  We study two-component bosons on the Harper-Hofstadter model with two legs.
The synthetic magnetic fields for the two types of bosons point to either the
same direction or opposite directions. The bosons have hardcore intra-species
interaction such that there can be no more than one boson of the same type on
each lattice site. For certain filling factors in the absence of inter-species
interaction, each component realizes a vortex Mott insulator with rung current
or a Meissner superfluid without rung current. The system undergoes phase
transitions to other phases as inter-species interaction is turned on, which
are characterized numerically using the density matrix renormalization group
method and supplemented with analytical studies when possible. The vortex Mott
insulator transits to a gapped Meissner phase without rung current and the
Meissner superfluid transits to a gapped vortex phase with rung current. In
both cases, we observe gapped spin density wave states that break certain
${\mathbb Z}_{2}$ symmetries.
