Finite particle number description of neutron matter using the unitary correlation operator and high

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1.Introduction

Properties of neutron matter play a crucial role in determining the structures of not only neutron stars [1-7] but also extremely isospin-asymmetric nuclear systems [8-11]. Though finite nuclei have provided much information about nuclear matter at sub- and around saturation densities [12-19], it is difficult to extrapolate the properties to a higher-density region. To obtain a reliable equation of state (EOS) of nuclear matter in the overall density region, microscopic many-body calculations can be conducted based on bare nucleon–nucleon ($ NN $) interactions. In a short-distance region, the central force of the $ NN $ interaction has a strong repulsion core, while in intermediate- and long-distance regions, there is a strong tensor force [20-22]. The high-momentum components in the nuclear system are mainly induced by the aforementioned two types of forces. The short-range repulsion can decrease the wave function amplitude of the relative motion at short-range distances for the two nucleons in a nucleon pair, while the tensor force can introduce the D-wave state because there can exist strong S–D couplings.
Different approaches have been proposed to treat the $ NN $ correlations originating from the $ NN $ interaction, such as the utilization of correlation functions introducing the Jastrow factor and the renormalization of the $ NN $ interaction using unitary transformation. In our recent variational approach proposed for finite nuclei, namely tensor-optimized antisymmetrized molecular dynamics (TOAMD) [23-27], the central- and tensor-operator-type correlation functions are employed to treat the short-range and tensor correlations, respectively. Within this method, the antisymmetrized-molecular-dynamics (AMD) [28] bases are used as the basis wave functions. With bare $ NN $ interactions, the two aforementioned correlations in s-shell nuclei, as well as their structure and properties, have been successfully investigated [24-27]. Recently, Yamada proposed another variational theory to study the properties of nuclear matter, namely, the tensor-optimized Fermi sphere (TOFS) method [29, 30]. In this method, the nuclear matter is described within a Fermi sphere, and the correlation functions multiplied with the Fermi-sphere state are used to treat the $ NN $ correlations in nuclear matter. The minimal energy of nuclear matter is searched to determine the parameters in the functions. With Argonne V4' (AV4') central $ NN $ potential [22], which can induce short-range correlation, the obtained EOS of symmetric nuclear matter within TOFS agrees with the benchmark results calculated with other theories [29, 30].
The short-range correlation can be also treated by the unitary correlation operator method (UCOM) by employing the unitary correlation operator [31-33]. The short-range repulsion results in decreasing the wave function amplitudes for the two nucleons in a nucleon pair for the relative motion at short-range distances. It has been shown that UCOM can effectively describe the short-range correlation in finite nuclei [31-35]. Under the relativistic and nonrelativistic frameworks within the Hartree-Fock (HF) approximation for a Fermi sphere [36, 37], Hu et al. employed UCOM to further describe the short-range correlation in neutron matter. The obtained EOS of neutron matter agrees well with those calculated by the Relativistic Brueckner–Hartree–Fock theory, which indicates that the short-range correlation in neutron matter can also be treated by UCOM [36, 37].
To describe the tensor correlation, besides employing correlation functions, other approaches have also been successfully proposed in our previous studies on finite nuclei. As is well known that the two-particle two-hole (2p2h) excitations can describe the strong tensor correlation originating from the tensor force in nucleon pairs, the tensor correlation can be described by optimizing the 2p2h configurations, as introduced in the tensor-optimized shell model (TOSM) [33-35]. In the TOSM, the total wave function is superposed by the standard shell-model state and sufficient 2p2h states. By optimizing the 2p2h configurations without truncation of the particle states, the tensor correlation can be treated based on the framework of the shell model. Within the extension of HF theory [38], the 2p2h configurations are employed to describe the high-momentum components of nuclear matter. The obtained results are similar to those of Brueckner–Hartree–Fock (BHF) theory, and the corresponding momentum distribution is found to have high-momentum components because of the 2p2h excitations. In our recently developed approach, namely high-momentum AMD (HM-AMD) [39-43], a high-momentum (HM) nucleon pair is introduced, in which both nucleons for a 2p2h excitation involve a large transfer momentum in an opposite direction [44]. Similar to the TOAMD approach, the AMD bases [28] are used as basis wave functions. However, the 2p2h excitations are employed in HM-AMD to treat the $ NN $ correlations instead of the correlation functions used in TOAMD. This new approach clearly describes the HM components in finite nuclei [39-43].
