粗字是注释或者自己加的理解、总结

The Chern number is defined through the integration of the Berry curvature in the Brillouin zone (BZ) as

通过对布里渊区(BZ)的贝里曲率的积分来定义Chern数为

is the non- vanishing component of the Berry curvature, with the periodic part of the Bloch wavefunction of the nth band.The Chern number is necessarily integer- valued, counting the net number of chiral edge modes that enter and leave a given band.

上式是Berry曲率的非消失分量，ku ()n为第n波段布洛赫波函数的周期部分。Chern数必须是整数值，计算进入和离开一个给定波段的手性边缘模式的净数量。

手性边缘态：

物理上外界磁场引起的时间反演对称破缺阻断了边界电子背散射路径，使其只能单向无耗散流动且方向取决于磁场取向，形成手性边缘态。边缘态手性模型近期还直接促成了对拓扑绝缘体的发现。由于此类材料可实现零磁场下边缘态电子的无耗散流动而具有重大应用价值，因此成为当前凝聚态物理研究的热点。迄今，边缘态手性在量子体系电子输运过程中扮演的重要角色已广为人知。

The chiral edge modes, which are responsible for the quantized Hall conductance, propagate along the edges unidirectionally, and are robust against gap- preserving disorder and defects.

手性边缘模负责量子化霍尔电导，沿边缘单向传播，对保留间隙的无序和缺陷具有鲁棒性。

Owing to undesired effects such as non- synchronous rotation and flow instabilities. To overcome these issues, an acoustic ring resonator lattice was designed with optimized structural parameters and a high- order mode with high quality factor, which considerably reduces the required airflow speed. Based on this design (fig. 2e), acoustic chiral edge modes were successfully observed.

由于不同步旋转和流动不稳定等不良影响。为了克服这些问题，设计了一种优化结构参数和高质量因子高阶模态的声学环形谐振腔晶格，大大降低了所需的气流速度。基于此设计(图2e)，成功观测到声手性边缘模。

在时间反演系统中，量子霍尔效应相不存在

声学中的赝自旋和t保留拓扑相

Pseudospins and T-preserved topological phases in acoustics

在本节中，我们讨论了通过利用自旋自由度和谷自由度实现t不变声学拓扑绝缘相的方法，或所谓的声学QSH和声学谷霍尔绝缘子。

量子自旋霍尔绝缘子的声学类似物。

Acoustic analogues of quantum spin Hall insulators.

The QSH insulator, which can be regarded as a system simultaneously supporting two time- reversed copies of the QH insulator, hosts ‘helical’ edge states. The term ‘helical’ refers to the fact that opposite spins counterpropagate along the edge, in contrast to chiral edge states that propagate unidirectionally.

QSH绝缘子可以被视为同时支持QH绝缘子的两个时间反转副本的系统，具有“螺旋”边缘状态。术语“螺旋”指的是相反自旋沿边缘反向传播，与手性边缘单向传播相反。

Unlike the QH phase, a QSH insulator does not break T symmetry but is instead protected by T symmetry. This is because T symmetry for an electron satisfies T**2 = − 1 , which enables Kramers doublets, where opposite spins are degenerate at T-invariant momenta. It is thus possible to achieve gapless edge states robust against T- preserving perturbations. However, for sound, which carries intrinsic spin-0, T symmetry satisfies T**2 = 1 .Such a fundamental difference poses a challenge to the implementation of an acoustic analogue.

