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 $

Fig. 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 $

Fig. 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} $为自变量的零模的拓扑相图; (b), (c) 以$ t_{3} $和γ为自变量的拓扑相图; (d), (e) 以$ t_{3} $和$ t_{1} $为自变量的拓扑相图; (f) 静态体系的拓扑相图. 共同参数取值为$ t_{2} = 0.4 $, $ \lambda = 1.2 $, $ \omega = 3 $. (a) $ t_{1} = 1 $, $ \gamma = 0.7 $; (b), (c) $ t_{1} = 1 $; (d)—(f) $ \gamma = 0.2 $

Fig. 3.  (a) Topological phase diagram of the zero mode; (b), (c) topological phase diagram with $ t_{3} $ and γ as the independent variables; (d), (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), (c) $ t_{1} = 1 $; (d)–(f) $ \gamma = 0.2 $.

图 4  (a), (b) $ t_{3} $的相位$ \theta = 0 $, 即$ {\mathrm{e}}^{{\mathrm{i}}\theta} = 1 $时的拓扑相图; (c), (d) $ t_{3} $的相位$ \theta = \dfrac{\pi}{2} $时的拓扑相图; (e), (f) 以$ {\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] $, 即$ {\mathrm{e}}^{{\mathrm{i}}\theta}\in[-1, 1] $; (g), (h) 以$ {\mathrm{e}}^{{\mathrm{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), (b) $ \theta = 0 $; (c), (d) $ \theta = \dfrac{\pi}{2} $; (e), (f) $ t_{3} = 0.5 $; (g), (h) $ t_{2} = 0.3 $

Fig. 4.  (a), (b) Topological phase diagrams with the phase $ \theta = 0 $, that is $ {\rm{e}}^{{\rm{i}}\theta} = 1 $; (c), (d) topological phase diagrams with the phase $ \theta = \dfrac{\pi}{2} $; (e), (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), (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), (b) $ \theta = 0 $; (c), (d) $ \theta = \dfrac{\pi}{2} $; (e), (f) $ t_{3} = $$ 0.5 $; (g), (h) $ t_{2} = 0.3 $.