  We study the effects of an artificial gauge field on the ground-state phases
of the Bose-Hubbard model on a checkerboard superlattice in two dimensions,
including the superfluid phase and the Mott and alternating Mott insulators.
First, we discuss the single-particle Hofstadter problem, and show that the
presence of a checkerboard superlattice gives rise to a magnetic
flux-independent energy gap in the excitation spectrum. Then, we consider the
many-particle problem, and derive an analytical mean-field expression for the
superfluid-Mott and superfluid--alternating-Mott insulator phase transition
boundaries. Finally, since the phase diagram of the Bose-Hubbard model on a
checkerboard superlattice is in many ways similar to that of the extended
Bose-Hubbard model, we comment on the effects of magnetic field on the latter
model, and derive an analytical mean-field expression for the
superfluid-insulator phase transition boundaries as well.
