# Dynamical Singularities of Floquet Higher-Order Topological Insulators

## TL;DR

This paper introduces a systematic method to create and analyze Floquet higher-order topological insulators with dynamically generated corner modes, using topological invariants derived from the full unitary return map.

## Contribution

It provides a versatile framework for generating Floquet higher-order topological phases from trivial Hamiltonians and introduces new dynamical topological invariants for their characterization.

## Key findings

- Demonstrated Floquet quadrupole and octupole insulators with protected corner modes
- Developed dynamical invariants from the full unitary return map
- Identified Weyl singularities in phase bands related to topological charges

## Abstract

We propose a versatile framework to dynamically generate Floquet higher-order topological insulators by multi-step driving of topologically trivial Hamiltonians. Two analytically solvable examples are used to illustrate this procedure to yield Floquet quadrupole and octupole insulators with zero- and/or $\pi$-corner modes protected by mirror symmetries. Furthermore, we introduce dynamical topological invariants from the full unitary return map and show its phase bands contain Weyl singularities whose topological charges form dynamical multipole moments in the Brillouin zone. Combining them with the topological index of Floquet Hamiltonian gives a pair of $\mathbb{Z}_2$ invariant $\nu_0$ and $\nu_\pi$ which fully characterize the higher-order topology and predict the appearance of zero- and $\pi$-corner modes. Our work establishes a systematic route to construct and characterize Floquet higher-order topological phases.