Quantum Field Theory of Correlated Bose-Einstein condensates: I. Basic Formalism
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Quantum Field Theory of Correlated Bose-Einstein condensates:I. Basic Formalism
Takafumi K ita
Department of Physics, Hokkaido University, Sapporo 060-0810, Japan
Quantum field theory of equilibrium and nonequilibrium Bose-Einstein condensates is formulatedso as to satisfy three basic requirements: the Hugenholtz-Pines relation; conservation laws; identitiesamong vertices originating from Goldstone’s theorem I. The key inputs are irreducible four-pointvertices, in terms of which we derive a closed system of equations for Green’s functions, three-and four-point vertices, and two-particle Green’s functions. It enables us to study correlated Bose-Einstein condensates with a gapless branch of single-particle excitations without encountering anyinfrared divergence. The single- and two-particle Green’s functions are found to share poles, i.e., thestructure of the two-particle Green’s functions predicted by Gavoret and Nozi`eres for a homogeneouscondensate at T = 0 is also shown to persist at finite temperatures, in the presence of inhomogeneity,and also in nonequilibrium situations. I. INTRODUCTION
The aim of this paper is to derive a closed system ofself-consistency equations for the single- and two-particleGreen’s functions of correlated Bose-Einstein conden-sates, which is formally exact and can also be used inpractical calculations to describe both equilibrium andnonequilibrium condensates. This will be performed insuch a way that it automatically meets the three exactrequirements: (i) the Hugenholtz-Pines relation predict-ing a branch of gapless single-particle excitations [1]; (ii)conservation laws [2–4]; (iii) identities among vertices[5, 6] originating from Goldstone’s theorem I, i.e., thefirst proof [7, 8].We have already made similar attempts in terms of(i) and (ii) [9–11]. However, the resulting self-consistentperturbation expansion has encountered an infrared di-vergence, similarly as in the case of simple perturba-tion expansion [12] starting from either the ideal gasor the Bogoliubov theory [13]. which has preventedus from performing practical calculations on correlatedBose-Einstein condensates. A key additional observationhere, which originates from our previous renormalization-group study [6, 14], is that the infrared divergence canonly be removed by extending the self-consistency proce-dure beyond the self-energies up to the four-point verticesso as to satisfy hierarchical identities among two-, three-,and four-point vertices [5, 6] as dictated by Goldstone’stheorem I [7, 8].The background of the present study is briefly sketchedas follows. Bogoliubov [13] pioneered a microscopic de-scription of interacting Bose-Einstein condensates to pre-dict that the quadratic energy-momentum relation of freeparticles should be changed upon switching on the in-teraction into a linear sound-wave-like dispersion, whosespeed is proportional to the square root of the bare in-teraction U . Beliaev [15] formulated a field-theoreticperturbation expansion in terms of Green’s functions.Hugenholtz and Pines [1] proved that single-particle ex-citations should have a gapless branch. Gavoret andNozi`eres [12] performed a structural analysis of the perturbation expansion for the single- and two-particleGreen’s functions to show that they have a commonbranch of poles. Nepomnyashchi˘i and Nepomnyashchi˘i[16, 17] used the identity between the two- and three-point vertices derived by Gavoret and Nozi`eres [12] toconclude that the anomalous self-energy should vanishin the low energy-momentum limit, contrary to the Bo-goliubov theory where it is finite and proportional tothe bare interaction U . These basic studies consideronly homogeneous Bose-Einstein condensates in equilib-rium at T = 0. The field-theoretic approach has also en-countered difficulties in practical applications such as theinfrared divergence mentioned above or the conserving-gapless dilemma [18, 19].The present formulation covers both equilibrium con-densates at finite temperatures and nonequilibrium ones.It will proceed by combining Schwinger’s functionalderivative method based on the generating functional[2, 20–22], the Legendre transformation to the effectiveaction [8, 21–24], the Luttinger-Ward functional [25], andconserving gapless condition [9, 11]. A similar approachwas adopted previously to analyze properties of two-particle Green’s functions [10], which however reachedan erroneous conclusion that the single- and two-particleGreen’s functions do not have common poles. It willbe reexamined here by (i) incorporating the identitiesamong the vertices and (ii) correcting the form of theperturbation. The resulting revised conclusion is thatthe single- and two-particle Green’s functions do sharepoles not only at T = 0 of a homogeneous condensate, aspredicted by Gavoret and Nozi`eres [12] and also restatedrecently by Watabe [26], but also at finite temperatures,in the presence of inhomogeneity, and also in nonequilib-rium situations. As a bonus, we will be able to clarifythe connections among the vertices which were not givenin the Gavoret-Nozi`eres study [12].This paper is organized as follows. Section II studiesproperties of the condensate wave function Ψ and Green’sfunctions G in equilibrium in terms of the effective ac-tion. Section III derives expressions of the three-pointand four-point (i.e., two-particle) Green’s functions basedon the functional derivative method. Section IV obtainsself-energies in terms of Ψ, G , and vertices. Section Vsummarizes the key equations derived and also supple-ment them with equations for the irreducible four-pointvertices to construct a closed system of equations. Sec-tion VI performs a nonequilibrium extension. SectionVII presents concluding remarks. II. EFFECTIVE ACTION AND GREEN’SFUNCTIONSA. System and Partition Function
