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

Semimetals exhibiting nodal lines or nodal surfaces represent a novel class of topological states of matter. While conventional Weyl semimetals exhibit momentum-space Dirac monopoles, these more exotic semimetals can feature unusual topological defects that are analogous to extended monopoles. In this work, we describe a scheme by which nodal rings and nodal spheres can be realized in synthetic quantum matter through well-defined periodic-driving protocols. As a central result of our work, we characterize these nodal defects through the quantum metric, which is a gauge-invariant quantity associated with the geometry of quantum states. In the case of nodal rings, where the Berry curvature and conventional topological responses are absent, we show that the quantum metric provides an observable signature for these extended topological defects. Besides, we demonstrate that quantum-metric measurements could be exploited to directly detect the topological charge associated with a nodal sphere. We discuss possible experimental implementations of Floquet nodal defects in few-level atomic systems, paving the way for the exploration of Floquet extended monopoles in quantum matter.