-grams correspond to an infinite number of terms in theΦ or Ψ-functional. In the new variables G and Γ wehave a corresponding functional Ξ[G,Γ] . In order toderive the Ξ-functional, which we will denote as the DeDominicis functional [28, 29, 30, 31, 43], we start withthe action functional

iY [u, V ] = − ln Tr U [u, V ](t0 − iβ, t0) (7.45)

72 THE RENORMALIZED FOUR-POINT VERTEX

which we will regard as a functional of u and V , wherewe defined

U [u, V ](t0 − iβ, t0) = TC exp(−iZdtH0(t)

−iZd1

Zd2 ψ†(x1)u(12)ψ(x2)

− i

4

Zd(1234)V (1234)

×ψ†(x1)ψ†(x2)ψ(x3)ψ(x4)) (7.46)

Here V (1234) is a general time-dependent two-body in-teraction which we require to have the following sym-metry properties

V (1234) = −V (2134) = −V (1243) = V (2143) (7.47)

This will guarantee that the Feynman rules of theHugenholtz diagram method are satisfied. Eventually,when we have derived our equations, we will set V equalto expression V0 of (7.30). To give precise meaning toexpression Eq.(7.46) we again have to specify how thetime-ordering is defined when we expand the exponent.This is done in Appendix B where we show that

iδY

δu(21)= G(12) (7.48)

iδY

δV (4321)= − i

4G2(1234) (7.49)

In Appendix B it is further demonstrated that theseone- and two-particle Green functions are related bythe equations of motion of Eq.(7.28) and (7.29). By aLegendre transform we can now construct a functionalof G and G2

F [G,G2] = iY [u[G,G2], V [G,G2]]

−Zd(12)u(21)G(12)

+i

4

Zd(1234)V (4321)G2(1234) (7.50)

where we now regard u and V as functionals of G andG2. This functional satisfies

δF

δG(12)= −u(21) (7.51)

δF

δG2(1234)=i

4V (4321). (7.52)

Therefore the functional

iY [G,G2] = F [G,G2]

+

Zd(12)u(21)G(12)

− i

4

Zd(1234)V (4321)G2(1234) (7.53)

G2(1234) = 4 3

1 2

+ 4 3

1 2

+ Γ

4

1

3

2

Figure 7.4: Definition of the renormalized fourvertexΓ

for fixed u and V is a stationary functional of G andG2, i.e.

iδY

δG(12)= 0 (7.54)

iδY

δG2(1234)= 0 (7.55)

where we will eventually be interested in the case u =0 and V = V0. We can now modify the functionalY [G,G2] such that, rather than the two-particle Greenfunction, we can use the renormalized fourvertex Γ asa basic variable. For this purpose we use Eq.(7.41)which is displayed pictorially in fig.7.4 and which givesG2[G,Γ] as an explicit functional of G and Γ. We thendefine the functional

H [G,Γ] = F [G,G2[G,Γ]] (7.56)

which is a functional of the Green function G and thefourvertex Γ. Then for fixed Γ we have

δH

δG(12)=

δF

δG(12)+

Zd(3456)

δF

δG2(3456)

δG2(3456)

δG(12)

= −u(21) − Σ(21) − ΣC(21) (7.57)

where we defined

Σ(12) = ΣHF (12) + ΣC(12) (7.58)

ΣHF (14) = −iZd(23)V (1234)G(32) (7.59)

ΣC(18) = − i

2

Zd(234567)V (1234)G(36)

×G(45)Γ(5678)G(72) (7.60)

ΣC(18) = − i

2

Zd(234567)Γ(1234)G(36)

×G(45)V (5678)G(72) (7.61)

From Eq.(7.33) we recognize these terms as selfenergydiagrams. They are displayed graphically in Fig. 7.5.We recognize the first term in Eq.(7.59) for V = V0

as the Hartree-Fock part of the self-energy. The sec-ond part ΣC of Eq.(7.60) involving the fourvertex Γdescribes the time-nonlocal correlation part of the self-energy. The third part ΣC on Eq.(7.61) is the adjointcorrelation part of the self-energy. As mentioned earlier

7.4. Construction of a variational functional 73

ΣHF = ΣC =Γ

ΣC =ΓFigure 7.5: Graphical display of the self-energy terms.The small dot denotes the bare vertex V and the bigsquare denotes the full four-vertex Γ.

we can show from the Kubo-Martin-Schwinger bound-ary conditions for a system initially in thermodynamicequilibrium that ΣC(12) = ΣC(12). However, in thefollowing we will keep the tilde on the self-energy tokeep track of the origin of this term. For fixed G wecan also calculate the derivative with respect to Γ forwhich we have

δH

δΓ(1234)=

Zd(5678)

δF

δG2(5678)

δG2(5678)

δΓ(1234)

=i

4V (4321) (7.62)

where we defined

V (1234) =

Zd(5678)G(15)G(26)V (5678)G(73)G(84)

(7.63)which is simply a bare vertex dressed with two ingoingand two outgoing dressed Green function lines. Usingthe functional H we can now regard the expression iYof Eq.(7.53) as a functional of G and Γ, i.e.

iY [G,Γ] = H [G,Γ]

+

Zd(12)u(21)G(12)

− i

4

Zd(1234)V (4321)G2[G,Γ](1234) (7.64)

which is a stationary functional of G and Γ for fixedu and V . We have thus achieved our first goal andexpressed the action iY as functional of G and Γ. Ournext step is to specify the functional H in more detail.

