Homotopy approach to the classification of band-structure nodes*

While most band-structure nodes are protected by a single topological charge, it was shown using homotopy theory that nodal lines protected by time-reversal and spatial inversion in the absence of spin-orbit coupling carry a pair of independent Z2 charges, which enhances their stability. We generalize the homotopy arguments, and show that in 3D the global symmetries together with spatial inversion are able to protect doubly charged nodes in four out of the ten Atland-Zirnbauer classes. These include nodal lines in classes AI and CI, and nodal surfaces in classes D and BDI. In all cases, the additional invariant leads to enhanced robustness of the nodes, when they cannot be gapped out individually but only by a pairwise annihilation, thus leading to an analogue of the Nielsen-Ninomiya doubling. These exotic nodes are relevant for various semimetallic as well as superconducting phases, including the recently discovered Bogoliubov-Fermi surfaces. We further generalize the homotopy approach to nodes protected by crystalline symmetries. In this way, we are able to understand the topological protection of most common species of nodes in a unified language.