  Whether long-range interactions allow for a form of causality in
non-relativistic quantum models remains an open question with far-reaching
implications for the propagation of information and thermalization processes.
Here, we study the out-of-equilibrium dynamics of the one-dimensional
transverse Ising model with algebraic long-range exchange coupling. Using a
state of the art tensor-network approach, complemented by analytic calculations
and considering various observables, we show that a weak form of causality
emerges, characterized by non-universal dynamical exponents. While the local
spin and spin correlation causal edges are sub-ballistic, the causal region has
a rich internal structure, which, depending on the observable, displays
ballistic or super-ballistic features. In contrast, the causal region of
entanglement entropy is featureless and its edge is always ballistic,
irrespective of the interaction range. Our results shed light on the
propagation of information in long-range interacting lattice models and pave
the way to future experiments, which are discussed.
