  We investigate the ground-state phase diagram of interacting binary Bose
gases trapped in two-dimensional optical lattices by means of quantum Monte
Carlo simulations. Our simulations reveal a magnetic phase transition from a
$x-y$ ferromagnetic-order to a spin insulator inside the Mott insulating phase
with two particles per site for quasi-balanced on-site inter- and
intra-particle interactions, i.e., $U_{\uparrow \downarrow} \lesssim U$. This
3D-XY transition is characterized by the establishment of a finite local
magnetic moment along the $z$-axis, ferromagnetic correlations in the $x-y$
plan and by counterflow superfluidity inside the Mott phase. When decreasing
$U_{\uparrow \downarrow}/U$, this transition merges with the Mott-superfluid
transition and becomes first-order. The merging of the two transitions is
investigated with respect to $U_{\uparrow \downarrow}/U$ parameter.
