Quantum-Number Projected Generator Coordinate Method for ²¹Ne with a Chiral Two-Nucleon-Plus-Three-Nucleon Interaction

Abstract

In this paper, we report a study of the low-lying states of deformed ²¹Ne within the framework of the quantum-number projected generator coordinate method (PGCM), starting from a chiral two-nucleon-plus-three-nucleon ($NN+3N$) interaction. The wave functions of states are constructed as a linear combination of a set of axially deformed Hartree–Fock–Bogliubov (HFB) wave functions with different quadrupole deformations. These HFB wave functions are projected onto different angular momenta and the correct neutron and proton numbers for ²¹Ne. The results of the calculations based on the effective Hamiltonians derived by normal-ordering the $3N$ interaction with respect to three different reference states, including the quantum-number projected HFB wave functions for ²⁰Ne, ²²Ne, and an ensemble of them with equal weights, are compared. This study serves as a key step towards ab initio calculations of odd-mass deformed nuclei with the in-medium GCM.

1. Introduction

Studying nuclear low-lying states, including energy spectra and electroweak transition strengths, is crucial for advancing our understanding of nuclear physics [1,2]. It also plays a key role in exploring new physics at the high-precision frontier, such as nonzero electric dipole moments [3,4], single-$\beta$ decay [5], and neutrinoless double-$\beta$ decay [6]. Modeling the low-lying states of light to heavy atomic nuclei directly from the fundamental interactions between nucleons is of great interest for this purpose. Compared to even-even nuclei, the low-lying states of odd-mass nuclei contain richer nuclear structure information because of the interplay of single-particle and collective motions, presenting a considerable challenge for nuclear theory.

The generator coordinate method (GCM) provides an efficient and flexible framework to describe the wave function of a quantum many-body system, represented as a superposition of a set of nonorthogonal basis functions, such as Slater determinants, generated by continuously changing parameters called generator coordinates [7,8]. In nuclear physics, the quantum-number projected GCM (PGCM) has been extensively employed in studies of the energies and transition rates of low-lying states. See, for instance, refs. [9,10,11]. In the recent decade, the PGCM has been implemented into ab initio methods for atomic nuclei. This idea has given birth to a new generation of ab initio methods, including the no-core Monte Carlo shell model [12], the in-medium generator coordinate method (IM-GCM) [13,14] and perturbative PGCM with second-order perturbation theory, abbreviated as PGCM-PT(2) [15,16,17].

In this paper, we extend the PGCM for the low-lying states of an odd-mass deformed nucleus ²¹Ne, starting from a Hamiltonian composed of two-plus-three-nucleon ($NN+3N$) interaction derived from chiral effective field theory (EFT). The PGCM has been extended for odd-mass nuclei based on different energy density functionals (EDFs) [18,19,20,21,22,23,24,25,26]. It is known that EDF-based PGCM approaches may suffer from spurious divergences and discontinuities [26,27,28,29]. In this work, we examine that this Hamiltonian-based framework is free of those problems as the same interaction is applied to both the particle-hole and particle-particle channels when computing the energy overlaps of Hamiltonian kernels. Additionally, we compare the energy spectra of the low-lying states from the PGCM calculations using the effective Hamiltonian normal-ordered with respect to the following three different reference states: ²⁰Ne, ²²Ne, and an ensemble with equal weights.

The article is arranged as follows: in  Section 2, we present the main formulas of PGCM for an odd-mass nucleus, including the generation of an effective Hamiltonian in the normal-ordering two-body (NO2B) approximation, and the construction of nuclear wave functions in the PGCM. The results of calculations for ²¹Ne are presented in Section 3. A short summary and outlook are provided in Section 4.

2. The PGCM for an Odd-Mass Nucleus

2.1. Nuclear Hamiltonian

We employ an intrinsic nuclear A-body Hamiltonian containing both $NN$ and $3N$ interactions,

$${\widehat{H}}_0=\left(1-\frac{1}{A}\right)T^{\left[1\right]}+\frac{1}{A}{T}^{\left[2\right]}+\sum _{i<j}{V}_{ij}^{\left[2\right]}+\sum _{i<j<k}{W}_{ijk}^{\left[3\right]},$$

(1)

where the kinetic term is composed of one- and two-body terms,

$$T^{\left[1\right]}=\sum _{i=1}^A\frac{{\mathbf{p}}_i^2}{2m_N},T^{\left[2\right]}=-\sum _{i<j}\frac{{\mathbf{p}}_i·{\mathbf{p}}_j}{m_N},$$

(2)

with $m_N$ being the mass of nucleon and ${\mathbf{p}}_i$ the momentum of the i-th nucleon.

The above Hamiltonian is normal-ordered with respect to a symmetry-conserving reference state $|\Psi \rangle$, and truncated up to NO2B terms. The resultant Hamiltonian ${\widehat{\mathcal{H}}}_0$ in the NO2B approximation can be written as

$$\begin{array}{c}\hfill {\widehat{\mathcal{H}}}_0=E_0+\sum _{pq}{f}_q^p:A_q^p:+{\displaystyle \frac{1}{4}}\sum _{pqrs}{\Gamma }_{rs}^{pq}:A_{rs}^{pq}:,\end{array}$$

(3)

where the strings of creation and annihilation operators are defined as

$$A_{stu\dots }^{pqr\dots }=a_p^{†}{a}_q^{†}{a}_r^{†}\dots a_u{a}_t{a}_s.$$

(4)

The expectation values of the normal-ordered operators, indicated by $:A_{q\dots }^{p\dots }:$, with respect to the reference state are zero. The zero-body piece of the ${\mathcal{H}}_0$ is just the energy of the reference state

$$\begin{array}{ccc}\hfill E_0\equiv \langle \Psi |{\widehat{H}}_0|\Psi \rangle & =& \sum _{pq}{\overline{t}}_q^p{\gamma }_q^p+{\displaystyle \frac{1}{4}}\sum _{pqrs}{\overline{v}}_{rs}^{pq}{\gamma }_{rs}^{pq}+{\displaystyle \frac{1}{36}}\sum _{pqrstu}{w}_{stu}^{pqr}{\gamma }_{stu}^{pqr},\hfill \end{array}$$

