Quantum phases of spinful Fermi gases in optical cavities

We propose the realization of a quantum fluid of spinful fermionic atoms interacting via a cavity-photon mediating mechanism, whose effective coupling g can be tuned in strength and sign after detuning the cavity frequency from the atomic transition. Cavity photons are able to drive strong correlations at all coupling strengths, with fermionic character for g>0 and bosonic character for g 0.