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It is of theoretical and experimental interest to engineer topological phaseswith very large topological invariants via periodic driving. As advocated bythis work, such Floquet engineering can be elegantly achieved by the particleswarm optimization (PSO) technique from the swarm intelligence family. With therecognition that conventional gradient-based optimization approaches are notsuitable for directly optimizing topological invariants as integers, the highlyeffective PSO route yields new promises in the search for exotic topologicalphases, requiring limited physical resource. Our results are especially timelyin view of two important insights from literature: low-frequency driving may bebeneficial in creating large topological invariants, but an open-endedlow-frequency driving often leads to drastic fluctuations in the obtainedtopological invariants. Indeed, using a simple continuously driven Harper modelwith three quasi-energy bands, we show that the Floquet-band Chern numbers canenjoy many-fold increase compared with that using a simple harmonic driving ofthe same period, without demanding more energy cost of the driving field. It isalso found that the resulting Floquet insulator bands are still well-gapped,with the maximized topological invariants in agreement with physicalobservations from Thouless pumping. The emergence of many edge modes under theopen boundary condition is also consistent with the bulk-edge correspondence.Our results are expected to be highly useful towards the optimization of manydifferent types of topological invariants in Floquet topological matter.

Floquet Engineering with Particle Swarm Optimization: Maximizing Topological Invariants