Floquet engineering with particle swarm optimization: Maximizing topological invariants

It is of theoretical and experimental interest to engineer topological phases with very large topological invariants via periodic driving. As advocated by this paper, such Floquet engineering can be elegantly achieved by the particle swarm optimization (PSO) technique from the swarm intelligence family. With the recognition that conventional gradient-based optimization approaches are not suitable for directly optimizing topological invariants as integers, the highly effective PSO route yields new promises in the search for exotic topological phases, requiring limited physical resources. Our results are especially timely in view of two important insights from literature: low-frequency driving may be beneficial in creating large topological invariants, but an open-ended low-frequency driving often leads to drastic fluctuations in the obtained topological invariants. Indeed, using a simple continuously driven Harper model with three quasienergy bands, we show that the Floquet-band Chern numbers can enjoy a many-fold increase compared with that using a simple harmonic driving of the same period, without demanding more energy cost of the driving field. It is also found that the resulting Floquet insulator bands are still well gapped with the maximized topological invariants in agreement with physical observations from Thouless pumping. The emergence of many edge modes under the open boundary condition is also consistent with the bulk-edge correspondence. Our results are expected to be highly useful towards the optimization of many different types of topological invariants in Floquet topological matter.