Topological phases in electronic structures contain a new type of topology, called fragile, which can arise, for example, when an elementary band representation (atomic limit band) splits into a particular set of bands. For the first time, we obtain a complete classification of the fragile topological phases, which can be diagnosed by symmetry eigenvalues, to find an incredibly rich structure that far surpasses that of stable or strong topological states. We find and enumerate hundreds of thousands of different fragile topological phases diagnosed by symmetry eigenvalues, and we link the mathematical structure of these phases to that of affine monoids in mathematics. Furthermore, for the first time, we predict and calculate (hundreds of realistic) materials where fragile topological bands appear, and we showcase the very best ones.

中文翻译：

---

脆弱相作为仿射类化合物：分类和材料实例

电子结构中的拓扑阶段包含一种称为脆弱的新型拓扑，例如，当基本能带表示形式（原子极限带）分裂成一组特定的能带时，就会出现这种拓扑。首次，我们获得了脆弱拓扑阶段的完整分类，可以通过对称特征值进行诊断，以找到一个远远超过稳定或强拓扑状态的难以置信的丰富结构。我们发现并枚举了通过对称特征值诊断的成千上万个不同的易碎拓扑阶段，并将这些阶段的数学结构与数学上的仿射类似半体联系起来。此外，我们第一次预测和计算（数百个现实的）易碎的拓扑带出现的材料，并展示了最好的材料。