Shell-model coupled-cluster method for open-shell nuclei

We present an approach to derive effective shell-model interactions from microscopic nuclear forces. The similarity transformed coupled-cluster Hamiltonian decouples the single-reference state of a closed-shell nucleus and provides us with a core for the shell model. We use a second similarity transformation to decouple a shell-model space from the excluded space. We show that the three-body terms induced by both similarity transformations are crucial for an accurate computation of ground and excited states. As a proof of principle we use a nucleon-nucleon interaction from chiral effective field theory, employ a ${}^4\mathrm{He}$ core, and compute low-lying states of ${}^{6–8}\mathrm{He}$ and ${}^{6–8}\mathrm{Li}$ in $p$-shell model spaces. Our results agree with benchmarks from full configuration interaction.

Figure 1

Diagrams of the operator $S$ that couple the valence space and the excluded space. The horizontal double bar is the $S$ operator, with incoming and outgoing particle lines as indicated by arrows. The double incoming line denotes a particle inside the model space. Diagram (a) is a one-body operator, while diagrams (b) and (c) are two-body operators. The solid particle line is a general particle (i.e., either from the excluded space or the model space), except for the outgoing two particles in diagram (b), where at least one of the outgoing particles must be from the excluded space.

Figure 2

Diagrammatic representation of ${[\chi ,S]}_{\text{1b}}$ multiplied with a $+$ or $-$ sign from the commutator as indicated below each diagram. The wiggly line refers to the intermediate $\chi$, and the horizontal double bar is $S$.

Figure 3

Diagrammatic representation of ${[\chi ,S]}_{\text{2b}}$ multiplied with a $+$ or $-$ sign from the commutator.

Figure 4

Two-body diagrams $[[\mathcal{H},T_2],T_2]$ in CCSD closed-shell calculation. (Here, the black double horizontal bar denotes the cluster operator $T$, while the wiggly line is the Hamiltonian $\mathcal{H}$.) The intermediate three-body $[{[\mathcal{H},T_2]}_{3b},T_2{]}_{2b}$ and one-body $[{[\mathcal{H},T_2]}_{1b},T_2{]}_{2b}$ terms inside the commutator result from making cuts at the locations $a$ and $b$, respectively, and are indicated by red double lines.

Figure 5

“Core polarization” diagrams that are neglected when all commutators are truncated at the two-body level. Cutting the diagrams at the red double bars reveals intermediate three-body terms.

Figure 6

Diagrammatic representation of ${[\chi ,S]}_{\text{3b}}$ multiplied with a $+$ or $-$ sign from the commutator as indicated. Here $\chi$ and $S$ are shown as wiggly and double horizontal lines, respectively.

Figure 7

Diagrams of ${[{[{\chi }_{2b},S_{2b}]}_{3b},S_{2b}]}_{2b}$ multiplied with a $+$ or $-$ sign from the commutator as indicated below each diagram. Here $\chi$ and $S$ are shown as wiggly and double horizontal lines, respectively.

Figure 8

Diagram representation of ${[{\overline{\overline{\mathcal{H}}}}_{2b},S_{3b}]}_{2b}$.

Figure 9

Diagram representation of ${[{\overline{\overline{\mathcal{H}}}}_{2b},S_{3b}]}_{1b}$.

Figure 10

Spectrum of ${}^6\mathrm{He}$ computed with various methods (FCI, EOM-CC, and IMSRG) and compared to the SMCC results.

Figure 11

Spectrum of ${}^6\mathrm{Li}$ computed with various methods (FCI and VS-IMSRG) and compared to the SMCC(3b) result. SMCC(3b) include both three-body interaction diagonal and off-diagonal parts.

Figure 12

Spectrum of ${}^7\mathrm{He}$ nucleus, computed with various methods (FCI, EOM-CC, and VS-IMSRG) and compared to the SMCC results.

Figure 13

Same as Fig. 12 but for ${}^7\mathrm{Li}$.

Figure 14

Spectrum of ${}^8\mathrm{He}$ computed with various methods (FCI, EOM-CC, CCEI, and IMSRG) and compared to SMCC. The SMCC(3b-od) computation employs induced three-body forces that contribute to two-body forces, while SMCC(3b) also employs explicit three-body forces in the shell-model calculation.

Figure 15

Same as Fig. 14 but for ${}^8\mathrm{Li}$.

Figure 16

Squared point proton radii and isotope shift $[{\delta }_{R_p^2}=R_p^2-R_p^2\left({}^4\mathrm{He}\right)]$ of selected $p$-shell nuclei calculated using the SMCC(3b), the Hellmann-Feynman theorem [based on SMCC(3b)], the VS-IMSRG, and FCI.