Spectra of open-shell nuclei with Padé-resummed degenerate perturbation theory

We apply degenerate many-body perturbation theory at high orders for the ab initio description of ground states and excitation spectra of open-shell nuclei using soft realistic nucleon-nucleon interactions. We derive a recursive formulation of standard degenerate many-body perturbation theory that enables us to evaluate order-by-order perturbative energy and state corrections up to the 30th order. We study ${}^{6,7}\mathrm{Li}$ as test cases using a similarity renormalization group (SRG) evolved nucleon-nucleon interaction from chiral effective field theory. The simple perturbation series exhibits a strong, often oscillatory divergence, as was observed previously for ground states of closed-shell nuclei. Even for very soft interactions resulting from SRG evolutions up to large flow parameter, i.e., low-momentum scales, the perturbation series still diverges. However, a resummation of the perturbation series via Padé approximants yields very stable and converged ground and excited-state energies in very good agreement with exact no-core shell-model calculations for the same model space.

Figure 1

Energy from DMBPT power series truncated at order $p$ for the energy levels of ${}^6\mathrm{Li}$ corresponding to the degenerate HO $n=0$ subspace using a $N_{\max }=8$ model space for $\hslash \Omega =20\mathrm{MeV}$. The indices $d$ are determined by diagonalization in the degenerate subspace, i.e., by the first-order energy corrections. The dashed lines correspond to the NCSM result in the same model space. Two SRG flow parameters are shown: $0.04{\mathrm{fm}}^4$ ($\Lambda \approx 2.24{\mathrm{fm}}^{-1}$; blue discs, blue dashed lines) and the softer $0.16{\mathrm{fm}}^4$ ($\Lambda \approx 1.58{\mathrm{fm}}^{-1}$; red diamonds, red dashed lines).

Figure 2

Padé approximation for energy of the ten states corresponding to the $n=0$ HO subspace of ${}^6\mathrm{Li}$. Shown are the diagonal $[L/L]$ (blue discs), superdiagonal $[L/L+1]$ (red diamonds), and subdiagonal $[L/L-1]$ (green triangles) Padé approximants. The dashed horizontal lines represent the exact energies obtained by a NCSM calculation. We used a $N_{\max }=8$ model space with HO frequency $20\mathrm{MeV}$ and the SRG parameter is $0.04{\mathrm{fm}}^4(\Lambda \approx 2.24{\mathrm{fm}}^{-1})$.

Figure 3

Low-lying spectrum of ${}^6\mathrm{Li}$ computed in an $N_{\max }=8$ model space for $\hslash \Omega =20\mathrm{MeV}$, using SRG transformed N${}^3$LO two-body interactions with flow parameter $0.04{\mathrm{fm}}^4(\Lambda \approx 2.24{\mathrm{fm}}^{-1})$ in (a) and $0.16{\mathrm{fm}}^4(\Lambda \approx 1.58{\mathrm{fm}}^{-1})$ in (b). Shown are from the left the experimental observed energies, the NCSM results, the Padé resummed results (see text for details) and results from the truncated power series of DMBPT at order $p=2$, 3, 4, and 8. Experimental values taken from Ref. [22].

Figure 4

Excitation energies of the nine energetically lowest states of the ${}^7\mathrm{Li}$ spectrum computed in an $N_{\max }=8$ model space for $\hslash \Omega =20\mathrm{MeV}$, using SRG transformed N${}^3$LO two-body interactions with flow parameter $0.04{\mathrm{fm}}^4(\Lambda \approx 2.24{\mathrm{fm}}^{-1})$ in (a) and $0.16{\mathrm{fm}}^4(\Lambda \approx 1.58{\mathrm{fm}}^{-1})$ in (b). Shown are from the left the experimental observed energies, the NCSM results, the Padé resumed results (see text for details), and results from the truncated power series of DMBPT at order $p=2$, 3, 4, and 8. Experimental values taken from Ref. [22].