  We study the phase diagram of the one-dimensional Bose-Fermi-Hubbard model at
unit filling for the scalar bosons and half filling for the $S=1/2$ fermions
using quantum Monte Carlo simulations. The bare interaction between the
fermions is set to zero. A central question of our study is what type of
interactions can be induced between the fermions by the bosons, for both weak
and strong interspecies coupling. We find that the induced interactions can
lead to competing instabilities favoring phase separation, superconducting
phases, and density wave structures, in many cases at work on length scales of
more than 100 sites. Marginal bosonic superfluids with a density matrix
decaying faster than what is allowed for pure bosonic systems with on-site
interactions, are also found.
