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

We give an exhaustive characterization of the topology of band structures in class AI, using nonsymmorphic space group 33 ($Pna2_1$) as a representative example where a great variety of symmetry protected line-nodal structures can be formed. We start with the topological classification of all line-nodal structures given through the combinatorics of valence irreducible representations (IRREPs) at a few high-symmetry points (HSPs) at a fixed filling. We decompose the total topology of nodal valence band bundles through the local topology of elementary (i.e. inseparable) nodal structures and the global topology that constrains distinct elementary nodal elements over the Brillouin zone (BZ). Generalizing from the cases of simple point nodes and simple nodal lines (NLs), we argue that the local topology of every elementary nodal structure is characterized by a set of poloidal-toroidal charges, one monopole, and one thread charge (when threading the BZ torus), while the global topology only allows pairs of nontrivial monopole and thread charges. We show that all these charges are given in terms of symmetry protected topological invariants, defined through quantized Wilson loop phases over symmetry constrained momentum loops, which we derive entirely algebraically from the valence IRREPs at the HSPs. In particular, we find highly connected line-nodal structures, line-nodal monopole pairs, and line-nodal thread pairs, that are all protected by the unitary crystalline symmetries only. Furthermore, we show symmetry preserving topological Lifshitz transitions through which independent NLs can be connected, disconnected, or linked. Our work constitutes a heuristic approach to the systematic topological classification and characterization of all momentum space line-nodal structures protected by space group symmetries in class AI.