Improving the t^{D−1} Lieb–Robinson scaling for the standard Bose–Hubbard model (p=2) in low dimensions

Determine whether, for the standard Bose–Hubbard model with quadratic on-site interaction (p = 2) in spatial dimensions D ≤ 3 and for broad classes of initial states, the known Lieb–Robinson bound with velocity scaling v ∼ t^{D−1} can be qualitatively improved to a strictly smaller time exponent (e.g., v ∼ t^α with α < D−1).

Background

The paper improves known bounds under higher-order on-site repulsion and translation invariance, but explicitly notes that the most relevant p=2 case lies beyond the current techniques. For p=2 in low dimensions, whether the general v ∼ t^{D−1} upper bound can be tightened remains unsettled.

Clarifying this would delineate the true light-cone structure for the standard Bose–Hubbard model in physically relevant dimensions and for broad initial-state classes.