Ab-initio description of excited states of a one-dimensional nuclear matter with the Hohenberg-Kohn-theorem-inspired functional-renormalization-group method

We demonstrate for the first time that a functional-renormalization-group aided density-functional theory (FRG-DFT) describes well the characteristic features of the excited states as well as the ground state of an interacting many-body system with infinite number of particles in a unified manner. The FRG-DFT is applied to a $(1+1)$-dimensional spinless nuclear matter. For the excited states, the density--density spectral function is calculated at the saturation point obtained in the framework of FRG-DFT, and it is found that our result reproduces a notable feature of the density--density spectral function of the non-linear Tomonaga-Luttinger liquid: The spectral function has a singularity at the edge of its support of the lower-energy side. These findings suggest that the FRG-DFT is a promising first-principle scheme to analyze the excited states as well as the ground states of quantum many-body systems starting from the inter-particle interaction.

We demonstrate for the first time that a functional-renormalization-group aided densityfunctional theory (FRG-DFT) describes well the characteristic features of the excited states as well as the ground state of an interacting many-body system with infinite number of particles in a unified manner. The FRG-DFT is applied to a (1 + 1)-dimensional spinless nuclear matter. For the excited states, the density-density spectral function is calculated at the saturation point obtained in the framework of FRG-DFT, and it is found that our result reproduces a notable feature of the density-density spectral function of the non-linear Tomonaga-Luttinger liquid: The spectral function has a singularity at the edge of its support of the lower-energy side. These findings suggest that the FRG-DFT is a promising first-principle scheme to analyze the excited states as well as the ground states of quantum many-body systems starting from the inter-particle interaction. Density-functional theory (DFT) has greatly contributed to our understanding of quantum manybody systems in various fields including quantum chemistry and atomic, molecular, condensedmatter, and nuclear physics; see Refs. [1][2][3][4][5][6] for some recent reviews. The DFT is founded by the Hohenberg-Kohn (HK) theorem [7]. The theorem states that the total energy of the system is a functional of the particle density which is a function of single variable x and that the variational principle with respect to the density gives the ground-state density and energy exactly. The HK theorem is, however, just an existence theorem, but the DFT or the HK theorem cannot tell us about the energy-density functional (EDF) that we need to minimize. The EDFs employed usually in the practical calculations are thus constructed phenomenologically, and improvement of the EDFs lies at the center in the studies based on DFT. Therefore, it is highly desirable to develop a systematic method to derive the EDF from the underlying microscopic Hamiltonian. Successful application to the ground state of interacting systems in conjunction with the Kohn-Sham theory [8] has stimulated attempts to describing excited states and dynamics in a framework of time-dependent DFT (TDDFT) [2,5,9]. Presently, the linear-response TDDFT has been successfully applied to the small-amplitude collective modes of excitation, and the real-time TDDFT has been developed to describe even the non-linear dynamics as an initial-value problem. The TDDFT, in principle, can describe the many-body dynamics exactly. It is, however, an open problem to develop a practical method to extract the information of excited states possessing the large-amplitude collective character.
In view of quantum field theory, the two-particle point-irreducible (2PPI) effective action formalism [10] gives the HK theorem naturally and further the foundation of TDDFT is given in a unified way [11,12]. Here, the starting point is a generating functional with a source coupled to the local composite density operatorψ † ( x)ψ( x). And then a functional Legendre transformation with respect to the source leads to an effective action of the density, which gives the quantum equation of motion. Therefore, the 2PPI effective action is considered to be a generalization of the EDF.
Let us call here the exact or functional renormalization group (FRG) method [13], which is established as a practical way to treat the effective action non-perturbatively and has been successfully applied to quantum many-body problems [14][15][16][17]. The FRG is based on a one-parameter flow equation for the parameter-dependent effective average action, which gives the effective action of a fully-interacting system eventually by taking the quantum fluctuation and correlation gradually starting from a bare system. The 2PPI effective action formalism combined with the FRG thus gives possibly a systematic construction of the EDF based on a microscopic Hamiltonian [18][19][20][21][22][23][24][25]; we call such an approach the functional-renormalization-group aided density-functional theory (FRG-DFT). This approach can be a promising scheme for solving the fundamental problems in DFT and providing further insights in understanding the many-body systems.
