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Classification of multipole superconductivity in multi-orbital systems and its implications

Motivated by a growing interest in multi-orbital superconductors with spin-orbit interactions, we perform the group-theoretical classification of various unconventional superconductivity emerging in symmorphic $\rm O$, $\rm D_4$, and $\rm D_6$ space groups. The generalized Cooper pairs, which we here call "multipole" superconductivity, possess spin-orbital coupled (multipole) degrees of freedom, instead of the conventional spin singlet/triplet in single-orbital systems. From the classification, we obtain the following key consequences, which have been overlooked in the long history of research in this field: (1) A superconducting gap function with $\varGamma_9\otimes\varGamma_9$ in $\rm D_6$ possesses nontrivial momentum dependence, different from the usual spin 1/2 classification. (2) Unconventional gap structure can be realized in the BCS approximation of purely local (on-site) interactions irrespective of attractive/repulsive. It implies the emergence of an electron-phonon (e-ph) driven unconventional superconductivity. (3) Reflecting symmetry of orbital basis functions, there appear not symmetry-protected but inevitable line nodes/gap minima, and thus, anisotropic $s$-wave superconductivity can be naturally explained without any competitive fluctuations.