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_ip 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.

References

This question remains unresolved even in the case of the standard Bose-Hubbard model with nearest-neighbor hopping and on-site quadratic interaction, underscoring its conceptual importance.

— Enhanced Lieb-Robinson bounds for a class of Bose-Hubbard type Hamiltonians  (2405.04672 - Kuwahara et al., 2024) in Introduction, Section 1 (after Question 1)

Our bounds give almost-ballistic propagation at fixed accuracy, with radii growing as $t{1+\eta}$ for every $\eta>0$. A finite-velocity exponential light cone remains open, since the constants grow with the moment order and do not permit taking $p\to\infty$ directly.

— Dissipation-Enabled Operator-Norm Locality Bounds for Bose-Hubbard Hamiltonians  (2610.01669 - Lemm et al., 1 Oct 2026) in Section Discussion