Disordered Lieb-Robinson bounds in one dimension

TL;DR

This paper refines Lieb-Robinson bounds for disordered one-dimensional spin chains, revealing how heavy-tailed disorder suppresses ballistic operator growth and introducing site-specific bounds to better understand complex dynamics.

Contribution

It provides new bounds for disordered systems, identifies the correct dynamical exponent under heavy-tailed disorder, and distinguishes between global and site-specific Lieb-Robinson bounds.

Findings

Heavy-tailed disorder suppresses ballistic growth

Site-specific bounds can differ dramatically from global bounds

The dynamical exponent identified is proven to be tight

Abstract

By tightening the conventional Lieb-Robinson bounds to better handle systems which lack translation invariance, we determine the extent to which "weak links" suppress operator growth in disordered one-dimensional spin chains. In particular, we prove that ballistic growth is impossible when the distribution of coupling strengths $\mu(J)$ has a sufficiently heavy tail at small $J$, and identify the correct dynamical exponent to use instead. Furthermore, through a detailed analysis of the special case in which the couplings are genuinely random and independent, we find that the standard formulation of Lieb-Robinson bounds is insufficient to capture the complexity of the dynamics -- we must distinguish between bounds which hold for all sites of the chain and bounds which hold for a subsequence of sites, and we show by explicit example that these two can have dramatically different behaviors. All the same, our result for the dynamical exponent is tight, in that we prove by counterexample that there cannot exist any Lieb-Robinson bound with a smaller exponent. We close by discussing the implications of our results, both major and minor, for numerous applications ranging from quench dynamics to the structure of ground states.