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

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{21}Ne 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

^{21}Ne. 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

^{20}Ne,

^{22}Ne, 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

^{21}Ne, 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:

^{20}Ne,

^{22}Ne, and an ensemble with equal weights.

^{21}Ne 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

^{20}Ne and

^{22}Ne, 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

#### 2.3. Evaluation of Norm and Hamiltonian Overlaps

- The contribution of the one-body term is simply given by$$\begin{array}{c}\hfill {E}^{\left(1B\right)}=\sum _{pq}{\delta}_{{j}_{p}{j}_{q}}{\widehat{j}}_{p}{\mathcal{F}}_{\left(qp\right)}^{0}{\tilde{\rho}}_{\left(qp\right)00},\end{array}$$$${\widehat{\rho}}_{\left(qp\right)00}\equiv \frac{{\left[{a}_{q}^{\u2020}{\tilde{a}}_{p}\right]}_{00}}{\sqrt{2{j}_{q}+1}}{\delta}_{{\xi}_{q}{\xi}_{p}}$$
- The energy by the two-body term consists of $pp$ term$$\begin{array}{c}\hfill {E}_{pp}^{\left(2B\right)}=-{\displaystyle \frac{1}{4}}\sum _{abcd,L}{\mathcal{V}}_{\left(ab\right)\left(cd\right)}^{L}\sum _{{M}_{L}}{(-1)}^{L-{M}_{L}}{\tilde{\kappa}}_{\left(ab\right)L{M}_{L}}^{\left(01\right)}{\tilde{\kappa}}_{\left(cd\right)L-{M}_{L}}^{\left(10\right)}\end{array}$$$$\begin{array}{ccc}\hfill {E}_{ph}^{\left(2B\right)}& =& {\displaystyle \frac{1}{2}}\sum _{abcd,L}{\mathcal{V}}_{\left(a\overline{b}\right)\left(c\overline{d}\right)}^{L}\sum _{{M}_{L}}{\tilde{\rho}}_{\left(ba\right)L{M}_{L}}{\tilde{\rho}}_{\left(dc\right)L-{M}_{L}}^{\u2020}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\tilde{\rho}}_{\left(ba\right)L{M}_{L}}& =& \sum _{{m}_{a}{m}_{b}}{s}_{b}\langle {j}_{a}{m}_{a}{j}_{b}-{m}_{b}|L{M}_{L}\rangle {\tilde{\rho}}_{b}^{a},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\tilde{\rho}}_{\left(dc\right)L-{M}_{L}}^{\u2020}& =& \sum _{{m}_{c}{m}_{d}}{s}_{d}\langle {j}_{c}{m}_{c}{j}_{d}-{m}_{d}|L-{M}_{L}\rangle {\left({\tilde{\rho}}_{d}^{c}\right)}^{\u2020},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\tilde{\kappa}}_{\left(ab\right)L{M}_{L}}^{\left(01\right)}& =& \sum _{{m}_{a}{m}_{b}}\langle {j}_{a}{m}_{a}{j}_{b}{m}_{b}|L{M}_{L}\rangle {\tilde{\kappa}}^{ab},\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\tilde{\kappa}}_{\left(cd\right)L-{M}_{L}}^{\left(10\right)}& =& {(-1)}^{L+{M}_{L}}\sum _{{m}_{c}{m}_{d}}\langle {j}_{c}{m}_{c}{j}_{d}{m}_{d}|L{M}_{L}\rangle {\left({\tilde{\kappa}}_{cd}\right)}^{\u2020}.\hfill \end{array}$$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],$$\begin{array}{c}\hfill {\mathcal{V}}_{\left(i\overline{j}\right)\left(k\overline{l}\right)}^{J}=-\sum _{L}{\widehat{L}}^{2}\left\{\begin{array}{ccc}{j}_{i}& {j}_{j}& J\\ {j}_{k}& {j}_{l}& L\end{array}\right\}{\mathcal{V}}_{\left(il\right)\left(kj\right)}^{L},\end{array}$$$$\begin{array}{ccc}\hfill {\mathcal{V}}_{\left(ij\right)\left(kl\right)}^{J}& =& \sum _{{m}_{i}{m}_{j}{m}_{k}{m}_{l}}\langle {j}_{i}{m}_{i}{j}_{j}{m}_{j}|JM\rangle \langle {j}_{k}{m}_{k}{j}_{l}{m}_{l}|JM\rangle {\mathcal{V}}_{kl}^{ij}.\hfill \end{array}$$

