# Wannier representation of Floquet topological states

## TL;DR

This paper extends the concept of Wannier functions to periodically driven systems, providing a unified topological framework for Floquet insulators and related states through hybrid Wannier center connectivity.

## Contribution

It introduces a generalized Wannier representation for Floquet topological states, revealing topological obstructions in driven systems not present in static insulators.

## Key findings

- Hybrid Wannier centers characterize Floquet topological phases.
- Time dependence of Wannier centers indicates topological obstructions.
- Obstructions prevent continuous deformation into trivial insulators.

## Abstract

A universal feature of topological insulators is that they cannot be adiabatically connected to an atomic limit, where individual lattice sites are completely decoupled. This property is intimately related to a topological obstruction to constructing a localized Wannier function from Bloch states of an insulator. Here we generalize this characterization of topological phases toward periodically driven systems. We show that nontrivial connectivity of hybrid Wannier centers in momentum space and time can characterize various types of topology in periodically driven systems, which include Floquet topological insulators, anomalous Floquet topological insulators with micromotion-induced boundary states, and gapless Floquet states realized with topological Floquet operators. In particular, nontrivial time dependence of hybrid Wannier centers indicates impossibility of continuous deformation of a driven system into an undriven insulator, and a topological Floquet operator implies an obstruction to constructing a generalized Wannier function which is localized in real and frequency spaces. Our results pave a way to a unified understanding of topological states in periodically driven systems as a topological obstruction in Floquet states.