Dynamics and correlations in Motzkin and Fredkin spin chains

The Motzkin and Fredkin quantum spin chains are described by frustration-free Hamiltonians recently introduced and studied because of their anomalous behaviors in the correlation functions and in the entanglement properties. In this paper we analyze their quantum dynamical properties, focusing in particular on the time evolution of the excitations driven by a quantum quench, looking at the correlations functions of spin operators defined along different directions, and discussing the results in relation with the cluster decomposition property. We show that, for such models, one can have absence of cone-like structure in the propagation of the excitations after a quantum quench, together with a violation of the cluster decomposition property for the ground state characterized by only a logarithmic violation of the area law for the entanglement entropy.