An effective mean field theory for the coexistence of anti-ferromagnetism and superconductivity: Applications to iron-based superconductors and cold Bose–Fermi atomic mixtures

• We study interplay between anti-ferromagnetism and superconductivity.
• We use a mean-field theory and Landau energy functional approach.
• We treat both iron-based superconductors and Bose–Fermi mixtures on optical lattices.
• We treat both s-wave and d-wave symmetries in the system.
• We consider that the attractive interactions are mediated by bosonic excitations; spin fluctuations and excitations of Bose–Einstein condensation (BEC).

We study an effective fermion model on a square lattice to investigate the cooperation and competition of superconductivity and anti-ferromagnetism. In addition to particle tunneling and on-site interaction, a bosonic excitation mediated attractive interaction is also included in the model. We assume that the attractive interaction is mediated by spin fluctuations and excitations of Bose–Einstein condensation (BEC) in electronic systems and Bose–Fermi mixtures on optical lattices, respectively. Using an effective mean-field theory to treat both superconductivity and anti-ferromagnetism at equal footing, we study a single effective model relevant for both systems within the Landau energy functional approach and a linearized theory. Within our approaches, we find possible co-existence of superconductivity and anti-ferromagnetism for both electronic and cold-atomic models. Our linearized theory shows while spin fluctuations favor d-wave superconductivity and BEC excitations favor s-wave superconductivity.