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Topological invariants and edge states in one-dimensional two-tile lattices

Lu Man-Xin Deng Wen-Ji

Topological invariants and edge states in one-dimensional two-tile lattices

Lu Man-Xin, Deng Wen-Ji
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  • The existence of robust conducting edge states is one of the most prominent properties of topological insulator, which is often simply illustrated as a consequence of bulk-boundary correspondence. Then here arises a new question whether similar robust edge states appear in some other topological-trivial systems, or rather, given a general answer of fundamental mathematics such as harmonic analysis or K-theory to this problem, we study one-dimensional two-tile lattices and show that the robust edge states can exist in topological-trivial complex lattices. Under the tight-binding approximation, all kinds of one-dimensional two-tile lattices with staggered hopping matrix elements can be described by the Su-Schrieffer-Heeger model or the Rice-Mele model, depending on their site energy. The site energy values of the Su-Schrieffer-Heeger model are equal, and often assumed to be zero, and the Rice-Mele model is constructed to describe the one-dimensional two-tile lattices having two different site energy values. With the help of the generalized Bloch theorem, the eigen-state problem of electrons in one-dimensional two-tile complex lattices are solved systematically, and the analytical expressions for the wavefunctions of the edge states in the corresponding finite lattice are obtained. The numerical and analytical results show that the edge states can also emerge in any of one-dimensional two-tile lattices beyond the Su-Schrieffer-Heeger lattice, i.e., provided that the magnitude of intracell hopping is less than the intercell hopping, a pair of edge states can also emerge in Rice-Mele lattice. Unlike the Su-Schrieffer-Heeger edge states, the two Rice-Mele edge states are locally distributed at one end of the finite lattice: one at the left and another one at right. The Zak phase is a topological invariant of the Su-Schrieffer-Heeger model, but it is no longer invariant for the Rice-Mele model because of the breaking of spatial inversion symmetry, and therefore the Rice-Mele lattices are topologically trivial. However, the Rice-Mele edge states are also robust to the non-diagonal disorder of the lattice. In addition, it is proven that the winding number can provide a general criterion for the existence of a couple of edge states in any one-dimensional two-tile lattice whether it is the Su-Schrieffer-Heeger lattice or not. These results lead to a conclusion that the topological invariant is not necessary for the robust edge states to occur.
      Corresponding author: Deng Wen-Ji, phwjdeng@scut.edu.cn
    [1]

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

    [2]

    Thouless D J 1983 Phys. Rev. B 27 6083

    [3]

    Berry M V 1984 Proc. R. Soc. Lond. 392 45

    [4]

    Thouless D J, Kohmoto M, Nightingale M P, Nijs M 1982 Phys. Rev. Lett. 49 405

    [5]

    Niu Q, Thouless D J 1984 J. Phys. A 17 2453

    [6]

    Kohmoto M 1985 Ann. Phys. 160 343

    [7]

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

    [8]

    Niu Q 1986 Phys. Rev. B 34 5073

    [9]

    Niu Q, Thouless D J, Wu Y S 1985 Phys. Rev. B 31 3372

    [10]

    Simon B 1983 Phys. Rev. Lett. 51 2167

    [11]

    Zak J 1989 Phys. Rev. Lett. 62 2747

    [12]

    Haldane F D M 2004 Phys. Rev. Lett. 93 206602

    [13]

    Murakami S, Nagaosa N, Zhang S C 2004 Phys. Rev. Lett. 93 156804

    [14]

    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802

    [15]

    Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757

    [16]

    Moore J E 2010 Nature 464 194

    [17]

    Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959

    [18]

    Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045

    [19]

    Nagaosa N, Tokura Y 2013 Nature Nano 8 899

    [20]

    Shen S Q 2012 Topological Insulators (Berlin: Springer) pp81−86

    [21]

    Bernevig B A, Hughes T L 2013 Topological Insulators and Topological Superconductors (Princeton: Princeton University Press) pp6−14

    [22]

    Asbóth J K, Oroszlány L, Pályi A 2016 A Short Course on Topological Insulators (Budapest: Springer) pp16−22

    [23]

    Guo H M 2016 Sci. China: Phys. Mech. Astron. 59 637401

    [24]

