In this paper we analize a family of one dimensional fully analytically solvable models, named the n-cluster models in a transverse magnetic field, in which a many-body cluster interaction competes with a uniform transverse magnetic field. These models, independently by the cluster size n + 2, exibit a quantum phase transition, that separates a paramagnetic phase from a cluster one, that corresponds to a nematic ordered phase or a symmetry-protected topological ordered phase for even or odd n respectively. Due to the symmetries of the spin correlation functions, we prove that these models have no genuine n+2-partite entanglement. On the contrary, a non vanishing concurrence arises between spins at the endpoints of the cluster, for a magnetic field strong enough. Due to their analyticity and peculiar entanglement properties, the n-cluster models in a transverse magnetic field serve as a prototype for studying non trivial-spin orderings and as a potential reference system for the applications of quantum information tasks.
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