Enhanced Lieb-Robinson bounds for commuting long-range interactions

Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In D spatial dimensions, interactions decaying as a power-law r−α with α > 2D + 1 exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold 2D + 1 is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form t r−α for any α > 0, and this scaling is sharp. In particular, the linear light cone occurs at α = 1 in any dimension. Part of our motivation stems from quantum error-correcting codes. As applications, we derive enhanced bounds on ground state correlations and an enhanced local perturbations perturb locally principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.