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摘要: 非厄米理论是研究开放系统动力学行为的一种理论框架, 其概念起源于量子力学, 借助该理论可以揭示出奇异点、手性模态转换、拓扑趋肤效应等新奇现象, 为反常波动与振动调控提供了新思路. 本文着重以力学的语言对非厄米系统的基础概念进行介绍, 阐明经典系统与非厄米系统的关联和扩展关系, 并介绍相关前沿研究进展. 首先介绍非厄米力学系统中的奇异点、宇称-时间反演对称性等基本概念, 并介绍奇异点微扰理论及其在高灵敏度传感等领域的应用机理, 之后介绍奇异点附近特征值曲面的拓扑结构, 以及环绕奇异点的模态演化规律, 最后介绍非厄米力学系统中的拓扑波动行为.Abstract: Non-Hermitian theory, originated from quantum mechanics, is a theoretical framework for investigating the dynamics of open systems. New phenomena can be revealed with this theory, including exceptional point, chiral mode switching, and topological skin effect, which provide novel concepts for unusual wave and vibration control. This review will provide a comprehensive introduction to basic concepts of non-Hermitian theory in terms of classical mechanical systems, clarify the relationship between classical and non-Hermitian systems, and summarize the cutting-edge research progress in this field. Exceptional points and parity-time symmetry in non-Hermitian systems are firstly introduced. Then, the perturbation theory near exceptional points and its application to enhanced sensitivity are presented. Subsequently, the eigenvalue topological structure near exceptional points and eigenmode evolution in the process of dynamical encircling of exceptional points are discussed. Finally, the topological phase property of non-Hermitian mechanical systems is introduced.
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图 7 具有奇异点行为的时变质量调制单自由度系统. 来自文献(Yuan et al. 2022)
图 8 基于状态空间法得到的特征频率(a)实部, (b)虚部以及(c)特征向量矩阵行列式|det(Us)|随ωm/ω0的变化; 基于谐波平衡法得到的特征频率(d)实部, (e)虚部以及(f)|det(UH)|随ωm/ω0的变化. 来自文献(Yuan et al. 2022)
图 9 (a)具有平衡损耗/增益性质的PT 对称声学模型(Zhu et al. 2014), (b) 基于电声耦合实验实现PT对称声学系统(Fleury et al. 2015), (c) 具有高阶奇异点的四能级声腔系统(Ding et al. 2016), (d) 具有奇异弧的三能级声腔系统(Tang et al. 2020), (e) 具有非对称反射行为的非厄米声学超表面(Wang et al. 2019)
图 10 (a) 具有分流压电材料的PT 对称弹性超材料梁(Wu et al. 2019), (b) PT对称压电弹性波超材料(Li & Wang 2023), (c) PT对称二维弹性体(Rosa et al. 2021), (d) 具有反馈响应外力系统的PT对称弹性梁(Cai et al. 2022)
