  We consider the Bose-Hubbard model on a two-leg ladder under an artificial
magnetic field, and investigate the superfluid-to-Mott insulator transition in
this setting. Recently, this system has been experimentally realized [M.Atala
\textit{et al.}, Nature Physics \textbf{10}, 588--593 (2014)], albeit in a
parameter regime that is far from the Mott transition boundary. Depending on
the strength of the magnetic field, the single-particle spectrum has either a
single ground state or two degenerate ground states. The transition between
these two phases is reflected in the many-particle properties. We first
investigate these phases through the Bogoliubov approximation in the superfluid
regime and calculate the transition boundary for weak interactions. For
stronger interactions the system is expected to form a Mott insulator. We
calculate the Mott transition boundary as a function of the magnetic field and
interleg coupling with mean-field theory, strong-coupling expansion and density
matrix renormalization group (DMRG). Finally, using the DMRG, we investigate
the particle-hole excitation gaps of this system at different filling factors
and find peaks at simple fractions indicating the possibility of correlated
phases.
