In this paper, we theoretically investigate quantum phase transitions (QPTs) in a modified Kane-Mele (MKM) model with $C_{2z}$ and mirror symmetries. The phase diagram of the MKM model uncovers, in addition to the ${\mathbb{Z}}_2$ topological insulator phase, a second-order topological insulator (SOTI) phase and three distinct inversion-symmetry-breaking two-dimensional Weyl metal ($\mathcal{P}$-breaking 2D WM) phases in response to the interplay of the anisotropic nearest-neighbor hopping and Rashba spin-orbit coupling. During the QPTs among these phases, the edges states evolve topologically and manifest as helical edge states, quasiflat localized edge states, and Fermi arc edge states, respectively, in association with the merging and splitting of band-touching points. To achieve second-order topological corner states, we design a rhombus-shaped nanodisk, where two corner states emerge due to a filling anomaly resulting from the fixed Wannier configuration and $C_{2z}$ symmetry. In this paper, we not only propose an approach to realize the SOTI phase in the MKM model but also establish a platform for investigating the topological properties of the $\mathcal{P}$-breaking 2D WM.

中文翻译：

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改进的 Kane-Mele 模型中的二维 Weyl 金属和二阶拓扑绝缘体相

在本文中，我们从理论上研究了改进的 Kane-Mele (MKM) 模型中的量子相变 (QPT)$C_{2z}$和镜像对称。 MKM 模型的相图除了揭示${\mathbb{Z}}_2$拓扑绝缘体相、二阶拓扑绝缘体 (SOTI) 相和三个不同的反演对称性破缺二维 Weyl 金属 ($\mathcal{磷}$-打破2D WM）相位响应各向异性最近邻跳跃和Rashba自旋轨道耦合的相互作用。在这些相的QPT期间，边缘态拓扑演化并分别表现为螺旋边缘态、准平坦局域边缘态和费米弧边缘态，与带接触点的合并和分裂相关。为了实现二阶拓扑角态，我们设计了一个菱形纳米盘，其中由于固定 Wannier 配置产生的填充异常而出现两个角态，$C_{2z}$对称。在本文中，我们不仅提出了一种在 MKM 模型中实现 SOTI 相的方法，而且还建立了一个研究 SOTI 相拓扑特性的平台。$\mathcal{磷}$-突破性的2D WM。