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摘要: 拓扑绝缘体起源于量子波动系统, 因其单向传输、能量无耗散等新奇物理性质, 近年逐渐被拓展到电磁波、声波、弹性波等经典波动领域, 为经典波的调控提供了新思路. 本文将系统介绍拓扑绝缘体理论及其在弹性波领域的相关研究进展. 首先以一维、二维离散点阵系统为例, 阐释拓扑物理研究中的基本数学、物理概念, 如狄拉克锥、能带翻转、贝里曲率、拓扑数等. 随后, 依次讨论弹性系统谷霍尔绝缘体、陈绝缘体、自旋霍尔绝缘体的设计思想及目前研究进展, 并讨论了近年来逐渐受关注的高阶拓扑现象. 最后, 讨论了静力学中拓扑孤立子、拓扑零能模式现象.Abstract: Topological insulators, originated in quantum systems, are recently generalized to electromagnetic, acoustic, and elastic wave fields due to interesting wave controlling provides, such as unidirectional and dissipationless energy transportation. This new kind of materials provide unprecedented possibilities for engineering wave flows. This paper will introduce the basic theory of topological insulators and summarize the research progress of topological insulators in elastic fields. Based on one-dimensional and two-dimensional discrete models, preliminary concepts of topological insulators, such as Dirac cone, band inversion, Berry curvatures, topological invariants, are introduced. The design and progress of valley Hall insulators, Chern insulators as well as spin Hall insulators are introduced afterward, followed by discussions on high-order topological insulators. The last part of this paper covers topological phenomena in static mechanics, including topological solitons and toplogical zero energy modes.
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图 1 拓扑基本概念与拓扑绝缘体性质. (a) 球形橡皮泥可光滑连续变形至圆饼, 表明球和圆饼在几何上拓扑等价. 球形橡皮泥被撕裂产生1个孔洞, 则拓扑发生变化, 此时其与甜甜圈在几何上拓扑等价. 孔洞数可以看作分类几何的拓扑数; (b) 拓扑绝缘体内部不导电(传波), 但边界导电(传波), 其边界导电性是一种拓扑性质, 不易受到杂质、缺陷等影响
图 2 (a) 示例二维周期系统; (b) 由于周期系统在实空间的周期性, 菱形布里渊区四条边两两等价, 将对应边界拼接起来后, 布里渊区在几何上与圆环表面拓扑等价. 圆环表面每一处有对应的波矢及波函数
图 3 (a) 一维双原子链模型; (b) 三种典型情况的频散曲线, 插图表示相应点处的模态相位. 箭头表示质点的运动方向. 波数q = π/a处, γ = 0.2的声学模态与γ = −0.2时光学模态一样, 且 γ = 0.2的光学模态与γ = −0.2时声学模态一样. 即参数γ由正变负时, 出现了模态顺序交换, 称之为能带翻转
