Topological invariants of Floquet systems: General formulation, special properties, and Floquet topological defects

Abstract

Periodically driven (Floquet) systems have been under active theoretical and experimental investigations. This paper aims at a systematic study in the following aspects of Floquet systems: (i) A systematic formulation of topological invariants of Floquet systems based on the cooperation of topology and symmetries. Topological invariants are constructed for the ten symmetry classes in all spatial dimensions, for both homogeneous Floquet systems (Floquet topological insulators and superconductors) and Floquet topological defects. Meanwhile, useful representative Dirac Hamiltonians for all the symmetry classes are obtained and studied. (ii) A general theory of Floquet topological defects, based on the proposed topological invariants. (iii) Models and proposals of Floquet topological defects in low dimensions. Among other defect modes, we investigate Floquet Majorana zero modes and Majorana Pi modes in vortices of topologically trivial superconductors under a periodic drive. In addition, we clarified several notable issues about Floquet topological invariants. Among other issues, we prove the equivalence between the effective-Hamiltonian-based band topological invariants and the frequency-domain band topological invariants.

54 pages, 13 figures. For higher resolution figures please see the published version

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• Floquet engineering of topological localization transitions and mobility edges in one-dimensional non-Hermitian quasicrystals
• Topological space-time crystal
• Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry
• Engineering of topological phases in driven thin topological insulator: Structure inversion asymmetry effect
• Universal driving protocol for symmetry-protected Floquet topological phases
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• Theory of Anomalous Floquet Higher-Order Topology: Classification, Characterization, and Bulk-Boundary Correspondence
• Designing adiabatic time-evolution from high frequency bichromatic sources
• Immutable quantized transport in Floquet chains