Generalized conditions for odd-frequency pairing in superconducting systems

Odd-frequency superconducting pairing has been predicted to arise in many systems and is known to lead to phenomena such as paramagnetic Meissner response and long-range superconducting proximity effect. Here, we provide generalized necessary and sufficient conditions for odd-frequency pairing appearing in any superconducting system, from bulk superconductors to superconducting hybrid structures. This generalizes an earlier first-order expression in multiband bulk superconductors to all superconducting systems and to all orders, in both order parameter and frequency. We then apply the derived conditions to several systems, including superconducting-ferromagnet and superconducting Josephson junctions, as well as a transition metal dichalcogenide monolayer proximitized by a conventional superconductor, where the generalized conditions are used to understand the properties of the superconducting state.