The violation of the Floquet version of the eigenstate thermalization hypothesis is rigorously discussed with realistic Hamiltonians. Our model is based on the PXP-type interactions without disorder. We rigorously prove the existence of many-body scar states in the Floquet eigenstates that appear only at specific periods of driving by showing the explicit expressions of the wave functions. This is equivalently the first rigorous proof of the breakdown of the Floquet eigenstate thermalization hypothesis. Through the exact expression of the scar states, the underlying physical mechanism is clarified. Using the underlying mechanism, various driven Hamiltonians with Floquet-scar states can be systematically engineered. We then discuss Floquet-scar states that can be checked through time evolution of observables in a chain of Rydberg atoms.
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