  We study the quantum phases of a Bose-Hubbard model with staggered magnetic
flux in two dimensions, as has been realized recently [Aidelsburger {\it et
al.}, PRL, {\bf 107}, 255301 (2011)]. Within mean field theory, we show how the
structure of the condensates evolves from weak to strong coupling limit,
exhibiting a tricritical point at the Mott-superfluid transition. Non-trivial
topological structures (Dirac points) in the quasi-particle (hole) excitations
in the Mott state are found within random phase approximation and we discuss
how interaction modifies their structures. Excitation gap in the Mott state
closes at different ${\bf k}$ points when approaching the superfluid states,
which is consistent with the findings of mean field theory.
