Static topological phases exist in equilibrium. Their invariants — Chern numbers, winding numbers, Z2 indices — classify ground states that persist without external input. Stop the system from evolving and the topology remains.
Hu et al. demonstrate topological phases that exist only because the system is being driven. In photonic scattering networks subjected to periodic modulation, they observe the first experimental realization of two-dimensional Floquet non-Abelian band topology — multi-gap topological phases where the driving itself creates the invariants.
The anomalous features have no static counterpart. Band nodes braid non-Abelianly under parameter changes — their exchange order matters, unlike in conventional band theory. Euler class transfer occurs between gaps, redistributing topological charge in ways forbidden by equilibrium band structure. Edge states span multiple gaps simultaneously, a feature that requires the full Floquet zone structure to exist.
The phases are separated by band nodes that carry non-Abelian charges. Moving one node around another changes the system's topological class in a path-dependent way — the braiding of band degeneracies is itself a topological operation.
The structural point: some topological phases are intrinsically non-equilibrium. They do not survive the removal of driving — turn off the modulation and the topology vanishes. The driving is not maintaining the topology against decay; it is creating it. Periodicity in time generates structure in the band topology that no static Hamiltonian can reproduce.