The variations in H are given by the expression

δH =

Zd(12)(−u(21) − Σ(21) − ΣC(21))δG(12)

+i

4

Zd(1234)V (4321)δΓ(1234)

=

Zd(12)(G−1(21) −G−1

0 (21) − ΣC(21))δG(12)

+i

4

Zd(1234)V (4321)δΓ(1234) (7.65)

and hence we see that it is convenient to split up H asfollows

H [G,Γ] = −tr˘ln(−G−1)

¯

−tr˘G−1

0 (G−G0)¯− Ξ[G,Γ] (7.66)

This equation defines a new functional Ξ[G,Γ] whichwill be the central object for the rest of the paper. InEq.(7.66) we further defined the trace tr (not to beconfused with the thermodynamic trace Tr) as

trA =

Zd1A(1, 1+) (7.67)

where 1+ denotes that time t1 is approached from aboveon the contour. The definition of the Ξ-functional inEq.(7.66) is convenient because then we have

δH

δG(12)= G−1(21) −G−1

0 (21) − δΞ

δG(12)(7.68)

δH

δΓ(1234)= − δΞ

δΓ(1234)(7.69)

and therefore from Eq.(7.65) we see that the functionalΞ[G,Γ] satisfies the equations

δΞ

δΓ(1234)= − i

4V (4321) (7.70)

δΞ

δG(12)= ΣC(21) (7.71)

The functional Ξ is therefore directly related to thecorrelation part of the self-energy. To describe the cor-relations in the system it is therefore necessary to fur-ther study the structure of the Ξ-functional, which wewill do in detail in the next section.

Note that in Eq.(7.66) we could also have writtenln(G−1) rather than ln(−G−1). These terms differ onlyby a (possibly infinite) constant and depend on the def-inition of the branch cut of the logarithm. However, theparticular definition here reduces properly to the grandpotential of the noninteracting system when the inter-actions are switched off [10]. The final De Dominicis

74 THE RENORMALIZED FOUR-POINT VERTEX

G2(1234) = 4 3

1 2

+ 4 3

1 2

+ 4 3

1 2

(a) (b) (c)

+1 2

4 3

+ 1 2

4 3

+ 1 2

4 3

+ . . .

(d) (e) (f)

Figure 7.6: Expansion of the 2-particle Green functionG2 in terms of the full G. The dot denotes the barevertex V .

functional (for u = 0 ) is thus given from Eq.(7.64) and(7.66) by

iY [G,Γ] = −tr˘ln(−G−1)

¯− tr

˘G−1

0 (G−G0)¯

−Ξ[G,Γ]

− i

4

Zd(1234)V0(1234)G2[G,Γ](4321) (7.72)

We can check that in the absence of interactions we haveiY = −tr ln−G−1

0 which yields the grand potential ofthe noninteracting system, as we will discuss in moredetail later. Let us now check the variational property.The derivatives of iY with respect to G and Γ are givenby

iδY

δΓ(1234)=i

4(V (4321) − V0(4321)) (7.73)

iδY

δG(12)= G−1(21) −G−1

0 (21) − ΣC(21; V )

+Σ(21; V0) + ΣC(21; V0) (7.74)

where we used that

− i

4

δ

δG(56)

Zd(1234)V0(1234)G2[G,Γ](4321)

= Σ(65; V0) + ΣC(65;V0)(7.75)

The variational equations that are obtained by puttingthe derivatives (7.73) and (7.74) equal to zero, are obvi-ously solved for the G and Γ that self-consistently solvethe equations

G−1 = G−10 − Σ[G,Γ] (7.76)

V0 = V [G,Γ] (7.77)

where Σ is calculated from Eqs.(7.59) and Eq.(7.60).Therefore the functional Y [G,Γ] is stationary when-ever the Dyson equation is obeyed and whenever the

electron-electron interaction expanded in G and Γ isequal to the specified interaction V0. Equation (7.72)for the variational functional Y [G,Γ] is the first basicresult of this work. However, before it can be usedin actual calculations we have, among others, to spec-ify the specific structure of the functional Ξ[G,Γ]. Wewill show that for several infinite series of diagrammaticterms contributing to this functional we can find ex-plicit expressions in terms of G and Γ. To do this wefirst have to study the functional V [G,Γ] of Eq.(7.77).This is the topic of the next section.

7.5 Structure of the Ξ-

functional

In this section we analyze in more detail the diagra-matic structure of the fourvertex Γ and the functionalV [G,Γ] of Eq.(7.77) which will allow us to obtain moreexplicit expressions for the functional Ξ. These quanti-ties can be directly obtained from a diagrammatic ex-pansion of the two-particle Green function. If we ex-press the diagrams in terms of the fully dressed Greenfunction G we only need to consider diagrams that donot contain any self-energy insertions. Since differentauthors use different definitions and drawing conven-tions for the two-particle Green function, it is impor-tant to be clear about them. We strictly follow thesign, loop rule and drawing conventions of reference [40]with the small difference that we use Hugenholtz dia-grams [24, 25, 26, 27]. For clarity our Feynman rulesare given in Appendix C. In fig. 7.6 we show the firstand second order Hugenholtz diagrams in terms of thefully dressed Green function G that contribute to thetwo-particle Green function G2. We see that we canwrite Γ as a sum of four classes of diagrams. There arethree classes of the form (ab, cd) which denote diagramswhich by removal of two internal Green function linescan separate the diagram into two parts, one part beingconnected to the external points ab and one part beingconnected to points cd. The class (12, 34) contains di-agrams of the particle-particle type, such as diagram(d) in fig.7.6, and will be denoted by Γpp. There aretwo classes of particle-hole type, namely (14, 32) and(13, 24) which will be denoted by ΓA

ph and ΓBph. Exam-

ples of diagrams of these types are diagrams (e) and(f) in fig.7.6. The remaining diagrams which do notfall into one of these classes are denoted by Γ0 (such asdiagram (c) in fig.7.6). We can therefore write

