Microscopic derivation of density functional theory for superfluid systems based on effective action formalism
Takeru Yokota, Haruki Kasuya, Kenichi Yoshida, Teiji Kunihiro
aa r X i v : . [ nu c l - t h ] A ug Prog. Theor. Exp. Phys. , 00000 (26 pages)DOI: 10.1093 / ptep / Microscopic derivation of density functional theoryfor superfluid systems based on e ff ective actionformalism Takeru Yokota , Haruki Kasuya , Kenichi Yoshida , and Teiji Kunihiro Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581,Japan Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Density-functional theory for superfluid systems is developed in the framework of thefunctional renormalization group based on the e ff ective action formalism. We introduce thee ff ective action for the particle-number and nonlocal pairing densities and demonstrate that theHohenberg–Kohn theorem for superfluid systems is established in terms of the e ff ective action.The flow equation for the e ff ective action is then derived, where the flow parameter runs from 0to 1, corresponding to the non-interacting and interacting systems. From the flow equation andthe variational equation that the equilibrium density satisfies, we obtain the exact expression forthe Kohn–Sham potential generalized to including the pairing potentials. The resultant Kohn–Sham potential has a nice feature that it expresses the microscopic formulae of the external,Hartree, pairing, and exchange-correlation terms, separately. It is shown that our Kohn–Shampotential gives the ground-state energy of the Hartree–Fock–Bogoliubov theory by neglectingthe correlations. An advantage of our exact formalism lies in the fact that it provides ways tosystematically improve the correlation part. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index A63, B32
1. Introduction
The purpose of the present paper is to develop a density-functional theory (DFT) for superfluidsystems based on the e ff ective action formalism, where the Hohenberg–Kohn theorem is readilyestablished for superfluid systems with the spontaneous symmetry breaking (SSB) being incorpo-rated. Moreover, it will be shown that the exact expression for the Kohn–Sham (KS) potential isderived by making use of the functional renormalization group (FRG).DFT has been widely applied and proven to be an e ffi cient and convenient framework to dealwith the many-body problems in various fields including quantum chemistry and atomic, molec-ular, condensed-matter, and nuclear physics; see Refs. [1–6] for some recent reviews. DFT restsupon the Hohenberg–Kohn theorem [7] which states that the ground state of a quantum many-bodysystem is obtained by minimizing the energy-density functional (EDF) with respect to the particledensity only. In a practical implementation of DFT, one usually relies on the KS scheme [8], wherethe ground-state density of the interacting system is reproduced by just solving the Hartree–Fock-type single-particle Schr¨odinger equation, called the KS equation, for the non-interacting referencesystem. c (cid:13) The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http: // creativecommons.org / licenses / by-nc / owever it is known to be clumsy for a naive application of the original DFT to describe thesystems with SSB such as magnetization or superfluidity solely with the particle density. So theorder parameter as given by an anomalous density is included explicitly as an additional density todescribe such a system with SSB [9–20].A great challenge in the current study of DFT is to develop a microscopic and systematicframework for constructing the EDF from an inter-particle interaction. The notion of the e ff ec-tive action [21, 22] has brought forth the nonperturbative framework for dealing with the quantummany-body problems, and the concepts and recipes developed and accumulated in the study ofquantum-field theory (QFT) can be brought into the study of many-body systems. In particular, thee ff ective action for composite fields [23–25] links DFT and concepts and recipes of QFT. Indeed thetwo-particle point-irreducible e ff ective action [26] for the density field is identified with the free-energy density functional containing all the information of not only the ground state but also theexcited states [27, 28]. Thus a promising strategy to overcome the perennial challenge has come out.In Ref. [29], what they call the inversion method was developed to calculate the ground-state energynonperturbatively, and it was further extended to the Fermion system with superfluidity [30]. Alongwith this line of method, the notion of e ff ective field theory is applied to construct the free-energydensity functional including superfluidity [31–34].Although the inversion method is certainly a well-founded and powerful nonperturbative scheme,the notions and techniques developed in QFT may not have been fully utilized. An establishedmethod is currently available for the calculation of e ff ective actions, which is called the functionalrenormalization group (FRG) [35–38]. The FRG provides a nonperturbative and systematic pro-cedure for the analyses of renormalization flows by solving one-parameter functional di ff erentialequations in a closed form of e ff ective actions. Recently, the application to the e ff ective actionfor composite fields aimed at an ab-initio construction of DFT, which we call the functional-renormalization-group aided density functional theory (FRG-DFT), has been developed: Afterthe proposal of the formalism in Refs. [39, 40], several approximation schemes have been pro-posed, whose performances were tested in (0 + + + + +
1) dimensions [46] and (3 +
1) dimensions [47] were recentlyachieved.The purpose of this paper is to extend the FRG-DFT formalism as developed in Refs. [44, 45] soas to be applicable to superfluid systems at finite temperature. To this end, we introduce the e ff ec-tive action for the particle-number and nonlocal pairing densities, whereby the Hohenberg–Kohntheorem in terms of the Helmholtz free energy is found to be readily established. We then derive theflow equation for the e ff ective action by introducing a flow parameter λ with which the strength ofthe inter-particle interaction is adiabatically increased as λ is changed from 0 to 1. The di ff erentia-tion of the e ff ective action with respect to λ gives the coupled flow equations for the particle-numberand pairing densities. Together with the variational equation which the equilibrium densities shouldsatisfy, the exact expression for the KS potential generalized to including the pairing potentials isderived. The resultant KS potential has a nice feature that it expresses the microscopic formulae ofthe external, Hartree, pairing, and exchange-correlation terms, separately. As a demonstration of thevalidity of our formalism, we shall show that a lowest-order approximation to our flow equationswith respect to the flow parameter reproduces the gap equation, the ground-state energy and pairing / ensity given in the BCS theory for the case of short-range interactions in the weak-coupling limit.Although we show the validity of our microscopic formulation by applying it to simple cases, anadvantage of our exact FRG-DFT formalism lies in the fact that it provides ways to systematicallyimprove the correlation part.This paper is organized as follows: In Sec. 2, we introduce the e ff ective action for the particle-number and pairing densities and derive the flow equation to calculate the e ff ective action. From theflow equation, the exact expression of the KS potential is derived. In Sec. 3, we apply the lowest-order approximation to our formalism and compare our results with those given by the BCS theory.Section 4 is devoted to a brief summary and the conclusions.
2. Formalism
