小字原文 ----- 粗字自己的理解

Topological sound

we discuss how spin and valley degrees of freedom appear as highly novel ingredients to tailor the flow of sound in the form of one-way edge modes and defect-immune protected acoustic waves.

讨论了自旋和谷自由度如何作为高度新颖的成分出现，以单向边缘模式和缺陷免疫保护声波的形式调整声音的流动。

首先说一下，什么是时间反演对称性：

再说一下，什么是霍尔效应，什么是量子霍尔效应：

量子霍尔效应（quantum Hall effect）是量子力学版本的霍尔效应，需要在低温强磁场的极端条件下才可以被观察到，此时霍尔电阻与磁场不再呈现线性关系，而出现量子化平台。霍尔效应在1879年被E.H.霍尔发现，它定义了磁场和感应电压之间的关系。当电流通过一个位于磁场中的导体的时候，磁场会对导体中的电子产生一个横向的作用力，从而在导体的两端产生电压差。

In condensed-matter physics, the distinctive phases of matter are characterized by their underlying symmetries that are spontaneously broken.

在凝聚态物理学中，物质的不同相的特征在于其潜在的对称性被自发打破。

In 1980, Von Klitzing found that a two-dimensional (2D) electron gas sample, subjected to low temperature and strong magnetic field, has a quantized Hall conductance, which is independent of sample size and immune to impurities.

1980年，Von Klitzing发现在低温强磁场作用下的二维(2D)电子气样具有量子化霍尔电导，该电导与样品大小无关，不受杂质影响。

It was later demonstrated that the state responsible for such phenomena is characterized by a completely different classification paradigm based on the notion of topological order3,4, which describes phases of matter beyond the symmetry breaking (that means two different phases can have the same symmetry), therefore opening a new research branch.

后来证明，导致这种现象的状态是由一种完全不同的分类范式所表征的，这种分类范式基于拓扑顺序3,4的概念，它描述了超越对称性破缺的物质相(这意味着两个不同的相可以具有相同的对称性)，因此开辟了一个新的研究分支。

就是量子霍尔效应（条件：低温强磁场）出来以后分类可以有不同了，两个不同的相可以有同种对称性

对于霍尔效应来说，霍尔电导是一个基本性质。霍尔电导的量子化来源于能带的非平凡拓扑性，即能带具有一个非零拓扑不变量----陈数（和能带相关）。

The Chern number characterizes the geometric phase (commonly known as the Berry phase) accumulation over the Brillouin zone, and thus is closely related with the behaviors of the energy bands in the momentum space.

Chern数表征了几何相位(通常称为Berry相位)在布里渊区上的积累，因此与动量空间中能量带的行为密切相关。

It has been shown that a periodic magnetic flux, which breaks the time-reversal symmetry (unlike the traditional definition of the phases of matter, this symmetry breaking itself does not define the topological order), is able to produce non-zero Chern number

已经证明，周期性磁通量打破了时间反转对称(与物质相的传统定义不同，这种对称打破本身并不定义拓扑顺序)，能够产生非零的陈数

The resulting topologically non-trivial system supports a gapless edge state in the bulk energy gap, exhibiting an interesting electronic property that is insulating in the bulk but conducting on the edge. This is essentially different from a normal insulator, where the Chern number is zero

由此产生的拓扑非平凡系统在体能隙中支持无间隙边缘状态，表现出一种有趣的电子特性，即在体中绝缘，但在边缘上导电。这与普通绝缘体本质上不同，普通绝缘体的Chern数为零。

1：普通绝缘体---chern数为0

黄色gapped--有间隙的   蓝色gapless---无间隙的

2：时间反转损坏QH绝缘子

除了施加磁场之外，材料固有的自旋-轨道耦合也能产生非平凡的拓扑相。

 in systems with spin-orbital coupling, a pair of gapless edge states emerges in the insulating band gap. 

在具有自旋-轨道耦合的系统中，在绝缘带隙中出现一对无隙边缘态。

The edge states carry conjugate electronic spins and exhibit spin-dependent propagation behaviors, as sketched in Box 1. This is the so-called quantum spin Hall effect (QSHE).

边缘态携带共轭电子自旋，并表现自旋相关的传播行为，如框1所示。这就是所谓的量子自旋霍尔效应(QSHE)。

3：时反不变QSH绝缘子

谷电子霍尔效应

Valley refers to the two energy extrema of the band structure in momentum space, at which the Berry curvature exhibits opposite signs and therefore its integral over the full Brillouin zone is zero, while the integral within each valley is nonzero.As a result, the system shows a valley-selective topologically non-trival property

谷是指动量空间中带结构的两个能量极值，在这两个极值处，贝里曲率呈现相反的符号，因此它在全布里渊区的积分为零，而每个谷内的积分为非零。结果表明，该系统在拓扑上具有谷选择性的非平凡性 

贝利曲率可以看成是参数空间中的“磁场强度”，贝利联络可以看成是“磁矢势”，而贝利相位则可以看成是“磁通量”。