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arXiv:1910.14477 · Physical Review X

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

Tomotaka Kuwahara, Keiji Saito

Published Jul 13, 2020• 112 citations
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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 > 2D þ 1 (D, the spatial dimension), where we obtain the Lieb-Robinson bound as k½O i ðtÞ; O j k ≲ t 2Dþ1 ðR -vtÞ -α with v ¼ Oð1Þ 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 < 2D þ 1.In the regime of α > 2D þ 1, our result characterizes the best general constraints on the information spreading.

DOI
10.1103/physrevx.10.031010

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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 þ 1 (D, the spatial dimension), where we obtain the Lieb-Robinson bound as k½O i ðtÞ; O j k ≲ t 2Dþ1 ðR -vtÞ -α with v ¼ Oð1Þ 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 < 2D þ 1.In the regime of α > 2D þ 1, our result characterizes the best general constraints on the information spreading.
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The Lieb-Robinson theorem states that information propagates with a finite velocity in quantum systems on a lattice with nearest-neighbor interactions. What are the speed limits on information propagation in quantum systems with power-law interactions, which decay as 1/r^{α} at distance r? Here, we present a definitive answer to this question for all exponents α>2d and all spatial dimensions d. Schematically, information takes time at least r^{min{1,α-2d}} to propagate a distance r. As recent state transfer protocols saturate this bound, our work closes a decades-long hunt for optimal Lieb-Robinson bounds on quantum information dynamics with power-law interactions.
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Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In DDD spatial dimensions, interactions decaying as a power-law r−αr^{-α}r−α with α>2D+1α> 2 D+1α>2D+1 exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold 2D+12D +12D+1 is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form t r−αt \, r^{-α}tr−α for any α>0α>0α>0, and this scaling is sharp. In particular, the linear light cone occurs at α=1α= 1α=1 in any dimension. Part of our motivation stems from quantum error-correcting codes. As applications, we derive enhanced bounds on ground state correlations and an enhanced local perturbations perturb locally (LPPL) principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.
In nonrelativistic quantum theories with short-range Hamiltonians, a velocity vvv can be chosen such that the influence of any local perturbation is approximately confined to within a distance rrr until a time $t\ensuremath{\sim}r/v$, thereby defining a linear light cone and giving rise to an emergent notion of locality. In systems with power-law ($1/{r}^{\ensuremath{\alpha}}$) interactions, when $\ensuremath{\alpha}$ exceeds the dimension DDD, an analogous bound confines influences to within a distance rrr only until a time $t\ensuremath{\sim}(\ensuremath{\alpha}/v)\mathrm{log}r$, suggesting that the velocity, as calculated from the slope of the light cone, may grow exponentially in time. We rule out this possibility; light cones of power-law interacting systems are bounded by a polynomial for $\ensuremath{\alpha}>2D$ and become linear as $\ensuremath{\alpha}\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results impose strong new constraints on the growth of correlations and the production of entangled states in a variety of rapidly emerging, long-range interacting atomic, molecular, and optical systems.