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arXiv:2611.00776

Lieb-Robinson Light Cones in Long-Range Bosonic Systems

Carla Rubiliani, Cheng Shang, Tomotaka Kuwahara

math-phquant-phSubmitted Oct 31, 2026• Announced Nov 3, 2026
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Abstract

Lieb-Robinson bounds formulate emergent causality in quantum many-body dynamics. In this Letter, we study interacting bosonic systems with long-range interactions, in which both the standard assumptions of short-range interactions and uniformly bounded local energy scales are absent. We derive upper and lower bounds whose effective light-cone exponents are governed by a nontrivial interplay between the decay exponent $α$ and the bosonic interaction order $p$. The propagation protocol concentrates bosons within blocks, creating a mediating sublattice with amplified interactions. For $p\ge4$ and $2d<α\le2d+1$, the resulting exponents coincide, determining the optimal leading light-cone power. The upper bound includes $α=2d+1$ with arbitrarily small exponent loss, using a fixed-$k$ extension of the bounded-interaction result. Our results reveal that the interplay between long-range couplings and unbounded bosonic degrees of freedom gives rise to a richer light-cone structure than in either long-range bounded systems or short-range bosonic systems alone.

Comments
10 pages, 1 figure; Supplementary Material: 75 pages, 6 figures, 1 table

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In this paper, we prove a family of Lieb-Robinson bounds for discrete spin systems with long-range interactions. Our results apply for arbitrary kkk-body interactions, so long as they decay with a power law greater than kdkdkd, where ddd is the dimension of the system. More precisely, we require that the sum of the norm of terms with diameter greater than or equal to RRR, acting on any one site, decays as a power law $1/{R}^{\ensuremath{\alpha}}$, with $\ensuremath{\alpha}>d$. These bounds allow us to prove that, at any fixed time, the spatial decay of a time evolved operator follows arbitrarily closely to $1/{r}^{\ensuremath{\alpha}}$. Moreover, we introduce an alternative light cone definition for power-law interacting quantum systems which captures the region of the system where changing the Hamiltonian can affect the evolution of a local operator. In short-range interacting systems, this light cone agrees with the conventional definition. However, in long-range interacting systems, our definition yields a stricter light cone, which is more relevant in most physical contexts.
Recent works have revealed the intricate effect of long-range interactions on information transport in quantum many-body systems: In D spatial dimensions, interactions decaying as a power-law r−α with α > 2D + 1 exhibit a Lieb-Robinson bound (LRB) with a linear light cone and the threshold 2D + 1 is sharp in general. Here, we observe that mutually commuting, long-range interactions satisfy an enhanced LRB of the form t r−α for any α > 0, and this scaling is sharp. In particular, the linear light cone occurs at α = 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 principle for which we adapt a recent subharmonicity argument of Wang-Hazzard. Similar enhancements hold for commuting interactions with stretched exponential decay.
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The dynamics of quantum systems strongly depends on the local structure of the Hamiltonian. For short-range interacting systems, the well-known Lieb–Robinson bound defines the effective light cone with an exponentially small error with respect to the spatial distance, whereas we can obtain only polynomially small errors for distance in long-range interacting systems. In this paper, we derive a qualitatively new bound for quantum dynamics by considering how many spins can correlate with each other after time evolution. Our bound characterizes the number of spins which support the many-body entanglement with exponentially small errors and is valid for a large class of Hamiltonians including long-range interacting systems. To demonstrate the advantage of our approach in quantum many-body systems, we apply our bound to prove several fundamental properties which have not been derived from the Lieb–Robinson bound.