  A Hamiltonian describing the collective behaviour of N interacting spins can
be mapped to a bosonic one employing the Holstein-Primakoff realisation, at the
expense of having an infinite series in powers of the boson creation and
annihilation operators. Truncating this series up to quadratic terms allows for
the obtention of analytic solutions through a Bogoliubov transformation, which
becomes exact in the limit N -> infinit. The Hamiltonian exhibits a phase
transition from single spin excitations to a collective mode. In a vicinity of
this phase transition the truncated solutions predict the existence of
singularities for finite number of spins, which have no counterpart in the
exact diagonalization. Renormalisation allows to extract from these divergences
the exact behaviour of relevant observables with the number of spins around the
phase transition, and relate it with the class of universality to which the
model belongs. In the present work a detailed analysis of these aspects is
presented for the Lipkin model.
