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Topological edge states with skin effect in a trimerized non-Hermitian lattice

Xu Nan Zhang Yan

Topological edge states with skin effect in a trimerized non-Hermitian lattice

Xu Nan, Zhang Yan
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  • In recent years, exploring new topological quantum model structures and in depth analyzing the novel physical properties in various multimerized topological lattices have become a hot topic in the field of quantum optics. Among the different model structures, the multimerized non-Hermitian lattice controlled by different parameters in the future research of topological quantum materials, we believe, can exhibit more meaningful novel topological properties. As one of the most classic topological models, the one-dimensional Aubry-André-Harper (AAH) model has received more and more attention in the study of multimerized lattices. In this paper, we focus on the novel topological properties of a trimerized non-Hermitian lattice, and extend the trimer model structure from a one-dimensional chain to a quasi-one-dimensional zigzag structure. The results show that firstly, if the nearest-neighbor forward coupling coefficient in the unit cell is not equal to the backward coupling coefficient, the chiral inversion symmetry of the system is destroyed. It can be observed that the bulk states and the edge states in the trimerization model will be localized on the same edge of the lattice, and the skin effect will appear in the system. With the increase of the nearest-neighbor coupling coefficient, the width of the edge state changes in which the lower edge state of the imaginary part of the spectrum is narrowed until it disappears. The degree of degeneracy of the system changes, and the number of edge states is reduced from four to two. Remarkably, the generalized bulk-boundary correspondence is shown in certain non-Hermitian topological systems. Secondly, when the trimerization model considers the influence of the next-nearest-neighbor coupling, the numerical results show that the upper and lower energy gaps in the energy spectrum and the edge states in the energy spectrum are asymmetrical as the next-nearest-neighbor coupling coefficient is modulated in an appropriate range. The upper energy gaps and the edge states are narrowed, and the edge states of the lower energy gaps are widened. At the same time, the novel topology features of the system can also be used to achieve the quantitative control of the energy spectrum edge states, and other interesting directions are worth exploring.
      Corresponding author: Zhang Yan, zhangy345@nenu.edu.cn
    [1]

    Klitzing K, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494

    [2]

    Klitzing K 1986 Rev. Mod. Phys. 58 519

    [3]

    Haldane F D M 1988 Phys. Rev. Lett. 61 2015

    [4]

    Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904

    [5]

    Raghu S, Haldane F D M 2008 Phys. Rev. A 78 033834

    [6]

    Chang C Z, Zhang J S, Feng X, Xue Q K 2013 Sci. Rep. 340 6129

    [7]

    Goldman N, Budich J C, Zoller P 2016 Nat. Phys. 12 639

    [8]

    孙晓晨, 何程, 卢明辉, 陈延峰 2017 物理学报 22 224203

    Sun X C, He C, Lu M H, Chen Y F 2017 Acta Phys. Sin. 22 224203

    [9]

    张卫锋, 李春艳, 陈险锋, 黄长明, 叶芳伟 2017 物理学报 22 220201

    Zhang W F, Li C Y, Chen X F, Huang C M, Ye F W 2017 Acta Phys. Sin. 22 220201

    [10]

    陈西浩, 王秀娟 2018 物理学报 19 190301

    Chen X H, Wang X J 2018 Acta Phys. Sin. 19 190301

    [11]

    Aubry S, Andreé G, Isr A 1980 Phys. Soc. 322 235

    [12]

    harper P G 1955 Proc. Phys. Soc. London Sect. A 68 874

    [13]

    Lahini Y, Pugatch R, Pozzi F, Sorel M, Morandotti R, Davidson N, Silberberg Y 2009 Phys. Rev. Lett. 103 013901

    [14]

    Biddle J, Wang B, Priour D J, Sarma S D 2009 Phys. Rev. A 80 021603 (R)

    [15]

    Ganeshan S, Sun K, Sarma S D 2013 Phys. Rev. Lett. 110 180403

    [16]

