# Efficient computation of the $W_3$ topological invariant and application to Floquet-Bloch systems

## TL;DR

This paper presents a fast, robust algorithm for computing the $W_3$ topological invariant in unitary maps, applicable to Floquet-Bloch systems, and demonstrates its effectiveness on a driven graphene model to predict edge states.

## Contribution

The paper introduces a new efficient algorithm for $W_3$ invariant computation that requires minimal manipulation and applies it to Floquet-Bloch systems, enabling accurate topological analysis.

## Key findings

- Algorithm converges rapidly on coarse grids.
- Successfully applied to irradiated graphene model.
- Predicts the number of anomalous edge states.

## Abstract

We introduce an efficient algorithm for the computation of the $W_3$ invariant of general unitary maps, which converges rapidly even on coarse discretization grids. The algorithm does not require extensive manipulation of the unitary maps, identification of the precise positions of degeneracy points, or fixing the gauge of eigenvectors. After construction of the general algorithm, we explain its application to the $2+1$ dimensional maps that arise in the Floquet-Bloch theory of periodically driven two-dimensional quantum systems. We demonstrate this application by computing the $W_3$ invariant for an irradiated graphene model with a continuously modulated Hamilton operator, where it predicts the number of anomalous edge states in each gap.