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By using a dual vortex method, we study phases such as superfluid, solids,supersolids and quantum phase transitions in a unified scheme in extended bosonHubbard models at and slightly away from half filling on bipartite opticallattices such as honeycomb and square lattice. We also map out its global phasediagram at $ T=0 $ of chemical potential versus the ratio of kinetic energyover the interaction. We stress the importance of the self-consistencecondition on the saddle point structure of the dual gauge fields in thetranslational symmetry breaking insulating sides, especially in the chargedensity wave side. We find that in the translational symmetry breaking side,different kinds of supersolids are generic possible states slightly away fromhalf filling. We propose a new kind of supersolid: valence bond supersolid(VB-SS). In this VB-SS, the density fluctuation at any site is very largeindicating its superfluid nature, but the boson kinetic energies on bondsbetween two sites are given and break the lattice translational symmetriesindicating its valence bound nature. Implications on possible future QMCsimulations in both bipartite lattices are given. All these phases and phasetransitions can be potentially realized in ultra-cold atoms loaded on opticalbipartite lattices.

Duality, Magnetic space group and their applications to quantum phases and phase transitions on bipartite lattices in several experimental systems