In this section, we develop a formalism to describe microscopically the superfluid systems at finitetemperature in the framework of FRG-DFT, where the e ff ective action given in the functional-integral form is fully utilized. The formalism starts with construction of the e ff ective action Γ [ ρ, κ, κ ∗ ]for the nonlocal pairing densities as well as particle-number ones, and then we show that Γ [ ρ, κ, κ ∗ ]can be identified with the free-energy density functional for which the Hohenberg–Kohn theoremholds. We then introduce a flow parameter λ with which the inter-particle interaction is switchedon gradually as λ is varied from 0 to 1. Di ff erentiating thus constructed e ff ective action Γ λ [ ρ, κ, κ ∗ ]with respect to λ , we obtain the coupled flow equations for the particle-number and pairing densi-ties. Thus the problem of deriving the governing equation of the superfluid systems in the context ofDFT is reduced to solving the flow equations. Inherently in the way of the introduction of the flowparameter, the governing equation thus obtained actually leads to the exact expression of the KSpotential. Moreover, because of the very exact nature of the governing equation thus obtained, theresultant equation can be a starting point for an introduction of various systematic approximations.We consider a system composed of non-relativistic fermions interacting via two-body interactionssubject to the external potential at finite temperature 1 /β with β being the inverse temperature. Theaction describing the system in the imaginary-time formalism is given as follows: S [ ψ, ψ ∗ ] = Z ξ ψ ∗ ( ξ + ǫ τ ) ∂ τ − ∆ + V ( ξ ) ! ψ ( ξ ) + Z ξ,ξ ′ U ( ξ, ξ ′ ) ψ ∗ ( ξ + ǫ τ ) ψ ∗ ( ξ ′ + ǫ τ ) ψ ( ξ ′ ) ψ ( ξ ) . (1)Here, the short-hand notations ξ = ( τ, x , a ), ξ ′ = ( τ ′ , x ′ , a ′ ) , ξ ′ + ǫ τ = ( τ ′ + ǫ, x ′ , a ′ ) and R ξ = P a R β d τ R d x have been introduced, where τ , x , and a represent the imaginary time, spatial coordi-nate, and index for internal degree of freedoms, respectively with ǫ being an infinitesimal constant.The fermion fields are represented by Grassmann variables ψ ( ξ ) and ψ ∗ ( ξ ). The static externalpotential and the instantaneous two-body interaction are denoted by V ( ξ ) = V a ( x ) and U ( ξ, ξ ′ ) = δ ( τ − τ ′ ) U aa ′ ( x − x ′ ), respectively. The shifted coordinate ξ + ǫ τ = ( τ + ǫ, x , a ) is introduced so thatthe corresponding Hamiltonian becomes normal-ordered [44]. ff ective action for superfluid systems Let us introduce the e ff ective action, which is to be identified with the free-energy density func-tional for superfluid systems. The construction of the e ff ective action [21, 22] starts by defining thegenerating functional of the correlation functions of density fields. In this work, we generalize theprevious works [27, 44, 45] and introduce the generating functional for the correlation functions of / article-number and (nonlocal) pairing densities as Z [ ~ J ] = Z D ψ D ψ ∗ e − S [ ψ,ψ ∗ ] + R ξ J ρ ( ξ ) ψ ∗ ( ξ + ǫ τ ) ψ ( ξ ) + R ξ,ξ ′ J κ ( ξ,ξ ′ ) ψ ( ξ ) ψ ( ξ ′ ) + R ξ,ξ ′ J κ ∗ ( ξ,ξ ′ ) ψ ∗ ( ξ ′ ) ψ ∗ ( ξ ) = Z D ψ D ψ ∗ e − S [ ψ,ψ ∗ ] + P i = ρ,κ,κ ∗ R ξ i J i ( ξ i )ˆ ρ i ( ξ i ) . (2) ~ J = ( J ρ ( ξ ρ ) , J κ ( ξ κ ) , J κ ∗ ( ξ κ ∗ )) denotes the external source coupled to the respective density fields: (cid:26) ˆ ρ i ( ξ i ) (cid:27) i = ρ,κ,κ ∗ = { ψ ∗ ( ξ + ǫ τ ) ψ ( ξ ) , ψ ( ξ ) ψ ( ξ ′ ) , ψ ∗ ( ξ ′ ) ψ ∗ ( ξ ) } . (3)with the following abbreviated notations for arguments ξ i ≡ ξ = ( τ, x , a ) ; i = ρ ( ξ, ξ ′ ) = ( τ, x , a , τ ′ , x ′ , a ′ ) ; i = κ ( ∗ ) . (4)We emphasize that the pairing fields are introduced as nonlocal ones with which the present for-malism is applicable for describing not only the conventional spin-singlet pairing but any types ofpairings as in Ref. [48] and furthermore the UV divergence, which appears in the case of localinteractions and local pairings, can be avoided.Now we define the generating functional for connected correlation functions W [ ~ J ] by W [ ~ J ] : = ln Z [ ~ J ] . (5)Then the e ff ective action Γ [ ~ρ = ( ρ, κ, κ ∗ )] for the particle-number density ρ ( ξ ) as well as nonlocalpairing densities κ ( ∗ ) ( ξ, ξ ′ ) is given by a Legendre transformation of W [ ~ J ]: Γ [ ~ρ ] = sup ~ J X i Z ξ i J i ( ξ i ) ρ i ( ξ i ) − W [ ~ J ] = : X i Z ξ i J i sup [ ~ρ ]( ξ i ) ρ i ( ξ i ) − W [ ~ J sup [ ~ρ ]] . (6)Here ~ J sup [ ~ρ ] denotes the source that gives the supremum of the quantity in the bracket of the firstline and gives the expectation value of the density operator ˆ ρ i δ W [ ~ J ] δ J i ( ξ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~ J = ~ J sup [ ~ρ ] = ρ i ( ξ i ) (7)on account of Eqs. (2) and (5). Now that having defined the e ff ective action by Eq. (6) for superfluid systems in a rather naturalway, we are in a position to show one of the most important points in this article, i.e., a proof of theHohenberg–Kohn theorem for superfluid systems with nonzero temperature [11] on the basis of thee ff ective action without recourse to any heuristic arguments. ◦ Variational principle for Eq. (6) . For a proper description of the superfluidity where the SSB is accompanied, it is most conve-nient to introduce a τ -independent artificial external potential µ κ ( ∗ ) ( ξ κ ( ∗ ) ) coupled to the pairingdensity κ ( ∗ ) ( ξ κ ( ∗ ) ), which will be, however, taken to be zero in the end of calculation. We shallconsider a generic case where the chemical potential µ a may depend on the internal degree of / ! !" ! !" " !" ! !" " ! !" !"! ! " !" Fig. 1
Schematic pictures of the κ -dependence of I in the case of (a) no external potential and (b)small nonzero external potential, where the ρ -dependence is not depicted. Two di ff erent equilibriumpairing densities are denoted by κ (1 , . δ is an infinitesimally small but nonzero number. Without theexternal potential, κ ave is not determined uniquely in the variational way since I has the same valuein the region shown in (a). With a small external field being applied, the equilibrium density [ κ (1)ave in(b)] may be obtained variationally since the flat region is slightly tilted.freedom a ; however, notice that the possible spatial dependence of µ a can be neglected withoutloss of generality because it can be absorbed away into the redefinition of V a ( x ).Now under the condition that the external potential δ i ρ V ( ξ ρ ) − µ i ( ξ i ) is applied to the system,consider the variational problem of the following functional: I [ ~ρ ] = Γ [ ~ρ ] − X i Z ξ i µ i ( ξ i ) ρ i ( ξ i ) , (8)where ~µ = ( µ ρ ( ξ ρ ) , µ κ ( ξ κ ) , µ κ ∗ ( ξ κ ∗ )) = ( µ a , µ κ ( ξ κ ) , µ κ ∗ ( ξ κ ∗ )).The solution ~ρ ave of the variational equation δ I [ ~ρ ] /δρ i ( ξ i ) = J i sup [ ~ρ ave ]( ξ i ) = µ i ( ξ i ),and thus one obtains ρ ave , i ( ξ i ) = Z [ ~µ ] Z D ψ D ψ ∗ ˆ ρ i ( ξ i ) e − S [ ψ,ψ ∗ ] + P j = ρ,κ,κ ∗ R ξ j µ j ( ξ j )ˆ ρ j ( ξ j ) , (9)by the use of Eqs. (2) and (7). Equation (9) shows that ~ρ ave is the average of the density in theequilibrium state in the presence of the external potential δ i ρ V ( ξ ρ ) − µ i ( ξ i ).A remark is in order here: Although the vanishing limit is formally taken for µ κ ( ∗ ) ( ξ κ ( ∗ ) ), thislimit should be taken with a care when SSB is concerned; it should be interpreted as 0 + inthe mathematical notation meaning an infinitesimally small but nonzero value to obtain ~ρ ave variationally. Figure 1 shows a schematic picture of I [ ~ρ ] solely as a function of κ , say, with ρ fixed. If we have two solutions κ (1)ave and κ (2)ave as the pairing density in equilibrium, I [ ~ρ ] with µ κ ( ∗ ) = κ = ακ (1)ave + (1 − α ) κ (2)ave , 0 ≤ α ≤
1] since Γ [ ~ρ ] is a convex functional as a result of the Legendre transformation. This prevents one fromdetermining the equilibrium density uniquely by a variational method. As long as µ κ ( ∗ ) is keptsmall but nonvanishing, however, the flat region of I [ ~ρ ] is slightly tilted, then one can uniquelyidentify the equilibrium density in the variational method.Furthermore, it follows immediately from Eq. (9) that the external potential determines theequilibrium density uniquely. On the other hand, it is easily demonstrated that the equilibrium / ensity in turn determines uniquely the external potential by observing that I [ ~ρ ave ] /β coincideswith the grand potential Ω [ ~µ ] = − W [ ~µ ] /β for ~ρ ave minimizing I [ ~ρ ]: I [ ~ρ ave ] = − W [ ~ J sup [ ~ρ ave ]] + X i Z ξ i [ J i sup [ ~ρ ave ]( ξ i ) − µ i ( ξ i )] ρ i ( ξ i ) = − W [ ~µ ] , (10)where we have used J i sup [ ~ρ ave ]( ξ i ) = µ i ( ξ i ). This argument is based on the fact that the functional I [ ~ρ ] is strictly convex as seen in Fig. 1(b). Therefore, one has a one-to-one mapping between theexternal potential δ i ρ V ( ξ ρ ) − µ i ( ξ i ) and the equilibrium density ~ρ ave . In particular, the existenceof the map from ~ρ ave to δ i ρ V ( ξ ρ ) − µ i ( ξ i ) can be seen more clearly through the fact that Γ [ ~ρ ] islinearly dependent on the external field, which we now show below. ◦ One-to-one map between the external fields and the densities.
From Eq. (6), we obtain Γ [ ~ρ ] = sup ~ J X i Z ξ i J i ( ξ i ) ρ i ( ξ i ) − W | V = [ ~ J − ( V , , = Z ξ V ( ξ ) ρ ( ξ ) + sup ~ J X i Z ξ i J i ( ξ i ) ρ i ( ξ i ) − W | V = [ ~ J ] = Z ξ V ( ξ ) ρ ( ξ ) + Γ | V = [ ~ρ ] , (11)where W | V = [ ~ J ] and Γ | V = [ ~ρ ] are defined by the action with vanishing external potential. Fromthe first to second line of Eq. (11), we have shifted ~ J as ~ J → ~ J + ( V , , Γ [ ~ρ ] can be decomposed into the universal part Γ | V = [ ~ρ ] independent of V ( ξ ) and the termdepending on V ( ξ ) linearly.Through Eq. (11) and the variational equation of I [ ~ρ ], the existence of the one-to-one mapfrom ~ρ ave to δ i ρ V ( ξ ρ ) − µ i ( ξ i ) is established. To show this, let us assume that two sets of theexternal potential and the chemical potential denoted by ( V , ~µ ) and ( V , ~µ ), respectively,give the same densities ~ρ ave . By use of the variational equation of I [ ~ρ ] and Eq. (11), however,we obtain δ Γ | V = [ ~ρ ave ] δρ i ( ξ i ) + δ i ρ V ( ξ ρ ) − µ i ( ξ i ) = ,δ Γ | V = [ ~ρ ave ] δρ i ( ξ i ) + δ i ρ V ( ξ ρ ) − µ i ( ξ i ) = . Subtracting these equations from each other, we have the equality δ i ρ V ( ξ ρ ) − µ i ( ξ i ) = δ i ρ V ( ξ ρ ) − µ i ( ξ i ) , (12)which proves that δ i ρ V ( ξ ρ ) − µ i ( ξ i ) is uniquely determined by ~ρ ave .Thus the Hohenberg–Kohn theorem for superfluid systems with nonzero temperature, which wasfirst demonstrated in Ref. [11], is established in terms of the e ff ective action. Furthermore it is also / vident that β − Γ [ ~ρ ] can be identified with Helmholtz free energy F H [ ~ρ ] at ~ρ = ~ρ ave as F H [ ~ρ ] = Γ [ ~ρ ] β . (13)In fact, in the limit of µ κ ( ∗ ) →
