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Microcanonical analysis of the Curie-Weiss anisotropic quantum Heisenberg model in a magnetic field

The anisotropic quantum Heisenberg model with Curie-Weiss-type interactions is studied analytically in several variants of the microcanonical ensemble. (Non)equivalence of microcanonical and canonical ensembles is investigated by studying the concavity properties of entropies. The microcanonical entropy s(e,m) is obtained as a function of the energy e and the magnetization vector m in the thermodynamic limit. Since, for this model, e is uniquely determined by m, the same information can be encoded either in s(m) or s(e,m1,m2). Although these two entropies correspond to the same physical setting of fixed e and m, their concavity properties differ. The entropy s_h(u), describing the model at fixed total energy u and in a homogeneous external magnetic field h of arbitrary direction, is obtained by reduction from the nonconcave entropy s(e,m1,m2). In doing so, concavity, and therefore equivalence of ensembles, is restored. s_h(u) has nonanalyticities on surfaces of co-dimension 1 in the (u,h)-space. Projecting these surfaces into lower-dimensional phase diagrams, we observe that the resulting phase transition lines are situated in the positive-temperature region for some parameter values, and in the negative-temperature region for others. In the canonical setting of a system coupled to a heat bath of positive temperatures, the nonanalyticities in the microcanonical negative-temperature region cannot be observed, and this leads to a situation of effective nonequivalence even when formal equivalence holds.