    Hatano N, Nelson D R 1997 Phys. Rev. B 56 8651

    [17]

    Hatano N, Nelson D R 1998 Phys. Rev. B 58 8384

    [18]

    Yurkevich I V, Lerner I V 1999 Phys. Rev. Lett. 82 5080

    [19]

    Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243

    [20]

    Dorey P, Dunning C, Tateo R 2001 J. Phys. A 34 5679

    [21]

    Mostafazadeh A 2002 J. Math. Phys. 43 205

    [22]

    Jones H F 2005 J. Phys. A 38 1741

    [23]

    Klaiman S, Günther U, Moiseyev N 2008 Phys. Rev. Lett. 101 080402

    [24]

    Znojil M 2008 Phys. Rev. D 78 025026

    [25]

    Jin L, Song Z 2009 Phys. Rev. A 80 052107

    [26]

    Rotter I 2009 J. Phys. A: Math. Theor. 42 124206

    [27]

    Moiseyev N 2011 Non-Hermitian Quantum Mechanics (Cambridge: Cambridge University) pp 211—247

    [28]

    Joqlekar Y N, Barnett J L 2011 Phys. Rev. A 84 024103

    [29]

    Longhi S, Valle G D 2012 Phys. Rev. A 85 012112

    [30]

    Longhi S 2013 Phys. Rev. A 88 052102

    [31]

    Longhi S 2016 Phys. Rev. A 94 022102

    [32]

    Jin L, Xin F 2017 Phys. Rev. A 97 012121

    [33]

    Zhu B G, Lü R, Chen S 2014 Phys. Rev. A 89 062102

    [34]

    Xing Y, Qi L, Cao J, Wang D Y, Bai C H, Wang H F, Zhu A D, Zhang S 2017 Phys. Rev. A 96 043810

    [35]

    Zhou Y H, Shen H Z, Zhang X Y, Yi X X 2018 Phys. Rev. A 97 043819

    [36]

    Schomerus H, Wiersig J 2014 Phys. Rev. A 90 053819

    [37]

    Cheng Q, Pan Y, Wang Q, Li T, Zhu S 2015 Laser Photonics Rev. 9 392

    [38]

    Longhi S, Gatti D, Della Valle G 2015 Phys. Rev. B 92 094204

    [39]

    Martinez Alvarez V M, Barrios Vargas J E, Foa Torres L E F 2018 Phys. Rev. B 97 121401(R)

    [40]

    Yao S Y, Wang Z 2018 Phys. Rev. Lett. 121 086803

    [41]

    Jin L 2017 Phys. Rev. A 96 032103

  • 图 1  扩展的三聚化非厄密晶格示意图

    Figure 1.  Schematic of the generalized trimerized non-Hermitian lattice.

    图 2  三聚化非厄密晶格的本征能谱 (a), (c), (e)和(g)分别为$\mu=0, 0.1, 0.15, 0.2$时本征能谱的实部; (b), (d), (f)和(h)分别为$\mu=0, 0.1, 0.15, 0.2$时本征能谱的虚部, 边缘态能带用绿色实线和棕色虚线表示

    Figure 2.  The eigen-energy spectrum of the trimerized non-Hermitian lattice. (a), (c), (e) and (g) show the real parts of the energy spectrum for $\mu=0, 0.1, 0.15, 0.2$, respectively; (b), (d), (f) and (h) show the imaginary parts of the energy spectrum for $\mu=0, 0.1, 0.15, 0.2$. Green solid lines and brown dash lines represent the energy band of the edge states.

    图 3  系统本征态的光子分布 (a) $\mu=0$; (b) $\mu=0.1$; (c) $\mu=0.2$; (d) $\mu=0.3$

    Figure 3.  Photon distributions for eigenstates of the system: (a) $\mu=0$; (b) $\mu=0.1$; (c) $\mu=0.2$; (d) $\mu=0.3$.

