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Periodically-driven or Floquet systems can realize anomalous topologicalphenomena that do not exist in any equilibrium states of matter, whoseclassification and characterization require new theoretical ideas that arebeyond the well-established paradigm of static topological phases. In thiswork, we provide a general framework to understand anomalous Floquethigher-order topological insulators (AFHOTIs), the classification of which hasremained a challenging open question. In two dimensions (2D), such AFHOTIs aredefined by their robust, symmetry-protected corner modes pinned at specialquasienergies, even though all their Floquet bands feature trivial bandtopology. The corner-mode physics of an AFHOTI is found to be genericallyindicated by 3D Dirac/Weyl-like topological singularities living in the phasespectrum of the bulk time-evolution operator. Physically, such a phase-bandsingularity is essentially a "footprint" of the topological quantumcriticality, which separates an AFHOTI from a trivial phase adiabaticallyconnected to a static limit. Strikingly, these singularities featureunconventional dispersion relations that cannot be achieved on any staticlattice in 3D, which, nevertheless, resemble the surface physics of 4Dtopological crystalline insulators. We establish the above higher-orderbulk-boundary correspondence through a dimensional reduction technique, whichalso allows for a systematic classification of 2D AFHOTIs protected by pointgroup symmetries. We demonstrate applications of our theory to two concrete,experimentally feasible models of AFHOTIs protected by $C_2$ and $D_4$symmetries, respectively. Our work paves the way for a unified theory forclassifying and characterizing Floquet topological matters.

Theory of Anomalous Floquet Higher-Order Topology: Classification, Characterization, and Bulk-Boundary Correspondence