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- Title
Microcanonical Analysis of the Curie-Weiss Anisotropic Quantum Heisenberg Model in a Magnetic Field.
- Authors
Olivier, Gerrit; Kastner, Michael
- Abstract
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 $$s(e,\varvec{m})$$ is obtained as a function of the energy $$e$$ and the magnetization vector $${\varvec{m}}$$ in the thermodynamic limit. Since, for this model, $$e$$ is uniquely determined by $${\varvec{m}}$$ , the same information can be encoded either in $$s(\varvec{m})$$ or $$s(e,m_1,m_2)$$ . Although these two entropies correspond to the same physical setting of fixed $$e$$ and $${\varvec{m}}$$ , their concavity properties differ. The entropy $$s_{{\varvec{h}}}(u)$$ , describing the model at fixed total energy $$u$$ and in a homogeneous external magnetic field $${\varvec{h}}$$ of arbitrary direction, is obtained by reduction from the nonconcave entropy $$s(e,m_1,m_2)$$ . In doing so, concavity, and therefore equivalence of ensembles, is restored. $$s_{{\varvec{h}}}(u)$$ has nonanalyticities on surfaces of co-dimension 1 in the $$(u,\varvec{h})$$ -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.
- Subjects
CURIE-Weiss law; QUANTUM Heisenberg model; MAGNETIC fields; MICROCANONICAL ensemble; PHASE diagrams
- Publication
Journal of Statistical Physics, 2014, Vol 157, Issue 3, p456
- ISSN
0022-4715
- Publication type
Article
- DOI
10.1007/s10955-014-1093-9