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.

• Received 25 December 2023
• Accepted 1 April 2024

DOI:https://doi.org/10.1103/PhysRevB.109.144108