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

TL;DR

This paper introduces two driving schemes to systematically generate and analyze hierarchical Floquet topological insulators in 2D and 3D, revealing tunable higher-order modes and phase transitions with analytical and numerical methods.

Contribution

It proposes new driving protocols to engineer and understand Floquet topological phases, including higher-order modes, in both two and three dimensions.

Findings

Floquet phases exhibit regular, anomalous, and hybrid $0$-$\pi$ modes.

Number of $0$ and $\pi$-modes can be tuned independently of frequency.

Frequency-driven topological phase transitions are observed in the mass kick protocol.

Abstract

After extensive investigation on the Floquet second-order topological insulator (FSOTI) in two dimension (2D), here we propose two driving schemes to systematically engineer the hierarchy of Floquet first-order topological insulator, FSOTI, and Floquet third-order topological insulator in three dimension (3D). Our driving protocols allow these Floquet phases to showcase regular $0$, anomalous $\pi$, and hybrid $0$-$\pi$-modes in a unified phase diagram, obtained for both 2D and 3D systems, while staring from the lower order topological or non-topological 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 $\pi$-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 $\pi$ mode) by suitable topological invariant in 2D and 3D cases.