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拓扑边界态因在带隙中的鲁棒性和无损耗的传输特性备受关注, 但在复杂系统中实现其稳定激发仍是一个挑战. 本文提出了一种利用亚对称性保护的边界态与长程非互易耦合系数, 实现具有拓扑选择性的非厄密趋肤效应 (Non-Hermitian Skin Effect, NHSE) 的方法. 该方法能够选择性地对平庸体态施加非厄密趋肤效应, 同时保持拓扑边界态不受影响, 从而实现拓扑模式与体态模式在空间上的有效分离, 并在能带密集的系统中实现鲁棒的边界态激发. 此外, 我们将该模型扩展到二维体系, 实现了角态与体态模式的有效分离. 我们通过紧束缚模型进行理论预测, 分析了该模型中非厄密效应对能谱和趋肤性质的调控机制, 并利用有限元仿真在光学耦合环中验证了这一机制的可行性, 研究了非厄密趋肤效应的本征态特性, 并实现了拓扑态的鲁棒激发. 该机制将非厄密物理与拓扑光子学相结合, 为提升光子系统中信号的稳定性提供了新的思路与方向.Topologically protected waveguides have attracted growing interest due to their robustness against disorder and defects. In parallel, the advent of non-Hermitian physics—with its inherent gain-and-loss mechanisms—has introduced new tools for manipulating wave localization and transport. However, most attempts to combine non-Hermitian effects with topological systems impose the non-Hermitian skin effect (NHSE) uniformly on all modes, lacking selectivity for topological states.In this work, we propose a scheme that realizes a topologically selective NHSE by combining sub-symmetry-protected boundary modes with long-range, non-reciprocal couplings. In a modified Su-Schrieffer-Heeger (SSH) chain, we analytically demonstrate that even in a spectrum densely populated with bulk states, a robust zero-energy edge mode can be preserved while the NHSE is selectively applied to the trivial bulk modes, achieving spatial separation between topological and bulk states. By tuning the long-range couplings, we observe a non-Hermitian phase transition in the complex energy spectrum: it evolves from a closed loop (circle), to an arc, and then to a loop with reversed winding direction. These transitions correspond to a leftward NHSE, the disappearance of the NHSE, and a rightward NHSE, respectively. Calculating the generalized Brillouin zone (GBZ), we confirm this transition by observing the GBZ crossing the unit circle, indicating a change in the NHSE direction.We further extend our model to a two-dimensional higher-order SSH lattice, where selective non-Hermitian modulation enables clear spatial separation between topological corner states and bulk modes. To quantify this, we compute the local density of states (LDOS) in the complex energy plane for site 0 (a topologically localized corner) and site 288 (a region exhibiting NHSE). The LDOS comparison reveals that the topological states are primarily localized at site 0, while bulk states affected by NHSE accumulate at site 288.To validate the theoretical predictions, we perform finite-element simulations of optical resonator arrays employing whispering-gallery modes. By tuning the coupling distances and incorporating gain/loss through refractive index engineering, we replicate the modified SSH model and confirm the selective localization of topological and bulk modes.Our results demonstrate a robust method for the selective excitation and spatial control of topological states in non-Hermitian systems, providing a foundation for future low-crosstalk, high-stability topological photonic devices.
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图 1 (a) 系统的紧束缚耦合示意图, 红色/黄色为A/B位点, 耦合系数为$ t_1, t_2, t_3, \gamma $, 箭头表示具有方向的耦合系数; (b) 系统的实能量分布随$ t_1 $的变化图; (c) 系统的复能量分布, 颜色的变化代表Bloch动量相位(ϕ)的变化; (d) 具有选择性非厄密趋肤效应的模式图, 横坐标为位点, 纵坐标为实能量, 白字标示了耦合参数大小; (e) 无拓扑模式非厄密趋肤效应的模式图, 横坐标为位点, 纵坐标为实能量, 白字标示了耦合参数大小; (f) 具有反方向非厄密趋肤效应的模式图, 横坐标为位点, 纵坐标为实能量, 白字表示了耦合参数大小
Fig. 1. (a) Schematic diagram of the system's tight-binding couplings. Red/yellow dots represent A/B sites, and the coupling coefficients are $ t_1 $, $ t_2 $, $ t_3 $, and γ. Arrows indicate direction-dependent coupling. (b) Plot of the system's real energy spectrum as $ t_1 $ changes from $ -3 $ to $ 3 $. (c) system's complex energy spectrum. The color gradient indicates the variation of the Bloch momentum phase (ϕ). (d) Mode distribution exhibiting topological selective non-Hermitian skin effect, white text denotes coupling coefficients. The horizontal axis represents site index, and the vertical axis represents real energy. (e) Mode distribution under non-Hermitian skin effect without topological modes. The horizontal axis represents site index, and the vertical axis represents real energy, white text denotes coupling coefficients. (f) Mode distribution exhibiting reverse non-Hermitian skin effect. The horizontal axis represents site index, and the vertical axis represents real energy, white text denotes coupling coefficients.
