We investigate a spin-degenerate Fermi gas coupled to a high-finesse optical cavity, where the competition between scalar and vectorial couplings is controlled by the relative polarization angle of the pump and cavity fields. We find that the phase transition threshold is synergistically determined by the scalar-vectorial coupling weight and Pauli blocking, with the latter dictating the critical pump lattice depth required for the onset of superradiance. For a two-component Fermi gas with opposite spins, the population ratio drives two distinct types of phase transitions corresponding to real-space phase separation: continuous and discontinuous. Nevertheless, the boundary of the phase transition remains fundamentally governed by the scalar-vectorial coupling competition. We clarify the impact of the relative polarization angle on phase transitions of the system; these results also apply to bosonic systems. Our results provide valuable theoretical insights for future experimental realizations.

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• Atomic & molecular processes in external fields
• Cavity quantum electrodynamics
• Cold atoms & matter waves
• Fermi gases