    Prodan E, Schulz-Baldes H 2016 Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics (Switzerland: Springer International Publishing) pp85−111

    [25]

    Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698

    [26]

    Rice M L, Mele E J 1982 Phys. Rev. Lett. 49 1455

    [27]

    RhimJ-W, Behrends J, Bardarson J H 2017 Phys. Rev. B 95 035421

    [28]

    邓伟胤, 朱瑞, 邓文基 2013 物理学报 62 067301

    Deng W Y, Zhu R, Deng W J 2013 Acta Phys. Sin. 62 067301

  • 图 1  (a) R-M紧束缚模型, ${t_v}$${t_w}$分别表示原胞内和原胞间电子的跃迁矩阵元; (b), (c)分别为无限SSH晶格和R-M晶格的电子能谱; 蓝色实线和红色点线分别取 ${t_v} = {t_w} = 1$${t_v} = 0.8$, ${t_w} = 1.2$, R-M晶格$V = 0.2$; (d), (e)分别展示包含15个原胞的有限SSH晶格和R-M晶格的能量本征值随跃迁矩阵元的变化, ${t_v} = 1 + \cos \theta $, ${t_w} = 1 - \cos \theta $; 当${t_v} < {t_w}$时, 两者都涌现出一对边缘态

    Figure 1.  (a) Schematic diagram of tight-bonding R-M model, ${t_v}$ and ${t_w}$ denote the intracellular and intercellular hopping elements, respectively; (b) and (c) are energy spectrum of electron in infinite SSH model and R-M model, where the solid blue lines are drawn for ${t_v} = \;{t_w} = 1$, and the red dot line for ${t_v} = 0.8$, ${t_w} = 1.2$, and $V = 0.2$ for R-M model; (d) and (e) shown the eigen energies of electron in finite (15 cells) SSH lattice and R-M lattice varying with the hopping elements, ${t_v} = 1 + \cos \theta $, ${t_w} = 1 - \cos \theta $; for both of them, a couple of edge states emerge when ${t_v} < {t_w}$.

    图 2  一维二元有限晶格的边缘态(其中30个原胞数包含60个格点, 参数取$\theta \; = \;0.58{\text{π}}$, 即${t_v} = 0.75,$${t_w} = 1.25$)(a)有限SSH晶格$V = 0$, 两个边缘态都同时出现在晶格的两端, 它们的本征能量都逼近于零; (b) 有限R-M晶格$V=0.2$, 红色空心圆和蓝色实心圆点分别表示逼近上能带底和下能带顶两个边缘态, 分别局域在晶格的左右端

    Figure 2.  Edge states of electron in one-dimensional two-tile finite lattice; the parameters are taken as 30 cells (60 atoms), $\theta \; = \;0.58{\text{π}}$, i.e. ${t_v} = 0.75,$${t_w} = 1.25$: (a) $V = \;0$ for SSH lattice, the eigen-energy values of the two edge states approach to zero, and each of the edge states appears at two ends of the lattice; (b) $V = 0.2$ for R-M lattice, the red hollow circles stand for the edge state near the upper band, the blue solid dot for the edge state near the lower band; one of them is localized at the left end, and the other at the right end.

    图 3  一维二元晶格的Zak相位随跃迁矩阵元的变化, 其中红线表示SSH模型的Zak相位, 且${\gamma _ + } = {\gamma _ - }$; 黑色实线和虚线分别是R-M晶格($V = 0.2$)上下能级的Zak相位${\gamma _ + }$${\gamma _ - }$

    Figure 3.  Zak phases variation with the hopping elements and site energy. The red line presents the Zak phase for SSH model, and ${\gamma _ + } = {\gamma _ - }$. For the R-M lattice with $V = 0.2$, the black solid line and dotted line show ${\gamma _ + }$ and ${\gamma _ - }$, respectively.