Γ(1234) = Γpp(1234) + ΓAph(1234)

+ΓBph(1234) + Γ0(1234) (7.78)

The simplest diagram in class Γ0 is simply the barevertex iV (1234) (i.e.diagram (c) in fig.7.6, the factor i

7.5. Structure of the Ξ-functional 75

follows from the Feynman rules in Appendix C). Sincethis diagram is special we separate it off from Γ0 anddefine the remaining diagrams Γ′

0 by the equation

Γ0(1234) = Γ′0(1234) + iV (1234) (7.79)

Using Eq.(7.78) we can then write

−iV (1234) = Γpp(1234) + ΓAph(1234)

+ΓBph(1234) + Γ′

0(1234) − Γ(1234) (7.80)

We will now first show how all the terms on the right-hand side of this equation can be constructed as a func-tional of Γ. When we have done this we can insert thisfunctional into Eq.(7.70) and perform the integrationwith respect to Γ and thereby construct our desiredfunctional Ξ[G,Γ].Let us start with the particle-particle diagrams Γpp.The contribution of all diagrams for Γpp can be writ-ten as sums of blocks of diagrams J connected withtwo parallel Green function lines (see fig.7.7 ). Each ofthese blocks J contains diagrams which cannot discon-nect points (12) and (34) by cutting two Green functionlines (such blocks are called simple with respect to (12)and (34) in the terminology of De Dominicis) and there-fore each J-block does not contain diagrams of the typeΓpp. We thus have

J(1234) = Γ(1234) − Γpp(1234) (7.81)

We introduce a convenient matrix notation

〈12|J |34〉 = J(1234) (7.82)

〈12|GG|34〉 = G(13)G(24) (7.83)

Within this notation we can, for instance, convenientlywrite C = AB instead of

C(1234) = 〈12|AB|34〉

=

Zd(56)〈12|A|56〉〈56|B|34〉

=

Zd(56)A(1256)B(5634) (7.84)

If we use this notation, then from the Feynman rulesin Appendix C one can readily convince oneself that inmatrix notation we simply have

Γ = J + Γpp

= J +1

2JGGJ + (

1

2)2JGGJGGJ + . . .

= J +1

2JGGΓ (7.85)

where for every pair of Green function lines we haveto add a factor of 1

2(see [25, 26, 27, 43]). This fol-

lows because for any diagram contributing to J , the

Γ(1234) = J1 2

4 3+J

J

4 3

21

+

J

J

J

4 3

21

+ . . .

Figure 7.7: Expression of Γ in terms of J-blocks

diagram with outgoing lines interchanged leads to thesame diagram for Γ (for the simple diagram iV in J itfollows from Feynman rule 5 in Appendix C). RelationEq.(7.85) allows us to express Γpp in terms of Γ. Wehave

J = Γ(1 +1

2GGΓ)−1 (7.86)

In combination with Eq.(7.81) this then gives

Γpp = Γ − Γ(1 +1

2GGΓ)−1 (7.87)

which expresses Γpp in terms of Γ. Let us now do thesame for the particle-hole diagrams. Since

ΓBph(1234) = −ΓA

ph(2134) (7.88)

we only need to construct ΓAph as a functional of Γ. For

the particle-hole diagrams ΓAph we can follow a similar

reasoning as for Γpp and we first write Γ in terms ofrepeated blocks I given by

I(1234) = Γ(1234) − ΓAph(1234). (7.89)

The expression for Γ in terms of I is displayed in fig.7.8.If we use the notation

〈41|I |23〉 = I(1234) (7.90)

〈12|dGG|34〉 = G(31)G(24) (7.91)

where in the first term we defined a new matrix I bya cyclic permutation of the indices, then (again usingthe Feynman rules of Appendix C) we have in matrixnotation

Γ = I + ΓAph

= I − IdGGI + IdGGIdGGI − . . .

= I − IdGGΓ (7.92)

where the alternating signs in Eq.(7.92) are related toFeynman rule 4 in Appendix C. As a remark we note

76 THE RENORMALIZED FOUR-POINT VERTEX

Γ(1234) = I1 2

4 3

+

I I

4

1

3

2

+

I I I

4

1

3

2+ . . .

Figure 7.8: Expression of Γ in terms of I-blocks

that from Eq.(7.92) and (7.44) we see that there is asimple relation between I and the self-energy:

I(1234) = − δΣ(14)

δG(32)(7.93)

One can indeed check, by iterating Hedin’s equations(7.42) and (7.44), that the term δΣ/δG only yields di-agrams that contribute to I . We can now express ΓA

ph

in terms of Γ. We have

I = Γ(1 − dGGΓ)−1 (7.94)

which gives

ΓAph = Γ − Γ(1 − dGGΓ)−1 (7.95)

Before discussing the last set of diagrams Γ′0 let us see

if we can integrate the functionals Γpp and ΓAph that we

obtained sofar. To do this we first make a general re-mark about functional derivatives. We consider a givenfourpoint function a[Γ](1234) that we want to integratewith respect to Γ to obtain a functional A, i.e.