We consider a system of identical bosons with mass m and spin 0 described by the dimensionless action [8, 21,22] S = S + S int , (1)with S = Z dx ψ ∗ ( x ) (cid:18) ∂∂τ + ˆ p m − µ (cid:19) ψ ( x ) , (2a) S int = 12 Z dx Z dx ′ U ( x, x ′ ) ψ ∗ ( x ) ψ ∗ ( x ′ ) ψ ( x ′ ) ψ ( x ) . (2b)Here ψ is the complex bosonic field and ψ ∗ its con-jugate, x ≡ ( r , τ ) specifies a space-“time” point with0 ≤ τ ≤ β ≡ ( k B T ) − ( k B : Boltzmann constant, T : tem-perature), ˆ p ≡ − i ~ ∇ is the momentum operator, µ isthe chemical potential, and U ( x, x ′ ) ≡ δ ( τ − τ ′ ) U ( | r − r ′ | ) (3)is the interaction potential. We regard ψ ( x ) and ψ ∗ ( x )as elements of a column vector, (cid:20) ψ ( x ) ψ ∗ ( x ) (cid:21) ≡ (cid:20) ψ ( x ) ψ ( x ) (cid:21) ≡ ~ψ ( x ) , (4)and will often express ψ j ( x ) ≡ ψ ( ξ ) with ξ ≡ ( j, x ) and j = 1 , Z JI ≡ Z [ J, I ] with extra source functions [20] J ( ξ ) and I ( ξ, ξ ′ )by Z JI ≡ Z D [ ψ ] exp (cid:20) − S − Z dξ ψ ( ξ ) J ( ξ ) − Z dξ Z dξ ′ ψ ( ξ ) ψ ( ξ ′ ) I ( ξ ′ , ξ ) (cid:21) , (5) with Z dξ ≡ X j Z dx . It satisfies − δ ln Z JI δJ ( ξ ) = h ψ ( ξ ) i JI ≡ Ψ JI ( ξ ) , (6a) − δ ln Z JI δJ ( ξ ) δJ ( ξ ′ ) = − h T τ ψ ( ξ ) ψ ( ξ ′ ) i JI + Ψ JI ( ξ )Ψ JI ( ξ ′ ) ≡ G JI ( ξ, ξ ′ ) , (6b) − δ ln Z JI δI ( ξ ′ , ξ ) = h T τ ψ ( ξ ) ψ ( ξ ′ ) i JI = − G JI ( ξ, ξ ′ ) + Ψ JI ( ξ )Ψ JI ( ξ ′ ) ≡ − G JI ( ξ, ξ ′ ) , (6c)where T τ is the “time”-ordering operator [27] and sub-script JI emphasizes that J and I are finite.Introduction of the two-point external source function I ( ξ, ξ ′ ), besides J ( ξ ) in the standard formalism [8, 21, 22],is one of the key ingredients here. Indeed, it enables us toexpress the effective action in terms of the renormalizedGreen’s function G ( ξ, ξ ′ ) instead of the bare propagator G ( ξ, ξ ′ ), as seen below. B. Effective Action
Let us perform a Legendre transformation from − ln Z JI into the effective action [8, 21–24]Γ JI ≡ − ln Z JI − Z dξ Ψ JI ( ξ ) J ( ξ )+ Z dξ Z dξ ′ G JI ( ξ, ξ ′ ) I ( ξ ′ , ξ ) , (7)which is a functional of (Ψ JI , G JI ). Its first derivativeswith respect to Ψ JI and G JI can be calculated by con-sidering their explicit dependences only; the implicit de-pendences through ( J, I ) cancel out because of Eq. (6).Thus, we obtain δ Γ JI δ Ψ JI ( ξ ) = − J ( ξ ) − Z dξ ′ [ I ( ξ, ξ ′ ) + I ( ξ ′ , ξ )] Ψ JI ( ξ ′ ) , (8a) δ Γ JI δG JI ( ξ ′ , ξ ) = I ( ξ ′ , ξ ) + I ( ξ, ξ ′ )2 , (8b)where we have incorporated the symmetry G JI ( ξ ′ , ξ ) = G JI ( ξ, ξ ′ ) in the second differentiation.Next, we introduce the functionalsΓ J ≡ Γ JI (cid:12)(cid:12) I =0 , (9a)Γ ≡ Γ J (cid:12)(cid:12) J =0 , (9b)and correspondingly, (Ψ J , G J ) and (Ψ , G ). FunctionalsΓ J and Γ satisfy Eq. (8) with I = 0 and I = J = 0,respectively. Thus, Γ = Γ[Ψ , G ] obeys δ Γ δ Ψ( ξ ) = 0 , (10a) δ Γ δG ( ξ ′ , ξ ) = 0 , (10b)which determine (Ψ , G ) in equilibrium. Indeed, Γ isconnected with the grand potential Ω in equilibrium byΓ = β Ω.It should be noted that G J in Γ J ( G in Γ) is a func-tional of Ψ J (Ψ), i.e., G J = G J [Ψ J ] ( G = G [Ψ]), unlikethe case of Γ JI where G JI is independent of Ψ JI . Onthe other hand, it also follows from Eq. (10) that G inΓ can be regarded as independent of Ψ up to the linearorder. Put it another way, the total derivative δ in Eq.(10) can be replaced by the partial derivative, which wewill express by ∂ .Let us expand Γ J = Γ J [Ψ J ] formally with respect to δ Ψ( ξ ) ≡ Ψ J ( ξ ) − Ψ( ξ ) (11)in the Taylor seriesΓ J = Γ + ∞ X n =1 Z dξ · · · Z dξ n Γ ( n ) ( ξ , · · · , ξ n ) n ! × δ Ψ( ξ ) · · · δ Ψ( ξ n ) . (12)It follows from Eq. (11) that Γ ( n ) can be written in termsof Γ in equilibrium byΓ ( n ) ( ξ , · · · , ξ n ) = δ n Γ δ Ψ( ξ ) · · · δ Ψ( ξ n ) . (13)Thus, Eq. (10a) is expressible alternatively asΓ (1) ( ξ ) = 0 . (14a)In addition, Γ (2) satisfies [8, 21, 22]Γ (2) ( ξ, ξ ′ ) = − G − ( ξ, ξ ′ ) . (14b)This can be shown by (i) starting from the chain rule Z dξ ′′ δJ ( ξ ′′ ) δ Ψ( ξ ) δδJ ( ξ ′′ ) = δδ Ψ( ξ ) , (ii) substituting Eq. (8a) with I = 0 into its numerator J ( ξ ′′ ), (iii) operating the resulting expression to Eq. (6a)with ξ → ξ ′ and I = 0, (iv) setting J = 0 subsequently,and (v) using Eqs. (6b) and (13).It is often convenient to regard G − ( ξ, ξ ′ ) = G − jj ′ ( x, x ′ )as the jj ′ element of the 2 × G − ( x, x ′ ) in theparticle-hole (i.e., Nambu) space. Let us divide ˆ G − intothe noninteracting part ˆ G − and the self-energy ˆΣ,ˆ G − ( x, x ′ ) = ˆ G − ( x, x ′ ) − ˆΣ( x, x ′ ) , (15) which is equivalent to the Dyson-Beliaev equation, asseen by multiplying both sides by ˆ G ( x ′ , x ′′ ) from theright-hand side and integrating over x ′ . It follows fromEq. (2a) that ˆ G − is given byˆ G − ( x, x ′ ) = (cid:20) i ˆ σ ∂∂τ − ˆ σ (cid:18) ˆ p m − µ (cid:19)(cid:21) δ ( x − x ′ ) , (16)where ˆ σ i ( i = 1 , ,
