  Systems of two coupled bosonic species are studied using Mean Field Theory
and Quantum Monte Carlo. The phase diagram is characterized both based on the
mobility of the particles (Mott insulating or superfluid) and whether or not
the system is magnetic (different populations for the two species). The phase
diagram is shown to be population balanced for negative spin-dependent
interactions, regardless of whether it is insulating or superfluid. For
positive spin-dependent interactions, the superfluid phase is always polarized,
the two populations are imbalanced. On the other hand, the Mott insulating
phase with even commensurate filling has balanced populations while the odd
commensurate filling Mott phase has balanced populations at very strong
interaction and polarizes as the interaction gets weaker while still in the
Mott phase.
