# Hopf characterization of two-dimensional Floquet topological insulators

## TL;DR

This paper introduces a Hopf insulator framework for 2D Floquet topological insulators, linking their topological invariants to knot linking numbers, and proposes an experimental measurement scheme in optical lattices.

## Contribution

It provides a novel topological characterization of Floquet insulators using Hopf maps and linking numbers, connecting theoretical invariants to measurable quantities.

## Key findings

- Hopf invariants correspond to winding numbers in Floquet systems.
- Topological invariants can be measured via linking numbers in experiments.
- The framework applies to driven two-band models in 2+1 dimensions.

## Abstract

We present a topological characterization of time-periodically driven two-band models in 2+1 dimensions as Hopf insulators. The intrinsic periodicity of the Floquet system with respect to both time and the underlying two-dimensional momentum space constitutes a map from a three dimensional torus to the Bloch sphere. As a result, we find that the driven system can be understood by appealing to a Hopf map that is directly constructed from the micromotion of the drive. Previously found winding numbers are shown to correspond to Hopf invariants, which are associated with linking numbers describing the topology of knots in three dimensions. Moreover, after being cast as a Hopf insulator, not only the Chern numbers, but also the winding numbers of the Floquet topological insulator become accessible in experiments as linking numbers. We exploit this description to propose a feasible scheme for measuring the complete set of their Floquet topological invariants in optical lattices.