Random phase approximation for superconducting states in multi-orbital Hubbard models with spin-orbit coupling

To better understand electronic and magnetic properties of superconducting (SC) states in 5d transition metal oxides, we derive an analytical formula of the dynamical correlation function for multi-orbital systems with the spin-orbit coupling (SOC), which is accessible to the experimental responses. Starting with a multi-orbital BCS Hamiltonian including the SC gap function, we introduce the dynamical correlation function in the non-interacting case. Based on the diagrammatic techniques, we derive the dynamical correlation function for the interacting case within the random phase approximation. With the help of a numerical calculation, we also show the temperature dependence of the magnetic susceptibility for two different SC states in the presence of the SOC.