Calculating critical temperatures for ferromagnetic order in two-dimensional materials

Magnetic order in two-dimensional (2D) materials is intimately coupled to\nmagnetic anisotropy (MA) since the Mermin-Wagner theorem implies that\nrotational symmetry cannot be spontaneously broken at finite temperatures in\n2D. Large MA thus comprises a key ingredient in the search for magnetic 2D\nmaterials that retains the magnetic order above room temperature. Magnetic\ninteractions are typically modeled in terms of Heisenberg models and the\ntemperature dependence on magnetic properties can be obtained with the Random\nPhase Approximation (RPA), which treats magnon interactions at the mean-field\nlevel. In the present work we show that large MA gives rise to strong\nmagnon-magnon interactions that leads to a drastic failure of the RPA. We then\ndemonstrate that classical Monte Carlo (MC) simulations correctly describe the\ncritical temperatures in the large MA limit and agree with RPA when the MA\nbecomes small. A fit of the MC results leads to a simple expression for the\ncritical temperatures as a function of MA and exchange coupling constants,\nwhich significantly simplifies the theoretical search for new 2D magnetic\nmaterials with high critical temperatures. The expression is tested on a\nmonolayer of CrI$_3$, which were recently observed to exhibit ferromagnetic\norder below 45 K and we find excellent agreement with the experimental value.\n