0, Eq. (13) at ~ρ = ~ρ ave becomes F H [ ~ρ ave ] = β Z ξ µ a ρ ave ( ξ ) − β W [ ~µ = ( µ a , , , (14)because of the relation ~ J sup [ ~ρ ave ] = ~µ and Eq. (6). In terms of words, since Z [ ~µ ] is the grand partitionfunction in the presence of the chemical potential ~µ , − W [ ~µ ] /β = − ln Z [ µ ] /β is the grand potential,and hence F H [ ~ρ ave ] in Eq. (14) is identified with the Helmholtz free energy.Finally, let us consider the case of zero-temperature limit. At the zero temperature limit β →∞ , F H [ ~ρ ave ] becomes the ground-state energy E gs because F H [ ~ρ ave ] can be written as F H [ ~ρ ave ] = − β − ln P n exp( − β E n ), where { E n } is the energy eigenvalues of the system and satisfies E gs = E < E < · · · . Therefore, we can identify the EDF with the e ff ective action as E [ ~ρ ] = lim β →∞ β Γ [ ~ρ ] . (15)In particular, the universal part of the EDF is obtained by substitution of Γ with Γ | V = . This concludesthe generalization of the correspondence between the EDF and the e ff ective action to the superfluidsystems [27, 28, 44]. The e ff ective action Γ [ ~ρ ] contains the whole contents of physics of the quantum system in a compactform of the functional integral. Although the compact and exact form of the action allows us todefine consistent approximations, the problem is developing a computational method to performthe functional integral in a systematic and hopefully exact way. In this subsection, we are going todevelop a practical method to calculate the e ff ective action in such a manner.A wise way originally developed by Wegner [35] and Wilson [36] is to convert the functionalintegral to a di ff erential equation, which is called a renormalization-group (RG) equation or flowequation, and solve the equation with some systematic approximation being adopted. In the caseof the e ff ective action solely for the particle-number density and accordingly without SSB, acalculational framework is developed based on an RG / flow equation in Refs. [39, 40]; see alsoRefs. [44, 45]. Here, we generalize this framework to the case with SSB for the calculation of Γ [ ~ρ ]containing the pairing density fields.Following Refs. [39, 40, 44, 45], we introduce a regulated two-body interaction U λ ( ξ, ξ ′ ) with aflow parameter λ and define the regulated action as S λ [ ψ, ψ ∗ ] = Z ξ ψ ∗ ( ξ + ǫ τ ) ∂ τ − ∆ + V ( ξ ) ! ψ ( ξ ) + Z ξ,ξ ′ U λ ( ξ, ξ ′ ) ψ ∗ ( ξ + ǫ τ ) ψ ∗ ( ξ ′ + ǫ τ ) ψ ( ξ ′ ) ψ ( ξ ) , (16)where U λ ( ξ, ξ ′ ) is defined by U λ ( ξ, ξ ′ ) = δ ( τ − τ ′ ) U λ, aa ′ ( x − x ′ ) , (17)with a constraint U λ = , aa ′ ( x − x ′ ) = , U λ = , aa ′ ( x − x ′ ) = U aa ′ ( x − x ′ ) / o that the action evolves from the free S λ = to the fully-interacting one S λ = as λ runs from 0 to 1.With this action, the λ -dependent generating functionals and e ff ective action are defined in the samemanner as in Sec. 2.1: W λ [ ~ J ] = ln Z λ [ ~ J ] = ln Z D ψ D ψ ∗ e − S λ [ ψ,ψ ∗ ] + P i = ρ,κ,κ ∗ R ξ i J i ( ξ i )ˆ ρ i ( ξ i ) , (18) Γ λ [ ~ρ ] = sup ~ J X i Z ξ i J i ( ξ i ) ρ i ( ξ i ) − W λ [ ~ J ] = : X i Z ξ i J i sup ,λ [ ~ρ ]( ξ i ) ρ i ( ξ i ) − W λ [ ~ J sup ,λ [ ~ρ ]] , (19)where J i sup ,λ [ ~ρ ]( ξ i ) satisfies δ W λ [ ~ J ] δ J i ( ξ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~ J = ~ J sup ,λ [ ~ρ ]( ξ ) = ρ i ( ξ i ) . (20)Now utilizing the λ dependence of the action (19), the flow equation of Γ λ [ ~ρ ] can be obtainedsimply by di ff erentiating it with respect to λ [39, 40, 43–45]. It turns out, however that the mostconvenient way to obtain the flow equation for Γ λ [ ~ρ ] is first to derive that for W λ [ ~ J ] instead of Γ λ [ ~ρ ].The di ff erentiation of W λ [ ~ J ] with respect to λ reads ∂ λ W λ [ ~ J ] = − Z ξ ρ : = ( τ , x , a ) ,ξ ρ : = ( τ , x , a ) ∂ λ U λ ( ξ ρ , ξ ρ ) h ψ ∗ ( ξ ρ + ǫ τ ) ψ ∗ ( ξ ρ + ǫ τ ) ψ ( ξ ρ ) ψ ( ξ ρ ) i λ, ~ J , (21)where hO [ ψ, ψ ∗ ] i λ, ~ J : = Z λ [ ~ J ] Z D ψ D ψ ∗ O [ ψ, ψ ∗ ] e − S λ [ ψ,ψ ∗ ] + P i = ρ,κ,κ ∗ R ξ i ~ J ( ξ i )ˆ ρ i ( ξ i ) , (22)with O [ ψ, ψ ∗ ] being some functional of ψ and ψ ∗ . Noting that U λ ( ξ ρ , ξ ρ ) defined in Eq. (17) involvesthe delta function of times, the integrand in Eq. (21) can be cast into the form of the densitycorrelation functions with the use of the commutation relation: ∂ λ U λ ( ξ ρ , ξ ρ ) h ψ ∗ ( ξ ρ + ǫ τ ) ψ ∗ ( ξ ρ + ǫ τ ) ψ ( ξ ρ ) ψ ( ξ ρ ) i λ, ~ J = δ ( τ − τ ) ∂ λ U λ, aa ′ ( x − x ′ ) h ψ ∗ ( ξ ρ + ǫ τ ) ψ ∗ ( ξ ρ + ǫ τ ) ψ ( ξ ρ ) ψ ( ξ ρ ) i λ, ~ J = δ ( τ − τ ) ∂ λ U λ, aa ′ ( x − x ′ ) h h ψ ∗ ( ξ ρ + ǫ ′ τ + ǫ τ ) ψ ( ξ ρ + ǫ ′ τ ) ψ ∗ ( ξ ρ + ǫ τ ) ψ ( ξ ρ ) i λ, ~ J − δ a , a h ψ ∗ ( ξ ρ + ǫ τ ) ψ ( ξ ρ ) i λ, ~ J δ ( x − x ) i , (23)where an infinitesimal number ǫ ′ τ plays the same role as ǫ τ but its zero limit should be taken afterthat of ǫ τ ; see Appendix in Ref. [44]. From Eq. (18), one can immediately see h ˆ ρ i ( ξ i ) i λ, ~ J = δ W λ [ ~ J ] δ J i ( ξ i ) , (24) h ψ ∗ ( ξ ρ + ǫ ′ τ + ǫ τ ) ψ ( ξ ρ + ǫ ′ τ ) ψ ∗ ( ξ ρ + ǫ τ ) ψ ( ξ ρ ) i λ, ~ J = δ W λ [ ~ J ] δ J ρ ( ξ ρ ) δ W λ [ ~ J ] δ J ρ ( ξ ρ ) + δ W λ [ ~ J ] δ J ρ ( ξ ρ + ǫ ′ τ ) δ J ρ ( ξ ρ ) . (25)Using these relations, one finds that Eq. (21) takes the form of the flow equation for W λ [ ~ J ] as follows: ∂ λ W λ [ ~ J ] = − Z ξ ρ ,ξ ρ ∂ λ U λ ( ξ ρ , ξ ρ ) × δ W λ [ ~ J ] δ J ρ ( ξ ρ ) δ W λ [ ~ J ] δ J ρ ( ξ ρ ) + δ W λ [ ~ J ] δ J ρ ( ξ ρ + ǫ ′ τ ) δ J ρ ( ξ ρ ) − δ W λ [ ~ J ] δ J ρ ( ξ ρ ) δ a , a δ ( x − x ) . (26) / ext we shall show that this flow equation is in turn converted to that for Γ λ [ ~ρ ] on account ofEq. (19). By di ff erentiating Eq. (20) with respect to ~ J sup ,λ [ ~ρ ], we have δ W λ [ ~ J sup ,λ [ ~ρ ]] δ J i ( ξ i ) δ J j ( ξ j ) = δρ i ( ξ i ) δ J j ( ξ j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~ J = ~ J sup ,λ [ ~ρ ] . (27)By taking the second derivative of Eq. (6) with respect to ~ρ , we have δ Γ λ [ ~ρ ] δρ i ( ξ i ) δρ j ( ξ j ) = δ J j sup ,λ [ ~ρ ]( ξ j ) δρ i ( ξ i ) . (28)One finds that Eq. (27) is the inverse of Eq. (28): δ W λ [ ~ J sup ,λ [ ~ρ ]] δ J i ( ξ i ) δ J j ( ξ j ) = δ Γ λ δ~ρδ~ρ ! − i j [ ~ρ ]( ξ i , ξ j ) . (29)Here, ( δ Γ λ /δ~ρδ~ρ ) − i j [ ~ρ ]( ξ i , ξ ′ j ) is defined as follows: X k = ρ,κ,κ ∗ Z ξ ′′ k δ Γ λ δ~ρδ~ρ ! − ik [ ~ρ ]( ξ i , ξ ′′ k ) δ Γ λ [ ~ρ ] δρ k ( ξ ′′ k ) δρ j ( ξ ′ j ) = δ ( ξ i , ξ ′ j ) , (30)and δ ( ξ i , ξ ′ j ) is defined by δ ( ξ i , ξ ′ j ) = δ ( ξ, ξ ′ ) ( i = j = ρ, ξ ρ = ξ, ξ ρ = ξ ′ ) δ ( ξ , ξ ′ ) δ ( ξ , ξ ′ ) ( i = j = κ ( ∗ ) , ξ κ ( ∗ ) = ( ξ , ξ ) , ξ ′ κ ( ∗ ) = ( ξ ′ , ξ ′ ))0 (otherwise) , with δ ( ξ = ( τ, x , a ) , ξ ′ = ( τ ′ , x ′ , a ′ )) = δ a , a ′ δ ( τ − τ ′ ) δ ( x − x ′ ). By taking the derivative of Eq. (6) withrespect to λ , we obtain ∂ λ Γ λ [ ~ρ ] = − ( ∂ λ W λ ) [ ~ J sup ,λ [ ~ρ ]] . (31)Substituting Eqs. (20), (29), and (31) into Eq. (26), we eventually arrive at the formal flow equationfor Γ λ [ ~ρ ]: ∂ λ Γ λ [ ~ρ ] = Z ξ ρ = ( τ , x , a ) ,ξ ρ = ( τ , x , a ) ∂ λ U λ ( ξ ρ , ξ ρ ) × ρ ( ξ ρ ) ρ ( ξ ρ ) + δ Γ λ δ~ρδ~ρ ! − ρρ [ ~ρ ]( ξ ρ + ǫ ′ τ , ξ ρ ) − ρ ( ξ ρ ) δ a , a δ ( x − x ) . (32)Now Eqs. (25) and (29) together with Eq. (20) tell us that ( δ Γ λ δ~ρδ~ρ ) − ρρ gives essentially the correlationfunctions of the densities ~ρ , which collectively denote the particle-number and pairing densities. Tosee more explicitly the physical contents of the flow equation (32), in particular, the contributionsof the pairing densities, we find it convenient to separate the λ = / here the particle operators are free ones: δ Γ λ δ~ρδ~ρ ! − ρρ [ ~ρ ]( ξ ρ + ǫ ′ τ , ξ ρ ) − ρ ( ξ ρ ) δ a , a δ ( x − x ) = h ψ ∗ ( ξ ρ + ǫ τ + ǫ ′ τ ) ψ ( ξ ρ + ǫ ′ τ ) ψ ∗ ( ξ ρ + ǫ τ ) ψ ( ξ ρ ) i λ, ~ J sup ,λ [ ~ρ ] − ρ ( ξ ρ + ǫ ′ τ ) ρ ( ξ ρ ) − ρ ( ξ ρ ) δ a , a δ ( x − x ) = h ψ ∗ ( ξ ρ + ǫ τ + ǫ ′ τ ) ψ ( ξ ρ + ǫ ′ τ ) ψ ∗ ( ξ ρ + ǫ τ ) ψ ( ξ ρ ) i λ = , ~ J sup ,λ = [ ~ρ ] − ρ ( ξ ρ + ǫ ′ τ ) ρ ( ξ ρ ) − ρ ( ξ ρ ) δ a , a δ ( x − x ) + O ( λ ) . By applying the Wick’s theorem, we obtain δ Γ λ δ~ρδ~ρ ! − ρρ [ ~ρ ]( ξ ρ + ǫ ′ τ , ξ ρ ) − ρ ( ξ ρ ) δ a , a δ ( x − x ) = κ ∗ ( ξ κ ∗ ) κ ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) − h ψ ∗ ( ξ ρ + ǫ τ + ǫ ′ τ ) ψ ( ξ ρ ) i λ = , ~ J sup ,λ = [ ~ρ ] h ψ ∗ ( ξ ρ + ǫ τ ) ψ ( ξ ρ + ǫ ′ τ ) i λ = , ~ J sup ,λ = [ ~ρ ] − ρ ( ξ ρ ) δ a , a δ ( x − x ) + O ( λ ) , (33)where we have used the equality h ˆ ρ i ( ξ i ) i λ, ~ J sup ,λ [ ~ρ ] = ρ i ( ξ i ) which follows from Eqs. (24) and (20). Onesees that the first term given by a product of the pairing condensates in Eq. (33) is the counter partto that of the