Lieb-Robinson bounds and out-of-time order correlators in a long-range spin chain
Luis Colmenarez, David J. Luitz
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 ) = [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 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 < 1 and to linear light cones for > 1. The asymptotic form of the tails of C(r, t ) t/r is described by short-time perturbation theory, which is valid at short times and long distances.