(5)

where the matrix element of the normal-ordered one-body operator (NO1B) is given by

$$\begin{array}{ccc}\hfill f_q^p& =& {\overline{t}}_q^p+\sum _{rs}{\overline{v}}_{rs}^{pq}{\gamma }_s^r+{\displaystyle \frac{1}{4}}\sum _{rstu}{w}_{qtu}^{prs}{\gamma }_{tu}^{rs},\hfill \end{array}$$

(6)

and that of the NO2B operator,

$$\begin{array}{ccc}\hfill {\Gamma }_{rs}^{pq}& =& {\overline{v}}_{rs}^{pq}+\sum _{tu}{w}_{rsu}^{pqt}{\gamma }_u^t.\hfill \end{array}$$

(7)

In the above equations, the symbols hatted with bars are introduced for the sake of brevity,

$$\overline{t}\equiv \left(1-\frac{1}{A}\right)T^{\left[1\right]},\overline{v}\equiv \frac{1}{A}{T}^{\left[2\right]}+V^{\left[2\right]},w\equiv W^{\left[3\right]}.$$

(8)

The last terms in (5)–(7) contributed by the $3N$ interaction are depicted schematically in Figure 1a–c, respectively. Here, we have introduced the density matrices of the (symmetry-conserving) correlated reference state $|\Psi \rangle$,

$$\begin{array}{ccc}\hfill {\gamma }_q^p& \equiv & \langle \Psi |A_q^p|\Psi \rangle ,\hfill \end{array}$$

(9a)

$$\begin{array}{ccc}\hfill {\gamma }_{rs}^{pq}& \equiv & \langle \Psi |A_{rs}^{pq}|\Psi \rangle ,\hfill \end{array}$$

(9b)

$$\begin{array}{ccc}\hfill {\gamma }_{stu}^{pqr}& \equiv & \langle \Psi |A_{stu}^{pqr}|\Psi \rangle .\hfill \end{array}$$

(9c)

Static correlations within the reference state are encoded in the corresponding irreducible density matrices

$$\begin{array}{ccc}\hfill {\lambda }_q^p& \equiv & {\gamma }_q^p,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \hfill \end{array}$$

(10a)

$$\begin{array}{ccc}\hfill {\lambda }_{rs}^{pq}& \equiv & {\gamma }_{rs}^{pq}-\mathcal{A}\left({\lambda }_r^p{\lambda }_s^q\right)={\gamma }_{rs}^{pq}-{\lambda }_r^p{\lambda }_s^q+{\lambda }_s^p{\lambda }_r^q,\hfill \end{array}$$

(10b)

$$\begin{array}{ccc}\hfill {\lambda }_{stu}^{pqr}& \equiv & {\gamma }_{stu}^{pqr}-\mathcal{A}({\lambda }_s^p{\lambda }_{tu}^{qr}+{\lambda }_s^p{\lambda }_t^q{\lambda }_u^r),\hspace{1em}\hspace{1em}\hspace{1em} \hfill \end{array}$$

(10c)

where the antisymmetrization operator $\mathcal{A}$ generates all possible permutations (each only once) of upper indices and lower indices. For a single-reference state, the two-body and three-body irreducible densities ${\lambda }_{rs}^{pq}$ and ${\lambda }_{stu}^{pqr}$ vanish. The expressions for the one-, two-, and three-body irreducible density matrices of a particle-number projected spherical HFB wave function have been given in Ref. [30].

The Hamiltonian ${\widehat{\mathcal{H}}}_0$ is subsequently rewritten into the unnormal-ordered form as follows:

$$\begin{array}{ccc}\hfill {\widehat{\mathcal{H}}}_0& =& {\mathcal{E}}_0+\sum _{pq}{\mathcal{F}}_q^p{A}_q^p+{\displaystyle \frac{1}{4}}\sum _{pqrs}{\mathcal{V}}_{rs}^{pq}{A}_{rs}^{pq},\hfill \end{array}$$

(11)

where the zero-body term is given by

$$\begin{array}{ccc}\hfill {\mathcal{E}}_0& =& E_0-\sum _{pq}\left(f_q^p-\sum _{rs}{\Gamma }_{qs}^{pr}{\gamma }_s^r\right){\gamma }_q^p-{\displaystyle \frac{1}{4}}\sum _{pqrs}{\Gamma }_{qs}^{pr}{\gamma }_{qs}^{pr}\hfill \\ & =& {\displaystyle \frac{1}{36}}\sum _{pqrstu}{w}_{stu}^{pqr}\left({\gamma }_{stu}^{pqr}+36{\gamma }_s^p{\gamma }_t^q{\gamma }_u^r-18{\gamma }_{st}^{pq}{\gamma }_u^r\right)\hfill \\ & =& {\displaystyle \frac{1}{36}}\sum _{pqrstu}{w}_{stu}^{pqr}\left(6{\lambda }_s^p{\lambda }_t^q{\lambda }_u^r-9{\lambda }_{st}^{pq}{\lambda }_u^r+{\lambda }_{stu}^{pqr}\right).\hfill \end{array}$$

(12)

The matrix elements of one-body read

$$\begin{array}{ccc}\hfill {\mathcal{F}}_q^p& =& f_q^p-\sum _{rs}{\Gamma }_{qs}^{pr}{\gamma }_s^r\hfill \\ & =& {\overline{t}}_q^p+{\displaystyle \frac{1}{4}}\sum _{rstu}{w}_{qtu}^{prs}\left({\lambda }_{tu}^{rs}-2{\lambda }_t^r{\lambda }_u^s\right),\hfill \end{array}$$

