  Deformed ground state of $^{32}$Mg is investigated using the axially deformed
relativistic Hartree-Fock-Bogoliubov (D-RHFB) model with the effective
Lagrangian PKA1, which provides coincident description with the experimental
measurements. It is illustrated that obvious breaking of the pseudo-spin
symmetry (PSS) given by PKA1, being consistent with the experimental
observation in nearby isotone $^{40}$Ca, is crucial for describing correctly
the deformed ground state by producing unique shape evolution of neutron orbit
$1/2_4^+$ in $^{32}$Mg. The PSS breaking is essentially determined by
characteristic in-medium balance between nuclear attractions and repulsions
that is manifested as unparalleled density dependent behaviors for coupling
strengths $g_\sigma$ and $g_\omega$ in dominant $\sigma$-scalar and
$\omega$-vector channels.
