Functional renormalization-group calculation of the equation of state of one-dimensional nuclear matter inspired by the Hohenberg--Kohn theorem
aa r X i v : . [ nu c l - t h ] S e p Functional renormalization-group calculation of the equation of state ofone-dimensional nuclear matter inspired by the Hohenberg–Kohn theorem
Takeru Yokota, ∗ Kenichi Yoshida, † and Teiji Kunihiro ‡ Department of Physics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
We present the first successful functional renormalization group(FRG)-aided density-functional(DFT) calculation of the equation of state (EOS) of an infinite nuclear matter (NM) in (1+1)-dimensions composed of spinless nucleons. We give a formulation to describe infinite matters inwhich the ’flowing’ chemical potential is introduced to control the particle number during the flow.The resultant saturation energy of the NM coincides with that obtained by the Monte-Carlo methodwithin a few percent. Our result demonstrates that the FRG-aided DFT can be as powerful as anyother methods in quantum many-body theory.
I. INTRODUCTION
The Hohenberg-Kohn (HK) theorem [1] tells us thatthe problems of quantum many-body systems can be for-mulated solely in terms of the particle density ρ ( x ) with-out the many-particle wave function. A formalism basedon this theorem is the density functional theory (DFT)utilizing the energy density functional (EDF) E [ ρ ]. TheDFT is used in various fields including quantum chem-istry, condensed matter physics, and nuclear physics:Thanks to the practical methods based on the Kohn-Sham formalism [2], DFT has become a powerful methodto analyze the properties of ground states; see Refs. [3–6]for an overview. Methods to investigate excited statessuch as the time-dependent density functional theory [7]have also been developed [8, 9].It should be, however, noted that the HK theorem onlyguarantees the existence of the EDF E [ ρ ] which couldbe minimized to obtain the exact ground-state densityand energy, but does not provide any theoretical pre-scription to construct E [ ρ ] itself. Thus most practicalcalculations utilize E [ ρ ] that is constructed in a semi-empirical way. Hence developing a systematic method toderive E [ ρ ] from microscopic Hamiltonian still remainsas a fundamental problem in the field of quantum many-body theory.A clue of this fundamental problem may be provided bythe notion of effective field theory developed in quantumfield theory. Indeed the two-particle point-irreducible(2PPI) effective action [10] can lead to an energy den-sity functional written in terms of ρ for which the HKtheorem naturally emerges [11, 12]. A nice point withthe effective-action approach is that an established pow-erful computational machinery is now available, whichis called the functional renormalization group (FRG)method [13–16]. In this method, the quantum fluctua-tions are gradually taken into account from an ultraviolet ∗ [email protected] † [email protected] ‡ [email protected] to infrared scale by solving the one-parameter flow equa-tion of the scale-dependent effective action, and hence acoarse-grained effective action is eventually obtained; seeRefs. [17–19] for reviews. Since the 2PPI effective ac-tion is a generalization of the energy density functional,the FRG method formulated for the 2PPI action pos-sibly gives a formal foundation to DFT and provides along-desired method for constructing the density func-tional from a microscopic Hamiltonian, as initiated byPolonyi, Sailer and Schwenk [11, 12]. Such an obser-vation has lead to a notion of DFT-RG or 2PPI-FRGmethod [20–24], which is a quite attractive scheme forsolving the fundamental problem in the field of the DFT:In a pioneering work [21], accurate ground state energiesof simple toy models in quantum mechanics are obtainedwithin the fourth-order truncation: A recent paper [24]proposed an efficient method to incorporate the higher-order correlations, and the method was applied to a 0-dimensional quartic model successfully. The subsequentanalysis of