Strictly Linear Light Cones in Long-Range Interacting Systems of Arbitrary Dimensions

In locally interacting quantum many-body systems, the velocity of information propagation is finitely bounded, and a linear light cone can be defined. Outside the light cone, the amount of information rapidly decays with distance. When systems have long-range interactions, it is highly nontrivial whether such a linear light cone exists. Herein, we consider generic long-range interacting systems with decaying interactions, such as R-α with distance R. We prove the existence of the linear light cone for α>2D+1 (D, the spatial dimension), where we obtain the Lieb-Robinson bound as ∥[Oi(t),Oj]∥≲t2D+1(R-v¯t)-α with v¯=O(1) for two arbitrary operators Oi and Oj separated by a distance R. Moreover, we provide an explicit quantum-state transfer protocol that achieves the above bound up to a constant coefficient and violates the linear light cone for α<2D+1. In the regime of α>2D+1, our result characterizes the best general constraints on the information spreading.