Spin-orbital physics for $t_{2g}$ electron in FCC lattice with application to ordered double perovskites

Ordered double perovskites A$_2$BB$'$O$_6$ are derived compounds from the usual perovskites ABO$_3$ by selectively replacing half of the B ions with magnetic ions B$'$ so that the B$'$ ions form an FCC lattice structure with a lattice constant twice the original cubic lattice of the B ions. Motivated by recent experiments on ordered double perovskites, we started from a microscopic nearest neighbor exchange Hamiltonian and relativistic spin-orbit interaction, and analyzed the spin-orbital physics for different regimes of coupling parameters. We found that when the spin-orbit coupling is weak compared to the exchange, the system develops real orbital order and weakly-coupled layered spin order primarily driven by the exchange. When the spin-orbit coupling is strong, we suggested three different candidate ground states. They are ferromagnetic state, complex orbital order with ordering wavevector ${\bf p} = 2\pi(001)$ and the spin-orbital liquid state. Realization of these states in different double perovskites are discussed.