We prove that the entanglement entropy of any state evolved under an arbitrary 1/rα long-range-interacting D-dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area for any α > D + 1. We also prove that for any α > 2D + 2, the ground state of such a Hamiltonian satisfies the entanglement area law if it can be transformed along a gapped adiabatic path into a ground state known to satisfy the area law. These results significantly generalize their existing counterparts for short-range interacting systems, and are useful for identifying dynamical phase transitions and quantum phase transitions in the presence of long-range interactions.

%B Physical Review Letters %V 119 %P 050501 %8 2017/07/31 %G eng %U https://arxiv.org/abs/1702.05368 %N 5 %R 10.1103/PhysRevLett.119.050501