Lieb-Robinson bounds and out-of-time order correlators in a long-range spin chain

Abstract

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)=[Ai(t),Bi+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/rα 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 α<1 and to linear light cones for α>1. The asymptotic form of the tails of C(r,t)∝t/rα is described by short-time perturbation theory, which is valid at short times and long distances.