与QH相不同，QSH绝缘子不破坏T对称，而是受到T对称的保护。这是因为电子的T对称性满足T**2 =−1，这使克雷默斯双态成为可能，其中相反自旋在T不变动量下简并。因此，有可能获得抗保T扰动的无间隙边缘状态。然而，对于具有自旋-0特性的声音，T对称满足T**2 = 1。这样的基本差异给声学模拟的实现带来了挑战。

克服这个问题的一个常用策略是将T修改为TU (U是一个通常由格对称生成的算子)，这样(UT)**2 =−1。

Γ（gamma）点是什么：

Γ 点处的三重简并态是偶然简并形成的，保持介质矩形柱的 ε 不变，改变边长 d ，使它偏离 0.3633a，三重简并态分裂成一个两重简并态和一个单态.当 d=0.353a 时，两重简并态的频率高于单态的频率，如图2(a)所示;当 d=0.373a 时，两重简并态的频率低于单态的频率，如图2(b)所示;当 d=0.3633a时，两重简并态与单态在 Γ 点处对应不同的 d 能带具有反转现象，因此在能带反转的两个不同 d 之间必然存在一个 d 使得三个能带简并，如图2(c)所示.对于二维电介质光子晶体，当波矢 k =0 处存在狄拉克点时，如果狄拉克点是由单极子和偶极子形成，那么在狄拉克点频率的光子晶体可以等效为介电常数和磁导率都为零的材料。

声谷霍尔绝缘子

Acoustic valley Hall insulators

具有量子化偶极矩和多极矩的声学拓扑相位

Acoustic topological phases with quantized dipole and multipole moments

In this section, we turn our focus to another large class of topological phases that are characterized by a quantized dipole moment, that is, the integration of the Berry connection, and by its generalizations, such as quantized quadrupole and octupole moments.

在前两节中，我们讨论了声学中的QH、QSH和valley Hall相位。这些拓扑相由切恩数及其导数表征，如自旋数和谷切恩数，它们是贝里曲率的积分。在本节中，我们将重点转向另一类拓扑相，它们的特征是一个量子化偶极矩，即Berry连接的积分，以及它的推广，如量子化四极矩和八极矩。

量子化四极矩和八极矩：

把带电体的电场分布关于带电体的尺寸和电荷分布展开，和带电体的电荷分布以及带电体尺寸的零阶项相当于是电荷中心的点电荷电场，一阶项相当于一个电偶极矩，二阶项相当于一个电四极矩，三阶项相当于一个电八极矩。。。

一维声学拓扑相

Acoustic topological phases in 1D are characterized by quantized bulk dipole polarization.

一维声学拓扑相位由量子化体偶极子极化表征。

In the modern theory of polarization, the bulk dipole moment is formulated through the Berry phase as

在现代偏振理论中，体偶极矩通过Berry相位被表示为

The dipole moment corresponds to the Wannier centre and in general can take any value within one unit cell. Importantly, the dipole moment can be quantized by symmetries such as mirror and chiral symmetries, making it eligible as a topological invariant. Note that the dipole moment by definition is a gauge- variant quantity that depends on the choice of a unit cell. Nevertheless, the Wannier centre positions are unambiguous. When a 1D system has a non- trivial dipole moment, there are fractional charges at boundaries. The quantized dipole moment, as well as the fractional boundary charge, cannot be removed by perturbations that preserve both the protective symmetry and the bandgap. In realistic acoustic systems, although there is no actual polarization, because there are no charged particles like electrons, the above picture still applies. The dipole polarization can be understood as the Wannier centre, and the fractional boundary charge can be interpreted as a ‘fractional boundary anomaly’.

偶极矩对应于瓦尼尔中心，通常可以取一个单元内的任何值。重要的是，偶极矩可以通过镜像对称性和手性对称性等量化，使其成为拓扑不变量。注意，根据定义，偶极矩是一个量纲变化的量，取决于单位单元的选择。然而，万尼尔的中心立场是明确的。当一维系统具有非平凡偶极矩时，在边界处有分数电荷。量子化偶极矩和分数阶边界电荷不能通过保持保护对称性和带隙的扰动而消除。在现实的声学系统中，虽然没有实际的极化，因为没有像电子一样的带电粒子，但上图仍然适用。偶极极化可以理解为瓦尼尔中心，分数阶边界电荷可以解释为“分数阶边界异常”。

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