Predicting Néel temperatures in helimagnetic materials

The critical temperature for magnetic order comprises a crucial property of any magnetic material and ranges from a few Kelvin in certain antiferromagnets to 1400 K in ferromagnetic Co. However, the prediction of critical temperatures based on, for example, a spin-wave dispersion is in general nontrivial. For ferromagnets and simple collinear antiferromagnets, estimates may be obtained from the Heisenberg model using either renormalized spin-wave theory or the Green's function random-phase approximation (RPA), but a systematic assessment of the accuracy of such approaches seems to be lacking in the literature. In this work, we propose generalizations of both renormalized spin-wave theory and RPA to calculate the critical temperatures of single-$Q$ helimagnetic ground states, which include ferromagnets and antiferromagnets as special cases. We compare the methods to classical Monte Carlo simulations and mean-field theory, using experimental exchange parameters for a wide range of materials; MnO and NiO (single site N\'eel ground states), ${\mathrm{MnF}}_{2}$ (altermagnet), ${\mathrm{Cr}}_{2}{\mathrm{O}}_{3}$ and ${\mathrm{Fe}}_{2}{\mathrm{O}}_{3}$ (two site N\'eel states), and ${\mathrm{Ba}}_{3}{\mathrm{NbFe}}_{3}{\mathrm{Si}}_{2}{\mathrm{O}}_{14}$ (incommensurate helimagnet). In all cases, we observe that predictions from RPA are in excellent agreement with experimental values and RPA thus constitutes a rather reliable all-purpose method for calculating critical temperatures.