Exotic topological phases induced by periodic driving

Topological phases are a novel class of physical phases, distinct from those defined by Landau symmetry-breaking theory and characterized by order parameters. The phase transitions of topological phases are typically unrelated to symmetry breaking, their properties being described by topological invariants. The important characteristic of topological phases is their topological protection: as long as the band structure remains open, continuous variations or perturbations of the Hamiltonian will not alter the topological properties of the system. Their uniqueness lies in boundary states being linked to topological invariants and exhibiting distinctive transport properties. Despite rapid advances in this field, material limitations persist, making the control of topological properties an urgent challenge. To address this challenge, Floquet engineering offers novel approaches. By introducing time as a tunable variable, this method transforms systems into periodic time-dependent regimes, thereby enabling precise control over topological properties and providing practical pathways for applications. This paper primarily explores novel topological phases induced by periodic driving in diverse systems, including phase transitions between distinct topological phases, topological phases governed by major topological invariants, and hybrid-order topological phases.