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

TL;DR

This paper investigates the spreading of information in long-range quantum spin chains by analyzing out-of-time-order correlators and Lieb-Robinson bounds, revealing different light cone behaviors depending on interaction decay.

Contribution

It provides a numerical study connecting Lieb-Robinson bounds with out-of-time order correlators in long-range spin chains, showing asymptotic behaviors and light cone structures.

Findings

Power law light cones for $ ext{α}<1$

Linear light cones for $ ext{α}>1$

Operator norms exhibit same asymptotic behavior

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 $1/r^\alpha$ 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.