In the study of nuclear matter, there are several other microscopic theories treating the $ NN $ correlations, starting from the bare $ NN $ interactions, such as BHF and Brueckner-Bethe-Goldstone (BBG) [45-47], self-consistent Green's function (SCGF) [48-51], Fermi hypernetted chain (FHNC) [52-55], auxiliary field diffusion Monte Carlo (AFDMC) [56-58], Green's function Monte Carlo (GFMC) [59, 60], and coupled cluster theory (CC) [61-63]. The BHF approach can be interpreted as the lowest order under the framework of the BBG theory. The ground-state energy of the latter is calculated by employing the linked cluster expansion by means of the G-matrix, which is ordered based on the number of independent hole lines. The diagram corresponding to the hole line with a number n describes the n-body correlations. In the BHF approach, the total energy is calculated within the truncation of two hole-line approximation, which only includes the two-body correlations. In the SCGF approach, the total energy is calculated from the in-medium one-body propagator, which is obtained from the Dyson equation by using the ladder diagram expansion. The FHNC approach is one of the variational methods different from the above nonperturbative ones. With a given trial wave function multiplied by correlation operators, the total energy can be evaluated within the cluster expansion by using the FHNC integral equations [52]. Besides, both the AFDMC and GFMC approaches are extended for nuclear systems under the framework of the quantum Monte Carlo (QMC) method, which has successfully described the ground state of infinite atomic systems [64, 65]. The difference between AFDMC and GFMC mainly exists in their treatments of the spin and isospin channels. The AFDMC samples the spin-isospin states according to the Hubbard-Stratonovich auxiliary fields, while the GFMC sums all the states. Compared to AFDMC, GFMC can treat nuclear systems with more accuracy but smaller mass number. The comparison of benchmark results obtained by Baldo et al. by applying the different above-mentioned theories for nuclear matter with several types of Argonne bare $ NN $ interactions [45] results in similar density dependence of the total energy per particle of nuclear matter. In detail, if only the central potential, including the short-range repulsion, is considered, the calculated results agree with each other. However, when the tensor and spin-orbit forces are additionally included in the $ NN $ potentials, there will be large differences among the density dependences, especially for the symmetric nuclear matter [45], which mainly result from the different treatments of the tensor force as tensor correlations. The intermediate- and long-range properties of the tensor correlation derived from the $ NN $ interaction can induce many-body correlations, which is important for nuclear matter studies. In particular, the saturation property of symmetric nuclear matter has a close relation to tensor correlation. Besides the bare two-body $ NN $ force, there are also studies discussing the effects of many-body forces on the properties of nuclear matter, especially three-body force (TBF) [66-71]. A detailed analysis with several different forms for TBF in Ref. [70] showed that the TBF can have a large impact on the EOS of nuclear matter at high densities. Considering the several uncertainties in many-body forces, we mainly concentrate on the bare two-body $ NN $ interaction to investigate the properties of neutron matter in this study.
In our recent paper, we proposed a new variational approach for nuclear matter description starting from the bare $ NN $ interaction [72]. Under a periodic boundary condition, the nuclear matter is described in a cubic box with finite size. This finite particle number approach has been successfully used in AFDMC, GFMC, CC, and electron systems [73]. Allowing for the different correlations originating from the $ NN $ interaction, UCOM is employed in the new variational method to treat the central short-range correlation induced by the short-range repulsion. In addition, the 2p2h excitations of nucleon pairs are further included to describe the HM components in nuclear matter. The 2p2h configurations are added into the total wave function in the same manner for finite nuclei [33-35]. Both nucleons in the nucleon pair for a 2p2h excitation obtain a transfer momentum with opposite directions, which results in a large relative momentum for the two nucleons of a 2p2h excitation. The aforementioned new framework for nuclear matter is named UCOM+HM [72].
We have previously validated the present UCOM+HM framework to study the neutron and symmetric nuclear matters [72]. From the calculation for nuclear matter with AV4' central potential involving the short-range repulsion, we confirm the applicability of UCOM to treat the central short-range correlation. Besides, the additional 2p2h excitations are found to contribute to the total energy around the normal nuclear density in nuclear matter by several MeVs per particle. The EOSs provided by the present method for the neutron and symmetric nuclear matters are consistent with those of other theories from low- to high-density regions [72], which indicates the reliability of this new variational approach.
In the present study, we concentrate on the effects of tensor correlation and spin-orbit coupling on the properties of neutron matter. The tensor correlation is well known to be important for the symmetric nuclear matter, especially the saturation property. It is also interesting to investigate the tensor correlation in neutron matter. With Argonne V6' (AV6') and V8' (AV8') $ NN $ interactions [22], the effects of not only tensor correlation but also spin-orbit coupling on the EOS of neutron matter are studied, and their respective contributions to the total energy of neutron matter are obtained. In addition, the corresponding EOSs of neutron matter calculated with different $ NN $ interactions are compared with those of several other microscopic many-body theories. In Section 2, a detailed formulism of this new variational framework is presented. In Section 3, the calculated results of neutron matter are discussed. A summary is provided in Section 4.

