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Floquet engineering is one of the most vigorous fields in periodically driven(Floquet) systems, with which we can control phases of matter usually byhigh-frequency drives. In this paper, with Floquet engineering by a combinationof high-frequency drives and resonant drives, we propose a way to realizenontrivial topological phases protected by a $\mathbb{Z}_2 \times \mathbb{Z}_2$symmetry only in the presence of a $\mathbb{Z}_2$ symmetry, using a robustemergent $\mathbb{Z}_2$ symmetry induced by the resonant drives. Moreover, thesymmetry protected topological (SPT) phases are switchable between nontrivialand trivial phases only by the direction of a static transverse field, and evenperturbations on the resonant drive can be utilized to realize richer SPTphases. We also discuss the real-time dynamics of the model, and find thatwhich topological phases the system lies in can be distinguished by a perioddoubling of a nonlocal order parameter, as with discrete time crystals. Arealization or a control of nontrivial SPT phases without the requiredsymmetries by resonant drives, proposed in this paper, would shed a new lighton the observation of topological phenomena in nonequilibrium setups.

Floquet engineering of topological phases protected by emergent symmetries under resonant drives