Scaling of the light-cone in the time evolution of long-range interacting quantum spin-chains

We provide a universal picture of correlation spreading in spin systems with long-range spin-exchange term. Using the quasi-particle picture, we demonstrate that the space-time correlation map exhibits a two-fold structure. While the outer structure determines the correlation edge, the inner structure determines the propagation of local extrema. The outer structure depends on the spectrum and on the observable and it is sub-ballistic for observable without a strong infrared divergence. This means that the spreading is slower than the one predicted by the Lieb-Robinson bounds for these models. On the other side, the inner structure depends just on the energy spectrum and it is ballistic for gapped Hamiltonians and faster-than-ballistic for gapless Hamiltonians. Our results shed new light on correlation spreading in quantum systems and provides a unified picture to interpret previous results present in the literature.