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

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