Floquet non-Abelian topological insulator and multifold bulk-edge correspondence

Topological phases characterized by non-Abelian charges are beyond the scope of the paradigmatic tenfold way and have gained increasing attention recently. Here we investigate topological insulators with multiple tangled gaps in Floquet settings and identify uncharted Floquet non-Abelian topological insulators without any static or Abelian analog. We demonstrate that the bulk-edge correspondence is multifold and follows the multiplication rule of the quaternion group Q8. The same quaternion charge corresponds to several distinct edge-state configurations that are fully determined by phase-band singularities of the time evolution. In the anomalous non-Abelian phase, edge states appear in all bandgaps despite trivial quaternion charge. Furthermore, we uncover an exotic swap effect-the emergence of interface modes with swapped driving, which is a signature of the non-Abelian dynamics and absent in Floquet Abelian systems. Our work, for the first time, presents Floquet topological insulators characterized by non-Abelian charges and opens up exciting possibilities for exploring the rich and uncharted territory of non-equilibrium topological phases. The authors propose an implementation of Floquet non-Abelian topological insulators in a 1D three-band system with parity-time symmetry. Furthermore, they demonstrate that the bulk-edge correspondence is multifold and follows the multiplication rule of a quaternion group.

Automated summary

This study investigates topological insulators with multiple tangled gaps in Floquet settings and uncovers uncharted Floquet non-Abelian topological insulators without any static or Abelian analog. It reveals that the bulk-edge correspondence follows the multiplication rule of the quaternion group and demonstrates an exotic swap effect that is absent in Floquet Abelian systems. This work presents Floquet topological insulators characterized by non-Abelian charges and opens up possibilities for exploring non-equilibrium topological phases.