The FRG-DFT method has been applied to a zero-dimensional model of anharmonic vibrator [22,26], and a one-dimensional quantum anharmonic vibrator [22] to show the feasibility and effectiveness. The estimation of uncertainty due to the truncation was given and an effective way to treat the higher-order correction was proposed [26]. The FRG-DFT method has also been applied to a (1+1)-dimensional many-body model simulating one-dimensional nucleons with a fixed particlenumber formalism [23], although the bound-state energy was underestimated by approximately 30% in comparison with the exact solution in two-particle system and over 17% in comparison with the results obtained using the Monte Carlo method [27] when the number of particles is no more than eight. In addition, the study of (1+1)-dimensional systems composed of finite number of spin-1/2 fermions interacting via a contact interaction has been reported [24,25].
Since the 2PPI effective action is an effective action which is by definition capable of describing time-dependent phenomena as well as static phenomena equally, the FRG-DFT method should be in principle applicable to not only the ground state but also the excited sates though there have been no attempts to demonstrate it as far as we are aware of. It should be noted here that the FRG has been applied to obtain the spectral functions in the O(N) model [28] and the quark-meson model [29][30][31][32]. Here, the analytic continuation is taken at each order before evaluation of the flow equations. This technique is thus much easier numerically than the standard method such as the maximum entropy method or the Padé approximation.
In this Letter, we demonstrate for the first time that our FRG-DFT works well for describing the excited states as well as the ground states of continuum matter. We are going to consider a (1+1)dimensional spinless nuclear matter [27] with infinite number of particles. After summarizing our result for the ground state energy [33] where the resultant equation of state was found to give the saturation energy compatible with that obtained using the Monte Carlo method [27], we show the numerical result for the density-density spectral function. We find that our density-density spectral function reproduces the existence of the peak at the edge of its support in the lower-energy side, 2/10 which is known as a notable feature of the non-linear Tomonaga-Luttinger liquid. Our result suggests that the FRG-DFT is a powerful way to analyze excited states as well as ground states of quantum many-body systems.
For self-containedness,we first recapitulate some part of our FRG-DFT formalism to analyze the ground state properties of infinite matters, which has been developed by the present authors in Ref. [33].
A microscopic input in our study is the inter-particle interaction. The interaction we adopted consists of the short-range repulsive core and the long-range attractive force of 'nucleons' both of which are given by a Gaussian : , where σ 1 > 0, σ 2 > 0 and g > 0. As given in Ref. [27], we chose g = 12, σ 1 = 0.2 and σ 2 = 0.8 in units such that the mass of a nucleon is 1. These values were determined under the assumptions that the relevant dimensionless values in one dimension are comparable with the empirical ones in three dimensions.
Although we are primarily interested in the system with zero-temperature in the present work, we found that the use of the finite-temperature imaginary-time formalism is most convenient. Then the action of the one-dimensional interacting spinless fermions reads where we have introduced the shorthands Since we are going to employ the techniques developed in the FRG method, we regulated the interaction between fermions by multiplying U 2b (χ, χ ′ ) by a regulator function R λ (τ, x, τ ′ , x ′ ) as introduced in Refs. [18,19,23], and the resulting regulated action is given as Here, R λ (χ, χ ′ ) was chosen so as to satisfy the following conditions: lim λ→0 R λ (χ, χ ′ ) = 0 and lim λ→1 R λ (χ, χ ′ ) = 1. Under these conditions, S λ becomes the action of free particles at λ = 0 and that of interacting particles at λ = 1, namely Eq. (1). Specifically, we chose R λ (χ 1 , χ 2 ) = λ for simplicity [18,19]. We then define the λ-dependent generating functional for density-density correlation functions as being the composite local number density operator. The generating functional for the connected density correlation functions Then the effective action Γ λ [ρ] of the local density ρ(χ) is obtained by the Legendre transformation [11], i.e. the ground state density ρ gs,λ (χ) and energy E gs,λ are obtained variationally from Γ λ [ρ]. When considering the variational problem under the constraint that the particle number is set to some value, we should minimize I λ [ρ] := Γ λ [ρ] − µ λ χ ρ(χ) with respect to ρ(χ). Here, we have introduced a λ-dependent chemical potential µ λ to control the change of the particle number during the flow caused by switching on of the interaction [33]. In this case, the ground state density ρ gs,λ (χ) satisfies the following stationary condition: (δΓ λ /δρ(χ))[ρ gs,λ ] = µ λ . Here, we should mention that the chemical potential depending on the RG parameter was studied [34,35] in the framework of the functional RGà la Wetterich [13] and the change of the chemical potential by the presence of interaction was discussed in the context of DFT [6].
