  We use density functional theory based first-principles methods to study the
magnetism in a 2D hexagonal BN sheet induced by the different concentrations of
oxygen and silicon atoms substituting for nitrogen (O$_\mathrm{N}$) and boron
(Si$_\mathrm{B}$) respectively. We demonstrate the possible formation of three
distinct phases based on the magnetization energy calculated self-consistently
for the ferromagnetic (ME$_{\mathrm{FM}}$) and antiferromagnetic
(ME$_{\mathrm{AFM}}$) states, i.e. the paramagnetic phase with
ME$_{\mathrm{FM}}$=ME$_{\mathrm{AFM}}$, the ferromagnetic phase with
ME$_{\mathrm{FM}}$$>$ME$_{\mathrm{AFM}}$ and finally the polarized itinerant
electrons with finite ME$_{\mathrm{FM}}$ but zero ME$_{\mathrm{AFM}}$. While
the O$_\mathrm{N}$ system was found to exist in all three phases, no tendency
towards the formation of the polarized itinerant electrons was observed for the
Si$_\mathrm{B}$ system though the existence of the other two phases was
ascertained. The different behavior of these two systems is associated with the
diverse features in the magnetization energy as a function of the oxygen and
silicon concentrations. Finally, the robustness of the polarized itinerant
electron phase is also discussed with respect to the O substitute atom
distributions and the applied strains to the system.
