  In this work, we investigate quantum phase transition (QPT) in a generic
family of spin chains using the ground-state energy, the energy gap, and the
geometric measure of entanglement (GE). In many of prior works, GE per site was
used. Here, we also consider GE per block with each block size being two. This
can be regarded as a coarse grain of GE per site. We introduce a useful
parameterization for the family of spin chains that includes the XY models with
n-site interaction, the GHZ-cluster model and a cluster-antiferromagnetic
model, the last of which exhibits QPT between a symmetry-protected topological
(SPT) phase and a symmetry-breaking antiferromagnetic phase. As the models are
exactly solvable, their ground-state wavefunctions can be obtained and thus
their GE can be studied. It turns out that the overlap of the ground states
with translationally invariant product states can be exactly calculated and
hence the GE can be obtained via further parameter optimization. The QPTs
exhibited in these models are detected by the energy gap and singular behavior
of geometric entanglement. In particular, the XzY model exhibits transitions
from the nontrivial SPT phase to a trivial paramagnetic phase. Moreover, the
halfway XY model exhibits a first-order transition across the Barouch-McCoy
circle, on which it was only a crossover in the standard XY model.
