Quantum-number projected generator coordinate method for Ne 21 with a chiral two-nucleon-plus-three-nucleon interaction

In this paper, we report a study of the low-lying states of deformed Ne 21 within the framework of 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 di ff erent quadrupole deformations. These HFB wave functions are projected onto di ff erent angular momenta and the correct neutron and proton numbers for Ne 21 . The results of calculations based on the e ff ective Hamiltonians derived by normal-ordering the 3N interaction with respect to three di ff erent reference states, including the quantum-number projected HFB wave functions for Ne 20 , Ne 22 , 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.


I. 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-β decay [5], and neutrinoless double-β 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 oddmass 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 [15][16][17].
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 three different reference states, i.e., particle-number projected Hartree-Fock-Bogliubov (PNP-HFB) wave functions for Ne

A. Nuclear Hamiltonian
We employ an intrinsic nuclear A-body Hamiltonian containing both NN and 3N interactions, V [2]  i j + i< j<k W [3]  i jk , where the kinetic term is composed of one-and two-body pieces, with m N being the mass of nucleon and p i the momentum of the i-th nucleon.
The above Hamiltonian is normal-ordered with respect to a symmetr-conserving reference state |Ψ⟩, and truncated up to NO2B terms.The resultant Hamiltonian Ĥ0 in the NO2B approximation can be written as The strings of creation and annihilation operators are defined as The expectation values of the normal-ordered operators, indicated by :A p... q... :, with respect to the reference state are zero.The zero-body piece of the H 0 is just the energy of the reference state The matrix element of the normal-ordered one-body operator (NO1B) is given by and that of the NO2B operator, The last terms in ( 5), ( 6) and ( 7) contributed by the 3N interaction are depicted schematically in Fig. 1(a), (b), and (c), respectively.Here, we have introduced the density matrices of the (symmetry-conserving) correlated reference state |Ψ⟩, Static correlations within the reference state are encoded in the corresponding irreducible density matrices where the antisymmetrization operator 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 λ pq rs and λ pqr stu vanish.The Hamiltonian Ĥ0 is subsequently rewritten into the unnormal-ordered form as follows, where the zero-body term is given by The matrix elements of one-body read and those of two-body terms In this work, the reference state |Ψ⟩ is chosen as a PNP-HFB state for Ne The expressions for the one-, two-, and three-body density matrices of a spherical PNP-HFB state have been given in Ref. [29].Subsequently, these Hamiltonians are employed into the PGCM calculations.

B. Nuclear wave functions
The wave functions of low-lying states for an odd-mass nucleus are constructed with the PGCM as follows, Here, α distinguishes the states with the same angular momentum J, and the symbol c is a collective label for the indices (K, κ, q).The basis function with correct quantum numbers (NZJπ) is given by where PJ MK and PN,Z are projection operators that select components with the angular momentum J, neutron number N and proton number Z [2], The operator PJ MK extracts the component of angular momentum along the intrinsic axis z defined by K.The Wigner D-function is defined as D J MK (Ω) ≡ ⟨JM| R(Ω) |JK⟩ = ⟨JM| e iϕ Ĵz e iθ Ĵy e iψ Ĵz |JK⟩, where Ω = (ϕ, θ, ψ) represents the three Euler angles.The N = k a † k a k is particle-number operator.The mean-field configurations |Φ (OA) κ (q)⟩ for odd-mass nuclei can be constructed as onequasiparticle excitations on even-even vacua [2], where |Φ (κ) (q)⟩ is a HFB state with even-number parity labeled with the collective coordinate q.The quasiparticle operators (α, α † ) are connected to single-particle operators (a, a † ) via the Bogoliubov transformation [2], where the U, V matrices are determined by the minimization of particle-number projected energy, 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 selfconsistently by simply exchanging the k-column of the U and V matrices in the HFB wave function [2]: where the index p = (τnℓ jm) p ≡ (n p , ξ 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 π , where K = |m p | with m p being the projection of angular momentum j p along z-axis, and parity π = (−1) ℓ p .The collective coordinate q is replaced with the dimensionless quadrupole deformation β 2 , The U and V matrices are determined from the HFB calculation within the scheme of variation after particle-number projection(VAPNP).For details, see, for instance, Ref. [30].We note that the Kramer's degeneracy is lifted due to the breaking of time-reversal invariance in the self-consistent HFB calculation.
The weight function f Jαπ c of the state ( 12) is determined by the variational principle, which leads to the following Hill-Wheeler-Griffin (HWG) equation [2,7], where the Hamiltonian kernel and norm kernel are defined by with the operator Ô representing Ĥ and 1, respectively.The parity π is defined by the quasiparticle configurations |Φ (OA) κ (q)⟩.
The HWG equation (20) for a given set of quantum numbers (NZJ) is solved in the standard way as discussed in Refs.[2,31].It is accomplished by diagonalizing the norm kernel N NZ Jπ cc ′ 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 J α and the mixing weights f Jαπ c of nuclear states |Ψ Jπ α ⟩.Since the basis functions |NZJ; c⟩ are nonorthogonal to each other, one usually introduces the collective wave function g Jπ α (K, q) as below which fulfills the normalization condition.The distribution of g Jπ α (K, q) over K and q reflects the contribution of each basis function to the nuclear state |Ψ Jπ α ⟩.

