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Is there any first principle method available to study phase transitions at the critical region for ferromagnetism (FM) or antiferromagnetism (AFM)?

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Yes, this topic is pretty mature I believe to FM and AFM by using quantum mechanics and specifically density functional theory. More specifically, the mean-field theory and Ising directions are essential to calculate phase transition temperature in ferromagnetic or anti-ferromagnetic materials. I'm not going to go through all the mathematical details here, which is beyond the scope of this SE I believe, but you could find the more solid theoretical background of this topic from outstanding work of Gyorffy et. al. that described how one could construct a self-consistent model based on Pauli equation and derive the magnetic momentum field from first-principle calculations.

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