    图 4  非对角无序对能谱和边缘态的影响(非对角无序的幅度为$\xi = 0.5$, 跃迁矩阵元${t_v} = 0.75$, ${t_w} = 1.25$) (a), (b)分别是包含15个原胞的有限SSH晶格和R-M晶格的能谱; (c), (d)分别是包含30个原胞的有限SSH晶格和R-M晶格的边缘态

    Figure 4.  Effects of non-diagonal disorder on the energy spectrum and the edge states. The strength of the off-diagonal disorder is taken of $\xi \; = \;{\rm{0.5}}$, hopping elements ${t_v} = 0.75$, ${t_w} = 1.25$. Panels (a) and (b) present the spectrums of finite SSH lattice and R-M lattice consist of 15 unit cells, respectively; (c) and (d) show the edge states of electron in the finite SSH lattice and the R-M lattice of 30 unit cells.

  • [1]

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

    [2]

    Thouless D J 1983 Phys. Rev. B 27 6083

    [3]

    Berry M V 1984 Proc. R. Soc. Lond. 392 45

    [4]

    Thouless D J, Kohmoto M, Nightingale M P, Nijs M 1982 Phys. Rev. Lett. 49 405

    [5]

    Niu Q, Thouless D J 1984 J. Phys. A 17 2453

    [6]

    Kohmoto M 1985 Ann. Phys. 160 343

    [7]

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

    [8]

    Niu Q 1986 Phys. Rev. B 34 5073

    [9]

    Niu Q, Thouless D J, Wu Y S 1985 Phys. Rev. B 31 3372

    [10]

    Simon B 1983 Phys. Rev. Lett. 51 2167

    [11]

    Zak J 1989 Phys. Rev. Lett. 62 2747

    [12]

    Haldane F D M 2004 Phys. Rev. Lett. 93 206602

    [13]

    Murakami S, Nagaosa N, Zhang S C 2004 Phys. Rev. Lett. 93 156804

    [14]

    Kane C L, Mele E J 2005 Phys. Rev. Lett. 95 146802

    [15]

    Bernevig B A, Hughes T L, Zhang S C 2006 Science 314 1757

    [16]

    Moore J E 2010 Nature 464 194

    [17]

    Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959

    [18]

    Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045

    [19]

    Nagaosa N, Tokura Y 2013 Nature Nano 8 899

    [20]

    Shen S Q 2012 Topological Insulators (Berlin: Springer) pp81−86

    [21]

    Bernevig B A, Hughes T L 2013 Topological Insulators and Topological Superconductors (Princeton: Princeton University Press) pp6−14

    [22]

    Asbóth J K, Oroszlány L, Pályi A 2016 A Short Course on Topological Insulators (Budapest: Springer) pp16−22

    [23]

    Guo H M 2016 Sci. China: Phys. Mech. Astron. 59 637401

    [24]

    Prodan E, Schulz-Baldes H 2016 Bulk and Boundary Invariants for Complex Topological Insulators: From K-Theory to Physics (Switzerland: Springer International Publishing) pp85−111

    [25]

    Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698

    [26]

    Rice M L, Mele E J 1982 Phys. Rev. Lett. 49 1455

    [27]

    RhimJ-W, Behrends J, Bardarson J H 2017 Phys. Rev. B 95 035421

    [28]

    邓伟胤, 朱瑞, 邓文基 2013 物理学报 62 067301

    Deng W Y, Zhu R, Deng W J 2013 Acta Phys. Sin. 62 067301

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  • Received Date:  19 February 2019
  • Accepted Date:  02 April 2019
  • Available Online:  01 June 2019
  • Published Online:  20 June 2019

Topological invariants and edge states in one-dimensional two-tile lattices

    Corresponding author: Deng Wen-Ji, phwjdeng@scut.edu.cn
  • School of Physics and Optoelectronics, South China University of Technology, Guangzhou 510641, China

