Abstract
Floquet higher-order topological insulators and superconductors (HOTI/SCs) with an order-two 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-two crystalline symmetry/antisymmetry, one is able to obtain the $K$ groups for the unitary loops and thus complete the classification of Floquet HOTI/SCs. Interestingly, we found that for every order-two nontrivial space-time symmetry/antisymmetry involving a half-period time translation, there exists a unique order-two 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 HOTI/SCs is provided, which can be used to understand the classification of Floquet HOTI/SCs from an intuitive and complimentary perspective.
Highlights
The interplay between symmetry and topology leads to various of topological phases
In this work we have completed the classification of the Floquet higher-order topological insulators (HOTIs) and SCs with an order-2 space-time symmetry/antisymmetry
By introducing a Hermitian map, we were able to map the unitary loops into Hermitian matrices and define bulk K groups as well as K subgroup series for unitary loops
Summary
The interplay between symmetry and topology leads to various of topological phases. For a translationally invariant noninteracting gapped system, the topological phase is characterized by the band structure topology, as well as the symmetries the system respects. This Hermitian map has advantages over the one used in earlier works [32,41], because it maps a unitary loop with a given order-2 space-time symmetry/antisymmetry to a static Hamiltonian of a topological crystalline insulator with an order-2 crystalline symmetry/antisymmetry This enable us to exploit the full machinery of K theory, to define K groups, as well as the K subgroup series introduced in Ref. IV–VII and understand the main results in terms of the frequency-domain formulation
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