  We combine model mapping, exact spectral bounds, and a quantum Monte Carlo
method to study the ground state phases of a mixture of ultracold bosons and
spin-polarized fermions in a one-dimensional optical lattice. The exact
boundary of the boson-demixing transition is obtained from the Bethe Ansatz
solution of the standard Hubbard model. We prove that along a symmetry plane in
the parameter space, the boson-fermion mixed phase is stable at all densities.
This phase is a two-component Luttinger liquid for weak couplings or for
incommensurate total density, otherwise it has a charge gap but retains a
gapless mode of mixture composition fluctuations. The static density
correlations are studied in these two limits and shown to have markedly
different features.
