We perform a numeric study (worm algorithm Monte Carlo simulations) of
ultracold two-component bosons in two- and three-dimensional optical lattices.
At strong enough interactions and low enough temperatures the system features
magnetic ordering. We compute critical temperatures and entropies for the
disappearance of the Ising antiferromagnetic and the xy-ferromagnetic order and
find that the largest possible entropies per particle are ~0.5kB. We also
estimate (optimistically) the experimental hold times required to reach
equilibrium magnetic states to be on a scale of seconds. Low critical entropies
and long hold times render the experimental observations of magnetic phases
challenging and call for increased control over heating sources.Comment: 6 pages, 6 figure