a one-dimensional system composed of a finitenumber of particles motivated by the nuclear saturationproblem [22, 23], however, showed that the second-ordertruncation only gives a 30% accuracy in comparison withthe result of the Monte-Carlo simulation [25]. Possibleimprovement of the result may be obtained by an in-corporation of the higher-order correlation functions assuggested in the demonstration in a 0-dimensional model[24].In this paper, we apply the DFT-RG scheme to an in-finite uniform system. Our point is that an infinite sys-tem with a definite particle density may be well describedwith first few correlation functions while rarefied systemsare interaction-dominating systems and higher correla-tion functions may play significant roles. Needless to say,an infinite uniform system is a fundamentally importantsystem for understanding many-body physics and indeedthe local density approximation for E [ ρ ] was found to beunexpectedly successful [2]. We give a DFT-RG formal-ism for infinite matters in which we introduce a ’flowing’chemical potential to control the flow of the particle num-ber caused by switching on the inter-particle interaction.Then we calculate the equation of state (EOS) of an infi-nite nuclear matter (NM) in (1+1)-dimensions composedof spinless nucleons as in Refs. [22, 25]. Starting from thetwo-’nucleon’ interaction constructed in Ref. [25] wherethe saturation curve of the one-dimensional NM is ob-tained by the Monte-Carlo simulation, we solve the flowequation for the 2PPI effective action with some reason-able truncation. We show that the resultant density func-tional E [ ρ ] nicely gives the saturation energy, i.e. theminimum of the energy derived by the EOS with respectto the density, that coincides with that of the MonteCarlo method [25].This paper is organized as follows: In Sec. II, our for-malism is shown. We introduce the flowing chemical po-tential to control the particle number during the flowand derive the DFT-RG flow equation for infinite uni-form systems with a definite particle number. In Sec. III,we apply our formalism to a (1+1)-dimensional spinlessnuclear matter. The results of the density dependenceof the ground state energy, i.e. the equation of state isshown in this section. Section IV is devoted to the con-clusion. II. FORMALISM
In this section, we show our formalism to analyzeground state energies of one-dimensional continuum mat-ters composed of spinless fermions in the framework ofDFT-RG.We consider (1+1)-dimensional spinless fermions witha two-body interaction U . We employ the imaginary-time finite-temperature formalism for convenience. Thenthe action in the units such that mass of a fermion is 1reads S [ ψ ∗ , ψ ] = Z χ ψ ∗ ( χ ǫ ) (cid:18) ∂ τ − ∂ x (cid:19) ψ ( χ )+ 12 Z χ,χ ′ U ( χ, χ ′ ) ψ ∗ ( χ ǫ ) ψ ∗ ( χ ′ ǫ ) ψ ( χ ′ ) ψ ( χ ) , (1)where χ := ( τ, x ), χ ǫ := ( τ + ǫ, x ) with a positive in-finitesimal ǫ , R χ := R τ R x := R β dτ R dx with an in-verse temperature β , and U ( χ, χ ′ ) := U ( χ − χ ′ ) := δ ( τ − τ ′ ) U ( x − x ′ ). The imaginary times of the fermionfields ψ and ψ ∗ at the same point are infinitesimally dif-ferent, which comes from the construction of the pathintegral formalism [26].Following the prescription in Ref. [11, 12], we intro-duce the regulated interaction U ,λ ( χ ) ( U λ (x)) such that U ,λ =0 ( χ ) = 0 and U ,λ =1 ( χ ) = U ( χ ) ( U λ =0 ( x ) = 0and U λ =1 ( x ) = U ( x )). Then the regulated action is de-fined in terms of U ,λ ( χ ): S λ [ ψ ∗ , ψ ] = Z χ ψ ∗ ( χ ) (cid:18) ∂ τ − ∂ x (cid:19) ψ ( χ )+ 12 Z χ,χ ′ U ,λ ( χ, χ ′ ) ψ ∗ ( χ ǫ ) ψ ∗ ( χ ′ ǫ ) ψ ( χ ′ ) ψ ( χ ) . (2) This action becomes that for free particles at λ = 0 andEq. (1) at λ = 1. The parameter λ ∈ [0 ,
