\begin{document}$ N=32, 34 $\end{document}. We systematically analyzed both neutron and proton shell evolutions by examining the excitation energies of the first \begin{document}$ 2^+ $\end{document} states and the effective single-particle energies. Our calculations show that the \begin{document}$ N=32 $\end{document} sub-shell gradually weakens as protons are removed from the doubly magic nucleus \begin{document}$ ^{52} {\rm{Ca}}$\end{document}, eventually disappearing in \begin{document}$ ^{46} {\rm{Si}}$\end{document}. Conversely, the strength of the \begin{document}$ N=34 $\end{document} sub-shell is enhanced with the removal of protons from \begin{document}$ ^{54} {\rm{Ca}}$\end{document}. Furthermore, our results indicate the existence of the proton \begin{document}$ Z=14 $\end{document} sub-shell in neutron-rich Si isotopes. These findings suggest that \begin{document}$ ^{48} {\rm{Si}}$\end{document} is a doubly magic nucleus, with the excitation energy of the first \begin{document}$ 2^+ $\end{document} state around 2.49 MeV, which is approximately 400 keV higher than that of \begin{document}$ ^{54} {\rm{Ca}}$\end{document}. This value is comparable to those of other well-known exotic doubly magic nuclei, such as \begin{document}$ ^{52} {\rm{Ca}}$\end{document} and \begin{document}$ ^{78} {\rm{Ni}}$\end{document}, which is of great interest for further experiments at RIB facilities. In addition, we predicted the low-lying spectra of neutron-rich Si, S, and Ar isotopes, providing new insights for future experiments."> <i>Ab initio</i> study of shell evolution in neutron-rich Si, S, Ar, and Ca isotopes near <i>N</i> = 32 and 34 -
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