Eight-fold classification of superconducting orders

We present a thorough symmetry-based classification of superconducting order parameters that is independent of their microscopic origins. Our approach involves classifying pairing states through the pairwise permutation of spatial, temporal, spin, and orbital indices, focusing on the parities of the relative coordinates and times of the paired fields. This framework leads to the emergence of new classes of superconducting states, which we visualize as a multidimensional cube representing the relative parities in space and time, referred to as the Berezinskii-Abrahams hypercube. While some predicted states, such as odd-frequency superconductors, have been previously discussed, our symmetry analysis also reveals states that have yet to be explored, including odd-frequency Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states. Furthermore, we draw parallels to the classification of particle-hole states, and reiterate the possibility of new states, such as odd-frequency density waves.