图 2 (a), (d), (g) $ t_3=0.4 $时的能带, 本征态分布和GBZ, (a)的横坐标与纵坐标分别为实/虚能量, 颜色图标代表Bloch动量的相位ϕ, (d)的横坐标/纵坐标为位点/能量, 颜色图标代表模式振幅, (g)横坐标/纵坐标为虚/实β, 灰色小球代表有限模型$ (n=100) $计算出的GBZ, 红星处$ \beta =1 $; (b), (e), (h) $ t_3=0.59 $时的能带, 本征态分布和GBZ; (c), (f), (i) $ t_3=0.8 $时的能带, 本征态分布和GBZ
Fig. 2. (a), (d), (g) energy bands, eigenstate and GBZ at $ t_3 = 0.4 $. In (a), the horizontal(vertical) axes is real(imaginary) parts of eigen energy, color bar indicates the phase of the Bloch momentum. In (d), the horizontal axis is the site index, the vertical axis is the energy, and the color bar represents the mode amplitude. In (g), the horizontal(vertical) axes is real(imaginary) parts of β, gray dots means GBZ calculated from finite model ($ n=100 $), the red star means where $ \beta=1 $. (b), (e), (h) energy bands, eigenstate distribution and GBZ at $ t_3 = 0.59 $. (c), (f), (i) energy bands, eigenstate distribution and GBZ at $ t_3 = 0.8 $.
图 3 (a) 耦合环示意图; (b) 有限元仿真的周期行边界复能带, x轴为实频率, y轴为复频率; (c) 有限元仿真的开放边界能带$ (n=10) $, x轴为解数, y轴为实频率, 红色五角星代表拓扑态; (d) 有限元仿真的向左的NHSE; (e) 有限元仿真的向右的NHSE; (f) 有限元仿真中的拓扑模式; (g) 在特定位点上激发的拓扑模式
Fig. 3. (a) Schematic diagram of the coupling rings. (b) complex energy band of finite element simulation under periodic boundary condition. (c) real energy band of finite element simulation under open boundary condition$ (n=10) $, the red star denotes topological states. (d) leftward non-Hermitian skin effect (NHSE) from finite element simulation. (e) rightward non-Hermitian skin effect (NHSE) from finite element simulation. (f) topological mode in the finite element simulation. (g) excited topological mode.
图 4 (a) 2维高阶SSH模型示意图; (b) 非互易耦合环中的二维高阶拓扑态; (c) 位点0处的复平面LDOS; (d) 二维模型的放大示意图; (e) $ B, C, D $三个位点非互易耦合导致的体态非厄密效应; (f) 位点288处的复平面LDOS
Fig. 4. (a) schematic diagram of the 2D higher-order SSH model. (b) 2D higher-order topological state in the presence of non-reciprocal coupling rings. (c) complex plane LDOS at finite slab site 0. (d) enlarged unit cell schematic of the 2D model. (e) non-Hermitian skin effect on bulk states induced by non-reciprocal coupling among sites B, C, and D. (f) complex plane LDOS at site 288.
图 A1 (a) 光学耦合环系统的本征能带; (b) 本文选取光学耦合环系统的本征能带
Fig. A1. (a) eigenenergy band of the optical coupling ring system; (b) enlarged chosen eigenenergy band of the optical coupling ring system.
图 A2 (a) 2D选择性非互易趋肤效应SSH模型的本征能带, 红色表示零能模式; (b) 2D选择性模式中不受到影响的零能模式; (c) 2D选择性非互易趋肤效应SSH模型的代表本征态之一; (d) 2D选择性非互易趋肤效应SSH模型的代表本征态之二; (e) 2D选择性非互易趋肤效应SSH模型的代表本征态之三
Fig. A2. (a) eigenenergy band of the 2D selective non-reciprocal skin effect SSH model, red denote zero energy mode; (b) 2D topological mode not affected by the selective skin effect; (c) representative eigenstate (1) of the 2D selective non-reciprocal skin; (d) representative eigenstate (2) of the 2D selective non-reciprocal skin; (e) representative eigenstate (3) of the 2D selective non-reciprocal skin.
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