δA =

Zd(1234) a[Γ](1234)δΓ(1234) (7.96)

In our case we want to do this for a[Γ] being Γpp, ΓAph,

ΓBph and Γ′

0. Because δΓ has the symmetry property ofEq.(7.47) this can also be written as

δA =1

4

Zd(1234) [a(1234) − a(2134)

+a(3412) − a(1243)]δΓ(1234) (7.97)

Therefore any part of a which is symmetric in the in-dices (12) or (34) (or anti-symmetric with respect tothe interchange of pair (12) and pair (34)) will not con-tribute to this variation. Therefore only certain (anti-)symmetric parts of a are uniquely determined as func-tional derivatives. This does not pose a problem if the

functional a we want to integrate already has the samesymmetry as Γ. This applies for instance to Γpp and Γ′

0.However, the function ΓA

ph(1234) is not anti-symmetricin the indices (12) and (34). However, the combination

ΓAph(1234)−ΓA

ph(2134) = ΓAph(1234)+ΓB

ph(1234) (7.98)

has this property and therefore

2

Zd(1234) ΓA

ph(1234)δΓ(1234) = (7.99)

Zd(1234) [ΓA

ph(1234) + ΓBph(1234)]δΓ(1234)

We can therefore obtain ΓAph + ΓB

ph as a functionalderivative by formally integrating ΓA

ph and multiplyingthe resulting functional by 2. With this in mind we cannow address the integration of V in the right hand sideof Eq.(7.70) with respect to Γ. Using Eq.(7.80) we canwrite

−iV (1234) = Γpp(1234)

+ ΓAph(1234) + ΓB

ph(1234)

+ Γ′0(1234) − Γ(1234) (7.100)

where the expressions with the tilde are defined as inEq.(7.63). Let us start by integrating Γpp with respectto Γ. Using Eq.(7.87) and taking into account the factor1/4 in Eq.(7.70) we have

1

4Γpp =

1

4GGΓppGG

=1

4GGΓ[1 − (1 +

1

2GGΓ)−1]GG

=1

4GGΓGG− 1

2[1 − (1 +

1

2GGΓ)−1]GG

=δLpp[G,Γ]

δΓ(7.101)

where we defined

Lpp[G,Γ] =1

8tr GGΓGGΓ − 1

2tr GGΓ

+tr

ln(1 +

1

2GGΓ)

ff(7.102)

In this expression the trace tr (not to be confused withthe thermodynamic trace Tr) for two-particle functionsis defined as

tr A =

Zd(12)〈12|A|12〉 (7.103)

The diagrammatic expansion of the functional Lpp isdisplayed in the upper part of fig.7.9. Let us now con-sider the particle-hole diagrams. Since

trn

dGGΓAph

dGGδΓo

= trn

ΓAphδΓ

o(7.104)

7.5. Structure of the Ξ-functional 77

Lpp[G, Γ] =1

3

1

23 −1

4

1

24 +1

5

1

25 − . . .

2 Lph[G, Γ] = −1

3 −1

4 −1

5 − . . .

Figure 7.9: Expansion of the functionals Lpp and Lph

in diagrams. The fourvertex Γ is denoted with a bigblack dot.

it is sufficient to integrate dGGΓAph

dGG with respect toΓ. We have using Eq.(7.95)

1

4˜ΓA

ph =1

4dGGΓA

phdGG

=1

4dGGΓ[1 − (1 − dGGΓ)−1]dGG

=1

4dGGΓA

phdGG+

1

4[1 − (1 − dGGΓ)−1]dGG

=1

2

δLph[G,Γ]

δΓ(7.105)

where we defined the functional

Lph[G,Γ] =1

4tr

ndGGΓdGGΓ

o+

1

2tr

ndGGΓ

o

+1

2tr

nln(1 − dGGΓ)

o(7.106)

The diagrammatic expansion of the functional Lph isdisplayed in the lower part of fig.7.9. Now since

trn(ΓA

ph + ΓBph)δΓ

o= 2 tr

nΓA

phδΓo

= 2 trn

dGGΓAph

dGGδΓo(7.107)

we obtain

1

4(ΓA

ph + ΓBph) =

δLph[G,Γ]

δΓ(7.108)

We now collect our results and define

L[G,Γ] = Lpp[G,Γ] + Lph[G,Γ]

−1

8tr GGΓGGΓ (7.109)

This functional L has the property

δL

δΓ=

1

4(Γpp + ΓA

ph + ΓBph − Γ) (7.110)

Using this functional we can now split up the functionalΞ further as

Ξ[G,Γ] = L[G,Γ] + L′[G,Γ] (7.111)

This defines a new functional L′[G,Γ]. Then fromEq.(7.70) we see that if we differentiate both sides ofEq.(7.111) with respect to Γ we obtain

δΞ

δΓ= − i

4V =

1

4(Γpp + ΓA

ph + ΓBph − Γ) +

δL′

δΓ(7.112)

We therefore see by comparing to Eq.(7.80) that thefunctional L′ must satisfy

1

4Γ′

0 =δL′[G,Γ]

δΓ(7.113)

This functional can not be written out explicitly, butsince Γ′

0 is well-defined diagrammatically the functionalL′ does have a diagrammatic expansion. The first termin this expansion is displayed in fig.7.10 together withits functional derivative. Note that the derivative yieldsfour diagrams in accordance with Eq.(7.97). We canfurther consider the functional derivative of the func-tional Ξ[G,Γ] with respect toG. According to Eq.(7.71)this yields self-energy diagrams, as is also seen fromthe diagrammatic expansion of Ξ. The G-derivativesof Lpp, Lph and L′ lead to correlation self-energy dia-grams ΣC,pp[G,Γ], ΣC,ph[G,Γ] and Σ′

C [G,Γ] in termsof G and Γ that fall into different topological classes.

