Entanglement area laws for long-range interacting systems

We prove that the entanglement entropy of any state evolved under an arbitrary $1\mathrm{/}{r}^{\alpha }$ long-range-interacting D-dimensional lattice spin Hamiltonian cannot change faster than a rate proportional to the boundary area for any $\alpha >D+1$. We also prove that for any $\alpha >2D+2$, 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.