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

Competing ground states of strongly correlated bosons in the Harper-Hofstadter-Mott model