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Entanglement in N-harmonium: bosons and fermions

The ground-state entanglement of a single particle of the N-harmonium system (i.e., a completely-integrable model of $N$ particles where both the confinement and the two-particle interaction are harmonic) is shown to be analytically determined in terms of $N$ and the relative interaction strength. For bosons, we compute the von Neumann entropy of the one-body reduced density matrix by using the corresponding natural occupation numbers. There exists a critical number $N_c$ of particles so that below it, for positive values of the coupling constant, the entanglement grows when the number of particles is increasing; the opposite occurs for $N > N_c$. For fermions, we compute the one-body reduced density matrix for the closed-shell spinned case. In the strong coupling regime, the linear entropy of the system decreases when $N$ is growing. For fixed $N$, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Moreover, the spatial and spin contributions to the total entanglement are found to be of comparable size.