3) is the i th Pauli matrix.It should be noted that the present arrangement of ˆ G =( G jj ′ ) in the Nambu space, which naturally results fromEq. (6b), differs from that of ˆ G prev used in the previousstudies [9–11]; they are connected by ˆ G prev = ˆ G ( − i ˆ σ ). C. Goldstone’s Theorem I
Functional Γ J is invariant under the gauge transforma-tion (Ψ J ( x ) , Ψ J ( x )) → (Ψ J ( x ) e iχ , Ψ J ( x ) e − iχ ), where χ is a constant. Thus δ Γ J /δχ = 0 holds, which can betransformed by using Eq. (11) into X j =1 δ Γ J δ Ψ( ξ ) ( − j − [Ψ( ξ ) + δ Ψ( ξ )] = 0 . Substituting Eq. (12) and setting the coefficients of n thorder equal to zero, we obtain (cid:2) ( − j − + · · · + ( − j n − (cid:3) Γ ( n ) ( ξ , · · · , ξ n )= − Z dξ Γ ( n +1) ( ξ , · · · , ξ n , ξ )( − j − Ψ( ξ ) . (17)Note that differentiation of Eq. (17) with respect toΨ( ξ n +1 ) yields the ( n +1)th identity by using Eq. (13).The case of n = 1 is expressible by substituting Eq.(14) and adopting the vector-matrix notation of Eqs. (4)and (15) as Z dx ′ ˆ σ h ˆ G − ( x, x ′ ) − ˆΣ( x, x ′ ) i ˆ σ ~ Ψ( x ′ ) = ~ , (18)which extends the Hugenholtz-Pines relation [1] to in-homogeneous systems. Next, we set n = 2 in Eq. (17),substitute Eq. (14b) with Eq. (15), and use [( − j − +( − j ′ − ] G − ( ξ, ξ ′ ) = 0 as seen from Eq. (16). The pro-cedure yields h ( − j − + ( − j ′ − i Σ( ξ, ξ ′ )= − Z dξ Γ (3) ( ξ, ξ ′ , ξ )( − j − Ψ( ξ ) , (19)which connects the anomalous self-energy Σ jj ( x, x ′ ) withthe three-point vertex.The n = 1 identity (18) has been presented as thekey result from Goldstone’s theorem I [7, 8, 22, 24]. Onthe other hand, higher-order identities have turned outequally important. Among them, the n = 2 identitywas obtained by Gavoret and Nozi`eres; see the secondequality of Eq. (5.4). Later, it was used by Nepom-nyashchi˘i and Nepomnyashchi˘i to show that the anoma-lous self-energy vanishes in the low energy-momentumlimit [16, 17]. Castellani et al . [5] derived and consid-ered the identities of n ≤ T = 0. The n = 2 identity (19) will play a crucialrole in the derivation of the two-particle Green’s functionbelow. D. Luttinger-Ward Functional
Following Luttinger and Ward [25], we formally writeΓ in terms of another unknown functional Φ as [9, 11]Γ = 12 ~ Ψ T ˆ σ ˆ G − ˆ σ ~ Ψ+ 12 Tr (cid:8) ln (cid:2)(cid:0) − i ˆ σ (cid:1)(cid:0) ˆ G − − ˆΣ (cid:1)(cid:3) + ˆΣ ˆ G (cid:9) + Φ , (20)where T denotes transpose, ˆΣ = ˆΣ[ ~ Ψ , ˆ G ] with ˆ G = ˆ G [ ~ Ψ],and integration over ξ ≡ ( j, x ) is implied. Then Γ (1) ( ξ ) = δ Γ /δ Ψ( ξ ) acquires the expressionΓ (1) ( ξ ) = Z dξ ′ G − ( ξ, ξ ′ )( − j + j ′ Ψ( ξ ′ ) + ∂ Φ ∂ Ψ( ξ ) , (21)where we have used Eq. (10b) to omit implicit depen-dences through G in the differentiation; see also the com-ment in the paragraph below Eq. (10) concerning the useof ∂ instead of δ . The right-hand side of Eq. (21) shouldbe identical with the left-hand side of Eq. (18). Thus, weobtain ∂ Φ ∂ Ψ( ξ ) = Z dξ ′ Σ( ξ, ξ ′ )( − j + j ′ − Ψ( ξ ′ ) . (22a)Similarly, substitution of Eq. (20) into Eq. (10b) yields ∂ Φ ∂G ( ξ ′ , ξ ) = −
12 Σ( ξ, ξ ′ ) , (22b)where we have used Eqs. (10a) and (15). These are thetwo basic relations concerning Φ. III. TWO-PARTICLE GREEN’S FUNCTIONS
We will derive expressions of two-particle Green’s func-tions based on the Dyson-Beliaev equation (15) andHugenholtz-Pines relation (18).
A. Variations δ~ Ψ and δ ˆ G under perturbation To this end, we switch on the infinitesimal perturba-tion given in terms of Eq. (5) by (
J, I ) = (0 , → (0 , δI )once again [2, 20]. Accordingly, Eqs. (15) and (18), which are expressible concisely as ( ˆ G − − ˆΣ) ˆ G = ˆ1 andˆ σ ( ˆ G − − ˆΣ)ˆ σ ~ Ψ = ~
0, are modified into (cid:16) ˆ G − − ˆΣ I − δ ˆ I (s) (cid:17) ˆ G I = ˆ1 , (23a) h ˆ σ (cid:16) ˆ G − − ˆΣ I (cid:17) ˆ σ + δ ˆ I (s) i ~ Ψ I = ~ , (23b)where δ ˆ I (s) is defined by δ ˆ I (s) ( x, x ′ ) ≡ δ ˆ I ( x, x ′ ) + δ ˆ I T ( x ′ , x ) . (24)Equation (23) together with ˆ σ ˆ G − ˆ σ = − ˆ G − impliesthat the perturbation gives rise to the direct variationˆ G − → ˆ G − − δ ˆ I (s) plus the implicit one through theself-energies, the total of which cannot be described bythe simple replacement ˆ G − → ˆ G − − δ ˆ I (s) , however.This point was overlooked in the previous study [10]; Eq.(23b) forms one of the main corrections.Let us collect terms of the first order in δI from Eq.(23). The resulting equations can be written in terms of δ ˆ G ≡ ˆ G I − ˆ G and δ ˆΣ ≡ ˆΣ I − ˆΣ as δ ˆ G = ˆ G (cid:16) δ ˆ I (s) + δ ˆΣ (cid:17) ˆ G, (25a)ˆ σ ˆ G − ˆ σ δ~ Ψ = (cid:16) − δ ˆ I (s) + ˆ σ δ ˆΣ ˆ σ (cid:17) ~ Ψ . (25b)Since ˆΣ = ˆΣ[Ψ , G ], moreover, we can express δ Σ as δ Σ( ξ , ξ ′ ) = ∂ Σ( ξ , ξ ′ ) ∂G ( ξ , ξ ′ ) δG ( ξ , ξ ′ ) + ∂ Σ( ξ , ξ ′ ) ∂ Ψ( ξ ) δ Ψ( ξ ) , (26)where integration over repeated arguments is implied.Substitution of Eq. (22b) into Eq. (26) yields δ Σ( ξ , ξ ′ ) = −
