Nucleon-Pair Shell Model within a Symmetry Broken Basis

: The nucleon-pair shell model (NPSM) within the framework of a broken symmetry basis is studied. The results demonstrate the validity of such a scheme, which in turn leads to a signiﬁcant reduction in the dimensionality of the multi-pair conﬁguration space. Speciﬁcally, in the case of an axially-deformed basis, the results yield a satisfactory description of the low-lying spectrum, even without the need for applying the projection procedure. In the case of a triaxially-deformed basis, the result of variation after angular momentum projection suggest a potential for using the NPSM to ﬁnd exact solutions in a semi-magic system.


Introduction
Atomic nuclei are self-bounded quantum many-body systems. The modeling of such systems is one of the crucial challenging problems in theoretical nuclear physics. Much progress has been made on this topic over the years, including the Green's function Monte Carlo method [1,2], various full configuration interaction (FCI) approaches, the so-called no-core shell model (NCSM) [3], the symmetry-adapted no-core shell model [4], etc. Most of these methods mentioned above are limited to cases for light nuclei. The coupledcluster method (CC) [5] has been applied to both light nuclei and medium mass nuclei around closed shells. The equation of motion method also been applied to various nuclear problems [6][7][8]. While a mean-field theory can in principle be applied across the whole chart of the nuclides [9], within such a framework there is no guarantee that all important correlations can be addressed equitably. Additionally, while valence space shell models (SM) play an important role in describing and understanding nuclear structures [10][11][12][13][14][15][16], complete SM studies remain out of reach for the heavy mass nuclei because of the huge number of configurations encountered [17]. In addition, several alternative models that use broken rotational symmetry bases have been advanced, with good angular momenta restored by various projection techniques, such as the projected Hartree-Fock [18,19], including variation after projection [18,20], the Hartree Fock-Bogoliubov theories [18,[21][22][23], and projected relativistic mean-field calculations [24], the Monte Carlo shell model [25,26], the projected shell model [27][28][29], a projected configuration interaction [30] as well as the so-called projected generator coordinate methods [31,32].
In 1993, a novel technique known as generalized Wick theory was proposed to calculate the commutators for coupled operators and fermion clusters by Chen et al. [33,34]. Based on this approach, a nucleon pair shell model (NPSM) was developed [35], in which the building blocks of the configuration space were nucleon-pairs. Due to successes of the interacting boson model (IBM) [36], the model space of the NPSM was truncated to the SD pair subspace, and hence this method is called the SD-pair shell model (SDPSM). Within the latter framework it has been shown that the collectivity of the low-lying states can be described well within the SD pair subspace [37][38][39]. Additionally, it has been found that within the framework of the M-scheme for NPSM, the computational time required for calculating the matrix elements can be reduced significantly [40,41].
However, at a more fundamental level, the truncation of the NPSM should be carefully re-examined because the convergence of the theory has not yet been demonstrated. To address this issue, the basis should be constructed for more kinds of collective pairs, with the pair structure coefficients examined using variation procedures [42][43][44][45]. However, the dimensionality of the many-pair model space increases rapidly with the number of collective pairs, hindering its further study. In the NPSM, the nucleon-pairs always have good angular momentum J and third component M, so the many-body basis maintains its the rotational invariance. Therefore, both the total and intermediate angular momentum J or third component M, depending upon the J-and M-scheme adopted, should be good quantum numbers. To work with a complete subspace, a group of multi-pair basis should be considered, and hence in this paper we introduce the concept of symmetry broken pairs within a NPSM framework, and include some early results for semi-magic nuclei.
The remainder of this paper is organized as follows: in Section 2, a review of the NPSM and the angular momentum projection schemes is presented. In Section 3, some initial applications of the concept of symmetry broken pairs within the NPSM framework are explored. Finally, in Section 4, a summary of current results is given along with a suggestion regarding potential next steps.

The Model
In this section we introduce the model and establish labeling conventions, etc. In the following sub-sections to this introduction we establish the overarching framework of the theory by introducing two novel sub-schemes to the NPSM; namely, the axially-deformed NPSM and a triaxially-deformed NPSM. Regarding applications, in this paper, which is a preliminary report, we only discuss semi-magic nuclei.
The general form of the shell model Hamiltonian between identical nucleon pairs is that which is adopted in [18], where the parameters y(j a j b J) are the pair-structure coefficients, which can be determined by variational methods. With this, the multi-pair basis can be constructed in both the M-scheme [40,41] and the J-scheme [35,39], respectively, and In the M-scheme, the multi-pair basis is constructed by taking the inner product of the pair operator, as given by Equation (3). In the J-scheme, on the other hand, the multipair basis is formed by successively coupling the pair operator to angular momentum ranging from L 2 to J. When considering specific collective pairs, a set of multi-pair bases is constructed, and these bases depend on the pair structure coefficients associated with each pair. Such many-pair bases are non-normal and non-orthogonal, the Hamiltonian kernel and norm kernel must be transformed into a ortho-normal basis [35]. The matrix elements of the Hamiltonian can be carried out by using the commutator relations between collective pairs [35,[39][40][41]. Finally, the Hamiltonian can be diagonalized in such a subspace, with the sum running over relatively few of the lowest-lying energies; specifically, minimizing the following average energy [46].

