  We study the problem of ensemble equivalence in spin systems with short-range
interactions under the existence of a first-order phase transition. The
spherical model with nonlinear nearest-neighbour interactions is solved exactly
both for canonical and microcanonical ensembles. The result reveals apparent
ensemble inequivalence at the first-order transition point in the sense that
the microcanonical entropy is non-concave as a function of the energy and
consequently the specific heat is negative. In order to resolve the paradox, we
show that an unconventional saddle point should be chosen in the microcanonical
calculation that represents a phase separation. The XY model with non-linear
interactions is also studied by microcanonical Monte Carlo simulations in two
dimensions to see how this model behaves in comparison with the spherical
model.