图 5 (a)含400个单胞(γ < 0, 即υ = 1)的点阵结构, 其固定边界上支持1支边界态模式, 如图(b)所示; (c) 含400个单胞(γ > 0, 即υ = 0)的点阵结构, 其固定边界上不支持边界态模式; (d) 图(b)所示边界态对应位移分布; (e) 右端固定的含400个单胞的点阵结构在不同刚度参数下的特征频谱
图 6 (a) 含界面的点阵结构, 其两侧各有200个单胞; (b) γ = −0.5时的界面模态; (c) γ = 0.5时的界面模态; (d) 两侧各有200个单胞的含界面点阵结构在不同刚度参数下的特征频谱
图 7 (a) 二维蜂窝质量弹簧系统示例, 每一个单胞包含两个质点, 最近邻质点之间由弹簧连接; (b) 第一布里渊区及角点K′和K. 图中三个K点相互之间通过倒格矢量b1和b2平移得到, 等价于同一个K点, 同理K′也类似. 来自文献(Chen et al. 2019)
图 8 二维蜂窝质量弹簧系统第2、第3支色散曲面. (a) 参数mp = mq = 1.0, t = 1.0, 第2、第3支色散曲面在K′和K处呈线性简并特征, 局部色散曲面在K′和K处呈现双锥, 称为狄拉克锥(Dirac cone); (b) 参数mp = 0.9, mq = 1.1, t = 1.0, 色散曲面在布里渊区角点K′和K处取不同极值, 系统存在完全禁带
图 9 常见二维拓扑绝缘体色散曲线及波传播规律. (a) 霍尔绝缘体色散曲线, 灰色区域对应禁带, 斜线为界面态色散曲线, 该界面态只能单向传播, 如图(d); (b) 自旋霍尔绝缘体带结构曲线, 禁带内存在斜率相反的两条色散曲线, 对应于沿界面相反方向传播的上、下自旋界面态, 如图(e); (c) 正负谷霍尔绝缘体组成的界面传播色散曲线, 禁带区域有斜率相反的两条色散曲线, 表明该界面支持双向传播的界面态
图 10 (a) 带结构曲线, 其中离散点表示精确解, 谷点K和K′附近的连续曲线为微扰模型得出的解析解; (b)(c) 胞元中两质量不等时(mp = 0.9, mq = 1.1, t = 1)的精确贝里曲率及由微扰模型得出的解析贝里曲率. 来自文献(Chen et al. 2019)
图 11 (a) 正负谷霍尔相组成的界面示例, 界面具体几何形式与方位角度θ有关; (b)(c) 锯齿形/扶手形界面对应于方位角θ = 0o/30o. 来自文献(Chen et al. 2019)
图 12 (a) 正负谷霍尔相组成的条状几何结构; (b) 条状几何结构沿y方向的波传播带结构曲线, 计算中沿y方向施加Bloch波连续条件; (c)(e) 图(b)中谷点K′上下两个体态对应的质点位移场, 颜色表示水平方向位移幅值, 红色代表幅值更大; (d) 图(b)中界面态对应的质点位移场. 来自文献(Chen et al. 2019)
图 13 (a)谷点K′/K界面态对应的质点位移场; (b)(c) 界面附近质点的运动轨迹, 黑色圆点代表质点的位置, 红色/蓝色代表绕平衡位置顺时针/逆时针旋转; (d)(e) q/p质点水平位移幅值沿着x方向的分布. 来自文献(Chen et al. 2019)
图 14 (a) 局域共振型谷霍尔绝缘体离散模型; (b) 对应的第一布里渊区以及两个布里渊区角点K′和K. 来自文献(Zhang et al. 2020)
图 15 (a) 单胞中两个局域振子完全相同时的频散曲线(灰色点线)以及不含局域振子的蜂窝点阵的频散曲线(红色实线); (b) 单胞中两个局域振子的质量存在小幅差异时的频散曲线; (c) 图(b)中前两条频散分支对应的贝里曲率. 来自文献(Zhang et al. 2020)
图 16 (a) 设计的局域共振型谷霍尔绝缘体微结构; (b) 单胞内两个局域振子完全相同时的频散曲线, 颜色表征极化模式, 值为1对应出平面极化振动; (c) 单胞内两个局域振子的质量存在小幅差异时的频散曲线; (d) 单胞内两个局域振子的质量存在小幅差异时的出平面等效密度. 来自文献(Zhang et al. 2020)
图 17 (a) 包含界面的条带状超胞; (b) 条带状超胞的频散曲线, 绿色标注的为界面态分支; (c) f = 2500 Hz时的特征模态; (d) (e) (f) f = 2500 Hz时三种有限尺寸结构(不包含界面, “直线”形界面路径, “Z”形界面路径)的稳态位移场; (g) 图(e)中标注的蓝色虚线上的振幅分布; (h) 三种有限尺寸结构(不包含界面的体态, “直线”形界面路径, “Z”形界面路径)的透射率曲线. 来自文献(Zhang et al. 2020)
