  We study the stability of a Bose-Fermi system loaded into an array of coupled
one-dimensional (1D) "tubes", where bosons and fermions experience different
dimensions: Bosons are heavy and strongly localized in the 1D tubes, whereas
fermions are light and can hop between the tubes. Using the 174Yb-6Li system as
a reference, we obtain the equilibrium phase diagram. We find that, for both
attractive and repulsive interspecies interaction, the exact treatment of 1D
bosons via the Bethe ansatz implies that the transitions between pure fermion
and any phase with a finite density of bosons can only be first order and never
continuous, resulting in phase separation in density space. In contrast, the
order of the transition between the pure boson and the mixed phase can either
be second or first order depending on whether fermions are allowed to hop
between the tubes or they also are strictly confined in 1D. We discuss the
implications of our findings for current experiments on 174Yb-6Li mixtures as
well as Fermi-Fermi mixtures of light and heavy atoms in a mixed dimensional
optical lattice system.
