Emergent locality in systems with power-law interactions

Locality imposes stringent constraints on the spreading of information in nonrelativistic quantum systems, which is reminiscent of a "light-cone," a casual structure arising in their relativistic counterparts. Long-range interactions can potentially soften such constraints, allowing almost instantaneous long jumps of particles, thus defying causality. Since interactions decaying as a power-law with distance, $r^{-\alpha}$, are ubiquitous in nature, it is pertinent to understand what is the fate of causality and information spreading in such systems. Using a numerically exact technique we address these questions by studying the out-of-time-order correlation function of a representative generic system in one-dimension. We show that while the interactions are long-range, their effect on information spreading is asymptotically negligible as long as $\alpha>1$. In this range we find a complex compound behavior, where after a short transient a fully local behavior emerges, yielding asymptotic "light-cones" virtually indistinguishable from "light-cones" in corresponding local models. The long-range nature of the interaction is only expressed in the power-law leaking of information from the "light-cone," with the same exponent as the exponent of the interaction, $\alpha$. Our results directly imply that all previously obtained rigorous bounds on information spreading in long-range interacting systems are not tight, and thus could be improved.