图 11 (a) 含损耗的三自由度质量弹簧系统示意图. 来自文献(Geng et al. 2021a), (b) 三模态非厄米力学系统示意图, (c) 具有纯虚数耦合的两态反PT对称系统示意图
图 13 (a) 具有间接耗散耦合的反 PT 对称光波导系统(Yang et al. 2017), (b) 电阻耦合的反PT 对称电路系统(Choi et al. 2018), (c) 反PT 对称光学谐振器系统(Zhang et al. 2020)
图 14 二阶奇异点(a) ωm/ω0=1.91和 (b) ωm/ω0=2.12处, 特征频率随微扰的变化关系. 来自文献(Yuan et al. 2022)
图 15 (a)基于时变质量奇异点的高灵敏缺陷检测传感器模型, (b) 奇异点传感器对缺陷响应的频谱特性, (c) 线性谐振式传感器对缺陷响应的频谱特性. 来自文献(Yuan et al. 2022)
图 17 (a) 可实现声波单向不可见的PT对称声学超表面(Li et al. 2019), (b) 紧凑型非对称吸声结构(Li et al. 2021), (c) 亚波长通风超表面结构(Zhu et al. 2022)
图 19 特征值(a)实部Re(λ)和(b)虚部Im(λ)在二维参数空间(κ2, γ2)的分布曲面. 来自文献(Geng et al. 2021b)
图 20 含时变调制刚度和阻尼机构的两态质量弹簧系统示意图. 来自文献(Geng et al. 2021b)
图 21 起始点位于对称相附近、不同输入态和不同环绕方向时(a) ~ (d)本征态在黎曼面上的演化轨迹以及(e) ~ (h)模态幅值随时间的演化曲线: (a)和(e)高损耗模态输入, 逆时针环绕; (b)和(f)低损耗模态输入, 逆时针环绕; (c)和(g)高损耗模态输入, 顺时针环绕; (d)和(h)低损耗模态输入, 顺时针环绕. 来自文献(Geng et al. 2021b)
图 22 起始点位于破缺相附近、不同输入态和不同环绕方向时 (a) ~ (d)本征态在黎曼面上的演化轨迹以及(e) ~ (h)模态幅值随时间的演化曲线: (a)和(e)高损耗模态输入, 逆时针环绕; (b)和(f)低损耗模态输入, 逆时针环绕; (c)和(g)高损耗模态输入, 顺时针环绕; (d)和(h)低损耗模态输入, 顺时针环绕. 来自文献(Geng et al. 2021b)
图 23 特征值实部(a)和虚部(b)在二维参数空间(κL, κR)的分布曲面, (c) 反PT对称相和反PT对称破缺相处, 模态1和模态2的模态场幅值比分布. 来自文献(Geng et al. 2021a)
图 24 含时变调制刚度机构的三态质量弹簧系统示意图. 来自文献(Geng et al. 2021a)
图 25 起始点位于破缺相附近、不同输入态和不同环绕方向时 (a) ~ (d)本征态在黎曼面上的演化轨迹以及(e) ~ (h)模态幅值随时间的演化曲线: (a)和(e)模态1输入, 逆时针环绕; (b)和(f)模态2输入, 逆时针环绕; (c)和(g)模态1输入, 顺时针环绕; (d)和(h)模态2输入, 顺时针环绕. 来自文献(Geng et al. 2021a)
图 26 起始点位于对称相附近、不同输入态和不同环绕方向时(a) ~ (d)本征态在黎曼面上的演化轨迹以及(e) ~ (h)模态幅值随时间的演化曲线: (a)和(e)模态1输入, 逆时针环绕; (b)和(f)模态2输入, 逆时针环绕; (c)和(g)模态1输入, 顺时针环绕; (d)和(h)模态2输入, 顺时针环绕. 来自文献(Geng et al. 2021a)
图 27 耦合声波导系统示意图. 来自文献(Duan et al. 2023)
图 28 特征值(a)虚部Im(λ)和(b)实部Re(λ)在二维参数空间(β1,κ)的分布曲面, 不同参数位置处系统的模态场分布: (c)对称相参数点; (d)破缺相参数点; (e)参数轨迹起始点. 来自文献(Duan et al. 2023)
图 29 不同输入态和不同环绕方向时模态幅值的数值解和理论解. (a)模态L输入, 逆时针环绕; (b)模态G输入, 逆时针环绕; (c)模态L输入, 顺时针环绕; (d)模态G输入, 顺时针环绕
图 32 不同双向刚度情况时波矢q随频率
$\varOmega $ 变化的频散曲线: (a)$ \varepsilon = 0 $ ; (b)$ \varepsilon = - 0.4 $ ; (c)$ \varepsilon = 0.4 $ 
