Boson Hubbard model with weakly coupled fermions*

Using an imaginary-time path integral approach, we develop the perturbation theory suited to the boson Hubbard model, and apply it to calculate the effects of a dilute gas of spin- polarized fermions weakly interacting with the bosons. The full theory captures both the static and the dynamic effects of the fermions on the generic superfluid-insulator phase diagram. We find that, in a homogenous system described by a single-band boson Hubbard Hamiltonian, the intrinsic perturbative effect of the fermions is to suppress the Mott insulating lobes and enhance the superfluid phase.