12 Γ (4i) ( ξ , ξ ′ ; ξ , ξ ′ ) δG ( ξ , ξ ′ )+ Γ (3i)T ( ξ , ξ ′ ; ξ ) δ Ψ( ξ ) , (27)where Γ (4i) is the irreducible four-point vertex defined byΓ (4i) ( ξ , ξ ′ ; ξ , ξ ′ ) ≡ ∂ Φ ∂G ( ξ ′ , ξ ) ∂G ( ξ , ξ ′ )= Γ (4i) ( ξ ′ , ξ ; ξ ′ , ξ ) . (28a)Similarly, we have introduced the irreducible three-pointvertex by Γ (3i)T ( ξ , ξ ′ ; ξ ) ≡ ∂ Σ( ξ , ξ ′ ) /∂ Ψ( ξ ), whichcan be transformed by using Eqs. (22) and (28a) intoΓ (3i)T ( ξ , ξ ′ ; ξ ) = Γ (4i) ( ξ , ξ ′ ; ξ ′ , ξ )( − j + j ′ − Ψ( ξ ′ )= Ψ( ξ ′ )( − j + j ′ − Γ (4i) ( ξ , ξ ′ ; ξ ′ , ξ ) ≡ Γ (3i) ( ξ ; ξ ′ , ξ ) . (28b)To proceed further, we adopt the notation [10] h ξ | ~ Ψ ≡ Ψ( ξ ) , (29a) h ξ | ˆ σ | ξ ′ i ≡ δ ( ξ, ξ ′ )( − j − , (29b) h ξ | ˆ G | ξ ′ i ≡ G ( ξ, ξ ′ ) , (29c) h ξ, ξ ′ | ~G ≡ G ( ξ, ξ ′ ) , (29d) h ξ , ξ ′ | GG | ξ , ξ ′ i ≡ G ( ξ , ξ ) G ( ξ ′ , ξ ′ ) , (29e) h ξ , ξ ′ | | ξ , ξ ′ i ≡ δ ( ξ , ξ ) δ ( ξ ′ , ξ ′ ) , (29f) h ξ | Ψ (3) | ξ , ξ ′ i ≡ δ ( ξ , ξ )Ψ( ξ ′ ) ≡ h ξ ′ , ξ | Ψ (3)T | ξ i , (29g) h ξ | Ψ (3 σ ) | ξ , ξ ′ i ≡ δ ( ξ , ξ )Ψ( ξ ′ )( − j ′ − ≡ h ξ ′ , ξ | Ψ (3 σ )T | ξ i , (29h) h ξ , ξ ′ | Γ (4i) | ξ , ξ ′ i ≡ Γ (4i) ( ξ , ξ ′ ; ξ , ξ ′ ) , (29i) h ξ | Γ (3i) | ξ , ξ ′ i ≡ Γ (3i) ( ξ ; ξ , ξ ′ ) ≡ h ξ ′ , ξ | Γ (3i)T | ξ i , (29j)together with δ~ Σ and δ~I which are defined similarly as ~G in Eq. (29d). Using the notation, we can express Eq.(28b) as Γ (3i) = − ˆ σ Ψ (3 σ ) Γ (4i) , (30a)Γ (3i)T = − Γ (4i) Ψ (3 σ )T ˆ σ , (30b)and Eq. (25) can be written alternatively as δ ~G = GG (cid:16) δ~I (s) + δ~ Σ (cid:17) , (31a)ˆ σ ˆ G − ˆ σ δ~ Ψ = − Ψ (3) δ~I (s) + ˆ σ Ψ (3 σ ) δ~ Σ . (31b)For example, we can reproduce Eq. (25a) from Eq. (31a)by multiplying the latter by h ξ , ξ ′ | from the left, in-serting | ξ , ξ ′ ih ξ , ξ ′ | after GG , and using G ( ξ , ξ ′ ) = G ( ξ ′ , ξ ). Similarly, Eq. (27) reads δ~ Σ = −
12 Γ (4i) δ ~G + Γ (3i)T δ~ Ψ . (32)The coupled equations (31) and (32) are solved as fol-lows. First, after substituting Eq. (32), we can solve Eq.(31a) formally in terms of δ ~G as δ ~G = g (4) (cid:16) δ~I (s) + Γ (3i)T δ~ Ψ (cid:17) , (33)with g (4) ≡ (cid:18) GG Γ (4i) (cid:19) − GG. (34)It is useful at this stage to introduce the full four- andthree-point vertices byΓ (4) ≡ (cid:18) (4i) GG (cid:19) − Γ (4i) , (35a)Γ (3) ≡ − ˆ σ Ψ (3 σ ) Γ (4) , (35b)Γ (3)T ≡ − Γ (4) Ψ (3 σ )T ˆ σ . (35c) Indeed, Eq. (33) is expressible alternatively in terms ofthe vertices as δ ~G = GG (cid:18) −
12 Γ (4) GG (cid:19) δ~I (s) + GG Γ (3)T δ~ Ψ , (36)where we have used the matrix identity (1 + A B ) − A = A (1+ B A ) − = A − A (1+ B A ) − B A and Eq. (30). With δ ~G = − GG δ ~G − which is equivalent to δ ˆ G = − ˆ G δ ˆ G − ˆ G ,we can transform Eq. (36) further into δ ~G − = − (cid:18) −
12 Γ (4) GG (cid:19) δ~I (s) − Γ (3)T δ~ Ψ . (37)Noting Eq. (14b), we can identify Γ (3)T above as Γ (3) of Eq. (13) defined for δI = 0. Thus, we conclude thatΓ (3)T ( ξ , ξ ′ ; ξ ) ≡ h ξ , ξ ′ | Γ (3)T | ξ i is symmetric with re-spect to any permutation of its arguments, andΓ (3)T = Γ (3) (38)holds. This observation will play a crucial role below. Itshould also be noted that the vertices may acquire asym-metry in practical studies of using approximate Φ in Eq.(28a). With this possibility in mind, we will proceed withkeeping the formal distinction between Γ (3) and Γ (3)T .Next, we focus on Eq. (31b) and substitute Eq. (32)with Eq. (33) into it. We then obtain (cid:20) ˆ σ ˆ G − ˆ σ − ˆ σ Ψ (3 σ ) (cid:18) −
12 Γ (4i) g (4) (cid:19) Γ (3i)T (cid:21) δ~ Ψ= − (cid:18) Ψ (3) + 12 ˆ σ Ψ (3 σ ) Γ (4i) g (4) (cid:19) δ~I (s) . (39)The prefactor of δ~ Ψ can be transformed asˆ σ ˆ G − ˆ σ − ˆ σ Ψ (3 σ ) (cid:18) −
12 Γ (4i) g (4) (cid:19) Γ (3i)T = ˆ σ ˆ G − ˆ σ − ˆ σ ˆΣˆ σ − ˆ σ Ψ (3 σ ) (cid:18) (4i) GG (cid:19) − Γ (3i)T = − ˆ G − − ˆ σ ˆΣˆ σ − ˆ σ Ψ (3 σ ) Γ (3)T = − ˆ G − − ˆ σ ˆΣˆ σ − ˆ σ (cid:0) − ˆ σ ˆΣ − ˆΣˆ σ (cid:1) = − ˆ G − , (40)where we have successively used Eqs. (15), (34), (16),(35), (38), and (19). Thus, the prefactor has been iden-tified as − ˆ G − , due mainly to the n = 2 identity (19)which has been incorporated additionally in the presentstudy. Equation (40) forms the second correction to theprevious study [10]. Indeed, the result will lead us to theconclusion that the single- and two-particle Green’s func-tions share poles, in agreement with the Gavoret-Nozi`erestheory [12].Let us substitute Eq. (40) into Eq. (39) and writeˆ σ Ψ (3 σ ) Γ (4i) g (4) = − Γ (3) GG based on Eqs. (30), (34),and (35). We thereby obtain δ~ Ψ = ˆ G (cid:18) Ψ (3) −
12 Γ (3) GG (cid:19) δ~I (s) , (41a)so that Eq. (36) now reads δ ~G = (cid:20) GG (cid:18) −
12 Γ (4) GG (cid:19) + GG Γ (3)T ˆ G (cid:18) Ψ (3) −