Abstract: The existence of robust conducting edge states is one of the most prominent properties of topological insulator, which is often simply illustrated as a consequence of bulk-boundary correspondence. Then here arises a new question whether similar robust edge states appear in some other topological-trivial systems, or rather, given a general answer of fundamental mathematics such as harmonic analysis or K-theory to this problem, we study one-dimensional two-tile lattices and show that the robust edge states can exist in topological-trivial complex lattices. Under the tight-binding approximation, all kinds of one-dimensional two-tile lattices with staggered hopping matrix elements can be described by the Su-Schrieffer-Heeger model or the Rice-Mele model, depending on their site energy. The site energy values of the Su-Schrieffer-Heeger model are equal, and often assumed to be zero, and the Rice-Mele model is constructed to describe the one-dimensional two-tile lattices having two different site energy values. With the help of the generalized Bloch theorem, the eigen-state problem of electrons in one-dimensional two-tile complex lattices are solved systematically, and the analytical expressions for the wavefunctions of the edge states in the corresponding finite lattice are obtained. The numerical and analytical results show that the edge states can also emerge in any of one-dimensional two-tile lattices beyond the Su-Schrieffer-Heeger lattice, i.e., provided that the magnitude of intracell hopping is less than the intercell hopping, a pair of edge states can also emerge in Rice-Mele lattice. Unlike the Su-Schrieffer-Heeger edge states, the two Rice-Mele edge states are locally distributed at one end of the finite lattice: one at the left and another one at right. The Zak phase is a topological invariant of the Su-Schrieffer-Heeger model, but it is no longer invariant for the Rice-Mele model because of the breaking of spatial inversion symmetry, and therefore the Rice-Mele lattices are topologically trivial. However, the Rice-Mele edge states are also robust to the non-diagonal disorder of the lattice. In addition, it is proven that the winding number can provide a general criterion for the existence of a couple of edge states in any one-dimensional two-tile lattice whether it is the Su-Schrieffer-Heeger lattice or not. These results lead to a conclusion that the topological invariant is not necessary for the robust edge states to occur.

    • 早在1980年代, 有关量子霍尔效应[1]、量子泵浦[2]和几何相位[3]的研究已经揭示了一些典型物理效用的拓扑特性. 霍尔电导是严格量子化的, 取决于磁场作用下二维电子气中电子能带的拓扑结构, 不依赖样品的大小、形状和载流子密度甚至迁移率[47], 泵浦电荷数由量子泵浦系统的陈数(Chern number)所决定[8,9], 它们本质上都是量子系统周期绝热演化过程中的几何相位, 又称为Berry 相位, 并具有规范不变性[3,10,11].

      最近十几年, 关于拓扑绝缘体的发现再一次引发了人们对量子系统拓扑特性的极大兴趣[1215], 拓扑相关的科学研究迅速成为不同学科的研究热点[1619], 边缘态(edge state)、体拓扑不变量(bulk topological invariant)和体边对应(bulk-edge correspondence)已成为拓扑物理学的基本概念[2023]. 存在受拓扑保护的边缘态被认为是拓扑绝缘体的主要特征, 而电子能带的拓扑不变量与边缘态的对应关系通常都被归结于系统的体边对应[16], 更深刻的认识则植根于调和分析与K-理论, 涉及现代数学的重要进展[24].

      在紧束缚近似下, 本文系统地求解了一维二元复式晶格电子的能量本征值问题. 如所周知[20], Su-Schrieffer-Heeger (SSH)模型[25]描述最简单的拓扑绝缘体. 不失一般性, 一维二元晶格都可以简化为Rice-Mele (R-M)模型[26], 而SSH模型只是座能量为零的特例. 由于空间反演对称性破缺, 座能量不为零的其他一维二元晶格都被认为是拓扑平庸的, 其电子能带的Zak相位不能作为描述拓扑相变和拓扑相的拓扑不变量[22,27]. 然而, 只要原胞内两原子之间的跃迁矩阵元小于最近邻原胞间跃迁矩阵元, 边缘态不仅可以存在于SSH拓扑晶格, 而且也可以存在于拓扑平庸的R-M晶格, 且两种类型的边缘态都对非对角(跃迁矩阵元)无序是鲁棒的.

      本文第2节在紧束缚近似下系统地研究了一维二元晶格中电子的能量本征态, 并根据推广的布洛赫定理[28]获得了有限晶格边缘态的解析表达式; 第3节计算了电子能带的Zak相位和缠绕数(winding number), 讨论了边缘态的存在性和鲁棒性(robust); 第4节是简单小结.

