Mean-field theory for Bose-Hubbard Model under a magnetic field

We consider the superfluid-insulator transition for cold bosons under an effective magnetic field. We investigate how the applied magnetic field affects the Mott transition within mean field theory and find that the critical hopping strength $(t/U)_c$, increases with the applied field. The increase in the critical hopping follows the bandwidth of the Hofstadter butterfly at the given value of the magnetic field. We also calculate the magnetization and superfluid density within mean field theory.