    图 4  次近邻耦合影响下系统的本征能谱和边缘态的光子分布 (a), (d)和(g)分别为$\nu=0.1, 0.15, 0.2$时本征能谱的实部; (b), (e)和(h)分别为$\nu=0.1, 0.15, 0.2$时上能隙中的边缘态; (c), (f)和(i)分别为$\nu=0.1, 0.15, 0.2$时下能隙的边缘态

    Figure 4.  The eigen-energy spectrum and photon distributions for edge states of the system under the influence of the next-nearest-neighbor coupling: (a), (d) and (g) show the real parts of the energy spectrum for $\nu=0.1, 0.15, 0.2$; (b), (e) and (h) show the edge states whose energy band are in the upper gaps for $\nu=0.1, 0.15, 0.2$; (c), (f) and (i) show the edge states whose energy bands are in the lower gaps for $\nu=0.1, 0.15, 0.2$.

  • [1]

    Klitzing K, Dorda G, Pepper M 1980 Phys. Rev. Lett. 45 494

    [2]

    Klitzing K 1986 Rev. Mod. Phys. 58 519

    [3]

    Haldane F D M 1988 Phys. Rev. Lett. 61 2015

    [4]

    Haldane F D M, Raghu S 2008 Phys. Rev. Lett. 100 013904

    [5]

    Raghu S, Haldane F D M 2008 Phys. Rev. A 78 033834

    [6]

    Chang C Z, Zhang J S, Feng X, Xue Q K 2013 Sci. Rep. 340 6129

    [7]

    Goldman N, Budich J C, Zoller P 2016 Nat. Phys. 12 639

    [8]

    孙晓晨, 何程, 卢明辉, 陈延峰 2017 物理学报 22 224203

    Sun X C, He C, Lu M H, Chen Y F 2017 Acta Phys. Sin. 22 224203

    [9]

    张卫锋, 李春艳, 陈险锋, 黄长明, 叶芳伟 2017 物理学报 22 220201

    Zhang W F, Li C Y, Chen X F, Huang C M, Ye F W 2017 Acta Phys. Sin. 22 220201

    [10]

    陈西浩, 王秀娟 2018 物理学报 19 190301

    Chen X H, Wang X J 2018 Acta Phys. Sin. 19 190301

    [11]

    Aubry S, Andreé G, Isr A 1980 Phys. Soc. 322 235

    [12]

    harper P G 1955 Proc. Phys. Soc. London Sect. A 68 874

    [13]

    Lahini Y, Pugatch R, Pozzi F, Sorel M, Morandotti R, Davidson N, Silberberg Y 2009 Phys. Rev. Lett. 103 013901

    [14]

    Biddle J, Wang B, Priour D J, Sarma S D 2009 Phys. Rev. A 80 021603 (R)

    [15]

    Ganeshan S, Sun K, Sarma S D 2013 Phys. Rev. Lett. 110 180403

    [16]

    Hatano N, Nelson D R 1997 Phys. Rev. B 56 8651

    [17]

    Hatano N, Nelson D R 1998 Phys. Rev. B 58 8384

    [18]

    Yurkevich I V, Lerner I V 1999 Phys. Rev. Lett. 82 5080

    [19]

    Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243

    [20]

    Dorey P, Dunning C, Tateo R 2001 J. Phys. A 34 5679

    [21]

    Mostafazadeh A 2002 J. Math. Phys. 43 205

    [22]

    Jones H F 2005 J. Phys. A 38 1741

    [23]

    Klaiman S, Günther U, Moiseyev N 2008 Phys. Rev. Lett. 101 080402

    [24]

    Znojil M 2008 Phys. Rev. D 78 025026

    [25]

    Jin L, Song Z 2009 Phys. Rev. A 80 052107

    [26]

    Rotter I 2009 J. Phys. A: Math. Theor. 42 124206

    [27]

    Moiseyev N 2011 Non-Hermitian Quantum Mechanics (Cambridge: Cambridge University) pp 211—247

    [28]

    Joqlekar Y N, Barnett J L 2011 Phys. Rev. A 84 024103

    [29]

