# Out of equilibrium higher-order topological insulator: Floquet engineering and quench dynamics

## TL;DR

This paper demonstrates the realization of a Floquet second-order topological insulator with corner modes in a quantum spin Hall insulator, analyzing its properties and dynamics under periodic driving and quench conditions.

## Contribution

It introduces a method to engineer a Floquet HOT insulator with quantized quadrupolar moment using periodic kicking of a QSHI, and studies the quench dynamics of corner modes.

## Key findings

- Realization of a Floquet second-order topological insulator with corner modes.
- Quantized Floquet quadrupolar moment $Q^{
m Flq}_{xy}=0.5$ achieved.
- Corner mode survival probability exhibits periodic revivals after a quench.

## Abstract

Higher-order topological~(HOT) states,~hosting topologically protected modes on lower-dimensional boundaries,~such as hinges and corners, have recently extended the realm of the static topological phases.~Here we demonstrate the possibility of realizing a two-dimensional \emph{Floquet} second-order topological insulator, featuring corner-localized zero quasienergy modes and characterized by quantized Floquet qudrupolar moment $Q^{\rm Flq}_{xy}=0.5$, by periodically kicking a quantum spin Hall insulator (QSHI) with a discrete fourfold ($C_4$) and time-reversal (${\mathcal T}$) symmetry breaking Dirac mass perturbation.~Furthermore, we show that $Q^{\rm Flq}_{xy}$ becomes independent of the choice of origin as the system approaches the thermodynamic limit.~We also analyze the dynamics of a corner mode after a sudden quench, when the $C_4$ and ${\mathcal T}$ symmetry breaking perturbation is switched off, and find that the corresponding survival probability displays periodic appearances of complete, partial and no revival for long time, encoding the signature of corner modes in a QSHI.~Our protocol is sufficiently general to explore the territory of dynamical HOT phases in insulators and gapless systems.