Finite-speed propagation for the standard Bose–Hubbard model (p=2)

Establish whether the standard Bose–Hubbard model with nearest-neighbor hopping and on-site quadratic interaction (two-body repulsion, p = 2), initialized in a bounded-density state, admits a finite, time-independent Lieb–Robinson velocity; equivalently, resolve whether information propagation has a finite maximal speed in this canonical case.

Background

While the paper obtains enhanced bounds under strong on-site repulsion n_i^p with large p and translation invariance, the most physically relevant case is the standard Bose–Hubbard model with quadratic on-site interaction (p=2).

The authors explicitly state that even this canonical case remains unresolved regarding whether a finite speed of information propagation holds for bounded-density initial states.