Disordered Lieb-Robinson bounds in one dimension

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.

23 pages, comments welcome

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Cited by in corpus (6)

• Many-Body Localization in the Age of Classical Computing
• Speed limits and locality in many-body quantum dynamics
• The dynamical $\alpha$-Rényi entropies of local Hamiltonians grow at most linearly in time
• Sub-ballistic operator growth in spin chains with heavy-tailed random fields
• Lieb-Robinson correlation function for the quantum transverse field Ising model
• Lieb-Robinson bounds with exponential-in-volume tails