Unconventional Floquet topological phases from a new bulk-boundary correspondence.

The conventional bulk-boundary correspondence fails for anomalous Floquet topological phases in which the boundary modes do not correspond to the bulk topology of Floquet bands. Here, we rebuild bulk-boundary correspondence for anomalous Floquet topological phases and predict new topological states through local information of Floquet bands in particular momentum subspaces called band inversion surfaces (BISs). We show that the local topological structure formed in each BIS uniquely determines the number and features of gapless boundary modes, which facilitates to realize novel Floquet topological phases by engineering BIS configurations. In particular, we predict a two-dimensional conventionally unclassifiable anomalous Floquet valley-Hall phase which carries trivial bulk topological invariants but features counter-propagating edge states in each quasienergy gap, and an anomalous chiral topological phase in higher dimensions. Our findings open a systematic way to realize and characterize unconventional Floquet topological phases beyond conventional characterization.