Ab initio large-scale calculations [8, 9] have recently revealed a remarkably ubiquitous and almost perfect symmetry, the Sp(3, R ) symplectic symmetry, in nuclei that naturally emerges from first principles up through the calcium region (anticipated to hold even stronger in heavy nuclei [14]). Since this symmetry does not mix nuclear shapes, this novel nuclear feature provides important insight from first principles into the physics of nuclei and their low-lying excitations as dominated by only one or two collective shapes—equilibrium shapes with their vibrations—that rotate (Figure 1A).

The SA-NCSM theory [8, 10, 15] capitalizes on these findings and exploits the idea that the infinite Hilbert space can be equivalently spanned by “microscopic” nuclear shapes and their rotations [or symplectic irreducible representations (irreps), subspaces that preserve the symmetry], where “microscopic” refers to the fact that these configurations track with the position and momentum coordinates of each particle. A collective nuclear shape can be viewed as an equilibrium (“static”) deformation and its vibrations (“dynamical” deformations) of the giant-resonance type, as illustrated in the β-γ plots of Figure 1A [8, 16]. A key ingredient of the SA concept is illustrated in Figure 1B, namely, while many shapes relevant to low-lying states are included in typical shell-model spaces (Figure 1B, top), the vibrations of largely deformed equilibrium shapes and spatially extended modes like clustering often lie outside such spaces. The selected model space in the SA-NCSM remedies this, and includes, in a well prescribed way, those configurations. Note that this is critical for enhanced deformation, since spherical and less deformed shapes, including relevant single-particle effects, easily develop in comparatively small model-space sizes.

In this study, we utilize the ab initio SA-NCSM theory [8–10] that is based on the NCSM concept [17, 18] with nuclear interactions typically derived from the chiral EFT (e.g. [2, 19–23]). We use SA-NCSM model spaces, which are reorganized to a correlated basis that respects the shape-preserving Sp (3, R ) symmetry and its embedded symmetry, the deformation-related SU(3) [8–10]. We note that while the model utilizes symmetry groups to construct the basis and calculate matrix elements, descriptions are not limited a priori to any symmetry and can account for significant symmetry breaking.

The SA-NCSM is reviewed in Refs. [9, 10] and has been applied to light and medium-mass nuclei using SU(3)- and Sp(3, R )-adapted bases. The many-nucleon basis states of the SA-NCSM are constructed using efficient group-theoretical algorithms and are labeled according to SU(3) × SU(2) by the proton, neutron and total intrinsic spins, S _p, S _n, and S, respectively, and (λ _ω μ _ω ) quantum numbers with λ _ω = N _z − N _x and μ _ω = N _x − N _y , where N _x + N _y + N _z = N ₀ + N, for a total of N ₀ + N HO quanta distributed in the x, y, and z directions ² . Here, N ₀ ℏΩ is the lowest total HO energy for all particles (“valence-shell configuration”) and NℏΩ (N ≤ N _max) is the additional energy of all particle-hole excitations. Thus, for example, (λ _ω μ _ω ) = (0 0), for which N _x = N _y = N _z , describes a spherical configuration, while N _z larger than N _x = N _y (μ _ω = 0) indicates prolate deformation. In addition, a closed-shell configuration has (0 0). Indeed, spherical shapes, or no deformation, are a part of the SA basis. However, most nuclei—from light to heavy—are deformed in the body-fixed frame, which for 0⁺ states appear spherical in the laboratory frame.

Furthermore, considering the embedding symmetry Sp(3, R ) ⊃SU(3), one can further organize SU(3) deformed configurations into subspaces that preserve Sp(3, R ) symmetry. Each of these subspaces (symplectic irrep, labeled by σ) is characterized by a given equilibrium shape, labeled by a single deformation N _σ (λ _σ μ _σ ). For example, the symplectic irrep N _σ (λ _σ μ _σ ) = 0(8 0) in ²⁰Ne consists of a prolate 0(8 0) equilibrium shape (static deformation) with λ _ω = 8 and μ _ω = 0 in the valence-shell 0p-0h (0-particle-0-hole) subspace, along with many other SU(3) deformed configurations or dynamical deformation (vibrations), such as N _ω (λ _ω μ _ω ) = 2(10 0), 2(6 2), and 8 (16 0), which include particle-hole excitations of the equilibrium shape to higher shells [8, 14, 16]. These vibrations are multiples of 2ℏΩ 1p-1h excitations of the giant-resonance monopole and quadrupole types, that is, induced by the monopole r 2 = ∑ i = 1 A r ⃗ i ⋅ r ⃗ i and quadrupole Q 2 = 16 π / 5 ∑ i = 1 A r i 2 Y 2 ( r ̂ i ) operators, respectively (for further details, see Refs. [10, 24]).

