Quantum Oscillation Signatures of $\mathbb{Z}_2$ Monopole Charge in Nodal-Ring Semimetals
Topological semimetals host band nodes characterized by quantized invariants that can appear in bulk responses, yet some invariants remain hidden from standard probes. In particular, band nodes can carry secondary topological charges whose transport signatures are still largely unexplored. Here we study three-dimensional nodal-line semimetals in which nodal rings carry both the Berry phase $w_1π$ and a $\mathbb{Z}_2$ monopole charge $w_2$. We show that magnetic quantum oscillations, usually treated as a probe of $w_1$, can directly diagnose $w_2$, with the relevant signal selected by the magnetic-field direction. For a field along the ring axis, the inner and outer extremal orbits of the toroidal Fermi surface both encircle the $w_2$-enforced thread and exhibit a topological phase shift $νw_2π$ in the $ν$th harmonic, which is accessible through standard phase-resolved quantum-oscillation analysis. By contrast, for a field applied perpendicular to the ring axis, the relevant extremal orbit exhibits the usual $π$ phase shift associated with the Berry phase $w_1π$, independent of $w_2$. For weak doping, three-dimensional ABC-stacked graphdiyne is predicted to exhibit the proposed oscillations in a field range accessible with present-day high-field facilities.