particle-number densities given in the first term of Eq. (32), while the terms in thesecond line and the remaining order- λ terms are identified with the exchange and correlation terms,respectively. Thus one finds that it is natural to represent Eq. (32) in a form where the terms withdi ff erent physical significance are written separately as, ∂ λ Γ λ [ ~ρ ] = Z ξ ρ = ( τ , x , a ) ,ξ ρ = ( τ , x , a ) ∂ λ U λ ( ξ ρ , ξ ρ ) × (cid:20) ρ ( ξ ρ ) ρ ( ξ ρ ) + κ ∗ ( ξ κ ∗ ) κ ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) + G (2)xc ,λ [ ~ρ ]( ξ ρ , ξ ρ ) (cid:21) , (34)where G (2)xc ,λ [ ~ρ ]( ξ ρ , ξ ρ ) denotes the exchange–correlation term given by G (2)xc ,λ [ ~ρ ]( ξ ρ , ξ ρ ) = G (2)x [ ~ρ ]( ξ ρ , ξ ρ ) + G (2)c ,λ [ ~ρ ]( ξ ρ , ξ ρ ) , (35)with G (2)x [ ~ρ ]( ξ ρ , ξ ρ ) = δ Γ λ = δ~ρδ~ρ ! − ρρ [ ~ρ ]( ξ ρ + ǫ ′ τ , ξ ρ ) − κ ∗ ( ξ κ ∗ ) κ ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) − ρ ( ξ ρ ) δ a , a δ ( x − x ) , (36) G (2)c ,λ [ ~ρ ]( ξ ρ , ξ ρ ) = δ Γ λ δ~ρδ~ρ ! − ρρ [ ~ρ ]( ξ ρ + ǫ ′ τ , ξ ρ ) − δ Γ λ = δ~ρδ~ρ ! − ρρ [ ~ρ ]( ξ ρ + ǫ ′ τ , ξ ρ ) . (37)We call G (2)x [ ~ρ ]( ξ ρ , ξ ρ ) and G (2)c ,λ [ ~ρ ]( ξ ρ , ξ ρ ) the exchange and correlation terms, respectively.We note that the flow equation gives the systematic way to calculate the higher-order contributionfor Γ λ [ ~ρ ], or G (2)c ,λ [ ~ρ ], since it is not only an exact but also closed equation for Γ λ [ ~ρ ]. Moreover, italso provides us with the basis for defining consistent approximations, say, respecting symmetries,if necessary. In this respect, some approximation schemes developed for the study of the normalphase such as the vertex expansion [39, 40] is expected to be utilized for the determination of Γ λ [ ~ρ ]also in the case of superfluid systems. / .4. Expressions of Helmholtz free energy and ground-state energy Having derived the flow equation (34) for the e ff ective action Γ λ [ ~ρ ], we here give a reduced formulafor the Helmholtz free energy which is given in terms of the e ff ective action.The Helmholtz free energy given in Eq. (13) now takes the following form F H [ ~ρ ave ] = Γ λ = [ ~ρ ave ] β , (38)where ~ρ ave is the equilibrium density of the fully-interacting ( λ =
1) system in the presence of agiven chemical potential ~µ : ρ ave , i ( ξ i ) = Z λ = [ ~µ ] Z D ψ D ψ ∗ ˆ ρ i ( ξ i ) e − S λ = [ ψ,ψ ∗ ] + P j = ρ,κ,κ ∗ R ξ j µ j ( ξ j )ˆ ρ j ( ξ j ) , (39)which is nothing but Eq. (9) with a replacement Z [ ~µ ] → Z λ = [ ~µ ]. The expression for Γ λ = [ ~ρ ave ] isobtained by integrating Eq. (34) with respect to λ . Inserting the expression into Eq. (38) and usingEq. (11) for Γ λ = , we have F H [ ~ρ ave ] = Γ λ = | V = [ ~ρ ave ] β + β Z ξ V ( ξ ) ρ ave ( ξ ) + β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) (cid:20) ρ ave ( ξ ρ ) ρ ave ( ξ ρ ) + κ ∗ ave ( ξ κ ∗ ) κ ave ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) (cid:21) + Z d λ β Z ξ ρ ,ξ ρ ∂ λ U λ ( ξ ρ , ξ ρ ) G (2)xc ,λ [ ~ρ ave ]( ξ ρ , ξ ρ ) . (40)The ground-state energy is obtained by taking the zero temperature limit in Eq. (40): E gs = lim β →∞ F H [ ~ρ ave ] = T [ ~ρ gs ] + lim β →∞ β Z ξ V ( ξ ) ρ ave ( ξ ) + lim β →∞ β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) (cid:20) ρ ave ( ξ ρ ) ρ ave ( ξ ρ ) + κ ∗ ave ( ξ κ ∗ ) κ ave ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) (cid:21) + lim β →∞ Z d λ β Z ξ ρ ,ξ ρ ∂ λ U λ ( ξ ρ , ξ ρ ) G (2)xc ,λ [ ~ρ ave ]( ξ ρ , ξ ρ ) , (41)where T [ ~ρ gs ] : = lim β →∞ Γ λ = | V = [ ~ρ ave ] /β with ~ρ gs ≔ lim β →∞ ~ρ ave being the ground-state density isinterpreted as the kinetic energy. Notice that T [ ~ρ gs ] is neither the kinetic energy of a free gas northat of the interacting system, but that of the non-interacting system with an appropriate potential,as may be utilized in KS formalism. Indeed, we shall show in the following subsection that the KStheory for superfluid systems at finite temperature emerges naturally in the present formalism. In our formulation of the FRG-DFT theory for superfluid systems, the starting action with λ = ~ J sup ,λ [ ~ρ ] is related to the KS potential ~ V KS [ ~ρ ] that is generalized so as to incorporate the anomalousor pair potentials besides the normal ones: The KS potential is defined by the derivative of the EDFsubtracted by the non-interacting kinetic energy part at zero temperature. In terms of the e ff ective / ction at finite temperature, the kinetic energy part T KS [ ~ρ ] may be identified with Γ λ = | V = , and thenwe naturally arrive at the definition of the KS potential as V i KS [ ~ρ ]( ξ i ) = δδρ i ( ξ i ) (cid:0) Γ λ = [ ~ρ ] − Γ λ = | V = [ ~ρ ] (cid:1) . (42)By di ff erentiating Eq. (19) with respect to the densities, we obtain δ Γ λ [ ~ρ ] δρ i ( ξ i ) = J i sup ,λ [ ~ρ ]( ξ i ) . (43)Using this relation and Eq. (11) for Γ = Γ λ = , Eq. (42) is rewritten as follows: V i KS [ ~ρ ]( ξ i ) = δ i ρ V ( ξ i ) + J i sup ,λ = [ ~ρ ] − J i sup ,λ = [ ~ρ ] . (44)The KS potential determines ~ρ ave for given chemical potentials ~µ : Since J i sup ,λ = [ ~ρ ave ]( ξ i ) = µ i ( ξ i ) issatisfied, Eq. (44) at ~ρ = ~ρ ave reads δ Γ λ = | V = [ ~ρ ave ] δρ i ( ξ i ) + V i KS [ ~ρ ave ]( ξ i ) = µ i ( ξ i ) , (45)which is equivalent to the variational equation in the KS DFT [49] and determines ~ρ ave from theKS potential (42). The present formulation does not refer to any single-particle orbitals for estab-lishing the KS DFT. Note that Eq. (45) does possess the property of a self-consistency in termsof the densities. In fact the equilibrium density of the full-interacting system with a given chem-ical potential ~µ is also the equilibrium density of the non-interacting system under the externalpotential ~µ − ~ V KS [ ~ρ ave ], as we will now show explicitly: First of all, we notice that Eq. (45) isalso expressed as J i sup ,λ = [ ~ρ ave ]( ξ i ) = δ i ρ V ( ξ i ) + µ i ( ξ i ) − V i KS [ ~ρ ave ]( ξ i ) on account of the relation J i sup ,λ = [ ~ρ ave ]( ξ i ) = µ i ( ξ i ) and Eq. (44). Then inserting this relation into Eqs. (18) and (20), one findsthat Eq. (20) tells us that the equilibrium density ~ρ ave in Eq. (39) also has the following expression ρ ave , i ( ξ i ) = Z λ = | V = [ ~µ − ~ V KS [ ~ρ ave ]] Z D ψ D ψ ∗ ˆ ρ i ( ξ i ) e − S λ = | V = [ ψ ∗ ,ψ ] + P j = ρ,κ,κ ∗ R ξ j (cid:18) µ j ( ξ j ) −V j KS [ ~ρ ave ]( ξ j ) (cid:19) ˆ ρ j ( ξ j ) , (46)where Z λ | V = [ ~ J ] is defined by the action S λ | V = [ ψ ∗ , ψ ] with no external potential in Eq. (18). Notethat Eq. (46) is a self-consistent equation for ~ρ ave since ~ V KS [ ~ρ ave ] in the right-hand side depends on ~ρ ave in the left-hand side, as noted before.Next, let us reduce the exact expression of the central quantity ~ V KS [ ~ρ ] in our formalism in a morecalculable form from the flow equation (34). After di ff erentiating Eq. (34) with respect to ~ρ and thenintegrating it with respect to λ , we have J i sup ,λ = [ ~ρ ]( ξ i ) = J i sup ,λ = [ ~ρ ]( ξ i ) + δ i ,ρ Z ξ ρ U λ = ( ξ ρ , ξ ρ ) ρ ( ξ ρ ) + δ i ,κ U λ = ( ξ κ ∗ ) κ ∗ ( ξ κ ∗ ) + δ i ,κ ∗ U λ = ( ξ κ ) κ ( ξ κ ) + Z d λ Z ξ ρ ,ξ ρ ∂ λ U λ ( ξ ρ , ξ ρ ) δ G (2)xc ,λ [ ~ρ ]( ξ ρ , ξ ρ ) δρ i ( ξ i ) , (47) / here U λ ( ξ κ ( ∗ ) : = ( ξ, ξ ′ )) = U λ ( ξ, ξ ′ ) and Eq. (43) has been used. With use of Eq. (44), we finallyarrive at the following expression for ~ V KS [ ~ρ ]: V i KS [ ~ρ ]( ξ i ) = δ i ρ V ( ξ ρ ) + δ i ,ρ Z ξ ρ U λ = ( ξ ρ , ξ ρ ) ρ ( ξ ρ ) + δ i ,κ U λ = ( ξ κ ∗ ) κ ∗ ( ξ κ ∗ ) + δ i ,κ ∗ U λ = ( ξ κ ) κ ( ξ κ ) ! + Z d λ Z ξ ρ ,ξ ρ ∂ λ U λ ( ξ ρ , ξ ρ ) δ G (2)xc ,λ [ ~ρ ]( ξ ρ , ξ ρ ) δρ i ( ξ i ) . (48)It is noteworthy that V i KS [ ~ρ ] is now naturally decomposed into four terms, which denote the exter-nal, Hartree, pairing, and exchange-correlation terms, respectively. In particular, a microscopicexpression of the exchange-correlation term V i KS , xc [ ~ρ ] is explicitly given in terms of G (2)xc ,λ [ ~ρ ]: V i KS , xc [ ~ρ ]( ξ i ) = Z d λ Z ξ ρ ,ξ ρ ∂ λ U λ ( ξ ρ , ξ ρ ) δ G (2)xc ,λ [ ~ρ ]( ξ ρ , ξ ρ ) δρ i ( ξ i ) . (49)We emphasize that our FRG-DFT formalism applied to superfluid systems has naturally led tothe self-consistent equation for the densities that are kept the same between the non-interacting andinteracting systems together with the microscopic expressions of the KS potential and the kineticenergy without recourse to notions of single-particle orbitals. It is also noted here that the e ff ectiveaction formalism with the inversion method was applied to a microscopic construction of the KSpotential for a normal system [50] and a superfluid system of the local singlet pairs [33]. However,the exact expression of the KS potential is not derived in the inversion method. Furthermore, thesingle-particle basis are introduced in developing the KS DFT. In the next section, we are going to show that Eq. (40) together with Eqs. (45) and (48) leads to theground-state energy in the Hartree–Fock–Bogoliubov approximation, which is furthermore reducedto the well-known gap equations of BCS theory in the case of short-range interaction.