图 18 制备的局域共振型谷霍尔绝缘体以及实验测试系统. 来自文献(Zhang et al. 2020)
图 19 (a) 图18中A点和B点的实验测得频响曲线; (b) (c) (d) 频率1500 Hz, 2045 Hz和2500 Hz时的均方根速度场. 来自文献(Zhang et al. 2020)
图 20 左侧: 图19(a)中的频响曲线; 右侧: 从实验带隙位置反演得到微结构梁的真实刚度后, 重新计算的图17(a)所示条带状超胞的带结构. 来自文献(Zhang et al. 2020)
图 21 (a) 单胞不含磁流体时, 布里渊区角点K处发生狄拉克简并; (b) 单胞其中一个空腔充满磁流体时, 空间反演对称性破坏, 使得狄拉克简并退化并形成带隙(蓝色频散分支); (c) 在图(b)中蓝色频散分支带隙范围内, 存在拓扑界面态传播模式; (d) 利用设计的可编程磁场可以控制每个单胞中磁流体的分布; (e) 因此, 通过改变单胞中磁流体的分布状态, 可以调整用于传播拓扑界面态的界面路径的形状; (f) 设计的单胞的几何参数. 来自文献(Zhang et al. 2019a)
图 22 (a) 条带状超胞; (b) 出平面极化振动模式频散曲线; (c) f = 1234 Hz下界面态频散分支上的特征模态; (d) (e) (f) f = 1234 Hz下三种界面路径的稳态位移场, 分别对应直线形界面路径、“L”形界面路径、“Z”形界面路径. 来自文献(Zhang et al. 2019a)
图 23 (a) 制备的16 × 16测试样件; (b)(c)(d) 实验测试的f = 1450 Hz下三种界面路径的稳态位移场, 分别对应直线形界面路径、“L”形界面路径、“Z”形界面路径. 来自文献(Zhang et al. 2019a)
图 24 设计的可编程磁铁升降阵列系统. (a) 可编程控制软件; (b) 16通道继电器开关; (c) 磁铁升降阵列, 绿色圆圈标注的磁铁已由软件控制升起; (d) 磁铁升降阵列的侧视图. 来自文献(Zhang et al. 2019a)
图 25 将离散蜂窝系统放置于旋转基座上可以打破时间反演对称, 实现霍尔绝缘体; 在与旋转平台固定的坐标系上, 质点受到科里奥利力及离心力, 科氏力与旋转基座角速度大小|Ω|线性相关, 离心力与角速度平方|Ω|2线性相关
图 26 (a) 拓扑相分布图, 横坐标与基座旋转强弱相关, 纵坐标与胞元内两质点质量差相关; (b) 当参数处于A和C相的公共边界时, 如图中黄色圆点所示, 系统对应的带结构曲线. 来自文献(Chen et al. 2019)
图 27 6种拓扑相组成的锯齿界面波传播色散曲线; 棕色代表体波色散曲线, 蓝色代表界面态色散曲线. 来自文献(Chen et al. 2019)
图 28 弹性波沿6种拓扑界面传播时瞬态结果; 颜色代表位移幅值大小, 红色表示位移更大; 数值模拟区域大致包含78 × 90个单胞. 来自文献(Chen et al. 2019)
图 29 (a) 夹杂六角排布的二维弹性陀螺复合结构示意图; (b) 单胞剖面图, 上下为滑移边界, 基体为弹性体, 夹杂为刚体, 其内部耦合一个陀螺转子; (c) 运动状态下的夹杂侧视图和顶视图. 来自文献(Zhao et al. 2020)
图 30 (a) 非互易瑞利波传播仿真结果, 激励形式为上下振动点激励; (b) 归一化波速及表面附近质点位移场, 当α = 3.0时, 仅支持左行波; (c) 表面质点极化轨迹曲线. 来自文献(Zhao et al. 2020)
图 31 能带折叠产生双重狄拉克锥. (a) 蜂窝排布质量弹簧点阵系统, 其中六边形超胞包含6个质点, 连接胞元内质点的弹簧的刚度为ti, 连接胞元间质点的弹簧的刚度为to; (b) 较小的六边形为图(a)中六边形超胞对应的第一布里渊区; 较大的六边形区域为刚度均匀分布时, 即ti = to对应第一布里渊区, 最简单胞为图(a)中菱形区域; (c) Γ点对应d和p特征模态的位移分布示意图, 蓝色圆圈表示质点振动最大位置, 空心圆为质点平衡时位置. 来自文献(Chen et al. 2019)
图 32 自旋霍尔相变. (a) ti > to时蜂窝质量弹簧系统的带结构曲线, 系统为平凡绝缘体; (b) ti = to对应的带结构曲线, Γ点带隙完全闭合, 系统处于拓扑相变临界状态; (c) ti < to对应的带结构曲线, 系统为自旋霍尔绝缘体; 颜色表示特征模态包含的p和d特征模态成分多少. 来自文献(Chen et al. 2019)
