Nearest-neighbor approximation in one-excitation state evolution along spin-1/2 chain governed by XX-Hamiltonian

The approximation of nearest neighbor interaction (NNI) is widely used in short-time spin dynamics with dipole-dipole interactions (DDI) when the intensity spin-spin $\sim 1/r^3$, where $r$ a distance between those spins. However, NNI can not approximate long time evolution such systems. We consider system 1/r^{\alpha}$, $\alpha\ge 3$, and find low boundary $\alpha_c$ applicability to an arbitrary one-excitation initial quantum state homogeneous chain governed by $XX$-Hamiltonian. obtain logarithmic dependence on length.