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A non-Hermitian system with long-range hopping under periodic driving is constructed in this work. The Hamiltonian has chiral symmetry, implying that a topological invariant can be determined. Using the non-Bloch band theory and the Floquet method, the relevant operators and topological number can be determined, thereby providing quantitative approaches for studying topological properties. For example, by calculating the non-Bloch time-evolution factor, the Floquet operator, etc., it can be found that the topological invariant is determined by changing the phase of $U^{+}_{\epsilon=0,\pi}(\beta)$ as it moves along the generalized Brillouin zone, corresponding to the emergence of quasi-energy zero mode and π mode. The results show that the topological structure of the static system can be significantly affected by periodic driving. The topological phase boundary of the zero mode can be changed. In the absence of periodic driving, energy spectrum does not exhibit π mode. After introducing periodic driving, a gap appears at the quasi-energy $\epsilon=\pi$, thereby inducing a non-trivial π-mode phase and enriching the topological phase diagram. Furthermore, the next nearest neighbor hopping has a unique effect in this system. It can induce large topological numbers. However, unlike the static system, large topological numbers only appear in specific parameter intervals under periodic driving. As the strength of the next nearest neighbor hopping increases, the large topological number phase disappears, reflecting the non-monotonic regulation characteristics of the Floquet system. In addition, introducing the phase of the next nearest neighbor hopping can change the topological phase boundary, providing new ideas for experimentally regulating topological states. This research is of significance in the field of topological phase transitions in non-Hermitian systems. Theoretically, it reveals the synergistic effect of long-range hopping and periodic driving, and improves the theoretical framework for the cross-research of long-range and dynamic regulation in non-Hermitian systems. From an application perspective, it provides theoretical support for experimentally realizing the controllable modulation of topological states, which is helpful in promoting the development of fields such as low energy consumption electronic devices and topological quantum computing. -
Keywords:
- non-Hermitian /
- periodic driving /
- long-range hopping /
- topological phase transition
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图 1 (a) 静态体系的开边界能谱. (b) Floquet驱动下开边界能谱, 这时系统激发π模(红点). (c) 静态体系的广义布里渊区(蓝线)和布里渊区(绿线). (d) Floquet驱动系统的广义布里渊区(蓝线)和布里渊区(绿线). (e) β沿广义布里渊区逆时针遍历时, $ U^{+}_{0}(\beta) $在复平面形成的曲线, 其没有环绕原点(绿点)表征体系具有平庸拓扑不变量$ W_{0} = 0 $. (f) β沿广义布里渊区逆时针遍历时, $ U^{+}_{\pi}(\beta) $在复平面形成的曲线, 显然其将原点(绿点)包含在内, 代表这时体系具有非平庸拓扑不变量$ W_{\pi} = 1 $. 参数为$ t_{1} = 1 $, $ t_{2} = 0.6 $, $ t_{3} = 0.1 $, $ \gamma = 0.5 $, $ \lambda = 0.8 $, $ \omega = 3 $. 开边界对应的系统尺寸为$ N = 200 $.
Figure 1. (a) Eigenvalues under open boundary of static system. (b) Eigenvalues under open boundary of Floquet system, demonstrating existence of π-modes (red dots). (c) Generalized Brillouin zone in equilibrium (blue curve) and Brillouin zone (green circle). (d) Generalized Brillouin zone (blue curve) under periodic driving and Brillouin zone (green circle). (e) Complex contour formed by $ U^{+}_{0}(\beta) $ during counterclockwise $ \beta $-traversal along Generalized Brillouin zone, whose origin not-encircling (green marker) characterizes the trivial winding number $ W_{0} = 0 $. (f) Trajectory of $ U^{+}_{\pi}(\beta) $ in complex plane, with unambiguous origin inclusion (green marker) confirming topologically protected π-mode via $ W_{\pi} = 1 $. Common parameters are $ t_{1} = 1 $, $ t_{2} = 0.6 $, $ t_{3} = 0.1 $, and $ \gamma = 0.5 $. Floquet parameters are $ \lambda = 0.8 $ and $ \omega = 3 $. System size under open-boundary is $ N = 200 $.