We now again collect our results and find fromEq.(7.72) that the final De Dominicis functional (foru = 0 ) is given by

iY [G,Γ] = −tr˘ln(−G−1)

¯− tr

˘G−1

0 (G−G0)¯

−L[G,Γ] − L′[G,Γ] − i

4tr V0G2[G,Γ](7.114)

We finally write the functional in a different form usingthe Dyson equation of Eq.(7.35)

iY [G,Γ] = −tr˘ln(Σ −G−1

0 )¯− tr ΣG

−L[G,Γ] − L′[G,Γ] − i

4tr V0G2[G,Γ](7.115)

We can readily check the variationally property of thisfunctional. We then find that

iδY = −tr˘[(Σ −G−1

0 )−1 +G]δΣ¯

−trn[Σ(V ) − Σ(V0) + ΣC(V ) − ΣC(V0)]δG

o

+i

4tr

n(V − V0)δΓ

o= 0 (7.116)

whenever V [G,Γ] = V0 for a self-consistent solution ofthe Dyson equation. The variational functional (7.115)together with the variational property (7.116) is thecentral result of this work. In the next sections wewill investigate the practical evaluation of this func-tional. It is important to note that although thefunctionals in Eq.(7.114) and (7.115) are equivalentwhen evaluated on the fully self-consistent G and Γ

78 THE RENORMALIZED FOUR-POINT VERTEX

L′[G, Γ] =1

5 + . . .

δL′

δΓ[G, Γ] =

1

41 2

4 3

+1

42 1

4 3

+1

42 1

3 4

+1

41 2

3 4

+ . . .

Figure 7.10: The first term in the expansion of the L′-functional and its functional derivative with respect toΓ. For clarity in drawing the diagrams for δL′/δΓ weinterchanged the endpoint labels rather than makingthe outgoing lines cross.

obtained from the Dyson equation and V [G,Γ] = V0

this is not true anymore when evaluated at approxi-mate G and Γ. In accordance with ref.[5] the func-tional forms in Eqs.(7.114) and (7.115) will be denotedas the Klein-form and Luttinger-Ward-form of the func-tional Y . It was demonstrated in the Φ-formalism thatthe Luttinger-Ward form of the functional is more sta-ble (has a smaller second derivative) when used for thecalculation of total energies [8]. We will therefore inthe following use the Luttinger-Ward form of the func-tional.

7.6 Ξ-derivable theories are

conserving

In this section we will show that any approximate Ξ-functional leads to a corresponding Φ-functional. Sincewe know from the work of Baym [36] that any Φ-derivable theory is conserving it follows that also Ξ-derivable theories are conserving, i.e. they respect themacroscopic conservation laws, such a momentum, en-ergy and particle number conservation and related con-straints such as the virial theorem [7]. Consider anyapproximate Ξ-functional. Then from the variationalequation

δΞ[G,Γ]

δΓ= − i

4V0 (7.117)

we can construct Γ[G, V0] as a functional of G and thebare interaction V0 (some examples of this procedure

are given in the next section). With the functionalΓ[G, V0] defined in this way we define the following Φfunctional

Φ[G, V0] = −Ξ[G,Γ[G, V0]] − i

4tr V0G2[G,Γ[G, V0]]

(7.118)and the action functional

iY [G,V0] = −tr˘ln(Σ −G−1

0 )¯− tr ΣG + Φ[G, V0]

(7.119)where in this expression self-energy Σ[G,Γ[G, V0]] mustalso be regarded as a functional of G and V0. From thedefinition of Φ it then follows directly that

δΦ = −trnΣCδG

o+i

4tr

nV0δΓ

o

+ trn(Σ + ΣC)δG

o− i

4tr

nV0δΓ

o

= tr ΣδG (7.120)

We therefore obtain the result

δΦ

δG(12)= Σ(21) (7.121)

We further have that the functional Y [G, V0] ofEq.(7.119) is stationary when

0 = −tr˘((Σ −G−1

0 )−1 +G)δΣ¯

−tr

(Σ − δΦ

δG)δG

ff(7.122)

i.e. whenever the Dyson equation is obeyed for a Φ-derivable self-energy. On the basis of the work ofBaym [36] we can therefore conclude that Ξ-derivabletheories are conserving.

7.7 Approximations using the

Ξ-functional

7.7.1 Practical use of the variational

property

After having discussed the general properties of thefunctional Y [G,Γ] we will discuss its use in practicalapproximations. For a given approximation to Ξ[G,Γ]the stationary point of the functional Y corresponds toan approximation for the self-energy and the fourver-tex obtained from a solution of the Dyson equation andof an equation of Bethe-Salpeter type, both of whichneed to be solved to self-consistency. The solution ofthese equations for general electronic systems is compu-tationally very expensive or impossible. However, if weuse the variational property of Y we can save greatlyin computational cost as the full self-consistency step

7.7. Approximations using the Ξ-functional 79

can then be skipped. To illustrate this we let G and Γbe self-consistent solutions to the variational equationsand we let G and Γ be approximations to G and Γ.Then we have that

Y [G, Γ] = Y [G,Γ] +1

2tr

δ2Y

δGδG∆G∆G

ff

+ tr

δ2Y

δGδΓ∆G∆Γ

ff

+1

2tr

δ2Y

δΓδΓ∆Γ∆Γ

ff+ . . . (7.123)

where ∆G = G − G and ∆Γ = Γ − Γ are the devi-ations from the Green function and fourvertex to theself-consistent ones. We see that the error we make inY is only of second order in ∆G and ∆Γ. We maytherefore obtain rather accurate energies from rathercrude inputs. These expectations were indeed borneout by our earlier calculations within the Φ-formalismon atoms and molecules [8]. Obviously the actual errorwe make also depends on how large the second deriva-tives of functional Y are. For this reason the Kleinand Luttinger-Ward forms of the functional performdifferently. In fact, experience within the Φ-functionalformalism has shown that the Luttinger-Ward is morestable than the Klein functional with respect to changesof the input Green function [8].

7.7.2 Approximate Ξ-functionals

In the following we study some approximate schemesusing the Ξ-functional in order to illustrate the for-malism discussed in the preceeding sections. We re-strict ourselves here to the two most simplest examples,the self-consistent second order approximation and theself-consistent T -matrix approximation. A more ad-vanced approximation, also involving the particle-holediagrams, is discussed in the section on the practicalevaluation of the Ξ-functional.

