Abstract
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 $R^{-\alpha}$ with distance $R$. We prove the existence of the linear light cone for $\alpha>2D+1$ ($D$: the spatial dimension), where we obtain the Lieb--Robinson bound as $\|[O_i(t),O_j]\|\lesssim{t}^{2D+1}(R-\bar{v}t)^{-\alpha}$ with $\bar{v}=\mathcal{O}(1)$ for two arbitrary operators $O_i$ and $O_j$ 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 $\alpha<2D+1$. In the regime of $\alpha>2D+1$, our result characterizes the best general constraints on the information spreading.
Highlights
In deep understanding of many-body physics, we necessarily encounter the question on how fast information propagates in the dynamics
We rigorously prove that a linear light cone is obtained in generic long-range interacting systems under the condition of α > 2D þ 1
We investigate the achievability of our LiebRobinson bound and the possibility to violate the linear light cone for α < 2D þ 1, by which we show that our bound is the best general upper bound
Summary
In deep understanding of many-body physics, we necessarily encounter the question on how fast information propagates in the dynamics. We rigorously prove that a linear light cone is obtained in generic long-range interacting systems under the condition of α > 2D þ 1. In onedimensional two-body interacting systems, the long-range Lieb-Robinson bound has been proved very recently in the form of k1⁄2OiðtÞ; Ojk ≲ t=R for α > 3 [112], which gives a nontrivial upper bound up to the time t 1⁄4 OðRÞ. We consider a quantum-state transfer protocol through the dynamics on a spin network and give an explicit example that achieves our Lieb-Robinson bound for α > 2D þ 1 and violates the linear light cone for α < 2D þ 1. Our protocol is applied to (1=2)-spin systems and comprises only the Ising-type longrange interactions and simple short-range interactions that generate the controlled NOT gate operation This example ensures the optimality of our results.
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