(13)

and those of two-body terms

$$\begin{array}{ccc}\hfill {\mathcal{V}}_{rs}^{pq}& =& {\Gamma }_{rs}^{pq}={\overline{v}}_{rs}^{pq}+\sum _{tu}{w}_{rstu}^{pqt}{\lambda }_u^t.\hfill \end{array}$$

(14)

In this work, the reference state is chosen as a particle-number projected HFB state for ²⁰Ne and ²²Ne, which are labeled with magic-Ne20 and magic-Ne22, respectively. The obtained effective Hamiltonians ${\mathcal{H}}_0$ are labeled as H0. For comparison, we also derive the Hamiltonian without the $3N$ interaction term in (1), and this Hamiltonian is labeled as H0 (w/o 3N). The expressions for the one-, two-, and three-body density matrices of the particle-number projected HFB state have been given in Ref. [30]. Subsequently, these Hamiltonians are employed into the PGCM calculations.

2.2. Nuclear Wave Functions

The wave functions of low-lying states for an odd-mass nucleus are constructed with the PGCM as follows:

$$|{\Psi }_{\alpha }^{J\pi }\rangle =\sum _c{f}_c^{J\alpha \pi }|NZJ\pi ;c\rangle ,$$

(15)

where $\alpha$ distinguishes the states with the same angular momentum J, and the symbol c is a collective label for the indices $(K,\kappa ,\mathbf{q})$. The basis function with correct quantum numbers ($NZJ\pi$) is given by

$$|NZJ\pi ;c\rangle ={\widehat{P}}_{MK}^J{\widehat{P}}^N{\widehat{P}}^Z|{\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)\rangle ,$$

(16)

where ${\widehat{P}}_{MK}^J$ and ${\widehat{P}}^{N,Z}$ are projection operators that select components with the angular momentum J, neutron number N, and proton number Z [2],

$$\begin{array}{cc}\hfill {\widehat{P}}_{MK}^J& =\frac{2J+1}{8{\pi }^2}\int d\mathsf{\Omega }{D}_{MK}^{J*}\left(\mathsf{\Omega }\right)\widehat{R}\left(\mathsf{\Omega }\right),\hfill \end{array}$$

(17a)

$$\begin{array}{cc}\hfill {\widehat{P}}^{N_{\tau }}& =\frac{1}{2\pi }{\int }_0^{2\pi }d{\phi }_{\tau }{e}^{i{\phi }_{\tau }({\widehat{N}}_{\tau }-N_{\tau })}.\hfill \end{array}$$

(17b)

The operator ${\widehat{P}}_{MK}^J$ extracts the component of angular momentum along the intrinsic axis z defined by K. The Wigner D-function is defined as $D_{MK}^J\left(\mathsf{\Omega }\right)\equiv \langle JM|\widehat{R}\left(\mathsf{\Omega }\right)|JK\rangle =\langle JM|e^{i\varphi {\widehat{J}}_z}{e}^{i\theta {\widehat{J}}_y}{e}^{i\psi {\widehat{J}}_z}|JK\rangle$, where $\mathsf{\Omega }=(\varphi ,\theta ,\psi )$ represents the three Euler angles. The $\widehat{N}={\sum }_k{a}_k^{†}{a}_k$ is particle-number operator. The mean-field configurations $|{\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)\rangle$ for odd-mass nuclei can be constructed as one-quasiparticle excitations on even-even vacua [2],

$$\begin{array}{c}\hfill |{\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)\rangle ={\alpha }_{\kappa }^{†}|{\mathsf{\Phi }}_{\left(\kappa \right)}\left(\mathbf{q}\right)\rangle ,{\alpha }_{\kappa }|{\mathsf{\Phi }}_{\left(\kappa \right)}\left(\mathbf{q}\right)\rangle =0,\end{array}$$

(18)

where $|{\mathsf{\Phi }}_{\left(\kappa \right)}\left(\mathbf{q}\right)\rangle$ is a HFB state with even-number parity labeled with the collective coordinate $\mathbf{q}$. The quasiparticle operators $(\alpha ,{\alpha }^{†})$ are connected to single-particle operators $(a,a^{†})$ via the Bogoliubov transformation [2],

$$\begin{array}{c}\hfill \left(\begin{array}{c}\alpha \\ {\alpha }^{†}\end{array}\right)=\left(\begin{array}{cc}{U}^{†}& V^{†}\\ V^T& U^T\end{array}\right)\left(\begin{array}{c}a\\ a^{†}\end{array}\right),\end{array}$$

(19)

where the $U,V$ matrices are determined by the minimization of particle-number projected energy,

$$\delta \frac{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|\widehat{\mathcal{H}}{\widehat{P}}^N{\widehat{P}}^Z|{\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)\rangle }{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|{\widehat{P}}^N{\widehat{P}}^Z|{\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)\rangle }=0.$$

(20)

Different from the recent study based on a covariant EDF in Ref. [26], where three different schemes were employed to construct the configurations for odd-mass nuclei within the BCS ansatz, in this work we obtain the configurations of one-quasiparticle states with odd-number parity self-consistently by simply exchanging the k-column of the U and V matrices in the HFB wave function [2]:

$$(U_{pk},V_{pk})⟷(V_{pk}^{*},U_{pk}^{*}),$$

(21)

where the index $p={(\tau n\ell jm)}_p\equiv (n_p,{\xi }_p)$ is a label for the spherical harmonic oscillator basis, and k the label for a quasiparticle state. For simplicity, axial symmetry is assumed. In this case, quasiparticle states are labeled with quantum numbers $K^{\pi }$, where $K=\left|m_p\right|$ with $m_p$ being the projection of angular momentum $j_p$ along z-axis, and parity $\pi ={(-1)}^{{\ell }_p}$. The collective coordinate $\mathbf{q}$ is replaced with the dimensionless quadrupole deformation ${\beta }_2$,

$${\beta }_2=\frac{4\pi }{3A{R}^2}\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|r^2{Y}_{20}|{\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)\rangle .$$

(22)

The U and V matrices are determined from the HFB calculation within the scheme of variation after particle-number projection (VAPNP), in which a gradient descent method is exploited to minimize the expectation value of the Hamiltonian with respect to the particle-number projected HFB state. For details, see, for instance, Refs. [2,31,32]. We note that the Kramers’ degeneracy is lifted due to the breaking of time-reversal invariance in the self-consistent HFB calculation.

