Quantum geometry and geometric entanglement entropy of one-dimensional Floquet topological matter

The geometry of quantum states could offer indispensable insights for characterizing the topological properties, phase transitions, and entanglement nature of many-body systems. In this work, we reveal the quantum geometry and the associated entanglement entropy (EE) of Floquet topological states in one-dimensional periodically driven systems. The quantum metric tensors of Floquet states are found to show nonanalytic signatures at topological phase transition points. Away from the transition points, the bipartite geometric EE of Floquet states exhibits an area-law scaling vs the system size, which holds for a Floquet band at any filling fractions. For a uniformly filled Floquet band, the EE further becomes purely quantum geometric. At phase transition points, the geometric EE scales logarithmically with the system size and displays cusps in the nearby parameter ranges. These discoveries are demonstrated by investigating typical Floquet models including periodically driven spin chains, Floquet topological insulators, and superconductors. Our findings uncover the rich quantum geometries of Floquet states, unveiling the geometric origin of EE for gapped Floquet topological phases, and introducing information-theoretic means of depicting topological transitions in Floquet systems.