In-medium similarity renormalization group for open-shell nuclei

We present a nonperturbative derivation of effective valence-shell Hamiltonians in the framework of the re-cently developed in-medium similarity renormalization group (IM-SRG). As a first application, we calculate the spectra of $p$- and $sd$-shell nuclei, ${}^6$Li and ${}^{18}$O, based on evolved chiral nucleon-nucleon interactions. For ${}^6$Li, the spectrum is in very good agreement with ab initio results. For ${}^{18}$O, the IM-SRG provides a new method for the shell model to systematically go beyond effective interaction techniques based on diagrammatic expansions.

Figure 1

Schematic illustration for the valence-space decoupling by the IM-SRG evolution from $H(s=0)\to H\left(\infty \right)$ for two valence nucleons.

Figure 2

The ground-state energy of ${}^6$Li versus harmonic-oscillator parameter $\hslash \omega$ obtained by a diagonalization of the IM-SRG(2) $H_{\mathrm{eff}}$ in the $p$ shell, using generators $H_1^{\mathrm{od}}$ and $H_2^{\mathrm{od}}$ (left and center panels). The flow equations are solved in the HF basis truncated to $e_{\max }=\max (2n+l)$ single-particle excitations. For comparison we show in the right panel the convergence with increasing $N_{\max }$ of the NCSM energy [15], where the dot-dashed band is the extrapolated result.

Figure 3

Convergence as a function of $e_{\max }$ of the excitation energies of ${}^6$Li obtained by diagonalizing the IM-SRG(2) $H_{\mathrm{eff}}$ in the $p$ shell using the $H_1^{\mathrm{od}}$ generator. The HF basis at a fixed $\hslash \omega =24\mathrm{MeV}$ is used for the IM-SRG(2) calculations. For comparison we show the convergence with $N_{\max }$ of the NCSM energies at the same $\hslash \omega$ value [15].

Figure 4

Left panel: Excitation energies of ${}^{18}$O obtained by diagonalizing the IM-SRG(2) $H_{\mathrm{eff}}$ in the $sd$ shell, compared with results obtained at first ($Q^{\left(1\right)}$), second ($Q^{\left(2\right)}$), and third ($Q^{\left(3\right)}$) order in the $Q$-box expansion [9] for effective valence-shell interactions (the filled symbols include higher-order folded-diagram contributions). All results are for $\hslash \omega =24\mathrm{MeV}$, but a HF basis is used. For comparison, we also show the experimental energies [18]. Right panel: Excitation energies of ${}^{18}$O vs $\hslash \omega$ calculated at the second-order $Q$-box level (plus folding) in harmonic-oscillator and HF bases. For the results of both panels, an $e_{\max }=8$ space was used, and to simplify the comparison, the calculations used single-particle energies from the USDb interaction [19] for the diagonalization in the $sd$ shell.