12 Γ (3) GG (cid:19)(cid:21) δ~I (s) . (41b)It is worth noting that ˆ G Ψ (3) δ~I (s) in Eq. (41a) is alsoexpressible as ˆ G Ψ (3) δ~I (s) = ˆ G δ ˆ I (s) ~ Ψ . (42)The equivalence can be seen easily by operating h ξ | fromthe left, inserting | ξ ih ξ | or | ξ ξ ′ ih ξ ξ ′ | appropriately,and using Eq. (29). B. Expressions of G (3) and G (4) Let us define the n -point Green’s function in terms ofEq. (5) with I = 0 by G ( n ) ( ξ , · · · , ξ n ) = ( − n − [ n ] δ n ln Z J δJ ( ξ ) · · · J ( ξ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J =0 , (43)where (cid:2) n (cid:3) denotes the largest integer that does notexceed n . Equation (43) is the n th cumulant com-posed of connected Feynman diagrams. The first twoof them are given in terms of the functions in Eq. (6)by G (1) ( ξ ) = Ψ( ξ ) and G (2) ( ξ , ξ ) = G ( ξ , ξ ). Not-ing G ( n +1) ( ξ , · · · , ξ n +1 ) ∝ δG ( n ) ( ξ , · · · , ξ n ) /δJ ( ξ n +1 ),we can obtain G (3) and G (4) successively from Eq. (6b).They are expressible concisely in terms of G ( n ) ( ξ , · · · , ξ n ) ≡ ( − n ] h T τ ψ ( ξ ) · · · ψ ( ξ n ) i . (44)with abbreviating G (4) ( ξ , ξ , ξ , ξ ) ≡ G (4)1234 , etc., as G (3)123 = G (3)123 − Ψ G − Ψ G − Ψ G + Ψ Ψ Ψ , (45a) G (4)1234 = G (4)1234 + Ψ G (3)234 + Ψ G (3)341 + Ψ G (3)412 + Ψ G (3)123 − G G − G G − G G + Ψ Ψ G + Ψ Ψ G + Ψ Ψ G + Ψ Ψ G + Ψ Ψ G + Ψ Ψ G − Ψ Ψ Ψ Ψ . (45b)The correctness of Eqs. (45a) and (45b) can be seen in thefact that terms with Ψ and G on their right-hand sidesappropriately remove all the disconnected contributionsfrom G ( n ) for n = 3 , ~ Ψ I and G I with respect to I by G (3)123 = δ Ψ δI (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I =0 − Ψ G − Ψ G , (46a) G (4)1234 = δG δI (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I =0 + Ψ G (3)124 + Ψ G (3)123 − G G − G G . (46b)Let us express δ Ψ( ξ ) = h ξ | δ~ Ψ in Eq. (46a), substituteEq. (41a) with Eq. (24), and perform the differentiation.Noting that Γ (3) and G are symmetric with respect tothe arguments, we obtain G (3) ( ξ , ξ , ξ )= − G ( ξ , ξ ′ ) G ( ξ , ξ ′ ) G ( ξ , ξ ′ )Γ (3) ( ξ ′ ; ξ ′ , ξ ′ ) . (47a)We also write δG ( ξ , ξ ) = h ξ , ξ | δ ~G in Eq. (46b), substi-tute Eq. (41b), perform the differentiation, and use Eq.(47a). The procedure yields G (4) ( ξ , ξ , ξ , ξ )= − G ( ξ , ξ ′ ) G ( ξ , ξ ′ ) G ( ξ , ξ ′ ) G ( ξ , ξ ′ ) × h Γ (4) ( ξ ′ , ξ ′ ; ξ ′ , ξ ′ ) + Γ (3)T ( ξ ′ , ξ ′ ; ξ ) G ( ξ , ξ ′ ) × Γ (3) ( ξ ′ ; ξ ′ , ξ ′ ) i , (47b)where Γ (4) , Γ (3)T , and Γ (3) are given in Eqs. (35).Functions G (3) and G (4) in Eq. (47) are both con-nected, as they should, and Eq. (47b) tells us clearly that G (4) shares poles with G , in agreement with the resultof Gavoret and Nozi`eres [12]. They should be symmetricwith respect to any permutation of its arguments in theexact treatment; the apparent asymmetry in Eq. (47)originates from that of the irreducible vertex Γ (4i) de-fined by Eq. (28a) with which we have constructed Γ (4) .It should also be noted that both G (3) and G (4) will ac-quire asymmetry in practical studies of using some ap-proximate Φ in Eq. (28a). IV. SELF-ENERGIES AND CONDENSATEWAVE FUNCTION
In this section, we derive (i) expressions of the self-energies in the Dyson-Beliaev equation and (ii) the equa-tion for the condensate wave function, i.e., the gen-eralized Gross-Pitaevski˘i equation, both in terms of( G, Γ (3) , Γ (4) ). Subsequently, we will see that conserva-tion laws are satisfied by the Dyson-Beliaev and Gross-Pitaevski˘i equations. A. Expressions of self-energies
The Heisenberg equation of motion for the field oper-ator ˆ ψ j ( x ) ≡ e τ ˆ H ˆ ψ j ( r ) e − τ ˆ H corresponding to Eq. (1) isgiven by [2] (cid:20) ( − j ∂∂τ − ˆ p m + µ (cid:21) ˆ ψ j ( x ) − Z dx U ( x, x ) ˆ ψ ( x ) ˆ ψ j ( x ) ˆ ψ ( x ) = 0 . (48)Taking its thermodynamic average yields (cid:20) ( − j ∂∂τ − ˆ p m + µ (cid:21) Ψ j ( x )+ Z dx U ( x, x ) G (3)2 j ( x , x, x ) = 0 , (49a)where G (3) is defined by Eq. (44). One can also showbased on Eq. (48) that G ≡ G (2) obeys [2] (cid:20) ( − j ∂∂τ − ˆ p m + µ (cid:21) G jj ′ ( x, x ′ )+ Z dx U ( x, x ) G (4)2 j j ′ ( x , x, x , x ′ ) = δ j, − j ′ δ ( x, x ′ ) . (49b)We can construct the equation of motion for G jj ′ ( x, x ′ ) = G jj ′ ( x, x ′ )+Ψ j ( x )Ψ j ( x ′ ) from Eq. (49). It is given with j → − j by (cid:20) ( − − j ∂∂τ − ˆ p m + µ (cid:21) G − j,j ′ ( x, x ′ )+ Z dx U ( x, x ) h G (4)2 , − j, ,j ′ ( x , x, x , x ′ )+ G (3)2 , − j, ( x , x, x )Ψ j ′ ( x ′ ) i = δ jj ′ δ ( x, x ′ ) . (50)Equation (50) should be identical with Eq. (15) that canbe written as ( ˆ G − − ˆΣ) ˆ G = ˆ1 in terms of ˆ G − in Eq.(16). Hence, we obtainΣ( ξ, ξ ′ )= − δδG ( ξ ′ , ξ ′′ ) Z dx U ( x, x ) h G (4)2 , − j, ,j ′′ ( x , x, x , x ′′ )+ G (3)2 , − j, ( x , x, x )Ψ j ′′ ( x ′′ ) i . (51)The terms in the square brackets of Eq. (51) can be trans-formed by using Eq. (45) into G (4)1234 + G (3)123 Ψ = G (4)1234 − Ψ G (3)234 − Ψ G (3)134 − Ψ G (3)124 + G G + G G + G G − Ψ Ψ G − Ψ Ψ G − Ψ Ψ G . (52)We use Eq. (52) in Eq. (51), substitute Eq. (47), performthe differentiation, and symmetrize the expression so thatΣ( ξ , ξ ′ ) = Σ( ξ ′ , ξ ) can be seen manifestly. We thereby obtainΣ( ξ, ξ ′ )= δ j, − j ′ δ ( x, x ′ ) U ( x, x ) (cid:2) Ψ ( x )Ψ( x ) − G ( x , x ) (cid:3) + U ( x, x ′ ) (cid:2) Ψ − j ( x )Ψ − j ′ ( x ′ ) − G − j, − j ′ ( x, x ′ ) (cid:3) + 12 U ( x, x ) n G − j,j ( x, x ) G j ( x , x ) G j ′ ( x , x ′ ) × (cid:2) Γ (4) ( ξ , ξ ′ ; ξ , ξ ′ ) + Γ (3)T ( ξ , ξ ′ ; ξ ) G ( ξ , ξ ′ ) × Γ (3) ( ξ ′ ; ξ , ξ ′ ) (cid:3) − G − j,j ( x, x ) (cid:2) G j ( x , x )Ψ ( x )+ Ψ ( x ) G j ( x , x ) (cid:3) Γ (3)T ( ξ , ξ ′ ; ξ ) − Ψ − j ( x ) G j ( x , x ) G j ′ ( x , x ′ )Γ (3) ( ξ ′ ; ξ , ξ ′ )+ ( ξ ↔ ξ ′ ) o , (53)where integration over repeated arguments is implied,and ( ξ ↔ ξ ′ ) denotes terms obtained from the preced-ing three terms in the curly brackets by exchanging ξ and ξ ′ . The first two terms on the right-hand side arethe Hartree and Fock terms that are