图 33 (a) 左侧自旋霍尔绝缘体与右侧平凡绝缘体组成的条状超胞, 界面沿y方向; (b) 超胞计算得到的带结构曲线, 红色/蓝色表示上/下自旋界面态, 灰色表示体态; (c) 上/下自旋界面态对应的质点位移幅值; (d)(e) 上/下自旋界面态中界面附近质点振动轨迹示意图, 红色/蓝色表示质点振动为顺/逆时针方向. 来自文献(Chen et al. 2019)
图 34 自旋霍尔绝缘体单向传播模拟结果. (a)(c) 施加上/下自旋激励时, 胞元内各质点振动轨迹示意图; (b)(d) 上/下自旋激励时弹性波传播瞬态模拟结果, 红色代表质点位移幅值较大, 模拟区域共包含78 × 90个胞元. 来自文献(Chen et al. 2019)
图 35 自旋霍尔绝缘体不同边界选取及其能带结构. (a) (b) 边界为完整胞元的条状超胞及其能带结构, 红/蓝色曲线对应自旋边界态; (c) (d) 边界为不完整胞元的条状超胞及其能带结构, 带结构中不包含边界态. 来自文献(Chen et al. 2019)
图 36 弹性波中的类自旋自由度. (a) 弹性体表面瑞利波的色散曲线, 红/蓝表示向右/左传播色散曲线, 箭头表示质点旋转方向; (b) 边界施加逆/顺时针圆极化激励激发向右/左传播的瑞利波. 来自文献(Long et al. 2018)
图 37 压电主动调节实现双重狄拉克锥. (a) 包含压电片的单胞; (b) 压电片可接入(c)负电容电路. (d) 电路为开路时带结构; (e) 接入负电容电路后的能带结构, 四条能带在Γ处简并; (f) 四个简并模态. 来自文献(Li et al. 2020)
图 38 弹性波赝自旋态的构建. (a) 实验样品由两块弹性波绝缘体拼接构成, 左侧和右侧绝缘体的带隙范围一致, 但顶带和底带对应的特征模态互为反转(两侧绝缘体的能带具有不同的拓扑不变量); (b) 在两侧绝缘体的体带隙范围内存在两条界面态模式; (c) 由四种简并态构建的赝自旋基矢(S与A); (d) 实验探测到的由下向上传输的界面态, 出面位移场呈现出“+S → +A → −S ···”的时域特征, 对应于赝自旋态S + iA; (e) 实验探测到的由上向下传输的界面态, 出面位移场呈现出“+S → − A → − S ···”的时域特征, 对应于赝自旋态S − iA. 来自文献(Yu et al. 2018)
图 39 对缺陷和拐角免疫的界面态传输功能. (a) 界面路径不含任何缺陷及拐角; (b) 界面路径上含有一个由孔洞缺失构成的“空位”缺陷; (c) 界面路径上含有一个由孔洞错位构成的“位错”缺陷; (d) 包含两个120°拐角的“Z”形界面路径; (e) 上述四种界面路径的传输率实验测试结果. 来自文献(Yu et al. 2018)
图 40 弹性拓扑环形谐振器. (a) 实验样品包含一条平直、一条闭合环形界面; (b) 在左侧平直界面下端激发的pseudospin+赝自旋态向上传输; (c) 环形谐振腔中的能量谱实验结果, 可观测到两个关于狄拉克频率对称分布的共振峰分布; (d)(e) 图(c)中两个谐振频率下的出面位移场及能流分布. 来自文献(Yu et al. 2018)
图 41 (a) 蜂窝点阵的复合元胞; (b) 蜂窝弹性声子晶体板的局部截图, 菱形框标记的单胞为原始单胞, 正六边形标记的单胞为复合元胞. 蓝色横梁表示胞间耦合梁, 定义为linter. 红色横梁表示胞内耦合梁, 定义为lintra; (c) linter = lintra时, 复合元胞的能带结构, 在1517 Hz处具有双狄拉克点; (d) lintra < linter时, 复合元胞的能带结构. 来自文献(Fan et al. 2019)
图 42 (a) 第一不可约布里渊区Γ点本征模态频率关于lintra/L的函数; (b) lintra/L = 0.836和lintra/L = 1.2时, 第一不可约布里渊区Γ点本征模态的位移场图. 来自文献(Fan et al. 2019)
