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.
Highlights
One of the most general concepts to study the dynamical properties of quantum many-body systems is the dynamics of quantum information, generalizing the spreading of all possible types of correlations in the system
Our analysis suggests that both the average and the largest singular value of C (r, t ) have the same asymptotic behavior: We find a linear growth in time of C (r, t ) 2 at short times, and a power-law decay with distance at long distances with the exponent α, which can be understood from perturbation theory
We analyze in detail the space-time profile of the operator norm of the commutator C (r, t ) 2 of the long-range XXX chain (2) and compare it to the case of the normalized Frobenius norm (OTOC), for which a very detailed analysis can be found in Ref. [13]
Summary
One of the most general concepts to study the dynamical properties of quantum many-body systems is the dynamics of quantum information, generalizing the spreading of all possible types of correlations in the system. While in relativistic systems the spreading of information is limited by the speed of light, there is no such strict limit in nonrelativistic quantum mechanics It was shown by Lieb and Robinson [4] that quantum systems with short-range interactions exhibit a similar, nonuniversal speed limit, implying a causal structure. Large but finite α is consistently found to exhibit asymptotically short-range behavior [13,19,21,23,34] It was argued by Gong et al that a linear light-cone structure persists for α > D [15], which is supported by numerical simulations [13,34]. Our analysis suggests that both the average and the largest singular value of C (r, t ) have the same asymptotic behavior: We find a linear growth in time of C (r, t ) 2 at short times, and a power-law decay with distance at long distances with the exponent α, which can be understood from perturbation theory
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