图 2 (a) 静态体系的非布洛赫拓扑不变量, 非零值对应开边界条件下零模的出现. (b) 当加入周期驱动时体系零模的拓扑相图. (c) 加入周期驱动后, 不但零模对应的拓扑不变量会出现改变, 还会导致π模的出现. (d) 静态体系的开边界能谱. (e) 周期驱动体系的开边界准能谱. (f) 当加入次近邻后, 零模对应的相图. 共同参数为$ \gamma = 0.5 $, $ \lambda = 0.8 $, $ \omega = 3 $. (a)—(e) $ t_{3} = 0 $, (d)—(e) $ t_{2} = $$ 1.2 $, (f) $ t_{3} = 0.4 $. 开边界对应的系统尺寸为$ N = 60 $
Figure 2. (a) Non-Bloch topological invariant of static system, where non-zero values correspond to the emergence of zero mode under open boundary conditions. (b) Topological phase diagram of the zero modes when the periodic driving is added. (c) After adding the periodic driving, not only will the topological invariant corresponding to the zero mode change, but it will also lead to the emergence of π modes. (d) Open boundary energy spectrum of the static system. (e) Open boundary quasi-energy spectrum of the periodically driven system. (f) When next-nearest neighbor is added, the phase diagram corresponding to the zero mode. Common parameters are $ \gamma = 0.5 $, $ \lambda = 0.8 $, $ \omega = 3 $, (a)–(e) $ t_{3} = 0 $, (d)–(e) $ t_{2} = 1.2 $, (f) $ t_{3} = 0.4 $. System size under open-boundary condition is $ N = 60 $.
图 3 (a) 以$ t_{3} $为自变量的零模的拓扑相图. (b)和(c) 以$ t_{3} $和γ为自变量的拓扑相图. (d)和(e) 以$ t_{3} $和$ t_{1} $为自变量的拓扑相图. 共同参数取值为$ t_{2} = 0.4 $, $ \lambda = 1.2 $, $ \omega = 3 $. (f) 静态体系的拓扑相图. (a) $ t_{1} = 1 $, $ \gamma = 0.7 $. (b)—(c) $ t_{1} = 1 $. (d)和(e) $ \gamma = 0.2 $
Figure 3. (a) Topological phase diagram of the zero mode. (b) and (c) Topological phase diagram with $ t_{3} $ and γ as the independent variables. (d) and (e) Topological phase diagrams with $ t_{3} $ and γ as the independent variables. (f) Toplogical phase diagram with $ t_{3} $ and γ as the independent variables. Common parameter values are $ t_{2} = 0.4 $, $ \lambda = 1.2 $, $ \omega = 3 $. (a) $ t_{1} = 1 $, $ \gamma = 0.7 $. (b) and (c) $ t_{1} = 1 $. (d), (e) and (f) $ \gamma = 0.2 $.
图 4 (a)和(b) $ t_{3} $的相位$ \theta = 0 $, 即$ e^{i\theta} = 1 $时的拓扑相图. (c)和(d) $ t_{3} $的相位$ \theta = \dfrac{\pi}{2} $时的拓扑相图. (e)和(f) 以$ {\rm{e}}^{{\rm{i}}\theta} $和$ t_{2} $为自变量的体系的拓扑相图, 其中$ \theta\in[-\pi, \pi] $, 即$ e^{i\theta}\in[-1, 1] $. (g)和(h) 以$ e^{i\theta} $和$ t_{3} $为自变量的体系的拓扑相图. 共同参数取值为$ t_{1} = 1 $, $ \gamma = 0.2 $, $ \lambda = 1.5 $, $ \omega = 3 $. (a)和(b) $ \theta = 0 $. (c)和(d) $ \theta = \dfrac{\pi}{2} $. (e)和(f) $ t_{3} = 0.5 $. (g)和(h) $ t_{2} = 0.3 $
Figure 4. (a) and (b) Topological phase diagrams with the phase $ \theta = 0 $, that is $ {\rm{e}}^{{\rm{i}}\theta} = 1 $. (c) and (d) Topological phase diagrams with the phase $ \theta = \dfrac{\pi}{2} $. (e) and (f) Topological phase diagram of the system with $ {\rm{e}}^{{\rm{i}}\theta} $ and $ t_{2} $ as the independent variables, where $ \theta\in[-\pi, \pi] $, that is $ {\rm{e}}^{{\rm{i}}\theta}\in[-1, 1] $. (g) and (h) Topological phase diagram of the system with $ {\rm{e}}^{{\rm{i}}\theta} $ and $ t_{3} $ as the independent variables. Common parameters are $ t_{1} = 1 $, $ \gamma = 0.2 $, $ \lambda = 1.5 $ and $ \omega = 3 $. (a) and (b) $ \theta = 0 $. (c) and (d) $ \theta = \dfrac{\pi}{2} $. (e) and (f) $ t_{3} = $$ 0.5 $. (g) and (h) $ t_{2} = 0.3 $.
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