1] is interpretedas the flow parameter from the free to interacting system.In this paper, we choose U ,λ ( χ, χ ′ ) = λU ( χ, χ ′ ).The EDF E λ [ ρ ] realizing the HK theorem can be de-fined in terms of the 2PPI effective action [27], which isdefined as follows:Γ λ [ ρ ] = sup J (cid:18)Z χ J ( χ ) ρ ( χ ) − W λ [ J ] (cid:19) , (3)where ρ ( χ ) is the density field, W λ [ J ] := ln Z λ [ J ] is thegenerating functional for the connected density correla-tion functions and Z λ [ J ] = R D ψ ∗ D ψ exp( − S λ [ ψ ∗ , ψ ] + R χ J ( χ ) ρ ψ ( χ )) is the generating functional for the corre-lation functions of the density field ρ ψ ( χ ) = ψ ∗ ( χ ǫ ) ψ ( χ ).To see the correspondence of the 2PPI effective action tothe EDF, we consider the variational problem of Γ λ [ ρ ]with a fixed number of particles. The stationary con-dition of the effective action determines the behavior ofthe expectation value of the density field. Under the con-straint that the number of particles is fixed, we shouldminimize I λ [ ρ ] := Γ λ [ ρ ] − µ λ R χ ρ ( χ ) with respect to ρ ( χ ).In general, the Fermi energy, and the particle number, de-pend on the interaction and change during the flow . Wethus have introduced a λ -dependent Lagrange multiplier,or physically a ’flowing’ chemical potential, µ λ to controlthe particle number during the flow. The ground statedensity ρ gs ,λ ( χ ) satisfies the stationary condition:Γ (1) λ [ ρ gs ,λ ]( χ ) = J sup [ ρ gs ,λ ]( χ ) = µ λ , (4)where J sup [ ρ ]( χ ) is J ( χ ) maximizing the right hand sideof Eq. (3) andΓ ( n ) λ [ ρ ]( χ , · · · , χ n ) := δ n Γ[ ρ ] δρ ( χ ) · · · δρ ( χ n ) . Because − W λ [ µ λ ] /β is the grand potential, we haveΓ λ [ ρ gs ,λ ] = µ λ R χ ρ gs ,λ ( χ ) − W λ [ µ λ ] = βF λ , where F λ is the Helmholtz free energy. At the zero tempera-ture limit β → ∞ , Γ λ [ ρ gs ,λ ] /β becomes the groundstate energy E gs ,λ because F λ can be written as F λ = − β − ln P n exp( − βE n,λ ), where { E n,λ } is the energyeigenvalues of the system and satisfies E gs ,λ = E ,λ The flow equation of Γ λ [ ρ ] is derived by differentiatingEq. (3) with respect to λ : ∂ λ Γ λ [ ρ ] = − ( ∂ λ W λ )[ J sup ,λ [ ρ ]] . The right-hand side of this flow equation becomes − ( ∂ λ W λ )[ J sup ,λ [ ρ ]]= 12 Z χ,χ ′ ˙ U ,λ ( χ, χ ′ ) h ψ ∗ ( χ ǫ ) ψ ∗ ( χ ′ ǫ ) ψ ( χ ′ ) ψ ( χ ) i ρ = 12 Z χ,χ ′ ˙ U ,λ ( χ, χ ′ ) (cid:16) h ρ ψ ( χ ) i ρ h ρ ψ ( χ ′ ) i ρ + W (2) λ [ J sup ,λ [ ρ ]]( χ, χ ′ ) − δ ( x − x ′ ) h ρ ψ ( χ ) i ρ (cid:17) , (6)where ˙ U ,λ ( χ, χ ′ ) := ∂ λ U ,λ ( χ, χ ′ ), h· · ·i ρ := Z D ψ ∗ D ψ · · · e − S λ [ ψ ∗ ,ψ ]+ R χ J sup ,λ [ ρ ]( χ ) ρ ψ ( χ ) Z λ [ J sup ,λ [ ρ ]] , and W ( n ) λ [ J ]( χ , · · · , χ n ) := δ n W λ [ J ] δJ ( χ ) · · · δJ ( χ n ) . To derive Eq. (6), we have used U ( χ, χ ′ ) ∼ δ ( τ − τ ′ ) and the canonical commutation rela-tion: h ψ ∗ ( τ + ǫ, x )( ψ ∗ ( τ + ǫ, x ′ ) ψ ( τ, x ′ ) + ψ ( τ + ǫ, x ′ ) ψ ∗ ( τ, x ′ )) ψ ( τ, x ) i ρ = δ ( x − x ′ ) h ρ ψ ( χ ) i ρ . We notethat h· · ·i ρ gives averages for imaginary-time-ordered op-erator products and that the density–density correlationfunction W (2) λ [ J sup ,λ [ ρ ]]( χ, χ ′ ) at τ = τ ′ should be inter-preted as lim τ → τ ′ lim ǫ → +0 ( h ψ ∗ ( χ ǫ ) ψ ( χ ) ψ ∗ ( χ ′ ǫ ) ψ ( χ ′ ) i ρ − h ψ ∗ ( χ ǫ ) ψ ( χ ) i ρ h ψ ∗ ( χ ′ ǫ ) ψ ( χ ′ ) i ρ ) where the limit τ → τ ′ istaken after the limit ǫ → +0, i.e. | τ ′ − τ | > ǫ . By use ofthe relations h ρ ψ ( χ ) i ρ = W (1) λ [ J sup ,λ ]( χ ) = ρ ( χ ) and Z χ ′ Γ (2) λ [ ρ ]( χ, χ ′ ) W (2) λ [ J sup ,λ [ ρ ]]( χ ′ , χ ′′ )= Z χ ′ δJ sup ,λ [ ρ ]( χ ) δρ ( χ ′ ) δρ ( χ ′ ) δJ sup ,λ [ ρ ]( χ ′′ ) = δ ( χ, χ ′′ )the flow equation can be written in term of Γ λ [ ρ ] [12, 22]: ∂ λ Γ λ [ ρ ] = 12 Z χ,χ ′ ˙ U ,λ ( χ, χ ′ ) ( ρ ( χ ) ρ ( χ ′ )+ Γ (2) − λ [ ρ ]( χ, χ ′ ) − ρ ( χ ) δ ( x − x ′ ) (cid:17) . (7)where Γ (2) − λ [ ρ ]( χ, χ ′ ) is the inverse of Γ (2) λ [ ρ ]( χ, χ ′ ).In principle, the functional flow equation (7) with theeffective action Γ [ ρ ] for the free fermions gives the effec-tive action Γ [ ρ ] for the interacting fermions. In general,however, some approximation is needed for the practicaluse of Eq. (7). Here, we employ the vertex expansion:Γ λ [ ρ ] =Γ λ [ ρ gs ,λ ] + ∞ X n =1 Z χ · · · Z χ n Γ ( n ) λ [ ρ gs ,λ ]( χ , · · · , χ n ) × ( ρ ( χ ) − ρ gs ,λ ( χ )) · · · ( ρ ( χ n ) − ρ gs ,λ ( χ n )) . Up to