图 33 具有双向刚度的弹簧在变形循环过程中系统势能的变化: (a) 不同工况下弹簧的变形状态; (b)
$\varepsilon = 0.4$ 时双向刚度弹簧经历(a)中变形循环后, 两侧外力及对应位移随弹簧变形的演化; (c) 弹簧双向刚度相同时, 系统总势能经过变形循环后保持不变; 弹簧双向刚度系数为(d)$\varepsilon = - 0.4$ 和 (e)$\varepsilon = 0.4$ 时, 非保守力做功, 系统总势能经过变形循环后发生改变
图 34 具有不同双向刚度系数弹簧链的拓扑相. 约束波数为实数时, 不同双向刚度系数(a) ε=0, (b) ε=−0.4和(c) ε=0.4时弹簧链的色散曲线. (d)和(e) 将色散曲线沿波矢增大方向映射到复频率空间的曲线: (d) 双向刚度系数为0时, 系统对应缠绕数为0; (e) 双向刚度系数为−0.4时, 系统对应缠绕数为−1; (f) 双向刚度系数为0.4时, 系统对应缠绕数为1, 其中黄色标记表示外激励频率点
图 36 二维非厄米周期晶格系统. (a) 具有非对称刚度的弹簧单元; (b) 由非对称弹簧组成的二维周期晶格系统, 蓝色箭头表示正向(k1)刚度, 橙色箭头表示负向(k2)刚度, 绿色虚线表示可以由黑线所示弹簧单元沿着晶格矢量周期性平移得到的双向刚度弹簧; (c) 周期晶格系统的第一布里渊区, 灰色区域表示最简布里渊区
图 37 双向刚度系数为
$\varepsilon = 0.4$ 时, 二维非厄米周期晶格系统的色散曲面: (a)和(c) 第一支和第二支色散曲面的实部; (b)和(d) 第一支和第二支色散曲面的虚部
图 38 由双向刚度系数为
$\varepsilon = 0.4$ 的弹簧组成的二维晶格色散及拓扑性质. (a) 二维非厄米周期晶格系统第一支色散曲面实部和虚部分量, 蓝色和灰色切面分别表示波矢沿$ - \pi /4$ 和$ - 3\pi /4$ 方向的色散切面; (b) 波矢沿$ - \pi /4$ 方向的色散切面; (c) 将(b)对应切面沿波矢增大方向映射到复频率空间中的曲线; (d) 波矢沿$ - 3\pi /4$ 方向的色散切面; (e) 将(d)对应切面沿波矢增大方向映射到复频率空间中的曲线
图 39 (a) 双向刚度系数为
$\varepsilon = 0.4$ 的弹簧组成的二维晶格拓扑数随波矢角度分布图, (b) 厄米晶格本征模态分布, (c)和(d) 非厄米晶格的局域化本征模态分布
图 40 (a) 奇弹性介质中四种变形模式示意图, (b)通过旋转-体膨胀非对称耦合实现非保守力做功与路径相关, (c)通过两种剪切变形非对称耦合实现非保守力做功与路径相关
图 41 (a)具有奇弹性模量的超材料梁结构 (Chen et al. 2021), (b)基于压电材料和反馈控制设计实现的奇弹性材料 (Cheng & Hu 2021), (c) 具有非对称二阶张量密度的非厄米力学系统(Wu et al. 2023), (d) 静态力学超材料中的非厄米拓扑效应(Wang et al. 2023)
图 43 (a) 一维双端口声学波导系统示意图, (b)散射体速度和体应变的演化工况, 在(b)所示工况下, 不同声学系统的做功情况: (c) 传统声学介质; (d) 声学Willis介质; (e) 非互易声学Willis介质
图 45 非互易声学Willis介质中的声趋肤效应. (a) 有限长非互易声学Willis介质模型图, 其中左右两侧为辐射边界, 背景介质为传统声学介质, 中间为有限长度的非互易声学Willis介质, 二者密度和体积模量相同, (b)基于传递矩阵法计算得到的声压场分布曲线 (红色圆圈标记) 和有限元计算结果 (实线)
图 46 非互易声学Willis介质的设计与声学特性. (a) 非局部声散射体示意图, (b)由空腔-非局部散射体-空腔组成的散射体单胞示意图, (c)基于参数反演计算得到的等效参数, (d)和(e)阵列结构局域化声场响应的数值模拟和连续介质理论预测结果
图 47 三种W1指向情况下, 缠绕数随W2指向Arg(W2)和空间方位角θ的分布云图. (a) Arg(W1)=π/4, (b) Arg(W1)=0, (c) Arg(W1)=−π/4. 来自文献(Cheng & Hu 2022)
图 48 (a)二维非局部声学散射体示意图; 不同耦合向量指向情况下, 基于连续介质理论和有源正方形晶格预测得到的声学趋肤效应(b) ~ (d), 以及相应远场辐射图(c) ~ (e). 来自文献(Cheng & Hu 2022)
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