Revealing effects of the local dimension on a variable-range interacting model by connecting Lieb-Robinson bounds and multipartite entanglement

A spin-$s$ variable-range interacting Ising model may display qualitatively different behaviors depending on the fall-off rate of the interactions, as already seen in equilibrium studies of spin-1/2 systems. We propose a dynamical method using weighted graph states, generated through time evolution that confirms the existence of the transition point in the fall-off rate for the spin-$s$ Ising model. Moreover, the dependence of local dimension on information spreading and multipartite entanglement profile in this model remains unclear, which we establish here. In particular, our analysis shows that the maximum of genuine multipartite entanglement (GME) with the fall-off rate serves as a clear indicator of the information spreading, which aligns with changes in the profile of the Lieb-Robinson bound. Further, in the case of an open chain, the spread of information is related to the divergence in the first derivative of GME. Additionally, we validate this signature by performing a scaling analysis of the time-averaged mutual information.