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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
Figure 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
Figure 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) $, x轴为解数, y轴为实频率, 红色五角星代表拓扑态; (d) 有限元仿真的向左的NHSE; (e) 有限元仿真的向右的NHSE; (f) 有限元仿真中的拓扑模式; (g) 在特定位点上激发的拓扑模式
Figure 3. (a) Schematic diagram of the coupling rings. (b) complex energy band of finite element simulation under periodic boundary condition. (c) real energy band of finite element simulation under open boundary condition$ (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
Figure 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
Figure 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
Figure 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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[1] Su W P, Schrieffer J R, Heeger A J 1979 Physical Review Letters 42 1698
Google Scholar
[2] Wang Z, Chong Y, Joannopoulos J D, Soljačić M 2009 Nature 461 772
Google Scholar
[3] Hafezi M, Demler E A, Lukin M D, Taylor J M 2011 Nature Physics 7 907
Google Scholar
[4] Fang K, Yu Z, Fan S 2012 Nature Photonics 6 782
Google Scholar
[5] Hafezi M, Mittal S, Fan J, Migdall A, Taylor J M 2013 Nature Photonics 7 1001
Google Scholar
[6] Khanikaev A B, Hossein Mousavi S, Tse W K, Kargarian M, MacDonald A H, Shvets G 2013 Nature Materials 12 233
Google Scholar
[7] Wu L H, Hu X 2015 Physical Review Letters 114 223901
Google Scholar
[8] Ma T, Shvets G 2016 New Journal of Physics 18 025012
Google Scholar
[9] 王子尧, 陈福家, 郗翔, 高振, 杨怡豪 2024 物理学报 73 064201
Google Scholar
Wang Z Y, Chen F J, Xi X, Gao Z, Yang Y H 2024 Acta Physica Sinica 73 064201
Google Scholar
[10] 王洪飞, 解碧野, 詹鹏, 卢明辉, 陈延峰 2019 物理学报 68 224206
Google Scholar
Wang H F, Xie B Y, Zhan P, Lu M H, Chen Y F 2019 Acta Physica Sinica 68 224206
Google Scholar
[11] 孙晓晨, 何程, 卢明辉, 陈延峰 2017 物理学报 66 224203
Google Scholar
Xiao-Chen S, Cheng H, Ming-Hui L, Yan-Feng C 2017 Acta Physica Sinica 66 224203
Google Scholar
[12] Xie B, Wang H X, Zhang X, Zhan P, Jiang J H, Lu M, Chen Y 2021 Nature Reviews Physics 3 520
Google Scholar
[13] Benalcazar W A, Noh J, Wang M, Huang S, Chen K P, Rechtsman M C 2022 Physical Review B 105 195129
Google Scholar
[14] Wang Q, Xue H, Zhang B, Chong Y 2020 Physical Review Letters 124 243602
Google Scholar
[15] 胡军容, 孔鹏, 毕仁贵, 邓科, 赵鹤平 2022 物理学报 71 054301
Google Scholar
Hu J R, Kong P, Bi R G, Deng K, Zhao H P 2022 Acta Physica Sinica 71 054301
Google Scholar
[16] 严忠波 2019 物理学报 68 226101
Google Scholar
Yan Z B 2019 Acta Physica Sinica 68 226101
Google Scholar
[17] Wang Z, Wang X, Hu Z, Bongiovanni D, Jukić D, Tang L, Song D, Morandotti R, Chen Z, Buljan H 2023 Nature Physics 1
[18] Bender C M, Boettcher S 1998 Physical Review Letters 80 5243
Google Scholar
[19] Bender C M 2007 Reports on Progress in Physics 70 947
Google Scholar
[20] Bender C, Boettcher S, Meisinger P 1999 Journal of Mathematical Physics 40 2201
Google Scholar
[21] Yao S, Wang Z 2018 Physical Review Letters 121 086803
Google Scholar
[22] Lee T E 2016 Physical Review Letters 116 133903
Google Scholar
[23] Okuma N, Kawabata K, Shiozaki K, Sato M 2020 Physical Review Letters 124 086801
Google Scholar
[24] Song W, Sun W, Chen C, Song Q, Xiao S, Zhu S, Li T 2019 Physical Review Letters 123 165701
Google Scholar
[25] Li Y, Liang C, Wang C, Lu C, Liu Y C 2022 Physical Review Letters 128 223903
Google Scholar
[26] Wang W, Wang X, Ma G 2022 Nature 608 50
Google Scholar
[27] Zhu W, Gong J 2022 Physical Review B 106 035425
Google Scholar
[28] Liu G G, Mandal S, Zhou P, Xi X, Banerjee R, Hu Y H, Wei M, Wang M, Wang Q, Gao Z, Chen H, Yang Y, Chong Y, Zhang B 2024 Physical Review Letters 132 113802
Google Scholar
[29] Song W, Lin Z, Ji J, Sun J, Chen C, Wu S, Huang C, Yuan L, Zhu S, Li T 2024 Physical Review Letters 132 143801
Google Scholar
[30] Chen J, Shi A, Peng Y, Peng P, Liu J 2024 Chin. Phys. Lett. 41
[31] Martinez Alvarez V M, Barrios Vargas J E, Foa Torres L E F 2018 Physical Review B 97 121401
Google Scholar
[32] Pan M, Zhao H, Miao P, Longhi S, Feng L 2018 Nature Communications 9 1308
Google Scholar
[33] Huang Q K L, Liu Y K, Cao P C, Zhu X F, Li Y 2023 Chinese Physics Letters 40 106601
Google Scholar
[34] Leykam D, Bliokh K Y, Huang C, Chong Y, Nori F 2017 Physical Review Letters 118 040401
Google Scholar
[35] Liu X, Lin Z, Song W, Sun J, Huang C, Wu S, Xiao X, Xin H, Zhu S, Li T 2024 Physical Review Letters 132 016601
Google Scholar
[36] Jackiw R, Rebbi C 1976 Physical Review D 13 3398
Google Scholar
[37] Qi X L, Wu Y S, Zhang S C 2006 Physical Review B 74 045125
Google Scholar
[38] Zhang K, Yang Z, Fang C 2020 Physical Review Letters 125 126402
Google Scholar
[39] Yang Z, Zhang K, Fang C, Hu J 2020 Physical Review Letters 125 226402
Google Scholar
[40] Wang Z, Wang X, Li A, Zhang K, Ji Y, Zhong M 2023 Chinese Physics B 32 064207
Google Scholar
[41] 陈云天, 王经纬, 陈伟锦, 徐竞 2020 物理学报 69 154206
Google Scholar
Chen Y T, Wang J W, Chen W J, Xu J 2020 Acta Physica Sinica 69 154206
Google Scholar
[42] Ozawa T, Price H M, Amo A, Goldman N, Hafezi M, Lu L, Rechtsman M C, Schuster D, Simon J, Zilberberg O, Carusotto I 2019 Reviews of Modern Physics 91 015006
Google Scholar
[43] Yan Q C, Ma R, Hu X Y, Gong Q H 2023 Chinese Physics B 33 010301
Google Scholar
[44] Xu Y X, Tang W J, Jiang L W, Wu D X, Wang H, Xu B C, Chen L 2024 Chinese Physics B 33 060306
Google Scholar
[45] Wu D, Chen J, Su W, Wang R, Wang B, Xing D Y 2023 Communications Physics 6 1
Google Scholar
[46] Xu X W, Li Y Z, Liu Z F, Chen A X 2020 Physical Review A 101 063839
Google Scholar
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