From closed shells to open shells: Coupled-cluster calculations of atomic nuclei

TL;DR

This work addresses open-shell nuclei from first principles using coupled-cluster theory and compares three CC-based strategies: equation-of-motion CC (2PR/2PA variants), Bogoliubov CC (BCCSD) with a particle-number–breaking reference, and CC on top of a deformed reference. It applies these methods to Ca and Ni isotopes with chiral two- and three-body forces, finding consistent predictions for bulk observables such as ground-state energies, the two-neutron separation energies $S_{2n}$, and the two-neutron shell gaps $ abla_{2n}$, with differences among CC variants smaller than estimated triple-excitation uncertainties. The results demonstrate CC theory as a robust tool for mid-mass open-shell nuclei and show that symmetry-broken references extend reach to mid-shell regions where low-rank EOM expansions struggle, while also highlighting the ongoing importance of triples and (in the future) symmetry restoration for spectroscopy. The study points toward a unified framework that combines deformation and pairing within CC theory and motivates symmetry restoration approaches to enable accurate spectroscopy in heavy open-shell nuclei.

Abstract

Coupled-cluster theory is a powerful tool for first-principles calculations of atomic nuclei, enabling accurate predictions of nuclear observables across the Segrè chart. While coupled-cluster computations are especially efficient at shell closures, extensions have been developed to tackle open-shell nuclei, by exploiting the equation-of-motion method or by expanding the coupled-cluster wave function on top of a symmetry-breaking (either deformed or superfluid) reference state. In this study, we provide a comprehensive comparison of these different formulations applied to the calcium and nickel isotopes using nuclear two- and three-body interactions from chiral effective field theory. Based on ground-state energies, two-neutron separation energies, and two-neutron shell gaps, different coupled-cluster computations - based on symmetry-broken reference states and equation-of-motion techniques - offer consistent descriptions of bulk properties across medium-mass isotopic chains.

Figures (5)

• Figure 1: Ground-state energies of even Ca isotopes as a function of the mass number $A$. The left and right panels display results obtained using the EM 1.8/2.0 MagicInteraction and $\Delta$NNLO$_\text{GO}$(394) DeltaGo2020 potentials, respectively. Calculations performed with CCSD (squares), BCCSD (circles), 2PA-EOM (upward triangles), and 2PR-EOM (downward triangles) are reported. The gray bands refer to an estimate of the theoretical uncertainties stemming from the model-space and many-body truncations (see text). Experimental energies (black) are taken from Ref. wang2021ame.
• Figure 2: Same as Fig. \ref{['fig:calciumenergies']} but for Ni isotopes.
• Figure 3: Ground-state energies for $^{48,50,52}$Ca (top) and $^{56,58,60}$Ni (bottom) isotopes as a function of the number of neutrons $N$. Calculations are performed with the $\Delta$NNLO$_\text{GO}$(394) interaction, and results are reported for CCSD (squares), BCCSD (circles), and, when available, 2PA-EOM (upward triangles) and 2PR-EOM (downward triangles). As in Fig. \ref{['fig:calciumenergies']}, the shaded bands account for an estimate of the theoretical uncertainties due to the model-space truncation and the contribution of the missing triples. Experimental ground-state energies are shown as black bars.
• Figure 4: Two-neutron separation energies $S_{2n}(N,Z)$ as a function of the mass number $A$ in the Ca and Ni isotopic chains. Results obtained with the EM 1.8/2.0 and $\Delta$NNLO$_\text{GO}$(394) potentials are shown as filled and empty symbols, respectively. Calculations performed with CCSD and BCCSD are denoted by squares and circles, respectively. Experimental data are shown as black bars. For clarity, the results for the $\Delta$NNLO$_\text{GO}$(394) interaction are offset horizontally by 0.3.
• Figure 5: Two-neutron shell gaps in Ca (left panel) and Ni (right panel) isotopes computed with the EM 1.8/2.0 interaction using the BCCSD and CCSD approaches.