  We investigate the quantum phases of mixed-dimensional cold atom mixtures. In
particular, we consider a mixture of a Fermi gas in a two-dimensional lattice,
interacting with a bulk Fermi gas or a Bose-Einstein condensate in a
three-dimensional lattice. The effective interaction of the two-dimensional
system mediated by the bulk system is determined. We perform a functional
renormalization group analysis, and demonstrate that by tuning the properties
of the bulk system, a subtle competition of several superconducting orders can
be controlled among $s$-wave, $p$-wave, $d_{x^2-y^2}$-wave, and
$g_{xy(x^2-y^2)}$-wave pairing symmetries. Other instabilities such as a
charge-density wave order are also demonstrated to occur. In particular, we
find that the critical temperature of the $d$-wave pairing induced by the
next-nearest-neighbor interactions can be an order of magnitude larger than
that of the same pairing induced by doping in the simple Hubbard model. We
expect that by combining the nearest-neighbor interaction with the
next-nearest-neighbor hopping (known to enhance $d$-wave pairing), an even
higher critical temperature may be achieved.
