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. A critical number, Nc, of particles exists, and below it, for positive values of the coupling constant, the entanglement grows when the number of particles increases; 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 grows. For fixed N, the entanglement is found (a) to decrease (increase) for negatively (positively) increases values of the coupling constant, and (b) to grow when the energy increases. Moreover, the spatial and spin contributions to the total entanglement are found to be of comparable size.