  We study the quantum phases of bosons with repulsive contact interactions on
a two-leg ladder in the presence of a uniform Abelian gauge field. The model
realizes many interesting states, including Meissner phases, vortex-fluids,
vortex-lattices, charge-density-waves and the biased-ladder phase. Our work
focuses on the subset of these states that break a discrete symmetry. We use
density matrix renormalization group simulations to demonstrate the existence
of three vortex-lattice states at different vortex densities and we
characterize the phase transitions from these phases into neighboring states.
Furthermore, we provide an intuitive explanation of the chiral-current reversal
effect that is tied to some of these vortex lattices. We also study a
charge-density-wave state that exists at 1/4 particle filling at large
interaction strengths and flux values close to half a flux quantum. By changing
the system parameters, this state can transition into a completely gapped
vortex-lattice Mott-insulating state. We elucidate the stability of these
phases against nearest-neighbor interactions on the rungs of the ladder
relevant for experimental realizations with a synthetic lattice dimension. A
charge-density-wave state at 1/3 particle filling can be stabilized for flux
values close to half a flux-quantum and for very strong on-site interactions in
the presence of strong repulsion on the rungs. Finally, we analytically
describe the emergence of these phases in the low-density regime, and, in
particular, we obtain the boundaries of the biased-ladder phase, i.e., the
phase that features a density imbalance between the legs. We make contact to
recent quantum-gas experiments that realized related models and discuss
signatures of these quantum states in experimentally accessible observables.
