We study the band topology and the associated linking structure of topological semimetals with nodal lines carrying $Z_2$ monopole charges, which can be realized in three-dimensional systems invariant under the combination of inversion $P$ and time reversal $T$ when spin-orbit coupling is negligible. In contrast to the well-known $PT$-symmetric nodal lines protected only by the $\pi$ Berry phase, in which a single nodal line can exist, the nodal lines with $Z_2$ monopole charges should always exist in pairs. We show that a pair of nodal lines with $Z_2$ monopole charges is created by a double band inversion process and that the resulting nodal lines are always linked by another nodal line formed between the two topmost occupied bands. It is shown that both the linking structure and the $Z_2$ monopole charge are the manifestation of the nontrivial band topology characterized by the second Stiefel-Whitney class, which can be read off from the Wilson loop spectrum. We show that the second Stiefel-Whitney class can serve as a well-defined topological invariant of a $PT$-invariant two-dimensional insulator in the absence of Berry phase. Based on this, we propose that pair creation and annihilation of nodal lines with $Z_2$ monopole charges can mediate a topological phase transition between a normal insulator and a three-dimensional weak Stiefel-Whitney insulator. Moreover, using first-principles calculations, we predict $ABC$-stacked graphdiyne as a nodal line semimetal (NLSM) with $Z_2$ monopole charges having the linking structure. Finally, we develop a formula for computing the second Stiefel-Whitney class based on parity eigenvalues at inversion-invariant momenta, which is used to prove the quantized bulk magnetoelectric response of NLSMs with $Z_2$ monopole charges under a $T$-breaking perturbation.

• Received 3 April 2018
• Revised 6 July 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.106403