  A novel way to produce quantum Hall ribbons in a cold atomic system is to use
M hyperfine states of atoms in a 1D optical lattice to mimic an additional
"synthetic dimension". A notable aspect here is that the SU(M) symmetric
interaction between atoms manifests as "infinite ranged" along the synthetic
dimension. We study the many body physics of fermions with attractive
interactions in this system. We use a combination of analytical field theoretic
and numerical density matrix renormalization group (DMRG) methods to reveal the
rich ground state phase diagram of the system, including novel phases such as
squished baryon fluids. Remarkably, changing the parameters entails unusual
crossovers and transitions, e. g., we show that increasing the magnetic field
(that produces the Hall effect) may convert a "ferrometallic" state at low
fields to a "squished baryon superfluid" (with algebraic pairing correlations)
at high fields. We also show that this system provides a unique opportunity to
study quantum phase separation in a multiflavor ultracold fermionic system.
