Topological phenomena in honeycomb Floquet metamaterials

We dedicate this paper to the topological analysis of subwavelength solutions in Floquet metamaterials. This work should be considered as a basis for further investigation on whether topological properties of the bulk materials are linked to the occurrence of edge modes. The subwavelength solutions being described by a periodically parameterized time-periodic linear ordinary differential equation $\left\{\frac{d}{dt}X = A_\alpha(t)X\right\}_{\alpha \in \mathbb{T}^d}$, we put ourselves in the general setting of periodically parameterized time-periodic linear ordinary differential equations and introduce a way to (topologically) classify a Floquet normal form $F,~P$ of the associated fundamental solution $\left\{X_\alpha(t) = P(\alpha,t)\exp(tF_\alpha)\right\}_{\alpha \in \mathbb{T}^d}$. This is achieved by analysing the topological properties of the eigenvalues and eigenvectors of the monodromy matrix $X_\alpha(T)$ and the Lyapunov transformation $P(\alpha,t)$. The corresponding topological invariants can then be applied to the setting of Floquet metamaterials. In this paper these general results are considered in the case of honeycomb Floquet metamaterials. We provide two interesting examples of topologically non-trivial time-modulated honeycomb structures.