the second order expansion, Eq. (7) is rewritten asthe following flow equations: ∂ λ Γ λ [ ρ gs ,λ ] = Z χ Γ (1) λ [ ρ gs ,λ ]( χ ) ∂ λ ρ gs ,λ + 12 Z χ,χ ′ ˙ U ,λ ( χ, χ ′ ) ( ρ gs ,λ ( χ ) ρ gs ,λ ( χ ′ )+ Γ (2) − λ [ ρ gs ,λ ]( χ, χ ′ ) − ρ gs ,λ ( χ ) δ ( x − x ′ ) (cid:17) , (8) ∂ λ Γ (1) λ [ ρ gs ,λ ]( χ ) = Z χ ′ Γ (2) λ [ ρ gs ,λ ]( χ, χ ′ ) ∂ λ ρ gs ,λ ( χ ′ ) + Z χ ′ ˙ U ,λ ( χ, χ ′ ) ρ gs ,λ ( χ ′ ) − 12 ˙ U λ (0) − Z χ ,χ ,χ ,χ ˙ U ,λ ( χ , χ )Γ (2) − λ [ ρ gs ,λ ]( χ , χ )Γ (3) λ [ ρ gs ,λ ]( χ , χ , χ )Γ (2) − λ [ ρ gs ,λ ]( χ , χ ) , (9) ∂ λ Γ (2) λ [ ρ gs ,λ ]( χ, χ ′ ) = Z χ Γ (3) λ [ ρ gs ,λ ]( χ, χ ′ , χ ) ∂ λ ρ gs ,λ ( χ ) + ˙ U ,λ ( χ, χ ′ ) − Z χ , ··· ,χ ˙ U ,λ ( χ , χ )Γ (2) − λ [ ρ gs ,λ ]( χ , χ )Γ (4) λ [ ρ gs ,λ ]( χ , χ , χ, χ ′ )Γ (2) − λ [ ρ gs ,λ ]( χ , χ )+ Z χ , ··· ,χ ˙ U ,λ ( χ , χ )Γ (2) − λ [ ρ gs ,λ ]( χ , χ )Γ (3) λ [ ρ gs ,λ ]( χ , χ , χ ) × Γ (2) − λ [ ρ gs ,λ ]( χ , χ )Γ (3) λ [ ρ gs ,λ ]( χ , χ , χ ′ )Γ (2) − λ [ ρ gs ,λ ]( χ , χ ) . (10)These flow equation can be simplified by rewriting in terms of the connected correlation functions: G ( n ) λ ( χ , · · · , χ n ) = W ( n ) λ [ J sup [ ρ gs ,λ ]]( χ , · · · , χ n ) . Γ ( n ) λ [ ρ gs ,λ ] is related to the connected correlation functions with the following relation:Γ ( n ) λ [ ρ gs ,λ ]( χ , · · · , χ n ) = n − Y i =1 Z χ ′ i W (2) − λ [ J ]( χ i , χ ′ i ) δδJ ( χ ′ i ) ! W (2) − λ [ J ]( χ n − , χ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) J = J sup [ ρ gs ,λ ] , which is derived from the following identity: δδρ ( χ ) = Z χ ′ δJ sup ,λ [ ρ ]( χ ′ ) δρ ( χ ) δδJ sup ,λ ( χ ′ ) = Z χ ′ W (2) − λ [ J sup ,λ [ ρ ]]( χ, χ ′ ) δδJ sup ,λ ( χ ′ ) . (11)The relations between Γ (2 , , λ [ ρ gs ,λ ] and the connected correlation functions readΓ (2) λ [ ρ gs ,λ ]( χ , χ ) = G (2) − λ ( χ , χ )Γ (3) λ [ ρ gs ,λ ]( χ , χ , χ ) = − Z χ ′ ,χ ′ ,χ ′ G (3) λ ( χ ′ , χ ′ , χ ′ ) G (2) − λ ( χ , χ ′ ) G (2) − λ ( χ , χ ′ ) G (2) − λ ( χ , χ ′ )Γ (4) λ [ ρ gs ,λ ]( χ , χ , χ .χ ) = − Z χ ′ ,χ ′ ,χ ′ .χ ′ G (4) λ ( χ ′ , χ ′ , χ ′ , χ ′ ) G (2) − λ ( χ χ ′ ) G (2) − λ ( χ , χ ′ ) G (2) − λ ( χ , χ ′ ) G (2) − λ ( χ , χ ′ )+ Z χ ′ ,χ ′ ,χ ′ G (2) − λ ( χ , χ ′ ) G (2) − λ ( χ , χ ′ ) G (3) λ ( χ ′ , χ ′ , χ ′ ) × G (2) − λ ( χ ′ , χ ′ ) G (3) λ ( χ ′ , χ ′ , χ ′ ) G (2) − λ ( χ , χ ′ ) G (2) − λ ( χ , χ ′ )+ Z χ ′ ,χ ′ ,χ ′ G (2) − λ ( χ , χ ′ ) G (2) − λ ( χ , χ ′ ) G (3) λ ( χ ′ , χ ′ , χ ′ ) × G (2) − λ ( χ ′ , χ ′ ) G (3) λ ( χ ′ , χ ′ , χ ′ ) G (2) − λ ( χ , χ ′ ) G (2) − λ ( χ , χ ′ )+ Z χ ′ ,χ ′ ,χ ′ G (2) − λ ( χ , χ ′ ) G (2) − λ ( χ , χ ′ ) G (3) λ ( χ ′ , χ ′ , χ ′ ) × G (2) − λ ( χ ′ , χ ′ ) G (3) λ ( χ ′ , χ ′ , χ ′ ) G (2) − λ ( χ , χ ′ ) G (2) − λ ( χ , χ ′ ) . By use of these relations, Eqs. (8)-(10) are rewritten as follows: ∂ λ Γ λ [ ρ gs ,λ ] = Z χ µ λ ∂ λ ρ gs ,λ ( χ ) + 12 Z χ,χ ′ ˙ U ,λ ( χ, χ ′ ) (cid:16) ρ gs ,λ ( χ ) ρ gs ,λ ( χ ′ ) + G (2) λ ( χ, χ ′ ) − ρ gs ,λ ( χ ′ ) δ ( x ′ − x ) (cid:17) , (12) ∂ λ ρ gs ,λ ( χ ) = − Z χ ,χ ˙ U ,λ ( χ , χ ) G (3) λ ( χ , χ , χ )+ Z χ G (2) λ ( χ, χ ) (cid:18) ∂ λ µ λ − Z χ ˙ U ,λ ( χ , χ ) ρ gs ,λ ( χ ) + 12 ˙ U λ (0) (cid:19) , (13) ∂ λ G (2) λ ( χ, χ ′ ) = − Z χ ,χ ˙ U ,λ ( χ , χ ) (cid:18) G (2) λ ( χ, χ ) G (2) λ ( χ , χ ′ ) + 12 G (4) λ ( χ , χ , χ, χ ′ ) (cid:19) + Z χ G (3) λ ( χ, χ ′ , χ ) (cid:18) ∂ λ µ λ − Z χ ˙ U ,λ ( χ , χ ) ρ gs ,λ ( χ ) + 12 ˙ U λ (0) (cid:19) . (14)Here, we have used Eqs. (4) and (5). These flow equationsdetermine the behavior of the free energy Γ λ [ ρ gs ,λ ] = βF λ , the ground state density ρ gs ,λ ( χ ) and the density–density correlation function G (2) λ ( χ, χ ′ ) under a given λ -dependent chemical potential µ λ .For the case of free fermion λ = 0, the ground stateof the system is homogeneous. In this paper, we assumethat the homogeneity of the system remains even if λ = 0,i.e. the transition to an inhomogeneous state does not occur even if the interaction is switched on. In general,the switching on of the interaction at λ = 0 changesthe density as represented in Eq. (13). However, we