  The SU(4)-symmetric spin-orbital model on the honeycomb lattice was recently
studied in connection to correlated insulators such as the $e_{g}$ Mott
insulator Ba$_{3}$CuSb$_{2}$O$_{9}$ and the insulating phase of magic-angle
twisted bilayer graphene at quarter filling. Here we provide a unified
discussion of these systems by investigating an extended model that includes
the effects of Hund's coupling and anisotropic, orbital-dependent exchange
interactions. Using a combination of mean-field theory, linear flavor-wave
theory, and variational Monte Carlo, we show that this model harbors a quantum
spin-orbital liquid over a wide parameter regime around the SU(4)-symmetric
point. For large Hund's coupling, a ferromagnetic antiferro-orbital ordered
state appears, while a valence-bond crystal combined with a vortex orbital
state is stabilized by dominant orbital-dependent exchange interactions.
