标题： 基于表示论的拓扑晶体绝缘体分类 显示英文标题

标题： Classification of topological crystalline insulators based on representation theory

摘要： 拓扑晶体绝缘体定义了一类新的拓扑绝缘体相，其具有由晶体对称性保护的无能隙表面态。 在本工作中，我们提出一种一般理论，基于空间群的表示理论来分类拓扑晶体绝缘体相。 我们的方法是直接在一个具有特定表面的半无限系统中识别可能的非平凡表面态，其对称性性质可以用17个二维空间群来描述。 我们复现了拓扑晶体绝缘体的现有结果，例如在$pm$或$pmm$群中的镜像陈绝缘体，$C_{nv}$群中的拓扑绝缘体，以及在$p4m$、$p31m$和$p6m$群中的拓扑非对称晶体绝缘体，还有在$pg$和$pmg$群中的拓扑非平移晶体绝缘体。 除了这些现有的结果外，我们还获得了以下新的结果：(1) 在$pm$群中存在两个整数镜像陈数 ($\mathbb{Z}^2$)，但在自旋无规和自旋有规情况下，$cm$或$p3m1$群中仅有一个 ($\mathbb{Z}$)；(2) 对于$pmm$($cmm$) 群，在自旋无规情况下没有拓扑分类，但在自旋有规情况下有$\mathbb{Z}^4$($\mathbb{Z}^2$) 分类；(3) 我们展示了$pg$群中的拓扑晶体绝缘体相如何与$pm$群中的相相关联；(4) 我们确定了自旋有规情况下$p4m$、$p31m$和$p6m$的拓扑分类；(5) 我们发现自旋有规情况下，拓扑非平移晶体绝缘体也存在于$pgg$和$p4g$群中，与$pg$和$pmg$群中的情况相比，它们表现出新的特性。 我们强调了在拓扑晶体相分类中，某些特定高对称动量处的不可约表示的重要性。 我们的理论可以作为在实际材料中寻找拓扑晶体绝缘体相的指导。 显示英文摘要

摘要： Topological crystalline insulators define a new class of topological insulator phases with gapless surface states protected by crystalline symmetries. In this work, we present a general theory to classify topological crystalline insulator phases based on the representation theory of space groups. Our approach is to directly identify possible nontrivial surface states in a semi-infinite system with a specific surface, of which the symmetry property can be described by 17 two-dimensional space groups. We reproduce the existing results of topological crystalline insulators, such as mirror Chern insulators in the $pm$ or $pmm$ groups, $C_{nv}$ topological insulators in the $p4m$, $p31m$ and $p6m$ groups, and topological nonsymmorphic crystalline insulators in the $pg$ and $pmg$ groups. Aside from these existing results, we also obtain the following new results: (1) there are two integer mirror Chern numbers ($\mathbb{Z}^2$) in the $pm$ group but only one ($\mathbb{Z}$) in the $cm$ or $p3m1$ group for both the spinless and spinful cases; (2) for the $pmm$ ($cmm$) groups, there is no topological classification in the spinless case but $\mathbb{Z}^4$ ($\mathbb{Z}^2$) classifications in the spinful case; (3) we show how topological crystalline insulator phase in the $pg$ group is related to that in the $pm$ group; (4) we identify topological classification of the $p4m$, $p31m$, and $p6m$ for the spinful case; (5) we find topological non-symmorphic crystalline insulators also existing in $pgg$ and $p4g$ groups, which exhibit new features compared to those in $pg$ and $pmg$ groups. We emphasize the importance of the irreducible representations for the states at some specific high-symmetry momenta in the classification of topological crystalline phases. Our theory can serve as a guide for the search of topological crystalline insulator phases in realistic materials.