3. Analysis in the lowest order approximation: neglect of correlations
We demonstrate that our formalism actually reproduces the results of the mean field theory at thelowest-order truncation, implying that G (2)xc ,λ does not have λ dependence: G (2)xc ,λ ≈ G (2)x . Applying the lowest-order truncation to Eq. (48), we have V i KS [ ~ρ ]( ξ i ) = δ i ,ρ V ( ξ ρ ) + Z ξ ρ U λ = ( ξ ρ , ξ ρ ) ρ ( ξ ρ ) ! + δ i ,κ U λ = ( ξ κ ) κ ∗ ( ξ κ ) + δ i ,κ ∗ U λ = ( ξ κ ∗ ) κ ( ξ κ ∗ ) + Z ξ ρ ,ξ ρ ∂ λ U λ ( ξ ρ , ξ ρ ) δ G (2)x [ ~ρ ]( ξ ρ , ξ ρ ) δρ i ( ξ i ) , (50) In Ref. [34], it was anticipated that the KS DFT may be constructed without single-particle orbitals in theFRG-DFT. 13 / or the KS potential. On account of Eq. (36), δ G (2)x [ ~ρ ]( ξ ρ , ξ ρ ) /δρ i ( ξ i ) is represented as follows: δ G (2)x [ ~ρ ]( ξ ρ , ξ ρ ) δρ i ( ξ i ) = − X i ′ , i ′ Z ξ i ′ ′ ,ξ i ′ ′ Γ (2) − λ = ,ρ i ′ [ ~ρ ]( ξ ρ + ǫ ′ τ , ξ i ′ ′ ) Γ (2) − λ = ,ρ i ′ [ ~ρ ]( ξ ρ , ξ i ′ ′ ) Γ (3) λ = , ii ′ i ′ [ ~ρ ]( ξ i , ξ i ′ ′ , ξ i ′ ′ ) − δ ( ξ i , ξ κ ) κ ∗ ( ξ κ ∗ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) − δ ( ξ i , ξ κ ∗ ) κ ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) − δ ( ξ i , ξ ρ ) δ a , a δ ( x − x ) . (51)Here, the following notation has been introduced: Γ ( n ) λ, i ··· i n [ ~ρ ]( ξ i , · · · , ξ i n n ) ≔ δ n Γ λ [ ~ρ ] δρ i ( ξ i ) · · · δρ i n ( ξ i n n ) , and Γ (2) − λ, i j [ ~ρ ]( ξ i , ξ ′ j ) is equivalent to ( δ Γ λ /δ~ρδ~ρ ) − i j [ ~ρ ]( ξ i , ξ ′ j ) defined in Eq. (30). Γ ( n ≥ λ = areexpressed in terms of the density correlation functions of the non-interacting system, which canbe further reduced to simpler forms using the Wick’s theorem. When the superfluidity is not consid-ered [43, 44], Γ ( n ≥ λ are also written in terms of the connected density correlation functions. We aregoing to extend them so as to incorporate the anomalous density-correlation functions as Γ (2) λ, i i [ ~ρ ]( ξ i , ξ i ) = G (2) − λ, i i [ ~ρ ]( ξ i , ξ i ) , (52) Γ (3) λ, i i i [ ~ρ ]( ξ i , ξ i , ξ i ) = − X i , i , i Z ξ i ,ξ i ,ξ i G (2) − λ, i i [ ~ρ ]( ξ i , ξ i ) G (2) − λ, i i [ ~ρ ]( ξ i , ξ i ) × G (2) − λ, i i [ ~ρ ]( ξ i , ξ i ) G (3) λ, i i i [ ~ρ ]( ξ i , ξ i , ξ i ) , (53)where G ( n ) λ, i ··· i n ( ξ i , · · · , ξ i n n ) is the connected correlation function given by G ( n ) λ, i ··· i n [ ~ρ ]( ξ i , · · · , ξ i n n ) = δ n W λ [ ~ J sup ,λ [ ~ρ ]] δ J i ( ξ i ) · · · δ J i n ( ξ i n n ) , and G (2) − λ = , i i ( ξ i , ξ i ) satisfies X i Z ξ i G (2) − λ, i i [ ~ρ ]( ξ i , ξ i ) G (2) λ, i i [ ~ρ ]( ξ i , ξ i ) = δ ( ξ i , ξ i ) . Using these relations, the last term in the right-hand side of Eq. (50) is rewritten as follows:12 Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) δ G (2)x [ ~ρ ]( ξ ρ , ξ ρ ) δρ i ( ξ i ) = Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) X i ′ Z ξ i ′ ′ G (2) − λ = , ii ′ [ ~ρ ]( ξ i , ξ i ′ ′ ) G (3) λ = , i ′ ρρ [ ~ρ ]( ξ i ′ ′ , ξ ρ + ǫ ′ τ , ξ ρ ) − δ ( ξ i , ξ κ ) κ ∗ ( ξ κ ∗ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) − δ ( ξ i , ξ κ ∗ ) κ ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) − δ ( ξ i , ξ ρ ) δ a , a δ ( x − x ) (cid:21) = X i ′ Z ξ i ′ ′ G (2) − λ = , ii ′ [ ~ρ ]( ξ i , ξ i ′ ′ ) Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) × (cid:20) G (3) λ = , i ′ ρρ [ ~ρ ]( ξ i ′ ′ , ξ ρ + ǫ ′ τ , ξ ρ ) − G (2) λ = , i ′ κ [ ~ρ ]( ξ i ′ ′ , ξ κ ) κ ∗ ( ξ κ ∗ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) − G (2) λ = , i ′ κ ∗ [ ~ρ ]( ξ i ′ ′ , ξ κ ∗ ) κ ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) − G (2) λ = , i ′ ρ ( ξ i ′ ′ , ξ ρ ) δ a , a δ ( x − x ) (cid:21) . (54) / y use of the Wick’s theorem, G (3) λ = , i ′ ρρ [ ~ρ ]( ξ i ′ ′ , ξ ρ + ǫ ′ τ , ξ ρ ) is evaluated as follows: G (3) λ = , i ′ ρρ [ ~ρ ]( ξ i ′ ′ , ξ ρ + ǫ ′ τ , ξ ρ ) = h ψ ∗ ( ξ ) ψ ∗ ( ξ + ǫ ′ τ ) i ~ρ (cid:18) h ψ ( ξ ) ψ ( ξ + ǫ ′ τ ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ( ξ ) ψ ( ξ + ǫ ′ τ ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) + h ψ ( ξ ) ψ ( ξ + ǫ ′ τ ) i ~ρ (cid:18) h ψ ∗ ( ξ + ǫ ′ τ ) ψ ∗ ( ξ ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ∗ ( ξ + ǫ ′ τ ) ψ ∗ ( ξ ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) − h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) i ~ρ (cid:18) h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) − h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) i ~ρ (cid:18) h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) = κ ∗ ( ξ κ ∗ ) G (2) λ = , i ′ κ [ ~ρ ]( ξ i ′ ′ , ξ κ ) | ξ κ ( ∗ ) = ( ξ ρ ,ξ ρ ) + κ ( ξ κ ) G (2) λ = , i ′ κ ∗ [ ~ρ ]( ξ i ′ ′ , ξ κ ∗ ) | ξ κ ( ∗ ) = ( ξ ρ ,ξ ρ ) − h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) i ~ρ (cid:18) h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) − h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) i ~ρ (cid:18) h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) . Here, we have introduced hO [ ψ, ψ ∗ ] i ~ρ : = hO [ ψ, ψ ∗ ] i λ = , ~ J sup ,λ = [ ~ρ ] , (55)in which the right-hand side is defined by Eq. (22) for a functional O [ ψ, ψ ∗ ]. Then the integral withrespect to ξ ρ and ξ ρ in Eq. (54) is evaluated as follows: Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) (cid:20) G (3) λ = , i ′ ρρ [ ~ρ ]( ξ i ′ ′ , ξ ρ + ǫ ′ τ , ξ ρ ) − G (2) λ = , i ′ κ [ ~ρ ]( ξ i ′ ′ , ξ κ ) κ ∗ ( ξ κ ∗ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) − G (2) λ = , i ′ κ ∗ [ ~ρ ]( ξ i ′ ′ , ξ κ ∗ ) κ ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) − G (2) λ = , i ′ ρ [ ~ρ ]( ξ i ′ ′ , ξ ρ ) δ a , a δ ( x − x ) (cid:21) = − Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) i ~ρ × (cid:18) h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) , (56)where we have used the equal-time commutation relation: U λ = ( ξ ρ , ξ ρ ) h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) i ~ρ = δ ( τ − τ ) U λ, a a ( x − x ) h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) i ~ρ = δ ( τ − τ ) U λ, a a ( x − x ) (cid:16) h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) i ~ρ − δ a , a δ ( x − x ) (cid:17) . Substituting Eq. (56) into Eq. (54), we have V i KS [ ~ρ ]( ξ i ) = δ i ,ρ V ( ξ ρ ) + Z ξ ρ U λ = ( ξ ρ , ξ ρ ) ρ ( ξ ρ ) ! + δ i ,κ U λ = ( ξ κ ) κ ∗ ( ξ κ ) + δ i ,κ ∗ U λ = ( ξ κ ∗ ) κ ( ξ κ ∗ ) − X i ′ Z ξ i ′ ′ G (2) − λ = , ii ′ [ ~ρ ]( ξ i , ξ i ′ ′ ) Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) i ~ρ × (cid:18) h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) . (57) / lthough the last term in the right-hand side may not be possible to represent in terms of the densitycorrelation functions in general, it can be further simplified for an arbitrary U λ . In the next sub-section, we shall discuss the case with short-range interactions and then show that the last term ofEq. (57) is approximately represented in terms of the density correlation functions.Finally, let us discuss the Helmholtz free energy. Applying the approximation G (2)xc ,λ ≈ G (2)x toEq. (40), we have F H [ ~ρ ave ] = Γ λ = | V = [ ~ρ ave ] β + β Z ξ V ( ξ ) ρ ave ( ξ ) + β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) (cid:20) ρ ave ( ξ ρ ) ρ ave ( ξ ρ ) + κ ∗ ave ( ξ κ ∗ ) κ ave ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) (cid:21) + β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) G (2)x [ ~ρ ave ]( ξ ρ , ξ ρ ) . (58)From Eqs. (33) and (36), G (2)x [ ~ρ ]( ξ ρ , ξ ρ ) is rewritten as