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Symmetry-Guided Ab Initio Approach to Light and Medium-Mass Nuclei

We use algorithms of computational group theory to perform ab initio configuration-interaction calculations in a SU(3)-based coupling scheme for p-shell nuclei. Details given for 6Li are reflective of similar results found for 8B, 8Be, 12C and 16O, all of which exhibit a strong preference for large quadrupole deformations and a narrow set of intrinsic spin quantum numbers. Our results suggest that a small subspace of symmetry-adapted configurations can very closely approximate the exact solutions while allowing for exact factorization of the center-of-mass degrees of freedom. This, in turns, promises to allows us to extend the reach of the ab initio framework for structure and reaction studies towards sd-shell nuclei and beyond.


Introduction
Theoretical advances achieved in recent years in the development of realistic nuclear potential models [1][2][3][4] along with progress in high performance computing have placed ab initio manyparticle approaches [5][6][7] at the frontier of nuclear structure explorations. The ab initio methods are built on fundamental principles and therefore hold promise to provide predictive capabilities essential for a description of the structure and reactions of unstable and exotic nuclei, many of which are of high interest, e.g. in nucleosynthesis, but remain inaccessible even to experiment.
The no-core shell model (NCSM) [5] is a prominent ab initio method that has achieved a good description of low-lying states and associated spectroscopic properties up through pshell nuclei [8][9][10]. The NCSM typically employs the Lanczos algorithm to solve the eigenvalue problem for a realistic Hamiltonian. Matrix elements of the Hamiltonian are calculated in a many-particle basis of m-scheme states, which are constructed as an antisymmetrized product of the harmonic oscillator single-particle wave functions, and carry the z-component of the total angular momentum along with the total parity as good quantum numbers. The main limitation of this approach, and the predictive power thereof, is inherently coupled with the combinatorial growth in the size of the many-particle model space with increasing nucleon numbers and expansion in the number of single-particle levels in the model space.
We developed an innovative ab initio model, the symmetry-adapted no-core shell model (SA-NCSM), which utilizes a many-particle basis that exploits the physically relevant SU(3)⊃SO(3) group-subgroup chain. The significance of the SU(3) group for a microscopic description of the nuclear collective dynamics can be readily seen from the fact that it is the symmetry group of the Elliott model [11], and a subgroup of the Sp(3, R) symplectic model of nuclear collective motion [12,13]. The concept of symmetry-adapted many-particle basis represents a powerful tool that allows one to winnow a model space to correlations indispensable for modeling important modes of nuclear collective dynamics, specifically nuclear deformation and cluster substructures, thereby overcoming the scale explosion bottleneck of ab initio nuclear structure computations. Hence, the SA-NCSM framework holds promise to expand dramatically the reach of current ab initio approaches toward describing heavier mass nuclei with unprecedented accuracy.

Ab initio Calculations in a SU(3) Symmetry-Adapted Basis
The basis states of the SA-NCSM are constructed in the proton-neutron formalism and are labeled by the physical SU(3)⊃SO(3) subgroup chain quantum numbers (λ µ)κL, and by proton, neutron, and total intrinsic spins S p , S n , and S. The orbital angular momentum L is coupled with S to the total angular momentum J and its projection M J . Each basis state is thus labeled in the SU(3)-scheme as where N signifies the number of harmonic oscillator quanta with respect to the minimal number for a given nucleus. The deformation-related (λ µ) set of quantum numbers labels SU(3) irreducible representations (irreps) and bring forward important information about nuclear shapes and deformation. For example, (00), (λ 0) and (0 µ) describe spherical, prolate and oblate shapes, respectively. The label κ distinguishes multiple occurrences of the same L value in the parent irrep (λ µ). The symbol α schematically denotes the additional quantum numbers needed to unambiguously distinguish between irreps carrying the same N (λ µ) S p S n S quantum numbers. These irreps compose a well-defined subspace with a unique feature that allows for the complete separation of intrinsic and center-of-mass degrees of freedom [14].

