Quantum mechanics

Gravitational singularity

Economics

Floquet theory

Mathematical optimization

Hamiltonian (control theory)

Physics

Nonlinear system

Topological order

Mathematics

Topological insulator

Homogeneous space

Geometry

Quantum

Combinatorics

Finance

Topology (electrical circuits)

Order (exchange)

Physical Review Letters

We propose a versatile framework to dynamically generate Floquet higher-order topological insulators by multistep 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 $\ensuremath{\pi}$-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 a Floquet Hamiltonian gives a pair of ${\mathbb{Z}}_{2}$ invariant ${\ensuremath{\nu}}_{0}$ and ${\ensuremath{\nu}}_{\ensuremath{\pi}}$ which fully characterize the higher-order topology and predict the appearance of zero- and $\ensuremath{\pi}$-corner modes. Our work establishes a systematic route to construct and characterize Floquet higher-order topological phases.

Dynamical Singularities of Floquet Higher-Order Topological Insulators