Predicting the N\'eel temperatures in general helimagnetic materials: a comparison between mean field theory, random phase approximation, renormalized spin wave theory and classical Monte Carlo simulations

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 non-trivial. 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), MnF$_2$ (altermagnet), Cr$_2$O$_3$ and Fe$_2$O$_3$ (two site N\'eel states) and Ba$_3$NbFe$_3$Si$_2$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.