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

Tomotaka Kuwahara; Keiji Saito

doi:10.1103/PhysRevX.10.031010

paper

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

arXiv:1910.14477 · doi:10.1103/PhysRevX.10.031010

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 $R^{- α}$ with distance $R$ . We prove the existence of the linear light cone for $α > 2 D + 1$ ( $D$ : the spatial dimension), where we obtain the Lieb--Robinson bound as $∥ [O_{i} (t), O_{j}] ∥ ≲ t^{2 D + 1} (R - v ˉ t)^{- α}$ with $v ˉ = O (1)$ 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 $α < 2 D + 1$ . In the regime of $α > 2 D + 1$ , our result characterizes the best general constraints on the information spreading.

arXiv comment11 pages + 53 pages, 10 figures. [v.2] Typos are corrected and readability is improved. The result is slightly improved for few-body Hamiltonians. A discussion on the optimality is added. [v.3] Readability is further improved. [v4] Published version

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

Abstract

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