  We study quantum Hall states for two-component particles (hardcore bosons and
fermions) loading in topological lattice models. By tuning the interplay of
interspecies and intraspecies interactions, we demonstrate that two-component
fractional quantum Hall states emerge at certain fractional filling factors
$\nu=1/2$ for fermions ($\nu=2/3$ for bosons) in the lowest Chern band,
classified by features from ground states including the unique Chern number
matrix (inverse of $\mathbf{K}$-matrix), the fractional charge and spin
pumpings, and two parallel propagating edge modes. Moreover, we also apply our
strategy to two-component fermions at integer filling factor $\nu=2$, where a
possible topological Neel antiferromagnetic phase is under intense debate very
recently. For the typical $\pi$-flux checkerboard lattice, by tuning the onsite
Hubbard repulsion, we establish a first-order phase transition directly from a
two-component fermionic $\nu=2$ quantum Hall state at weak interaction to a
topologically trivial antiferromagnetic insulator at strong interaction, and
therefore exclude the possibility of an intermediate topological phase for our
system.