The very simplest nontrivial approximation to the wecan make to the Ξ-functional is to take Lpp = Lph =L′ = 0. This yields the functional

iY2[G,Γ] = −tr˘ln(Σ −G−1

0 )¯− tr ΣG

+1

8tr GGΓGGΓ − i

4tr V0G2[G,Γ] (7.124)

which we will denote by Y2 since it only involves secondorder diagrams. The variational equations yield

G−1 = G−10 − Σ[G,Γ] (7.125)

0 =1

4Γ − i

4V0 (7.126)

which simply implies that Γ = iV0 and that

Σ[G, V0](11′) = −i

Zd(23)V0(1231

′)G(32)

+1

2

Zd(234567)V0(1234)G(36)G(45)

×V0(5671′)G(72) (7.127)

This amounts to a self-consistent solution of the Dysonequation with only second order diagrams. A fully self-consistent solution of these equations for molecules wasrecently carried out by us [7]. One of the next simplestapproximations is obtained by taking L′ = Lph = 0which yields the functional

iYpp[G,Γ] = −tr˘ln(Σ −G−1

0 )¯− tr ΣG

−Lpp[G,Γ] +1

8tr GGΓGGΓ

− i

4tr V0G2[G,Γ] (7.128)

The variational equations correspond to

G−1 = G−10 − Σ[G,Γ] (7.129)

0 = − δLpp

δΓ+

1

4Γ − i

4V0 (7.130)

where Σ is calculated from Eqs.(7.59) and Eq.(7.60).The second variational Eq.(7.130) corresponds to

iV0 = Γ(1 +1

2GGΓ)−1. (7.131)

This equation can be inverted to give

Γ = (iV0)(1 − 1

2GG(iV0))

−1 (7.132)

and expresses the renormalized four-vertex as a sum ofparticle-particle (direct and exchange) ladder diagramsin terms of the bare potential V0. The correspondingself-energy is then readily obtained from Eqs.(7.59) and(7.60) by inserting the Γ of Eq.(7.132) in Eq.(7.60).This approximation is equivalent to the self-consistentT -matrix approximation. It is clear that the set ofapproximations can be made more and more advancedby using more sophisticated approximations for the Ξ-functional. In the following sections we will discuss thenumerical evaluation of iY . We will then among otherthings, consider an approximate fourvertex obtainedfrom the T -matrix approximation as an approximateinput for the evaluation of the energy functional iY ata more sophisticated level of perturbation theory.

80 THE RENORMALIZED FOUR-POINT VERTEX

7.8 Practical evaluation of the

functional

7.8.1 Evaluation of the traces

In this section we discuss the how to evaluate the func-tional Y [G,Γ] in actual applications. Our goal is toevaluate Y for an equilibrium system in which case alltwo-time quantities depend on relative time variableson the vertical stretch of the Keldysh contour. In thatcase it is convenient to go over to a Matsubara represen-tation (we use the notation of Kadanoff and Baym [44])

A(t− t′) =i

β

X

z

e−iz(t−t′)A(z) (7.133)

A(z) =

Z −iβ

0

dtA(t− t′)eiz(t−t′) (7.134)

where the times are imaginary ( t = −iτ for 0 ≤ τ ≤ β)and where z = inπ/β are the Matsubara frequencieswhich run over even or odd integers n depending onwhether A is a bosonic or fermionic function. In thisway the equation of motion for the Green function sim-ply attains the form

(z − h(x1))G(x1x2, z) = δ(x1,x2)

+

Zdx3Σ(x1x3, z)G(x3x2, z) (7.135)

For the traces of two-point functions we have the ex-pression

trA =

Z −iβ

0

dtdxA(1, 1+)

= limη→0+

X

z

Zdx eηzA(x,x, z) (7.136)

For the various traces in the functional Y it is furtherconvenient to introduce a one-particle basis, such thatwe can write

A(x1,x2, z) =X

ij

Aij(z)ϕi(x1)ϕ∗j (x2) (7.137)

Then we have, for instance, that

trAB = limη→0+

X

ij,z

eηzAij(z)Bji(z) (7.138)

If we choose the orbitals to be eigenfunctions of theone-particle Hamiltonian h,

h(x)ϕi(x) = eiϕi(x) (7.139)

then the equation of motion of the Green function at-tains the form

(z − ei)Gij(z) = δij +X

k

Σik(z)Gkj(z) (7.140)

and we see immediately that the noninteracting Greenfunction G0 is given by

G0,ij(z) =δij

z − ei(7.141)

Consequently the grand potential for the noninteractingsystem is given by Ω0 = iY0/β where [10, 27, 45]

Ω0 = − 1

βtr ln

˘−G−1

0

¯

= − 1

βlim

η→0+

X

i

X

z

eηz ln(ei − z)

= − 1

β

X

i

ln(1 + e−βei) (7.142)

In the zero-temperature limit β → ∞ this simply gives

limβ→∞

Ω0 =

NX

i=1

ei (7.143)

where the sum runs over the N occupied electron or-bitals. Note that the chemical potential µ is includedin h such that ei = ǫi − µ where ǫi are the eigenvaluesof the one-body part of the Hamiltonian.

