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.

Topological invariants for Floquet-Bloch systems with chiral, time-reversal, or particle-hole symmetry

“…(2) also fulfills the standard chiral symmetry relation but possesses an additional symmetry that protects the topological phases and boundary states. Without the k π -shift, chiral symmetry does not allow for the symmetry-protected boundary states observed here [27][28][29] . Because of the T − t argument on the right hand side, the symmetry relation (2) does not extend to U (k, t) but only to the time-symmetrized propagator …”

“…(2) chiral symmetry is defined with a k → k+k π momentum shift, which differs from the standard definition in the literature [27][28][29] , that does not involve a momentum shift. The origin of the momentum shift in Eq.…”

“…Note that the symmetry relation (2) differs from the standard definition of chiral symmetry [27][28][29] , which does not contain the momentum shift k π . The inclusion of the momentum shift k π is crucial for the existence of symmetry-protected boundary states and of the Z 2 -invariant defined below.…”

“…Topological invariants for Floquet-Bloch systems with and without additional symmetries have been introduced before 7,11,[24][25][26]28,29 , and our constructions resemble some of them [24][25][26] in various aspects. However, most constructions in the literature differ for each symmetry.…”

Floquet analysis of excitations in materials

“…Still using the effective Hamiltonian formulation, all such Floquet-topological phases have been systematically classified [58]. It has also been used to propose the Floquet topological Magnon [59][60][61].…”