- 1Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA, United States
- 2Department of Physics, Chalmers University of Technology, Gothenburg, Sweden
- 3Nuclear Physics Institute, Academy of Sciences of the Czech Republic, Řež, Czech Republic
We discuss emulators from the ab initio symmetry-adapted no-core shell-model framework for studying the formation of alpha clustering and collective properties without effective charges. We present a new type of an emulator, one that utilizes the eigenvector continuation technique but is based on the use of symplectic symmetry considerations. This is achieved by using physically relevant degrees of freedom, namely, the symmetry-adapted basis, which exploits the almost perfect symplectic symmetry in nuclei. Specifically, we study excitation energies, point-proton root-mean-square radii, along with electric quadrupole moments and transitions for 6Li and 12C. We show that the set of parameterizations of the chiral potential used to train the emulators has no significant effect on predictions of dominant nuclear features, such as shape and the associated symplectic symmetry, along with cluster formation, but slightly varies details that affect collective quadrupole moments, asymptotic normalization coefficients, and alpha partial widths up to a factor of two. This makes these types of emulators important for further constraining the nuclear force for high-precision nuclear structure and reaction observables.
Introduction
Ab initio approaches to nuclear structure and reactions (for an overview, see Ref. [1]) aim to provide accurate predictions based on few-nucleon forces, such as the ones derived from chiral effective field theory (EFT) [for a review, see e.g. Ref. [2] and references therein]. To achieve this, it is imperative to utilize high-precision nuclear forces that accurately describe nuclear correlations, from short- to long-range correlations, as well as to quantify uncertainties that arise from the nuclear force and the controlled approximations in solving the many-body Schrödinger equation [3]. Such developments use statistical tools, including, for example, Bayesian analysis [4], global sensitivity methods [5], and uncertainty estimates based on regression [6, 7], that sometimes require a large number of computationally intensive calculations which often poses a challenge.
In this paper, we seek to overcome some of these difficulties by combining the symmetry-adapted no-core shell model (SA-NCSM) framework [8–10] with the methodology of eigenvector continuation (EVC) [5, 11, 12]. The SA-NCSM uses a physically relevant basis that, in manageable model spaces, achieves descriptions of light to medium-mass nuclei, including challenging nuclear features, such as collectivity, clustering, and related continuum effects. Similarly, EVC further reduces the sizes of Hamiltonian matrices by mapping them onto much smaller matrices referred to as emulators, low-dimensional manifolds built upon a set of characteristic solutions to the many-body Schrödinger equation. The proposed symmetry-adapted eigenvector continuation (SA-EVC) method opens the door to calculations up through the medium-mass region and studies of collective and clustering nuclear features that otherwise might be computationally infeasible.
With a view toward inferring new knowledge of the nuclear forces relevant to structure and reaction observables, we construct novel SA-EVC emulators to study collective and clustering nuclear properties in 6Li and 12C (an emulator for the 6Li binding energy is validated in Ref. [13]). Because this study focuses on the method validity, we utilize SA-NCSM calculations for a single harmonic oscillator (HO) strength ℏΩ, for which and for a specific parameterization of the chiral potential we show that the observables under consideration converge with the number of HO excitations, including point-proton root-mean-square (rms) radii and E2 transitions. The SA-NCSM utilizes a symplectic Sp(3,
Theoretical methods
Ab initio symmetry-adapted no-core shell model
Ab initio large-scale calculations [8, 9] have recently revealed a remarkably ubiquitous and almost perfect symmetry, the Sp(3,
FIGURE 1. (A) Emergent symplectic symmetry in nuclei: Contribution of the most dominant shape to the 0+ ground state of 20Ne and its rotational band (2+, 4+, 6+, and 8+), as well as to excited 0+ states, pointing to a fragmented giant monopole resonance [8]; for selected states, the deformation distribution within a shape is shown in terms of the shape parameters, the average deformation β and triaxiality angle γ (based on ab initio SA-NCSM calculations with NNLOopt in a model space of 11 HO shells with ℏΩ = 15 MeV inter-shell distance). (B) Schematic illustration of the SA concept shown for 8Be: a smaller model space (square) includes all possible shapes (labeled as “All”) and yields spatially compressed wave functions (top); a larger model space (rectangle in lower panel) accommodates, in a well prescribed way, spatially extended modes (“SA selection”) that are neglected in smaller model spaces. Figure from Ref. [9] under the terms of its CC BY license.