expressible as Fig.1(a)-(d), whereas the third one represents correlation ef-fects given diagrammatically by Fig. 1(e)-(h).It should be noted that there is arbitrariness in express-ing the correlation term of Eq. (53) in terms of Γ (4) andΓ (3) , which are symmetric in the exact theory but mayacquire asymmetry in approximate treatments. We haveremoved it here so that the two Green’s functions enter-ing and leaving x of the bare interaction vertex U ( x, x )in Eq. (53) are linked with the latter two arguments ofΓ (4) , i.e., ( ξ , ξ ′ ). The advantage of this choice is thatthe density fluctuation mode is naturally incorporated inΓ (4) even in approximate treatments.Using Eq. (19) and following the argument of Nepom-nyashchi˘i and Nepomnyashchi˘i [16, 17], one can con-firm oneself that diagrams (g) and (h) in Fig. 1 makethe anomalous self-energy vanish in the low energy-momentum limit for homogeneous systems. Thus, theNepomnyashchi˘i identity is naturally satisfied in our for-mulation to remove the infrared divergence, thereby mak-ing practical calculations possible. B. Equation for the condensate wave function
Let us express G (3) in Eq. (49a) in terms of ( G (3) , G, Ψ)by using Eq. (45a) and substitute Eq. (47a) subsequently.We then obtain (cid:20) ( − j ∂∂τ − ˆ p m + µ (cid:21) Ψ j ( x ) = η j ( x ) , (54)with η j ( x ) ≡ U ( x, x ) (cid:2) G jj ( x, x ) G j ( x , x ) G j ′ ( x , x ′ ) × Γ (3) ( x ; x , x ′ ) − Ψ j ( x ) G ( x , x ) − Ψ ( x ) G j ( x, x ) − Ψ ( x ) G j ( x , x )+ Ψ j ( x )Ψ ( x )Ψ ( x ) (cid:3) , (55)where integrations over ( x , ξ , ξ ′ , ξ , ξ ′ ) are implied.Equation (54) generalizes the Gross-Pitaevski˘i equation[28, 29] so as to incorporate the quasiparticle contribu-tion and correlation effects in η j ( x ). It is equivalent toEq. (18), i.e., the generalized Hugenholtz-Pines relation,in the exact theory. However, they will be different inapproximate treatments. We prefer Eq. (54) to Eq. (18),because conservation laws are satisfied as seen below.Adopting Eq. (54), we should determine the chemical po-tential so as to reproduce a branch of gapless excitationsin the single-particle channel. C. Conservation Laws
We follow the argument of Kadanoff and Baym [2, 3]to confirm that the number-, momentum-, and energy-conservation laws are satisfied in our formulation.First, Eq. (53) satisfies Σ( ξ, ξ ′ ) = Σ( ξ ′ , ξ ) and sodo Green’s functions determined by the Dyson-Beliaevequation (15). Hence, criterion A of Kadanoff andBaym ( ˆ G − − ˆΣ) ˆ G = ˆ G ( ˆ G − − ˆΣ) = ˆ1 is met, therebyensuring the number-conservation law. Second, weconsider their criterion B on the momentum- andenergy-conservation laws. The Dyson-Beliaev equa-tion (15) with Eq. (53) can be written alternativelyas Eq. (50). Moreover, Eq. (54) is equivalent to Eq.(49a). Hence, our G ( ξ, ξ ′ ) = G ( ξ, ξ ′ ) − Ψ( ξ )Ψ( ξ ′ ) obeysEq. (49b), where criterion B of Kadanoff and Baym G (4)2211 ( x , x, x , x ) = G (4)2211 ( x, x , x, x ) holds manifestly.Thus, the momentum- and energy-conservation laws arealso fulfilled. V. SYSTEM OF EQUATIONSA. Derived equations
Let us summarize our system of equations for easy ref-erence. First, Green’s functions G ( ξ, ξ ′ ) and the conden-sate wave function Ψ( ξ ) obey the Dyson-Beliaev equationand generalized Gross-Pitaevski˘i equation given by (cid:2) G − ( ξ, ξ ) − Σ( ξ, ξ ) (cid:3) G ( ξ , ξ ′ ) = δ ( ξ, ξ ′ ) , (56a) (cid:20) ( − j ∂∂τ − ˆ p m + µ (cid:21) Ψ( ξ ) = η ( ξ ) , (56b)respectively, where G − is defined by Eq. (16), ξ ≡ ( j, x )with j = 1 , x = ( r , τ ), and integration over therepeated argument ξ is implied. Second, the self-energy Σ and source function η in Eq.(56) are given in terms of (Ψ , G, Γ (3) , Γ (4) ) byΣ( ξ, ξ ′ )= δ j, − j ′ δ ( x, x ′ ) U ( x, x ) (cid:2) Ψ ( x )Ψ( x ) − G ( x , x ) (cid:3) + U ( x, x ′ ) (cid:2) Ψ − j ( x )Ψ − j ′ ( x ′ ) − G − j, − j ′ ( x, x ′ ) (cid:3) + 12 U ( x, x ) n G − j,j ( x, x ) G j ( x , x ) G j ′ ( x , x ′ ) × (cid:2) Γ (4) ( ξ , ξ ′ ; ξ , ξ ′ ) + Γ (3)T ( ξ , ξ ′ ; ξ ) G ( ξ , ξ ′ ) × Γ (3) ( ξ ′ ; ξ , ξ ′ ) (cid:3) − G − j,j ( x, x ) (cid:2) G j ( x , x )Ψ ( x )+ Ψ ( x ) G j ( x , x ) (cid:3) Γ (3)T ( ξ , ξ ′ ; ξ ) − Ψ − j ( x ) G j ( x , x ) G j ′ ( x , x ′ )Γ (3) ( ξ ′ ; ξ , ξ ′ )+ ( ξ ↔ ξ ′ ) o , (57a) η ( ξ ) ≡ U ( x, x ) (cid:2) G jj ( x, x ) G j ( x , x ) G j ′ ( x , x ′ ) × Γ (3) ( x ; x , x ′ ) − Ψ j ( x ) G ( x , x ) − Ψ ( x ) G j ( x, x ) − Ψ ( x ) G j ( x , x )+ Ψ j ( x )Ψ ( x )Ψ ( x ) (cid:3) . (57b)Equation (57a) is expressible diagrammatically as Fig. 1.Third, the vertices Γ (4) and Γ (3) are defined by Eqs.(35a) and (35b), i.e., Γ (4) obeysΓ (4) = Γ (4i) −
12 Γ (4i) GG Γ (4) , (58a)and Γ (3) ( ξ ; ξ , ξ ′ ) ≡ h ξ | Γ (3) | ξ , ξ ′ i is given in terms ofΓ (4) ( ξ , ξ ′ ; ξ , ξ ′ ) ≡ h ξ , ξ | Γ (4) | ξ , ξ ′ i byΓ (3) ( ξ ; ξ , ξ ′ ) ≡ Ψ( ξ ′ )( − j + j ′ − Γ (4) ( ξ , ξ ′ ; ξ , ξ ′ )= Γ (4) ( ξ ′ , ξ ; ξ ′ , ξ )( − j + j ′ − Ψ( ξ ′ ) ≡ Γ (3)T ( ξ ′ , ξ ; ξ ) . (58b)Equations (58a) and (58b) are expressible diagrammati-cally as Fig. 2. (a) (b) (c) (d)(e) (f) (g) (h) FIG. 1. Diagramatic expression of the self-energy. A wavyand dotted lines denote U and Ψ j , respectively. = += FIG. 2. Diagramatic expressions of the four- and three-pointvertices. The dotted line denotes Ψ.
