  Cold atom experiments can now realize mixtures where different components
move in different spatial dimensions. We investigate a fermion mixture where
one species is constrained to move along a one-dimensional lattice embedded in
a two-dimensional lattice populated by another species of fermions, and where
all bare interactions are contact interactions. By focusing on the
one-dimensional fermions, we map this problem onto a model of fermions with
non-local interactions on a chain. The effective interaction is mediated by the
two-dimensional fermions and is both attractive and retarded, the form of which
can be varied by changing the density of the two-dimensional fermions. By using
the functional renormalization group in the weak-coupling and adiabatic limit,
we show that the one-dimensional fermions can be controlled to be in various
density-wave, or spin-singlet or triplet superconducting phases.
