  We study the von Neumann and R\'enyi bipartite entanglement entropies in the
thermodynamic limit of many-body quantum states with spin-s sites, that possess
full symmetry under exchange of sites. It turns out that there is essentially a
one-to-one correspondence between such thermodynamic states and probability
measures on CP^{2s}. Let a measure be supported on a set of possibly fractal
real dimension d with respect to the Study-Fubini metric of CP^{2s}. Let m be
the number of sites in a subsystem of the bipartition. We give evidence that in
the limit where m goes to infinity, the entanglement entropy diverges like
(d/2)log(m). Further, if the measure is supported on a submanifold of CP^{2s}
and can be described by a density f with respect to the metric induced by the
Study-Fubini metric, we give evidence that the correction term is simply
related to the entropy associated to f: the geometric entropy of geometric
quantum mechanics. This extends results obtained by the authors in a recent
letter where the spin-1/2 case was considered. Here we provide more examples as
well as detailed accounts of the ideas and computations leading to these
general results. For special choices of the state in the spin-s situation, we
recover the scaling behaviour previously observed by Popkov et al., showing
that their result is but a special case of a more general scaling law.