The weight function $f_c^{J\alpha \pi }$ of the state (15) is determined by the variational principle, which leads to the following Hill–Wheeler–Griffin (HWG) Equation [2,7]:

$$\begin{array}{c}\hfill \sum _{c^{\prime }}\left[{\mathcal{H}}_{c{c}^{\prime }}^{NZJ\pi }-E_{\alpha }^J{\mathcal{N}}_{c{c}^{\prime }}^{NZJ\pi }\right]f_{c^{\prime }}^{J\alpha \pi }=0,\end{array}$$

(23)

where the Hamiltonian kernel and norm kernel are defined by

$$\begin{array}{ccc}\hfill {\mathcal{O}}_{c{c}^{\prime }}^{NZJ\pi }& =& \langle NZJ\pi ;c|\widehat{O}|NZJ\pi ;c^{\prime }\rangle \hfill \\ & =& \frac{2J+1}{8{\pi }^2}\int d\mathsf{\Omega }{D}_{K{K}^{\prime }}^{J*}\left(\mathsf{\Omega }\right){\int }_0^{2\pi }d{\phi }_n\frac{e^{-iN{\phi }_n}}{2\pi }{\int }_0^{2\pi }d{\phi }_p\frac{e^{-iZ{\phi }_p}}{2\pi }\hfill \\ & & \times \langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|\widehat{O}\widehat{R}\left(\mathsf{\Omega }\right)e^{i\widehat{Z}{\phi }_p}{e}^{i\widehat{N}{\phi }_n}|{\mathsf{\Phi }}_{{\kappa }^{\prime }}^{\left(\mathrm{OA}\right)}\left({\mathbf{q}}^{\prime }\right)\rangle ,\hfill \end{array}$$

(24)

with the operator $\widehat{O}$ representing $\widehat{\mathcal{H}}$ and 1, respectively. The parity $\pi$ is defined by the quasiparticle configurations $|{\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)\rangle$.

The HWG Equation (23) for a given set of quantum numbers $\left(NZJ\right)$ is solved in the standard way as discussed in Refs. [2,33]. It is accomplished by diagonalizing the norm kernel ${\mathcal{N}}_{c{c}^{\prime }}^{NZJ\pi }$ first. A new set of basis is constructed using the eigenfunctions of the norm kernel with eigenvalue larger than a pre-chosen cutoff value to remove possible redundancy in the original basis. The Hamiltonian is diagonalized in this new basis. In this way, one is able to obtain the energies $E_{\alpha }^J$ and the mixing weights $f_c^{J\alpha \pi }$ of nuclear states $|{\Psi }_{\alpha }^{J\pi }\rangle$. Since the basis functions $|NZJ;c\rangle$ are nonorthogonal to each other, one usually introduces the collective wave function $g_{\alpha }^{J\pi }(K,\mathbf{q})$ as below

$$g_{\alpha }^{J\pi }(K,\mathbf{q})=\sum _{c^{\prime }}{\left({\mathcal{N}}^{1/2}\right)}_{c,c^{\prime }}^{NZJ\pi }{f}_{c^{\prime }}^{J\alpha \pi },$$

(25)

which fulfills the normalization condition. The distribution of $g_{\alpha }^{J\pi }(K,\mathbf{q})$ over K and $\mathbf{q}$ reflects the contribution of each basis function to the nuclear state $|{\Psi }_{\alpha }^{J\pi }\rangle$.

2.3. Evaluation of Norm and Hamiltonian Overlaps

The energy overlap is defined as the ratio of Hamiltonian overlap to the norm overlap,

$$\begin{array}{ccc}\hfill E(\kappa \mathbf{q},{\kappa }^{\prime }{\mathbf{q}}^{\prime };g)& \equiv & \frac{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|\widehat{\mathcal{H}}\widehat{R}\left(\mathsf{\Omega }\right)e^{i\widehat{Z}{\phi }_p}{e}^{i\widehat{N}{\phi }_n}|{\mathsf{\Phi }}_{{\kappa }^{\prime }}^{\left(\mathrm{OA}\right)}\left({\mathbf{q}}^{\prime }\right)\rangle }{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|\widehat{R}\left(\mathsf{\Omega }\right)e^{i\widehat{Z}{\phi }_p}{e}^{i\widehat{N}{\phi }_n}|{\mathsf{\Phi }}_{{\kappa }^{\prime }}^{\left(\mathrm{OA}\right)}\left({\mathbf{q}}^{\prime }\right)\rangle }\hfill \\ & =& \mathcal{E}+\sum _{pq}{\mathcal{F}}_q^p{\tilde{\rho }}_q^p(\kappa \mathbf{q},{\kappa }^{\prime }{\mathbf{q}}^{\prime };g)+{\displaystyle \frac{1}{4}}\sum _{pqrs}{\mathcal{V}}_{rs}^{pq}{\tilde{\rho }}_{rs}^{pq}(\kappa \mathbf{q},{\kappa }^{\prime }{\mathbf{q}}^{\prime };g),\hfill \end{array}$$

