Revisiting Electromagnetic response of superconductors in mean-field approximation

In the standard mean-field treatment of superconductors, the electron-electron interactions are assumed to be written in terms of local density operators. However, more general interactions, such as pair-hopping interactions, may exist or may be generated in a low-energy effective Hamiltonian. In this work, we study the effect of correlated hopping interactions toward the electromagnetic response of superconductors. When only the Hamiltonian after the mean-field approximation is provided, one cannot unambiguously determine its electromagnetic response whenever such interactions are allowed. Our research demonstrates that these interactions induce additional terms in the current operator, leading to modifications in the Meissner weight and optical conductivities, deviating from conventional expectations. These modifications underscores the importance of providing a microscopic model with U(1) symmetry when introducing U(1) gauge fields to the BdG Hamiltonian. As a result, our findings underscore the need for caution when incorporating gauge fields into the BdG Hamiltonian, as exemplified in this work.