An advantage of the SA-NCSM is that the SA model space can be down-selected from the corresponding ultra-large N _max complete model space to a subset of SA basis states that describe static and dynamical deformation, and within this SA model space the spurious center-of-mass motion can be factored out exactly [25, 26]. Another benefit is the use of group theory for constructing the basis and calculating matrix elements, including the Wigner-Eckart theorem, which allows for calculations with SU(3) reduced matrix elements that depend only on (λ μ), along with computationally efficacious group-theoretical algorithms and data structures, as detailed in Refs. [27–31]. A third advantage is that deformation and collectivity are examined and treated in the approach without the need for breaking and restoring rotational symmetry. The reason is that basis states utilize the SU(3)_{(λ μ)} ⊃ SO(3) _L reduction chain that has a good orbital angular momentum L, whereas all SU(3) reduced matrix elements can be calculated in the simpler canonical SU(3)_{(λ μ)} ⊃ SU(2) _I reduction chain (for details, see Refs. [32, 33]). The canonical reduction chain provides a natural reduction to the x and y degrees of freedom, it is simple to work with, and most importantly, provides a complete labeling of a basis state that includes the single-shell quadrupole moment eigenvalue that measures the deformation along the body-fixed symmetry z-axis [34]. SU(3) reduced matrix elements calculated within this scheme yield, in turn, matrix elements for the SA-NCSM basis by invoking the Wigner-Eckart theorem with the appropriate SU(3)_{(λ μ)} ⊃ SO(3) _L Clebsch-Gordan coefficients that are readily available [32].

We emphasize that all basis states are kept up to some N max C , yielding results equivalent to the corresponding N max C NCSM calculations. Building upon this complete N max C model space, we expand the model space to N _max by adding selected basis states to include only the necessary vibrations of largely deformed equilibrium shapes that lie outside this N max C (such SA-NCSM model spaces are denoted as ⟨ N max C ⟩ N max ).

As introduced in Ref. [11], the EVC method utilizes the fact that if a Hamiltonian is a smooth function of some real-valued parameters, its eigenvectors will also be well-behaved functions of those parameters. In practice, this means that one can use a relatively small number of known wave functions to construct an accurate emulator well-approximated by a low-dimensional manifold, and with it accurately predict observables for an arbitrary chiral potential parameterization [12]. To compute these initial wave functions from first principles, it is advantageous to use SA model spaces that can accommodate deformation, including spatially expanded modes, as well as medium-mass regions.

An advantage of the EVC method is that solutions are achieved by diagonalizing matrices with sizes that are many orders of magnitude smaller than those used in exact calculations. This results in a drastically reduced computational time with practically no discrepancies from the exact results. EVC thus provides a means of generating large samples of nuclear observables from variations in the Hamiltonian parameters. This, in turn, makes computationally intensive statistical analyses, such as sensitivity studies [5, 12], possible. It also allows for a reduced computational load for quantifying uncertainties of ab initio predictions.

In this study, we construct emulators capable of probing collective and clustering features by employing the EVC method with SA model spaces. As illustrated in Table 1, the SA-NCSM reduces the sizes of Hamiltonian matrices by up to four orders of magnitude, or equivalently by more than 97%. The application of EVC to these SA spaces results in an additional reduction of up to 3 more orders of magnitude, or as much as 99%. In this combined framework, the final size of the resulting matrices are as much as 10^–5 times smaller than they would be in the corresponding N _max complete spaces. As the first step, we consider a chiral EFT nucleon-nucleon (NN) interaction truncated at next-to-next-to-leading order (NNLO), which depends on 14 low-energy constants (LECs). It turns out that we can write the chiral Hamiltonian as H ( c ⃗ ) = ∑ i = 0 14 c i h i , where c ⃗ is a vector representing a unique combination of the LECs, h _i are the constituent chiral potentials, h ₀ is the LEC-independent part of the chiral potential plus relative kinetic energy and the Coulomb interaction, and c ₀ = 1.

