In-medium similarity renormalization group with chiral two- plus three-nucleon interactions

We use the recently proposed in-medium similarity renormalization group (IM-SRG) to carry out a systematic study of closed-shell nuclei up to ${}^{56}\mathrm{Ni}$, based on chiral two- plus three-nucleon interactions. We analyze the capabilities of the IM-SRG by comparing our results for the ground-state energy to coupled cluster calculations, as well as to quasiexact results from the importance-truncated no-core shell model. Using chiral two- plus three-nucleon Hamiltonians whose resolution scales are lowered by free-space SRG evolution, we obtain good agreement with experimental binding energies in ${}^4\mathrm{He}$ and the closed-shell oxygen isotopes, while the calcium and nickel isotopes are somewhat overbound.

Figure 1

Schematic representation of the initial and final Hamiltonians, $H\left(0\right)$ and $H\left(\infty \right)$, in the many-body Hilbert space spanned by particle-hole excitations of the reference state.

Figure 2

Schematic illustration of the energy flow equation (14) for the White generator (20) in terms of Hugenholtz diagrams (see text). The gray vertices correspond to $H\left(s\right)$, and the double lines indicate energy denominators calculated with $f\left(s\right)$. On the right-hand side, the flow equation is expanded in terms of $H(s-\delta s)$ (simple black vertex) and the corresponding energy denominators from $f(s-\delta s)$ (single lines). The dots indicate omitted terms from the expansion of the Epstein-Nesbet energy denominator, as well as additional diagrams generated by the integration step $s-\delta s\to s$.

Figure 3

Convergence of the IM-SRG(2) ground-state energy of closed-shell nuclei for the $NN+3N$-full Hamiltonian (see text), evolved to $\lambda =2.0{\text{fm}}^{-1}$. The energies indicated by dashed lines are obtained with the extrapolation method proposed in Ref. [39].

Figure 4

IM-SRG(2) ground-state energy of ${}^{40}\mathrm{Ca}$ as a function of the flow parameter $s$ for the bare $NN$ Hamiltonian (top) and the $NN+3N$-induced Hamiltonian with $\lambda =2.0{\text{fm}}^{-1}$ (bottom). Shown are $E$(s) () and $E\left(s\right)$ plus second- () and third-order () MBPT corrections for $H\left(s\right)$. The dashed line indicates the final value $E\left(\infty \right)$. All calculations were done with $e_{\text{Max}}=10$, at optimal $\hslash \Omega =32\text{MeV}$ (top) and $24\text{MeV}$ (bottom). For $s>1.5$ we only show every tenth numerical data point to reduce clutter.

Figure 5

Ground-state energies of stable closed-shell nuclei as a function of the resolution scale $\lambda$ for the SRG-evolved $NN+3N$-induced (left column) and $NN+3N$-full Hamiltonians (right column), using different many-body methods: IM-SRG(2) (), CCSD (), $\Lambda$-CCSD(T) (), and IT-NCSM (). IM-SRG and CC results are obtained with $e_{\text{Max}}=14$, and the IT-NCSM results have been extrapolated to infinite model space, with error bars indicating the uncertainties of the importance truncation and the model space extrapolation (for ${}^4\mathrm{He}$, the error bars are smaller than the symbols). The gray solid line for ${}^4\mathrm{He}$ in the top left panel is the result of a converged NCSM calculation with the bare $NN$ Hamiltonian, $E=-25.39\text{MeV}$ (see, e.g., Ref. [4]). Dashed black lines are experimental values from Ref. [44].

Figure 6

Ground-state energies of stable closed-shell nuclei for the bare NN Hamiltonian ($\lambda =\infty$), using IM-SRG(2) (), CCSD (), and $\Lambda$-CCSD(T) (). IM-SRG and CC results are obtained with $e_{\text{Max}}=14$. The gray solid line for ${}^4\mathrm{He}$ indicates the result of a converged NCSM calculation, $E=-25.39\text{MeV}$ (see, e.g., Ref. [4]). The open and solid orange arrows are converged CCSD and $\Lambda$-CCSD(T) results from Refs. [6, 45].

Figure 7

IM-SRG(2) ground-state energy per nucleon of closed-shell nuclei for $NN+3N$-induced (top) and $NN+3N$-full Hamiltonians (bottom) at different resolution scales $\lambda$. Energies are determined at optimal $\hslash \Omega$ for $e_{\text{Max}}=14$. Experimental energies (black bars) are taken from Ref. [44].