Entanglement entropy in collective models

We discuss the behaviour of the entanglement entropy of the ground state in various collective systems. Results for general quadratic two-mode boson models are given, yielding the relation between quantum phase transitions of the system (signalled by a divergence of the entanglement entropy) and the excitation energies. Such systems naturally arise when expanding collective spin Hamiltonians at leading order via the Holstein–Primakoff mapping. In a second step, we analyse several such models (the Dicke model, the two-level Bardeen–Cooper–Schrieffer model, the Lieb–Mattis model and the Lipkin–Meshkov–Glick model) and investigate the properties of the entanglement entropy over the whole parameter range. We show that when the system contains gapless excitations the entanglement entropy of the ground state diverges with increasing system size. We derive and classify the scaling behaviours that can be met.