  The shape phase transition for certain isotope or isotone chains, associated
with the quantum phase transition of finite nuclei, is an intriguing phenomenon
in nuclear physics. A notable case is the Xe isotope chain, where the structure
transits from a $\gamma$-soft rotor to a spherical vibrator, with the
second-order shape phase transition occurring in the vicinity of
$^{128-130}$Xe. In this letter, we focus on investigating the $\gamma$-soft
deformation of $^{129}$Xe associated with the second-order shape phase
transition by constructing novel correlators for ultra-relativistic
$^{129}$Xe+$^{129}$Xe collisions. In particular, our iEBE-VISHNU model
calculations show that the $v_2^2-[p_T]$ correlation $\rho_{2}$ and the mean
transverse momentum fluctuation $\Gamma_{p_T}$, which were previously
interpreted as the evidence for the rigid triaxial deformation of $^{129}$Xe,
can also be well explained by the $\gamma$-soft deformation of $^{129}$Xe. We
also propose two novel correlators $\rho_{4,2}$ and $\rho_{2,4}$, which carry
non-trivial higher-order correlations and show unique capabilities to
distinguish between the $\gamma$-soft and the rigid triaxial deformation of
$^{129}$Xe in $^{129}$Xe+$^{129}$Xe collisions at the LHC. The present study
also provides a novel way to explore the second-order shape phase transition of
finite nuclei with ultra-relativistic heavy ion collisions.
