  [Background] $\beta$-decay half-life is sensitive to the shell structure near
the Fermi levels. Nuclear deformation thus impacts the $\beta$-decay
properties. [Purpose] A first-order shape-phase transition in neutron-rich Zr
isotopes is predicted by some models. We investigate the $\beta$-decay
half-lives of neutron-rich nuclei around $^{110}$Zr, where the shape-phase
transition is predicted to occur, to see if the $\beta$-decay half-life can be
an indicator of the shape changes. [Method] The proton-neutron quasiparticle
random-phase approximation (RPA) is adopted to calculate the Gamow-Teller
transitions. In addition, we apply the quasiparticle phonon-vibrational
coupling (PVC) to consider the phonon couplings. [Results] The spherical and
oblate configurations give similar half-lives but shorter ones than the prolate
configuration at the RPA level. The PVC effect further reduces the half-lives
in general, but the effect is smaller for the deformed configuration than that
for the spherical one. As a result, it makes the shape change from the oblate
configuration to the spherical configuration visible. Therefore, a sudden
shortening of $\beta$-decay half-lives is always found at the nuclear shape
changes. [Conclusions] $\beta$-decay half-life is an indicator of the
shape-phase transition. The shape mixing and the roles of the triaxial
deformation are subject to study in the future.
