Dynamic bulk-boundary correspondence for anomalous Floquet topology from dimensional hierarchy

Periodically driven systems with internal and spatial symmetries can exhibit a variety of anomalous boundary behaviors at both the zero and π quasienergies despite the trivial bulk Floquet bands. These phenomena are called anomalous Floquet topology (AFT) as they are unconnected from their static counterpart, arising from the winding of the time evolution unitary rather than the bulk Floquet bands at the end of the driving period. In this paper, we systematically derive the first and inversion-symmetric second-order AFT bulk-boundary correspondence for Altland-Zirnbauer (AZ) classes BDI, D, DIII, AII. For each AZ class, we start the dimensional hierarchy with the parent dimension having Z-classification, then use it as a interpolating map to classify the lower-dimensional descendants. From the Atiyah-Hierzebruch spectral sequence (AHSS), we identify the subspace that contains the topological information and faithfully derive the AFT bulk-boundary correspondence for both the parent and descendants. Our theory provides analytic tools for Floquet topological phenomena.