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.
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