Lieb-Robinson bounds in long range interacting spin chains

Lieb-Robinson bounds quantify the maximal speed of information spreading in nonrelativistic quantum systems.  We discuss the relation of Lieb-Robinson bounds to out-of-time order correlators, which correspond to different norms of commutators $C(r,t)=[A_i(t),B_{i+r}]$ of local operators.                                                                            

Using an exact Krylov space-time evolution technique, we calculate these two different norms of such commutators for the spin-1/2 Heisenberg chain with interactions decaying as a power law $1/r^\alpha$ with distance $r$. Our numerical analysis shows that both norms (operator norm and normalized Frobenius norm) exhibit the same asymptotic behavior, namely, a linear growth in time at short times and a power-law decay in space at long distance, leading asymptotically to power-law light cones for $\alpha<1$ and to linear light cones for $\alpha>1$. The asymptotic form of the tails of $C(r,t)\propto t/r^\alpha$ is described by short-time perturbation theory, which is valid at short times and long distances.