Mott phases and quantum phase transitions of extended Bose–Hubbard models in 2+1 dimensions

We review the recent theoretical developments toward understanding the Mott phases and quantum phase transition of extended Bose–Hubbard models on lattices in two spatial dimensions. We focus on the description of these systems using the dual vortex picture and point out the crucial role played by the geometry of underlying lattices in determining the nature of both the Mott phases and the quantum phase transitions. We also briefly compare the results of dual vortex theory with quantum Monte Carlo results.