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In this work, we prove a new family of Lieb-Robinson bounds for lattice spinsystems with long-range interactions. Our results apply for arbitrary $k$-bodyinteractions, so long as they decay with a power-law greater than $kd$, where$d$ is the dimension of the system. More precisely, we require that the sum ofthe norm of terms with diameter greater than or equal to $R$, acting on any onesite, decays as a power-law $1/R^\alpha$, with $\alpha > d$. These new boundsallow us to prove that, at any fixed time, the spatial decay of quantuminformation follows arbitrarily closely to $1/r^{\alpha}$. Moreover, we definea new light-cone for power-law interacting quantum systems, which captures theregion of the system where changing the Hamiltonian can affect the evolution ofa local operator. In short-range interacting systems, this light-cone agreeswith the conventional definition. However, in long-range interacting systems,our definition yields a stricter light-cone, which is more relevant in mostphysical contexts.

An improved Lieb-Robinson bound for many-body Hamiltonians with power-law interactions