    2.   紧束缚模型及其电子态
    • 考虑无限长的理想一维二元晶格, 每个原胞包含A和B两个原子. 在紧束缚近似下, 电子的能量本征方程写作[20]

      其中E为能量本征值; ${\varepsilon _\alpha }$表示格点α的座能量, 对A和B两种不同原子, 可以分别取不同的数值${\varepsilon _{\rm{A}}}$${\varepsilon _{\rm{B}}}$; ${t_{\alpha \beta }}$标记电子由格点α到格点β的跃迁矩阵元; ${\varphi _\alpha }$是电子波函数, α遍历所有格点, 对β的求和只包含格点α的左右最近邻. 不失一般性, A和B原子的座能量可设为${\varepsilon _{\rm{A}}} = V = - {\varepsilon _{\rm{B}}}$; 跃迁矩阵元取正实数, 原胞内和原胞间最近邻原子的跃迁矩阵元分别记作${t_v}$${t_w}$, 如图1(a)所示; 为简单起见, 以原胞尺寸为长度单位, 即相邻的两个同类原子的间距为$1$. 所以一维二元复式晶格的紧束缚近似总是简化为R-M模型[26]; 特别地, 当A和B原子的座能量相同, 即$V = 0$时, 退化为SSH模型[25].

      Figure 1.  (a) Schematic diagram of tight-bonding R-M model, ${t_v}$ and ${t_w}$ denote the intracellular and intercellular hopping elements, respectively; (b) and (c) are energy spectrum of electron in infinite SSH model and R-M model, where the solid blue lines are drawn for ${t_v} = \;{t_w} = 1$, and the red dot line for ${t_v} = 0.8$, ${t_w} = 1.2$, and $V = 0.2$ for R-M model; (d) and (e) shown the eigen energies of electron in finite (15 cells) SSH lattice and R-M lattice varying with the hopping elements, ${t_v} = 1 + \cos \theta $, ${t_w} = 1 - \cos \theta $; for both of them, a couple of edge states emerge when ${t_v} < {t_w}$.

      根据布洛赫定理, 一维二元复式晶格中电子的能量本征函数可普遍地假设为布洛赫行波, 即

      其中n为长链中的原胞序数; ${\varphi _{\rm{A}}}$${\varphi _{\rm{B}}}$分别表示A和B子晶格上布洛赫波的复数振幅; k为布洛赫波矢, 且可以限定$k \subset \left[ {0,2{\text{π}}} \right)$. 代入(1)式可得

      能量本征值和本征函数分别为

      值得指出的是, 这一能量-波矢共同本征态波函数具有一个相位不确定性.

      $V = 0$, R-M模型简化为SSH模型, 能量本征值和本征函数(4)和(5)式分别简化为

      其中幺模复数

      由(4)式可知, R-M晶格中电子的能带结构受${t_v}$, ${t_w}$V三个参数的调控, 上下能带之间的带隙为

      只有当$V = 0$, 即退化为SSH晶格时才能通过调节两个跃迁矩阵元${t_v}$${t_w}$的相对取值闭合能隙, 实现量子拓扑相变, 如图1(b)图1(c)所示.

    • 实际样品均为有限晶格, 拓扑能带理论只考虑包含N个完整原胞的情形. 如果忽略晶格的表面效应, 将这样一段原子链首尾相连, 它与无限晶格的主要差别就是采用了周期性边条件, 即$\varphi _{N + 1}^{\rm{A}} = $$ { \varphi _0^{\rm{A}}},\;{\varphi _{N + 1}^{\rm{B}} = \varphi _0^{\rm{B}}} $, 从而波矢取N个分离值, 即

      并以热力学极限$N \to \infty $近似描述晶格的电子结构. 由于存在受对称性保护的边缘态是拓扑绝缘体的主要特征, 通常采取自然边条件, 假设电子被完全限制在有限晶格内, 即

      此时晶格的平移对称性被破坏, 电子的能量本征态不再是简单的布洛赫波.

      在采用推广的布洛赫定理求解有限晶格能量本征态的新方案中[28], 仍然假定波函数的试探解为(2)式的形式, 但不再预设波矢取实数, 即更一般地假设$k = \beta + {\rm{i}}\alpha $. 代入本征方程(3)可得

      显然, 为了确保能量本征值为实数, 复波矢的允许值分成两大类: 或者波矢只有实部β, 虚部$\alpha = 0$; 或者波矢可以具有非零的虚部α, 但实部只能取特殊的值, 即$\beta = 0$或者${\text{π}}$.

