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We extend the standard Bose-Hubbard model to capture arbitrarily strongon-site correlations. In addition to being important for quantitativelymodeling experiments, for example, with Rubidium atoms, these correlations mustbe included to describe more exotic situations. Two such examples are when theinteractions are made large via a Feshbach resonance, or when each site rotatesrapidly, making a coupled array of quantum Hall puddles. Remarkably, even themean field approximation to our model includes all on-site correlations. Wedescribe how these on-site correlations manifest themselves in the system'sglobal properties: modifying the phase diagram and depleting the condensate.

On-site correlations in optical lattices: band mixing to coupled quantum Hall puddles