  We construct and analyze a microscopic model for insulating rock salt ordered
double perovskites, with the chemical formula A$_2$BB'O$_6$, where the magnetic
ion B' has a 4d$^2$ or 5d$^2$ electronic configuration and forms a face
centered cubic (fcc) lattice. For these B' ions, the combination of the
triply-degenerate antisymmetric two-electron orbital states and strong
spin-orbit coupling forms local quintuplets with an effective spin moment
$j=2$. Moreover, due to strongly orbital-dependent exchange, the effective
spins have substantial biquadratic and bicubic interactions (fourth and sixth
order in the spins, respectively). This leads, at the mean field level, to a
rich ground state phase diagram which includes seven different phases: a
uniform ferromagnetic phase with an ordering wavevector ${\bf p} = {\bf 0}$ and
uniform magnetization along $[111]$ direction, four two-sublattice phases with
an ordering wavevector ${\bf p} = 2\pi(001)$ and two four-sublattice
antiferromagnetic phases. Amongst the two-sublattice phases there is a
quadrupolar ordered phase which preserves time reversal symmetry. Extending the
mean field theory to finite temperatures, we find ten different magnetization
processes with different magnetic thermal transitions. In particular, we find
that thermal fluctuations stabilize the two-sublattice quadrupolar ordered
phase in a large portion of phase diagram. Existing and possible future
experiments are discussed in light of these theoretical predictions.
