We propose a general principle for realizing second-order topological corner states in the modified Kane-Mele model with magnetization. It is demonstrated that the sign of the edge Dirac mass depends on the magnetization of the edge sublattice termination. By adjusting the directions of magnetization according to the type of sublattice at the termination of two edges, a mass domain wall can be induced in the presence of topological corner states at an arbitrary position. All previous work on introducing magnetization in the Kane-Mele model to realize second-order topological corner states can be explained by the presence of the Dirac mass domain wall with opposite signs. Applying this principle, we design square-shaped and armchair-type hexagon-shaped graphene nanoflakes with edge magnetization, allowing for the emergence of second-order topological corner states. Our findings serve as a general theory, demonstrating that the realization of second-order topological corner states is not limited by boundary type or nanoflake shape.

中文翻译：

---

Kane-Mele 模型中磁化引起的二阶拓扑角态背后的一般原理

我们提出了在带有磁化的修正 Kane-Mele 模型中实现二阶拓扑角态的一般原理。结果表明，边缘狄拉克质量的符号取决于边缘亚晶格终止的磁化强度。通过根据两个边缘终止处的亚晶格类型调整磁化方向，可以在任意位置存在拓扑角态的情况下诱导质量畴壁。之前所有在 Kane-Mele 模型中引入磁化以实现二阶拓扑角态的工作都可以通过具有相反符号的狄拉克质量畴壁的存在来解释。应用这一原理，我们设计了具有边缘磁化的方形和扶手椅式六边形石墨烯纳米片，允许出现二阶拓扑角态。我们的研究结果作为一般理论，证明二阶拓扑角态的实现不受边界类型或纳米片形状的限制。