Altermagnetic Weyl node-network metals protected by spin symmetry

Symmetry protected topology has been studied extensively since the mid-2000s, but the topology protected by spin symmetry has just begun to be studied. In this work, based on spin-symmetry analysis, we propose that a class of Weyl nodal line metals is protected by the spin symmetry. Then, by the first-principles electronic structure calculations, we predict that both altermagnetic ${\mathrm{Nb}}_{2}{\mathrm{FeB}}_{2}$ and ${\mathrm{Ta}}_{2}{\mathrm{FeB}}_{2}$ are node-network metals protected by the spin symmetry. Moreover, both altermagnetic ${\mathrm{Nb}}_{2}{\mathrm{FeB}}_{2}$ and ${\mathrm{Ta}}_{2}{\mathrm{FeB}}_{2}$ have nodal rings protected by the mirror symmetry and Dirac points protected by nonsymmorphic spin symmetry. Furthermore, both altermagnetic ${\mathrm{Nb}}_{2}{\mathrm{FeB}}_{2}$ and ${\mathrm{Ta}}_{2}{\mathrm{FeB}}_{2}$ transform node-network metal phase into Weyl metal phase when considering spin-orbit coupling. Therefore, our work not only enriches the topological phases protected by spin symmetry but also provides an excellent material platform to investigate the exotic physical properties of multiple altermagnetic topological metal phases in experiment.