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,
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,
Furthermore, considering the embedding symmetry Sp(3,
An advantage of the SA-NCSM is that the SA model space can be down-selected from the corresponding ultra-large Nmax 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
Eigenvector continuation method in the symmetry-adapted framework
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 Nmax 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
TABLE 1. Model space dimensions (labeled as “Dim”), excitation energy EX, point-proton rms radius rrms, electric quadrupole moment Q, and B (E2 ↑) transition strengths from the ground state (g.s.) to the first excited state of 6Li and12C, calculated with NNLOopt and ℏΩ = 15 MeV in SA and complete model spaces. ⟨2All⟩813 denotes an Nmax = 2 model space with all symplectic irreps (complete), 13 Sp (3,
A state
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
Results and discussions
The results presented in this paper use the SA-NCSM in an Sp(3,
For the EVC calculations, we use NT = 32 training points within the 14-dimensional parameter space for NNLO. We restrict the ranges of the LECs to lie within ±10% of their values for NNLOopt [21] and adopt the regularization for NNLOopt. We sample training points using a randomly seeded latin hypercube design, and validate the emulators for 256 points that are different from the training points but within the same range of the LECs.
The SA-EVC results start with SA model spaces that are reduced by three to four orders of magnitude compared to the corresponding Nmax complete model space (or, equivalently, NCSM calculations), as outlined in Table 1. Moreover, the associated observables are in good agreement for SA and complete model spaces, with differences that are typically comparable to differences resulting from varying ℏΩ (see Ref. [8], supplemental material). Specifically, for the example of NNLOopt, we report in Table 1 excitation energies, point-proton rms radii, electric quadrupole moments, and B (E2 ↑) transition strengths between the two lowest energy states of 6Li and 12C. We also show that for the SA spaces used to train the emulators all of the above observables are converged with Nmax (Figure 2).
FIGURE 2. Convergence with Nmax of the quadrupole moments Q(J), point-proton rms radii rrms(J), excitation energies EX, and B (E2 ↑) transition strengths for the two lowest-lying states in (A) 6Li and (B) 12C. Observables are computed with the NNLOopt parameterization for ℏΩ = 15 MeV in SA model spaces reported in Table 1.
Thus, for example, as shown in Table 1, collectivity-driven observables agree within 0.3%–2.9%, and radii agree at the sub-percent level. The largest deviation is observed for the 6Li 1+ quadrupole moment, however, it is important that its sign and very small magnitude are reproduced in both calculations. Furthermore, such differences are expected to decrease in richer model spaces; indeed, in a series of benchmark studies for light nuclei such as 4He, 6Li, 12C, and 16O (reviewed in Ref. [9]), we have shown that the SA-NCSM uses significantly smaller model spaces in comparison to the corresponding large complete Nmax model spaces without compromising the accuracy for various observables (including electron scattering form factors [39] and sum rules [35]), as well as for effective inter-cluster potentials [31]. Reference [9] has also shown that for light nuclei, the SA-NCSM is in reasonable agreement with other ab initio approaches, such as hyperspherical harmonics [40, 41], the NCSM [17, 18], and quantum Monte Carlo [42].
Collectivity and clustering of training wave functions
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 NNLOopt in the corresponding Nmax 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.
FIGURE 3. The largest SU (3) probability amplitudes (solid lines) as a function of emulator training LECs sets for (A) 6Li 1+ ground state and (B) 6Li
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,
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 3S1 partial wave, we observe about 20% variations in the calculated asymptotic normalization coefficients (C0 = 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 NNLOopt ANC for this particular channel is C0 = 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 NNLOopt case, only its position slightly varies with the LECs.
FIGURE 4. α + d (A) 3S1-wave and (B) 3D3-wave as functions of the relative distance r, computed from the 6Li training wave functions for SA model spaces reported in Table 1. The spread of the curves is given by the ± 10% variation in the LECs. The case for NNLOopt is shown in black.
While the 3D3 spectroscopic factors (SF = 0.73–0.92, with 0.90 for NNLOopt) vary approximately at the 15% level (Figure 4B), which is practically the same as for the 3S1 partial wave, α widths of the 3+ state range from Γα = 6.34 keV–14.05 keV, which is about ±40% from Γα = 9.81 keV calculated for this particular channel with NNLOopt (similarly to the ANCs, we use the experimental threshold energy). We note that the NNLOopt values for C0 and Γα 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,
Validation of the symmetry-adapted eigenvector continuation
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.
FIGURE 5. Exact vs. SA-EVC observables in 6Li (blue circles) for the quadrupole moment Q of (A) the 1+ ground state and (B) the first excited 3+ state, as well as (C) for the excitation energy EX of the 3+ state, in ⟨2All⟩813 SA model spaces and for ℏΩ = 15 MeV. Also shown is the agreement between the exact and emulated values to guide the eye (red line), and experimental results (vertical green line) where available. Insets show 5%-regions surrounding reported experimental data [45] or the NNLOopt result where data is not available [a 50%-region is used for the very small Q in (A)].