2.Formulism

2.1.Bare nucleon-nucleon ($ NN $) interaction

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2.1.Bare nucleon-nucleon ($ NN $) interaction

The Argonne V18 (AV18) potential is, currently, one of the most accurate bare $ NN $interactions [20-22]. Because of its sophisticated operatorial structure for some many-body schemes, several simplified versions of AV18 potential have been devised for benchmark purposes, such as AV4', AV6', and AV8' $ NN $ interactions [22]. The AV8' potential can be written as the summation of the first eight components of AV18 potential:

$ V_{ij} = \sum\limits_{k = 1,...,8}v_k(r_{ij})O^k_{ij}. $

(1)

The other six quadratic spin-orbit and four charge-dependent components of AV18 potential are removed for the AV8' version, and the radial functions $ v_k(r_{ij}) $ are readjusted to preserve experimental data on the phase shifts and properties of deuterons. The eight operators $ O^k_{ij} $ in Eq. (1) include the spin, isospin, tensor, and spin-orbit coupling components of nuclear force [22]:

$ \begin{aligned}[b] O^{k = 1,...,8}_{ij} = &1,\ { \sigma}_i\cdot{ \sigma}_j,\ { \tau}_i\cdot{ \tau}_j,\ ({ \sigma}_i\cdot{ \sigma}_j)({ \tau}_i\cdot{ \tau}_j),\\ &S_{ij},\ S_{ij}({ \tau}_i\cdot{ \tau}_j),\\ &{ L}\cdot{ S},\ { L}\cdot{ S}({ \tau}_i\cdot{ \tau}_j). \end{aligned} $

(2)

By removing the last two spin-orbit coupling components and refitting the radial functions, the first six components in Eq. (2) constitute the AV6' potential. Similarly, without considering the tensor and spin-orbit coupling components, the AV4' central potential only includes the first four components of Eq. (2), but the radial functions have been refitted by the deuteron binding energy. With these simplified $ NN $ interactions, namely AV4', AV6', and AV8', the EOSs of neutron matter will be microscopically calculated in the present study and the effects of the tensor correlation and spin-orbit coupling on the results will be discussed.
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2.2.Wave function

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2.2.Wave function

As introduced in our recent study [72], the 0p0h state of neutron matter is defined by the Slater determinant as

$ |{0p0h}\rangle = \frac{1}{\sqrt{N!}}\mathrm{det}\left\{\mathop \prod \limits_{i = 1}^N \phi_{\alpha_i}({ r}_i)\right\}, $

(3)

$ \phi_{\alpha}({ r}) = \frac{1}{\sqrt{L^3}}{\rm e}^{{\rm i}{ k}_{\alpha}\cdot{ r}}\chi^{\sigma}_{\alpha}, $

(4)

$ \langle\phi_{\alpha}|\phi_{\alpha'}\rangle = \delta_{\alpha,\alpha'},$

(5)