(26)

where g stands for the set of parameters $\{\mathsf{\Omega },{\phi }_n,{\phi }_p\}$. The matrix elements of the mixed one-body densities and pairing tensors, hatted with the symbol ∼, are defined as

$$\begin{array}{ccc}\hfill {\tilde{\rho }}_q^p(\kappa \mathbf{q},{\kappa }^{\prime }{\mathbf{q}}^{\prime };g)& \equiv & \frac{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|a_p^{†}{a}_q\widehat{R}\left(\mathsf{\Omega }\right)e^{i\widehat{Z}{\phi }_p}{e}^{i\widehat{N}{\phi }_n}|{\mathsf{\Phi }}_{{\kappa }^{\prime }}^{\left(\mathrm{OA}\right)}\left({\mathbf{q}}^{\prime }\right)\rangle }{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|\widehat{R}\left(\mathsf{\Omega }\right)e^{i\widehat{Z}{\phi }_p}{e}^{i\widehat{N}{\phi }_n}|{\mathsf{\Phi }}_{{\kappa }^{\prime }}^{\left(\mathrm{OA}\right)}\left({\mathbf{q}}^{\prime }\right)\rangle },\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill {\tilde{\kappa }}^{pq}(\kappa \mathbf{q},{\kappa }^{\prime }{\mathbf{q}}^{\prime };g)& \equiv & \frac{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|a_p^{†}{a}_q^{†}\widehat{R}\left(\mathsf{\Omega }\right)e^{i\widehat{Z}{\phi }_p}{e}^{i\widehat{N}{\phi }_n}|{\mathsf{\Phi }}_{{\kappa }^{\prime }}^{\left(\mathrm{OA}\right)}\left({\mathbf{q}}^{\prime }\right)\rangle }{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|\widehat{R}\left(\mathsf{\Omega }\right)e^{i\widehat{Z}{\phi }_p}{e}^{i\widehat{N}{\phi }_n}|{\mathsf{\Phi }}_{{\kappa }^{\prime }}^{\left(\mathrm{OA}\right)}\left({\mathbf{q}}^{\prime }\right)\rangle },\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill {\tilde{\kappa }}_{rs}(\kappa \mathbf{q},{\kappa }^{\prime }{\mathbf{q}}^{\prime };g)& \equiv & \frac{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|a_s{a}_r\widehat{R}\left(\mathsf{\Omega }\right)e^{i\widehat{Z}{\phi }_p}{e}^{i\widehat{N}{\phi }_n}|{\mathsf{\Phi }}_{{\kappa }^{\prime }}^{\left(\mathrm{OA}\right)}\left({\mathbf{q}}^{\prime }\right)\rangle }{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|\widehat{R}\left(\mathsf{\Omega }\right)e^{i\widehat{Z}{\phi }_p}{e}^{i\widehat{N}{\phi }_n}|{\mathsf{\Phi }}_{{\kappa }^{\prime }}^{\left(\mathrm{OA}\right)}\left({\mathbf{q}}^{\prime }\right)\rangle }.\hfill \end{array}$$

(29)

The matrix elements of the mixed two-body density are determined by the generalized Wick theorem [34],

$$\begin{array}{ccc}\hfill {\tilde{\rho }}_{rs}^{pq}(\kappa \mathbf{q},{\kappa }^{\prime }{\mathbf{q}}^{\prime };g)& \equiv & \frac{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|a_p^{†}{a}_q^{†}{a}_s{a}_r\widehat{R}\left(\mathsf{\Omega }\right)e^{i\widehat{Z}{\phi }_p}{e}^{i\widehat{N}{\phi }_n}|{\mathsf{\Phi }}_{{\kappa }^{\prime }}^{\left(\mathrm{OA}\right)}\left({\mathbf{q}}^{\prime }\right)\rangle }{\langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|\widehat{R}\left(\mathsf{\Omega }\right)e^{i\widehat{Z}{\phi }_p}{e}^{i\widehat{N}{\phi }_n}|{\mathsf{\Phi }}_{{\kappa }^{\prime }}^{\left(\mathrm{OA}\right)}\left({\mathbf{q}}^{\prime }\right)\rangle }\hfill \\ & =& {\tilde{\rho }}_r^p{\tilde{\rho }}_s^q-{\tilde{\rho }}_s^p{\tilde{\rho }}_r^q+{\tilde{\kappa }}^{pq}{\tilde{\kappa }}_{rs}.\hfill \end{array}$$

(30)

With the above relation, the energy overlap can be rewritten as below,

$$\begin{array}{ccc}\hfill E(\kappa \mathbf{q},{\kappa }^{\prime }{\mathbf{q}}^{\prime };g)& =& \mathcal{E}+\sum _{pq}{\mathcal{F}}_q^p{\tilde{\rho }}_q^p+{\displaystyle \frac{1}{2}}\sum _{pq}\left({\tilde{\Gamma }}_q^p{\tilde{\rho }}_q^p+{\tilde{\mathsf{\Delta }}}^{pq}{\tilde{\kappa }}_{pq}\right),\hfill \end{array}$$

(31)

where the matrix elements of the mixed particle-hole field $\tilde{\Gamma }$ and particle-particle field $\tilde{\mathsf{\Delta }}$ are defined as

$${\tilde{\Gamma }}_q^p\equiv \sum _{pqrs}{\mathcal{V}}_{qs}^{pr}{\tilde{\rho }}_s^r,{\tilde{\mathsf{\Delta }}}^{pq}\equiv \frac{1}{2}\sum _{rs}{\mathcal{V}}_{pq}^{rs}{\tilde{\kappa }}^{rs}.$$

(32)

It is efficient to compute the energy overlap directly in the J-coupled scheme.