      首先关注波矢为实数的情形. 仍然得到(4)式的正负两支能带和(5)式中的对应波函数. 对于有限晶格, 还需保证波函数满足边条件(11)式. 注意到能量本征值是波矢的偶函数, 相向而行的两个布洛赫波是能量简并的, 所以它们适当的线性叠加才可能给出有限晶格中拥有同一本征值的能量本征态. 设

      其中原胞序数$n = 0,1,2, \cdots,N + 1.$ 代入(11)式可得

      若取$C_{ - \beta }^ \pm = - C_\beta ^ \pm $, 则要求波矢满足

      ${t_v} = {t_w}$, 它在区间$\left[ {0,2{\text{π}}} \right)$内共有$2\left( {N + 1} \right)$个实数解, 其中仅有开区间$\left( {0,{\text{π}}} \right)$N个实波矢态可按(13)式叠加成有限晶格的2N个能量本征态, 即

      类似地, 若${t_v} > {t_w}$, 仍然可以按(15)式在开区间$\left( {0,\;{\text{π}}} \right)$内求得N个实波矢并按(13)式叠加成有限晶格的2N个能量本征态; 但是, 若${t_v} < {t_w}$, 则方程(15)在开区间$\left( {0,\;{\text{π}}} \right)$内只有$N - 1$个实数解, 且只能按(13)式叠加形成有限晶格的$2(N - 1)$个驻波形式的能量本征态, 缺失的两个能量本征态的波矢必定具有非零的虚部.

      考虑$k = {\text{π}} +{\rm{ i}}\alpha $的情形, 则由(12)式可得正负两支能带为

      注意到${E_ \pm }\left( {{\text{π}} - {\rm{i}}\alpha } \right) = {E_ \pm }\left( {{\text{π}} +{\rm{ i}}\alpha } \right)$, 为了得到满足边条件(11)式的能量本征态, 可设

      其中$n = 0,1,2, \cdots,N + 1.$ 代入(11)式可得

      若取$D_{ - {\rm{i}}\alpha }^ \pm = - D_{{\rm{i}}\alpha }^ \pm $, 则要求波矢满足

      显然, 在$0 < {t_v} < {t_w}$的前提下, 这个关于α的超越方程的确存在一对实数解, 并由(18)式得出一对能量本征态, 即

      其中$D_{ - {\rm{i}}\alpha }^ \pm $为归一化常数. 类似地研究$k = \pm {\rm{i}}\alpha $的情形, 不难证明它们不能叠加形成有限晶格中满足自然边条件(11)式的能量本征态.

      作为对比, 图1(d)图1(e)分别展示了有限SSH晶格($V = 0$)和R-M晶格($V = 0.2$)中电子的能量本征值随原胞内跃迁矩阵元${t_v}$和原胞间跃迁矩阵元${t_w}$的变化方式. 为图中曲线不要过分密集, 晶格只包含了15个完整原胞, 仅30个原子. 注意到这个二能带结构取决于两个跃迁矩阵元的比值${{{t_v}} /{{t_w}}}$, 可采用一个参量$0 \leqslant \theta \leqslant {\text{π}}$描述全部可能情形, 即${t_v} = 1 + \cos \theta $, ${t_w} = 1 - \cos \theta $. 值得关注的是, 无论是SSH拓扑晶格还是反演对称性破缺的R-M拓扑平庸晶格, 当$\theta $变化跨越$\theta = {{\text{π}} / 2}$的临界点都涌现一对边缘态, 即在射线${t_v} = {t_w}$${t_v} - \;{t_w}$平面分割为不同的两相, 若$0 \leqslant {t_v} < {t_w}$则存在边缘态, 若$0 \leqslant {t_w} < {t_v}$则无边缘态. 图2进一步给出了有限R-M晶格中典型能量本征态波函数的数值计算结果, 它们与根据推广的布洛赫定理求得的解析解完全吻合.