Fourth, Γ (3) is connected with the anomalous self-energy by Eq. (19), i.e.,Σ jj ( x, x ′ ) = 12 Ψ j ( x )( − j + j − Γ (3) j jj ( x ; x, x ′ ) . (59)Combined with Eq. (57a), we can conclude that theanomalous self-energy of homogeneous systems vanishin the low energy-momentum limit due to processes (g)and (h) of Fig. 1, as first shown by Nepomnyashchi˘i andNepomnyashchi˘i[16, 17]. B. Equation for Γ (4i) Equations (56)–(59) are formally exact, which still in-clude the irreducible four-point vertices Γ (4i) as unknownfunctions. Hence, it is necessary for performing practicalmicroscopic studies to supplement them with equationsto determine Γ (4i) . Incidentally, we seek an alternativepossibility in the following paper [30] of constructing phe-nomenological parameters in terms of Γ (4i) to describelow-energy properties, like the Landau theory of Fermiliquids [31–34].To derive the equations for Γ (4i) , we approximate thefunctional Φ in the conserving-gapless form of satisfyingEq. (22) [9, 11]. To be specific, our Φ is given in termsof an unknown effective two-body potential ˜ U ( x , x ) =˜ U ( x , x ) by [11]Φ = 12 ˜ U ( x , x ) [ ρ ( x , x ) ρ ( x , x ) + ρ ( x , x ) ρ ( x , x ) − ˜ ρ ( x , x )˜ ρ ( x , x )] , (60)where ρ and ˜ ρ jj are defined by ρ ( x , x ) ≡ Ψ ( x )Ψ ( x ) − G ( x , x ) + G ( x , x )2 , (61a) ρ jj ( x , x ) ≡ Ψ j ( x )Ψ j ( x ) + G jj ( x , x ) + G jj ( x , x )2 . (61b)The irreducible vertices Γ (4i) are obtained from this func-tional by Eq. (28a). The basic finite elements are given byΓ (4i)12;12 ( x , x ; x ′ , x ′ ) = ˜ U ( x , x ′ ) [ δ ( x , x ) δ ( x ′ , x ′ )+ δ ( x , x ′ ) δ ( x , x ′ )] , (62a)Γ (4i)11;22 ( x , x ; x ′ , x ′ ) = − ˜ U ( x , x ) [ δ ( x , x ′ ) δ ( x , x ′ )+ δ ( x , x ′ ) δ ( x , x ′ )] . (62b)The other finite elements can be found easily by usingthe symmetries Γ (4i) ( ξ , ξ ; ξ ′ , ξ ′ ) = Γ (4i) ( ξ , ξ ; ξ ′ , ξ ′ ) =Γ (4i) ( ξ , ξ ; ξ ′ , ξ ′ ) = Γ (4i) ( ξ ′ , ξ ′ ; ξ , ξ ). The correspond-ing Γ (3i) is obtained by Eq. (28b). The finite elementsare given byΓ (3i) j ; j, − j ( x ; x ′ , x ) = Γ (3i)3 − j ; jj ( x ; x , x ′ )= ˜ U ( x , x ′ ) [ δ ( x , x )Ψ − j ( x ′ ) + δ ( x ′ , x )Ψ − j ( x )] . (63)Thus, Eqs. (62a) and (62b) yield an identical expression,as they should.We determine the unknown function ˜ U ( x , x ′ ) so asto satisfy Eq. (59). Noting that Σ ( x, x ′ ) = Σ ∗ ( x, x ′ )holds, we realize that the number of unknown variables,i.e., ˜ U ( x, x ′ ), is equal to the number of constraints to besatisfied, i.e., Eq. (59). Especially in the weak-couplingcases, we can impose the condition that ˜ U ( x, x ′ ) ap-proaches the bare interaction potential U ( x, x ′ ) in thehigh energy-momentum limit. VI. EXTENSION TO NONEQUILIBRIUMSYSTEMS
The formulation of Sects. II-IV can be extended tononequilibrium systems by (i) performing the inverseWick rotation τ = it/ ~ and (ii) changing the Matsubaracontour τ ∈ [0 , β ] into the round-trip Keldysh contour C that extends over t ∈ [ −∞ , ∞ ] [35, 36]. We sketch it with(a) modifying the definitions of functions and (b) trans-forming every integral on C into that over t ∈ [ −∞ , ∞ ]in the second half. A. Equations on C We introduce the partition function by Z JI ≡ Z D [ ψ ] exp (cid:26) i ~ (cid:20) S C − Z dξ ψ ( ξ ) J ( ξ ) − Z dξ Z dξ ′ ψ ( ξ ) ψ ( ξ ′ ) I ( ξ ′ , ξ ) (cid:21)(cid:27) . (64)Here S C is obtained from Eq. (1) by S C ≡ i ~ S ( τ = it/ ~ ) with an adiabatic factor on S int [36], variable ξ nowdenotes ξ ≡ ( j, x ) with x ≡ ( r , t ), and every time integral0extends over C . It satisfies i ~ δ ln Z JI δJ ( ξ ) = h ψ ( ξ ) i JI ≡ Ψ JI ( ξ ) , (65a) i ~ δ ln Z JI δJ ( ξ ) δJ ( ξ ′ ) = − i ~ [ h T C ψ ( ξ ) ψ ( ξ ′ ) i JI − Ψ JI ( ξ )Ψ JI ( ξ ′ )] ≡ G JI ( ξ, ξ ′ ) , (65b) i ~ δ ln Z JI δI ( ξ ′ , ξ ) = h T C ψ ( ξ ) ψ ( ξ ′ ) i JI = i ~ G JI ( ξ, ξ ′ ) + Ψ JI ( ξ )Ψ JI ( ξ ′ ) ≡ i ~ G JI ( ξ, ξ ′ ) , (65c)where T C arranges operators according to their chrono-logical order on C from right to left.Let us perform the Legendre transformation [8, 21–24]Γ JI ≡ i ~ ln Z JI − Z dξ Ψ JI ( ξ ) J ( ξ ) − Z dξ Z dξ ′ i ~ G JI ( ξ, ξ ′ ) I ( ξ ′ , ξ ) . (66)Using it, we can follow every step from Eq. (8) to Eq.(19), where the only modification necessary is to add thefactor i/ ~ on the left-hand side of Eq. (8b).We now express Γ in terms of another functional Φ,Γ = 12 ~ Ψ T ˆ σ ˆ G − ˆ σ ~ Ψ − i ~ n ln h(cid:0) − i ˆ σ (cid:1)(cid:0) ˆ G − − ˆΣ (cid:1)i + ˆΣ ˆ G o + Φ , (67)similarly as Eq. (20) for equilibrium systems. Accord-ingly, Eq. (22) is replaced by ∂ Φ ∂ Ψ( ξ ) = Z dξ ′ Σ( ξ, ξ ′ )( − j + j ′ − Ψ( ξ ′ ) , (68a) ∂ Φ ∂G ( ξ ′ , ξ ) = i ~ ξ, ξ ′ ) . (68b)To derive two-particle Green’s functions, we can alsofollow every step from Eq. (23) to Eq. (26). Equation(27) is then modified into δ Σ( ξ , ξ ′ ) = i ~ (4i) ( ξ , ξ ′ ; ξ , ξ ′ ) δG ( ξ , ξ ′ )+ Γ (3i)T ( ξ , ξ ′ ; ξ ) δ Ψ( ξ ) , (69)where Γ (4i) is now defined byΓ (4i) ( ξ , ξ ′ ; ξ , ξ ′ ) ≡ i ~ ) ∂ Φ ∂G ( ξ ′ , ξ ) ∂G ( ξ , ξ ′ ) , (70)and Γ (3i)T ( ξ , ξ ′ ; ξ ) ≡ δ Σ( ξ , ξ ′ ) /δ Ψ( ξ ) is expressiblein terms of this Γ (4i) as Eq. (28b). With the differencebetween Eqs. (27) and (69) in mind, we can proceed inexactly the same way as from Eq. (29) to Eq. (41). In-deed, the only modification necessary is to replace every prefactor 1 / − i ~ /
