Does a time-independent Bose-Hubbard Hamiltonian have a finite propagation speed for bounded-density initial states?

Determine whether a time-independent Bose-Hubbard type Hamiltonian of the form H = ∑_{i∼j} J_{i,j}(b_i^† b_j + b_j^† b_i) + ∑_{i} w(n_i), when prepared in a bounded-density initial state ρ, exhibits a finite speed of information propagation; equivalently, establish whether there exists a Lieb–Robinson bound with a time-independent velocity for such systems.

Background

Recent results show that for Bose–Hubbard-type models, Lieb–Robinson bounds can feature time-growing velocities (e.g., v ∼ t^{D−1}) for general bounded-density initial states, raising the possibility of accelerated information spreading. In contrast, for certain structured initial states or additional assumptions, velocities can be time-independent.

The paper frames the existence of a finite velocity for general bounded-density states as a central open question. Resolving this would clarify whether bosonic systems governed by time-independent Bose–Hubbard Hamiltonians necessarily possess a finite information propagation speed or can fundamentally exhibit acceleration.