Symmetry-enhanced Lieb-Robinson bounds for a class of Bose-Hubbard type Hamiltonians

Several recent works have derived Lieb-Robinson bounds (LRBs) for Bose-Hubbard-type Hamiltonians. For certain structured initial states, e.g., vacuum perturbations or near-stationary states, information propagates with velocity $v \leq C$ . However, for general bounded-density initial states, it was shown by the first author, Vu, and Saito that the velocity can grow in time as $v \sim t^{D-1}$, where $D$ is the spatial dimension -- demonstrating the possibility of accelerated information spreading in bosonic systems. In this work, we introduce a new perspective on this phenomenon: we show that translation invariance combined with local $p$-body repulsion ($n^p$ with $p > D+1$) qualitatively alters the propagation behavior, leading to a bound of the form $v \sim t^{\frac{D}{p - D - 1}}$ for general bounded-energy-density initial states. In particular, this establishes for an almost-linear light cone at large $p$, in stark contrast to the previously found accelerated regimes. Our result identifies symmetry-driven constraints as a new mechanism for suppressing propagation speed in bosonic systems and thereby reframes the scope of what types of LRBs can hold. We further provide matching examples showing that, under the given assumptions, this bound is sharp -- no further improvement in the power of $t$ is possible without invoking additional dynamical constraints.