Strictly linear light cones in long-range interacting systems of arbitrary dimensions

TL;DR

This paper establishes the existence of linear light cones in long-range interacting quantum systems for decay exponents greater than 2D+1, providing bounds on information propagation and a protocol that violates these bounds for smaller exponents.

Contribution

It proves the existence of linear light cones in long-range systems for >2D+1 and offers an explicit protocol that surpasses these bounds when <2D+1.

Findings

Linear light cone exists for >2D+1 with specific Lieb-Robinson bounds.

An explicit quantum-state transfer protocol violates the light cone for <2D+1.

Characterizes optimal constraints on information spreading in long-range systems.

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.