Symmetry Broken Basis
Taking into consideration the broken rotational symmetry of nucleon pairs, the collective pairs with good angular momentum J are mixed. To circumvent this issue, a new model can be constructed using two different approaches. In the following two sub-sections, we introduce these two models. Specifically, in first part of this section, the concept of an axiallydeformed NPSM is introduced, while the second part focuses on a triaxially-deformed NPSM.

Axially-Deformed Basis
First, we introduce what is meant by axially-deformed pairs. In particular, it is determined by summing over collective pairs with the same quantum number M but different angular momenta J, as defined in Equation (3).
where the axial-deformed pairs P M maintain the third component of angular momentum as a good quantum number. A set of multi-pair basis states with fixed total third component M is constructed, after the collective pairs are introduced, The Hamiltonian can then be diagonalized in a subspace with the fixed total third component M.
The B(E2) transitions are given by with e n the effective charge with the reduced transition matrix elements are given by The transition probability T f i is given by [47] T where E γ is the energy of the transition. The half-life of the transition is then simply In this model, no angular momentum projection is needed as the angular momentum J is determined by its third component, namely M.

Triaxially-Deformed Basis
The triaxially-deformed pairs can be constructed by mixing the collective pairs. Unlike the axially-deformed pairs, the triaxially-deformed pairs involve summation over both the angular momentum J as well as the third component M of a set of collective pairs.
where both the angular momentum J and third component M are mixed, and hence break the rotational invariance. The multi-pair basis is given by Since all rotational symmetry is broken, only one basis exists for a given set of collective pairs. In this case, the triaxially-deformed pairs P † i are chosen to be different from each other, with the pair structure coefficients independent for different triaxially-deformed pairs.
The rotational symmetry can be restored by a standard projection process; namely, by introducing a quadrature over orthogonal functions. For an arbitrary rotation operation applied to a state with a well-defined angular momentum state |J M , it only mixes the third component of angular momentum M, and not the J value itself [18].
where D M, K (α, β, γ) is a Wigner D matrix,R(α, β, γ) is the rotation operator over the Euler angles Ω = (α, β, γ) whereinĴ z andĴ y are the generators of rotations about the z and y axes, respectively. The Wigner D functions are the matrix elements of the rotation operator in a basis with good angular momentum quantum numbers, and the Wigner D functions compose of a complete orthogonal set, To carry out the angular momentum projection procedure, an expression for the rotation of basis states is required. For a given rotation applied to a triaxially-deformed basis state, this can be expressed as the inner product of the rotated collective pairs.
where the rotated triaxially-deformed collective pairs are given bŷ In particular, the rotation of symmetry broken basis states can be interpreted in another way. For any basis that breaks rotational symmetry, such as a triaxially-deformed basis |φ , it can be expanded in terms of states with good angular momenta, denoted as |φ; J M i .
where K indicates the quantum number in the initial state, and M denotes the third component in the rotated state. Therefore, the norm kernel and Hamiltonian kernel with a good angular momentum quantum number can be obtained by using the orthogonal relation of the Wigner D function, as given by Equation (18). The norm matrix elements are projected out by the standard equation, The Hamiltonian matrix elements are given Finally, the generalized eigenvalue problems for every J are solved, where the solutions are labeled by r, The B(E2) transitions are given by wherein the reduced transition matrix elements are given by where P J M −µ,M is the angular momentum projection operator. In this work, we only perform variation after projection on a triaxially-deformed basis.

Numerical Results
In this section, we use two real cases to demonstrate the advantages of the symmetry broken basis. In the first sub-section, the low-lying structure of 138 Sn is studied within an axially-deformed basis. In the second half sub-section, the structure of 20 O is studied with the triaxially-deformed basis.