where N is the neutron number of neutron matter, $ \phi_{\alpha}({ r}) $ is the plane wave function of a neutron, $ { k}_{\alpha} $ is the momentum, and $ \chi^{\sigma}_{\alpha} $ is the spin component, which can be up or down. The index $ \alpha $ represents the quantum number for both momentum and spin. The neutron matter is described in a cubic box with finite size L, as given in Eq. (4), which is determined by wave function normalization. With the periodic boundary condition $ \phi_{\alpha}({ r}+L\hat{{ x}}) = \phi_{\alpha}({ r}) $ employed in the description of neutron matter, the neutron momentum is discretized by the gap $ \Delta k = \dfrac{2\pi}{L} $. By using an integer vector $ { n} = (n_x, n_y, n_z) $, the momentum of each neutron can be calculated by $ { k} = \dfrac{2\pi}{L}{ n} $. Within a lattice in momentum space corresponding to the cubic box in coordinate space, as shown in our previous study [72], the momentum eigenstate of each neutron can be represented by the grid point.
On account of the periodic boundary condition and symmetry of the wave function in Eq. (3), there exist magic particle numbers corresponding to the shell closures, where the corresponding grid points are $ N_g = 1, 7, 19, 27, 33, 57, ... $. For each grid number $ N_g $, besides the single neutron wave function with momentum $ { k}_i $ ($ i = 1, ..., N_g $) in the 0p0h state, the total particle number becomes $ N = 2N_g $ because of the spin.
For the 2p2h excitations of neutron matter, the 2p2h configurations can be written as

$ |{2p2h}\rangle = |mn;i^{-1}j^{-1}\rangle = a^{\dagger}_ma^{\dagger}_na_ia_j|{0p0h}\rangle, $

(6)

where the indices i and j ($ i, j = 1, ..., N $) represent hole states from lower magnitude of momenta, and the indices m and n ($ m, n>N $) are particle states in 2p2h configurations. The total momentum between two holes and two particles is conserved under the following condition:

${ k}_i+{ k}_j = { k}_m+{ k}_n, $

(7)

${ k}_m = { k}_i+{ q},\quad { k}_n = { k}_j-{ q}. $

(8)

The quantity $ { q} = \dfrac{2\pi}{L}{ n}_q $ is the transfer momentum to link the two particles and two holes in a 2p2h configuration, which are correlated with the relative momentum of the two particle states. Besides, the periodic boundary condition leads to the transfer momentum also being discretized with the mode $ { n}_q = (n_{qx}, n_{qy}, n_{qz}) $. If the transfer momentum is large enough, the HM components in neutron matter can be naturally induced by 2p2h excitations. For example, if the neutron number $ N = 66 $ and normal density $ \rho = 0.17\; \mathrm{fm}^{-3} $, the magnitude of the transfer momentum is approximately $ |{ q}| = \dfrac{2\pi}{L}|{ n}_q| = 5.2\; \mathrm{fm}^{-1} $ with the integer mode $ |{ n}_q| = 6 $. Compared to the empirical Fermi momentum $ k_F = 1.4\; \mathrm{fm}^{-1} $, it is large enough to excite nucleons to HM regions. The HM component is closely related to the descriptions of both short-range and tensor correlations [40-43]. In the present study, a maximum integer for the magnitude of the transfer mode $ { n}_q $ as $ n^{\mathrm{max}}_q $ is employed under the condition $ n^{\mathrm{max}}_q\geqslant|{ n}_q| $. The total number of 2p2h configurations is determined by the quantity $ n^{\mathrm{max}}_q $, which affects the basis space for the present calculation. By increasing this parameter, we first check the energy convergence of neutron matter at normal density.
By superposing the 0p0h and 2p2h configurations, the total wave function $ \Phi $ of neutron matter can be written as [72]

$\Phi = {C_0}|{{0p0h}}\rangle + \sum\limits_{p = 1}^{{N_{{{2p2h}}}}} {{C_p}} |{{2p2h}},p\rangle , $

(9)

where $ N_{{2p2h}} $ is the total number of 2p2h configurations and $ \{C_p\} $ are the configuration amplitudes, which can be variationally determined. The index p represents each configuration, where $ p = 0 $ corresponds to the 0p0h state. In Fig. 1, by taking the neutron magic numbers $ N = 14 $ and $ N = 66 $ as examples, we show the variation in the number $ N_{{2p2h}} $ with the maximum transfer mode $ n^{\mathrm{max}}_q $. As shown in Fig. 1, the total number of 2p2h configurations approximately exponentially increases with the maximum transfer mode. That is, the mode space of the present calculation will increase sharply with the maximum transfer mode.

Figure1. (color online) Total number of 2p2h configurations $N_{{2p2h}}$ varied with the maximum mode $n^{\mathrm{max}}_q$ of the transfer momentum in neutron matter by taking neutron magic numbers $N = 14$ and $N = 66$ as examples.

