Beyond the random phase approximation for calculating Curie temperatures in ferromagnets: application to Fe, Ni, Co and monolayer CrI₃

The magnetic properties of solids are typically analyzed in terms of Heisenberg models where the electronic structure is approximated by interacting localized spins. However, even in such models the evaluation of thermodynamic properties constitutes a major challenge and is usually handled by a mean field decoupling scheme. The random phase approximation (RPA) comprises a common approach and is often applied to evaluate critical temperatures although it is well known that the method is only accurate well below the critical temperature. In the present work we compare the performance of the RPA with a different decoupling scheme proposed by Callen as well as the mean field decoupling of interacting Holstein-Primakoff (HP) magnons. We consider three-dimensional (3D) as well as two-dimensional (2D) model systems where the Curie temperature is governed by anisotropy. In 3D, the Callen method is the most accurate in the classical limit, and we show that the Callen decoupling (CD) produces the best agreement with experiments for bcc Fe, fcc Ni and fcc Co with exchange interactions obtained from first principles. In contrast, for low spin systems where a quantum mechanical treatment is pertinent, the HP and RPA methods are superior to the CD. In 2D systems with magnetic order driven by single-ion anisotropy, it is shown that HP fails rather dramatically and both RPA and Callen approaches severely overestimates Curie temperatures. The most accurate approach is then constructed by combining RPA with the CD of single-ion anisotropy, which yields the correct lack of order for S = 1 / 2 . We exemplify this by the case of monolayer CrI₃ using exchange constant extracted from experiments.