Low-energy monopole strength in spherical and deformed nuclei : cluster and soft modes

Background : Several recent experiments report significant low-energy isoscalar monopole strength, below the giant resonance, in various nuclei. In light $\alpha$-conjugate nuclei, these low-energy resonances were recently interpreted as cluster vibration modes. However, the nature of these excitations in neutron-rich nuclei remain elusive. Purpose : The present work provides a systematic analysis of the low-energy monopole strength in isotopic chains, from Neon to Germanium, in order to monitor and understand its nature and conditions of emergence. Methods : We perform covariant quasiparticle random phase approximation (QRPA) calculations, formulated within the finite amplitude method (FAM), on top of constrained relativistic Hartree-Bogoliubov (RHB) reference states. Results : Neutron excess leads to the appearance of low-energy excitations according to a systematic pattern reflecting the single-particle features of the underlying RHB reference state. With the onset of deformation, these low-energy resonances get split and give rise to more complex patterns, with possible mixing with the giant resonance. At lower energy, cluster-like excitations found in $N=Z$ systems survive in neutron-rich nuclei, with valence neutrons arranging in molecular-like orbitals. Finally, at very low energy, pair excitations are also found in superfluid nuclei, but remain negligible in most of the cases. Conclusions : The low-energy part of the monopole strength exhibits various modes, from cluster vibrations ($\sim$ 5-10 MeV) to components of the giant resonance downshifted by the onset of deformation, including soft modes ($\sim$ 10-15 MeV) as well as pair excitation ($<$ 5 MeV), with possible mixing, depending on neutron-excess, deformation, and pairing energy.