We need to evaluate the momentum integrals such as d p ′Ũ (p ′ )/p ′ , which appear in the second term in the right-hand side of Eq. (8). The integrand apparently has a singular point at p ′ = 0, which is actually absent becauseŨ(0) = 0 in the present case. In order to avoid a division-by-zero operation, we rewrote the integrand by using the Maclaurin expansion ofŨ(p ′ ) at small p ′ to a manifestly regular form for the numerical calculation.
The results for the equation of state and the saturation energy, and the comparison with the Monte Carlo simulation [27] were shown in Ref. [33]. A remark is in order here: In the Monte Carlo simulation, the saturation energy was mere an extrapolated energy at a given density n = 1.16, which is considered to be close [27] but not equal to the saturation density; moreover the extrapolation was made from the results for finite particle systems with a particle number up to 12. In contrast, our FRG-DFT calculation was made for the system with infinite number of particles, and the density was varied continuously. The saturation energy derived from FRG-DFT is quite close to the result of the Monte Carlo simulation: We found that the discrepancy between the saturation energy given by FRG-DFT and that by the Monte Carlo simulation is 2.7%. We pointed out that such an accuracy was acquired with little computational time or resources in our framework of FRG-DFT.
This successful application to the ground-state properties of a many-body system rouses one's interest in extension of FRG-DFT to describing excited states. Then we are going to describe our  way to calculate the density-density spectral function. We define the density-density spectral function ρ d,λ (ω, p) as ρ d,λ (ω, p) = −2ImG (2) R,λ (ω, p), whereG (2) R,λ (ω, p) is the retarded two-point density correlation function, which is obtained from the analytic continuation ofG (2) λ (ω I , p):G (2) R,λ (ω, p) = −G (2) ana,λ (ω + iǫ, p). Here, ǫ is a positive infinitesimal andG (2) ana,λ (z, p) is a complex function of z ∈ C which is regular in the upper-half plane of z and satisfiesG (2) ana,λ (−iω I , p) =G (2) λ (ω I , p) for ω I ∈ R. The analytic continuation to obtainG (2) ana,λ (z, p) fromG (2) λ (ω I , p) is often an obstruction for a numerical analysis. In our case, however, the analytic continuation can be performed in the level of the flow equations as in Refs. [28][29][30][31][32], which is much easier numerically than the standard procedures such as the maximum entropy method or the Páde approximation. Under the approximation Eq. (9), one finds that the second term of Eq. (8) is regular in the upper-half plane of z when P 0 is simply replaced with −iz. ThenG (2) λ (P)| P 0 →−iz also keeps regular in the upper-half of z during the flow. Therefore, we have the flow equation forG (2) R,λ (ω, p): .
The initial condition of this flow equationG (2) R,0 (ω, p) is given by the replacement of P 0 with −i(ω + iǫ) in Eq. (10). As discussed below, the contribution from the second term in the right-hand side of the flow equation (13) is important to capture the feature of the spectral function in (1+1) dimensions. If the second term of the right-hand side of Eq. (13) is neglected, this flow equation can be solved analytically: We haveG (2) R,λ=1 (ω, p) = (G (2) R,0 (ω, p) −1 − U(p)) −1 , which is equivalent to that derived in the random phase approximation (RPA).