cancompensate this effect of the interaction by choosing anappropriate µ λ and realize ∂ λ ρ gs ,λ = 0 in the case ofthe homogeneous system . We employ the momentum Our idea can be extended to the case of inhomogeneous systems representation for convenience to discuss how to choose µ λ . In the momentum representation, Eqs. (12)-(14) inthe case of homogeneous states read ∂ λ Γ λ [ ρ gs ,λ ] βV = µ λ ∂ λ ρ gs ,λ + 12 ˜ U (0) ρ ,λ + 12 Z p ˜ U ( p ) T X ω ˜ G (2) λ ( P ) − ρ gs ,λ ! , (15) ∂ λ ρ gs ,λ = − Z P ˜ U ( p ) ˜ G (3) λ ( P, − P ) + ˜ G (2) λ (0) (cid:18) ∂ λ µ λ − ˜ U (0) ρ gs ,λ + 12 U (0) (cid:19) , (16) ∂ λ ˜ G (2) λ ( P ) = − ˜ U ( p ) G (2) λ ( P ) − Z P ′ ˜ U ( p ′ ) ˜ G (4) λ ( P ′ , − P ′ , P ) + G (3) λ ( P, − P ) (cid:18) ∂ λ µ λ − ˜ U (0) ρ gs ,λ + 12 U (0) (cid:19) . (17)Here we have used U ,λ ( χ, χ ′ ) = λU ( x − x ′ ) δ ( τ − τ ′ )and introduced the volume of the system V andthe Fourier transformations ˜ U ( p ) := R x U ( x ) e − ipx and (2 π ) δ ( P + · · · + P n ) ˜ G ( n ) λ ( P , · · · , P n − ) := R χ , ··· ,χ n e − i ( P · χ + ··· + P n · χ n ) G ( n ) λ ( χ , · · · , χ n ), where P i := ( ω i , p i ) is a vector of a Matsubara frequency anda momentum. We have introduced the short hands R p := R dp/ (2 π ) and R P := R p T P ω . Then ∂ λ ρ gs ,λ = 0 isrealized if the flow of µ λ is set as follows: ∂ λ µ λ = ˜ U (0) ρ gs ,λ − U (0)2 + Z P ˜ U ( p ) ˜ G (3) λ ( P, − P )2 ˜ G (2) λ (0) . (18) We note that ˜ G (2) λ (0) should be interpreted as the p limit of ˜ G (2) λ ( P ): ˜ G (2) λ (0) = lim p → ˜ G (2) λ (0 , p ), becausethe Matsubara frequency ω is discrete. The p limitof the ˜ G (2) λ ( P ) is the static particle-density susceptibil-ity and generally nonzero, while lim p → ˜ G (2) λ ( P ) = 0with a finite frequency [31–33]. This is in contrast tothe case of a finite number of particles in a finite box[22], where density correlation functions with vanish-ing frequency and momentum were interpreted as the ω limit, i.e., not only R dx i G ( n ) λ ( χ , · · · , χ n ) = 0 but also R χ i G ( n ) λ ( χ , · · · , χ n ) = 0 with i ∈ { , · · · , n } were usedto derive the flow equations.In this paper, we focus on the zero temperature limit.In the zero temperature limit with the condition Eq. (18),Eqs. (15) and (17) becomes as follows: ∂ λ E gs ,λ = ρ gs , U (0) + 12 ρ gs , Z p ˜ U ( p ) (cid:18)Z ω ˜ G (2) λ ( P ) − ρ gs , (cid:19) , (19) ∂ λ ˜ G (2) λ ( P ) = − ˜ U ( p ) ˜ G (2) λ ( P ) − Z P ′ ˜ U ( p ′ ) ˜ G (4) λ ( P ′ , − P ′ , P ) + Z P ′ ˜ U ( p ′ ) ˜ G (3) λ ( P ′ , − P ′ ) ˜ G (3) λ ( P, − P )2 ˜ G (2) λ (0) , (20)where we have introduced the energy per particle E gs ,λ =lim β → Γ λ [ ρ gs ,λ ] / ( β R x ρ gs , ) and the shorthand R ω = R dω/ (2 π ).The flow equation for G ( n ) λ generally depends on G ( m ≤ n +2) λ , which means that an infinite series ofcoupled flow equations emerges. We avoid to treatsuch an infinite series of coupled flow equations byignoring the flows of G (3 ≤ n ) λ . However, we do notsimply substitute ˜ G (3 , λ for ˜ G (3 , in Eq. (20). Such a by use of the x -dependent chemical potential µ λ ( x ). However, itwould be impossible to fix ρ gs ,λ ( χ ) to an arbitral density suchas those not satisfying the v-representability [28–30]. simple replacement breaks a constraint for multi-particledistribution functions imposed by the Pauli blocking:By use of the canonical commutation relation, the n -particle distribution function f n,λ ( x , · · · , x n ) =lim ǫ → +0 h ψ ∗ ( ǫ, x ) · · · ψ ∗ ( ǫ, x n ) ψ (0 , x n ) · · · ψ (0 , x ) i ρ gs ,λ is related to the connected correlation functions G ( m ≤ n ) λ .Because of the Pauli blocking, the distribution functionsatisfies f n,λ ( x , · · · , x n ) = 0 if x i = x j for i = j . Inthe case of n = 2, the relation between f ,λ ( x , x )and G (2) λ is derived in the same same manner as theaforementioned derivation of Eq. (6): f ,λ ( x , x )= h ρ ψ (0 , x ) ρ ψ (0 , x ) i ρ gs ,λ − δ ( x − x ) h ρ ψ (0 , x ) i ρ gs ,λ = G (2) λ (0 , x , , x ) + ρ gs ,λ ( x ) ρ gs ,λ ( x ) − δ ( x − x ) ρ gs ,λ ( x ) . Then the condition imposed by the Pauli blocking readslim x → x G (2) λ (0 , x , , x )= lim x → x ( δ ( x − x ) ρ gs ,λ ( x ) − ρ gs ,λ ( x ) ρ gs ,λ ( x )) . In our case, the right-hand side of this condition does notdepend on λ : lim x → x ∂ λ G (2) λ (0 , x , , x ) = 0, because ∂ λ ρ gs ,λ = 0. Therefore, from Eq. (20), the following con-dition should be satisfied: − Z P ˜ U ( p ) ˜ G (2) λ ( P ) − Z P,P ′ ˜ U ( p ′ ) ˜ G (4) λ ( P ′ , − P ′ , P )+ Z P,P ′ ˜ U ( p ′ ) ˜ G (3) λ ( P ′ , − P ′ ) ˜ G (3) λ ( P, − P )2 ˜ G (2) λ (0) = 0 (21)This condition, however, is broken by the simple substi-tution ˜ G (3 , λ for ˜ G (3 , . To respect the condition Eq. (21),we approximate the second and third terms in the right-hand side of Eq. (20) as follows [22]: − Z P ′ ˜ U ( p ′ ) ˜ G (4) λ ( P ′ , − P ′ , P )+ Z P ′ ˜ U ( p ′ ) ˜ G (3) λ ( P ′ , − P ′ ) ˜ G (3) λ ( P, − P )2 ˜ G (2) λ (0) ≈ c λ (cid:18) − Z P ′ ˜ U ( p ′ ) ˜ G (4)0 ( P ′ , − P ′ , P )+ Z P ′ ˜ U ( p ′ ) ˜ G (3)0 ( P ′ , − P ′ ) ˜ G (3)0 ( P, − P )2 ˜ G (2)0 (0) ! (22)where c λ is the factor to preserve the condition Eq. (21): c λ = Z P ˜ U ( p ) ˜ G (2) λ ( P ) × (cid:18) − Z P ′ ,P ′′ ˜ U ( p ′ ) ˜ G (4)0 ( P ′ , − P ′ , P ′′ )+ Z P ′ ,P ′′ ˜ U ( p ′ ) ˜ G (3)0 ( P ′ , − P ′ ) ˜ G (3)0 ( P ′′ , − P ′′ )2 ˜ G (2)0 (0) ! − . At λ = 0, we have c = 1. In our approximation, thecontribution of diagrams such as multi-pair creations isnot included. Such diagrams are important to investi-gate the spectral properties of one-dimensional fermionsystems [34–36], which is beyond the scope of this paper.We specify the initial conditions E gs ,λ =0 , ρ gs ,λ =0 , and˜ G (2 , , λ =0 . We denote ρ gs , as n , which is the ground state ˜ G (2)0 ( P ) = P P ˜ G (3)0 ( P , P ) = P P − P + P P − P ˜ G (4)0 ( P , P , P ) = P − P − P P + P − P − P P + P − P − P P + P − P − P P + P − P − P P + P − P − P P FIG. 1. The diagrammatic representations of ˜ G (2 , , . Thesolid line is the free fermion propagator ˜ G (2)F , ( P ). In thediagrams for ˜ G (3)0 and ˜ G (4)0 , P = − P − P and P = − P − P − P , respectively. density during the flow, and in particular at λ = 1, be-cause ρ gs ,λ ( χ ) = ρ gs , . The fermi momentum and fermienergy are defined as p F = πn and E F = p / 2, respec-tively. E gs ,λ =0 is the ground state energy per particle ofa one-dimensional free Fermi gas: E gs , = E F / 3. ˜ G (2 , , λ =0 are the density correlation functions for free fermions:˜ G (2)0 ( P ) = − Z P ′ ˜ G (2)F , ( P ′ ) ˜ G (2)F , ( P + P ′ ) , ˜ G (3)0 ( P , P ) = − X σ ∈ S Z P ′ ˜ G (2)F , ( P ′ ) ˜ G (2)F , ( P σ (1) + P ′ ) × ˜ G (2)F , ( P σ (1) + P σ (2) + P ′ ) , ˜ G (4)0 ( P , P , P ) = − X σ ∈ S Z P ′ ˜ G (2)F , ( P ′ ) ˜ G (2)F , ( P σ (1) + P ′ ) × ˜ G (2)F , ( P σ (1) + P σ (2) + P ′ ) × ˜ G (2)F , ( P σ (1) + P σ (2) + P σ (3) + P ′ ) . Here S and S are the symmetric groups of order twoand three, respectively, and ˜ G (2)F , ( P ) is the two-pointpropagator of free fermions: ˜ G (2)F , ( P ) = 1 / ( iω − ξ ( p )),where ξ ( p ) := p / − E F . The diagrammatic representa-tions of these correlation functions are shown in Fig. 1.The explicit forms of ˜ G (2)0 ( P ) and the second and thirdterms of the right-hand side of Eq. (20) after the fre-quency integral under the approximation Eq. (22) be-comes as follows:˜ G (2)0 ( P ) = Z p ′ θ ( ξ ( p + p ′ )) θ ( − ξ ( p ′ )) ( ξ ( p + p ′ ) − ξ ( p ′ )) ω + [ ξ ( p + p ′ ) − ξ ( p ′ )] , (23) − c λ Z P ′ ˜ U ( p ′ ) ˜ G (4)0 ( P ′ , − P ′ , P ) + c λ Z P ′ ˜ U ( p ′ ) ˜ G (3)0 ( P ′ , − P ′ ) ˜ G (3)0 ( P, − P )2 ˜ G (2)0 (0)= 2 c λ Z p ′ ,p ′′ ˜ U ( p ′ ) θ ( − ξ ( p ′ + p ′′ ))( θ ( − ξ ( p ′′ + p )) − θ ( − ξ ( p ′′ ))) (cid:20) ( ξ ( p ′′ + p ) − ξ ( p ′′ )) − ω ( ω + ( ξ ( p ′′ + p ) − ξ ( p ′′ )) ) + 2 θ ( ξ ( p ′′ + p + p ′ )) ξ ( p ′′ + p ′ + p ) − ξ ( p ′′ + p ) − ξ ( p ′′ + p ′ ) + ξ ( p ′′ ) ξ ( p ′′ + p + p ′ ) − ξ ( p ′′ + p ′ ) ω + ( ξ ( p ′′ + p + p ′ ) − ξ ( p ′′ + p ′ )) (cid:21) . (24)The flow equations (19) and (20) under the approxima-tion Eq. (22) with