G (2)x [ ~ρ ]( ξ ρ , ξ ρ ) = −h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~ρ h ψ ∗ ( ξ + ǫ τ ) ψ ( ξ + ǫ ′ τ ) i ~ρ − ρ ( ξ ρ ) δ a , a δ ( x − x ) . Then the integral with respect to ξ ρ and ξ ρ in the third line of Eq. (58) is evaluated as follows: Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) G (2)x [ ~ρ ave ]( ξ ρ , ξ ρ ) = − Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~ρ ave h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~ρ ave , (59)where we have used the equal-time commutation relation: U λ = ( ξ ρ , ξ ρ ) h ψ ∗ ( ξ + ǫ τ ) ψ ( ξ + ǫ ′ τ ) i ~ρ = δ ( τ − τ ) U λ, a a ( x − x ) h ψ ∗ ( ξ + ǫ τ ) ψ ( ξ + ǫ ′ τ ) i ~ρ = δ ( τ − τ ) U λ, a a ( x − x ) (cid:16) h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~ρ − δ a , a δ ( x − x ) (cid:17) . Substituting Eq. (59) into Eq. (58), we have F H [ ~ρ ave ] = Γ λ = | V = [ ~ρ ave ] β + β Z ξ V ( ξ ) ρ ave ( ξ ) + β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) h ρ ave ( ξ ρ ) ρ ave ( ξ ρ ) − h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~ρ ave h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~ρ ave i + β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) κ ∗ ave ( ξ κ ∗ ) κ ave ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) . (60)At the zero temperature limit β → ∞ , it gives the ground-state energy: E gs = T [ ~ρ gs ] + lim β →∞ β Z ξ V ( ξ ) ρ ave ( ξ ) + lim β →∞ β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) h ρ ave ( ξ ρ ) ρ ave ( ξ ρ ) − h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~ρ ave h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~ρ ave i + lim β →∞ β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) κ ∗ ave ( ξ κ ∗ ) κ ave ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) . The third term in the right-hand side is interpreted as the Hartree–Fock energy; The fourth term isthe contribution from the pairing condensate. This formula of the ground-state energy is identical tothat given in the Hartree–Fock–Bogoliubov approximation [51–53]. / .2. The case of short-range interactions in the weak coupling Let us take the case where the inter-particle interaction is short-range in the weak coupling limit ofthe paring force as in the BCS theory [54]. Then it will be found that Eq. (57) is reduced to a simpleand familiar form.In the case of short-range interactions, the integral in the last term of Eq. (57) is dominated bythe contribution from the region where x is close to x . Therefore, we can make the followingapproximation for short-range interactions that holds exactly for the zero-range interaction: Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) i ~ρ (cid:18) h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) ≈ Z ξ ρ = ( τ , x , a ) ,ξ ρ = ( τ , x , a ) U λ = ( ξ ρ , ξ ρ ) ρ a a ( τ , x ) × (cid:18) h ψ ∗ ( τ , x , a ) ψ ( τ + ǫ ′ τ , x , a ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ∗ ( τ , x , a ) ψ ( τ + ǫ ′ τ , x , a ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) , (61)where we have used U λ = ( ξ ρ : = ( τ , x , a ) , ξ ρ : = ( τ , x , a )) = δ ( τ − τ ) U λ = ( τ , x , a ; τ , x , a ) , and introduced ρ a a ( τ, x ) = h ψ ∗ ( τ + ǫ ′ τ , x , a ) ψ ( τ, x , a ) i ~ρ .We evaluate ρ a a ( τ, x ) in the weak coupling limit leading to small κ in comparison with the particledensity ρ . In the non-interacting case λ =
0, the vanishing external field J κ ( ∗ ) sup ,λ = [ ~ρ ] = κ ( ∗ ) =
0. Conversely, we have lim κ,κ ∗ → J κ ( ∗ ) sup ,λ = [ ~ρ ] = κ ( ∗ ) uniquely determines the external field.Therefore, by use of Eqs. (55) and (22), ρ a a ( τ, x ) is approximated in the weak coupling as ρ a a ( τ, x ) ≈ Z λ = [( J ρ sup ,λ = [ ~ρ ] , , Z D ψ D ψ ∗ ψ ∗ ( τ + ǫ ′ τ , x , a ) ψ ( τ, x , a ) e − S λ = [ ψ,ψ ∗ ] + R ξ J ρ sup ,λ = [ ~ρ ]( ξ )ˆ ρ ( ξ ) . (62)Notice that there is no term causing a mixing between ψ ( ∗ ) ( τ, x , a ) with di ff erent a ’s in the exponent S λ = [ ψ, ψ ∗ ] − R ξ J ρ sup ,λ = [ ~ρ ]( ξ ) ˆ ρ ( ξ ) any more. Thus we have ρ a a ( τ, x ) ≃ a , a in the weakcoupling limit. In other words, ρ a a ( τ, x ) with a , a is negligible in comparison with the diagonalcomponent ρ a a ( τ, x ) = ρ ( τ, x , a ) in the weak coupling limit or small κ ( ∗ ) . Thus, we can make theapproximation for ρ a a ( τ, x ) as follows: ρ a a ( τ, x ) ≈ δ a , a ρ ( τ, x , a ) . (63)Then, Eq. (61) is reduced as follows, Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) h ψ ∗ ( ξ + ǫ ′ τ ) ψ ( ξ ) i ~ρ (cid:18) h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ∗ ( ξ ) ψ ( ξ + ǫ ′ τ ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) ≈ Z ξ ρ = ( τ , x , a ) ,ξ ρ = ( τ , x , a ) U λ = ( ξ ρ , ξ ρ ) δ a a ρ ( τ , x , a ) × (cid:18) h ψ ∗ ( τ , x , a ) ψ ( τ + ǫ ′ τ , x , a ) ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ − h ψ ∗ ( τ , x , a ) ψ ( τ + ǫ ′ τ , x , a ) i ~ρ h ˆ ρ i ′ ( ξ i ′ ′ ) i ~ρ (cid:19) = Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) δ a , a ρ ( ξ ρ ) G (2) λ = ,ρ i ′ [ ~ρ ]( ξ ρ , ξ i ′ ′ ) . (64) / pplying this approximation to Eq. (57), we arrive at the following expression of the basic equationfor the KS theory for the case of short-range interactions in the weak coupling as V i KS [ ~ρ ]( ξ i ) = δ i ,ρ V ( ξ ρ ) + Z ξ ρ (1 − δ a , a ) U λ = ( ξ ρ , ξ ρ ) ρ ( ξ ρ ) ! + δ i ,κ U λ = ( ξ κ ) κ ∗ ( ξ κ ) + δ i ,κ ∗ U λ = ( ξ κ ∗ ) κ ( ξ κ ∗ ) . (65) In this subsection, we are going to discuss a homogeneous system with V ( ξ ) = µ has nospin dependence. We also assume that the two-body interaction is a spin-independent short-rangeone U ab ( x ) = U ( x ) and consider the weak-coupling limit so that the approximated equation Eq. (64)is applicable. Although we treat the spin-singlet condensate in the present investigation exclusivelyin the present article, the extension to the spin-triplet case is straightforward by considering thevector pairing densities.Here, we introduce the following notation: V i KS ( ξ i ) : = V i KS [ ~ρ ave ]( ξ i ). Owing to the translationalsymmetry, the coordinate dependence of V ρ KS ( ξ ρ ) is simplified as follows: V ρ KS ( ξ ρ : = ( τ, x , a )) = V ρ KS (0 , , a ) ≕ V ρ KS , a , V κ ( ∗ ) KS ( ξ : = ( τ, x , a ) , ξ ′ : = ( τ ′ , x ′ , a ′ )) = V κ ( ∗ ) KS ( τ − τ ′ , x − x ′ , a , , , a ′ ) ≕ V κ ( ∗ ) KS , aa ′ ( τ − τ ′ , x − x ′ ) . Similarly, we have ρ ave ( ξ : = ( τ, x , a )) = ρ ave ( τ = , x = , a ) ≕ ρ a ave ,κ ( ∗ )ave ( ξ : = ( τ, x , a ) , ξ ′ : = ( τ ′ , x ′ , a ′ )) = κ ( ∗ )ave ( τ − τ ′ , x − x ′ , a , , , a ′ ) ≕ κ aa ′ ( ∗ )ave ( τ − τ ′ , x − x ′ ) . Since the spin-saturated system under consideration has the time-reversal symmetry, we have V κ ∗ KS , aa ′ ( τ, x ) = V κ KS , a ′ a ( τ, x ) ∗ . Furthermore, the following relation should be satisfied because of the spin symmetry: ρ ↑ ave = ρ ↓ ave ≕ ρ ave / , V ρ KS , ↑ = V ρ KS , ↓ ≕ V ρ KS ,κ ↑↓ ave ( τ, x ) = − κ ↓↑ ave ( τ, x ) ≕ κ save ( τ, x ) , V κ KS , ↑↓ ( τ, x ) = − V κ KS , ↓↑ ( τ, x ) ≕ V κ, sKS ( τ, x ) . Here, ρ ave = ρ ↑ ave + ρ ↓ ave is the total particle-number density. Other components for κ ab ave ( τ, x ) and V κ KS , ab ( τ, x ) are assumed to be zero since we focus on the systems with the spin-singlet condensate.For convenience for the description of the homogeneous system, let us move to the momentumrepresentation. Then Eq. (65) is reduced to the following two equations for the particle and pairing / ensities, respectively, V ρ KS =
