  Exact solutions are obtained for the mean-field spherical model, with or
without an external magnetic field, for any finite or infinite number N of
degrees of freedom, both in the microcanonical and in the canonical ensemble.
The canonical result allows for an exact discussion of the loci of the Fisher
zeros of the canonical partition function. The microcanonical entropy is found
to be nonanalytic for arbitrary finite N. The mean-field spherical model of
finite size N is shown to be equivalent to a mixed isovector/isotensor
sigma-model on a lattice of two sites. Partial equivalence of statistical
ensembles is observed for the mean-field spherical model in the thermodynamic
limit. A discussion of the topology of certain state space submanifolds yields
insights into the relation of these topological quantities to the thermodynamic
behavior of the system in the presence of ensemble nonequivalence.