C. Evaluation of norm and Hamiltonian overlaps
The energy overlap is defined as the ratio of Hamiltonian overlap to the norm overlap, where g stands for the set of parameters {Ω, φ n , φ p }.The matrix elements of the mixed one-body densities and pairing tensors, hatted with the symbol ∼, are defined as κpq (κq, κ ′ q ′ ; g) ≡ ⟨Φ (OA) κrs (κq, κ ′ q ′ ; g) ≡ ⟨Φ (OA) The matrix elements of the mixed two-body density are determined by the generalized Wick theorem [32], With the above relation, the energy overlap can be rewritten as below, where the matrix elements of the mixed particle-hole field Γ and particle-particle field ∆ are defined as 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 where ĵp ≡ 2 j p + 1.The reduced matrix element is defined as F 0 qp = ⟨q| |F 0 | |p⟩, and the one-body density operator with the two angular momenta coupled to zero [13] ρ(qp)00 ≡ a † q ãp 00 2 j q + 1 δ ξ q ξ p (31) with ãnl jm ≡ (−1) j+m a nl j−m .
• The energy by the two-body term consists of pp term and ph term where the J-coupled mixed density and pairing density are defined as, where the unnormalized pp-type two-body matrix elements in the J-coupled form are related to those in M-scheme as follows The norm overlap of the HFB wave functions with odd-number parity is computed with the Pfaffian formula in Ref. [34].