• The contribution of the one-body term is simply given by

  $$\begin{array}{c}\hfill E^{\left(1B\right)}=\sum _{pq}{\delta }_{j_p{j}_q}{\widehat{j}}_p{\mathcal{F}}_{\left(qp\right)}^0{\tilde{\rho }}_{\left(qp\right)00},\end{array}$$

  (33)

  where ${\widehat{j}}_p\equiv \sqrt{2j_p+1}$. The reduced matrix element is defined as ${\mathcal{F}}_{qp}^0=\langle q|\left|{\mathcal{F}}_0\right||p\rangle$, and the one-body density operator with the two angular momenta coupled to zero [13]

  $${\widehat{\rho }}_{\left(qp\right)00}\equiv \frac{{\left[a_q^{†}{\tilde{a}}_p\right]}_{00}}{\sqrt{2j_q+1}}{\delta }_{{\xi }_q{\xi }_p}$$

  (34)

  with ${\tilde{a}}_{nljm}\equiv {(-1)}^{j+m}{a}_{nlj-m}$.
• The energy by the two-body term consists of $pp$ term

  $$\begin{array}{c}\hfill E_{pp}^{\left(2B\right)}=-{\displaystyle \frac{1}{4}}\sum _{abcd,L}{\mathcal{V}}_{\left(ab\right)\left(cd\right)}^L\sum _{M_L}{(-1)}^{L-M_L}{\tilde{\kappa }}_{\left(ab\right)L{M}_L}^{\left(01\right)}{\tilde{\kappa }}_{\left(cd\right)L-M_L}^{\left(10\right)}\end{array}$$

  (35)

  and $ph$ term

  $$\begin{array}{ccc}\hfill E_{ph}^{\left(2B\right)}& =& {\displaystyle \frac{1}{2}}\sum _{abcd,L}{\mathcal{V}}_{\left(a\overline{b}\right)\left(c\overline{d}\right)}^L\sum _{M_L}{\tilde{\rho }}_{\left(ba\right)L{M}_L}{\tilde{\rho }}_{\left(dc\right)L-M_L}^{†}\hfill \end{array}$$

  (36)

  where the J-coupled mixed density and pairing density are defined as,

  $$\begin{array}{ccc}\hfill {\tilde{\rho }}_{\left(ba\right)L{M}_L}& =& \sum _{m_a{m}_b}{s}_b\langle j_a{m}_a{j}_b-m_b|L{M}_L\rangle {\tilde{\rho }}_b^a,\hfill \end{array}$$

  (37a)

  $$\begin{array}{ccc}\hfill {\tilde{\rho }}_{\left(dc\right)L-M_L}^{†}& =& \sum _{m_c{m}_d}{s}_d\langle j_c{m}_c{j}_d-m_d|L-M_L\rangle {\left({\tilde{\rho }}_d^c\right)}^{†},\hfill \end{array}$$

  (37b)

  $$\begin{array}{ccc}\hfill {\tilde{\kappa }}_{\left(ab\right)L{M}_L}^{\left(01\right)}& =& \sum _{m_a{m}_b}\langle j_a{m}_a{j}_b{m}_b|L{M}_L\rangle {\tilde{\kappa }}^{ab},\hfill \end{array}$$

  (37c)

  $$\begin{array}{ccc}\hfill {\tilde{\kappa }}_{\left(cd\right)L-M_L}^{\left(10\right)}& =& {(-1)}^{L+M_L}\sum _{m_c{m}_d}\langle j_c{m}_c{j}_d{m}_d|L{M}_L\rangle {\left({\tilde{\kappa }}_{cd}\right)}^{†}.\hfill \end{array}$$

  (37d)
  Here, we introduce the symbol $s_b\equiv {(-1)}^{j_b-m_b}$. The symmetry of Clebsch–Gordan coefficient $\langle j_a{m}_a{j}_b-m_b|L{M}_L\rangle$ implies the relation ${\tilde{\rho }}_{\left(ba\right)L{M}_L}={(-1)}^{L-(j_a+j_b)+1}{\tilde{\rho }}_{\left(ab\right)L{M}_L}$. The $ph$-type two-body interaction matrix elements in the J-coupled form are related to those of $pp$-type via the Pandya transformation [35],

  $$\begin{array}{c}\hfill {\mathcal{V}}_{\left(i\overline{j}\right)\left(k\overline{l}\right)}^J=-\sum _L{\widehat{L}}^2\left\{\begin{array}{ccc}{j}_i& j_j& J\\ j_k& j_l& L\end{array}\right\}{\mathcal{V}}_{\left(il\right)\left(kj\right)}^L,\end{array}$$

  (38)

  where the unnormalized $pp$-type two-body matrix elements in the J-coupled form are related to those in M-scheme as follows:

  $$\begin{array}{ccc}\hfill {\mathcal{V}}_{\left(ij\right)\left(kl\right)}^J& =& \sum _{m_i{m}_j{m}_k{m}_l}\langle j_i{m}_i{j}_j{m}_j|JM\rangle \langle j_k{m}_k{j}_l{m}_l|JM\rangle {\mathcal{V}}_{kl}^{ij}.\hfill \end{array}$$

  (39)

The norm overlap of the HFB wave functions with odd-number parity is computed with the Pfaffian formula in Refs. [36,37].

