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Recent experimental advances in Floquet engineering and controllingdissipation in open systems have brought about unprecedented flexibility intailoring novel phenomena without any static and Hermitian analogues. It can beepitomized by the various Floquet and non-Hermitian topological phases.Topological classifications of either static/Floquet Hermitian or staticnon-Hermitian systems based on the underlying symmetries have been wellestablished in the past several years. However, a coherent understanding andclassification of Floquet non-Hermitian (FNH) topological phases have not beenachieved yet. Here we systematically classify FNH topological bands for 54-foldgeneralized Bernard-LeClair (GBL)symmetry classes and arbitrary spatialdimensions using $K$-theory. The classification distinguishes two differentscenarios of the Floquet operator's spectrum gaps [dubbed as Floquet operator(FO) angle-gapped and FO angle-gapless]. The results culminate into twoperiodic tables, each containing 54-fold GBL symmetry classes. Our schemereveals FNH topological phases without any static/Floquet Hermitian and staticnon-Hermitian counterparts. And our results naturally produce the periodictables of Floquet Hermitian topological insulators and Floquet unitaries. Theframework can also be applied to characterize the topological phases of bosonicsystems. We provide concrete examples of one andtwo-dimensionalfermionic/bosonic systems. And we elucidate the meaning of thetopological invariants and their physical consequences. Our paper lays thefoundation for a comprehensive exploration of FNH topological bands. And itopens a broad avenue toward uncovering unique phenomena and functionalitiesemerging from the synthesis of periodic driving, non-Hermiticity, and bandtopology.

Symmetry and topological classification of Floquet non-Hermitian systems