Hilbert space fragmentation in a long-range system

We study the role of the interaction range on the Hilbert-space fragmentation and many-body scar states considering a spin-1/2 many-body Hamiltonian describing a generalized Fredkin spin chain. We show that both scar states and weak fragmentation of the Hilbert space survive for almost any range of the coupling. Moreover, when the interaction range is small enough, there are sectors with definite symmetries such that the ratio between the dimension of their largest fragment and their dimension decays algebraically with the system size. Finally we investigate the effects of such structures of the Hilbert space on the out-of-equilibrium dynamics, triggered by certain initial states, characterized by either local persistent oscillations or nonuniform stationary profile of the magnetization.