An improved Lieb-Robinson bound for many-body Hamiltonians with power-law interactionsDominic V. Else, Francisco Machado, Chetan Nayak, Norman Y. Yao
In this work, we prove a new family of Lieb-Robinson bounds for lattice spin systems with long-range interactions. Our results apply for arbitrary \(k\)-body interactions, 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 of the norm of terms with diameter greater than or equal to \(R\), acting on any one site, decays as a power-law \(1/R^\alpha\), with \(\alpha > d\). These new bounds allow us to prove that, at any fixed time, the spatial decay of quantum information follows arbitrarily closely to \(1/r^\alpha\). Moreover, we define a new light-cone 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.