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 of spin-spin interaction is $\sim 1\mathrm{/}{r}^3$, where $r$ is a distance between those spins. However, NNI can not approximate the long time evolution in such systems. We consider the system with the intensity of the spin-spin interaction $\sim 1\mathrm{/}{r}^{\alpha }$, $\alpha \ge 3$, and find the low boundary ${\alpha }_c$ of applicability of the NNI to the evolution of an arbitrary one-excitation initial quantum state in the homogeneous spin chain governed by the $XX$-Hamiltonian. We obtain the logarithmic dependence of ${\alpha }_c$ on the chain length.