12 ˜ U ( ) ρ ave ˜ V κ, sKS ( P ) = Z Q ˜ U ( q − p )˜ κ save ( Q ) ∗ . (66)Here, we have introduced the notations as P = ( p , p ) and Q = ( q , q ) with the Matsubara fre-quencies p and q , the spacial momenta p and q , and ˜ V κ, sKS ( P ), ˜ U ( p ) and ˜ κ save ( P ) are the Fouriertransforms of V κ, sKS ( τ, x ), U ( x ), and κ save ( τ, x ), respectively. The following abbreviation is also usedfor the Q -‘integrations’: Z Q = β − X q Z d q / (2 π ) . Notice that the right-hand side of Eq. (66) and hence ˜ V κ, sKS ( P ) in the left-hand side are independentof p . Thus we write as ˜ V κ, sKS ( P ) = ˜ V κ, sKS ( p ) from now on.Now we calculate the equilibrium densities in the KS system via the variational equation (45) withthe obtained KS potential Eq. (66). However, we have already seen that the solution ~ρ ave of Eq. (45)satisfies the self-consistent equation Eq. (46), which is now simplified to ρ ave , i ( ξ i ) = Z λ = [ ~µ − ~ V KS ] Z D ψ D ψ ∗ ˆ ρ i ( ξ i ) e − S λ = [ ψ ∗ ,ψ ] + P j = ρ,κ,κ ∗ R ξ j (cid:18) µ j −V j KS ( ξ j ) (cid:19) ˆ ρ j ( ξ j ) . Furthermore, since the exponent in the integral becomes a bilinear form when Eq. (66) is inserted,we can perform the path integral analytically to give ρ ave = Z p − ε ( p ) E ( p ) tanh β E ( p )2 ! , (67)˜ κ save ( P ) = − V κ, sKS ( p ) ∗ p + E ( p ) , (68)where R p = R d p / (2 π ) , ε ( p ) = p / + V ρ KS − µ , and E ( p ) = q ε ( p ) + | V κ, sKS ( p ) | , as is detailed inAppendix. Substitution of Eq. (68) into the second equation of Eq. (66) in turn leads to ˜ V κ, sKS ( p ) = − Z q ˜ U ( q − p ) ˜ V κ, sKS ( q )2 q ε ( q ) + | V κ, sKS ( q ) | tanh β q ε ( q ) + | V κ, sKS ( q ) | ε ( p ) = p +
12 ˜ U ( ) ρ ave − µ . (69)Equation (68) implies that ˜ V κ, sKS ( p ) is identified with the energy gap ∆ s ( p ) in the BCS theory as ∆ s ( p ) = − V κ, sKS ( p ) ∗ . With this identification, we find that Eq. (69) is nothing else than the celebratedgap equation in the BCS theory [54].As is also shown in the Appendix, the Helmholtz free energy is calculated from Eq. (60) to yield F H [ ~ρ ave ] V = Γ λ = | V = [ ~ρ ave ] β V +
12 ˜ U ( ) ρ − Z p , q ˜ U ( p − q ) − ε ( p ) E ( p ) tanh β E ( p )2 ! − ε ( q ) E ( q ) tanh β E ( q )2 ! + Z p , q ˜ U ( p − q ) ˜ V κ, s KS ( p ) ˜ V κ, s KS ( q ) ∗ E ( p ) E ( q ) tanh β E ( p )2 tanh β E ( q )2 . (70)where V = R d x is the volume of the system. / aking the zero temperature limit β → ∞ in Eq. (70), the ground-state energy is obtained as E gs V = T [ ~ρ gs ] V + ρ ˜ U ( ) − Z p , q ˜ U ( p − q ) − ε ( p ) E ( p ) ! − ε ( q ) E ( q ) ! + Z p , q ˜ U ( p − q ) ∆ s ( p ) ∆ s ( q ) ∗ E ( p ) E ( q ) , (71)Our ground-state energy is identical to that obtained in the BCS theory with the identification ∆ s ( p ) = − V κ, sKS ( p ) ∗ [51, 53, 54, 56].
4. Conclusion
On the basis of the e ff ective action formalism, we have developed a generalized density-functionaltheory (DFT) for superfluid systems at finite temperature in a rigorous way. The rigorous formu-lation is combined with the renormalization group method by introducing a scale parameter that ismultiplied to the inter-particle potential, which is reminiscent of the adiabatic connection method. Apossible di ffi culty that the spontaneous symmetry breaking (SSB) is not described when symmetry-breaking term is absent is nicely circumvented by a suitable choice of the external fields by makinguse of the functional renormalization group (FRG).Then we have established the Hohenberg–Kohn theorem with the SSB being taken into account.By introducing flow-parameter-dependent source terms appropriately fixing the ground-state densi-ties during the flow, we have derived the flow equation for the e ff ective action for the particle-numberand nonlocal pairing densities where the flow parameter λ runs from 0 to 1, corresponding to thenon-interacting and interacting systems, respectively.Integrating this flow equation and using the variational equation for the thermal equilibrium den-sities, we have arrived at the exact self-consistent equation to determine the Kohn–Sham potential V i KS [ ~ρ ] generalized to including the pairing potentials at finite temperature without introductionof single-particle orbitals. The resultant Kohn–Sham potential has a nice feature that it expressesthe microscopic formulae of the external, Hartree, pairing, and exchange-correlation terms, sep-arately. In particular, the exchange-correlation term V i KS , xc [ ~ρ ] is given explicitly in terms of thecorresponding density-density correlation functions G (2)xc ,λ [ ~ρ ].As a demonstration of the validity of our formulation as a microscopic theory of DFT forsuperfluidity, we have shown that our self-consistent equation with Kohn–Sham potential leadsto the ground-state energy of the Hartree–Fock–Bogoliubov theory when the correlations areneglected. And the self-consistent equation is further reduced for short-range interaction and in theweak-coupling limit to reproduce the gap equation of the BCS theory for the spin-singlet pairing.Although we have shown the validity of our microscopic formulation of DFT based on FRG byapplying it to simple cases, an advantage of our exact FRG-DFT formalism lies in the fact thatit contains the equation (37) with which the correlation part can be improved systematically. Oneof the most e ff ective systematic schemes used in the FRG-DFT is the vertex expansion, where theTaylor expansion around the equilibrium densities are applied to the flow equation for the e ff ectiveaction. In fact, it is shown [45] for the system without superfluidity that the vertex expansion onlyup to the second order gives an approximate scheme superior to the random phase approximation(RPA). Therefore it is naturally expected that a simple vertex expansion applied to the basic equationin the present work would lead to an approximation scheme that incorporates correlations beyondthose given by the quasi-particle RPA. We hope to report on such an attempt near future. / ur formulation was given to general cases, i.e. any spatial profiles and the dependence of theinteraction on the internal degrees of freedom are not imposed. Such a general formulation may behelpful when applying to the superfluidity in various systems. Introduction of the nonlocal particledensity is an interesting extension of the present formalism, with which we can describe the particle-hole and particle-particle correlations on the same footing. Furthermore, our general formulation fornonlocal pairing should provide us with a sound basis for describing pairing phenomena with anysymmetry such as spin-triplet superfluidity and / or the paring with spin-orbit coupling incorporated.We hope that there will be a chance to report on such investigations. Acknowledgements
T. Y. was supported by the Grants-in-Aid for JSPS fellows (Grant No. 20J00644). This workis supported in part by the Grants-in-Aid for Scientific Research from JSPS (Nos. JP19K03872and JP19K03824), the Yukawa International Program for Quark-hadron Sciences (YIPQS), and byJSPS-NSFC Bilateral Program for Joint Research Project on “Nuclear mass and life for unravelingmysteries of the r-process.”
A. Equilibrium densities and Helmholtz energy of homogeneous systems
In this appendix we shall derive the equilibrium densities and Helmholtz free energy of homoge-neous systems.
