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The pairing interactions between electrons play an essential role indetermining the properties in superconducting states. Recently, a plethora ofunconventional superconducting states has been extensively explored, whichoften emerge owing to multipole fluctuations in the vicinity of multipoleorders. We classify such superconducting states from the viewpoint of themultipole degrees of freedom by extending its representation to Nambu space. Weclarify that arbitrary Cooper pairs between electrons with any angular momentaare systematically classified by four types of multipoles, electric, magnetic,magnetic toroidal, and electric toroidal multipoles, under the crystallographicpoint group. As examples, we apply our formulation to an $sp$-orbital electronsystem, which potentially exhibits exotic Cooper pairs under polar and axialpoint groups. Our systematic classification will be useful in characterizingunconventional superconducting states in multi-orbital systems.

Classification of Multi-Orbital Superconducting State based on Augmented Multipoles