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In this work, we investigate the delocalization-localization transition of Floquet eigenstates in a driven chain with an incommensurate Aubry-André (AA) on-site potential and a small non-reciprocal hopping term that is driven periodically in time. The driving protocol is chosen such that the Floquet Hamiltonian corresponds to a localized phase in the high-frequency limit and a delocalized phase in the low-frequency limit. By numerically calculating 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 $\omega>\omega_{\mathrm{c}}$, 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 exhibit 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 contrast, for $\omega<\omega_{\mathrm{c}}$, we identify the presence of a mobility edge over a range of driving frequencies, separating localized states (above the edge) from multifractal and extended states (below the edge). Notably, multifractal states are observed in the Floquet eigenspectrum across a broad frequency range. Importantly, we highlight that the non-driven, non-reciprocal AA model does not support multifractal states nor a mobility edge in its spectrum. Thus, our findings reveal unique dynamical signatures that do not exist in the non-driven non-Hermitian scenario, providing 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 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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图 1 一维非互易模型示意图. 红线和蓝线代表不同跃迁强度, 其中$ \mathcal{J}(t) $由式(1)所示, $ h $ 代表非互易强度, $ j $代表格点
Fig. 1. Schematic diagram of the one-dimensional non-reciprocal model. The red and blue lines represent different hopping amplitudes, $ h $ is non-reciprocal amplitudes and $ j $ is the site index.
图 2 (a)逆参与率$ I^{(2)}_{n} $随着能级指标$ n/L $和频率$ \omega $的变化情况. 这里, 能量实部升序排列. (b)分形维度$ D_{2} $随着能级指标$ \tilde{n}/L $和频率$ \omega $的变化情况, 以及复能量占比$ f_{\mathrm{Im}} $(黑色实线)随频率$ \omega $的变化图, 其中能级指标$ \tilde{n} $以逆参与率值的大小升序排列. 这里$ L=2048 $
Fig. 2. (a) Plot of $ I^{(2)}_{n} $ as a function of the normalized eigenfunction index $ n/L $ and $ \omega $. Here, the real part of the eigenvalues is ordered in ascending order. (b) Plot of the fractal dimension $ D_{2} $ as a function of $ \tilde{n}/L $ and $ \omega $, and $ f_{\mathrm{Im}} $ (solid black line) as a function of $ \omega $, where the energies sort in increasing order of the inverse participation ratio. Here, $ L=2048 $.
图 3 (a)、(e)和(g) [(b)、(f)和(h)]分别为$ h=0.1 $, 0和$ -0.1 $时, 位于能级$ n/L\approx0.24 $ ($ n/L\approx0.76 $)处的密度分布$ \rho_j $. (c)和(d)分别为$ h=0.1 $时, 能级$ n/L\approx1/3 $和$ 1/2 $处的密度分布$ \rho_j $. (i) $ I_n^{(2)} $随着频率$ \omega $变化的情况. (j) 能量实部$ \mathrm{Re}(\varepsilon_n^{{\mathrm{F}}}) $随着能级指标$ n/L $的分布情况. 这里选取 $ L=2048 $和$ \omega=\pi $, 并且能级指标按照能量实部升序排列
Fig. 3. (a), (e), and (g) [(b), (f), and (h)] Plot of the density distributions $ \rho_j $ at $ n/L\approx0.24 $ ($ n/L\approx0.76 $) with $ h=0.1 $, 0, and $ -1 $, respectively. (c) and (d) Plot of $ \rho_j $ with $ h=0.1 $ at $ n/L\approx1/3 $ and $ 1/2 $, respectively. (i) $ I_n^{(2)} $ as a function of $ \omega $. (j) Plot of $ \mathrm{Re}(\varepsilon_n^{{\mathrm{F}}}) $ as a function of $ n/L $. Here, $ L=2048 $, $ \omega=\pi $, and the real part of eigenvalues is ordered in ascending order.
图 4 (a)—(c)分别展示了在频率$ \omega=0.235\pi $时, 处于能级$ \tilde{n}/L\approx 0.995 $、$ 0.7 $和$ 0.2 $处对应于不同$ q $的分形维度$ D_q $随着尺寸$ 1/\ln{(L)} $的变化情况. 这里能级按照逆参与率值的升序排列
Fig. 4. (a)–(c) Plot of the fractal dimensions $ D_q $ as a function of $ 1/\ln{(L)} $ with different $ q $ and $ \omega=0.235\pi $ at $ \tilde{n}/L\approx 0.995 $, $ 0.7 $, and $ 0.2 $, respectively. Here, the energies sort in increasing order of the inverse participation ratio.
图 5 (a) 在频率$ \omega=0.132\pi $时, $ \tilde{n}/L=0.75 $处波函数的密度分布$ \rho_j $. (b) 不同频率处, MIPR的标度分析. (c) 在图(b)频率下, 在参数$ h=0.1 $和$ \mu=0.05 $附近随机选取20个数进行无序平均后, MIPR的标度分析. (d)和(e) 在频率$ \omega=0.132\pi $时, $ I_{\tilde{n}}^{(2)}\cdot L^{0.51} $与$ I_{\tilde{n}}^{(2)}\cdot L $随不同尺寸的缩放图. 这里能级按照逆参与率值的升序排列
Fig. 5. (a) The density distribution $ \rho_j $ with $ \omega=0.132\pi $ at $ \tilde{n}/L=0.75 $. (b) The scaling of the MIPR for different $ \omega $. (c) The scaling of MIPR after averaging random 20 parameters near $ h = 0.1 $ and $ \mu = 0.05 $ with the frequency in Fig(b). (d) and (e) The scaling of $ I_{\tilde{n}}^{(2)}\cdot L^{0.51} $ and $ I_{\tilde{n}}^{(2)}\cdot L $ as a function of $ L $ with $ \omega=0.132\pi $. Here, the energies sort in increasing order of the inverse participation ratio.
图 6 频率$ \omega=0.132\pi $(a)和$ \omega=0.5\pi $(b)时在复空间的能谱图. (c)在开边界条件下, 当频率$ \omega=0.132\pi $时环上 随机5个能级所对应的密度分布$ \rho_j $. (d)在开边界条件下, 当频率$ \omega=0.5\pi $时随机5个能级所对应的密度分布$ \rho_j $, 这里$ L=2048 $
Fig. 6. The energy spectrum with $ \omega=0.132\pi $ (a) and $ \omega=0.5\pi $ (b) in complex space. (c) The density distribution $ \rho_j $ for 5 random energy levels on the ring under open boundary conditions with frequency $ \omega = 0.132\pi $. (d) The density distribution $ \rho_j $ for 5 random energy levels under open boundary conditions with frequency $ \omega = 0.5\pi $. Here, $ L=2048 $.
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