    Longhi S, Valle G D 2012 Phys. Rev. A 85 012112

    [30]

    Longhi S 2013 Phys. Rev. A 88 052102

    [31]

    Longhi S 2016 Phys. Rev. A 94 022102

    [32]

    Jin L, Xin F 2017 Phys. Rev. A 97 012121

    [33]

    Zhu B G, Lü R, Chen S 2014 Phys. Rev. A 89 062102

    [34]

    Xing Y, Qi L, Cao J, Wang D Y, Bai C H, Wang H F, Zhu A D, Zhang S 2017 Phys. Rev. A 96 043810

    [35]

    Zhou Y H, Shen H Z, Zhang X Y, Yi X X 2018 Phys. Rev. A 97 043819

    [36]

    Schomerus H, Wiersig J 2014 Phys. Rev. A 90 053819

    [37]

    Cheng Q, Pan Y, Wang Q, Li T, Zhu S 2015 Laser Photonics Rev. 9 392

    [38]

    Longhi S, Gatti D, Della Valle G 2015 Phys. Rev. B 92 094204

    [39]

    Martinez Alvarez V M, Barrios Vargas J E, Foa Torres L E F 2018 Phys. Rev. B 97 121401(R)

    [40]

    Yao S Y, Wang Z 2018 Phys. Rev. Lett. 121 086803

    [41]

    Jin L 2017 Phys. Rev. A 96 032103

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  • Received Date:  21 January 2019
  • Accepted Date:  11 March 2019
  • Available Online:  01 May 2019
  • Published Online:  20 May 2019

Topological edge states with skin effect in a trimerized non-Hermitian lattice

    Corresponding author: Zhang Yan, zhangy345@nenu.edu.cn
  • School of Physics, Northeast Normal University, Changchun 130024, China

Abstract: In recent years, exploring new topological quantum model structures and in depth analyzing the novel physical properties in various multimerized topological lattices have become a hot topic in the field of quantum optics. Among the different model structures, the multimerized non-Hermitian lattice controlled by different parameters in the future research of topological quantum materials, we believe, can exhibit more meaningful novel topological properties. As one of the most classic topological models, the one-dimensional Aubry-André-Harper (AAH) model has received more and more attention in the study of multimerized lattices. In this paper, we focus on the novel topological properties of a trimerized non-Hermitian lattice, and extend the trimer model structure from a one-dimensional chain to a quasi-one-dimensional zigzag structure. The results show that firstly, if the nearest-neighbor forward coupling coefficient in the unit cell is not equal to the backward coupling coefficient, the chiral inversion symmetry of the system is destroyed. It can be observed that the bulk states and the edge states in the trimerization model will be localized on the same edge of the lattice, and the skin effect will appear in the system. With the increase of the nearest-neighbor coupling coefficient, the width of the edge state changes in which the lower edge state of the imaginary part of the spectrum is narrowed until it disappears. The degree of degeneracy of the system changes, and the number of edge states is reduced from four to two. Remarkably, the generalized bulk-boundary correspondence is shown in certain non-Hermitian topological systems. Secondly, when the trimerization model considers the influence of the next-nearest-neighbor coupling, the numerical results show that the upper and lower energy gaps in the energy spectrum and the edge states in the energy spectrum are asymmetrical as the next-nearest-neighbor coupling coefficient is modulated in an appropriate range. The upper energy gaps and the edge states are narrowed, and the edge states of the lower energy gaps are widened. At the same time, the novel topology features of the system can also be used to achieve the quantitative control of the energy spectrum edge states, and other interesting directions are worth exploring.

