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局域化是物理学中一个基础且极具潜力的研究领域. 基于广义Su-Schrieffer-Heeger模型, 我们针对其非厄米项以准周期、非对角形式出现的特点, 提出了一种新的分析框架, 旨在分别探讨体态与边缘态的局域化特性. 对于体态, 它可以经历由准无序诱导的扩展—共存—局域—共存—局域的转变, 或者是由非厄米特性引起的共存—局域—共存—局域的转变. 同时边缘态可以被破坏和恢复, 且其拓扑相变与局域化转变完全同步. 最后, 我们发现在局域化转变点处归一化参与率的导数展现出明显的不连续性. 我们的结果不仅展示了体态和边缘态局域化性质的多样性, 而且为局域化研究开辟了一个新的研究视角.
The localization is one of the active and fundamental research areas in topology physics. In this field, a comprehensive understanding of how wave functions distribute within a system is crucial. The work delves into this topic by proposing a novel systematic method based on a generalized Su-Schrieffer-Heeger (SSH) model. This model incorporates a quasiperiodic non-Hermitian term that emerges in an off-diagonal location, adding a layer of complexity to the traditional SSH framework. By utilizing this model, we analyze the localization behaviors of both bulk and edge states independently. For the bulk states, the analysis reveals a fascinating transition sequence. Specifically, the bulk states can undergo an extended-coexisting-localized-coexisting-localized transition, which is induced by the introduction of quasidisorder. This transition is not arbitrary but is rather conformed by the inverse participation ratio (IPR), a metric that quantifies the degree of localization of a wave function. As quasidisorder increases, the bulk states initially remain extended, but gradually, some states begin to localize. A coexistence region emerges where both extended and localized states are present. Further increase in quasidisorder leads to a complete localization of all bulk states. However, remarkably, within a certain range of quasidisorder strengths, the localized states can once again transition back to an extended state, creating another coexistence region. This complex behavior demonstrates the rich and diverse localization properties of the bulk states in non-Hermitian quasiperiodic systems. In addition to the IPR, other metrics such as the normalized participation ratio (NPR) and the fractal dimension of the eigenstates also play important roles in characterizing the localization behavior. These metrics provide a more nuanced understanding of the transition process and help to confirm the existence of the coexistence regions. Overall, we presents a comprehensive analysis of the localization behaviors of bulk and edge states in non-Hermitian quasiperiodic systems based on a generalized SSH model. The proposed systematic method offers new insights into the complex interplay between quasidisorder, non-Hermiticity, and localization properties in topological physics. -
Keywords:
- non-Hermitian /
- quasidisorder /
- reentrant-localized /
- topological phase transition
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图 1 (a) ${\rm{IPR}} ^{B} $ (红) 和${\rm{NPR}} ^{B} $ (蓝) 随$ W_{1} $的变化, 灰色区域代表共存状态. (b)和(c) 体能谱的实部和虚部, 附着色代表本征态的${\rm{IPR}} ^{|\Psi^{n}\rangle} $值. (d) 拓扑零能$ |E|^{0} $与对应的$ \frac{\text{IPR}^{0}}{5} $和${\rm{NPR}} ^{0} $. (e) 拓扑不变量. 参数为$ t_{1} = 1 $, $ t_{2} = 2.5 $, $ \gamma = 0.2 $, $ L = 2000 $和$ W_{2} = -2\cos(3 W_{1})+2 $
Fig. 1. (a) ${\rm{IPR}} ^{B} $ (red) and ${\rm{NPR}} ^{B} $ (blue) versus $ W_{1} $. Shaded regions stand for coexisting regimes. (b) and (c) Real and imaginary parts of bulk energy spectrum, where the dressed colors stand for different values of ${\rm{IPR}} ^{|\Psi^{n}\rangle} $ for each bulk eigenstates. (d) Topological edge modes $ |E|^{0} $, accompanied by the corresponding $ \frac{\text{IPR}^{0}}{5} $ and ${\rm{NPR}} ^{0} $. (e) Topological invariant as a function of $ W_{1} $. Common parameters are $ t_{1} = 1 $, $ t_{2} = 2.5 $, $ \gamma = 0.2 $, $ L = 2000 $ and $ W_{2} = -2\cos(3 W_{1})+2 $.
图 2 (a1)—(e1) 复平面能谱及其${\rm{IPR}} ^{|\Psi^{n}\rangle} $. (a2)—(e2) 蓝色和红色曲线代表一些体态的局域性质, 绿色条代表边缘态的局域性质. 为了显示清楚, 这些物理量已按比例缩放. (a1)—(a2) $ W_{1} = 0.15 $, 所有体态均为扩展态. (b1)—(b2) $ W_{1} = 0.35 $, 一些态仍为扩展态, 但一些态转为局域态. (c1)—(c2) $ W_{1} = 1 $, 所有的体态均转变为局域态. (d1)—(d2) $ W_{1} = 2.02 $, 部分已经局域的本征态会再转变为扩展态. (e1)—(e2) $ W_{1} = 3.8 $, 所有的体态再次全部转变为局域态. 其它参数和图1取值一致
Fig. 2. (a1)–(e1) Energy spectrum on the complex plane with corresponding ${\rm{IPR}} ^{|\Psi^{n}\rangle} $. (a2)–(e2) Both the blue and red lines stand for the localized properties of some bulk eigenstates, and the zero-energy state is shown as the green bar. For a better visibility, the values have been scaled proportionally. (a1)–(a2) $ W_{1} = 0.15 $, all bulk eigenstates are extended. (b1)–(b2) $ W_{1} = 0.35 $, some eigenstates are still extended while some are localized. (c1)–(c2) $ W_{1} = 1 $, all bulk eigenstates are transformed into localized. (d1)–(d2) $ W_{1} = 2.02 $, some already localized states will change into extended. (e1)–(e2) $ W_{1} = 3.8 $, all bulk eigenstates are transformed into localized again. Other parameters are the same as the ones in Fig. 1.
