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There has been a recent resurgence of experimental efforts to quantitativelydetermine the phase diagram of the Bose Hubbard model by carefully analyzingexperiments with ultracold bosonic atoms on an optical lattice. In manyrealizations of these experiments, the hopping amplitudes are not homogeneousthroughout the lattice, but instead, the lattice has an anisotropy wherehopping along one direction is not exactly equal to hopping along aperpendicular direction. In this contribution, we examine how an anisotropy inthe hopping matrix elements affects the Mott lobes of the Bose Hubbard model.For weak anisotropy, we find the phase diagram is only slightly modified whenexpressed in terms of the average hopping, while for strong anisotropy, oneexpects to ultimately see dimensional crossover effects.

Effect of anisotropic hopping on the Bose Hubbard model phase diagram: strong-coupling perturbation theory on a square lattice