the expression Eq. (24) are found to bethe same as those obtained from the continuum limit ofthe system with finite number of particles in a finite boxpresented in [22]. Finally we comment on the derivationof Eq. (24). Both terms in the left-hand side of Eq. (24)have delta functions in the momentum integrals, whichcomes from the derivatives of the distribution functionsof fermions such as θ ′ ( − ξ ( p ′ )) = − δ ( ξ ( p ′ )). These con-tributions from the delta functions, however, cancel eachother out and do not appear in the final expression inEq. (24). III. DEMONSTRATION INONE-DIMENSIONAL SPINLESS NUCLEARMATTER In this section, we demonstrate the calculation of theground state energy as a function of the density, i.e. theequation of state (EOS) in a (1 + 1)-dimensional spinlesscontinuum nuclear matter [25]. A. One-dimensional spinless nuclear matter We consider a (1 + 1)-dimensional continuum mattercomposed of spinless fermions with the following two-body interaction [25]: U ( r ) = g √ π (cid:18) σ e − r σ − σ e − r σ (cid:19) where σ > σ > g > 0. In this model, the short-range repulsive force and long-range attractive force be-tween nucleons are represented with the superposition ofthe two Gaussians. Following Ref. [25], we choose g = 12, σ = 0 . σ = 0 . B. Numerical procedure We mention some details of our numerical analysis tosolve the flow equations.Equations (24) has a seemingly singular point at p ′ =0 in the term proportional to ( ξ ( p ′′ + p ′ + p ) − ξ ( p ′′ + p ) − ξ ( p ′′ + p ′ ) + ξ ( p ′′ )) − ∼ p ′− in the integrand. Thissingularity, however, vanishes because ˜ U ( p ′ = 0) = 0 forour interaction. To avoid the division-by-zero operation,we rewrite the integrand as a manifestly regular formfor the numerical calculation by use of the Maclaurinexpansion of ˜ U ( p ′ ).In order to calculate ˜ G (2) λ ( ω, p ) on the ( ω, p )-plane,we discretize ω and p . We change ω to ¯ ω =(2 /π ) arctan( ω/s ), where s is an arbitral number, anddiscretize ¯ ω in the domain [ − , p F , 1) for momentum p . We have checkedthat the result hardly depends on Λ even if it is set tolarger values. C. Ground state properties We show our result of the EOS together with the en-ergy of the free Fermi gas E gs , and the contribution fromthe inter-nucleon potential E gs , − E gs , in Fig. 2. Thefree case shows an increase with respect to n , which re-flects that the average kinetic energy increases becausethe fermi sphere becomes larger as n increases. On theother hand, the contribution from the inter-particle po-tential reduces the energy because it has a long-rangeattractive part. The competition between these contri- TABLE I. The saturation energies E s derived from the first-order perturbation, DFT-RG with c λ = 1, DFT-RG, and the MCsimulation [25]. Their relative errors compared to the result of the Monte Carlo simulation E s , MC and the saturation densities ρ s are also shown. The units for E s and ρ s are such that the mass of a nucleon is 1.1st order DFT-RG ( c λ = 1) DFT-RG Monte Carlo [25] (cid:12)(cid:12) E s (cid:12)(cid:12) (cid:12)(cid:12)(cid:0) E s − E s , MC (cid:1) /E s , MC (cid:12)(cid:12) (%) 10.0 4.6 2.7 - ρ s . . . . . n − − − − E g s DFT-RGFreepotential energyMonte Carlo FIG. 2. Energy per particle E gs as a function of density n in the case of DFT-RG (solid red line). The energy of freefermion (green dotted line), the contribution from the inter-nucleon potential (purple dashed line) and the result of theextrapolated energy given by a MC simulation [25] at a den-sity close but not equal to saturation density (black cross)are also shown. The red point is the saturation point derivedfrom the DFT-RG calculation. The units for E gs and n aresuch that the mass of a nucleon is 1. butions results in the emergence of a minimum point ata finite n , i.e. the saturation point.To discuss the quantitative aspects of our result, weshow our result of the EOS near the saturation point inFig. 