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In this work, we investigate the delocalization-localization transition of Floquet eigenstates in a driven chain with an incommensurate Aubry-Andr\'e (AA) on-site potential and a small non-reciprocal hopping term which is driven periodically in time. The driving protocol is chosen such that the Floquet Hamlitonian corresponds a localized phase in the high-frequency limit and a delocaized phase in the low-frequency limit. By numerically ecaluating the inverse participation ratio and the fractal dimension $D_q$, we identify a clear delocalization-localization transition of the Floquet eigenstates at a critical frequency $\omega_{c}\approx0.318\pi$. This transition aligns with the real-to-complex spectrum transition of the Floquet Hamiltonian. For the driven frequency $\omega>\omega_c$, the system resides in a localized phase, and we observe the emergence of CAT states-linear superposition of localized single particle states-in the Floquet spectrum. These states exhibits weight distributions concentrated around a few nearby sites of the chain, forming two peaks of unequal weight due to the non-reciprocal effect, distinguishing them from the Hermitic case. In constrast, for $\omega<\omega_c$, we identidfy the presence of a mobility edge over a range of driving frequencies, separateing localized states (above the edge) from mulitfractal and extended states (below the edge). Notablely, multifractal states are observed in the Floquet eigenspectrum across over a broad frequency range. Importantly, we highlight that the non-driven, non-reciprocal AA model does not support either multifractal states or a mobility edge in its spectrum. Thus, our findings reveal unique dynamical signatures absent in the non-driven non-Hermitian scenario, offering a fresh perspective on the localization properties of periodically driven systems. Finally, we provide a possible circuit experiment scheme for the periodically driven non-reciprocal AA model. In the following work, we will extend our research direction to clean systems, such as Stark models, to explore the influence of periodic driving on their localization properties.
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Keywords:
- periodically driven system /
- localization /
- mobility edge /
- non-reciprocal system
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