2. Thus, Eq. (41) is replaced by δ~ Ψ = ˆ G (cid:18) Ψ (3) + i ~ (3) GG (cid:19) δ~I (s) , (71a) δ ~G = (cid:20) GG (cid:18) i ~ (4) GG (cid:19) + GG Γ (3)T ˆ G (cid:18) Ψ (3) + i ~ (3) GG (cid:19)(cid:21) δ~I (s) . (71b)Let us introduce the n -point Green’s functions by G ( n ) ( ξ , · · · , ξ n ) = ( i ~ ) n − [ n ] δ n ln Z J δJ ( ξ ) · · · J ( ξ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J =0 . (72)The first two functions are expressible in terms of thosegiven in Eq. (65) by G (1) ( ξ ) = Ψ( ξ ) and G (2) ( ξ , ξ ) = G ( ξ , ξ ). Accordingly, Eq. (46) is modified into G (3)123 = δ Ψ δI (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I =0 − Ψ G − Ψ G , (73a) G (4)1234 = δG δI (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I =0 + i ~ (cid:16) Ψ G (3)124 + Ψ G (3)123 (cid:17) − G G − G G . (73b)Substituting Eq. (71) into Eq. (73), we obtain G (3) ( ξ , ξ , ξ )= i ~ G ( ξ , ξ ′ ) G ( ξ , ξ ′ ) G ( ξ , ξ ′ )Γ (3) ( ξ ′ ; ξ ′ , ξ ′ ) , (74a) G (4) ( ξ , ξ , ξ , ξ )= i ~ G ( ξ , ξ ′ ) G ( ξ , ξ ′ ) G ( ξ , ξ ′ ) G ( ξ , ξ ′ ) × h Γ (4) ( ξ ′ , ξ ′ ; ξ ′ , ξ ′ ) + Γ (3)T ( ξ ′ , ξ ′ ; ξ ) G ( ξ , ξ ′ ) × Γ (3) ( ξ ′ ; ξ ′ , ξ ′ ) i . (74b)Expressions of the self-energies can also be obtained byfollowing the procedure of Sect. IV. We thereby obtainΣ( ξ , ξ ′ )= δ j − j ′ δ ( x , x ′ ) U ( x , x ) (cid:2) Ψ ( x )Ψ ( x )+ i ~ G ( x , x ) (cid:3) + U ( x , x ′ ) (cid:2) Ψ − j ( x )Ψ − j ′ ( x ′ )+ i ~ G − j − j ′ ( x , x ′ ) (cid:3) + 12 U ( x , x ) (cid:8) ( i ~ ) G − j j ( x , x ) G j ( x , x ) × G j ′ ( x , x ′ ) (cid:2) Γ (4) ( ξ , ξ ′ ; ξ , ξ ′ ) + Γ (3)T ( ξ , ξ ′ ; ξ ) × G ( ξ , ξ ′ )Γ (3) ( ξ ′ ; ξ , ξ ′ ) (cid:3) + i ~ G − j j ( x , x ) (cid:2) G j ( x , x )Ψ ( x )+ Ψ ( x ) G j ( x , x ) (cid:3) Γ (3)T ( ξ , ξ ′ ; ξ )+ i ~ Ψ − j ( x ) G j ( x , x ) G j ′ ( x , x ′ )Γ (3) ( ξ ′ ; ξ , ξ ′ )+ ( ξ ↔ ξ ′ ) (cid:9) . (75)1 B. Equations on t ∈ [ −∞ , ∞ ] Every integral over C can be transformed into that of t ∈ [ −∞ , ∞ ] through the procedure [35, 36] Z ∞−∞ dt C = Z ∞−∞ dt + Z −∞∞ dt = Z ∞−∞ dt − Z ∞−∞ dt , (76)where t ( t ) denotes a time on the outward (return)path. We also introduce the argument x i = ( r , t i ) ( i =1 , G i i j j ( x , x ) ≡ G j j ( x i , x i ) , (77a)Σ i i j j ( x , x ) ≡ ( − i + i Σ j j ( x i , x i ) . (77b)Let us define the matrices in the Nambu-Keldysh space(i.e., the j - i space) byˇ G ( x, x ′ ) ≡ (cid:20) ˆ G ( x, x ′ ) ˆ G ( x, x ′ )ˆ G ( x, x ′ ) ˆ G ( x, x ′ ) (cid:21) , (78a)ˇΣ( x, x ′ ) ≡ (cid:20) ˆΣ ( x, x ′ ) ˆΣ ( x, x ′ )ˆΣ ( x, x ′ ) ˆΣ ( x, x ′ ) (cid:21) , (78b)ˇ G − ( x, x ′ ) ≡ (cid:20) ˆ G − ( x, x ′ ) 00 − ˆ G − ( x, x ′ ) (cid:21) , (78c)ˇ1 ≡ (cid:20) ˆ1 ˆ0ˆ0 ˆ1 (cid:21) , ˇ σ ≡ (cid:20) ˆ σ ˆ0ˆ0 ˆ σ (cid:21) , (78d)where each quantity with a hat on it is a 2 × G − ( x, x ′ ) is given by Eq. (16)with τ = it/ ~ . The factor ( − i + i in Eq. (77b) and the − sign in the 22 element of Eq. (78c) have been added toour previous definitions [36]. Because of these modifica-tions, ˇ G now obeys ( ˇ G − − ˇΣ) ˇ G = ˇ1 . (79)Similarly, the generalized Hugenholtz-Pines relation isgiven by ˇ σ ( ˇ G − − ˇΣ)ˇ σ ~ Ψ = ~ , (80) where ~ Ψ and ~ ~ Ψ( x ) ≡ (cid:20) ~ Ψ( x ) ~ Ψ( x ) (cid:21) . (81)With these quantities, derivation of the two-particleGreen’s functions on t ∈ [ −∞ , ∞ ] proceeds in exactlythe same way as that of Sect. VI A. Indeed, we only needto redefine ξ by ξ ≡ ( i, j, x ), replace every ˆ A by ˇ A , anduse δ ˇ I ( x, x ′ ) ≡ (cid:20) δ ˆ I ( x, x ′ ) ˆ0ˆ0 − δ ˆ I ( x, x ′ ) (cid:21) , (82)in place of δ ˆ I ( x, x ′ ). Equation (71) is thereby replacedby δ~ Ψ = ˇ G (cid:18) Ψ (3) + i ~ (3) GG (cid:19) δ~I (s) , (83a) δ ~G = (cid:20) GG (cid:18) i ~ (4) GG (cid:19) + GG Γ (3)T ˇ G (cid:18) Ψ (3) + i ~ (3) GG (cid:19)(cid:21) δ~I (s) . (83b) VII. CONCLUDING REMARKS
We have derived a system of equations for the con-densate wave function, Green’s functions, and three- andfour-point Green’s functions, which are summarized inSect. V. The four-point (i.e., two-particle) Green’s func-tions are confirmed to share poles with Green’s functions,as seen in Eq. (47b). One of the other key results hereis Eq. (41), which will be used in the following paper[30] to derive the Ward-Takahashi identities. They willenable us to describe low-temperature properties of cor-related Bose-Einstein condensates in terms of low-energyGreen’s functions and vertices.
ACKNOWLEDGMENT
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