  Monolayer MnO$_2$ is one of the few predicted two-dimensional (2D)
ferromagnets that has been experimentally synthesized and is commercially
available. The Mermin-Wagner theorem states that magnetic order in a 2D
material cannot persist unless magnetic anisotropy (MA) is present and
perpendicular to the plane, which permits a finite critical temperature.
Previous computational studies have predicted the magnetic ordering and Curie
temperature of 2D MnO$_2$ with DFT+U (Density Funtional Theory + Hubbard U
correction), with the results having a strong dependence on the Hubbard U
parameter. Diffusion Monte Carlo (DMC) is a correlated electronic structure
method that has had demonstrated success for the electronic and magnetic
properties of a variety of 2D and bulk systems since it has a weaker dependence
on the starting Hubbard parameter and density functional. In this study, we
used DMC and DFT+U to calculate the magnetic properties of monolayer MnO$_2$.
We found that the ferromagnetic ordering is more favorable than
antiferromagnetic and determined a statistical bound on the magnetic exchange
parameter ($J$). In addition, we performed spin-orbit MA energy calculations
using DFT+U and using our DMC and DFT+U parameters along with the analytical
model of Torelli and Olsen, we estimated an upper bound of 28.8 K for the
critical temperature of MnO$_2$. These QMC results intend to serve as an
accurate theoretical benchmark, necessary for the realization and development
of future 2D magnetic devices. These results also demonstrate the need for
accurate methodologies to predict magnetic properties of correlated 2D
materials.