3. For comparison, the results with the first-orderperturbation and with ignoring the running of c λ , i.e. c λ = 1, are also shown. In our formalism, the first-orderperturbation is reproduced when ˜ G (2) λ ( P ) in Eq. (19) isreplaced with ˜ G (2)0 ( P ). In Fig. 3, one finds that the sat-uration energy derived from DFT-RG is closer to the re-sult of the MC simulation than any other methods. Weagain note that the density used in the MC result shownin Fig. 3 is not a calculated but given density which isassumed to be close but not equal to the saturation den-sity. Therefore, the density used in the MC simulationshould not be used as a reference of the benchmark ofthe saturation density.Table I shows the numerical results of the saturationenergy E s and density ρ s derived from each method. Asshown in this table, the deviation of the saturation en- . . . . . . . . n − . − . − . − . − . − . − . − . − . . E g s c λ = 1 )DFT-RGMonte Carlo FIG. 3. Energy per particle E gs as a function of density n near the saturation point. The result of DFT-RG (solid redline), first-order perturbation (blue dashed line), DFT-RGwith c λ = 1 (green dotted–dashed line), and a MC simula-tion [25] (black cross) are shown. The density n = 1 . 16 usedin the MC simulation is an assumed one which is close butnot equal to saturation density. The units for E gs and n aresuch that the mass of a nucleon is 1. ergy between DFT-RG and the MC simulation is only2.7%. Compared to the result of the first-order pertur-bation, the accuracy of E s is largely improved by use ofthe DFT-RG scheme. Moreover, the introduction of therunning of the factor c λ also contributes to the improve-ment of the accuracy. Although there is no reference forthe saturation density ρ s , it seems to converge to a valuenear 1 . IV. CONCLUSION In this paper, we have presented the functional renor-malization group (FRG)-aided density-functional theory(DFT) calculation of the equation of state of an infinitenuclear matter in (1+1)-dimensions composed of spinlessnucleons. We have shown a novel formalism to treat infi-nite matters in which the flow of the chemical potential istaken into account to control the particle number duringthe flow. The resultant saturation energy coincides withthat obtained from the Monte-Carlo simulation within afew percent. Thus one sees that DFT-RG scheme workswell for the infinite homogeneous nuclear system aroundthe saturation point in comparison with the case of finitenuclear system [22] where the same approximation forthe flow equations was employed: A truncation includ-ing up to the second-order vertex expansion was used andan approximation was made for the three- and four-pointcorrelation functions so that the Pauli blocking effect istaken into account.Our result together with its numerical feasibilityclearly demonstrate that the DFT-RG scheme is apromising nonperturbative method to analyze quantummany-body systems, at least, composed of infinite num-ber of particles. One of the next steps is applying ourmethod to higher-dimensional systems which should beeasy in the present grand canonical formalism. The ex-tension to a finite-temperature case can be done as well.The extension to particles with internal degrees of free-dom [23] is also an important future direction.Another interesting extension of the present work iscalculation of dynamical quantities. One of the basicquantities for analyzing the dynamical properties is thespectral functions. Recently, the FRG has been devel-oped so that the spectral functions can be calculatedwhich describe the dynamical properties of the system[37–41]. It is intriguing to extend the DFT-RG scheme so that the spectral function can be calculated. We havealready calculated the spectral function of the density–density correlation function in the DFT-RG scheme forthe one-dimensional model used in this work. We willreport the analysis of the spectral function in a separatepaper, where some relevance to the Tomonaga-Luttingerliquid with a non-linear dispersion relation [36] will bediscussed. 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