**Abstract** : The possibility that an unconventional depletion (referred to as a " bubble ") occurs in the center of the charge density distribution of certain nuclei due to a purely quantum mechanical effect has attracted theoretical and experimental attention in recent years. Based on a mean-field rationale, a correlation between the occurrence of such a semibubble and an anomalously weak splitting between low angular-momentum spin-orbit partners has been further conjectured. Energy density functional and valence-space shell model calculations have been performed to identify and characterize the best candidates, among which $^{34}$Si appears as a particularly interesting case. While the experimental determination of the charge density distribution of the unstable $^{34}$Si is currently out of reach, (d,p) experiments on this nucleus have been performed recently to test the correlation between the presence of a bubble and an anomalously weak 1/2$^−$$_−$3/2$^−$ splitting in the spectrum of $^{35}$Si as compared to $^{37}$S.
Purpose: We study the potential bubble structure of $^{34}$Si on the basis of the state-of-the-art ab initio self-consistent Green's function many-body method.
Methods: We perform the first ab initio calculations of $^{34}$Si and $^{36}$S. In addition to binding energies, the first observables of interest are the charge density distribution and the charge root-mean-square radius for which experimental data exist in $^{36}$S. The next observable of interest is the low-lying spectroscopy of $^{35}$Si and $^{37}$S obtained from (d,p) experiments along with the spectroscopy of $^{33}$Al and $^{35}$P obtained from knockout experiments. The interpretation in terms of the evolution of the underlying shell structure is also provided. The study is repeated using several chiral effective field theory Hamiltonians as a way to test the robustness of the results with respect to input internucleon interactions. The convergence of the results with respect to the truncation of the many-body expansion, i.e., with respect to the many-body correlations included in the calculation, is studied in detail. We eventually compare our predictions to state-of-the-art multireference energy density functional and shell model calculations. Results: The prediction regarding the (non)existence of the bubble structure in $^{34}$Si varies significantly with the nuclear Hamiltonian used. However, demanding that the experimental charge density distribution and the root-mean-square radius of $^{36}$S be well reproduced, along with $^{34}$Si and $^{36}$S binding energies, only leaves the NNLO$_{sat}$ Hamiltonian as a serious candidate to perform this prediction. In this context, a bubble structure, whose fingerprint should be visible in an electron scattering experiment of $^{34}$Si, is predicted. Furthermore, a clear correlation is established between the occurrence of the bubble structure and the weakening of the 1/2$^−$$_−$3/2$^−$ splitting in the spectrum of $^{35}$Si as compared to $^{37}$S. Conclusions: The occurrence of a bubble structure in the charge distribution of $^{34}$Si is convincingly established on the basis of state-of-the-art ab initio calculations. This prediction will have to be reexamined in the future when improved chiral nuclear Hamiltonians are constructed. On the experimental side, present results act as a strong motivation to measure the charge density distribution of $^{34}$Si in future electron scattering experiments on unstable nuclei. In the meantime, it is of interest to perform one-neutron removal on $^{34}$Si and $^{36}$S in order to further test our theoretical spectral strength distributions over a wide energy range.