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Recent development in exact classification of a superconducting gap haselucidated various unconventional gap structures, which have not been predictedby the classification of order parameter based on the point group. One of theimportant previous results is that all symmetry-protected line nodes arecharacterized by nontrivial topological numbers. Another intriguing discoveryis the gap structures depending on the angular momentum $j_z$ of normal Blochstates on threefold and sixfold rotational-symmetric lines in the Brillouinzone. Stimulated by these findings, we classify irreducible representations ofthe Bogoliubov-de Gennes Hamiltonian at each $\boldsymbol{k}$ point on ahigh-symmetry $n$-fold ($n = 2$, $3$, $4$, and $6$) axis for centrosymmetricand paramagnetic superconductors, by using the combination of group theory and$K$ theory. This leads to the classification of all crystal symmetry-protectednodes (including $j_z$-dependent nodes) on the axis that crosses a normal-stateFermi surface. As a result, it is shown that the classification by group theorycompletely corresponds with the topological classification. Based on theobtained results, we discuss superconducting gap structures in SrPtAs,CeCoIn$_5$, UPt$_3$, and UCoGe.

Classification of topological crystalline superconducting nodes on high-symmetry lines: Point nodes, line nodes, and Bogoliubov Fermi surfaces