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

## TL;DR

This study uses relativistic Hartree-Fock theory to analyze the restoration and violation of pseudo-spin symmetry in $N=32$ and $34$ isotones, revealing the influence of density profiles and tensor forces.

## Contribution

It demonstrates the systematic restoration of PSS in certain isotonic chains and identifies the dominant role of Hartree terms and density profile evolution in this process.

## Key findings

- PSS is restored from sulphur to nickel isotones.
- Distinct PSS violation occurs near the drip lines, especially in silicon.
- Hartree terms dominate the PSS systematics, linked to density profile changes.

## Abstract

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