# Eigenvalue crossings in Floquet topological systems

## TL;DR

This paper interprets the topological invariant of Floquet systems through local eigenvalue crossings, linking global winding numbers to local data and classifying Floquet unitaries on manifolds with boundary.

## Contribution

It provides a local interpretation of the winding number in Floquet topological systems and extends the classification of Floquet unitaries using local eigenvalue crossing data.

## Key findings

- Eigenvalue crossings are finite and non-degenerate up to homotopy.
- Each crossing has a local Chern number that sums to the global winding number.
- The classification of Floquet unitaries is achieved via local indices and homotopy theory.

## Abstract

The topology of electrons on a lattice subject to a periodic driving is captured by the three-dimensional winding number of the propagator that describes time-evolution within a cycle. This index captures the homotopy class of such a unitary map. In this paper, we provide an interpretation of this winding number in terms of local data associated to the the eigenvalue crossings of such a map over a three dimensional manifold, based on an idea from Nathan and Rudner, New Journal of Physics, 17(12) 125014, 2015. We show that, up to homotopy, the crossings are a finite set of points and non degenerate. Each crossing carries a local Chern number, and the sum of these local indices coincides with the winding number. We then extend this result to fully degenerate crossings and extended submanifolds to connect with models from the physics literature. We finally classify up to homotopy the Floquet unitary maps, defined on manifolds with boundary, using the previous local indices. The results rely on a filtration of the special unitary group as well as the local data of the basic gerbe over it.