A state | ψ ( c ⃗ ) 〉 can be well-approximated as a linear combination of known “training” wave functions ∑ j N T α j ( c ⃗ ) | ψ ( c T ⃗ j ) 〉 , where each | ψ ( c T ⃗ j ) 〉 in this study is the lowest-energy eigenvector of H ( c T ⃗ j ) for a given J ^π , c T ⃗ corresponds to a training point in the LEC parameter space, and N _T is the number of training points. The chiral Hamiltonian matrices h _i are constructed in the representation of the training wave functions. These N _T × N _T matrices are used to emulate the wave function for any set of LECs c ⃗ by solving the Schrödinger equation for the unknown α j ( c ⃗ ) as a generalized eigenvalue problem that uses the norm matrix for the training wave functions, M i j = ⟨ ψ ( c T ⃗ i ) | ψ ( c T ⃗ j ) ⟩ .

The new features here are that we generate the emulator for the electric quadrupole moment Q by constructing the Q matrix in the representation of the training eigenvectors (as done for rms radii in Ref. [5]), and that these are calculated using SA model spaces. The quadrupole moment is then approximated by computing ⟨ ψ ( c ⃗ ) | Q | ψ ( c ⃗ ) ⟩ = ∑ i j α i ( c ⃗ ) α j ( c ⃗ ) ⟨ ψ ( c T ⃗ i ) | Q | ψ ( c T ⃗ j ) ⟩ .

An important feature of the training wave functions is that the dominant deformed configurations, or the SU(3) content of the states under consideration, remain practically the same for all of the training wave functions (Figure 3). In addition, the SU(3) content agrees with the probabilities obtained with NNLO_opt in the corresponding N _max complete model space, also shown in Figure 3. This ensures that the same static and dynamical deformed modes govern the physics for all LECs sets under considerations, thereby justifying the use of the same SA selection for all the training wave functions.

Specifically, we find that one SU(3) irrep dominates the dynamics of each state at the 50%–60% level, with several additional configurations each contributing from 1% to 20% depending on the LECs set. Moreover, when the basis states are further organized into Sp(3, R ) irreps, we find that a single symplectic irrep—which contains the dominant SU(3) configurations—contributes at practically the same level from one training wave function to another. For example, the (2 0) symplectic irrep in ⁶Li accounts for 83%–88% of each 1⁺ training wave function, whereas the (2 0) contributes at the 85%–88% level in the case of the 3⁺, out of thirteen available different irreps. Similarly, the probability of the (0 4) irrep in each of the ¹²C training ground states is between 80%–88%, and between 82%–94% for the first 2⁺ states. This is a strong indicator that the emulators are trained on wave functions that retain the symmetry-preserving and symmetry-breaking patterns that are observed in nuclei [8] and that the SA model spaces used in this study are sufficient to capture nuclear collectivity. Indeed, the fact that the Sp(3, R ) symmetry remains a near perfect symmetry for each of the training wave functions, retaining the same shape from one wave function to another, further supports the use of SA selections in the EVC method, or otherwise, the SA model spaces would need to be re-examined.

Another important feature of the training wave functions is that cluster formation is largely unaffected by the choice of interaction parameters. To study this, we project the ⁶Li states onto the α + d system, following Ref. [43]: we use a ground state for each cluster that is renormalized to the most dominant SU(3) configuration, and we adopt R-matrix theory to match the amplitude of the cluster wave function and its derivative to those of the exact Coulomb eigenfunctions at large distances. We note that we are primarily interested in the effect of the LECs on the correlations in the training wave functions; hence, we fix the threshold energy to the experimental one. For the ³ S ₁ partial wave, we observe about 20% variations in the calculated asymptotic normalization coefficients (C ₀ = 1.45–2.07 fm^−1/2) around their average value and 10% variations in the spectroscopic factor, namely, SF = 0.75–0.90 (Figure 4A). This tracks with the ±10% variation in the LECs. For comparison, the NNLO_opt ANC for this particular channel is C ₀ = 1.77 fm^−1/2 with SF = 0.87. Interestingly, the height of the second peak, which is located near the nuclear surface and informs the probability of cluster formation, remains fixed for all the parameterizations and coincides with the one for the NNLO_opt case, only its position slightly varies with the LECs.

