Floquet higher-order topological insulators and superconductors with space-time symmetries

Floquet higher-order topological insulators (HOTIs) and superconductors (SCs) with an order-2 space-time symmetry or antisymmetry are classified. This is achieved by considering unitary loops, whose nontrivial topology leads to the anomalous Floquet topological phases, subject to a space-time symmetry/antisymmetry. By mapping these unitary loops to static Hamiltonians with an order-2 crystalline symmetry/antisymmetry, one is able to obtain the K groups for the unitary loops and thus complete the classification of Floquet HOTIs and SCs. Interestingly, we find that for every order-2 nontrivial space-time symmetry/antisymmetry involving a half-period time translation, there exists a unique order-2 static crystalline symmetry/antisymmetry such that the two symmetries/antisymmetries give rise to the same topological classification. Moreover, by exploiting the frequency-domain formulation of the Floquet problem, a general recipe that constructs model Hamiltonians for Floquet HOTIs and SCs is provided, which can be used to understand the classification of Floquet HOTIs and SCs from an intuitive and complementary perspective.