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In this work, we investigate quantum phase transition (QPT) in a genericfamily of spin chains using the ground-state energy, the energy gap, and thegeometric measure of entanglement (GE). In many of prior works, GE per site wasused. Here, we also consider GE per block with each block size being two. Thiscan be regarded as a coarse grain of GE per site. We introduce a usefulparameterization for the family of spin chains that includes the XY models withn-site interaction, the GHZ-cluster model and a cluster-antiferromagneticmodel, the last of which exhibits QPT between a symmetry-protected topological(SPT) phase and a symmetry-breaking antiferromagnetic phase. As the models areexactly solvable, their ground-state wavefunctions can be obtained and thustheir GE can be studied. It turns out that the overlap of the ground stateswith translationally invariant product states can be exactly calculated andhence the GE can be obtained via further parameter optimization. The QPTsexhibited in these models are detected by the energy gap and singular behaviorof geometric entanglement. In particular, the XzY model exhibits transitionsfrom the nontrivial SPT phase to a trivial paramagnetic phase. Moreover, thehalfway XY model exhibits a first-order transition across the Barouch-McCoycircle, on which it was only a crossover in the standard XY model.

Geometric Entanglement and Quantum Phase Transition in Generalized Cluster-XY models