Structure of Nuclear Wave Functions
Here we use the SA-NCSM with the bare JISP16 N N interaction [1] to calculate binding energies and determine low-lying eigenstates of 6 Li, 8 Be, 12 C, and 16 O nuclei. The resulting wavefunctions are used to determine values or physical observables such as point-particle root-mean-square (rms) matter radii, electric quadrupole moments, magnetic dipole moments, reduced electromagnetic B(E2) and B(M 1) transition strengths. The expansion of calculated wave functions in a physically relevant SU(3)-scheme basis is illuminating salient features that emerge from the complex dynamics of strongly interacting many-particle systems. To explore the nature of the most important correlations, the probability distribution of intrinsic spins (S p S n S) and deformation-related (λ µ) quantum numbers of SU(3) for the lowest-lying T = 0 states of 6 Li were analyzed. Figure 1a shows the probability distribution of intrinsic spins across their Pauli-allowed deformations in the ground state of 6 Li. This figure illustrates a facet common to low-energy solutions considered: a highly structured and coherent mix of intrinsic spins and SU(3) spatial quantum numbers that has heretofore gone unrecognized in other ab initio studies. These results clearly corroborate the much earlier phenomenological work carried out within the context of the Elliott SU(3) model [11].
Specifically, we found that over 99% of the SA-NCSM eigenstates are accounted for by a small fraction of intrinsic spin combinations. For instance, the lowest-lying eigenstates in 6 Li are almost entirely realized in terms of configurations characterized by the following intrinsic spin (S p S n S) quantum numbers: , with the last one carrying over 90% of the eigenstate. Likewise, the same spin components as in the case of 6 Li are found to dominate the ground state and the lowest 1 + , 3 + , and 0 + excited states of 8 B (Table 1). Similarly, the ground state band of 8 Be and 12 C along with the ground state of 16 O are found to be dominated by many-particle configurations carrying total intrinsic spin of the protons  Figure 1. Probability distribution of proton, neutron, and total intrinsic spins (S p S n S) across their Pauli-allowed (λ µ) deformations (horizontal axis) in the ground 1 + state of 6 Li (a) and the ground 0 + state of 8 Be (b) obtained with JISP16 bare interaction forhΩ = 20 MeV in N max = 10 (a) and N max = 8 (b) full model spaces. The area of each circle is proportional to the total probability of N (λ µ) S p S n S states normalized with respect to the total probability of N (2 + N 0) 1 2 1 2 1 and N (4 + N 0) 0 0 0 stretched-states, respectively. and neutrons equal to zero and one, with the largest contribution due to (S p S n S) = (0 0 0) configurations, This is as illustrated, e.g., in Figure 1b for the ground state of 8 Be.
The mixing of (λ µ) spatial quantum numbers, induced by the SU(3) symmetry breaking terms of realistic interactions, exhibits a remarkably simple coherent pattern. One of its key features is the preponderance of a single 0hΩ SU(3) irrep, the so-called leading irrep, that is, the one characterized by the largest value of the second order SU(3) Casimir invariant,Ĉ 2 , and hence corresponding to a large intrinsic quadrupole deformation [15]. For instance, the low-lying T = 0 states of 6 Li project at 40%-70% level onto the prolate-like 0hΩ SU(3) irrep (2 0). For the considered states of 8 B, 8 Be, 12 C, and 16 O, qualitatively similar dominance of the leading 0hΩ SU(3) irreps is observed -(2 1), (4 0), (0 4), and (0 0) irreps, associated with triaxial, prolate, oblate, and spherical shapes, respectively. Such a clear dominance of the largest 0hΩ deformation within the low-lying states of p-shell nuclei points to the fact that the effective quadrupole-quadrupole interaction of the Elliott SU(3) model of nuclear rotations [11] is realized naturally within the framework of modern realistic interactions.
The analysis reveals that the dominant SU(3)-scheme states at each NhΩ subspace are typically those with (λ µ) quantum numbers such that where λ 0 and µ 0 denote labels of a leading SU(3) irrep at the 0hΩ (N = 0) subspace (Table  1). We conjecture that this coherent pattern of SU(3) quantum numbers reflects the presence of an underlying symplectic Sp(3, R) symmetry of microscopic nuclear collective motion [12] that governs the low-energy structure of both even-even and odd-odd p-shell nuclei. This can be seen from the fact that configurations with a (λ µ) shape that satisfies condition (2) can be determined from the leading SU(3) irrep (λ 0 µ 0 ) through a successive application of a specific subset of the Sp(3, R) symplectic 2hΩ raising operators. This subset is composed of the three operators,Â zz ,Â zx , andÂ xx , that distribute two oscillator quanta in z and x directions, but none in y direction, thereby inducing SU(3)-scheme configurations with ever-increasing values of the Casimir invariantĈ 2 . These three operators are the generators of the Sp(2, R) ⊂ Sp(3, R) subgroup [16], and give rise to a hierarchy of deformed shapes that are energetically favored by an attractive quadrupole-quadrupole interaction [13]. Furthermore, there is an apparent hierarchy among states that fulfill condition (2). In particular, the NhΩ configurations with (λ 0 +N µ 0 ), the so-called stretched states, carry a noticeably higher intensity than the others. For instance, the (2+N 0) stretched states contribute at the 85% level to the ground state of 6 Li. The sequence of the stretched-states, that is, the states with the highest possible deformations, can be formed from many-nucleon correlations of a leading SU(3) irrep by application of thê A zz operator, which is the generator of Sp(1, The revealed pattern of intrinsic spin and deformation mixing supports a symmetry guided truncation of the N max model space. Clearly, one can take advantage of the physical relevance of the SU(3)-scheme basis to winnow the full space down to the most relevant configurations that support the strongest many-nucleon correlations of the system using the underlying Sp(1, R) ⊂ Sp(2, R) ⊂ Sp(3, R) symmetry considerations. As noted previously, this truncation, while significantly reducing the size of the model space, also preserves the ability to factor out exactly the spurious center-of-mass degrees of freedom.