Axially-Deformed Basis
As a preliminary report, the low-lying spectra of 138 Sn are studied for the axiallydeformed basis. We assume that the nucleus 132 Sn is a closed core and the 82-126 shell (0h 9/2 , 1f 7/2 , 1f 5/2 , 2p 3/2 , 2p 1/2 , and 0i 13/2 ) is chosen as the valence neutron shell. As regards the single-particle energies, they are taken from the experiment and given in Table 1. In particular, the low-lying spectra of and 133 Sn are used to fix the neutron SP energies. The only exception is neutron i13/2, whose corresponding levels are still missing. Their values are taken from [48], within which one can find how these can be determined. The two-body matrix elements of the effective interaction are derived from the CD-Bonn NN potential [49]. The strong short-range repulsion term is renormalized by integrating out the high-momentum components above a certain cutoff momentum Λ, see [50]. A smooth potential V low−k , keeping the physics of bare nucleon-nucleon interaction up to Λ, is constructed, and can be used in calculations of shell model effective interactions. In this work, Λ is fixed as 2.2 fm −1 . Then the shell model effective interaction, with the Coulomb force for protons, is carried out within the framework ofQ-box folded-diagram expansion [51][52][53]. In this paper, theQ-box is calculated up to the second order in V low−k .
Usually, the collective pairs with the lowest angular momenta J, such as S (J = 0), D (J =2) and G (J = 4), are considered in the NPSM. For the low-lying states, these collective pairs should be dominant. The dimension of the multi-pair basis increases dramatically when more types of collective pairs are included. Originally those many-pair basis are nonnormal and non-orthogonal, the Hamiltonian matrix elements should be transformed to a normal orthogonal basis. To this end, one must to diagonalize the norm matrix, which is very difficult for large configuration spaces. Alternatively, the dimension of the configuration space will be reduced when the rotation symmetry is broken. For the semi-magic nucleus 138 Sn, the shell model space with M = 0 is 31,124. However, when considering open-shell cases in this region, the dimension of the model space increases dramatically, surpassing the computational capabilities of our current supercomputer facility. In the following results, we specifically limit our configurations to SDG collective pairs. Although the angular momentum J of each collective pair is mixed, the third component M is a good quantum number and the total M of each multi-pair basis is also a good quantum number. The effective Hamiltonian is diagonalized in this subspace and the pair-structure coefficients are determined by searching for the minimum energy.
The lowest energy levels in each axially-deformed subspace with a different third component M in the NPSM are shown in Figure 2, where the shell model calculations are carried out using KSHELL code [54]. It is seen that the results from the NPSM are close to the SM. The shift of 2 + level from the NPSM to SM is 0.059 MeV, and the shift of 4 + is 0.095 MeV. The B(E2) transition probability can be evaluated using the Equation (9), where an effective neutron charge of e n = 0.65e was used. The B(E2; 6 + 1 → 4 + 1 ) is 0.22 W.u., whereas the exact solution given by the shell model is 0.34 W.u. The experimental value is 0.36 (8) W.u. [55]. The corresponding predicted half-life of the 6 + state is 979.66 ns according to NPSM by using Equation (13), and 633.90 ns according to SM. The experimental half-life is 210(45) ns [55]. As we can see, the scheme can provide a satisfied spectrum, but the B(E2) transition is smaller than the exact solution. These results also suggest the low-lying levels of 138 Sn can be described in the SDG pair subspace. Overall, the results suggest that the NPSM can reproduce the spectrum very well with the axially-deformed basis.

Triaxially-Deformed Basis
The NPSM was also studied with the triaxially-deformed basis, which entails rotation symmetry breaking. In general, the symmetry is restored numerically. There are two ways to recover the rotational symmetry: variation before projection (VBP) and variation after projection (VAP). Here, the VAP method is used. The low-lying levels of the 20 O nucleus were studied with the USDA interaction [56]. Only one basis is used for each level in our variation procedure. The results given in Figure 3 show that the NPSM is consistent with the SM. For example, the first 2 + level from SM is −21.70088 MeV, the one from NPSM is also −21.70088 MeV, 2 + level from the SM is the same as for the NPSM, with any difference falling outside the accuracy limitation of the computational facility that was used in the analysis. To generate more levels with the same angular momenta J, more bases should be used in the configuration space. For example, if we want to obtain the second 2 + states, the first 2 + level should be calculated first, which can be described by one basis, after determining the parameters for the first basis, the second basis is considered in the model space, then the parameters of the second basis are fixed by a variation procedure.
As shown in Figure 4, the lowest five levels with J = 2 and J = 4 are calculated in the SM and NPSM with the triaxially-deformed basis. We found that the difference between the NPSM and SM is negligible within the machine precision limit. For the decay 2 + 1 → 0 + 1 , experimental data is available [57] , indicating a B(E2) value of 1.80 W.u. We can also compare the transition probability from our model with the SM one. The shell model predicts a transition probability of 1.52 W.u., while the VAP approach yields 1.50 W.u. The corresponding half-lives are 7.3(3) ps for the experimental data, 8.9 ps for the shell model, and 9.0 ps for the VAP approach. Those results suggest that if the VAP is adopted, there may be no significant difference between the NPSM and SM.

Summary and Discussion
In this paper, the validity of the symmetry broken basis concept in the NPSM was examined. The results show that the NPSM with an axially-deformed basis can reproduce the low-lying spectra of a semi-magic nucleus well.
Further, if the triaxially-deformed basis and variation after projection method are used for a semi-magic nucleus, the difference between the NPSM and SM seems to be negligible. These results suggest that the axially-deformed basis and triaxially-deformed basis may provide a satisfactory description for studying low-lying spectra. The case for a proton-neutron coupled system will be studied in a following publication.
The work was carried out at National Supercomputer Center in Tianjin, and the calculations were performed on TianHe-1(A).

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.

Conflicts of Interest:
The authors declare no conflict of interest.