      Figure 2.  Edge states of electron in one-dimensional two-tile finite lattice; the parameters are taken as 30 cells (60 atoms), $\theta \; = \;0.58{\text{π}}$, i.e. ${t_v} = 0.75,$${t_w} = 1.25$: (a) $V = \;0$ for SSH lattice, the eigen-energy values of the two edge states approach to zero, and each of the edge states appears at two ends of the lattice; (b) $V = 0.2$ for R-M lattice, the red hollow circles stand for the edge state near the upper band, the blue solid dot for the edge state near the lower band; one of them is localized at the left end, and the other at the right end.

    3.   拓扑不变性与边缘态
    • 拓扑学主要研究几何对象在连续变换下保持不变的特性, 即拓扑不变量. 拓扑能带理论以绝缘体哈密顿量的绝热变型(adiabatic deformation)过程中保持不变的物理量如陈数、几何相位或缠绕数等来区分和定义系统不同的拓扑相; 受对称性保护的边缘态也常常被认为是拓扑相的标志[1622].

    • Zak相位是Berry相位在晶格电子能带问题的重要推广, 能带n的Zak相位定义为[11,22]

      其中${\phi _n}\left( k \right) \equiv {\varphi _n}\left( k \right){{\rm e}^{ - {\rm{i}}kn}}$$\phi _n^\dagger \left( k \right)$分别表示布洛赫波的调制振幅及其厄米共轭, 波矢$k$的积分覆盖布里渊区, 包含该能带中的全部能量本征态. Zak相位可用于描述一维周期晶格能带的拓扑特性, 通常认为Zak相位取$0$的电子能带是拓扑平庸的, 若非零且为${\text{π}}$的整数倍才是拓扑能带; 并且无限晶格的拓扑能带总伴随着有限晶格的边缘态.

      将(7)式代入(22)式, 计算可得SSH晶格上下能带的Zak相位相等, 即

      其中, $ \pm $分别表示上下能带, $\Delta \arg $$\left( {{d_k}} \right)$表示在波矢k$0$$2{\text{π}}$的变化过程中, 复数${d_k}$幅角的增量. 注意到这一增量正比于复平面内复数${d_k}$围绕原点的缠绕数, 不难得到

      ${t_w} = {t_v}$, 则SSH晶格的二聚化消失, 退化为简单原子链.

      将(5)式的布洛赫波函数代入(22)式, 可得R-M晶格上下能带Zak相位, 即

      并且

      但由于空间反演对称性破缺, 不能保证R-M晶格Zak相位的取值为0或${\text{π}}$的整数倍. 如图3所示, SSH晶格的Zak相位只取$0$${\text{π}}$两个分离值, 作为拓扑不变量分别标志拓扑平庸相和拓扑相. 与此不同, 随着跃迁矩阵元相对取值${{{t_v}} /{{t_w}}} = {\cot ^2}\left( {\theta /2} \right)$的变化, R-M晶格的${\gamma _+}$${\gamma _ - }$可以连续改变; 若以Zak相位作为拓扑不变量, 则座能量变化的一维二元晶格都将被认为是拓扑平庸的, 尽管${\gamma _+}$${t_v} = $$ {t_w}$临界点发生不连续跃变, 且伴随着能隙内一对分别靠近导带底和价带顶的边缘态的涌现.

      Figure 3.  Zak phases variation with the hopping elements and site energy. The red line presents the Zak phase for SSH model, and ${\gamma _ + } = {\gamma _ - }$. For the R-M lattice with $V = 0.2$, the black solid line and dotted line show ${\gamma _ + }$ and ${\gamma _ - }$, respectively.

    • 缠绕数是刻画SSH模型电子能带拓扑性质的另一个常用的拓扑不变量[20,22]. 由能量本征方程(3)可得R-M模型的体哈密顿量, 即

      其中, 三维矢量${{d}}( k ) \equiv ( {{t_v} + {t_w}\cos k,\;{t_w}\sin k,{V} )}$; ${{\sigma}} \equiv $$ \left( {{\sigma _x},\;{\sigma _y},\;{\sigma _z}} \right)$为赝自旋的泡利矩阵, 描写原胞内A和B格点的自由度; 若$V = 0$则退化为SSH模型. (27)式中哈密顿量的能量本征态是赝自旋空间中沿${{d}}\left( k \right)$方向(或反方向)的单位矢量, 即$\tilde {{d}} = {{{d}} /{\left| {{d}} \right|}}$, 而它绕赝自旋空间$z$轴的缠绕数可以定量地表述为[22]