As a next step we will discuss how to evaluate thefunctional on an approximate Green function G and anapproximate vertex Γ. The input Green function willin practice not be a fully interacting Green function butrather one obtained from a local density approximation(LDA) or from a Hartree-Fock approximation. Withapproximate inputs G and Γ the first term in Eq.(7.115)can be written in a computationally convenient formas [5]

−tr lnnΣ[G, Γ] −G−1

0

o

= −tr˘ln(−G−1)

¯− tr

nln(1 − GΣC [G, Γ])

o(7.144)

where we defined

ΣC [G, Γ] = Σ[G, Γ] − ΣHF [G] (7.145)

and the Green function G from the Dyson equation

G = G0 +G0ΣHF [G]G (7.146)

The Green function G therefore presents the first itera-tion towards the Hartree-Fock Green function startingfrom G. Therefore G = GHF when we take G = GHF

as an input Green function. The term ΣC representsthe correlation part of the self-energy evaluated at anapproximate G and Γ. The reason for introducing G isthat by doing this we have in the last term of Eq.(7.144)eliminated a static part of the self-energy, which makesthis term well defined without a convergence factor and

7.8. Practical evaluation of the functional 81

also makes it decay much faster for large frequencieswhich is computationally advantageous as was shownin Ref. [5]. The first term in Eq.(7.144) can be evalu-ated analytically to give

iY0 = −tr˘ln(−G−1)

¯= −

X

i

ln(1 + e−βei) (7.147)

where ei = ǫi−µ and ǫi are the eigenvalues the Hartree-Fock equations with a nonlocal self-energy ΣHF [G]. Inpractice (for instance for LDA input Green functions)these eigenvalues are close to the true Hartree-Fockeigenvalues. Now the functional Y [G, Γ] can be writ-ten as

iY [G, Γ] = iY0

−trnln(1 − GΣC [G, Γ])

o− tr

nΣ[G, Γ]G

o

−L[G, Γ] − L′[G, Γ] − i

4tr

nV0G2[G, Γ]

o(7.148)

The second term can be evaluated by diagonalizationof GΣC since for a matrix A(z) we have

tr ln(1 − A) = limη→0+

X

z,i

eηz ln(1 − λi(z)) (7.149)

where λi(z) are the eigenvalues of A(z). This completesone part of the evaluation of the functional Y .

7.8.2 Evaluation of the L′ = 0-functional

Let us now discuss the evaluation of the L[G,Γ] andL′[G,Γ] functionals. The evaluation of even the lowestorder term of the L′-functional will be computationallyvery difficult in practice. The first term in the expan-sion of L′ is the pentagon of Fig.(7.10) containing fivefourvertices Γ. Since every fourvertex depends on fourspace-time coordinates the pentagon is (apart from thespin summations) formally an 80-dimensional integral.Fortunately, even the approximation L′ = 0 representsa very sophisticated many-body approximation. Wewill therefore in the following concentrate on this caseand consider the evaluation of the functional

iY [G,Γ] = iY0 − tr˘ln(1 − GΣC [G,Γ])

¯− tr ΣG

−L[G,Γ] − i

4tr V0G2[G,Γ] (7.150)

The evaluation of the first terms in this expression hasbeen discussed in the preceding subsection and we willtherefore concentrate on evaluation of L[G,Γ]. In thisterm the trace is taken over two-particle functions andits evaluation will therefore be slightly different fromthe case discussed above.

As our approximate Γ we will take the sum of allparticle-particle and exchange ladders in terms of V0 forwhich we will eventually take the zero-frequency limit.This is the approximate T -matrix used in Ref. [22].This approximate Γ we will denote as Γ. This approx-imate Γ will be expressed in terms of our approximateGreen function which we will denote with G. Thenfrom Eq.(7.85) we have

Γ = iV0 +i

2V0GGΓ (7.151)

If we write

V0(1234) = δ(t1, t2)δ(t1, t4)δ(t2, t3)V0(x1x2x3x4)(7.152)

with V0(x1x2x3x4) explicitly given in Eq.(7.23), we seethat we can write

Γ(1234) = δ(t1, t2)δ(t3, t4)γ(x1x2x3x4; t1t3) (7.153)

If we further expand γ in a basis as

γ(x1x2x3x4; t1t3) =X

ijkl

γijkl(t1t3)

× ϕ∗i (x1)ϕ

∗j (x2)ϕk(x3)ϕl(x4) (7.154)

then from Eq.(7.151) we see that γ satisfies

γijkl(t1t3) = iδ(t1, t3)V0,ijkl

+i

2

X

pqrs

Z −iβ

0

dt2V0,ijpq

×Gqr(t1, t2)Gps(t1, t2)γrskl(t2t3) (7.155)

which in frequency space attains the form

γijkl(z) = iV0,ijkl −1

2β

X

z1

X

pqrs

V0,ijpq

×Gqr(z1)Gps(z − z1)γrskl(z) (7.156)

(note that for γ we have to sum over the even Mat-subara frequencies). For simple approximate Greenfunctions G of Hartree-Fock or local density type thefrequency sum over z1 is readily evaluated. We arenow ready to evaluate the functionals Lpp[G, Γ] andLph[G, Γ]. They are given by the expressions

Lpp = tr ln(1 + A) − tr A +1

2tr

˘A2

¯(7.157)

Lph =1

2tr ln(1 −B) +

1

2tr B +

1

4tr

˘B2

¯(7.158)

where A = GGΓ and B = dGGΓ. Therefore in order tocalculate Lpp and Lph we have to diagonalize A and B

82 THE RENORMALIZED FOUR-POINT VERTEX

in a two-particle basis. Let us start by the calculationof A. We have for our approximate Γ and G :

Aijkl(t1t2t3t4) = δ(t3, t4)X

pq

Z −iβ

0

dt5Gip(t1, t5)

×Gjq(t2, t5)γpqkl(t5t3) (7.159)

Because of the equal-time delta function in Eq.(7.159)we find that

tr An =

=

Zd(11′ . . . nn′)〈11′|A|22′〉 . . . 〈nn′|A|11′〉

=X

p1...pn

Z −iβ

0

d(t1 . . . tn)Ap1p2(t1, t2) . . . Apnp1(tn, t1)