图 3 (a) $ L = 12000 $, 随着γ变化的${\rm{IPR}} ^{B} $ (红) 和${\rm{NPR}} ^{B} $ (蓝), 两个灰色区域代表共存状态. (b) $ \gamma = 4.66946 $, ${\rm{NPR}} ^{B} $的有限尺寸效应分析, 系统尺寸分别为$ L = 20000 $, $ 30000 $和$ 40000 $. (c) 体能谱实部, 不同附着色代表$ \eta^{|\Psi^{n}\rangle} $的不同取值, $ L = 6000 $. (d) $ |E|^{0} $与对应的${\rm{IPR}} ^{0} $和${\rm{NPR}} ^{0} $随着γ的变化趋势, $ L = 2000 $. (e) 系统尺寸$ L = 12000 $时${\rm{NPR}} ^{0} $对γ的导数. (f) 拓扑不变量随参数γ的变化趋势, $ L = 2000 $. 其它参数为$ t_{1} = 9 $, $ t_{2} = 1 $, $ W_{1} = 0.0039 $和$ W_{2} = 1.563 $
Fig. 3. (a) $ L = 12000 $, ${\rm{IPR}} ^{B} $ (red) and ${\rm{NPR}} ^{B} $ (blue) evolving with γ. Two shaded regions stand for intermediate regimes. (b) Finite size scaling analysis of ${\rm{NPR}} ^{B} $ at $ \gamma = 4.66946 $ for $ L = 20000 $, $ 30000 $ and $ 40000 $, respectively. (c) Real parts of bulk energy spectrum for $ L = 6000 $, where different colors stand for different values of $ \eta^{|\Psi^{n}\rangle} $. (d) Evolution of $ |E|^{0} $ versus γ associated with ${\rm{IPR}} ^{0} $ and ${\rm{NPR}} ^{0} $, $ L = 2000 $. (e) Derivative of ${\rm{NPR}} ^{0} $ with respect to γ for $ L = 12000 $. (f) Topological invariant versus parameter γ for $ L = 2000 $. Common parameters are $ t_{1} = 9 $, $ t_{2} = 1 $, $ W_{1} = 0.0039 $ and $ W_{2} = 1.563 $.
图 4 所有归一化体态的空间分布, 为了表示清楚, 系统尺寸取$ L = 1000 $, 并适当缩减能量值范围 (a) $ \gamma = 0.2 $, 扩展态与局域态一起出现. (b) $ \gamma = 1 $, 所有扩展态均转变为局域态. (c) $ \gamma = 4.66946 $, 一些局域态会转回扩展态. (d) $ \gamma = 5.5 $, 所有态再次变为局域态. 其它参数和图3 一致
Fig. 4. Spatial distributions of normalized bulk eigenstates with the change of eigenvalues. For enhanced visibility, the system size is $ L = 1000 $, and the values of the energies have been cut down appropriately. (a) $ \gamma = 0.2 $, the extended and localized eigenstates appearing together. (b) $ \gamma = 1 $, all extended eigenstates being transformed into the localized. (c) $ \gamma = 4.66946 $, some localized eigenstates turning to the extended. (d) $ \gamma = 5.5 $, all extended eigenstates being transformed into the localized once again. Other parameters are the same as the ones in Fig. 3.
图 5 (a) 拓扑不变量随$ W_{1} $的变化趋势. (b) $ |E|^{0} $与对应的${\rm{IPR}} ^{0} $和${\rm{NPR}} ^{0} $随$ W_{1} $的演化. (c) 图为 (b) 其中一部分的详细信息. (d) ${\rm{NPR}} ^{0} $的导数, 显然其中存在一些不连续的点. (e) 以$ W_{1} $为自变量的${\rm{IPR}} ^{B} $ (红) 和${\rm{NPR}} ^{B} $ (蓝), 阴影区域表示扩展态和局域态共存. (f) 体能谱的实部, 附着在其上的不同颜色代表不同的${\rm{IPR}} ^{|\Psi^{n}\rangle} $值. (a), (b), (e) 和 (f) $ L = 2000 $, (c) 和 (d) $ L = 10000 $, 其它参数为$ t_{1} = 1 $, $ t_{2} = 1.3 $, $ \gamma = 0.05 $和$ W_{2} = W_{1} $
Fig. 5. (a) Topological invariant versus $ W_{1} $. (b) Evolution of $ |E|^{0} $ via $ W_{1} $, associated with ${\rm{IPR}} ^{0} $ and ${\rm{NPR}} ^{0} $. (c) Detailed information on the localization of the eigenstates with the lowest energy. (d) Derivative of ${\rm{NPR}} ^{0} $, where there exist some discontinuity points. (e) ${\rm{IPR}} ^{B} $ (red) and ${\rm{NPR}} ^{B} $ (blue) being plotted as a function of $ W_{1} $. The shaded region indicates the coexisting regime, with extended and localized eigenstates being coexistence. (f) Real part of bulk energy spectrum, where different colors stand for different values of ${\rm{IPR}} ^{|\Psi^{n}\rangle} $. For (a), (b), (e) and (f) $ L = 2000 $, and for (c) and (d) $ L = 10000 $. Same parameters are $ t_{1} = 1 $, $ t_{2} = 1.3 $, $ \gamma = 0.05 $ and $ W_{2} = W_{1} $.
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