Lieb-Robinson bounds with exponential-in-volume tails

TL;DR

This paper extends Lieb-Robinson bounds to include exponential-in-volume tails, providing tighter constraints on operator growth and correlations in many-body quantum systems, with applications to simulation complexity and phase diagnostics.

Contribution

It introduces a new bound capturing volume-filling operator suppression, bridging cluster expansion intuition and Lieb-Robinson bounds, with practical implications for quantum simulation and phase analysis.

Findings

Volume-filling operators are suppressed by ^{-(r-vt)^d/(vt)^{d-1}} for r > vt.

Simulation resources scale as ^{O(t^{d-1})} for small error psilon.

Disorder operators exhibit volume-law suppression near the Ising point.

Abstract

Lieb-Robinson bounds demonstrate the emergence of locality in many-body quantum systems. Intuitively, Lieb-Robinson bounds state that with local or exponentially decaying interactions, the correlation that can be built up between two sites separated by distance $r$ after a time $t$ decays as $\exp(vt-r)$, where $v$ is the emergent Lieb-Robinson velocity. In many problems, it is important to also capture how much of an operator grows to act on $r^d$ sites in $d$ spatial dimensions. Perturbation theory and cluster expansion methods suggest that at short times, these volume-filling operators are suppressed as $\exp(-r^d)$ at short times. We confirm this intuition, showing that for $r > vt$, the volume-filling operator is suppressed by $\exp(-(r-vt)^d/(vt)^{d-1})$. This closes a conceptual and practical gap between the cluster expansion and the Lieb-Robinson bound. We then present two very different applications of this new bound. Firstly, we obtain improved bounds on the classical computational resources necessary to simulate many-body dynamics with error tolerance $\epsilon$ for any finite time $t$: as $\epsilon$ becomes sufficiently small, only $\epsilon^{-O(t^{d-1})}$ resources are needed. A protocol that likely saturates this bound is given. Secondly, we prove that disorder operators have volume-law suppression near the "solvable (Ising) point" in quantum phases with spontaneous symmetry breaking, which implies a new diagnostic for distinguishing many-body phases of quantum matter.