A.1. Free propagators
As was done in Sec. 3.3, we consider a system with the translational, time-reversal, and spin sym-metries. The generating functional for non-interacting fermions in the presence of τ -independentexternal sources coupled to the total particle-number density and spin-singlet pairing density reads: Z [ ~ J ] = Z D ψ D ψ ∗ e − S [ ψ,ψ ∗ ] + J ρ R X ˆ ρ ( X ) + R X , X ′ J κ s ( x − x ′ ) δ ( τ − τ ′ ) ˆ κ s ( X , X ′ ) + R X , X ′ J κ s ( x − x ′ ) ∗ δ ( τ − τ ′ ) ˆ κ s ∗ ( X , X ′ ) , (A1)where X = ( τ, x ), R X ≔ R β d τ R d x , ~ J = ( J ρ , J κ s ( x ) , J κ s ( x ) ∗ ), and S [ ψ, ψ ∗ ] = X s = ↑↓ Z X ψ ∗ s ( X + ǫ τ ) ∂ τ − ∆ ! ψ s ( X ) , (A2)ˆ ρ ( X ) = X s = ↑↓ ψ ∗ s ( X + ǫ τ ) ψ s ( X ) , (A3)ˆ κ s ( X , X ′ ) =
12 ( ψ ↑ ( X ) ψ ↓ ( X ′ ) − ψ ↓ ( X ) ψ ↑ ( X ′ )) , (A4)ˆ κ s ∗ ( X , X ′ ) =
12 ( ψ ∗↓ ( X ′ ) ψ ∗↑ ( X ) − ψ ∗↑ ( X ′ ) ψ ∗↓ ( X )) . (A5)Here, X + ǫ τ = ( τ + ǫ, x ), and ǫ ( >
0) is the time step size of the path integral, which will be takento be zero in the end of the calculation. We first note that the time derivative of the Grassmann field ∂ τ ψ s ( X ) in Eq. (A2) is defined as ∂ τ ψ s ( X ) ≔ ψ s ( X + ǫ τ ) − ψ s ( X ) ǫ . (A6)By the Fourier transformation ψ s ( X ) = Z P e iP · X ˜ ψ s ( P ) = Z P e − i ω n τ + i p · x ˜ ψ s ( P ) , ψ ∗ s ( X ) = Z P e − iP · X ˜ ψ ∗ s ( P ) , J κ s ( x ) = Z p e i p · x ˜ J κ s ( p ) , / here P = ( ω n , p ) with the Matsubara frequency ω n = (2 n + /β , R P ≔ β − P ω n R d p / (2 π ) , and R p ≔ R d p / (2 π ) , the generating functional is rewritten in the momentum representation as: Z [ ~ J ] = Z D ˜ ψ D ˜ ψ ∗ exp ( X s Z P ˜ ψ ∗ s ( P ) " e i ω n ǫ − ǫ − e i ω n ǫ p − J ρ ! ˜ ψ s ( P ) + Z P ˜ J κ s ( p ) h ˜ ψ ↑ ( − P ) ˜ ψ ↓ ( P ) − ˜ ψ ↓ ( − P ) ˜ ψ ↑ ( P ) i + Z P ˜ J κ s ( p ) ∗ h ˜ ψ ∗↓ ( P ) ˜ ψ ∗↑ ( − P ) − ˜ ψ ∗↑ ( P ) ˜ ψ ∗↓ ( − P ) i ) . (A7)The usual treatment of the exponent in the first term may be to make a replacement ( e i ω n ǫ − /ǫ → i ω n while leaving e i ω n ǫ ( p / − J ρ ). Indeed, nothing is wrong with this practical treatment and it willgive the correct result. Instead here we leave all ǫ s until just before the end of the calculation for aninstructive purpose.Introducing ˜ Ψ ( P ) ≔ h ˜ ψ ↑ ( P ) ˜ ψ ∗↓ ( − P ) i T , the exponent of the right-hand side of Eq. (A7) isorganized in a matrix form as in the Nambu–Gor’kov formalism: Z [ ~ J ] = Z D ˜ ψ D ˜ ψ ∗ exp − Z P ˜ Ψ † ( P ) − e i ω n ǫ ǫ + ε ( p ; J ρ ) e i ω n ǫ J κ s ( p ) ∗ J κ s ( p ) − − e − i ω n ǫ ǫ − ε ( p ; J ρ ) e − i ω n ǫ ˜ Ψ ( P ) , (A8)where ε ( p ; J ρ ) ≔ p / − J ρ . By adding external source terms P a = , hR P ˜ η ∗ a ( P ) ˜ Ψ a ( P ) + R P ˜ Ψ ∗ a ( P ) ˜ η a ( P ) i to the exponent of the right-hand side of Eq. (A8) with ˜ η and ˜ η ∗ being Grassmann variables, and dif-ferentiating the generating functional with respect to ˜ η and ˜ η ∗ , the propagators are obtained in amatrix form: * ˜ ψ ↑ ( P ) ˜ ψ ∗↑ ( P ) ˜ ψ ↑ ( P ) ˜ ψ ↓ ( − P )˜ ψ ∗↓ ( − P ) ˜ ψ ∗↑ ( P ) ˜ ψ ∗↓ ( − P ) ˜ ψ ↓ ( − P ) + ~ J = − e i ω n ǫ ǫ + ε ( p ; J ρ ) e i ω n ǫ J κ s ( p ) ∗ J κ s ( p ) − − e − i ω n ǫ ǫ − ε ( p ; J ρ ) e − i ω n ǫ − = √ − ǫε ( p ; J ρ ) ǫ sin ω n ǫ ! + E ( p ; ~ J ) ie − i ω n ǫ ǫ sin ω n ǫ + ε ( p ; J ρ ) e − i ω n ǫ J κ s ( p ) ∗ J κ s ( p ) ie i ω n ǫ ǫ sin ω n ǫ − ε ( p ; J ρ ) e i ω n ǫ , (A9)where E ( p ; ~ J ) ≔ q ε ( p ; J ρ ) + (cid:12)(cid:12)(cid:12) J κ s ( p ) (cid:12)(cid:12)(cid:12) , and hO [ ψ, ψ ∗ ] i ~ J is defined as hO [ ψ, ψ ∗ ] i ~ J ≔ Z [ ~ J ] Z D ψ D ψ ∗ O [ ψ, ψ ∗ ] e − S [ ψ,ψ ∗ ] + J ρ R X ˆ ρ ( X ) + R X , X ′ J κ s ( x − x ′ ) δ ( τ − τ ′ ) ˆ κ s ( X , X ′ ) + R XX ′ J κ ( x − x ′ ) ∗ δ ( τ − τ ′ ) ˆ κ s ∗ ( X , X ′ ) . (A10)with an arbitrary functional O [ ψ, ψ ∗ ] of ψ and ψ ∗ including the Fourier transforms of them. A.2. Equilibrium densities
From Eq. (46) the equilibrium densities satisfy ρ ave = h ˆ ρ ( X ) i ~µ − ~ V KS , (A11) κ s ( ∗ )ave ( X − X ′ ) = h ˆ κ s ( ∗ ) ( X , X ′ ) i ~µ − ~ V KS . (A12) / he equilibrium total particle-number density is calculated as follows: ρ ave = X s h ψ ∗ s ( X + ǫ τ ) ψ s ( X ) i ~µ − ~ V KS = Z P e i ω n ǫ X s h ˜ ψ ∗ s ( P ) ˜ ψ s ( P ) i ~µ − ~ V KS = Z P − ie i ω n ǫ ǫ sin ω n ǫ − ε ( p ) (cid:18) √ − ǫε ( p ) ǫ sin ω n ǫ (cid:19) + E ( p ) , (A13)where ε ( p ) ≔ ε ( p ; µ − V ρ KS ) = p / + V ρ KS − µ , and E ( p ) ≔ E ( p ; ~µ − ~ V KS ) = q ε ( p ) + (cid:12)(cid:12)(cid:12) V κ, s KS ( p ) (cid:12)(cid:12)(cid:12) .To calculate the Matsubara frequency summation in Eq. (A13), let us introduce a complex function h ( z ) = − ie i z ǫ ǫ sin z ǫ − ε (cid:18) √ − ǫεǫ sin z ǫ (cid:19) + E and a weighting function g ( z ) = − β n F ( z ) = − β + e β z . It is observed that the product h ( − iz ) g ( z ) decays su ffi ciently fast as | z | → ∞ , and h ( − iz ) has simplepoles at z = ± ǫ sinh − ǫ E √ − ǫε . Therefore,1 β X ω n h ( ω n ) = − β Res h ( − iz ) g ( z ) , z = ǫ sinh − ǫ E √ − ǫε ! − β Res h ( − iz ) g ( z ) , z = − ǫ sinh − ǫ E √ − ǫε ! = E + ε E n F ( E ) + E − ε E n F ( − E ) = (cid:18) − ε E tanh β E (cid:19) . (A14)In the second to last line, we have taken the limit ǫ → ρ ave = Z p − ε ( p ) E ( p ) tanh β E ( p )2 ! . (A15)In the same way, the equilibrium spin-singlet pairing density is calculated as follows: κ s ave ( X − X ′ ) = h ψ ↑ ( X ) ψ ↓ ( X ′ ) − ψ ↓ ( X ) ψ ↑ ( X ′ ) i ~µ − ~ V KS = Z P e iP · ( X − X ′ ) h h ˜ ψ ↑ ( P ) ˜ ψ ↓ ( − P ) i ~µ − ~ V KS − h ˜ ψ ↓ ( P ) ˜ ψ ↑ ( − P ) i ~µ − ~ V KS i = Z P e iP · ( X − X ′ ) − V κ, s KS ( p ) ∗ (cid:18) √ − ǫε ( p ) ǫ sin ω n ǫ (cid:19) + E ( p ) = Z p e i p · ( x − x ′ ) − ˜ V κ, s KS ( p ) ∗ E ( p ) e − E ( p ) | τ − τ ′ | + e − β E ( p ) − e E ( p ) | τ − τ ′ | + e β E ( p ) ! . (A16)In particular, the local spin-singlet pairing density in equilibrium is given by κ s ave (0) = Z p − ˜ V κ, s KS ( p ) ∗ E ( p ) tanh β E ( p )2 . (A17) / .3. Helmholtz free energy From Eq. (60) the Helmholtz energy in the lowest order approximation is given by F H [ ~ρ ave ] = Γ λ = [ ~ρ ave ] β + β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) ρ ave ( ξ ρ ) ρ ave ( ξ ρ ) − β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~µ − ~ V KS h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~µ − ~ V KS + β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) κ ∗ ave ( ξ κ ∗ ) κ ave ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) . (A18)The second term of the right-hand side of Eq. (A18) is evaluated as follows:12 β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) ρ ave ( ξ ρ ) ρ ave ( ξ ρ ) = Z x , x U ( x − x ) X s = ↑↓ ρ s ave X s = ↑↓ ρ s ave = V U ( ) ρ , (A19)where V = R d x is the volume and ˜ U ( p ) = R d x e − i p · x U ( x ). The third term is evaluated as follows: − β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~µ − ~ V KS h ψ ∗ ( ξ + ǫ τ + ǫ ′ τ ) ψ ( ξ ) i ~µ − ~ V KS = − Z x , x U ( x − x ) X s , s = ↑↓ h h ψ ∗ s ( X + ǫ τ + ǫ ′ τ ) ψ s ( X ) i ~µ − ~ V KS h ψ ∗ s ( ξ + ǫ τ + ǫ ′ τ ) ψ s ( ξ ) i ~µ − ~ V KS i τ = τ = − V Z p , q ˜ U ( p − q ) X s = ↑↓ β X m e i ω n ( ǫ + ǫ ′ ) h ˜ ψ ∗ s ( ω m , p ) ˜ ψ s ( ω m , p ) i ~µ − ~ V KS × β X n e i ω n ( ǫ + ǫ ′ ) h ˜ ψ ∗ s ( ω n , q ) ˜ ψ s ( ω n , q ) i ~µ − ~ V KS = − V Z p , q ˜ U ( p − q ) − ε ( p ) E ( p ) tanh β E ( p )2 ! − ε ( q ) E ( q ) tanh β E ( q )2 ! . (A20)The fourth term is evaluated as follows:12 β Z ξ ρ ,ξ ρ U λ = ( ξ ρ , ξ ρ ) κ ∗ ave ( ξ κ ∗ ) κ ave ( ξ κ ) | ξ κ ( ∗ )12 = ( ξ ρ ,ξ ρ ) = Z x , x U ( x − x ) h κ ↑↓∗ ave ( X , X ) κ ↑↓ ave ( X , X ) + κ ↓↑∗ ave ( X , X ) κ ↓↑ ave ( X , X ) i τ = τ = Z x , x U ( x − x ) κ s ∗ ave ( X , X ) (cid:12)(cid:12)(cid:12) τ = τ κ s ave ( X , X ) (cid:12)(cid:12)(cid:12) τ = τ = V Z p , q ˜ U ( p − q ) ˜ V κ, s KS ( p ) ˜ V κ, s KS ( q ) ∗ E ( p ) E ( q ) tanh β E ( p )2 tanh β E ( q )2 . (A21)Consequently, we have F H [ ~ρ ave ] V = Γ λ = | V = [ ~ρ ave ] β V +
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