While the ³ D ₃ spectroscopic factors (SF = 0.73–0.92, with 0.90 for NNLO_opt) vary approximately at the 15% level (Figure 4B), which is practically the same as for the ³ S ₁ partial wave, α widths of the 3⁺ state range from Γ _α = 6.34 keV–14.05 keV, which is about ±40% from Γ _α = 9.81 keV calculated for this particular channel with NNLO_opt (similarly to the ANCs, we use the experimental threshold energy). We note that the NNLO_opt values for C ₀ and Γ _α are reported for a single channel without taking excitations of the clusters into account (e.g., see Ref. [44]) and should not be compared directly to experiment. Of particular interest for this study is that the LECs sets induce a change in both the location and magnitude of the peak, to which the probability for alpha decay is typically sensitive to.

To summarize, the behavior of the surface peaks in both channels and the nuclear shapes of the 1⁺ and 3⁺ states in ⁶Li (as well as the shapes of the 0⁺ and 2⁺ states in ¹²C) are relatively consistent. This suggests that the terms of the nuclear potential that are independent of the LECs, including parts of the long-range interaction, are largely responsible for cluster formation, along with the development of the nuclear shape [equivalently, almost perfect Sp(3, R ) symmetry]. In contrast, the LECs, which capture the unresolved short-ranged interactions between nucleons, fine-tune collective and clustering features, and affect the associated observables by only a factor, namely, 1.4 for the 1 g . s . + ANCs, 2.2 for the 3 1 + alpha width, and 1.4 for the 3 1 + quadrupole moment in ⁶Li. Similarly, the quadrupole moment for the 2 1 + in ¹²C is affected by a factor of 2.1. While the clustering features are explored in this study for the training points only, the SA-EVC approach—the validation of which is discussed next—enables uncertainty quantification of such collective and reaction observables if the probability distributions for the LECs are available.

To validate the SA-EVC approach, we show that for the quadrupole moments of the ⁶Li 1⁺ ground state and first excited 3⁺ state, as well as for the 3⁺ excitation energy, the emulators provide very accurate results compared to the exact outcomes (Figure 5). The average relative errors over all 256 validation LECs sets are respectively 6.91 × 10⁻², 7.70 × 10⁻⁴, and 1.20 × 10⁻⁴. It is clear that any deviations of the emulators from the expected values are negligible, especially considering that, as mentioned above, the SA selection reduces the Hamiltonian dimension by more than 97%, and the EVC projection by an additional 99% or more.

It is worth noting that the average error for the ground state quadrupole moment is two orders of magnitude larger than that of the 3⁺ state. We note that Q(1⁺) of ⁶Li is very similar in nature to the deuteron quadrupole moment. The extremely small value in both nuclei results from a small mixing of an L = 2 component into the ground state of ⁶Li (and of the deuteron), which is not collective in essence like, e.g., the quadrupole moments of the 3⁺ state in ⁶Li or the 2⁺ state in ¹²C (discussed below). Indeed, the results of Figure 5A reflect the high sensitivity of the underlying NN interaction (and likely 3N forces [46]) to the L = 2 mixing in the ground state wave function.

Similar to ⁶Li, the SA-EVC emulated 2 1 + quadrupole moment and excitation energy for ¹²C are in very close agreement to the exact results (Figure 6). Namely, the average relative errors are given by 1.02 × 10⁻⁴ and 6.72 × 10⁻⁵, respectively. Compared to the average errors reported above for the 3 1 + quadrupole moment and excitation energy for ⁶Li, we find eight and two times improvement in the emulator’s predictions for ¹²C, respectively. The reason is likely related to the much smaller SA selection in ¹²C and the stronger collective nature observed in the low-lying states of ¹²C. Specifically, in ⁶Li the SA-EVC uses thousands of basis states, whereas in ¹²C only hundreds of basis states (see Table 1). We therefore expect the mixing of configurations to exert a more noticeable effect on ⁶Li than on ¹²C. The result is that the eigenvectors of ¹²C vary in fewer directions than those of ⁶Li, suggesting that more training points for ⁶Li may be beneficial to improve errors. While this warrants further study, this speaks to an advantage of merging the SA and EVC frameworks.