  Building on the work of Fisher et al. (Phys. Rev. B 40, 546 (1989)), we
develop a framework for perturbation theory in the Bose-Hubbard model and apply
it to calculate the effects of a degenerate gas of spin-polarized fermions
interacting by contact interactions with the constituent bosons. For the
single-band Bose-Hubbard model, the only non-trivial effect of the fermions is
to induce an effective space- and time-dependent density-density interaction
among the bosons. Using a path integral formulation, we develop the appropriate
theory describing the perturbative effects of this fermion-mediated interaction
(fermionic screening) on the generic superfluid-insulator phase diagram. For
the single-band Bose-Hubbard model, we find that the net effect of the fermions
is to inherently suppress the Mott-insulating lobes and enhance the area
occupied by the superfluid phase in the phase diagram. For the more general
multi-band Bose-Hubbard model, we find that, in addition to the fermion
screening of the boson interactions, the virtual excitations of the bosons to
the higher Bloch bands coupled with the contact interactions with the fermions
result in an effective increase of the boson on-site repulsion. If this
renormalization of the boson on-site potential is dominant over the fermion
screening of the boson interactions, the area of the Mott insulating lobes of
the Bose-Hubbard phase diagram will be enhanced for either sign of the
boson-fermion interactions, as seen in recent experiments.
