Approximate symmetries in atomic nuclei from a large-scale shell-model perspective

In this paper, we review recent developments that aim to achieve further understanding of the structure of atomic nuclei, by capitalizing on exact symmetries as well as approximate symmetries found to dominate low-lying nuclear states. The findings confirm the essential role played by the Sp(3, ℝ) symplectic symmetry to inform the interaction and the relevant model spaces in nuclear modeling. The significance of the Sp(3, ℝ) symmetry for a description of a quantum system of strongly interacting particles naturally emerges from the physical relevance of its generators, which directly relate to particle momentum and position coordinates, and represent important observables, such as, the many-particle kinetic energy, the monopole operator, the quadrupole moment and the angular momentum. We show that it is imperative that shell-model spaces be expanded well beyond the current limits to accommodate particle excitations that appear critical to enhanced collectivity in heavier systems and to highly-deformed spatial structures, exemplified by the second ⁰⁺ state in ¹²C (the challenging Hoyle state) and ⁸Be. While such states are presently inaccessible by large-scale no-core shell models, symmetry-based considerations are found to be essential.