Keiji Saito

In locally interacting quantum many-body systems, a velocity of the information propagation is finitely bounded and the linear light cone can be defined. Outside the light cone, amount of the information propagation rapidly decays with the distance. When systems have long-range interactions, it is highly nontrivial whether such a linear light cone exists or not. We herein consider generic long-range interacting systems with decaying interactions as $R^{-\alpha}$ with the distance $R$. We rigorously prove the existence of the linear light cone for $\alpha>2D+1$ ($D$: the spatial dimension), where we obtain the Lieb-Robinson bound as $\|[O_i(t),O_j]\| < t^{2D+1}(R-\bar{v}t)^{-\alpha}$ with $\bar{v}=O(1)$ for arbitrary two operators $O_i$ and $O_j$ separated by a distance $R$. Moreover, we give an explicit quantum-state transfer protocol that achieves the above bound up to a constant coefficient and violates the linear light cone for $\alpha<2D+1$. This implies that our Lieb-Robinson bound is the best general upper bound in the regime of $\alpha>2D+1$.