We study the von Neumann and R\'enyi bipartite entanglement entropies in thethermodynamic limit of many-body quantum states with spin-s sites, that possessfull symmetry under exchange of sites. It turns out that there is essentially aone-to-one correspondence between such thermodynamic states and probabilitymeasures on CP^{2s}. Let a measure be supported on a set of possibly fractalreal dimension d with respect to the Study-Fubini metric of CP^{2s}. Let m bethe number of sites in a subsystem of the bipartition. We give evidence that inthe 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 theStudy-Fubini metric, we give evidence that the correction term is simplyrelated to the entropy associated to f: the geometric entropy of geometricquantum mechanics. This extends results obtained by the authors in a recentletter where the spin-1/2 case was considered. Here we provide more examples aswell as detailed accounts of the ideas and computations leading to thesegeneral results. For special choices of the state in the spin-s situation, werecover the scaling behaviour previously observed by Popkov et al., showingthat their result is but a special case of a more general scaling law.
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