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.
Categories: cond-mat.quant-gas