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We give an exhaustive characterization of the topology of band structures inclass AI, using nonsymmorphic space group 33 ($Pna2_1$) as a representativeexample where a great variety of symmetry protected line-nodal structures canbe formed. We start with the topological classification of all line-nodalstructures given through the combinatorics of valence irreduciblerepresentations (IRREPs) at a few high-symmetry points (HSPs) at a fixedfilling. We decompose the total topology of nodal valence band bundles throughthe local topology of elementary (i.e. inseparable) nodal structures and theglobal topology that constrains distinct elementary nodal elements over theBrillouin zone (BZ). Generalizing from the cases of simple point nodes andsimple nodal lines (NLs), we argue that the local topology of every elementarynodal structure is characterized by a set of poloidal-toroidal charges, onemonopole, and one thread charge (when threading the BZ torus), while the globaltopology only allows pairs of nontrivial monopole and thread charges. We showthat all these charges are given in terms of symmetry protected topologicalinvariants, defined through quantized Wilson loop phases over symmetryconstrained momentum loops, which we derive entirely algebraically from thevalence IRREPs at the HSPs. In particular, we find highly connected line-nodalstructures, line-nodal monopole pairs, and line-nodal thread pairs, that areall protected by the unitary crystalline symmetries only. Furthermore, we showsymmetry preserving topological Lifshitz transitions through which independentNLs can be connected, disconnected, or linked. Our work constitutes a heuristicapproach to the systematic topological classification and characterization ofall momentum space line-nodal structures protected by space group symmetries inclass AI.

Bulk topology of line-nodal structures protected by space group symmetries in class AI