Medium-Mass Nuclei with Normal-Ordered Chiral $NN\mathbf{+}3N$ Interactions

We study the use of truncated normal-ordered three-nucleon interactions in nuclear structure calculations starting from chiral two- plus three-nucleon Hamiltonians evolved consistently with the similarity renormalization group. We present three key developments: (i) a rigorous benchmark of the normal-ordering approximation in the importance-truncated no-core shell model for ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$; (ii) a direct comparison of the importance-truncated no-core shell model results with coupled-cluster calculations at the singles and doubles level for ${}^{16}\mathrm{O}$; and (iii) first applications of similarity renormalization group-evolved chiral $NN+3N$ Hamiltonians in coupled-cluster calculations for medium-mass nuclei ${}^{16,24}\mathrm{O}$ and ${}^{40,48}\mathrm{Ca}$. We show that the normal-ordered two-body approximation works very well beyond the lightest isotopes and opens a path for studies of medium-mass and heavy nuclei with chiral two- plus three-nucleon interactions. At the same time we highlight the predictive power of chiral Hamiltonians.

Figure 1

IT-NCSM ground-state energies for ${}^4\mathrm{He}$ and ${}^{16}\mathrm{O}$ as a function of $N_{\max }$ for the $NN+3N$-induced and the $NN+3N$-full Hamiltonians for a range of flow parameters: $\alpha =0.04{\mathrm{fm}}^4$ (●), $0.05{\mathrm{fm}}^4$ (◆), $0.0625{\mathrm{fm}}^4$ (▴), $0.08{\mathrm{fm}}^4$ (▪). Solid symbols correspond to the exact $3N$ interaction, open symbols to the NO2B approximation. Error bars indicate the uncertainties of the threshold extrapolations of the IT-NCSM. Data points are connected by straight lines to guide the eye, beyond the largest $N_{\max }$ an exponential extrapolation fitted to the last four data points is shown.

Figure 2

Anatomy of the $\mathrm{NO}n\mathrm{B}$ approximation of the ground-state energies of ${}^4\mathrm{He}$, ${}^{16}\mathrm{O}$, and ${}^{40}\mathrm{Ca}$. The bar charts show the expectation values of the $3N$ interaction computed at different levels of the normal-ordering approximation, i.e., NO0B, NO1B, NO2B, and exact $3N$. We employ the $NN+3N$-induced and $NN+3N$-full Hamiltonians, each with two values of $\alpha$ (see labels). We use the eigenstates obtained for the exact $3N$ interaction in $N_{\max }=10$ for ${}^4\mathrm{He}$ and ${}^{16}\mathrm{O}$ and $N_{\max }=8$ for ${}^{40}\mathrm{Ca}$, all at $\hslash \Omega =20\mathrm{MeV}$.

Figure 3

Comparison of the ground-state energies of ${}^{16}\mathrm{O}$ obtained in IT-NCSM and CCSD including $3N$ interactions at the NO2B level for the $NN+3N$-induced and the $NN+3N$-full Hamiltonians with $\alpha =0.04{\mathrm{fm}}^4$ (●), $0.05{\mathrm{fm}}^4$ (◆), $0.0625{\mathrm{fm}}^4$ (▴), and $0.08{\mathrm{fm}}^4$ (▪).

Figure 4

CCSD ground-state energies for ${}^{16}\mathrm{O}$ and ${}^{24}\mathrm{O}$ as a function of $e_{\max }$ for the three types of Hamiltonians (see column headings) using the NO2B approximation for a range of flow parameters: $\alpha =0.04{\mathrm{fm}}^4$ (●), $0.05{\mathrm{fm}}^4$ (◆), $0.0625{\mathrm{fm}}^4$ (▴), and $0.08{\mathrm{fm}}^4$ (▪). The filled symbols for the $NN+3N$-full Hamiltonian are for the standard chiral $3N$ interaction with cutoff 500 MeV, the open symbols for a modified $3N$ interaction with cutoff 400 MeV (see text).

Figure 5

Same as Fig. 4 for ${}^{40}\mathrm{Ca}$ and ${}^{48}\mathrm{Ca}$.