3. Results and Discussion

Effective Hamiltonians

In this work, the $NN$ interaction $V_{ij}^{\left(2\right)}$ in Equation (1) is chosen as the chiral N³LO interaction by Entem and Machleidt [38], denoted as “EM”. We utilize the free-space SRG [39] to evolve the EM interaction to a resolution scale of $\lambda =1.8$ fm${ }^{-1}$. The $3N$ interaction $W_{ijk}^{\left(3\right)}$ is directly constructed with a cutoff of $\mathsf{\Lambda }=2.0$ fm${ }^{-1}$. The Hamiltonian is referred to as EM$\lambda$/$\mathsf{\Lambda }$, i.e., EM1.8/2.0, which was fitted to $NN$ scattering phase shifts, the binding energy of ³H, and the charge radius of ⁴He, see Ref. [40] for details. For the $3N$ interaction, we discard all matrix elements involving states with $e_1+e_2+e_3>14$, where $e_i=2n_i+{\ell }_i$ denotes the number of oscillator quanta in state i. The maximal value of $e_i$ is labeled with $e_{\max }$, and the frequency of the harmonic oscillator basis is chosen as $\hslash \omega =20$ MeV. Starting from the chiral $NN+3N$ interaction, we produce three sets of effective Hamiltonians labeled as magic-Ne20, magic-Ne22, and magic-ENO/EW, respectively. These Hamiltonians are generated by normal-ordering the $3N$ interaction with respect to the reference states of spherical particle-number projected HFB states for ²⁰Ne, ²²Ne, and their ensemble with equal weights. The residual normal-ordered three-body term, c.f. Figure 1d, is neglected.

Table 1. The expectation value (in MeV) of each term in the effective Hamiltonians ${\mathcal{H}}_0$ in (11), derived from the nuclear chiral interaction EM1.8/2.0 on the basis of the spherical particle-number projected HFB state for ²⁰Ne, ²²Ne, and their ensemble with equal weights (ENO/EW), respectively, where $e_{\max }=6$, and $\hslash \omega =20$ MeV.

Interactions	${\mathit{e}}_{\mathbf{max}}$	${\mathit{E}}_0$	$\langle \mathcal{F}\rangle$	$\langle \mathcal{V}\rangle$	${\mathcal{E}}_0$
Ne20	6	$-96.931$	211.205	$-358.229$	50.093
ENO/EW	6	$-101.781$	225.067	$-381.555$	54.706
Ne22	6	$-109.034$	242.241	$-408.614$	57.339
Ne20(w/o 3N)	6	$-177.269$	506.122	$-683.391$	0

lists the expectation value of each term in the three types of effective Hamiltonians ${\mathcal{H}}_0$ in (11) with respect to the corresponding reference state. One can see that in the case without the $3N$ interaction, the unnormal-ordering form of the Hamiltonian ${\mathcal{H}}_0$ returns back to the original Hamiltonian ${\widehat{H}}_0$.

For the case with $e_{\max }=6$, the relative contribution of each term in different effective Hamiltonians to the energy is compared in Table 1. The contribution of the $3N$ interaction to energy, c.f. Figure 1a, is given by

$$\begin{array}{ccc}\hfill E_0^{\left(3\right)}& =& \frac{1}{36}\sum _{pqrstu}{w}_{rsu}^{pqt}\left(6{\lambda }_s^p{\lambda }_t^q{\lambda }_u^r+9{\lambda }_s^p{\lambda }_{tu}^{qr}+{\lambda }_{rsu}^{pqt}\right).\hfill \end{array}$$

(40)

Comparing the $E_0$ value in the third row of Table 1, labeled by Ne20 with the $E_0$ value in the last row, labeled by Ne20 (w/o 3N), one finds the contribution of the $3N$ interaction to the energy $E_0^{\left(3\right)}=80.338$ MeV. On the other hand, the zero-point energy ${\mathcal{E}}_0$ in (12) of the unnormal-ordered Hamiltonian in the first row

$$\begin{array}{ccc}\hfill {\mathcal{E}}_0& =& {\displaystyle \frac{1}{36}}\sum _{pqrstu}{w}_{stu}^{pqr}\left(6{\lambda }_s^p{\lambda }_t^q{\lambda }_u^r-9{\lambda }_{st}^{pq}{\lambda }_u^r+{\lambda }_{stu}^{pqr}\right)\hfill \end{array}$$

(41)

is 50.093 MeV. Their difference gives

$$\frac{1}{2}\sum _{pqrstu}{w}_{stu}^{pqr}{\lambda }_{st}^{pq}{\lambda }_u^r=30.245\mathrm{MeV}.$$

(42)

Since the term depending on ${\lambda }_{stu}^{pqr}$ is much smaller than the other terms, we drop this term out and find the term,

$$\frac{1}{6}\sum _{pqrstu}{w}_{rsu}^{pqt}{\lambda }_s^p{\lambda }_t^q{\lambda }_u^r=65.215\mathrm{MeV}$$

(43)

depending solely on the one-body density, provides the predominant contribution to energies $E_0^{\left(3\right)}$ and ${\mathcal{E}}_0$. Subsequently, we carry out PGCM calculations for low-lying states of ²¹Ne using the above effective Hamiltonians.

Figure 2 and Figure 3 show the change in the effective single-particle energies (ESPEs) with the quadrupole deformation ${\beta }_2$ from the PNP+HFB (VAPNP) calculation for the HFB states, where the ESPE ${\epsilon }_k$ is obtained from the diagonalization of the single-particle Hamiltonian,

$$\begin{array}{ccc}\hfill h_q^p& =& {\mathcal{F}}_q^p+\sum _{rs}{\mathcal{V}}_{qs}^{pr}{\rho }_s^r\hfill \\ & =& {\overline{t}}_q^p+\sum _{rs}{\overline{v}}_{qs}^{pr}{\rho }_s^r+{\displaystyle \frac{1}{4}}\sum _{rstu}{w}_{qsu}^{prt}{\gamma }_{su}^{rt}+\sum _{rstu}{w}_{qsu}^{prt}{\gamma }_s^r({\rho }_u^t-{\gamma }_u^t),\hfill \end{array}$$

(44)

where ${\gamma }_u^t$ is the one-body density of the correlated state, and ${\rho }_s^r$ is the one-body density of mean-field state $|{\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)\rangle$ defined by

$${\rho }_s^r\equiv \langle {\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)|a_r^{†}{a}_s|{\mathsf{\Phi }}_{\kappa }^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)\rangle .$$

(45)

It is shown in Figure 2 that the neutron partner states with the same value of $\left|m\right|$, related by the time-reversal operator, are not degenerate in the HFB states for ²¹Ne with odd-number parity. A comparison is made between the ESPEs obtained by the effective Hamiltonians magic-Ne20 and magic-Ne22. The lifting of Kramers’ degeneracy in the HFB states for ²¹Ne results in non-degeneracy among time-reversal states with identical values of $\left|m\right|$. For clarity, only the energy of one of the time-reversal states is depicted in Figure 3. It is observed that the ESPEs from the two effective Hamiltonians are difficult to distinguish.

