Evidence for Symplectic Symmetry in Ab Initio No-Core Shell Model Results for Light Nuclei

Clear evidence for symplectic symmetry in low-lying states of ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$ is reported. Eigenstates of ${}^{12}\mathrm{C}$ and ${}^{16}\mathrm{O}$, determined within the framework of the no-core shell model using the $J$-matrix inverse scattering potential with $A\le 16$ (JISP16) nucleon-nucleon ($NN$) realistic interaction, typically project at the 85%–90% level onto a few of the most deformed symplectic basis states that span only a small fraction of the full model space. The results are nearly independent of whether the bare or renormalized effective interactions are used in the analysis. The outcome confirms Elliott’s SU(3) model which underpins the symplectic scheme, and above all, points to the relevance of a symplectic no-core shell model that can reproduce experimental $B(E2)$ values without effective charges as well as deformed spatial modes associated with clustering phenomena in nuclei.

Figure 1

NCSM space dimension as a function of the maximum $\hslash \Omega$ excitations, $N_{\max }$, compared to that of the $\mathrm{Sp}(3,\mathbb{R})$ subspace: (a) $J=0$, 2, and 4 for ${}^{12}\mathrm{C}$, and (b) $J=0$ for ${}^{16}\mathrm{O}$.

Figure 2

Projection of the $S=0$ [left-hand bars, blue] [and $S=1$ (right-hand bars, red)] $\mathrm{Sp}(3,\mathbb{R})$ irreps onto the corresponding significant spin components of the NSCM wave functions for (a) $0_{\mathrm{gs}}^{+}$, (b) $2_1^{+}$, and (c) $4_1^{+}$ in ${}^{12}\mathrm{C}$, and (d) $0_{\mathrm{gs}}^{+}$ in ${}^{16}\mathrm{O}$, for effective interaction for different $\hslash \Omega$ oscillator strengths and bare interaction.

Figure 3

Ground $0^{+}$ state probability distribution over $0\hslash \Omega$ (lowest, blue) to $6\hslash \Omega$ (highest, green) subspaces for the most dominant $0p\mathrm{\text{−}}0h+2\hslash \Omega$ $2p\mathrm{\text{−}}2h$ $\mathrm{Sp}(3,\mathbb{R})$ irrep case (left) and NCSM (right) together with the leading irrep contribution (black diamonds), (0 4) for ${}^{12}\mathrm{C}$ (a) and (0 0) for ${}^{16}\mathrm{O}$ (b), as a function of the $\hslash \Omega$ oscillator strength, $N_{\max }=6$.