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We study one-dimensional systems of $N$ particles in a one-dimensionalharmonic trap with an inverse power law interaction $\sim|x|^{-d}$. Within theframework of the harmonic approximation we derive, in the strong interactionlimit, the Schmidt decomposition of the one-particle reduced density matrix andinvestigate the nature of the degeneracy appearing in its spectrum.Furthermore, the ground-state asymptotic occupancies and their natural orbitalsare derived in closed analytic form, which enables their easy determination fora wide range of values of $N$. A closed form asymptotic expression for the vonNeumann entanglement entropy is also provided and its dependence on $N$ isdiscussed for the systems with $d=1$ (charged particles) and with $d=3$(dipolar particles).

The von Neumann entanglement entropy for Wigner-crystal states in one dimensional N-particle systems