Before presenting the projected energy curves with different angular momenta, we examine the issues of singularity and finite steps found in the MR-EDF [27,28,29]. Figure 4 displays the energies (normalized to the converged values) of particle-number projected HFB states for ²¹Ne with $K^{\pi }=1/2^{+}$ and quadrupole deformation ${\beta }_2=0.0$, as a function of the number $N_{\phi }$ of meshpoints in the gauge angle $\phi$. The Fomenko expansion method [41] is used for the particle-number projection, where the k-th gauge angle ${\phi }_k$ is chosen as $2\pi (k/N_{\phi })$. It is observed that the energy remains constant for $N_{\phi }\ge 5$, regardless of whether $N_{\phi }$ is an even or odd number. For comparison, we also show the results from the calculations by artificially multiplying a factor of $1.1$ to the two-body interaction matrix elements for the mixed particle-hole field. In this case, dips are indeed observed at $N_{\phi }=20,40,60,\dots$, corresponding to the situation where the gauge angle ${\phi }_k=\pi /2$ is chosen at the meshpoints with $k=5,10,15,\dots$, respectively. It demonstrates numerically that one should use the same interaction matrix elements for both the particle-hole and particle-particle channels, in which case one will be free of the problem of singularity.

Figure 5 displays the energy curves of particle-number projected HFB states for ²¹Ne with $K^{\pi }=3/2^{+}$ and $1/2^{+}$, respectively. The HFB wave functions are obtained from the PNP-HFB (VAPNP) calculations using the Hamiltonian ${\widehat{\mathcal{H}}}_0$, with the $3N$ interaction normal-ordered with respect to the references of ²⁰Ne, ²²Ne, and their ensemble with equal weights, respectively. It can be observed that the global energy minima of all three curves are located in prolate states with quadrupole deformation ${\beta }_2$ between 0.4 and 0.5. The configurations with $K^{\pi }=3/2^{+}$ are globally lower than those with $K^{\pi }=1/2^{+}$. Furthermore, the configurations based on different Hamiltonians are systematically shifted from each other in energy by less than one MeV.

Figure 6 displays the energies of states with projection onto the correct particle numbers and $J^{\pi }=3/2^{+}$, $5/2^{+}$, and $7/2^{+}$ for ${ }^{21}$Ne with $K^{\pi }=3/2^{+}$ and $K^{\pi }=1/2^{+}$, respectively. The effective Hamiltonians used are H0 with and without the $3N$ interaction. It is shown that the quadrupole deformation parameter ${\beta }_2$ of the prolate energy-minimal state by the H0 (w/o 3N) is smaller than the other two cases. Additionally, the energy curve with the increase in ${\beta }_2$ is also stiffer than that with the $3N$ interaction.

Figure 7 shows a comparison of the energy spectra for ²¹Ne from configuration-mixing calculations with different Hamiltonians. The states with the same $K^{\pi }$ are grouped into the same column. The main features of the two bands with $K^{\pi }=3/2^{+}$ and $1/2^{+}$ are reproduced, although the excitation energies of the states belonging to the $1/2^{+}$ band are systematically overestimated. The mixing of quasiparticle excitation configurations is expected to lower the entire $K^{\pi }=1/2^{+}$ band. In Figure 7c, one can observe that the energy spectra from the magic-Ne20 and magic-Ne22 Hamiltonians are very close to each other. The high-lying states from magic-Ne22 are slightly lower than those from magic-Ne20. In Figure 7b, the energy spectra become more stretched when the $3N$ interaction is turned off. We note that the ground-state energy from the pure PGCM calculation with the chiral $NN+3N$ interaction is systematically underestimated. According to Ref. [14], one may gain more correlation energy by implementing the multi-reference in-medium similarity renormalization group (MR-IMSRG) [42] and increasing the value of $e_{\max }$.

The collective wave functions of the low-lying states with different $J^{\pi }$, and $K^{\pi }=3/2^{+}$ and $1/2^{+}$, by the magic-Ne20 effective Hamiltonian, are displayed in Figure 8. It is shown that in all cases, the wave functions are peaked around ${\beta }_2=0.4$ and do not change significantly with the increase in angular momentum, implying the stability of the shapes in the low-lying states.

4. Conclusions

We have extended PGCM for the low-lying states of ²¹Ne, starting from a chiral two-plus-three-nucleon interaction, and compared the results obtained using effective Hamiltonians derived with the three-nucleon interaction normal-ordered with the following three different reference states: particle-number projected HFB states for ²⁰Ne, ²²Ne, and an ensemble with equal weights. The topology of the potential energy surfaces shows no significant differences among the three effective Hamiltonians, even though they exhibit a systematic energy shift of less than one MeV. The excitation energies of the low-lying states of ²¹Ne by the effective Hamiltonian based on the reference state of ²⁰Ne are slightly larger than those by the effective Hamiltonian of ²²Ne. Furthermore, we demonstrate that the three-nucleon interaction notably affects the low-lying states, i.e., the energy spectrum becomes stretched and the quadrupole collectivity is reduced.

Next, we will extend the in-medium GCM [13,14], namely, the combination of PGCM with the ab initio method of MR-IMSRG, to study the low-lying states of odd-mass nuclei using the consistently evolved operators. The results of this study will be published elsewhere, separately.