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Complex classes of periodically driven topological lattice systems

Periodically driven (Floquet) crystals are described by their quasi-energy spectrum. Their topological properties are characterized by invariants attached to the gaps of this spectrum. In this article, we define such invariants in all space dimensions, both in the case where no symmetry is present and in the case where the unitary chiral symmetry is present. When no symmetry is present, a $\mathbb{Z}$-valued invariant can be defined in each gap in all even space dimensions. This invariant does not capture all the properties of a system where chiral symmetry is present. In even space dimension, chiral symmetry puts constraints on its values in different gaps. In odd space dimension, chiral symmetry also enables to define a $\mathbb{Z}$-valued invariant, only in the chiral gaps $0$ and $π$. We relate both gap invariants to the standard invariants characterizing the quasi-energy bands of the system. Examples in one and three dimensions are discussed.