In a seminal paper Haldane conjectured that topological phenomena are not particular to quantum systems, and indeed experiments realized unidirectional, backscattering-free edge modes with electromagnetic waves. This raises two immediate questions: (1) Are there other topological effects in electromagnetic media? And (2) is Haldane's Quantum Hall Effect for light really analogous to the Quantum Hall Effect? We conclusively answer both of these questions by classifying topological photonic crystals according to material (as opposed to crystallographic) symmetries. It turns out there are four topologically distinct types of media, of which only one, gyrotropic media, is topologically non-trivial in $d = 2 , 3$. That means there are no as-of-yet undiscovered topological effects; in particular, there is no analog of the Quantum Spin Hall Effect in classical electromagnetism. Moreover, at least qualitatively, Haldane's Quantum Hall Effect for light is analogous to the Quantum Hall Effect from condensed matter physics as both systems as in the same topological class, class A. Our ideas are directly applicable to other classical waves.

中文翻译：

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拓扑光子晶体的对称性分类

在一篇开创性的论文中，霍尔丹推测拓扑现象并不是量子系统所特有的，实际上，实验实现了单向、无反向散射的电磁波边缘模式。这提出了两个紧迫的问题：（1）电磁介质中是否存在其他拓扑效应？(2) 霍尔丹的光量子霍尔效应真的类似于量子霍尔效应吗？我们通过根据材料（与晶体学相反）对称性对拓扑光子晶体进行分类，最终回答了这两个问题。事实证明，有四种拓扑不同的介质类型，其中只有一种，回旋介质，在 $d = 2 , 3$ 中在拓扑上是非平凡的。这意味着没有尚未发现的拓扑效应；特别是，经典电磁学中没有量子自旋霍尔效应的类似物。此外，至少在定性上，霍尔丹的光量子霍尔效应类似于凝聚态物理学中的量子霍尔效应，因为这两个系统都属于同一拓扑类 A 类。我们的想法直接适用于其他经典波。