First-principles studies of spin-orbital physics in pyrochlore oxides

The pyrochlore oxides $A_2B_2$O$_7$ exhibit a complex interplay between geometrical frustration, electronic correlations, and spin-orbit coupling, due to the lattice structure and active charge, spin, and orbital degrees of freedom. Understanding the properties of these materials is a theoretical chalenge, because their intricate nature depends on material-specific details and quantum many-body effects. Here we review our recent studies based on first-principles calculations and quantum many-body theories for 4$d$ and 5$d$ pyrochlore oxides with $B$=Mo, Os, and Ir. In these studies, the spin-orbit coupling and local electron correlations are treated within the LDA+$U$ and LDA+dynamical mean-field theory formalisms. We also discuss the technical aspects of these calculations.