图 43 (a) 膨胀复合元胞组成的正六边形样件; (b) 收缩复合元胞组成的正六边形样件; (c) 带有缺陷的收缩复合元胞组成的样件, 红色虚线框标记为缺陷; (d) 膨胀复合元胞组成的正六边形样件的本征频率; (e) 收缩复合元胞组成的正六边形样件的本征频率, 绿色、红色、蓝色和黑色圆点分别表示带隙边缘模态、拓扑角模态、平庸角模态和体模态; (f) 带有缺陷的收缩复合元胞组成的样件的本征频率; (g) ~ (j) 体模态(1750.2 Hz)、拓扑角模态(1555.8 Hz)、平庸角模态(1529.1 Hz)和带隙边缘模态(1600.1 Hz)的位移场图. 来自文献(Fan et al. 2019)
图 44 (a) 平庸正六边形样件的体(黑色)、边缘(绿色)和角(红色)传输谱; (b) 拓扑正六边形样件的体(黑色)、边缘(绿色)和角(红色)传输谱. 来自文献(Fan et al. 2019)
图 45 (a) 无缺陷正三角形样件图; (b) 含缺陷的正三角形样件图; (c) 无缺陷正三角形样件的本征频率; (d) 含缺陷的正三角形样件的本征频率, 绿色、蓝色和黑色圆点为带隙边缘模态、角模态和体模态; (e) ~ (g) 带隙边缘(1610.4 Hz)、角(1498.9 Hz)和体(1719.1 Hz)模态的位移场图; (h) 无缺陷正三角形样件(红色)和含缺陷的正三角形样件(黑色)的角传输谱. 来自文献(Fan et al. 2019)
图 46 (a) 正三边形结构π/3锐角的四个零模态; (b) 正六边形结构2π/3钝角的三个零模态; 绿色和紫色圆点表示的手征价(chiral charge)为+1和−1. 来自文献(Fan et al. 2019)
图 47 (a) Scott摆链系统中拓扑孤子(kink/antikink)的激发构型; (b) 类摆链系统中拓扑孤子与孤子晶格的激发构型; (c) 在摆链系统多重简并基态的能谱结构中表征的拓扑与非拓扑孤子; (d) 类摆链系统拥有双重简并基态的能谱结构和拓扑孤子(孤子晶格), 其中antikink须紧随kink而激发. 来自文献(Zhang et al. 2019c)
图 48 (a)(b) 软质力学超材料实验模型及元胞; (c) 准静态位移载荷下超材料的初始构型, 应变率约为έy = 3.1 × 10−5 s−1; (d)(e) 实验和有限元模拟中激发的静态孤子晶格, kink与antikink周期性地交替呈现于两种机械极化区域之间. 对应的宏观压缩应变约为εy = −0.11. 来自文献(Zhang et al. 2019c)
图 49 (a) 代表性元胞几何及嵌刻其中的简化模型, 圆点表示两类颈弹簧; (b) 元胞简化模型(初始倾角θ0)及其变形构型(当前倾角θ), 转角α = θ − θ0; (c) 几何空间参数κ对原位势Ucell的基态和对称性的分类; (d) 准1D超材料结构(Nx × 1)及其简化模型(κ = 1时, θ0 = 0, α = θ); (e) 1D“球−链”机理模型, 弹簧链与演化的原位势场(单基态→双基态)相互作用, 以刻画超材料承受位移压缩过程; (f) 实验、模拟与φ4理论解相互论证了静态孤子晶格的激发. 来自文献(Zhang et al. 2019c)
图 50 (a) 基于等效原位势特征的超材料分类相图; (b) 非平凡超材料等效原位势随应变的演化特征; (c) 超材料经历结构相变、对称性破缺至静态孤子激发的普适物理框架; (d) 基于普适性框架, 在“方块” 、 “杆系”与“圆−椭圆”多孔超材料中激发出静态拓扑孤子; (e) 由超材料序参量表征的拓扑孤子的实验、模拟与理论结果(εy = −0.01). 来自文献(Zhang et al. 2019c)
图 51 杆数nb = 9, 节点数ns = 6, 满足Maxwell准则的有限桁架结构. (a)等静定桁架结构, N0 = 3, M = 0, NSS = 0; (b) 非等静定桁架结构, N0 = 4, M = 1, NSS = 1
图 52 (a) 规则Kagome结构, 同时显示了周期边界条件下机构模式和自应力模式; (b) 规则Kagome结构第一支色散曲面的零频率等频线; (c) 扭曲Kagome结构; (d) 扭曲Kagome结构第一支色散曲面的零频率等频线; (e) 一般Kagome结构的参数化; (f) Kagome结构连续变形的拓扑相图
图 53 (a) 具有不同拓扑相的Kagome桁架结构界面处的零能模式, 箭头表示机构模式的无限小位移, 红绿线条显示自应力边界态, 红、绿色分别代表拉、压内力; (b) 沿界面平行方向的色散关系. 来自文献(Kane et al. 2014)
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