= limη→0+

X

z

eηzAp1p2(z) . . . Apnp1(z) (7.160)

where pk = (ikjk) are multi-indices and where we de-fined

Aijkl(t1t3) =X

pq

Z −iβ

0

dt5Gip(t1, t5)

×Gjq(t1, t5)γpqkl(t5t3) (7.161)

which in frequency space attains the form

Aijkl(z) =i

β

X

z1,pq

Gip(z1)Gjq(z − z1)γpqkl(z)(7.162)

From diagonalization of Apq(z) where p = (ij) and q =(kl) we then immediately obtain

Lpp[G, Γ] =X

z,p

(ln(1 + λp(z)) − λp(z) +1

2λ2

p(z)) (7.163)

where λp(z) are the eigenvalues of A(z). Let us finallyconcentrate on the evaluation of B. This expression isgiven by

Bijkl(t1t2t3t4) =X

pq

Gqi(t4, t1)

×Gjp(t2, t3)γpklq(t3t4) (7.164)

This expression depends on three relative times whichmakes it awkward to evaluate the logarithm. We there-fore follow reference [22] and replace in frequency spaceγijkl(z) by its zero-frequency limit γijkl(0),

γijkl(t3t4) = γijkl(0)δ(t3, t4) (7.165)

such that

Bijkl(t1t2t3t4) = δ(t3t4)X

pq

Gqi(t4, t1)

×Gjp(t2, t3)γpklq(0) (7.166)

Then, similarly as for the quantity A we have

tr Bn = limη→0+

X

z

eηzBp1p2(z) . . . Bpnp1(z)(7.167)

where

Bijkl(z) =i

β

X

z1,pq

Gqi(z1)Gjp(z1 + z)γpklq(0) (7.168)

Now B(z) is readily diagonalized with respect to itstwo-particle indices to give

Lph[G, Γ] =X

z,p

(1

2ln(1 − λp(z)) +

1

2λp(z) +

1

4λ2

p(z))(7.169)

where λp(z) are the eigenvalues of B(z). The full func-tional L[G, Γ] is then constructed as

L[G, Γ] = Lpp[G, Γ] + Lph[G, Γ] − 1

8tr

˘A2¯

(7.170)

where the last term is easily found by summing thesquares of the eigenvalues of A and performing a fre-quency sum. It finally remains to calculate an explicitexpression for Σ[G, Γ] and to evaluate the last term inEq.(7.150). The self-energy is readily calculated fromEqs.(7.59) and (7.60) to be

Σij(z) = ΣHFij + Σij,C(z) (7.171)

where

ΣHFij =

1

βlim

η→0+

X

z

V0,ipqjGqp(z) (7.172)

and

Σij,C(z) =i

2β

X

z1,z2

X

pqrstu

V0,ipqrGrs(z1)Gqt(z2)

×Gup(z1 + z2 − z)γstuj(0) (7.173)

This expression is, of course, considerably simplifiedwhen we use a diagonal input Green function. Thisfinally concludes the discussion on the practical evalu-ation of the functional.

In summary: evaluation of the functional Y inpractice therefore essentially involves the diagonaliza-tion of the one-particle matrix A(z) of Eq.(7.149) andthe diagonalization of the matrices A(z) and B(z) ofEqns.(7.161) and (7.168) in a two-particle basis followedby a frequency summation. This is, for instance withinthe DMFT approach used by Katsnelson and Lichten-stein [22], a numerically quite feasible procedure.

7.9. Conclusions 83

7.9 Conclusions

In this work we studied variational functionals of theGreen function and the renormalized fourvertex in or-der to calculate total energies for strongly correlatedsystems. The variational functionals were derived byLegendre transform techniques starting from an expres-sion of the action (or grand potential) defined on theKeldysh contour. The structure of the functionals wasfurther analyzed by means of diagrammatic techniques.We finally gave a detailed discussion of the practical useand evaluation of these for different approximate func-tionals. Future applications along the lines describedare intended.

Finally we comment on further applications of thevariatonal functionals. It was found that the Φ andthe Ψ-formalism could be succesfully used to derive ex-pressions for response functions within time-dependentdensity-functional theory (TDDFT) [37]. This wasdone by inserting approximate Green functions G[v],coming from a noninteracting system with a local po-tential v, into the variational functionals. Then thepotentials were optimized by requiring that δY/δv =0. Due to the one-to-one correspondence between thedensity and the potential (as follows from the time-dependent generalization of the Hohenberg-Kohn the-orem [2]) this then implies that we are optimizing atime-dependent density functional. The optimized po-tentials are then to be interpreted as Kohn-Sham po-tentials. In this way one obtains a density functionalfor every diagrammatic expression from the Φ- or Ψ-functional. A similar procedure can now be carried outfor the Ξ-functional.

A further point of future investigation is concernedwith finding the variationally most stable functional. Itwas already mentioned that the Klein and Luttinger-Ward (LW) forms of the functional lead to differentresults. The Luttinger-Ward form was found to be morestable. This is probably due to the fact that the secondderivatives of the LW functional are smaller than thoseof the Klein functional. However, it is very well possiblethat one could derive a better functional that wouldmake the second derivatives even smaller or make themvanish. In that case the errors we make would be onlyto third order in the deviation ∆G of the input Green tothe true self-consistent one. This still remains an issuefor future investigations. Finally we mention that workon implementation of the formalism discussed here is inprogress.

We like to thank Prof. M. I. Katsnelson and Prof.A. I. Lichtenstein for useful discussions and for interestin this work.

84 THE RENORMALIZED FOUR-POINT VERTEX