  We investigated the electronic and magnetic properties of the pyrochlore
oxide Cd$_2$Os$_2$O$_7$ using the density-functional theory plus on-site
repulsion ($U$) method, and depict the ground-state phase diagram with respect
to $U$. We conclude that the all-in/all-out non-collinear magnetic order is
stable in a wide range of $U$. We also show that the easy-axis anisotropy
arising from the spin-orbit (SO) coupling plays a significant role in
stabilizing the all-in/all-out magnetic order. A \textit{pseudo gap} was
observed near the transition between the antiferromagnetic metallic and
insulating phases. Finally, we discuss possible origins of the peculiar
low-temperature($T$) properties observed in experiments.

Strongly Correlated Electrons;Materials Science;Statistical Mechanics