# Floquet Hopf Insulators

## TL;DR

This paper predicts a new three-dimensional Floquet topological insulator called the Floquet Hopf insulator, characterized by two topological invariants, leading to unique edge states and expanding the classification of Floquet phases.

## Contribution

It introduces the Floquet Hopf insulator with two distinct topological invariants, linking topological defects in time evolution to novel bulk-boundary phenomena.

## Key findings

- Existence of a Floquet Hopf insulator with two invariants
- Topologically protected edge modes at 0- or π-quasienergy
- Novel bulk-boundary correspondence in Floquet systems

## Abstract

We predict the existence of a novel Floquet topological insulator in three-dimensional two-band systems, the Floquet Hopf insulator, which possesses two distinct topological invariants. One is the Hopf $\mathbb{Z}$ invariant, a linking number characterizing the (non-driven) Hopf topological insulator. The second invariant is an intrinsically Floquet $\mathbb{Z}_2$ invariant, and represents a condensed matter realization of the topology underlying the Witten anomaly in particle physics. Both invariants arise from topological defects in the system's time-evolution, subject to a process in which defects at different quasienergy exchange even amounts of topological charge. Their contrasting classifications lead to a measurable physical consequence, namely, an unusual bulk-boundary correspondence where gapless edge modes are topologically protected, but may exist at either 0- or $\pi$-quasienergy. Our results represent a phase of matter beyond the conventional classification of Floquet topological insulators.