  We establish the phase diagram of the strongly-interacting Bose-Hubbard model
defined on a two-leg ladder geometry in the presence of a homogeneous flux. Our
work is motivated by a recent experiment [Atala et al., Nature Phys. 10, 588
(2014)], which studied the same system, in the complementary regime of weak
interactions. Based on extensive density matrix renormalization group
simulations and a bosonization analysis, we fully explore the parameter space
spanned by filling, inter-leg tunneling, and flux. As a main result, we
demonstrate the existence of gapless and gapped Meissner and vortex phases,
with the gapped states emerging in Mott-insulating regimes. We calculate
experimentally accessible observables such as chiral currents and vortex
patterns.
