  The recent realization of mixed-dimensional systems of cold atoms has
attracted much attention from both experimentalists and theorists. Different
effective interactions and novel correlated quantum many-body phases may be
engineered in these systems, with the different phases being tunable via
external parameters. In this article we investigate a two-species Fermi atom
mixture: one species of atom exists in two hyperfine states and is confined to
move in a two-leg ladder, interacting with an on-site interaction, and the
other moves freely in a two dimensional square lattice that contains the
two-leg ladder. The two species of atoms interact via an on-site interaction on
the ladder. In the limit of weak inter-species interactions, the
two-dimensional gas can be integrated out, leading to an effective long-range
mediated interaction in the ladder, generated by to the on-site inter-species
interaction. We show that the form of the mediated interaction can be
controlled by the density of the two-dimensional gas and that it enhances the
charge density wave instability in the two-leg ladder after the renormalization
group transformation. Parameterizing the phase diagram with various
experimentally controllable quantities, we discuss the possible tuning of the
macroscopic quantum many-body phases of the two-leg ladder in this
mixed-dimensional fermionic cold atom system.