## 3. Results and Discussion

#### Effective Hamiltonians

^{3}LO 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

^{3}H, and the charge radius of

^{4}He, 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 ${e}_{i}=2{n}_{i}+{\ell}_{i}$ denotes the number of oscillator quanta in state i. The maximal value of ${e}_{i}$ is labeled with ${e}_{\mathrm{max}}$, 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

^{20}Ne,

^{22}Ne, and their ensemble with equal weights. The residual normal-ordered three-body term, c.f. Figure 1d, is neglected. Table 1 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 ${\widehat{H}}_{0}$.

`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

^{21}Ne using the above effective Hamiltonians.

^{21}Ne 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

^{21}Ne 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.

^{21}Ne 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.

^{21}Ne 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

^{20}Ne,

^{22}Ne, 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.

`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.

^{21}Ne 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 ${e}_{\mathrm{max}}$.

`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

^{21}Ne, 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

^{20}Ne,

^{22}Ne, 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

^{21}Ne by the effective Hamiltonian based on the reference state of

^{20}Ne are slightly larger than those by the effective Hamiltonian of

^{22}Ne. 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.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EFT | Effective field theory |

IMSRG | In-medium similarity renormalization group |

PGCM | Projected generator coordinate method |

IM-GCM | In-medium generator coordinate method |

MR-CDFT | Multi-reference covariant density functional theory |

HWG | Hill–Wheeler–Griffin |

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**Figure 1.**Schematic illustration of the three-nucleon interaction W (red squares), normal-ordered to (

**a**) zero-body, (

**b**) one-body, (

**c**) two-body, and (

**d**) three-body terms with a reference state. The density matrices $\gamma $ of the reference state are represented with black circles.

**Figure 2.**The effective single-particle energies of neutron states with $m>0$ (solid lines) and $m<0$ (dashed lines) as a function of quadrupole deformation ${\beta}_{2}$ from the PNP + HFB (VAPNP) calculation for the HFB states with ${K}^{\pi}=3/{2}^{+}$ using the effective Hamiltonians

`magic-Ne20`. The states with $\left|m\right|=1/2$, $|m=3/2|$ and $|m=5/2|$ are plotted in black, red and green colors, respectively.

**Figure 3.**The effective single-particle energies of neutron states (with $m>0$) from the PNP+HFB (VAPNP) calculation for the HFB states with ${K}^{\pi}=3/{2}^{+}$ (

**a**) and ${K}^{\pi}=1/{2}^{+}$ (

**b**), where the effective Hamiltonians

`magic-Ne20`(solid) and

`magic-Ne22`(dashed lines) are employed, respectively. The Fermi energies are indicated with dots. The states with $\left|m\right|=1/2$, $|m=3/2|$ and $|m=5/2|$ are plotted in black, red and green colors, respectively.

**Figure 4.**The energies of particle-number projected HFB states for

^{21}Ne with ${K}^{\pi}=1/{2}^{+}$ as a function of quadrupole deformation ${\beta}_{2}$, where the number ${N}_{\phi}$ of meshpoints in the gauge angle $\phi $ is chosen as $20,30$, and 40, respectively. The results from the calculations by multiplying a factor of $1.1$ artificially to the two-body interaction matrix elements for the mixed field $\tilde{\Gamma}$ are given for comparison. The inset shows the energy of spherical state normalized to the converged value as a function of ${N}_{\phi}$.

