Scrambling and entanglement spreading in long-range spin chains

We study scrambling in connection with multipartite entanglement dynamics in regular and chaotic long-range spin chains, characterized by a well-defined semi-classical limit. For regular dynamics, scrambling and entanglement dynamics are found to be very different: up to the Ehrenfest time, they rise side by side, departing only afterward. Entanglement saturates and becomes extensively multipartite, while scrambling, characterized by the dynamic of the square commutator of initially commuting variables, continues its growth up to the recurrence time. Remarkably, the exponential growth of the latter emerges not only in the chaotic case but also in the regular one, when the dynamics occurs at a dynamical critical point.