  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 $\sim 1/r^3$, 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 $\sim 1/r^{\alpha}$, $\alpha\ge 3$, 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.

Quantum Physics