**Figure 5.**The energies of mean-field states $|{\mathsf{\Phi}}_{\kappa}^{\left(\mathrm{OA}\right)}\left(\mathbf{q}\right)\rangle $ for

^{21}Ne with ${K}^{\pi}=1/{2}^{+},3/{2}^{+}$ as a function of intrinsic quadrupole deformation ${\beta}_{2}$ from the PNP-HFB (VAPNP) calculation, where the three types of Hamiltonians ${\widehat{\mathcal{H}}}_{0}$, i.e.,

`magic-Ne20, magic-Ne22, and magic-ENO/EW`, are employed. The harmonic oscillator basis is chosen as ${e}_{\mathrm{max}}=6,\phantom{\rule{3.33333pt}{0ex}}\hslash \omega =20$ MeV. See the main text for details.

**Figure 6.**The energies of states with projection onto particle numbers ($N=11,Z=10$) and spin-parity ${J}^{\pi}=3/{2}^{+},5/{2}^{+}$ and $7/{2}^{+}$ for ${}^{21}$Ne with quantum numbers ${K}^{\pi}=3/{2}^{+}$ (

**left panels**) and ${K}^{\pi}=1/{2}^{+}$ (

**right panels**) as a function of the quadrupole deformation parameter ${\beta}_{2}$. The results of calculations without the $3N$ interaction are given in (

**a**,

**c**), and those with the $3N$ interaction are given in (

**b**,

**d**). See the main text for details.

**Figure 7.**The energy spectra of low-lying states in

^{21}Ne with ${K}^{\pi}=3/{2}^{+}$ and $1/{2}^{+}$. Experimental data from Ref. [43] are shown in (

**a**). The results by the Hamiltonians

`H0`with and without the $3N$ interaction based on the reference state of

^{20}Ne are displayed in (

**b**). The results by the Hamiltonians

`H0`based on the reference state of

^{20}Ne and

^{22}Ne are compared in (

**c**). The total energy of ground state in each case is also provided. See the main text for details.

**Figure 8.**The distribution of collective wave functions $|{g}_{\alpha}^{J\pi}{|}^{2}$, defined in (25), as a function of quadrupole deformation ${\beta}_{2}$ for the low-lying states of ${}^{21}$Ne with ${K}^{\pi}=3/{2}^{+}$ (

**left panels**) and ${K}^{\pi}=1/{2}^{+}$ (

**right panels**), respectively. The energy of the ground-state in each case is also provided. The results of calculations without the $3N$ interaction are given in (

**a**,

**c**), and those with the $3N$ interaction are given in (

**b**,

**d**). See the main text for details.

**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

^{20}Ne,

^{22}Ne, and their ensemble with equal weights (ENO/EW), respectively, where ${e}_{\mathrm{max}}=6$, 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 |

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**MDPI and ACS Style**

Lin, W.; Zhou, E.; Yao, J.; Hergert, H.
Quantum-Number Projected Generator Coordinate Method for ^{21}Ne with a Chiral Two-Nucleon-Plus-Three-Nucleon Interaction. *Symmetry* **2024**, *16*, 409.
https://doi.org/10.3390/sym16040409

**AMA Style**

Lin W, Zhou E, Yao J, Hergert H.
Quantum-Number Projected Generator Coordinate Method for ^{21}Ne with a Chiral Two-Nucleon-Plus-Three-Nucleon Interaction. *Symmetry*. 2024; 16(4):409.
https://doi.org/10.3390/sym16040409

**Chicago/Turabian Style**

Lin, Wei, Enfu Zhou, Jiangming Yao, and Heiko Hergert.
2024. "Quantum-Number Projected Generator Coordinate Method for ^{21}Ne with a Chiral Two-Nucleon-Plus-Three-Nucleon Interaction" *Symmetry* 16, no. 4: 409.
https://doi.org/10.3390/sym16040409