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Restoration of pseudo-spin symmetry (PSS) along the $N=32$ and $34$ isotonicchains and the physics behind are studied by applying the relativisticHartree-Fock theory with effective Lagrangian PKA1. Taking the protonpseudo-spin partners $(\pi2s_{1/2},\pi1d_{3/2})$ as candidates, systematicrestoration of PSS along both isotonic chains is found from sulphur (S) tonickel (Ni), while distinct violation from silicon (Si) to sulphur isdiscovered near the drip lines. The effects of the tensor-force componentsintroduced naturally by the Fock terms are investigated, which can only partlyinterpret the systematics from calcium to nickel, but fail for the overalltrends. Further analysis following the Schr\"{o}dinger-like equation of thelower component of Dirac spinor shows that the contributions from the Hartreeterms dominate the overall systematics of the PSS restoration, and such effectscan be self-consistently interpreted by the evolution of the proton centraldensity profiles along both isotonic chains. Specifically the distinct PSSviolation is found to tightly relate with the dramatic changes from thebubble-like density profiles in silicon to the central-bumped ones in sulphur.

Restoration of pseudo-spin symmetry in $N=32$ and $34$ isotones described by relativistic Hartree-Fock theory