    • 拓扑绝缘体是一种具有全新量子特性的物质态, 在量子理论的发展过程中具有重要意义. 从1980年Klitzing等[1,2]在二维电子气系统中发现了量子整数霍尔效应, 到Haldane[3]发现了量子反常霍尔效应, 以及之后的十年, 拓扑性质相关的理论实验研究都得到了极大的发展[4-10]. 而作为其中最为经典的拓扑模型之一, 一维的Aubry-André-Harper (AAH)模型在研究多聚化晶格方面得到了越来越多的关注与重视. 该模型最初被用来研究类周期系统的局域性转变[11-15]. AAH晶格的本征能谱由三条能带、两条能隙组成, 两条能隙中分别存在可由参数调制的边缘态. 因此, 能谱可分为两种不同的相位: 有边缘态的非平庸相和没有边缘态的平庸相. 然而, 在实际的拓扑实验中, 由于总是与环境相互作用, 所以需考虑耗散效应对系统的影响. 其中, 典型的例子便是针对非厄密系统(引入增益和损耗)的研究[16-27]. 尤其在近几年, 均匀的增益或损耗被引入到一些非厄密的拓扑绝缘体理论或者实验中, 如Su-Schrieffer-Heeger (SSH)模型[28,29]和AAH模型[30-33]的理论研究, 以及一维耦合微腔[34,35]和一维耦合光波导的实验论证[36,37], 都得到了很多厄密系统中没有的新奇现象. 实验中通过设计一维耦合环腔, 在虚数规范场中调制辅助腔的增益和损耗[38], 使得二聚化模型系统中最近邻正反耦合系数出现差值, 从而产生趋肤效应[39,40], 该效应在近年获得极大关注. 区别于同一耦合的正反向系数一致的拓扑系统具有一般性的体边对应关系, 非厄密系统中通过改变晶格正反向耦合系数差, 破坏系统的手性反演对称, 从而展现了扩展的体边对应关系. 主要表现为包含系统的拓扑保护边缘态和体态在内的所有本征态都局域于开放系统单侧边缘附近的趋肤效应. 然而, 三聚化非厄密系统的趋肤效应的研究还不充分. 因此, 本文旨在研究一维三聚化的非厄密晶格的具有趋肤效应的拓扑特性. 将三聚化模型结构从一维链状拓展到类一维的锯齿形结构, 通过适当调制胞内最近邻正反耦合系数差和次近邻耦合系数, 观察三聚化模型中的拓扑特性变化. 发现正反向系数差不为零的最近邻耦合对非厄密系统的能谱和拓扑保护边缘态都有显著影响, 并引发新奇的趋肤效应; 次近邻耦合则对系统能谱的上下能隙中的边缘态能带的宽度和系统本征态的局域效果有影响. 最后, 进行了总结并展望了拥有丰富结构模型的各种新型拓扑材料的发展前景.

    2.   理论模型与能谱分布
    • 图1所示, 本文考虑的是类一维的非厄密三聚化晶格, 并为实现可调的耦合, 将晶胞结构设计为锯齿形. 晶胞内最近邻正向耦合系数为$ g_{1}+\mu $, 反向系数为$ g_{1}-\mu $, 其中$ 2\mu $为最近邻正反向耦合系数差, 次近邻耦合系数为$ \nu $, 胞间耦合系数为$ g_{2} $. 在该非厄密系统中, 在每个晶胞内添加均匀的增益损耗—添加势能$ {\rm i}\gamma $于增益格点$ A $、零势能于中性格点$ B $、势能$ -{\rm i}\gamma $于损耗格点$ C $. 这些参数条件构成了本研究所需的扩展的一维非厄密的AAH模型. 其中, 令$ a_{n} $, $ b_{n} $$ c_{n} $表示第$ n $个晶胞内格点的湮灭算符, 总晶胞数为$ \cal{N} $. 则该扩展的AAH模型的哈密顿量为

      Figure 1.  Schematic of the generalized trimerized non-Hermitian lattice.

