Quantum phases of two-component bosons on the Harper-Hofstadter ladder

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.