Quantized non-Abelian helicity of flat bands in 2D Floquet topological photonic insulators

Flat-band states in topological systems provide a unique platform for investigating strongly correlated phenomena and many-body physics. However, in two-dimensional (2D) static tight-binding systems, perfectly flat bands can only exist in the topologically trivial phase, as characterized by a zero Chern number. Here, we show that by introducing periodic driving into a 2D photonic Lieb lattice composed of coupled microring resonators, the resulting Floquet topological insulator can host perfectly flat bands with nontrivial topology. In particular, by tracking the evolution of the flat-band modes over each cycle, we show that the non-Abelian displacements of the flat-band modes are characterized by a nontrivial quantized helicity even though the quasienergy bands have zero Chern number. The helical motion of the flat-band modes can be described by a braiding of the world lines of their trajectories, with a nontrivial winding number directly connected to the helicity. We also propose a scheme to experimentally measure the quantized non-Abelian helicity in a microring lattice subject to a synthetic magnetic field. These results suggest that Floquet topological photonic insulators based on coupled microring resonators can provide a versatile platform for investigating non-Abelian topological physics and strongly correlated phenomena in photonic flat-band systems.