Efficacy of the SU(3) Basis
To probe the adequacy of the symmetry-adapted approach for the ab initio modelling of nuclear structure, we used the calculated eigenstates to determine spectroscopic properties of low-lying T = 0 states of 6 Li using a model space winnowed through symmetry considerations, and compared the outcomes with the corresponding results obtained in the full N max = 12 space.  In the study, Coulomb and bare JISP16 N N interactions were used forhΩ oscillator strengths ranging from 17.5 up to 25 MeV. The selected model space, depicted in Fig. 3 and denoted by 12 [6], incorporates all the configurations carrying up to 6 excitations of oscillator quanta (labeled by [6]) and only a subset of the shapes and a few intrinsic spin components to realize the leading modes of nuclear collective motion for the higher 8hΩ, 10hΩ and 12hΩ configurations. The model space 12 [6] constitutes a small percentage of the full space. For example, the full N max = 12 model space dimension is 4.9 × 10 7 whereas the dimensions of the 12 [6] subspaces with total angular momenta J = 1; J = 2; and J = 3, are 4.3 × 10 5 (0.87%); 6.5 × 10 5 (1.32%); and 8.3 × 10 5 (1.70%), respectively.
The ground state binding energies calculated in 12 [6] for oscillator energyhΩ ranging from 17.5 to 25 MeV represent from 98% up to 98.7% of the full-space binding energy. Furthermore, the excitation energies of 3 + 1 , 2 + 1 , and 1 + 2 states calculated in 12 [6] model space differ only by 20 keV to a few hundreds of keV from the corresponding full-space results, see Fig. 3.
As illustrated in Table 2   less than about 0.3%, or 5% for the µ(2 + 1 0). As the dipole moment is a short-range operator, the results suggest that it may suffice to include all low-lyinghΩ states up to a fixed limit, e.g. N max = 6 for 6 Li, to account for the most important short-range correlations.
To explore how closely one comes to reproducing the important long-range correlations of the full N max = 12 space in terms of nuclear collective excitations within the more restricted 12 [6] space, we compared observables that are sensitive to the tails of the wavefunctions; specifically, the point-particle rms matter radii, the electric quadrupole moments and the reduced electromagnetic B(E2) transition strengths. The results for the rms matter radii, listed in Table 2 forhΩ = 17.5 MeV, agree to within 1%. Similarly, the 12 [6] eigensolutions yield results for these quantities that track very closely with their full [12] space counterparts for all values of hΩ, as can be seen in Fig. 4 and 5. Also, as the B(E2) strengths almost doubles upon increasing the basis space from N max = 6 to N max = 12 -a result that suggests that further expansion of the basis space may be needed to reach the experimental value of 21.8(4.8) e 2 fm 4 , the close correlation between the N max = 12 and 12 [6] results is even more impressive.

Conclusion
We have developed a novel approach that capitalizes on advances being made in ab initio methods while exploiting exact and partial symmetries of nuclear many-body system. Using this approach we have demonstrated that the low-lying eigenstates of 6 Li, 8 Be, 12 C, and 16 O, which were obtained using the JISP16 N N interaction, exhibit a strong dominance of few intrinsic spin