Quantum simulation of competing orders with fermions in quantum optical lattices

Ultracold Fermi atoms confined in optical lattices coupled to quantized modes of an optical cavity are an ideal scenario to engineer quantum simulators in the strongly interacting regime. The system has both short-range and cavity-induced long-range interactions. We propose such a scheme to investigate the coexistence of superfluid pairing, density order, and quantum domains having antiferromagnetic or density order in the Hubbard model in a high-finesse optical cavity at T = 0. We demonstrate that those phases can be accessed by properly tuning the linear polarizer of an external pump beam via the cavity back-action effect, while modulating the system doping. This allows one to emulate the typical scenarios of analog strongly correlated electronic systems.