Before presenting our result of the density-density spectral function ρ d (ω, p), we briefly mention the expected behavior of the density-density spectral function of (1+1)-dimensional interacting fermions. First, let us consider the free fermion case. If the particle-hole excitation with an energy ω and a momentum p is kinematically forbidden, ρ d (ω, p) is zero. For (1+1)-dimensional free fermions, a particle-hole excitation with an energy ω and a momentum p is kinematically allowed if the following condition is satisfied: ω − (p) ≤ ω ≤ ω + (p), where ω − (p) := |p 2 − 2p F p|/2 and ω + (p) := |p 2 + 2p F p|/2; see Fig. 1(a). Therefore ρ d (ω, p) has its support as shown in Fig. 1(b). On this support, the strength of ρ d (ω, p) does not depend on ω: ρ d (ω, p) = p −1 . Then we consider the interacting fermions. To analyze ρ d (ω, p) for interacting fermions, the inclusion of the nonlinearity of the fermionic dispersion relation is crucial [39], which is not taken into account in the Tomonaga-Luttinger (TL) model [40,41]. The bosonization scheme taking the nonlinearity into account has been developed [39,42,43] and predicted that the qualitative behavior of ρ d (ω, p) drastically deviates from that in the case of free particles: First, ρ d (ω, p) has powerlaw singularities at the edge of its support of the lower-energy side ω = ω − (p). These singularities emerge due to the same mechanism as the singularity appearing in the X-ray absorption rate of metals [44,45], which is caused by the proliferation of low-energy particle-hole pairs. Second, ρ d (ω, p) exhibits power-law suppression at ω = ω + (p) and the support is broaden to ω > ω + (p) because of the contribution from the multi-pair productions. The expected shape of the strength of ρ d (ω, p) is schematically illustrated in Fig. 1(c).
Let us discuss our numerical results of ρ d (ω, p). We set the density to that at the saturation point derived from FRG-DFT: n = ρ s = 1.20 [33]. Figure 2 shows the contour map of ρ d (ω, p) on the (ω, p)-plane. Our spectral function has the same support as that for the free fermions, which is in contradiction to the expectation that ρ d (ω, p) has support in ω > ω + (p). This is possibly due to the approximation made in Eq. (9), where the contribution from the multi-pair diagrams is discarded. To include the contribution from the multi-pair diagrams, the flow of the four point correlation function is needed to be considered, though which is beyond the scope of this Letter.  Fig. 3 The ω dependence of the density-density spectral function when the momentum is fixed to p = p F . The results of the FRG-DFT (solid red line), free particles (green dotted line) and the RPA (blue dashed line) are shown. Fig. 3 is the spectral function ρ d (ω, p) at a fixed momentum p = p F . The results of the RPA and the case of free fermions are also shown for comparison. The spectral function obtained by FRG-DFT reveals the existence of a peak at ω = ω − (p F ) in contrast to that derived from the RPA. A key ingredient for such a peak to emerge is the contribution from the second term in the right-hand side in Eq. (13), which is not included in the RPA. This term has singularities at ω = ω − (p F ), which gives the peak structure in ρ d (ω, p). In the very close region to ω = ω − (p), we found that the spectral function is split into some peaks with slightly different energies, which is different from a simple power-law singularity.

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Summarizing the paper, we demonstrated how the FRG-DFT analysis of the ground and excited states works in a one-dimensional continuum spinless nuclear matter. We obtained the saturation energy from the resultant equation of state, which differs from that obtained using the Monte Carlo simulation by only 2.7%. Moreover, we reproduced a notable feature of the one-dimensional fermion system that the density-density spectral function has singularities at the edge of its support of the lower-energy side. Therefore, our result suggests that the FRG-DFT is a promising way for the analysis of not only ground states but also excited states of the quantum many-body systems. The FRG-DFT is expected to be adapted to various systems because our formalism can be naturally extended to higher-dimensional systems, and systems with internal degrees of freedom [24,25], superfluidity, and finite temperature.
There showed up, however, some unexpected behaviors in our result of the density-density spectral function. At higher-energy side, the broadening of the support of the spectral due to the multi-pair production has not appeared in the present framework. This would be because we miss the 8/10 contribution from the multi-pair production by ignoring the flow of the four-point correlation function. In addition, the spectral function was unexpectedly split into some peaks with slightly different energies in the very close region to the edge of its support of the lower-energy side. The inclusion of the flows of higher-order correlation functions or the use of other approximation schemes such as the KS-FRG [26] is an important future work to see whether these are due to the approximations and/or truncations employed.
Describing the non-uniform systems is another interesting direction. The introduction of a non-uniform chemical potential can realize such systems. Our flow equations (4)-(6) are straightforwardly extended to the case of non-uniform chemical potential and can be used to analyze non-uniform systems.