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Semimetals exhibiting nodal lines or nodal surfaces represent a novel classof topological states of matter. While conventional Weyl semimetals exhibitmomentum-space Dirac monopoles, these more exotic semimetals can featureunusual topological defects that are analogous to extended monopoles. In thiswork, we describe a scheme by which nodal rings and nodal spheres can berealized in synthetic quantum matter through well-defined periodic-drivingprotocols. As a central result of our work, we characterize these nodal defectsthrough the quantum metric, which is a gauge-invariant quantity associated withthe geometry of quantum states. In the case of nodal rings, where the Berrycurvature and conventional topological responses are absent, we show that thequantum metric provides an observable signature for these extended topologicaldefects. Besides, we demonstrate that quantum-metric measurements could beexploited to directly detect the topological charge associated with a nodalsphere. We discuss possible experimental implementations of Floquet nodaldefects in few-level atomic systems, paving the way for the exploration ofFloquet extended monopoles in quantum matter.

Floquet-engineering of nodal rings and nodal spheres and their characterization using the quantum metric