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

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 spacetime 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. CONTENTS 32 A. Equivalent classification with symmetrized evolution operators 32 B. Decomposition of time evolution operators 32 C. Order of HOTI/SCs and symmetry-breaking mass terms 32 References 33

Time-induced second-order topological superconductors

“…To further elaborate the above argument, we shall compare our construction with that of Refs. [79][80][81], which at first glance might look similar to ours (i.e., due to the use of monochromatic time-periodic drives). In Refs.…”

“…Before ending this section, it is necessary to compare this paper with relevant earlier literature on Floquet topological phases [47,[72][73][74][75][76][77][78][79][80][81]. First, Ref.…”

“…Second, some of Refs. [72][73][74][75][76][77][78][79][80][81] demonstrate the generation of higher-order topologically nontrivial phases by applying appropriate time-periodic drives to a static topologically trivial system. However, the latter may already possess the necessary requirements to host such higher-order topological phases on its own, such as accomplished by either tuning some system parameters or adding appropriate mass terms.…”

“…In Refs. [79][80][81], the time-periodic drives are designed such that the resulting Hamiltonian obeys a time-glide symmetry, which can then be viewed as an effective reflection symmetry in the enlarged Hilbert (Sambe) space. In this case, the role of the timeperiodic drives is to effectively create a symmetry necessary for the formation of second-order topological phases, whereas the underlying static Hamiltonian already contains the necessary topological structure.…”

Floquet Second-Order Topological Phases in Momentum Space

“…In recent years, the study of HOTPs has been generalized to nonequilibrium systems, such as those subject to time-periodic drivings [ 82 , 83 , 84 , 85 , 86 , 87 , 88 , 89 , 90 , 91 , 92 , 93 , 94 ] or non-Hermitian effects [ 95 , 96 , 97 , 98 , 99 , 100 , 101 , 102 , 103 ]. The motivation behind the exploration of HOTPs in periodically driven systems is threefold.…”