Engineering subharmonic responses beyond prethermalization via Floquet scar states

In this work we propose a new scheme to engineer subharmonic responses via scar states in a generalized PXP model. We first show that the generalized PXP model also possesses a band of scar states like the pristine PXP model does. In addition, we reveal that a generalized forward scattering approximation (FSA) still works for these scar states. We further argue that the FSA subspace exhibits an $SO(3)$ symmetry, which enables an $SO(3)$-FSA approach for the scar states. When such a model is placed under periodic driving, nontrivial Floquet scar states emerge in the quasienergy spectrum. One appealing feature of such Floquet scar states is that they can support subharmonic responses akin to the discrete time crystalline phase. In particular, such subharmonic responses can exist beyond the conventional prethermalization regime, where either large frequencies or large driving amplitudes are required.