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
FIGURE 6. The same as in Figure 5 but for (A) the quadrupole moment Q and (B) the excitation energy of the first 2+ state in 12C, calculated in 63 SA model spaces and for ℏΩ = 15 MeV. Insets show 5%-regions surrounding reported experimental data [47].
Conclusion
We have for the first time combined the framework of the SA-NCSM with the EVC procedure into the SA-EVC method for studies of collective and clustering observables. This builds upon earlier SA-NCSM explorations that have shown that an Sp(3,
Further, we demonstrate that SA-EVC emulators trained on SA model spaces are capable of accurately predicting such observables as the LECs are varied, while further reducing the dimensions of operator matrices by an additional 2-3 orders of magnitude. Combined with the initial reduction provided by the SA-NCSM, the emulator matrices have a dimension as much as 10–5 times smaller than the corresponding Nmax complete model spaces. They are small enough to perform linear algebra operations using a single CPU thread on a standard laptop without difficulty. Moreover, the SA-EVC approach will be critical for nuclei beyond the lightest systems; thus, e.g. in 20Ne, the complete Nmax = 8 model space has dimension of 1.52 × 1011, while the ab initio SA-NCSM solutions are achieved when using 112 million basis states for Jπ = 0+, 2+, 4+. This can be further reduced to emulators of dimension 102 especially given the predominance of a single symplectic irrep in the ground-state rotational band of this nucleus. Comparing the emulator results to exact calculations performed in the same SA spaces, we find that the average relative errors are typically 10–4. A larger error
In addition to validating the SA-EVC procedure, we show that the symmetry patterns and clustering features in the emulator training wave functions do not respond strongly to variations in the LECs. Across all of the training wave functions, there is a single nuclear shape (approximate symplectic symmetry) that accounts for 81%–94% of the total probability. Furthermore, the dominance of important SU(3) configurations is preserved from one training wave function to another. Projecting the training wave functions for 6Li onto the α + d system, we find that the likelihood of cluster formation in both the 3S1- and 3D3-wave channels is largely unaffected by the choice of LECs. Spectroscopic factors, ANCs and α-widths extracted from the cluster wave functions all vary within relatively narrow ranges around their average values, ranges that track reasonably well with the 10% variation of the LECs. This suggests that the part of the nuclear potential that is independent of the LECs and is practically the same for all chiral potentials (up to the regularization and related cutoffs employed) provides the dominant features of the wave function, such as Sp(3,
In order to better understand the relationships between collectivity and clustering explored in this study, and how both relate to the underlying nuclear forces, sensitivity analyses are required. As we enter the era of high-precision nuclear physics, this is also an important step towards constructing accurate interactions, with quantified uncertainties. We note that properly accounting for clustering features is important for the ab initio modeling of nuclear reactions, and related processes from fusion to fission. The SA-EVC method provides a clear and now verified framework for generating the huge number of chiral parameterizations required for such analyses. Hence, the door is now open to perform ab initio calculations with quantified uncertainties that emerge from the interaction and the controlled many-body approximations, from exotic light nuclei up to medium-mass isotopes, as well from spherical to highly enhanced collective and clustering modes.
Data availability statement
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.
Author contributions
All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Acknowledgments
We acknowledge invaluable discussions with Jerry P. Draayer, George Rosensteel, David Rowe, and Daniel Langr. This work was supported in part by the U.S. National Science Foundation (PHY-1913728, PHY-2209060), the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation program (Grant agreement No. 758027), the Czech Science Foundation (22-14497S). KSB greatly appreciates the financial support of a research fellowship from the Louisiana Board of Regents. This work benefited from high performance computational resources provided by LSU (www.hpc.lsu.edu), the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility operated under Contract Nos. DE-AC02-05CH11231, as well as the Frontera computing project at the Texas Advanced Computing Center, made possible by National Science Foundation award OAC-1818253.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s note
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Footnotes
1Throughout the paper, we will refer to the selected SA-NCSM model spaces as SA model spaces.
2We follow the notations of Ref. [15].
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Keywords: ab initio symmetry-adapted no-core shell model, nuclear collectivity, nuclear clustering, eigenvector continuation, emulators, 6Li, 12C
Citation: Becker KS, Launey KD, Ekström A and Dytrych T (2023) Ab initio symmetry-adapted emulator for studying emergent collectivity and clustering in nuclei. Front. Phys. 11:1064601. doi: 10.3389/fphy.2023.1064601
Received: 08 October 2022; Accepted: 13 January 2023;
Published: 01 March 2023.
Edited by:
Maria Piarulli, Washington University in St. Louis, United StatesReviewed by:
Heiko Hergert, Michigan State University, United StatesPraveen C Srivastava, Indian Institute of Technology Roorkee, India
Copyright © 2023 Becker, Launey, Ekström and Dytrych. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: K. S. Becker, a2JlY2sxM0Bsc3UuZWR1