Propagation of information in long-range interacting lattice systems

Propagation of information in short-range interacting lattice systems is restricted to within a linear ``light cone,'' as demonstrated by the well-known Lieb-Robinson bound, thus ensuring a well defined notion of maximum propagation velocity. Whether long-ranged interactions can lead to a different shape of this light cone, and the divergence of the associated velocity, is an important but largely unexplored question. We prove that for a wide class of long-range interacting lattice systems, a linear light cone still exists for certain regions of space and time, and for some experimentally relevant classes of models this linear light cone persists in the entire space-time. We then give counter-examples showing that, for well-engineered lattice system, long range interactions can indeed give rise to a sub-linear ``light cone,'' and thus a divergent speed of information propagation.