Systematic generation of the cascade of anomalous dynamical first- and higher-order modes in Floquet topological insulators

After extensive investigation of the Floquet second-order topological insulator (FSOTI) in two dimensions, here we propose two driving schemes to systematically engineer the hierarchy of the Floquet first-order topological insulator, the FSOTI, and the Floquet third-order topological insulator in three dimensions. Our driving protocols allow these Floquet phases to showcase regular 0, anomalous it, and hybrid 0-it-modes in a unified phase diagram, obtained for both two- and three-dimensional (2D and 3D) systems, while starting from the lower-order topological or nontopological phases. Both the step drive and the mass kick protocols exhibit the analogous structure of the evolution operator around the high symmetry points. These eventually enable us to understand the Floquet phase diagrams analytically and the Floquet higher-order modes numerically based on finite-size systems. The number of 0 and it modes can be tuned irrespective of the frequency in the step drive scheme, while we observe frequency-driven topological phase transitions for the mass kick protocol. We topologically characterize some of these higher-order Floquet phases (harboring either 0 or anomalous it mode) by a suitable topological invariant in 2D and 3D cases.

Automated summary

In this study, we propose two driving schemes to systematically engineer the hierarchy of Floquet first-order topological insulators, FSOTIs, and Floquet third-order topological insulators in three dimensions. The step drive and mass kick protocols exhibit analogous evolution operator structures and allow for the understanding of Floquet phase diagrams and higher-order modes.