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2.3.UCOM

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2.3.UCOM

For the short-range repulsion in $ NN $ interaction, UCOM is further used to treat the corresponding central short-range correlation in neutron matter. Within UCOM, the correlated wave function $ \Psi $ considering the short-range correlation in the nuclear system can be obtained by multiplying a unitary correlation operator $ C_r $ with the uncorrelated one $ \Phi $: $ \Psi = C_r\Phi $, where the unitary correlation operator $ C_r $ is defined as [31, 32]

$ C_r = \exp\left(-{\rm i}\sum\limits_{i<j}g_{ij}\right) = \mathop \prod \limits^N_{i<j}c_{r,ij}, $

(10)

where g is a pair-type Hermite generator and $ c_{r,ij} $ is a nucleon pair. The specific form of operator g can be written as

$ g = \frac{1}{2}\{p_rs(r)+s(r)p_r\}, $

(11)

where $ p_r $ is the parallel relative momentum between the nucleons and $ s(r) $ is the variation in the relative wave function. By using the transformation $ \Psi = C_r\Phi $ under UCOM, the transformed Schr?dinger equation for $ \Phi $ can be written as $ \tilde{H}\Phi = E\Phi $, where $ \tilde{H} $ is the Hamiltonian transformed from the original,$ H = T+V $ as

$ \tilde{H}= C^{\dagger}_rHC_r = C^{\dagger}_rTC_r+C^{\dagger}_rVC_r = \tilde{T}+\tilde{V}, $

(12)

$ \tilde{T}= \mathop \sum \limits^N_{i = 1}t_i+\mathop \sum \limits^N_{i<j}u_{ij},\quad \tilde{V} = \mathop \sum \limits^N_{i<j}\tilde{v}_{ij}. $

(13)

With the unitary correlation operator $ C_r $, we first derive the transformed Hamiltonian $ \tilde{H} $ within UCOM. Then, by using the total wave function $ \Phi $ given in Eq. (9), the Hamiltonian matrix $ {\cal H} $ can be calculated for different Hamiltonian components as

$ \begin{align} {\cal H} = \left( \begin{array}{ccc} \langle0|\tilde{H}|0\rangle & \langle0|\tilde{H}|2,p = 1\rangle & ... \\ \langle2,p' = 1|\tilde{H}|0\rangle & \langle2,p' = 1|\tilde{H}|2,p = 1\rangle & ... \\ ... & ... & ... \end{array} \right), \end{align} $

(14)

where $ |0\rangle $ and $ |2,p\rangle $ denote the configurations $ |{0p0h}\rangle $ and $ |{2p2h},p\rangle $, respectively. Within the power method [74-76], we can solve the energy eigenvalue problem for the Hamiltonian matrix with $ N_{{2p2h}}+1 $ dimensions. Then, the configuration amplitudes $ \{C_p\} $ in Eq. (9) can be variationally determined by minimizing the total energy as well as each Hamiltonian component energy. In fact, Eq. (10) shows that as the operator $ C_r $ is a many-body operator, we also confront a many-body problem regarding the transformed Hamiltonian $ \tilde{H} $. However, it is reasonable to take the two-body operator truncation for the short-range correlation case, as discussed in previous studies [31-35]. For the transformed kinetic part $ \tilde{T} $ in Eq. (13), there are two terms, the uncorrelated one-body term $ t_i $ and the correlated two-body term $ u_{ij} $, where the latter originates from the short-range correlation between nucleons and is closely related to both the momentum and angular momentum of relative motion:

$ u(r) = w(r)+\frac{1}{2}\left[p^2_r\frac{1}{2\mu_r(r)}+\frac{1}{2\mu_r(r)}p^2_r\right]+\frac{{ L}^2}{2\mu_{\Omega}(r)r^2}, $

(15)

where the forms for the functions $ w(r) $, $ \mu_r(r) $, and $ \mu_{\Omega}(r) $ are, respectively, [32]

$ w(r) = \frac{\hbar^2}{m}\left(\frac{7}{4}\frac{R^{''2}_{+}(r)}{R^{'4}_{+}(r)}-\frac{1}{2}\frac{R^{'''}_{+}(r)}{R^{'3}_{+}(r)}\right), $

(16)

$ \frac{1}{2\mu_r(r)} = \frac{1}{m}\left(\frac{1}{R^{'2}_{+}(r)}-1\right), $

(17)

