Dynamic bulk-boundary correspondence for anomalous Floquet topology

Periodically driven systems with internal and spatial symmetries can exhibit a variety of anomalous boundary behaviors at both the zero and $\ensuremath{\pi}$ quasienergies despite the trivial bulk Floquet bands. These phenomena are called anomalous Floquet topology (AFT) as they are unconnected from their static counterpart, emerging 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, and AII. For each AZ class, we start a dimensional hierarchy with a parent dimension having $\mathbb{Z}$ classification, then use it as an interpolating map to classify the lower-dimensional descendants. From the Atiyah-Hirzebruch spectral sequence, we identify the subspace that contains topological information and faithfully derive the AFT bulk-boundary correspondence for both the parent and descendants. Our theory provides analytic tools for out-of-equilibrium topological phenomena.