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Weyl points in three spatial dimensions are characterized by a$\mathbb{Z}$-valued charge -- the Chern number -- which makes them stableagainst a wide range of perturbations. A set of Weyl points can mutuallyannihilate only if their net charge vanishes, a property we refer to asrobustness. While nodal loops are usually not robust in this sense, it hasrecently been shown using homotopy arguments that in the centrosymmetricextension of the $\textrm{AI}$ symmetry class they nevertheless develop a$\mathbb{Z}_2$ charge analogous to the Chern number. Nodal loops carrying anon-trivial value of this $\mathbb{Z}_2$ charge are robust, i.e. they can begapped out only by a pairwise annihilation and not on their own. As this is anadditional charge independent of the Berry $\pi$-phase flowing along the banddegeneracy, such nodal loops are, in fact, doubly charged. In this manuscript,we generalize the homotopy discussion to the centrosymmetric extensions of allAtland-Zirnbauer classes. We develop a taylored mathematical framework dubbedthe AZ+$\mathcal{I}$ classification and show that in three spatial dimensionssuch robust and multiply charged nodes appear in four of such centrosymmetricextensions, namely AZ+$\mathcal{I}$ classes $\textrm{CI}$ and $\textrm{AI}$lead to doubly charged nodal lines, while $\textrm{D}$ and $\textrm{BDI}$support doubly charged nodal surfaces. We remark that no further crystallinesymmetries apart from the spatial inversion are necessary for their stability.We provide a description of the corresponding topological charges, and developsimple tight-binding models of various semimetallic and superconducting phasesthat exhibit these nodes. We also indicate how the concept of robust andmultiply charged nodes generalizes to other spatial dimensions.

Robust doubly charged nodal lines and nodal surfaces in centrosymmetric systems