$ \frac{1}{2\mu_{\Omega}(r)} = \frac{1}{m}\left(\frac{r^2}{R^{2}_{+}(r)}-1\right), $

(18)

where the function $ R_+(r) $ is usually employed to replace the previous function $ s(r) $ for UCOM calculations. For the two-body potential part $ \tilde{V} $ in Eq. (13), the potential energy $ \tilde{v} $ is also related to the function $ R_+(r) $ and is usually calculated by $ v(R_+(r)) $. The function $ R_+(r) $ indicates the transformed relative distance from the original one r [31, 32]. The relation between the two functions $ R_+(r) $ and $ s(r) $ is

$ \frac{{\rm d}R_+(r)}{{\rm d}r} = \frac{s[R_+(r)]}{s(r)}, $

(19)

$ c^{\dagger}_rrc_r = R_+(r).$

(20)

The function $ R_+(r) $ can decrease the wave function amplitude of the relative motion of the two nucleons in a nucleon pair at short-range distances, which can simulate the effect of the short-range correlation. The specific forms of $ R_+(r) $ for the even (odd) channel with positive (negative) parity are given as [31, 32]

$ R^{\mathrm{even}}_+(r) = r+\alpha\left(\frac{r}{\beta}\right)^{\gamma}\exp\left[-\exp\left(\frac{r}{\beta}\right)\right], $

(21)

$ R^{\mathrm{odd}}_+(r) = r+\alpha\left[1-\exp\left(-\frac{r}{\delta}\right)\right]\exp\left[-\exp\left(\frac{r}{\beta}\right)\right],$

(22)

where $ \alpha $, $ \beta $, $ \gamma $, and $ \delta $ are the parameters variationally determined by minimizing the total energy per particle of the 0p0h state within UCOM for neutron matter, which corresponds to the 0p0h+UCOM calculation. The values of the parameters $ \alpha $, $ \beta $, $ \gamma $, and $ \delta $ are listed in Table 1. For neutron matter, the parameters of $ R_+(r) $ are naturally determined to be the same for AV4', AV6', and AV8' $ NN $ interactions. This is because, for the three aforementioned $ NN $ interactions, the differences among them exist in the isospin-singlet channels $ ^3E $ and $ ^1O $, which are absent in neutron matter. For the two active channels $ ^1E $ and $ ^3O $ in neutron matter, the corresponding central parts are identical. In addition, the tensor and spin-orbit forces are not included in the process of determining the values of the three parameters, i.e., only the 0p0h+UCOM calculation is involved. As the number of 2p2h configurations increases very sharply with the maximum transfer mode, as shown in Fig. 1, it is difficult to determine the parameters with UCOM+HM calculation. Besides, we find that the small changes in the parameters slightly affect the total energy. Hence, we determine the values of the parameters at the 0p0h+UCOM level without much loss of accuracy.

	$\alpha$/fm	$\beta$/fm	$\gamma$	$\delta$/fm
$^1E$	1.33	1.00	0.31
$^3O$	0.65	1.39		0.24

Table1.Values of the parameters $\alpha$, $\beta$, $\gamma$, and $\delta$ in the function $R_+(r)$ for UCOM with $^1E$ and $^3O$ channels in neutron matter.

For the UCOM transformation, the wave function contains two-body correlations. In addition, as shown in Eq. (9), the 2p2h configurations with HM components are included in the total wave function. Physically, there are up to 4p4h correlations involved in the present approach. This new method is called UCOM+HM hereafter.

	$n^{\mathrm{max}}_q$	$E(N=14)$/MeV		$E(N=66)$/MeV
0p0h+UCOM+HM	1	17.629	17.505	17.311		17.766	17.759	17.749
2	17.473	17.298	16.691		17.579	17.542	17.455
3	17.274	17.062	16.109		17.432	17.332	16.933
4	17.212	16.988	15.980		17.323	17.196	16.600
5	17.162	16.932	15.914		17.241	17.100	16.388
6	17.136	16.903	15.884		17.196	17.047	16.293
7	17.124	16.890	15.871		17.168	17.015	16.250
8	17.118	16.884	15.865		17.145	16.988	16.221
$\Delta_{\mathrm{UCOM}}$		24.101	24.101	24.101		23.995	23.995	23.995
$\Delta_{\mathrm{HM}}$		0.827	1.060	2.079		0.682	0.831	1.585

