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Hartree-Fock emulators for nuclei: Application to charge radii of $^{48,52}$Ca

Margarida Companys Franzke, Alexander Tichai, Kai Hebeler, Achim Schwenk

Abstract

Understanding the emergence of complex structures of nuclei from chiral effective field theory (EFT) is a central challenge. The large number of low-energy couplings (LECs) in the EFT expansion and the significant cost of $\textit{ab initio}$ many-body calculations render large-scale sensitivity studies of many-body observables computationally prohibitive, necessitating the use of emulators as low-cost surrogates. In this work, we study a Hartree-Fock emulator based on eigenvector continuation to investigate trends in nuclear charge radii of neutron-rich calcium isotopes. We systematically vary the five LECs entering the leading three-nucleon (3N) interactions, and demonstrate the precision of the emulator through cross-validation over a wide parameter space. Our findings indicate that large charge radius increase from $^{48}$Ca to $^{52}$Ca is likely not explained by variations of the leading 3N couplings. This suggests that other effects, such as sensitivities to chiral two-nucleon interactions or neglected many-body effects, e.g., associated with nuclear collectivity, play an important role.

Hartree-Fock emulators for nuclei: Application to charge radii of $^{48,52}$Ca

Abstract

Understanding the emergence of complex structures of nuclei from chiral effective field theory (EFT) is a central challenge. The large number of low-energy couplings (LECs) in the EFT expansion and the significant cost of many-body calculations render large-scale sensitivity studies of many-body observables computationally prohibitive, necessitating the use of emulators as low-cost surrogates. In this work, we study a Hartree-Fock emulator based on eigenvector continuation to investigate trends in nuclear charge radii of neutron-rich calcium isotopes. We systematically vary the five LECs entering the leading three-nucleon (3N) interactions, and demonstrate the precision of the emulator through cross-validation over a wide parameter space. Our findings indicate that large charge radius increase from Ca to Ca is likely not explained by variations of the leading 3N couplings. This suggests that other effects, such as sensitivities to chiral two-nucleon interactions or neglected many-body effects, e.g., associated with nuclear collectivity, play an important role.

Paper Structure

This paper contains 17 sections, 29 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Emulator framework
  3. Eigenvector continuation
  4. Transition densities
  5. Operator kernels
  6. Emulator for general ground-state observables
  7. Emulator construction
  8. Computational details
  9. Symmetric vs. asymmetric formulation
  10. Training sets
  11. Cross-validation
  12. Regularization of the eigenvalue problem
  13. Application to calcium isotopes
  14. Distribution of LEC samples
  15. Variation of the NN interaction
  16. ...and 2 more sections

Figures (11)

  • Figure 1: Emulated ground-state energies (upper left panel) and charge radii (upper right panel) together with the corresponding emulator errors (lower panels) for symmetric (blue line) and asymmetric (orange line) EC as a function of $c_D$ values. The training points are indicated by crosses. All other 3N LECs are fixed to the 1.8/2.0 (EM) interaction.
  • Figure 2: Distribution of different ranges for $\delta \langle R_\text{ch}^2 \rangle^{48,52}$ as a function of $c_1$.
  • Figure 3: Distribution of the relative error of the emulated ground-state energies compared to exact HF calculations (upper panel) and the relative error of the charge radii (lower panel). The blue part of the histograms corresponds to the interactions restricted to the energy and radius constraints from Eq. (\ref{['eq:ER_constraints']}).
  • Figure 4: Distribution of the LECs for the energy and radius constraints from Eq. (\ref{['eq:ER_constraints']}). Darker shades correspond to a higher number of interactions. The lines correspond to the 39.3% and the 86.5% levels, and the numbers for the one-dimensional distributions give the mean and 68% confidence intervals.
  • Figure 5: Same as Fig. \ref{['fig:Crossvalidation']}, but with all training vectors restricted to the energy and radius constraints from Eq. (\ref{['eq:ER_constraints']}).
  • ...and 6 more figures