      式内各项分别为AAH模型哈密顿量$H_{\rm {AAH}} =$$ \sum_{N}[(g_{1}\!+\!\mu)b^{\dagger}_{n}a_{n}+(g_{1}\!+\!\mu)c^{\dagger}_{n}b_{n}+ g_{2}a^{\dagger}_{n+1}c_{n}\!+\!(g_{1}-\mu)$$a^{\dagger}_{n}b_{n}+(g_{1}$$ -\mu)b^{\dagger}_{n}c_{n}+g_{2}c^{\dagger}_{n}a_{n+1}] $, 次近邻项的哈密顿量$ H_{\rm {NNN}} = \sum_{N}\nu c^{\dagger}_{n}a_{n}+{\rm {h.c.}} $, 和非厄密项的哈密顿量$ U ={\rm i}\gamma\sum_{N}[a^\dagger_{n}a_{n}-c^ \dagger_{n}c_{n}] $. 然后将哈密顿量代入薛定谔方程[41], 便可得:

      结合(1)式与(2)式, 得到可描述该三聚化模型的本征能谱, 如图2所示; 并可对与最近邻耦合有关的趋肤效应和与次近邻耦合有关的拓扑保护边缘态进行深入的研究, 如图3图4所示. 这里设相关参数为: $ {\cal{N}} = 30 $, $ g_1 = 1 $, $ \gamma = 0.2 $, $ g_2\equiv g_1- $$\delta\cos\theta$ ($ \delta = 0.5 $, $ \theta\in \{0, 2{\text{π}}\} $).

      Figure 2.  The eigen-energy spectrum of the trimerized non-Hermitian lattice. (a), (c), (e) and (g) show the real parts of the energy spectrum for $\mu=0, 0.1, 0.15, 0.2$, respectively; (b), (d), (f) and (h) show the imaginary parts of the energy spectrum for $\mu=0, 0.1, 0.15, 0.2$. Green solid lines and brown dash lines represent the energy band of the edge states.

      Figure 3.  Photon distributions for eigenstates of the system: (a) $\mu=0$; (b) $\mu=0.1$; (c) $\mu=0.2$; (d) $\mu=0.3$.

      Figure 4.  The eigen-energy spectrum and photon distributions for edge states of the system under the influence of the next-nearest-neighbor coupling: (a), (d) and (g) show the real parts of the energy spectrum for $\nu=0.1, 0.15, 0.2$; (b), (e) and (h) show the edge states whose energy band are in the upper gaps for $\nu=0.1, 0.15, 0.2$; (c), (f) and (i) show the edge states whose energy bands are in the lower gaps for $\nu=0.1, 0.15, 0.2$.

      首先, 通过调节胞内最近邻正反向耦合系数差, 对系统本征能谱进行分析. 图2(a), 图2(c), 图2(e)图2(g)为系统本征能谱的实部, 能带中包含上、下两个非平庸能隙, 共四条边缘态. 上(下)能隙中存在两条边缘态的本征值, 分别用棕色虚线和绿色实线表示, 且这两条边缘态简并. 图2(b), 图2(d), 图2(f)图2(h)为本征能谱的虚部, 同样用绿色实线和棕色虚线表示对应边缘态的虚部. 系统本征值的实部描述的是能谱中系统能带的位置; 本征值的虚部描述的是能谱中能级的宽度.

      此处, 不考虑次近邻耦合($ \nu = 0 $), 体态的能谱具有对称特征. 如图2(a)图2(b)所示, 当胞内最近邻正反向耦合相同($ \mu = 0 $)时, 体态和边缘态的本征能谱均关于$ E = 0 $$ \theta = {\text{π}} $对称, 能谱的三条能带在$ 0.5{\text{π}} $$ 1.5{\text{π}} $附近区域结合, 上下能隙中的四条边缘态具有相同长度. 然而, 如图2(c)—图2(f)所示, 随着$ \mu $逐渐增大, 虽然体态能谱仍保持原有对称性, 但是用棕色虚线表示边缘态能谱的实部和虚部均会缩短, 而用绿色表示的两条边缘态的实部和虚部均不变化. 当最近邻正反向耦合差继续增大至$ \mu>0.2 $时, 如图2(g)图2(h)所示, 棕色虚线表示的两条边缘态消失, 只存在绿色实线表示的两条边缘态. 这表明边缘态的简并情况发生改变, 边缘态数量由四条缩减到两条, 该三聚化晶格产生了新的拓扑特性. 这对应边缘态在晶格的分布情况就是晶格一侧的两条边缘态消失, 如图3所示, 此为边缘态的趋肤效应.

