  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.
