Does accelerated information propagation occur in Bose–Hubbard models with bounded-density initial states?

Determine whether accelerated information propagation truly occurs for time-independent Bose–Hubbard Hamiltonians prepared in bounded-density initial states; specifically, decide if any valid Lieb–Robinson bound must involve a velocity that grows with time, implying the absence of a time-independent maximal propagation speed.

Background

The work of Kuwahara, Vu, and Saito established Lieb–Robinson bounds with time-growing velocities v ∼ t^{D−1} for general bounded-density initial states, suggesting possible acceleration, while other regimes (e.g., special initial states or added constraints) yield time-independent velocities.

The paper emphasizes that it is unclear whether such acceleration actually manifests in Bose–Hubbard dynamics starting from bounded-density states, highlighting a gap between upper bounds and realizable dynamics.