  Motivated by the experiments on the rare-earth double perovskites, we propose
a generalized Kitaev-Heisenberg model to describe the generic interaction
between the spin-orbit-entangled Kramers' doublets of the rare-earth moments.
We carry out a systematic analysis of the mean-field phase diagram of this new
model. In the phase diagram, there exist large regions with a continuous U(1)
or O(3) degeneracy. Since no symmetry of the model protects such a continuous
degeneracy, we predict that the quantum fluctuation lifts the continuous
degeneracy and favors various magnetic orders in the phase diagram. From this
order by quantum disorder mechanism, we further predict that the magnetic
excitations of the resulting ordered phases are characterized by nearly gapless
pseudo-Goldstone modes. We find that there exist Weyl magnon excitations for
certain magnetic orders. We expect our prediction to inspire further study of
Kitaev physics, the order by quantum disorder phenomenon and topological spin
wave modes in the rare-earth magnets and the systems alike.