      如所周知, 随波矢k由0连续变化增大到$2{\text{π}}$, 平面矢量${{{d}}_{xy}}\left( k \right) \equiv \left( {{t_v} + {t_w}\cos k,\;{t_w}\sin k,{\rm{ 0}}} \right)$绕原点的缠绕数可以描述SSH能带的不同拓扑相: 若${t_v} > {t_w}$则缠绕数为$0$, 能带是拓扑平庸的; 若${t_v} < {t_w}$则缠绕数为$1$, 能带是拓扑的. 与采用Zak相位给出一致的结论.

      方程(15)和(20)是否存在虚数波矢解完全由(8)式中的幺模复数${d_k}$所决定, R-M晶格存在边缘态的充分必要条件仍然是${t_v} < {t_w}$, 与座能量V的取值无关. 所以平面矢量${{{d}}_{xy}}\left( k \right)$绕原点的缠绕数, 或者矢量${{d}}\left( k \right)$绕赝自旋空间z轴的缠绕数可作为R-M晶格存在边缘态的判据: 若缠绕数由$0$变到1, 则有限一维二元晶格带隙中涌现一对逼近导带底和价带顶的边缘态.

    • 拓扑稳定性是边缘态的重要性质. 考虑杂质或无序对电子态的影响和改变, 是检验其鲁棒性的简单方法. 注意到一维二元有限晶格总可以涌现一对边缘态, 有必要比较杂质或无序对SSH边缘态和R-M边缘态的不同影响.

      系统的非对角无序表现在跃迁矩阵元受到随机调制, 即${t_v} \to {t_v}\left( {1 + \underline \xi } \right)$, ${t_w} \to {t_w}\left( {1 + \underline \xi } \right)$, 其中$\underline \xi $表示在区间$\left[ { - \xi ,\;\xi } \right]$的随机数. 图4显示了有关边缘态鲁棒性的典型结果. 与图1对比, 图4(a)图4(b)分别展示了幅度$\xi = 0.5$的非对角无序对包含15个原胞30个格点的有限SSH晶格和R-M晶格的影响. 虽然$V = 0.2$导致R-M晶格反演对称性破缺, 但是无序对它的能量本征值的影响与对相应的SSH能量本征值的影响并无二致. 与图2对比, 图4还展示了非对角无序对包含30个原胞60个格点的有限SSH晶格和R-M晶格两类边缘态的影响. 不难发现, 非对角无序对边缘态的影响都是微弱的, 而且R-M边缘态也具有SSH边缘态对非对角无序同样的鲁棒性.

      Figure 4.  Effects of non-diagonal disorder on the energy spectrum and the edge states. The strength of the off-diagonal disorder is taken of $\xi \; = \;{\rm{0.5}}$, hopping elements ${t_v} = 0.75$, ${t_w} = 1.25$. Panels (a) and (b) present the spectrums of finite SSH lattice and R-M lattice consist of 15 unit cells, respectively; (c) and (d) show the edge states of electron in the finite SSH lattice and the R-M lattice of 30 unit cells.

    4.   结 论
    • 结合数值与解析的方法求解了一维二元复式晶格电子能量本征值问题, 包括得到有限晶格边缘态的解析表达式. 在有限晶格中, 电子的布洛赫波矢不必一定为实数, 其能量本征态是能量简并的布洛赫态的叠加态, 它们大多仍然是遍布体内的驻波态, 但在${t_v} < {t_w}$条件下将涌现一对局域于晶格端点的边缘态.

      与SSH有限晶格一样, 在${t_v} < {t_w}$条件下, R-M有限晶格也可以涌现出一对边缘态. 尽管与SSH边缘态对称地局域分布于晶格两端有所不同, 两个R-M边缘态分别局域分布于有限R-M晶格的两端, 但它们对非对角无序具有类似的鲁棒性. 虽然空间反演对称性破缺的R-M晶格是拓扑平庸的, 其Zak相位和缠绕数都不再是拓扑不变量, 但缠绕数仍然可以作为其对应的一维二元晶格存在边缘态的一般判据.

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