  Using an efficient cluster approach, we study the physics of two-dimensional
lattice bosons in a strong magnetic field in the regime where the tunneling is
much weaker than the on-site interaction strength. We study both dilute, hard
core bosons at filling factors much smaller than unity occupation per site, and
the physics in the vicinity of the superfluid-Mott lobes as the density is
tuned away from unity. For hardcore bosons, we carry out extensive numerics for
a fixed flux per plaquette $\phi=1/5$ and $\phi = 1/3$. At large flux, the
lowest energy state is a strongly correlated superfluid, analogous to He-$4$,
in which the order parameter is dramatically suppressed, but non-zero. At
filling factors $\nu=1/2,1$, we find competing incompressible states which are
metastable. These appear to be commensurate density wave states. For small
flux, the situation is reversed, and the ground state at $\nu = 1/2$ is an
incompressible density-wave solid. Here, we find a metastable lattice
supersolid phase, where superfluidity and density-wave order coexist. We then
perform careful numerical studies of the physics near the vicinity of the Mott
lobes for $\phi = 1/2$ and $\phi = 1/4$. At $\phi = 1/2$, the superfluid ground
state has commensurate density-wave order. At $\phi = 1/4$, incompressible
phases appear outside the Mott lobes at densities $n = 1.125$ and $n = 1.25$,
corresponding to filling fractions $\nu = 1/2$ and $1$ respectively. These
phases, which are absent in single-site mean-field theory are metastable, and
have slightly higher energy than the superfluid, but the energy difference
between them shrinks rapidly with increasing cluster size, suggestive of an
incompressible ground state. We thus explore the interplay between Mott
physics, magnetic Landau levels, and superfluidity, finding a rich phase
diagram of competing compressible and incompressible states.