    3.   体态与边缘态的趋肤效应
    • 在上一部分中, 通过对动力学方程的求解, 讨论了本征能谱在最近邻耦合影响下边缘态能带宽度以及数量发生的变化. 本小节通过观察本征态的演变, 继续讨论三聚化非厄密系统中的边缘态和体态的趋肤效应. 图3为该三聚化模型包括体态(蓝点)和拓扑保护边缘态(绿线)在内的所有本征态的分布. 如图3(a)所示, 当$ \mu = 0 $时, 体态均匀分布, 边缘态则对称地局域在晶格的两端. 如图3(b)所示, 随着$ \mu $增大, 所有的本征态都开始向晶格的同一端附近局域, 展现了趋肤效应. 如图3(c)所示, 当$ \mu = 0.2 $时, 本征态的局域性更好; 体态继续向一端局域; 边缘态完全消失, 只存在于左端, 数量也由四条变为两条. 也就是说实现了边缘态数量的动态调节. 如图3(d)所示, 当最近邻正反耦合差继续增大, 体态也完全局域在左端, 也就是说三聚化非厄密晶格本征态的趋肤效应愈加显著.

      接下来, 在趋肤效应下, 继续观察次近邻耦合对本征能谱和边缘态的影响. 在图4中, 第一列描述的是受次近邻耦合影响的本征能谱的实部, 第二列代表上能隙中边缘态的光子分布, 第三列代表下能隙中边缘态的光子分布. 当$ \mu = 0.5 $时, 次近邻耦合系数在适当增加, 如图4(a), 图4(d)图4(g)所示, 能谱的实部仅关于$ \theta = {\text{π}} $对称, 而不再关于$ E = 0 $对称. 值得注意的是, 非平庸的上能隙开始变窄, 导致非平庸拓扑区域缩小; 非平庸的下能隙开始变宽, 导致非平庸相位区域扩大. 通过比较图4(b)图4(c)发现, 相对应的边缘态不同于体态的普通振荡, 而是呈急速衰减. 随着上下能隙中的边缘态能带数量的缩减, 只在上下能隙中各存在一条. 如图4(b), 图4(e)图4(h)所示, 随着次近邻耦合的增强, 由于趋肤效应而局域在左侧的上能隙边缘态的峰值在逐渐减小, 局域性减弱; 而由于趋肤效应而局域在左侧的下能隙边缘态变化正相反, 峰值逐渐增大, 局域效果逐渐增强. 此外, 如果次近邻耦合进一步增强, 本征能谱上下能隙及其边缘态会继续呈现相反的变化趋势; 最终上边缘态能带会缩短直至消失, 下边缘态峰值逐渐增大, 局域效果会明显增强.

    4.   结论与展望
    • 本文探讨和比较了扩展的非厄密三聚化晶格的能谱和边缘态的演化和局域效果. 首先, 特殊调制晶胞内的最近邻耦合系数, 即最近邻正反向耦合存在差异, 会导致系统本征态发生趋肤效应. 包括体态和边缘态在内的所有本征态都局域于系统的同一端附近; 并且, 边缘态的二重简并态消失, 边缘态数量由四条减少为两条. 其次, 次近邻耦合对系统能谱的演化有着显著的影响. 次近邻耦合会使系统非平庸能隙和相对应的具有趋肤效应的边缘态发生不对称变化, 且会影响所有本征态的局域效果. 未来, 通过设计多样的多聚化非厄密晶格, 并应用于拓扑量子材料的研究中, 一定会发现更多有意义的、新奇的拓扑物理性质.

      特别感谢东北师范大学物理学院吴金辉教授对文章写作的建议.

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