A. Effective Hamiltonians
In this work, the NN interaction V (2)  i j in Eq.( 1) is chosen as the chiral N 3 LO interaction by Entem and Machleidt [35], denoted as "EM".We utilize the free-space SRG [36] to evolve the EM interaction to a resolution scale of λ = 1.8 fm −1 .The 3N interaction W (3)  i jk is directly constructed with a cutoff of Λ = 2.0 fm −1 .The Hamiltonian is referred to as EMλ/Λ, i.e., EM1.8/2.0, which was fitted to NN scattering phase shifts, the binding energy of H 3 , and the charge radius of He 4 .See Ref. [37] 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 + ℓ i denotes the number of harmonic oscillator major shells for the i-th state.The maximal value of e i is labeled with e max , and the frequency of the harmonic oscillator basis is chosen as ℏω = 20 MeV.In this work, e max = 6, and ℏω = 20 MeV are employed.
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 PNP-HFB states for Ne , and their ensemble with equal weights, respectively.The residual normal-ordered three-body term, c.f. Fig. 1(d), is neglected.Table I lists the expectation value of each term in the three types of effective Hamiltonians H 0 in (10) 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 H 0 returns back to the original Hamiltonian Ĥ0 .
The relative contribution of each term in different effective Hamiltonians to the energy is compared in Tab I.The contribution of the 3N interaction to energy, c.f. Fig. 1(a), is given by Comparing the E 0 value in the third row of Tab I, 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 (3) 0 = 80.338 MeV.On the other hand, the zero-point energy E 0 in (11a) of the unnormal-ordered Hamiltonian in the first row Since the term depending on λ pqr stu is much smaller than the other terms, we drop this term out and find the term, 1 6 pqrstu w pqr stu λ p s λ q t λ r u = 65.215MeV, which depends solely on the one-body density, provides the predominant contribution to energies E (3)  0 and E 0 .It implies that the terms depend on higher-order of irreducible densities λ are less important.Subsequently, we carry out PGCM calculations for low-lying states of Ne 21 using the above effective Hamiltonians.Both Fig. 2 and Fig. 3 show the change of the effective single-particle energies (ESPEs) with the quadrupole deformation β 2 from the PNP-HFB (VAPNP) calculation for the HFB states with different K π , where the PNP is carried before variation.The ESPE is obtained from the diagonalization of the single-particle Hamiltonian, where γ t u is the one-body density of the correlated state, and ρ r s is the one-body density of meanfield state |Φ (OA) κ (q)⟩ defined by 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].spectrum becomes stretched, and the quadrupole collectivity is notably reduced.
This study provides a solid basis to extend the framework of IM-GCM [13,14], namely, the combination of PGCM with ab initio method of multi-reference in-medium similarity renormalization group (MR-IMSRG) [40], for the low-lying states of odd-mass nuclei based on consistentlyevolved operators.The results of this study will be published elsewhere, separately.

20 , Ne 22 ,
and an ensemble of them with equal weights.The article is arranged as follows.In Sec.II, 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 Ne 21 are presented in Sec.III.A short summary and outlook are provided in Sec.IV.

FIG. 1 :
FIG. 1: Schematic illustration of the three-nucleon interaction W (red squares), normal-ordered to (a) zerobody, (b) one-body, (c) two-body and (d) three-body terms with a reference state.The n-bdy density matrices γ [n] of the reference state, defined in (8), are represented with black circles.
with equal weights, which are labeled with magic-Ne20, magic-Ne22, and magic-ENO/EW, respectively.The obtained effective Hamiltonians 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).
(01) (ab)LM L = m a m b ⟨ j a m a j b m b |LM L ⟩κ ab , (34c) κ(10) (cd)L−M L = (−1) L+M L m c m d ⟨ j c m c j d m d |LM L ⟩(κ cd ) † .(34d) Here, we introduce the symbol s b ≡ (−1) j b −m b .The symmetry of Clebsch-Gordan coefficient ⟨ j a m a j b − m b |LM L ⟩ implies the relation ρ(ba)LM L = (−1) L−( j a + j b )+1 ρ(ab)LM L .The ph-type twobody interaction matrix elements in the J-coupled form are related to those of pp-type by Pandya transformation [33],

FIG. 2 :
FIG.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 β 2 from the PNP-HFB (VAPNP) calculation for the HFB states with K π = 3/2 + using the effective Hamiltonians magic-Ne20.

Figure 6 21 .Figure 7
Figure6displays the energies of states with projection onto correct particle numbers and J π = 3/2 + , 5/2 + , and 7/2 + for21 Ne with K π = 3/2 + and K π = 1/2 + , respectively.The effective Hamiltonians used are H0 with and without the 3N interaction.It is shown that the quadrupole deformation parameter β 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 of β 2 is also stiffer than that with the 3N interaction.In other words, the 3N interaction helps the development of quadrupole collectivity in Ne 21

FIG. 7 :
FIG. 7: The energy spectra of low-lying states in Ne 21

FIG. 8 :
FIG. 8: The distribution of collective wave functions |g Jπα | 2 , defined in(22), as a function of quadrupole deformation β 2 for the low-lying states of21 Ne with K π = 3/2 + (left panels) and K π = 1/2 + (right panels), respectively.The energy of the ground-state in each case is also provided.

TABLE I :
The expectation value (in MeV) of each term in the three different effective Hamiltonians H 0 with resect to corresponding reference state.