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