Linking structure of centrosymmetric superconductors in 3D momentum space

In topological semimetals and nodal superconductors, band crossings between occupied and unoccupied bands form stable nodal points/lines/surfaces carrying quantized topological charges. In particular, in centrosymmetric systems, some nodal structures at the Fermi energy carry two distinct topological charges, and thus they are called doubly charged nodes. We show that doubly charged nodal surfaces of centrosymmetric superconductors in three-dimensions always develop peculiar linking structures with nodal points or lines formed between occupied bands below the Fermi energy. Such linking structures can naturally explain the inherent relationship between the charge of the node below the Fermi energy and the two charges of the nodal surfaces at the Fermi energy. Based on the Altland-Zirnbauer (AZ)-type ten-fold classification of nodes with additional inversion symmetry I, which is called the AZ+I classification, we provide the complete list of linking structures of doubly charged nodes in centrosymmetric systems. The linking structures of doubly charged nodes clearly demonstrate that not only the local band structure around the node but also the global band structure play a critical role in characterizing the nodes of gapless topological phases.