  We present a perturbative method to compute the ground state entanglement
entropy for interacting systems. We apply it to a collective model of mutually
interacting spins in a magnetic field. At the quantum critical point, the
entanglement entropy scales logarithmically with the subsystem size, the system
size, and the anisotropy parameter. We determine the corresponding scaling
prefactors and evaluate the leading finite-size correction to the entropy. Our
analytical predictions are in perfect agreement with numerical results.
