Geometric entanglement and quantum phase transition in generalized cluster-XY models

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.