$\rho$/$\mathrm{fm}^{-3}$	AV4'/MeV		AV6'/MeV	AV8'/MeV
0.03	5.93	13.42	?7.49	5.91	13.42	?7.47	?0.04	5.87	13.43	?7.43	?0.05	?0.09
0.05	7.91	19.14	?11.23	7.87	19.13	?11.19	?0.08	7.74	19.15	?11.07	?0.09	?0.25
0.10	11.97	31.23	?19.26	11.88	31.23	?19.17	?0.18	11.49	31.28	?18.84	?0.18	?0.77
0.17	17.20	46.07	?28.87	17.05	46.09	?28.75	?0.30	16.29	46.22	?28.20	?0.28	?1.45
0.20	19.47	52.08	?32.62	19.30	52.12	?32.48	?0.34	18.40	52.27	?31.85	?0.31	?1.70
0.30	27.52	71.35	?43.82	27.30	71.41	?43.66	?0.45	26.01	71.63	?42.81	?0.40	?2.41
0.40	36.48	89.86	?53.38	36.21	89.94	?53.19	?0.54	34.62	90.23	?52.19	?0.46	?2.96
0.50	46.38	107.98	?61.60	46.07	108.08	?61.39	?0.61	44.24	108.44	?60.27	?0.52	?3.41

4.Summary

With our recently proposed variational approach, UCOM+HM, for describing nuclear matter, we calculate the EOS of neutron matter with the AV4', AV6', and AV8' $ NN $ interactions. The neutron matter is described under a finite particle number approach with neutron magic number $ N = 66 $ under a periodic boundary condition. UCOM is used to treat the short-range correlation induced by the short-range repulsion in the $ NN $ interaction. In addition, the 2p2h excitations as HM nucleon pairs are included to describe the high-momentum components of neutron matter, where the two nucleons in a 2p2h configuration involve a large relative momentum. The 2p2h excitations contribute toward treating the non-central tensor and spin-orbit forces in neutron matter.
Under the present UCOM+HM framework, the total energies per particle of neutron matter for the three aforementioned potentials are all well-converged with the increasing 2p2h configurations. By comparing the total energies for neutron matter calculated with the 0p0h, 0p0h+UCOM, and 0p0h+UCOM+HM wave functions, the same conclusion is found, where for neutron matter, the majority correlations originating from the $ NN $ interaction can be treated by UCOM and the residual part is described by the HM pairs, such as tensor correlation and the spin-orbit effect. Besides, as the tensor and spin-orbit forces can induce additional excited HM pairs, the effect of HM pairs is successively increased for the AV4', AV6', and AV8' potentials.
We also obtain the density dependence of all Hamiltonian components as well as the total energy per particle for the three aforementioned potentials. The results indicate that the kinetic and central parts of potential energies mainly contribute to the total energy per particle for neutron matter, while both the tensor and spin-orbit parts of potential energy are relatively small. Besides, the effect of short-range correlation is found to increase with the density of neutron matter.
The obtained EOSs of neutron matter calculated under the present UCOM+HM framework are also compared with those of other microscopic many-body theories with the same bare interactions. The comparison shows that the calculated total energy per particle of neutron matter is similar to those of other approaches, and it is especially consistent with that concerning AFMDC. The absence of the isospin-singlet channels $ ^3E $ and $ ^1O $ in neutron matter leads to the effect of the tensor correlation being small. When the spin-orbit effect is included in the calculations, a more attractive EOS of neutron matter is obtained.
In the future, we will extend the investigations to the properties of symmetric nuclear matter to study both the tensor correlation and spin-orbit effect, because of the significance of the tensor correlation. In addition, it will be interesting to study the effect of a three-body force on EOSs of both neutron and symmetric nuclear matters.

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2.1.Bare nucleon-nucleon ($ NN $) interaction

2.2.Wave function

2.3.UCOM

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Single-particle resonant states with Green’s function method

Formation region of emitted α and heavier particles inside radioactive nuclei

Exact solution of U(5)–O(6) transitional description in interacting boson model with two-particle an

Collision of spinning particles near BTZ black holes

Cosmological scenario based on particle creation and holographic equipartition

Dynamics of scalar fields in an expanding/contracting cosmos at finite temperature

Single-particle space-momentum angle distribution effect on two-pion HBT correlation in high-energy