  We study equilibrium states of quantum spin systems with non-additive
long-range interactions by adopting an appropriate scaling of the interaction
strength, i.e., the so called Kac prescription. In classical spin systems, it
is known that the equilibrium free energy is obtained by minimizing the free
energy functional over the coarse-grained magnetization. Here we show that it
is also true for quantum spin systems. From this observation, it is found that
when the canonical ensemble and the microcanonical ensemble are not equivalent
in some parameter region, it is not necessarily justified to replace the actual
long-range interaction by the infinite-range interaction (Curie-Weiss type
interaction). On the other hand, in the parameter region where the two
ensembles are equivalent, this replacement is always justified. We examine the
Heisenberg XXZ model as an illustrative example, and discuss the relation to
experiments.
